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4,400	Two is better than one: distinct roles for familiarity and recollection in retrieving palimpsest memories Cristina Savin1 cs664@cam.ac.uk Peter Dayan2 dayan@gatsby.ucl.ac.uk M´at´e Lengyel1 m.lengyel@eng.cam.ac.uk 1Computational & Biological Learning Lab, Dept. of Engineering, University of Cambridge, UK 2Gatsby Computational Neuroscience Unit, University College London, UK Abstract Storing a new pattern in a palimpsest memory system comes at the cost of interfering with the memory traces of previously stored items. Knowing the age of a pattern thus becomes critical for recalling it faithfully. This implies that there should be a tight coupling between estimates of age, as a form of familiarity, and the neural dynamics of recollection, something which current theories omit. Using a normative model of autoassociative memory, we show that a dual memory system, consisting of two interacting modules for familiarity and recollection, has best performance for both recollection and recognition. This ﬁnding provides a new window onto actively contentious psychological and neural aspects of recognition memory. 1 Introduction Episodic memory such as that in the hippocampus acts like a palimpsest – each new entity to be stored is overlaid on top of its predecessors, and, in turn, is submerged by its successors. This implies both anterograde interference (existing memories hinder the processing of new ones) and retrograde interference (new memories overwrite information about old ones). Both pose important challenges for the storage and retrieval of information in neural circuits. Some aspects of these challenges have been addressed in two theoretical frameworks – one focusing on anterograde interference through the interaction of novelty and storage [1]; the other on retrograde interference in individual synapses [2]. However, neither fully considered the critical issue of retrieval from palimpsests; this is our focus. First, [1] made the critical observation that autoassociative memories only work if normal recall dynamics are suppressed on presentation of new patterns that need to be stored. Otherwise, rather than memorizing the new pattern, the memory associated with the existing pattern that most closely matches the new input will be strengthened. This suggests that it is critical to have a mechanism for assessing pattern novelty or, conversely, familiarity, a function that is often ascribed to neocortical areas surrounding the hippocampus. Second, [2] considered the palimpsest problem of overwriting information in synapses whose efﬁcacies have limited dynamic ranges. They pointed out that this can be at least partially addressed through allowing multiple internal states (for instance forming a cascade) for each observable synaptic efﬁcacy level. However, although [2] provide an attractive formalism for analyzing and optimizing synaptic storage, a retrieval mechanism associated with this storage is missing. 1 potentiation depression a b Figure 1: a. The cascade model. Internal states of a synapse (circles) can express one of two different efﬁcacies (W, columns). Transitions between states are stochastic and can either be potentiating, or depressing, depending on pre- and postsynaptic activities. Probabilities of transitions between states expressing the same efﬁcacy p and between states expressing different efﬁcacies, q, decrease geometrically with cascade depth. b. Generative model for the autoassociative memory task. The recall cue ˜x is a noisy version of one of the stored patterns x. Upon storing pattern x synaptic states changed from V0 (sampled from the stationary distribution of synaptic dynamics) to V1. Recall occurs after the presentation of t −1 intervening patterns, when synapses are in states Vt, with corresponding synaptic efﬁcacies Wt. Only Wt and ˜x are observed at recall. Although these pieces of work might seem completely unrelated, we show here that they are closely linked via retrieval. The critical fact about recall from memory, in general, is to know how the information should appear at the time of retrieval. In the case of a palimpsest, the trace of a memory in the synaptic efﬁcacies depends critically on the age of the memory, i.e., its relative familiarity. This suggests a central role for novelty (or familiarity) signals during recollection. Indeed, we show retrieval is substantially worse when familiarity is not explicitly represented than when it is. Dual system models for recognition memory are the topic of a heated debate [3, 4]. Our results could provide a computational rationale for them, showing that separating a perirhinal-like network (involved in familiarity) from a hippocampal-like network can be beneﬁcial even when the only task is recollection. We also show that the task of recognition can also be best accomplished by combining the outputs of both networks, as suggested experimentally [4]. 2 Storage in a palimpsest memory We consider the task of autoassociative recall of binary patterns from a palimpsest memory. Specifically, the neural circuit consists of N binary neurons that enjoy all-to-all connectivity. During storage, network activity is clamped to the presented pattern x, inducing changes in the synapses’ ‘internal’ states V and corresponding observed binary efﬁcacies W (Fig. 1a). At recall, we seek to retrieve a pattern x that was originally stored, given a noisy cue ˜x and the current weight matrix W. This weight matrix is assumed to result from storing x on top of the stationary distribution of the synaptic efﬁcacies coming from the large number of patterns that had been previously stored, and then subsequently storing a sequence of t −1 other intervening patterns with the same statistics on top of x (Fig. 1b). In more detail, a pattern to be stored has density f, and is drawn from the distribution: Pstore(x) = Y i Pstore(xi) = Y i f xi · (1 −f)1−xi (1) The recall cue is a noisy version of the original pattern, modeled using a binary symmetric channel: Pnoise(˜x\|x) = Y i Pnoise( ˜xi\|xi) (2) Pnoise( ˜xi\|xi) = (1 −r)xi · r1−xi ˜ xi · rxi · (1 −r)1−xi1−˜ xi (3) where r deﬁnes the level of input noise. 2 The recall time t is assumed to come from a geometric distribution with mean ¯t: Precall(t) = 1 ¯t · 1 −1 ¯t t−1 (4) The synaptic learning rule is local and stochastic, with the probability of an event actually leading to state changes determined by the current state of the synapse Vij and the activity at the pre- and post-synaptic neurons, xi and xj. Hence, learning is speciﬁed through a set of transition matrices M (xi, xj), with M (xi, xj)l′l = P(V ′ ij = l′\|Vij = l, xi, xj). For convenience, we adopted the cascade model [2] (Fig. 1a), which assumes that the probability of potentiation and depression decays with cascade depth i as a geometric progression, q± i = χi−1, with q± n = χn−1 1−χ to compensate for boundary effects. The transition between metastates is given by p± i = ς± χi 1−χ, with the correction factors ς+ = 1−f f and ς−= f 1−f ensuring that different metastates are equally occupied for different pattern sparseness values f [2]. Furthermore, we assume synaptic changes occur only when the postsynaptic neuron is active, leading to potentiation if the presynaptic neuron is also active and to depression otherwise. The speciﬁc form of the learning rule could inﬂuence the memory span of the network, but we expect it not to change the results below qualitatively. The evolution of the distribution over synaptic states after encoding can be described by a Markov process, with a transition matrix M given as the average change in synaptic states expected after storing an arbitrary pattern from the prior Pstore(x), M = P xi,xj Pstore(xi)·Pstore(xj)·M(xi, xj). Additionally, we deﬁne the column vectors πV(xi, xj) and πW(xi, xj) for the distribution of the synaptic states and observable efﬁcacies, respectively, when one of the patterns stored was (xi, xj), such that πW l (xi, xj) = P(Wij = l\|xi, xj) and πV l (xi, xj) = P(Vij = l\|xi, xj). Given these deﬁnitions, we can express the ﬁnal distribution over synaptic states as: πV(xi, xj) = X t Precall(t) · M t−1 · M(xi, xj) · π∞ (5) where we start from the stationary distribution π∞(the eigenvector of M for eigenvalue 1), encode pattern (xi, xj) and then t −1 additional patterns from the same distribution. The corresponding weight distribution is πW(xi, xj) = T · πV(xi, xj), where T is a 2 × 2n matrix deﬁning the deterministic mapping from synaptic states to observable efﬁcacies. The fact that the recency of the pattern to be recalled, t, appears in equation 5 implies that pattern age will strongly inﬂuence information retrieval. In the following, we consider two possible solutions to this problem. We ﬁrst show the limitations of recall dynamics that involve a single, monolithic module which averages over t. We then prove the beneﬁts of a dual system with two qualitatively different modules, one of which explicitly represents an estimate of pattern age. 3 A single module recollection system 3.1 Optimal retrieval dynamics Since information storage by synaptic plasticity is lossy, the recollection task described above is a probabilistic inference problem [5,6]. Essentially, neural dynamics should represent (aspects of) the posterior over stored patterns, P (x\|˜x, W), that expresses the probability of any pattern x being the correct response for the recall query given a noisy recall cue, ˜x, and the synaptic efﬁcacies W. In more detail, the posterior over possible stored patterns can be computed as: P (x\|W, ˜x) ∝Pstore(x) · Pnoise(˜x\|x) · P(W\|x) (6) where we assume that evidence from the weights factorizes over synapses1, P (W\|x) = Q ij P (Wij\|xi, xj). 1This assumption is never exactly true in practice, as synapses that share a pre- or post- synaptic partner are bound to be correlated. Here, we assume the intervening patterns cause independent weight changes and ignore the effects of such correlations. 3 Previous Bayesian recall dynamics derivations assumed learning rules for which the contribution of each pattern to the ﬁnal weight were the same, irrespective of the order of pattern presentation [5,6]. By contrast, the Markov chain behaviour of our synaptic learning rule forces us to explicitly consider pattern age. Furthermore, as pattern age is unknown at recall, we need to integrate over all possible t values (Eq. 5). This integral (which is technically a sum, for discrete t) can be computed analytically using the eigenvalue decomposition of the transition matrix M. Alternatively, if the value of t is known during recall, the prior is replaced by a delta function, Precall(t) = δ(t −t∗). There are several possible ways of representing the posterior in Eq.6 through neural dynamics without reifying t. For consistency, we assume neural states to be binary, with network activity at each step representing a sample from the posterior [7, 8]. An advantage of this approach is that the full posterior is represented in the network dynamics, such that higher decision modules can not only extract the ‘best’ pattern (for the mean squared error cost function considered here, this would be the mean of the posterior) but also estimate the uncertainty of this solution. Nevertheless, other representations, for example representing the parameters of a mean-ﬁeld approximation to the true posterior [5,9,10], would also be possible and similarly informative about uncertainty. In particular, we use Gibbs sampling, as it allows for neurally plausible recall dynamics [7]. This results in asynchronous updates, in which the activity of a neuron xi changes stochastically as a function of its input cue ˜xi, the activity of all other neurons, x\i, and neighbouring synapses, Wi,· and W·,i. Speciﬁcally, the Gibbs sampler results in a sigmoid transfer function, with the total current to the neuron given by the log-odds ratio: Irec i = log P(xi = 1\|x\i, W, ˜xi) P(xi = 0\|x\i, W, ˜xi) = Irec,in i + Irec,out i + a˜xi + b (7) with the terms Iin/out rec deﬁning the evidence from the incoming and outgoing synapses of neuron i, and the constants a and b determined by the prior over patterns and the noise model.2 The terms describing the contribution from recurrent interactions, have a similar shape: Irec,in i = X j cin 1 · Wij xj + cin 2 · Wij + cin 3 · xj + cin 4 (8) Irec,out i = X j cout 1 · Wji xj + cout 2 · Wji + cout 3 · xj + cout 4 (9) The parameters cin/out k , uniquely determined by the learning rule and the priors for x and t, rescale the contribution of the evidence from the weights as a function of pattern age (see supplementary text). Furthermore, these constants translate into a unique signal, giving a sort of ‘sufﬁcient statistic’ for the expected memory strength. Note that the optimal dynamics include two homeostatic processes, corresponding to global inhibition, P j xj, and neuronal excitability regulation, P j Wij, that stabilize network activity during recall. 3.2 Limitations Beside the effects of assuming a factorized weight distribution, the neural dynamics derived above should be the best we can do given the available data (i.e. recall cue and synaptic weights). How well does the network fare in practice? Performance is as expected when pattern age is assumed known: as the available information from the weights decreases, so does performance, ﬁnally converging to control levels, deﬁned by the retrieval performance of a network without plastic recurrent connections, i.e. when inference uses only the recall cue and the prior over stored patterns (Fig. 2a, green). When t is unknown, performance also deteriorates with increasing pattern age, however this time beneath control levels (Fig. 2a, blue). Intuitively, one can see that relying on the prior over t is similar to assuming t ﬁxed to a value close 2Real neurons can only receive information from their presynaptic partners, so cannot estimate Iout rec . We therefore ran simulations without this term in the dynamics and found that although it did decrease recall performance, this decrease was similar to that obtained by randomly pruning half of the connections in the network and keeping this term in the dynamics (not shown). This indicated that performance is mostly determined by the number of available synapses used for inference, and not so much by the direction of those synapses. Hence, in the following we use both terms and leave the systematic study of connectivity for future work. 4 a b t known t known Gibbs tempered dual system 0 5 10 15 single module control transitions error (%) t error (%) 0 50 100 150 0 10 20 30 40 t unknown control Figure 2: a. Recall performance for a single module memory system. b. Average recollection error comparison for the single and dual memory system. Black lines mark control performance, when ignoring the information from the synaptic weights. to the mean of this prior. When the pattern that was actually presented is older than this estimate, the resulting memory signal is weaker than expected, suggesting that the initial pattern was very sparse (since a pair of inactive elements does not induce any synaptic changes according to our learning rule). However, less reasonable is the fact that averaging over the prior distribution of recall times t (Eq. 4), performance is worse than this control (Fig. 2b). One possible reason for this failure is that the sampling procedure used for inference might not work in certain cases. Since Gibbs samplers are known to mix poorly when the shape of the posterior is complex (with strong correlations, as in frustrated Ising models), perhaps our neural dynamics are unable to sample the desired distribution effectively. We conﬁrmed this hypothesis by implementing a more sophisticated sampling procedure using tempered transitions [11] (details in supplementary text). Indeed, with tempered transitions performance becomes signiﬁcantly better than control, even for the cases where Gibbs sampling fails (Fig. 2b). Unfortunately, there has yet to be a convincing suggestion as to how tempering dynamics (or in fact any other sampling algorithm that works well with correlated posteriors) can be represented neurally since, for example, they require a global acceptance decision to be taken at the end of each temperature cycle. It is worth noting that with more complex synaptic dynamics (e.g. deeper cascades) simple Gibbs sampling works reasonably well (data not shown), probably because the posterior is smoother and hence easier to sample. 4 A dual memory system An alternative to implicitly marginalizing over the age of the pattern throughout the inference process is to estimate it at the same time as performing recollection. This suggests the use of dual modules that together estimate the joint posterior P (x, t\|˜x, W), with sampling proceeding in a loop: the familiarity module generates a sample from the posterior over the age of the currently estimated pattern, P(t\|x, ˜x, W); and the recollection module uses this estimated age to compute a new sample from the distribution over possible stored patterns given the age, P (x\|˜x, W, t) (Fig. 3a). The module that computes familiarity can also be seen as a palimpsest, with each pattern overlaying, and being overlaid by, its predecessors and successors. Formally, it needs to compute the probability P(t\|x, ˜x, W), as the system continues to implement a Gibbs sampler with t as an additional dimension. As a separate module, the neural network estimating familiarity cannot however access the weights W of the recollection module. A biologically plausible approximation is to assume that the familiarity module uses a separate set of weights, which we call Wfam. Also, it is clear from Fig. 1b that t is independent of ˜x conditioned on x, thus the conditioning on ˜x can be dropped when computing the posterior over t, that is, external input need only feed directly into the recollection but not the familiarity module (Fig. 3a). In particular, we assume a feedforward network structure in the familiarity module, with each neuron receiving the output of the recollection module as inputs through synapses Wfam. These synaptic 5 familiarity recollection a b 0 50 100 150 10−1 100 101 t familiarity signal 1 100 200 0 0.05 neuron index activation cue Figure 3: a. An overview of the dual memory system. The familiarity network has a feedforward structure, with the activity of individual neurons estimating the probability of the true pattern age being a certain value t, see example in inset. The estimated pattern age translates into a familiarity signal, which scales the contribution of the recurrent inputs in the network dynamics. b. Dependence of the familiarity signal on the estimated pattern age. weights change according to the same cascade rule used for recollection.3 For simplicity, we assume that the familiarity neurons are always activated during encoding, so that synapses can change state (either by potentiation or depression) with every storage event. Concretely, the familiarity module consists of Nfam neurons, each corresponding to a certain pattern age in the range 1–Nfam (the last unit codes for t ≥Nfam). This forms a localist code for familiarity. The total input to a neuron is given by the log-posterior Ifam i = log P(t = i\|x, Wfam) which translates into a simple linear activation function: Ifam i = X j cfam 1,i W fam ij xj + cfam 2,i W fam ij + cfam 3,i xj + cfam 4,i + log P(t) −log(Z) (10) where the constants cfam k,i are similar to parameters cin/out before (albeit different for each neuron because of their tuning to different values of t), and Z is the unknown partition function. As mentioned above, we treat the activity of the familiarity module as a sample from the posterior over age t. This representation requires lateral competition between different units such that only one can become active at each step. Dynamics of this sort can be implemented using a softmax operator, P(xfam i = 1) = eIi P j eIj (thus rendering the evaluation of the partition function Z unnecessary), and are a common feature of a range of neural models [12,13]. Critically, this familiarity module is not just a convenient theoretical construct associated with retrieval. First, as we mentioned before, the assessment of novelty actually plays a key part in memory storage – in making the decision as to whether a pattern that is presented is novel, and so should be stored, or familiar, and so should have its details be recalled. This venerable suggestion [1] has played a central part in the understanding of structure-function relationships in the hippocampus. The graded familiarity module that we have suggested is an obvious extension of this idea; the use for retrieval is new. Second, it is in general accord with substantial data on the role of perirhinal cortex and the activity of neurons in this structure [3]. Recency neurons would be associated with small values of t; novelty neurons with large or effectively inﬁnite values of t [14], although perirhinal cortex appears to adopt a population coding strategy for age, rather than just one-of-n. The recollection module has the same dynamics as before, with constants ci computed assuming t ﬁxed to the output of the familiarity module. Thus we predict that familiarity multiplicatively modulates recurrent interactions in the recollection module during recall. Since there is a deterministic mapping between t and this modulatory factor (Fig. 3b), it can be computed using a linear unit pooling the outputs of all the neurons in the familiarity module, with weights given by the corresponding values for cfam i (t). 3There is nothing to say that the learning rule that optimizes the recollection network’s ability to recall patterns should be equally appropriate for assessing familiarity. Hence, the familiarity module could have their own learning rule, optimized for its speciﬁc task. 6 0 0.04 0.08 0.12 0 10 20 30 40 recollection: average entropy familiarity: estimated t 0.1 0.3 0.5 0.7 0.9 novel familiar a c d 0 0.2 0.4 0.6 0.8 1 0.5 0.6 0.7 0.8 0.9 1 false alarms hits b both rec fam 0 50 100 * * fam rec 80 90 100 * Figure 4: a. Decision boundaries for the recognition module. b. Corresponding ROC curve. c. Performance comparison when the decision layer uses signals from the familiarity module, the recollection module, or both. d. Same comparison, when data is restricted to recent stimuli. Note that difference between fam and rec became signiﬁcant compared to c. In order to compare single and dual module systems fairly, the computational resources employed by each should be the same. We therefore reduced the overall connectivity in the dual system such that the two have the same total number of synapses. Moreover, since elements of Wfam are correlated, the effective number of connections is in fact somewhat lower in the dual system. Regardless, the dual memory system performs signiﬁcantly better than the single module system (Fig. 2b). 5 Recognition memory We have so far considered familiarity merely as an instrument for effective recollection. However, there are many practical and experimental tasks in which it is sufﬁcient to make a binary decision about whether a pattern is novel or familiar rather than recalling it in all its gory detail. It is these tasks that have been used to elucidate the role of perirhinal cortex in recognition memory. In the dual module system, information about recognition is available from both the familiarity module (patterns judged to have young ages are recognized) and the recollection module (patterns recalled with higher certainty are recognized). We therefore construct an additional decision module which takes the outputs of the familiarity and recollection modules and maps them into a binary behavioral response (familiar vs. novel). Speciﬁcally, we use the average of the entropies associated with the activities of neurons in the recollection module and the mean estimate of t from the familiarity module. Since the palimpsest property implicitly assumes that all patterns have been presented at some point, we deﬁne a pattern to be familiar if its age is less than a ﬁxed threshold tth. We train the decision module using a Gaussian process classiﬁer4 [15], which yields as outcome the probability of a hit, P(familiar\|t∗, x∗), shown in Fig. 4a. The shape of the resulting discriminator, that it is not parallel to either axis, suggests that the output of both modules is needed for successful recognition, as suggested experimentally [4,16]. The fact that a classiﬁer trained using only one of the two dimensions cannot match the recognition performance of that using both conﬁrms this observation (Fig. 4c). Moreover, the ROC curve produced by the classiﬁer, plotting hit rates against false alarms as relative losses are varied, has a similar shape to those obtained for human behavioral data: it has a so-called ‘curvi-linear’ character because of the apparent intersect at a ﬁnite hit probability for 0 false alarm rate [17] (Fig. 4b). Lastly, as recognition is known to rely more on familiarity for relatively recent patterns [18], we estimate recognition performance for recent patterns, which we deﬁne as having age t ≤tth 2 . To determine the contribution of each module in recognition outcomes in this case, we estimate performance of classiﬁers trained on single input dimensions for this test data. Consistent with experimental data, our analysis reveals that the familiarity signal gives a more reliable estimate of novelty, compared to the recollection output for relatively recent items (Fig. 4d). 4The speciﬁc classiﬁer was chosen as it allows for an easy estimation of the ROC curves. Future work should explore analytical decision rules. 7 6 Conclusions and discussion Knowing the age of a pattern is critical for retrieval from palimpsest memories, a consideration that has so far eluded theoretical inquiry. We showed that a memory system could either treat this information implicitly, by marginalizing over all possible ages, or it could estimate age explicitly as a form of familiarity. In principle, both solutions should have similar performance, given the same resources. In practice, however, a system involving dual modules is signiﬁcantly better. In our model, the posterior over possible stored patterns was represented in neural activities via samples. We showed that a complex, biologically-questionable sampling procedure would be necessary for the implicit, single module, system. Instead, a dual memory system with two functionally distinct but closely interacting modules, yielded the best performance both for efﬁcient recollection and for recognition. Importantly, though Gibbs sampling and tempered transitions provide a useful framework for understanding the performance differences between different memory systems, the presented results are not restricted to a sampling-based implementation. Since age and identity are tightly correlated, a mean ﬁeld solution that use factorized distributions [5] shows very similar behavior (see supplementary text). Similarly, the speciﬁc details of the familiarity module are not critical for these effects, which should be apparent for any alternative implementation correctly estimating pattern age. Representing pattern age, t, explicitly essentially amounts to implementing an auxiliary variable for sampling the space of possible patterns, x more efﬁciently. Such auxiliary variable methods are widely used to increase sampling efﬁciency when other, simpler methods fail [19]. Moreover, since t in our case speciﬁcally modulates the correlated components of the posterior it can be seen as a ‘temperature’ parameter, and so we can understand the advantages brought about by the dual system as due to implementing a form of ‘simulated tempering’ – a class of methods known to help mixing in strongly correlated posteriors. Our proposal provides a powerful new window onto the contentious debate about the neural mechanisms of recognition and recall. The rationale for our familiarity network was improving recollection; however, the form of the network was motivated by the substantial experimental data [14] on recognition, and indeed standard models of perirhinal cortex activity [20]. These, for instance, also rely on some form of inhibition to mediate interactions between different familiarity neurons. Nevertheless, our model is the ﬁrst to link the computational function of familiarity networks to recall; it is distinct also in that it considers palimpsest synapses, as previous models use purely additive learning rules [20]. Although we only considered pattern age as the basis of familiarity here, the principle of the interaction between familiarity and recollection remains the same in an extended setting, when familiarity characterizes the expected strength of the memory trace more completely, including the effects of retention interval, number of repetitions, and spacing between repetitions. Future work with the extended model should allow us to address familiarity, novelty, and recency neurons in the perirhinal cortex, and indeed provide a foundation for new thinking about this region. In our model familiarity interacts with recollection by multiplicatively (or divisively) modulating the contribution of recurrent inputs in the recollection module. Neurally, this effect could be mediated by shunting inhibition via speciﬁc classes of hippocampal interneurons which target the dendritic segment corresponding to recurrent connections, thus rescaling the relative contribution of external versus recurrent inputs [21]. Whether pathways reaching CA3 from perirhinal cortex through entorhinal cortex preserve a sufﬁcient amount of input speciﬁcity of feed-forward inhibition is unknown. Our theory predicts important systems-level aspects of memory from synaptic-level constraints. In particular, by optimizing our dual system solely for memory recall we also predicted non-trivial ROC curves for recognition that are in at least broad qualitative agreement with experiments. Future work will be needed to explore whether the ROC curves in our model show similar dissociations in response to speciﬁc lesions of the two modules to those found in recent experiments [22,23] and the relation to other recognition memory models [24]. Acknowledgements This work was supported by the Wellcome Trust (CS, ML) and the Gatsby Charitable Foundation (PD). 8 References [1] Hasselmo, M.E. The role of acetylcholine in learning and memory. Current opinion in neurobiology 16, 710–715 (2006). [2] Fusi, S., Drew, P.J. & Abbott, L.F. Cascade models of synaptically stored memories. Neuron 45, 599–611 (2005). [3] Brown, M.W. & Aggleton, J.P. Recognition memory: What are the roles of the perirhinal cortex and hippocampus? Nature Reviews Neuroscience 2, 51–61 (2001). [4] Wixted, J.T. & Squire, L.R. The medial temporal lobe and the attributes of memory. Trends in Cognitive Sciences 15, 210–217 (2011). [5] Sommer, F.T. & Dayan, P. 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4,401	Automated Reﬁnement of Bayes Networks’ Parameters based on Test Ordering Constraints Omar Zia Khan & Pascal Poupart David R. Cheriton School of Computer Science University of Waterloo Waterloo, ON Canada {ozkhan,ppoupart}@cs.uwaterloo.ca John Mark Agosta∗ Intel Labs Santa Clara, CA, USA johnmark.agosta@gmail.com Abstract In this paper, we derive a method to reﬁne a Bayes network diagnostic model by exploiting constraints implied by expert decisions on test ordering. At each step, the expert executes an evidence gathering test, which suggests the test’s relative diagnostic value. We demonstrate that consistency with an expert’s test selection leads to non-convex constraints on the model parameters. We incorporate these constraints by augmenting the network with nodes that represent the constraint likelihoods. Gibbs sampling, stochastic hill climbing and greedy search algorithms are proposed to ﬁnd a MAP estimate that takes into account test ordering constraints and any data available. We demonstrate our approach on diagnostic sessions from a manufacturing scenario. 1 INTRODUCTION The problem of learning-by-example has the promise to create strong models from a restricted number of cases; certainly humans show the ability to generalize from limited experience. Machine Learning has seen numerous approaches to learning task performance by imitation, going back to some of the approaches to inductive learning from examples [14]. Of particular interest are problemsolving tasks that use a model to infer the source, or cause of a problem from a sequence of investigatory steps or tests. The speciﬁc example we adopt is a diagnostic task such as appears in medicine, electro-mechanical fault isolation, customer support and network diagnostics, among others. We deﬁne a diagnostic sequence as consisting of the assignment of values to a subset of tests. The diagnostic process embodies the choice of the best next test to execute at each step in the sequence, by measuring the diagnostic value among the set of available tests at each step, that is, the ability of a test to distinguish among the possible causes. One possible implementation with which to carry out this process, the one we apply, is a Bayes network [9]. As with all model-based approaches, provisioning an adequate model can be daunting, resulting in a “knowledge elicitation bottleneck.” A recent approach for easing the bottleneck grew out of the realization that the best time to gain an expert’s insight into the model structure is during the diagnostic process. Recent work in “QueryBased Diagnostics” [1] demonstrated a way to improve model quality by merging model use and model building into a single process. More precisely the expert can take steps to modify the network structure to add or remove nodes or links, interspersed within the diagnostic sequence. In this paper we show how to extend this variety of learning-by-example to include also reﬁnement of model parameters based on the expert’s choice of test, from which we determine constraints. The nature of these constraints, as shown herein, is derived from the value of the tests to distinguish causes, a value referred to informally as value of information [10]. It is the effect of these novel constraints on network parameter learning that is elucidated in this paper. ∗J. M. Agosta is no longer afﬁliated with Intel Corporation 1 Conventional statistical learning approaches are not suited to this problem, since the number of cases available from diagnostic sessions is small, and the data from any case is sparse. (Only a fraction of the tests are taken.) But more relevant is that one diagnostic sequence from an expert user represents the true behavior expected of the model, rather than a noisy realization of a case generated by the true model. We adopt a Bayesian approach, which offers a principled way to incorporate knowledge (constraints and data, when available) and also consider weakening the constraints, by applying a likelihood to them, so that possibly conﬂicting constraints can be incorporated consistently. Sec. 2 reviews related work and Sec. 3 provides some background on diagnostic networks and model consistency. Then, Sec. 4 describes an augmented Bayesian network that incorporates constraints implied by an expert’s choice of tests. Some sampling techniques are proposed to ﬁnd the Maximum a posterior setting of the parameters given the constraints (and any data available). The approach is evaluated in Sec. 5 on synthetic data and a real world manufacturing diagnostic scenario. Finally, Sec. 6 discusses some future work. 2 RELATED WORK Parameter learning for Bayesian networks can be viewed as searching in a high-dimensional space. Adopting constraints on the parameters based on some domain knowledge is a way of pruning this search space and learning the parameters more efﬁciently, both in terms of data needed and time required. Qualitative probabilistic networks [17] allow qualitative constraints on the parameter space to be speciﬁed by experts. For instance, the inﬂuence of one variable on another, or the combined inﬂuence of multiple variables on another variable [5] leads to linear inequalities on the parameters. Wittig and Jameson [18] explain how to transform the likelihood of violating qualitative constraints into a penalty term to adjust maximum likelihood, which allows gradient ascent and Expectation Maximization (EM) to take into account linear qualitative constraints. Other examples of qualitative constraints include some parameters being larger than others, bounded in a range, within ϵ of each other, etc. Various proposals have been made that exploit such constraints. Altendorf et al. [2] provide an approximate technique based on constrained convex optimization for parameter learning. Niculescu et al. [15] also provide a technique based on constrained optimization with closed form solutions for different classes of constraints. Feelders [6] provides an alternate method based on isotonic regression while Liao and Ji [12] combine gradient descent with EM. de Campos and Ji [4] also use constrained convex optimization, however, they use Dirichlet priors on the parameters to incorporate any additional knowledge. Mao and Lebanon [13] also use Dirichlet priors, but they use probabilistic constraints to allow inaccuracies in the speciﬁcation of the constraints. A major difference between our technique and previous work is on the type of constraints. Our constraints do not need to be explicitly speciﬁed by an expert. Instead, we passively observe the expert and learn from what choices are made and not made [16]. Furthermore, as we shall show later, our constraints are non-convex, preventing the direct application of existing techniques that assume linear or convex functions. We use Beta priors on the parameters, which can easily be extended to Dirichlet priors like previous work. We incorporate constraints in an augmented Bayesian network, similar to Liang et al. [11], though their constraints are on model predictions as opposed to ours which are on the parameters of the network. Finally, we also use the notion of probabilistic constraints to handle potential mistakes made by experts. 3 BACKGROUND 3.1 DIAGNOSTIC BAYES NETWORKS We consider the class of bipartite Bayes networks that are widely used as diagnostic models, though our approach can be used for networks with any structure. The network forms a sparse, directed, causal graph, where arcs go from causes to observable node variables. We use upper case to denote random variables; C for causes, and T for observables (tests). Lower case letters denote values in the domain of a variable, e.g. c ∈dom(C) = {c, ¯c}, and bold letters denote sets of variables. A set of marginally independent binary-valued node variables C with distributions Pr(C) represent unobserved causes, and condition the remaining conditionally independent binary-valued test vari2 able nodes T. Each cause conditions one or more tests; likewise each test is conditioned by one or more causes, resulting in a graph with one or more possibly multiply-connected components. The test variable distributions Pr(T\|C) incorporate the further modeling assumption of Independence of Causal Inﬂuence, the most familiar example being the Noisy-Or model [8]. To keep the exposition simple, we assume that all variables are binary and that conditional distributions are parametrized by the Noisy-Or; however, the algorithms described in the rest of the paper generalize to any discrete non-binary variable models. Conventionally, unobserved tests are ranked in a diagnostic Bayes network by their Value Of Information (VOI) conditioned on tests already observed. To be precise, VOI is the expected gain in utility if the test were to be observed. The complete computation requires a model equivalent to a partially observable Markov decision process. Instead, VOI is commonly approximated by a greedy computation of the Mutual Information between a test and the set of causes [3]. In this case, it is easy to show that Mutual Information is in turn well approximated to second order by the Gini impurity [7] as shown in Equation 1. GI(C\|T) = ∑ t Pr(T = t) [ ∑ c Pr(C = c\|T = t)(1 −Pr(C = c\|T = t)) ] (1) We will use the Gini measure as a surrogate for VOI, as a way to rank the best next test in the diagnostic sequence. 3.2 MODEL CONSISTENCY A model that is consistent with an expert would generate Gini impurity rankings consistent with the expert’s diagnostic sequence. We interpret the expert’s test choices as implying constraints on Gini impurity rankings between tests. To that effect, [1] deﬁnes the notion of Cause Consistency and Test Consistency, which indicate whether the cause and test orderings induced by the posterior distribution over causes and the VOI of each test agree with an expert’s observed choice. Assuming that the expert greedily chooses the most informative test T ∗(i.e., test that yields the lowest Gini impurity) at each step, then the model is consistent with the expert’s choices when the following constraints are satisﬁed: GI(C\|T ∗) ≤GI(C\|Ti) ∀i (2) We demonstrate next how to exploit these constraints to reﬁne the Bayes network. 4 MODEL REFINEMENT Consider a simple diagnosis example with two possible causes C1 and C2 and two tests T1 and T2 as shown in Figure 1. To keep the exposition simple, suppose that the priors for each cause are known (generally separate data is available to estimate these), but the conditional distribution of each test is unknown. Using the Noisy-OR parameterizations for the conditional distributions, the number of parameters are linear in the number of parents instead of exponential. Pr(Ti = true\|C) = 1 −(1 −θi 0) ∏ j\|Cj=true (1 −θi j) (3) Here, θi 0 = Pr(Ti = true\|Cj = false ∀j) is the leak probability that Ti will be true when none of the causes are true and θi j = Pr(Ti = true\|Cj = true, Ck = false ∀k ̸= j) is the link reliability, which indicates the independent contribution of cause Cj to the probability that test Ti will be true. In the rest of this section, we describe how to learn the θ parameters while respecting the constraints implied by test consistency. 4.1 TEST CONSISTENCY CONSTRAINTS Suppose that an expert chooses test T1 instead of test T2 during the diagnostic process. This ordering by the expert implies that the current model (parametrized by the θ’s) must be consistent with the constraint GI(C\|T2)−GI(C\|T1) ≥0. Using the deﬁnition of Gini impurity in Eq. 1, we can rewrite 3 Figure 1: Network with 2 causes and 2 tests Figure 2: Augmented network with parameters and constraints Figure 3: Augmented network extended to handle inaccurate feedback the constraint for the network shown in Fig. 1 as follows: ∑ t1 ( Pr(t1) − ∑ c1,c2 (Pr(t1\|c1, c2) Pr(c1) Pr(c2))2 Pr(t1) ) − ∑ t2 ( Pr(t2) − ∑ c1,c2 (Pr(t2\|c1, c2) Pr(c1) Pr(c2))2 Pr(t2) ) ≥0 (4) Furthermore, using the Noisy-Or encoding from Eq. 3, we can rewrite the constraint as a polynomial in the θ’s. This polynomial is non-linear, and in general, not concave. The feasible space may consist of disconnected regions. Fig. 4 shows the surface corresponding to the polynomial for the case where θi 0 = 0 and θi 1 = 0.5 for each test i, which leaves θ1 2 and θ2 2 as the only free variables. The parameters’ feasible space, satisfying the constraint consists of the two disconnected regions where the surface is positive. 4.2 AUGMENTED BAYES NETWORK Our objective is to learn the θ parameters of diagnostic Bayes networks given test constraints of the form described in Eq. 4. To deal with non-convex constraints and disconnected feasible regions, we pursue a Bayesian approach whereby we explicitly model the parameters and constraints as random variables in an augmented Bayes network (see Fig. 2). This allows us to frame the problem of learning the parameters as an inference problem in a hybrid Bayes network of discrete (T, C, V ) and continuous (Θ) variables. As we will see shortly, this augmented Bayes network provides a unifying framework to simultaneously learn from constraints and data, to deal with possibly inconsistent constraints, and to express preferences over the degree of satisfaction of the constraints. We encode the constraint derived from the expert feedback as a binary random variable V in the Bayes network. If V is true the constraint is satisﬁed, otherwise it is violated. Thus, if V is true then Θ lies in the positive region of Fig. 4, and if V is false then Θ lies in the negative region. We model the CPT for V as Pr(V \|Θ) = max(0, π), where π = GI(C\|T1) −GI(C\|T2). Note that the value of GI(C\|T) lies in the interval [0,1], so the probability π will always be normalized. The intuition behind this deﬁnition of the CPT for V is that a constraint is more likely to be satisﬁed if the parameters lie in the interior of the constraint region. We place a Beta prior over each Θ parameter. Since the test variables are conditioned on the Θ parameters that are now part of the network, their conditional distributions become known. For instance, the conditional distribution for Ti (given in Eq. 3) is fully deﬁned given the noisy-or parameters θi j. Hence the problem of learning the parameters becomes an inference problem to compute posteriors over the parameters given that the constraint is satisﬁed (and any data). In practice, it is more convenient to obtain a single value for the parameters instead of a posterior distribution since it is easier to make diagnostic predictions based on one Bayes network. We estimate the parameters by computing a maximum a posteriori (MAP) hypothesis given that the constraint is satisﬁed (and any data): Θ∗= arg maxΘ Pr(Θ\|V = true). 4 Algorithm 1 Pseudo Code for Gibbs Sampling, Stochastic Hill Climbing and Greedy Search 1 Fix observed variables, let V = true and randomly sample feasible starting state S 2 for i = 1 to #samples 3 for j = 1 to #hiddenV ariables 4 acceptSample = false; k = 0 5 repeat 6 Sample s′ from conditional of jth hidden variable Sj 7 S′ = S; Sj = s′ 8 if Sj is cause or test, then acceptSample = true 9 elseif S′ obeys constraints V∗ 10 if algo == Gibbs 11 Sample u from uniform distribution, U(0,1) 12 if u < p(S′) Mq(S′) where p and q are the true and proposal distributions and M > 1 13 acceptSample = true 14 elseif algo == StochasticHillClimbing 15 if likelihood(S′) > likelihood(S), then acceptSample = true 16 elseif algo == Greedy, then acceptSample = true 17 elseif algo == Greedy 18 k = k + 1 19 if k == maxIterations, then s′ = Sj; acceptSample = true 20 until acceptSample == true 21 Sj = s′ 4.3 MAP ESTIMATION Previous approaches for parameter learning with domain knowledge include modiﬁed versions of EM or some other optimization techniques that account for linear/convex constraints on the parameters. Since our constraints are non-convex, we propose a new approach based on Gibbs sampling to approximate the posterior distribution, from which we compute the MAP estimate. Although the technique converges to the MAP in the limit, it may require excessive time. Hence, we modify Gibbs sampling to obtain more efﬁcient stochastic hill climbing and greedy search algorithms with anytime properties. The pseudo code for our Gibbs sampler is provided in Algorithm 1. The two key steps are sampling the conditional distributions of each variable (line 6) and rejection sampling to ensure that the constraints are satisﬁed (lines 9 and 12). We sample each variable given the rest according to the following distributions: ti ∼Pr(Ti\|c, θi) ∀i (5) cj ∼Pr(Cj\|c −cj, t, θ) ∝ ∏ j Pr(Cj) ∏ i Pr(ti\|c, θi) ∀j (6) θi j ∼Pr(Θi j\|Θ −Θi j, t, c, v) ∝Pr(v\|t, Θ) ∏ i Pr(ti\|cj, θi) ∀i, j (7) The tests and causes are easily sampled from the multinomials as described in the equations above. However, sampling the θ’s is more difﬁcult due to the factor Pr(v\|Θ, t) = max(0, π), which is a truncated mixture of Betas. So, instead of sampling θ from its true conditional, we sample it from a proposal distribution that replaces max(0, π) by an un-truncated mixture of Betas equal to π + a where a is a constant that ensures that π + a is always positive. This is equivalent to ignoring the constraints. Then we ensure that the constraints are satisﬁed by rejecting the samples that violate the constraints. Once Gibbs sampling has been performed, we obtain a sample that approximates the posterior distribution over the parameters given the constraints (and any data). We return a single setting of the parameters by selecting the sampled instance with the highest posterior probability (i.e., MAP estimate). Since we will only return the MAP estimate, it is possible to speed up the search by modifying Gibbs sampling. In particular, we obtain a stochastic hill climbing algorithm by accepting a new sample only if its posterior probability improves upon that of the previous sample 5 0 0.2 0.4 0.6 0.8 1 0 0.5 1 −0.1 −0.05 0 0.05 0.1 Link Reliability of Test 2 and Cause 2 Link Reliability of Test 2 and Cause 1 Difference in Gini Impurity Figure 4: Difference in Gini impurity for the network in Fig. 1 when θ1 2 and θ2 2 are the only parameters allowed to vary. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 Link Reliability of Test 2 and Cause 1 Link Reliability of Test 2 and Cause 1 Posterior Probability Figure 5: Posterior over parameters computed through calculation after discretization. Figure 6: Posterior over parameters calculated through Sampling. (line 15). Thus, each iteration of the stochastic hill climber requires more time, but always improves the solution. As the number of constraints grows and the feasibility region shrinks, the Gibbs sampler and stochastic hill climber will reject most samples. We can mitigate this by using a Greedy sampler that caps the number of rejected samples, after which it abandons the sampling for the current variable to move on to the next variable (line 19). Even though the feasibility region is small overall, it may still be large in some dimensions, so it makes sense to try sampling another variable (that may have a larger range of feasible values) when it is taking too long to ﬁnd a new feasible value for the current variable. 4.4 MODEL REFINEMENT WITH INCONSISTENT CONSTRAINTS So far, we have assumed that the expert’s actions generate a feasible region as a consequence of consistent constraints. We handle inconsistencies by further extending our augmented diagnostic Bayes network. We treat the observed constraint variable, V , as a probabilistic indicator of the true constraint V ∗as shown in Figure 3. We can easily extend our techniques for computing the MAP to cater for this new constraint node by sampling an extra variable. 5 EVALUATION AND EXPERIMENTS 5.1 EVALUATION CRITERIA Formally, for M ∗, the true model that we aim to learn, the diagnostic process determines the choice of best next test as the one with the smallest Gini impurity. If the correct choice for the next test is known (such as demonstrated by an expert), we can use this information to include a constraint on the model. We denote by V+ the set of observed constraints and by V∗the set of all possible constraints that hold for M ∗. Having only observed V+, our technique will consider any M + ∈M+ as a possible true model, where M+ is the set of all models that obey V +. We denote by M∗the set of all models that are diagnostically equivalent to M ∗(i.e., obey V ∗and would recommend the same steps as M ∗) and by M MAP V+ the particular model obtained by MAP estimation based on the constraints V+. Similarly, when a dataset D is available, we denote by M MAP D the model obtained by MAP estimation based on D and by M MAP DV+, the model based on D and V+. Ideally we would like to ﬁnd the true underlying model M ∗, hence we will report the KL divergence between the models found and M ∗. However, other diagnostically equivalent M ∗may recommend the same tests as M ∗and thus have similar constraints, so we also report test consistency with M ∗ (i.e., # of recommended tests that are the same). 5.2 CORRECTNESS OF MODEL REFINEMENT Given V∗, our technique for model adjustment is guaranteed to choose a model M MAP ∈M∗by construction. If any constraint V ∗∈V∗is violated, the rejection sampling step of our technique 6 Figure 7: Mean KLdivergence and one standard deviation for a 3 cause 3 test network on learning with data, constraints and data+constraints. 1 2 3 4 5 6 7 0 10 20 30 40 50 60 70 80 90 100 Number of constraints used Percentage of tests correctly predicted Data Only Constraints Only Data+Constraints Figure 8: Test Consistency for a 3 cause 3 test network on learning with data, constraints and data+constraints. 10 0 10 1 10 2 10 3 −20 −18 −16 −14 −12 −10 −8 Elapsed Time (plotted on log scale from 0 to 1500 seconds) Negative Log Likelihood of MAP Estimate Comparing convergence of Different Techniques Gibbs Sampling Stochastic Hill Climbing Greedy Sampling Figure 9: Convergence rate comparison. would reject that set of parameters. To illustrate this, consider the network in Fig. 2. There are six parameters (four link reliabilities and two leak parameters). Let us ﬁx the leak parameters and the link reliability from the ﬁrst cause to each test. Now we can compute the posterior surface over the two variable parameters after discretizing each parameter in small steps and then calculating the posterior probability at each step as shown in Fig. 5. We can compare this surface with that obtained after Gibbs sampling using our technique as shown in Fig. 6. We can see that our technique recovers the posterior surface from which we can compute the MAP. We obtain the same MAP estimate with the stochastic hill climbing and greedy search algorithms. 5.3 EXPERIMENTAL RESULTS ON SYNTHETIC PROBLEMS We start by presenting our results on a 3-cause by 3-test fully-connected bipartite Bayes network. We assume that there exists some M ∗∈M∗that we want to learn given V+. We use our technique to ﬁnd M MAP. To evaluate M MAP, we ﬁrst compute the constraints, V∗for M ∗to get the feasible region associated with the true model. Next, we sample 100 other models from this feasible region that are diagnostically equivalent. We compare these models with M MAP (after collecting 200 samples with non-informative priors for the parameters). We compute the KL-divergence of M MAP with respect to each sampled model. We expect KLdivergence to decrease as the number of constraints in V+ increases since the feasible region becomes smaller. Figure 7 conﬁrms this trend and shows that M MAP DV+ has lower mean KL-divergence than M MAP V+ , which has lower mean KL-divergence than M MAP D . The data points in D are limited to the results of the diagnostic sessions needed to obtain V+. As constraints increase, more data is available and so the results for the data-only approach also improve with increasing constraints. We also compare the test consistency when learning from data only, constraints only or both. Given a ﬁxed number of constraints, we enumerate the unobserved trajectories, and then compute the highest ranked test using the learnt model and the sampled true models, for each trajectory. The test consistency is reported as a percentage, with 100% consistency indicating that the learned and true models had the same highest ranked tests on every trajectory. Figure 8 presents these percentatges for the greedy sampling technique (the results are similar for the other techniques). It again appears that learning parameters with both constraints and data is better than learning with only constraints, which is most of the times better than learning with only data. Figure 9 compares the convergence rate of each technique to ﬁnd the MAP estimate. As expected, Stochastic Hill Climbing and Greedy Sampling take less time than Gibbs sampling to ﬁnd parameter settings with high posterior probability. 5.4 EXPERIMENTAL RESULTS ON REAL-WORLD PROBLEMS We evaluate our technique on a real-world diagnostic network collected and reported by Agosta et al. [1], where the authors collected detailed session logs over a period of seven weeks in which the 7 Figure 10: Diagnostic Bayesian network collected from user trials and pruned to retain sub-networks with at least one constraint 6 8 10 12 14 16 18 20 22 1 2 3 4 5 6 7 8 Number of constraints used KL−divergence of when computing joint over all tests Data Only Constraints Only Data+Constraints Figure 11: KL divergence comparison as the number of constraints increases for the real world problem. entire diagnostic sequence was recorded. The sequences intermingle model building and querying phases. The model network structure was inferred from an expert’s sequence of positing causes and tests. Test-ranking constraints were deduced from the expert’s test query sequences once the network structure is established. The 157 sessions captured over the seven weeks resulted in a Bayes network with 115 tests, 82 root causes and 188 arcs. The network consists of several disconnected sub-networks, each identiﬁed with a symptom represented by the ﬁrst test in the sequence, and all subsequent tests applied within the same subnet. There were 20 sessions from which we were able to observe trajectories with at least two tests, resulting in a total of 32 test constraints. We pruned our diagnostic network to remove the sub-networks with no constraints to get a Bayes network with 54 tests, 30 root causes, and 67 parameters divided in 7 sub-networks, as shown in Figure 10, on which we apply our model reﬁnement technique to learn the parameters for each sub-network separately. Since we don’t have the true underlying network and the full set of constraints (more constraints could be observed in future diagnostic sessions), we treated the 32 constraints as if they were V∗ and the corresponding feasible region M∗as if it contained models diagnostically equivalent to the unknown true model. Figure 11 reports the KL divergence between the models found by our algorithms and sampled models from M∗as we increase the number of constraints. With such limited constraints and consequently large feasible regions, it is not surprising that the variation in KL divergence is large. Again, the MAP estimate based on both the constraints and the data has lower KL divergence than constraints only and data only. 6 CONCLUSION AND FUTURE WORK In summary, we presented an approach that can learn the parameters of a Bayes network based on constraints implied by test consistency and any data available. While several approaches exist to incorporate qualitative constraints in learning procedures, our work makes two important contributions: First, this is the ﬁrst approach that exploits implicit constraints based on value of information assessments. Secondly it is the ﬁrst approach that can handle non-convex constraints. We demonstrated the approach on synthetic data and on a real-world manufacturing diagnostic problem. Since data is generally sparse in diagnostics, this work makes an important advance to mitigate the model acquisition bottleneck, which has prevented the widespread application of diagnostic networks so far. In the future, it would be interesting to generalize this work to reinforcement learning in applications where data is sparse, but constraints may be inferred from expert interactions. Acknowledgments This work was supported by a grant from Intel Corporation. 8 References [1] John Mark Agosta, Omar Zia Khan, and Pascal Poupart. 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4,402	Structure Learning for Optimization Shulin (Lynn) Yang Department of Computer Science University of Washington Seattle, WA 98195 yang@cs.washington.edu Ali Rahimi Red Bow Labs Berkeley, CA 94704 ali@redbowlabs.com Abstract We describe a family of global optimization procedures that automatically decompose optimization problems into smaller loosely coupled problems. The solutions of these are subsequently combined with message passing algorithms. We show empirically that these methods produce better solutions with fewer function evaluations than existing global optimization methods. To develop these methods, we introduce a notion of coupling between variables of optimization. This notion of coupling generalizes the notion of independence between random variables in statistics, sparseness of the Hessian in nonlinear optimization, and the generalized distributive law. Despite its generality, this notion of coupling is easier to verify empirically, making structure estimation easy, while allowing us to migrate well-established inference methods on graphical models to the setting of global optimization. 1 Introduction We consider optimization problems where the objective function is costly to evaluate and may be accessed only by evaluating it at requested points. In this setting, the function is a black box, and have no access to its derivative or its analytical structure. We propose solving such optimization problems by ﬁrst estimating the internal structure of the black box function, then optimizing the function with message passing algorithms that take advantage of this structure. This lets us solve global optimization problems as a sequence of small grid searches that are coordinated by dynamic programming. We are motivated by the problem of tuning the parameters of computer programs to improve their accuracy or speed. For the programs that we consider, it can take several minutes to evaluate these performance measures under a particular parameter setting. Many optimization problems exhibit only loose coupling between many of the variables of optimization. For example, to tune the parameters of an audio-video streaming program, the parameters of the audio codec could conceivably be tuned independently of the parameters of the video codec. Similarly, to tune the networking component that glues these codecs together it sufﬁces to consider only a few parameters of the codecs, such as their output bit-rate. Such notions of conditional decoupling are conveniently depicted in a graphical form that represents the way the objective function factors into a sum or product of terms each involving only a small subset of the variables. This factorization structure can then be exploited by optimization procedures such as dynamic programming on trees or junction trees. Unfortunately, the factorization structure of a function is difﬁcult to estimate from function evaluation queries only. We introduce a notion of decoupling that can be more readily estimated from function evaluations. At the same time, this notion of decoupling is more general than the factorization notion of decoupling in that functions that do not factorize may still exhibit this type of decoupling. We say that two variables are decoupled if the optimal setting of one variable does not depend on the setting of the other. This is formalized below in a way that parallels the notion of conditional decoupling between random variables in statistics. This parallel allows us to migrate much of the machinery developed 1 for inference on graphical models to global optimization . For example, decoupling can be visualized with a graphical model whose semantics are similar to those of a Markov network. Analogs of the max-product algorithm on trees, the junction tree algorithm, and loopy belief propagation can be readily adapted to global optimization. We also introduce a simple procedure to estimate decoupling structure. The resulting recipe for global optimization is to ﬁrst estimate the decoupling structure of the objective function, then to optimize it with a message passing algorithm that utilises this structure. The message passing algorithm relies on a simple grid search to solve the sub-problems it generates. In many cases, using the same number of function evaluations, this procedure produces solutions with objective values that improve over those produced by existing global optimizers by as much as 10%. This happens because knowledge of the independence structure allows this procedure to explore the objective function only along directions that cause the function to vary, and because the grid search that solves the sub-problems does not get stuck in local minima. 2 Related work The idea of estimating and exploiting loose coupling between variables of optimization appears implicitly in Quasi-Newton methods that numerically estimate the Hessian matrix, such as BFGS (Nocedal & Wright, 2006, Chap. 6). Indeed, the sparsity pattern of the Hessian indicates the pairs of terms that do not interact with each other in a second-order approximation of the function. This is strictly a less powerful notion of coupling than the factorization model, which we argue below, is in turn less powerful than our notion of decoupling. Others have proposed approximating the objective function while simultaneously optimizing over it Srinivas et al. (2010). The procedure we develop here seeks only to approximate decoupling structure of the function, a much simpler task to carry out accurately. A similar notion of decoupling has been explored in the decision theory literature Keeney & Raiffa (1976); Bacchus & Grove (1996), where decoupling was used to reason about preferences and utilities during decision making. In contrast, we use decoupling to solve black-box optimization problems and present a practical algorithm to estimate the decoupling structure. 3 Decoupling between variables of optimization A common way to minimize an objective function over many variables is to factorize it into terms, each of which involves only a small subset of the variables Aji & McEliece (2000). Such a representation, if it exists, can be optimized via a sequence of small optimization problems with dynamic programming. This insight motivates message passing algorithms for inference on graphical models. For example, rather than minimizing the function f1(x, y, z) = g1(x, y)+g2(y, z) over its three variables simultaneously, one can compute the function g3(y) = minz g2(y, z), then the function g4(x) = miny g1(x, y) + g3(y), and ﬁnally minimizing g4 over x. A similar idea works for the function f2(x, y, z) = g1(x, y)g2(y, z) and indeed, whenever the operator that combines the factors is associative, commutative, and allows the “min” operator to distribute over it. However, it is not necessary for a function to factorize for it to admit a simple dynamic programming procedure. For example, a factorization for the function f3(x, y, z) = x2y2z2 + x2 + y2 + z2 is elusive, yet the arguments of f3 are decoupled in the sense that the setting of any two variables does not affect the optimal setting of the third. For example, argminx f3(x, y0, z0) is always x = 0, and similarly for y and z. This decoupling allows us to optimize over the variables separately. This is not a trivial property. For example, the function f4(x, y, z) = (x −y)2 + (y −z)2 exhibits no such decoupling between x and y because the minimizer of argminx f4(x, y0, z0) is y0, which is obviously a function of the second argument of f. The following deﬁnition formalizes this concept: Deﬁnition 1 (Blocking and decoupling). Let f : ⌦! R be a function on a compact domain and let X ⇥Y ⇥Z ✓⌦be a subset of the domain. We say that the coordinates Z block X from Y under f if the set of minimizers of f over X does not change for any setting of the variables Y given a setting of the variables Z: 8 Y12Y,Y22Y Z2Z argmin X2X f(X, Y1, Z) = argmin X2X f(X, Y2, Z). We will say that X and Y are decoupled conditioned on Z under f, or X ?f Y !!Z, if Z blocks X from Y and Z blocks Y from X under f at the same time. 2 We will simply say that X and Y are decoupled, or X ?f Y, when X ?f Y !!Z, ⌦= X ⇥Y ⇥Z, and f is understood from context. For a given function f(x1, . . . , xn), decoupling between the variables can be represented graphically with an undirected graph analogous to a Markov network: Deﬁnition 2. A graph G = ({x1, . . . , xn}, E) is a coupling graph for a function f(x1, . . . , xn) if (i, j) /2 E implies xi and xj are decoupled under f. The following result mirrors the notion of separation in Markov networks and makes it easy to reason about decoupling between groups of variables with coupling graphs (see the appendix for a proof): Proposition 1. Let X, Y, Z be groups of nodes in a coupling graph for a function f. If every path from a node in X to a node in Y passes through a node in Z, then X ?f Y !!Z. Functions that factorize as a product of terms exhibit this type of decoupling. For subsets of variables X, Y, Z, we say X is conditionally separated from Y by Z by factorization, or X ?⌦Y !!Z, if X and Y are separated in that way in the Markov network induced by the factorization of f. The following is a generalization of the familiar result that factorization implies the global Markov property (Koller & Friedman, 2009, Thm. 4.3) and follows from Aji & McEliece (2000): Theorem 1 (Factorization implies decoupling). Let f(x1, . . . , xn) be a function on a compact domain, and let X1, . . . , XS, X, Y, Z be subsets of {x1, . . . , xn}. Let ⌦be any commutative associative semi-ring operator over which the min operator distributes. If f factorizes as f(x1, . . . , xn) = ⌦S s=1 gs(Xs), then X ?f Y !!Z whenever X ?⌦Y !!Z. However decoupling is strictly more powerful than factorization. While X ?⌦Y implies X ?f Y, the reverse is not necessarily true: there exist functions that admit no factorization at all, yet whose arguments are completely mutually decoupled. Appendix B gives an example. 4 Optimization procedures that utilize decoupling When a cost function factorizes, dynamic programming algorithms can be used to optimize over the variables Aji & McEliece (2000). When a cost function exhibits decoupling as deﬁned above, the same dynamic programming algorithms can be applied with a few minor modiﬁcations. The algorithms below refer to a function f whose arguments are partitioned over the sets X1, . . . , Xn. Let X⇤ i denote the optimal value of Xi 2 Xi. We will take simplifying liberties with the order of the arguments of f when this causes no ambiguity. We will also replace the variables that do not participate in the optimization (per decoupling) with an ellipsis. 4.1 Optimization over trees Suppose the coupling graph between some partitioning X1, . . . , Xm of the arguments of f is treestructured, in the sense that Xi ?f Xj unless the edge (i, j) is in the tree. To optimize over f with dynamic programming, deﬁne X0 arbitrarily as the root of the tree, let pi denote the index of the parent of Xi, and let C1 i , C2 i , . . . denote the indices of its children. At each leaf node `, construct the functions ˆX` (Xp`) := argmin X`2X` f(X`, Xp`). (1) By decoupling, the optimal value of X` depends only on the optimal value of its parent, so X⇤ ` = ˆX`(X⇤ p`). For all other nodes i, deﬁne recursively starting from the parents of the leaf nodes the functions ˆXi(Xpi) = argmin Xi2Xi f(Xi, Xpi, ˆXC1 i (Xi), ˆXC2 i (Xi), . . .) (2) Again, the optimal value of Xi depends only on the optimal setting of its parent, X⇤ pi , and it can be veriﬁed that X⇤ i = ˆXi(X⇤ pi). In our implementation of this algorithm, to represent a function ˆXi(X), we discretize its argument into a grid, and store the function as a table. To compute the entries of the table, a subordinate global optimizer computes the minimization that appears in the deﬁnition of ˆXi. 3 4.2 Optimization over junction trees Even when the coupling graph for a function is not tree-structured, a thin junction tree can often be constructed for it. A variant of the above algorithm that mirrors the junction tree algorithm can be used to efﬁciently search for the optima of the function. Recall that a tree T of cliques is a junction tree for a graph G if it satisﬁes the following three properties: there is one path between each pair of cliques; for each clique C of G there is some clique A in T such that C ✓A; for each pair of cliques A and B in T that contain node i of G, each clique on the unique path between A and B also contains i. These properties guarantee that T is tree-structured, that it covers all nodes and edges in G, and that two nodes v and u in two different cliques Xi and Xj are decoupled from each other conditioned on the union of the cliques on the path between u and v in T. Many heuristics exist for constructing a thin junction tree for a graph Jensen & Graven-Nielsen (2007); Huang & Darwiche (1996). To search for the minimizers of f, using a junction tree for its coupling graph, denote by Xij := Xi \Xj the intersection of the groups of variables Xi and Xj and by Xi\j = Xi \Xj the set of nodes in Xi but not in Xj. At every leaf clique ` of the junction tree, construct the function ˆX` (X`,p`) := argmin X`\p`2X`\p` f(X`). (3) For all other cliques i, compute recursively starting from the parents of the leaf cliques ˆXi(Xi,pi) = argmin Xi,pi2Xi\pi f(Xi, ˆXC1 i (Xi,C1 i ), ˆXC2 i (Xi,C2 i ), . . .). (4) As before, decoupling between the cliques, conditioned on the intersection of the cliques, guarantees that ˆXi(X⇤ i,pi) = X⇤ i . And as before, our implementation of this algorithm stores the intermediate functions as tables by discretizing their arguments. 4.3 Other strategies When the cliques of the junction tree are large, the subordinate optimizations in the above algorithm become costly. In such cases, the following adaptations of approximate inference algorithms are useful: • The algorithm of Section 4.1 can be applied to a maximal spanning tree of the coupling graph. • Analogously to Loopy Belief Propagation Pearl (1997), an arbitrary neighbor of each node can be declared as its parent, and the steps of Section 4.1 can be applied to each node until convergence. • Loops in the coupling graph can be broken by conditioning on a node in each loop, resulting in a tree-structured coupling graph conditioned on those nodes. The optimizer of Section 4.1 then searches for the minima conditioned on the value of those nodes in the inner loop of a global optimizer that searches for good settings for the conditioned nodes. 5 Graph structure learning It is possible to estimate decoupling structure between the arguments of a function f with the help of a subordinate optimizer that only evaluates f. A straightforward application of deﬁnition 1 to assess empirically whether groups of variables X and Y are decoupled conditioned on a group of variables Z would require comparing the minimizer of f over X for every possible value of Z and Y. This is not practical because it is at least as difﬁcult as minimizing f. Instead, we rely on the following proposition, which follows directly from 1: Proposition 2 (Invalidating decoupling). If for some Z 2 Z and Y0, Y1 2 Y, we have argminX2X f(X, Y0, Z) 6= argminX2X f(X, Y1, Z), then X 6?f Y\|Z. Following this result, an approximate coupling graph can be constructed by positing and invalidating decoupling relations. Starting with a graph containing no edges, we consider all groupings X = 4 {xi}, Y = {xj}, Z = ⌦\{xi, xj}, of variables x1, . . . , xn. We posit various values of Z 2 Z, Y0 2 Y and Y1 2 Y under this grouping, and compute the minimizers over X 2 X of f(X, Y0, Z) and f(X, Y1, Z) with a subordinate optimizer. If the minimizers differ, then by the above proposition, X and Y are not decoupled conditioned on Z, and an edge is added between xi and xj in the graph. Algorithm 1 summarizes this procedure. Algorithm 1 Estimating the coupling graph of a function. input A function f : X1 ⇥· · · Xn ! R, with Xi compact; A discretization ˆ Xi of Xi; A similarity threshold ✏> 0; The number of times, NZ, to sample Z. output A coupling graph G = ([x1, . . . , xn], E). E ; for i, j 2 [1, . . . , n]; y0, y1 2 ˆ Xj; 1 . . . NZ do Z ⇠U( ˆ X1 ⇥· · · ⇥ˆ Xn \ ˆ Xi ⇥ˆ Xj) ˆx0 argminx2 ˆ Xi f(x, y0, Z); ˆx1 argminx2 ˆ Xi f(x, y1, Z) if kˆx0 −ˆx1k ≥✏then E E [ {(i, j)} end if end for In practice, we ﬁnd that decoupling relationships are correctly recovered if values of Y0 and Y1 are chosen by quantizing Y into a set ˆY of 4 to 10 uniformly spaced discrete values and exhaustively examining the settings of Y0 and Y1 in ˆY. A few values of Z (fewer than ﬁve) sampled uniformly at random from a similarly discretized set ˆZ sufﬁce. 6 Experiments We evaluate a two step process for global optimization: ﬁrst estimating decoupling between variables using the algorithm of Section 5, then optimizing with this structure using an algorithm from Section 4. Whenever Algorithm 1 detects tree-structured decoupling, we use the tree optimizer of Section 4.1. Otherwise we either construct a junction tree and apply the junction tree optimizer of Section 4.2 if the junction tree is thin, or we approximate the graph with a maximum spanning tree and apply the tree solver of Section 4.1. We compare this approach with three state-of-the-art black-box optimization procedures: Direct Search Perttunen et al. (1993) (a deterministic space carving strategy), FIPS Mendes et al. (2004) (a biologically inspired randomized algorithm), and MEGA Hazen & Gupta (2009) (a multiresolution search strategy with numerically computed gradients). We use a publicly available implementation of Direct Search 1, and an implementation of FIPS and MEGA available from the authors of MEGA. We set the number of particles for FIPS and MEGA to the square of the dimension of the problem plus one, following the recommendation of their authors. As the subordinate optimizer for Algorithm 1, we use a simple grid search for all our experiments. As the subordinate optimizer for the algorithms of Section 4, we experiment with grid search and the aforementioned state-of-the-art global optimizers. We report results on both synthetic and real optimization problems. For each experiment, we report the quality of the solution each algorithm produces after a preset number of function calls. To vary the number of function calls the baseline methods invoke, we vary the number of time they iterate. Since our method does not iterate, we vary the number of function calls its subordinate optimizer invokes (when the subordinate optimizer is grid search, we vary the grid resolution). The experiments demonstrate that using grid search as a subordinate strategy is sufﬁcient to produce better solutions than all the other global optimizers we evaluated. 1Available from http://www4.ncsu.edu/˜ctk/Finkel_Direct/. 5 Table 1: Value of the iterates of the functions of Table 2 after 10,000 function evaluations (for our approach, this includes the function evaluations for structure learning). MIN is the ground truth optimal value when available. GR is the number of discrete values along each dimension for optimization. Direct Search (DIR), FIPS and MEGA are three state-of-the-art algorithms for global optimization. Function (n=50) min GR Ours DIR FIPS MEGA Colville 0 100 0 3e-6 2e-14 3.75 Levy 0 400 0.013 2.80 4.20 3.22 Michalewics n/a 400 -48.9 -18.2 -18.4 -1.3e-3 Rastrigin 0 400 0 0 23.6 4.2e-3 Schwefel 0 400 8.6 1.9e4 1.6e4 1.4e4 Dixon&Price 0 20 1 0.667 16.8 0.914 Rosenbrock 0 20 0 2.9e4 5.7e4 48.4 Trid n/a 20 -2.2e4 -185 3.3e4 -41 Powell 0 6 19.4 324 121 0.014 6.1 Synthetic objective functions We evaluated the above strategies on a standard benchmark of synthetic optimization problems 2 shown in Appendix A. These are functions of 50 variables and are used as black-box functions in our experiments. In these experiments, the subordinate grid search of Algorithm 1 discretized each dimension into four discrete values. The algorithms of Section 4 also used grid search as a subordinate optimizer. For this grid search, each dimension was discretized into GR = ⇣ Emax Nmc ⌘ 1 Smc discrete values where Emax is a cap on the number of function evaluations to perform, Smc is the size of the largest clique in the junction tree, and Nmc is the number of nodes in the junction tree. Figure 1 shows that in all cases, Algorithm 1 recovered decoupling structure exactly even for very coarse grids. Values of NZ greater than 1 did not improve the quality of the recovered graph, justifying our heuristic of keeping NZ small. We used NZ = 1 in the remainder of this subsection. Table 1 summarizes the quality of the solutions produced by the various algorithms after 10,000 function evaluations. Our approach outperformed the others on most of these problems. As expected, it performed particularly well on functions that exhibit sparse coupling, such as Levy, Rastrigin, and Schwefel. In addition to achieving better solutions given the same number of function evaluations, our approach also imposed lower computational overhead than the other methods: to process the entire benchmark of this section takes our approach 2.2 seconds, while Direct Search, FIPS and MEGA take 5.7 minutes, 3.7 minutes and 53.3 minutes respectively. Colville Grid resolution 4.9e3 1.1e4 1.9e4 3.1e4 4.4e4 2 3 4 5 6 100% 080% 060% 040% 020% 000% Number of evaluations Levy Grid resolution 100% 080% 060% 040% 020% 000% Number of evaluations 4.9e3 1.1e4 1.9e4 3.1e4 4.4e4 2 3 4 5 6 Rosenbrock Grid resolution 100% 080% 060% 040% 020% 000% Number of evaluations 4.9e3 1.1e4 1.9e4 3.1e4 4.4e4 2 3 4 5 6 Powell Grid resolution 100% 080% 060% 040% 020% 000% Number of evaluations 4.9e3 1.1e4 1.9e4 3.1e4 4.4e4 2 3 4 5 6 Figure 1: Very coarse gridding is sufﬁcient in Algorithm 1 to correctly recover decoupling structure. The plots show percentage of incorrectly recovered edges in the coupling graph on four synthetic cost functions as a function of the grid resolution (bottom x-axis) and the number of function evaluations (top x-axis). NZ = 1 in these experiments. 6.2 Experiments on real applications We considered the real-world problem of automatically tuning the parameters of machine vision and machine learning programs to improve their accuracy on new datasets. We sought to tune the 2Acquired from http://www-optima.amp.i.kyoto-u.ac.jp/member/student/hedar/ Hedar_files/go.htm. 6 parameters of a face detector, a document topic classiﬁer, and a scene recognizer to improve their accuracy on new application domains. Automatic parameter tuning allows a user to quickly tune a program’s default parameters to their speciﬁc application domain without tedious trial and error. To perform this tuning automatically, we treated the accuracy of a program as a black box function of the parameter values passed to it. These were challenging optimization problems because the derivative of the function is elusive and each function evaluation can take minutes. Because the output of a program tends to depend in a structured way on its parameters, our method achieved signiﬁcant speedups over existing global optimizers. 6.2.1 Face detection The ﬁrst application was a face detector. The program has ﬁve parameters: the size, in pixels, of the smallest face to consider, the minimum distance, in pixels, between detected faces; a ﬂoating point subsampling rate for building a multiresolution pyramid of the input image; a boolean ﬂag that determines whether to apply non-maximal suppression; and the choice of one of four wavelets to use. Our goal was to minimize the detection error rate of this program on the GENKI-SZSL dataset of 3, 500 faces 3. Depending on the parameter settings, evaluating the accuracy of the program on this dataset takes between 2 seconds and 2 minutes. Algorithm 1 was run with a grid search as a subordinate optimizer with three discrete values along the continuous dimensions. It invoked 90 function evaluations and produced a coupling graph wherein the ﬁrst three of the above parameters formed a clique and where the remaining two parameter were decoupled of the others. Given this coupling graph, our junction tree optimizer with grid search (with the continuous dimensions quantized into 10 discrete values) invoked 1000 function evaluations, and found parameter settings for which the accuracy of the detector was 7% better than the parameter settings found by FIPS and Direct Search after the same number of function evaluations. FIPS and Direct Search fail to improve their solution even after 1800 evaluations. MEGA fails to improve over the initial detection error of 50.84% with any number of iterations. To evaluate the accuracy of our method under different numbers of function invocations, we varied the grid resolution between 2 to 12. See Figure 2. These experiments demonstrate how a grid search can help overcome local minima that cause FIPS and Direct Search to get stuck. 0 500 1000 1500 20 40 60 80 Number of evaluations Face detector classification error (%) Junction tree solver with grid search Direct search FIPS Figure 2: Depending on the number of function evaluations allowed, our method produces parameter settings for the face detector that are better than those recovered by FIPS or Direct Search by as much as 7%. 6.2.2 Scene recognition The second application was a visual scene recognizer. It extracts GIST features Oliva & Torralba (2001) from an input image and classiﬁes these features with a linear SVM. Our task was to tune the six parameters of GIST to improve the recognition accuracy on a subset of the LabelMe dataset 4, which includes images of scenes such as coasts, mountains, streets, etc. The parameters of the recognizer include a radial cut-off frequency (in cycles/pixel) of a circular ﬁlter that reduces illumination effects, the number of bins in a radial histogram of the response of a spatial spacial ﬁlter, and the number of image regions in which to compute these histograms. Evaluating the classiﬁcation error under a set of parameters requires extracting GIST features with these parameters on a training set, training a linear SVM, then applying the extractor and classiﬁer to a test set. Each evaluation takes between 10 and 20 minutes depending on the parameter settings. 3Available from http://mplab.ucsd.edu. 4Available from http://labelme.csail.mit.edu. 7 Algorithm 1 was run with a grid search as the subordinate optimizer, discretizing the search space into four discrete values along each dimension. This results in a graph that admits no thin junction tree, so we approximate it with a maximal spanning tree. We then apply the tree optimizer of Section 4.1 using as subordinate optimizers Direct Search, FIPS, and grid search (with ﬁve discrete values along each dimension). After a total of roughly 300 function evaluations, the tree optimizer with FIPS produces parameters that result in a classiﬁcation error of 29.17%. With the same number of function evaluations, Direct Search and FIPS produce parameters that resulted in classiﬁcation errors of 33.33% and 31.13% respectively. The tree optimizer with Direct Search and grid search as subordinate optimizers resulted in error rates of 31.72% and 33.33%. In this application, the proposed method enjoys only modest gains of ⇠2% because the variables are tightly coupled, as indicated by the denseness of the graph and the thickness of the junction tree. 6.2.3 Multi-class classiﬁcation The third application was to tune the hyperparameters of a multi-class SVM classiﬁer on the RCV1v2 text categorization dataset 5. This dataset consists of a training set of 23,149 documents and a test set of 781,265 documents each labeled with one of 101 topics Lewis et al. (2004). Our task was to tune the 101 regularization parameters of the 1 vs. all classiﬁers that comprise a multi-class classiﬁer. The objective was the so-called macro-average F-score Tague (1981) on the test set. The F score for one category is F = 2rp/(r + p), where r and p are the recall and precision rates for that category. The macro-average F score is the average of the F scores over all categories. Each evaluation requires training the classiﬁer using the given hyperparameters and evaluating the resulting classiﬁer on the test set, and takes only a second since the text features have been precomputed. Algorithm 1 with grid search as a subordinate optimizer with a grid resolution of three discrete values along each dimension found no coupling between the hyperparameters. As a result, the algorithms of Section 4.1 reduce to optimizing over each one-dimensional parameter independently. We carried out these one-dimensional optimizations with Direct Search, FIPS, and grid search (discretizing each dimension into 100 values). After roughly 100,000 evaluations, these resulted in similar scores of F = 0.6764, 0.6720, and 0.6743, respectively. But with the same number of evaluations, off-theshelf Direct Search and FIPS result in scores of F = 0.6324 and 0.6043, respectively, nearly 11% worse. The cost of estimating the structure in this problem was large, since it grows quadratically with the number of classes, but worth the effort because it indicated that each variable should be optimized independently, ultimately resulting in huge speedups 6. 7 Conclusion We quantiﬁed the coupling between variables of optimization in a way that parallels the notion of independence in statistics. This lets us identify decoupling between variables in cases where the function does not factorize, making it strictly stronger than the notion of decoupling in statistical estimation. This type of decoupling is also easier to evaluate empirically. Despite these differences, this notion of decoupling allows us to migrate to global optimization many of the message passing algorithms that were developed to leverage factorization in statistics and optimization. These include belief propagation and the junction tree algorithm. We show empirically that optimizing cost functions by applying these algorithms to an empirically estimated decoupling structure outperforms existing black box optimization procedures that rely on numerical gradients, deterministic space carving, or biologically inspired searches. Notably, we observe that it is advantageous to decompose optimization problems into a sequence of small deterministic grid searches using this technique, as opposed to employing existing black box optimizers directly. 5Available from http://trec.nist.gov/data/reuters/reuters.html. 6After running these experiments, we discovered a result of Fan & Lin (2007) showing that optimizing the macro-average F-measure is equivalent to optimizing per-category F-measure, thereby validating decoupling structure recovered by Algorithm 1. 8 References Aji, S. and McEliece, R. The generalized distributive law and free energy minimization. IEEE Transaction on Informaion Theory, 46(2), March 2000. Bacchus, F. and Grove, A. 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4,403	Multiclass Boosting: Theory and Algorithms Mohammad J. Saberian Statistical Visual Computing Laboratory, University of California, San Diego saberian@ucsd.edu Nuno Vasconcelos Statistical Visual Computing Laboratory, University of California, San Diego nuno@ucsd.edu Abstract The problem of multi-class boosting is considered. A new framework, based on multi-dimensional codewords and predictors is introduced. The optimal set of codewords is derived, and a margin enforcing loss proposed. The resulting risk is minimized by gradient descent on a multidimensional functional space. Two algorithms are proposed: 1) CD-MCBoost, based on coordinate descent, updates one predictor component at a time, 2) GD-MCBoost, based on gradient descent, updates all components jointly. The algorithms differ in the weak learners that they support but are both shown to be 1) Bayes consistent, 2) margin enforcing, and 3) convergent to the global minimum of the risk. They also reduce to AdaBoost when there are only two classes. Experiments show that both methods outperform previous multiclass boosting approaches on a number of datasets. 1 Introduction Boosting is a popular approach to classiﬁer design in machine learning. It is a simple and effective procedure to combine many weak learners into a strong classiﬁer. However, most existing boosting methods were designed primarily for binary classiﬁcation. In many cases, the extension to Mary problems (of M > 2) is not straightforward. Nevertheless, the design of multi-class boosting algorithms has been investigated since the introduction of AdaBoost in [8]. Two main approaches have been attempted. The ﬁrst is to reduce the multiclass problem to a collection of binary sub-problems. Methods in this class include the popular “one vs all” approach, or methods such as “all pairs”, ECOC [4, 1], AdaBoost-M2 [7], AdaBoost-MR [18] and AdaBoostMH [18, 9]. The binary reduction can have various problems, including 1) increased complexity, 2) lack of guarantees of an optimal joint predictor, 3) reliance on data representations, such as adding one extra dimension that includes class numbers to each data point [18, 9], that may not necessarily enable effective binary discrimination, or 4) using binary boosting scores that do not represent true class probabilities [15]. The second approach is to boost an M-ary classiﬁer directly, using multiclass weak learners, such as trees. Methods of this type include AdaBoost-M1[7], SAMME[12] and AdaBoost-Cost [16]. These methods require strong weak learners which substantially increase complexity and have high potential for overﬁtting. This is particularly problematic because, although there is a uniﬁed view of these methods under the game theory interpretation of boosting [16], none of them has been shown to maximize the multiclass margin. Overall, the problem of optimal and efﬁcient M-ary boosting is still not as well understood as its binary counterpart. In this work, we introduce a new formulation of multi-class boosting, based on 1) an alternative deﬁnition of the margin for M-ary problems, 2) a new loss function, 3) an optimal set of codewords, and 4) the statistical view of boosting, which leads to a convex optimization problem in a multidimensional functional space. We propose two algorithms to solve this optimization: CD-MCBoost, which is a functional coordinate descent procedure, and GD-MCBoost, which implements functional gradient descent. The two algorithms differ in terms of the strategy used to update the multidimensional predictor. CD-MCBoost supports any type of weak learners, updating one component of the predictor per boosting iteration, GD-MCBoost requires multiclass weak learners but updates all 1 components simultaneously. Both methods directly optimize the predictor of the multiclass problem and are shown to be 1) Bayes consistent, 2) margin enforcing, and 3) convergent to the global minimum of the classiﬁcation risk. They also reduce to AdaBoost for binary problems. Experiments show that they outperform comparable prior methods on a number of datasets. 2 Multiclass boosting We start by reviewing the fundamental ideas behind the classical use of boosting for the design of binary classiﬁers, and then extend these ideas to the multiclass setting. 2.1 Binary classiﬁcation A binary classiﬁer, F(x), is a mapping from examples x ∈X to class labels y ∈{−1, 1}. The optimal classiﬁer, in the minimum probability of error sense, is Bayes decision rule F(x) = arg miny∈{−1,1}PY \|X(y\|x). (1) This can be hard to implement, due to the difﬁculty of estimating the probabilities PY \|X(y\|x). This difﬁculty is avoided by large margin methods, such as boosting, which implement the classiﬁer as F(x) = sign[f ∗(x)] (2) where f ∗(x) : X →R is the continuous valued predictor f ∗(x) = arg min f R(f) (3) that minimizes the classiﬁcation risk R(f) = EX,Y {L[y, f(x)]} (4) associated with a loss function L[., .]. In practice, the optimal predictor is learned from a sample D = {(xi, yi)}n i=1 of training examples, and (4) is approximated by the empirical risk R(f) ≈ n i=1 L[yi, f(xi)]. (5) The loss L[., .] is said to be Bayes consistent if (1) and (2) are equivalent. For large margin methods, such as boosting, the loss is also a function of the classiﬁcation margin yf(x), i.e. L[y, f(x)] = φ(yf(x)) (6) for some non-negative function φ(.). This dependence on the margin yf(x) guarantees that the classiﬁer has good generalization when the training sample is small [19]. Boosting learns the optimal predictor f ∗(x) : X →R as the solution of minf(x) R(f) s.t f(x) ∈span(H). (7) where H = {h1(x), ...hp(x)} is a set of weak learners hi(x) : X →R, and the optimization is carried out by gradient descent in the functional space span(H) of linear combinations of hi(x) [14]. 2.2 Multiclass setting To extend the above formulation to the multiclass setting, we note that the deﬁnition of the classiﬁcation labels as ±1 plays a signiﬁcant role in the formulation of the binary case. One of the difﬁculties of the multiclass extension is that these labels do not have an immediate extension to the multiclass setting. To address this problem, we return to the classical setting, where the class labels of a M-ary problem take values in the set {1, . . . , M}. Each class k is then mapped into a distinct class label yk, which can be thought of as a codeword that identiﬁes the class. In the binary case, these codewords are deﬁned as y1 = 1 and y2 = −1. It is possible to derive an alternative form for the expressions of the margin and classiﬁer F(x) that depends explicitly on codewords. For this, we note that (2) can be written as F(x) = arg max k ykf ∗(x) (8) 2 and the margin can be expressed as yf = f if k = 1 −f if k = 2 = 1 2(y1f −y2f) if k = 1 1 2(y2f −y1f) if k = 2 = 1 2(ykf −max l̸=k ylf). (9) The interesting property of these forms is that they are directly extensible to the M-ary classiﬁcation case. For this, we assume that the codewords yk and the predictor f(x) are multi-dimensional, i.e. yk, f(x) ∈Rd for some dimension d which we will discuss in greater detail in the following section. The margin of f(x) with respect to class k is then deﬁned as M(f(x), yk) = 1 2[< f(x), yk > −max l̸=k < f(x), yl >] (10) and the classiﬁer as F(x) = arg maxk < f(x), yk >, (11) where < ., . > is the standard dot-product. Note that this is equivalent to F(x) = arg max k∈{1,...,M} M(f(x), yk), (12) and thus F(x) is the class of largest margin for the predictor f(x). This deﬁnition is closely related to previous notions of multiclass margin. For example, it generalizes that of [11], where the codewords yk are restricted to the binary vectors in the canonical basis of Rd, and is a special case of that in [1], where the dot products < f(x), yk > are replaced by a generic function of f, x, and k. Given a training sample D = {(xi, yi)}n i=1, the optimal predictor f ∗(x) minimizes the risk RM(f) = EX,Y {LM[y, f(x)]} ≈ n i=1 LM[yi, f(xi)]} (13) where LM[., .] is a multiclass loss function. A natural extension of (6) and (9) is a loss of the form LM[y, f(x)] = φ(M(f(x), y)). (14) To avoid the nonlinearity of the max operator in (10), we rely on LM[y, f(x)] = M k=1 e−1 2 [<f(x),y>−<f(x),yk>]. (15) which is shown, in Appendix A, to upper bound 1+e−M(f(x),y). It follows that the minimization of the risk of (13) encourages predictors of large margin M(f ∗(xi), yi), ∀i. For M = 2, LM[y, f(x)] reduces to L2[y, f(x)] = 1 + e−yf(x) (16) and the risk minimization problem is identical to that of AdaBoost [8]. In appendices B and C it is shown that RM(f) is convex and Bayes consistent, in the sense that if f ∗(x) is the minimizer of (13), then < f ∗(x), yk >= log PY \|X(yk\|x) + c ∀k (17) and (11) implements the Bayes decision rule F(x) = arg maxkPY \|X(yk\|x). (18) 2.3 Optimal set of codewords From (15), the choice of codewords yk has an impact in the optimal predictor f ∗(x), which is determined by the projections < f ∗(x), yk >. To maximize the margins of (10), the difference between these projections should be as large as possible. To accomplish this we search for the set of M distinct unit codewords Y = {y1, . . . , yM} ∈Rd that are as dissimilar as possible ⎧ ⎪ ⎨ ⎪ ⎩ maxd,y1,...yM [mini̸=j \|\|yi −yj\|\|2] s.t \|\|yk\|\| = 1 ∀k = 1..M. yk ∈Rd ∀k = 1..M. (19) 3 0 5 1 1.5 -1 -0.5 0 0.5 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 0.5 1 1.5 -1 -0.5 0 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 0 1 -1 0 1 -1 0 1 -1 (M = 2) (M = 3) (M = 4) Figure 1: Optimal codewords for M = 2, 3, 4. To solve this problem,we start by noting that, for d < M, the smallest distance of (19) can be increased by simply increasing d, since this leads to a larger space. On the other hand, since M points y1, ...yM lie in an, at most, M −1 dimensional subspace of Rd, e.g. any three points belong to a plane, there is no beneﬁt in increasing d beyond M −1. On the contrary, as shown in Appendix D, if d > M −1 there exits a vector v ∈Rd with equal projection on all codewords, < yi, v >=< yj, v > ∀i, j = 1, .., M. (20) Since the addition of v to the predictor f(x) does not change the classiﬁcation rule of (11), this makes the optimal predictor underdetermined. To avoid this problem, we set d = M −1. In this case, as shown in Appendix E, the vertices of a M −1 dimensional regular simplex1 centered at the origin [3] are solutions of (19). Figure 1 presents the set of optimal codewords when M = 2, 3, 4. Note that in the binary case this set consists of the traditional codewords yi ∈{+1, −1}. In general, there is no closed form solution for the vertices of a regular simplex of M vectors. However, these can be derived from those of a regular simplex of M −1 vectors, and a recursive solution is possible [3]. 3 Risk minimization We have so far deﬁned a proper margin loss function for M-ary classiﬁcation and identiﬁed an optimal codebook. In this section, we derive two boosting algorithms for the minimization of the classiﬁcation risk of (13). These algorithms are both based on the GradientBoost framework [14]. The ﬁrst is a functional coordinate descent algorithm, which updates a single component of the predictor per boosting iteration. The second is a functional gradient descent algorithm that updates all components simultaneously. 3.1 Coordinate descent In the ﬁrst method, each component f ∗ i (x) of the optimal predictor f ∗(x) = [f ∗ 1 (x), ..f ∗ M−1(x)], is the linear combination of weak learners that solves the optimization problem minf1(x),...,fM−1(x) R([f1(x), ..., fM−1(x)]) s.t fj(x) ∈span(H) ∀j = 1..M −1. (21) where H = {h1(x), ...hp(x)} is a set of weak learners, hi(x) : X →R. These can be stumps, regression trees, or member of any other suitable model family. We denote by f t(x) = [f t 1(x), ..., f t M−1(x)] the predictor available after t boosting iterations. At iteration t + 1 a single component fj(x) of f(x) is updated with a step in the direction of the scalar functional g that most decreases the risk R[f t 1, ..., f t j + α∗ jg, ..., f t M−1]. For this, we consider the functional derivative of R[f(x)] along the direction of the functional g : X →R, at point f(x) = f t(x), with respect to the jth component fj(x) of f(x) [10], δR[f t; j, g] = ∂R[f t + ϵg1j] ∂ϵ ϵ=0 , (22) 1A regular M −1 dimensional simplex is the convex hull of M normal vectors which have equal pair-wise distances. 4 where 1j ∈Rd is a vector whose jth element is one and the remainder zero, i.e. f t + ϵg1j = [f t 1, .., f t j + ϵg, ..f t M−1]. Using the risk of (13), it is shown in Appendix F that −δR[f t; j, g] = n i=1 g(xi)wj i , (23) with wj i = 1 2e−1 2 <f t(xi),yi> M k=1 < 1j, yi −yk > e 1 2 <f t(xi),yk>. (24) The direction of greatest risk decrease is the weak learner g∗ j (x) = arg max g∈H n i=1 g(xi)wj i , (25) and the optimal step size along this direction α∗ j = arg min α∈R R[f t(x) + αg∗ j (x)1j]. (26) The classiﬁer is thus updated as f t+1 = f t(x) + α∗ jg∗ j (x)1j = [f t 1, ..., f t j + α∗ jg∗ j , ..., f t M−1] (27) This procedure is summarized in Algorithm 1-left and denoted CD-MCBoost. It starts with f 0(x) = 0 ∈Rd and updates the predictor components sequentially. Note that, since (13) is a convex function of f(x), it converges to the global minimum of the risk. 3.2 Gradient descent Alternatively, (13) can be minimized by learning a linear combination of multiclass weak learners. In this case, the optimization problem is minf(x) R[f(x)] s.t f(x) ∈span(H), (28) where H = {h1(x), ..., hp(x)} is a set of multiclass weak learners, hi(x) : X →RM−1, such as decision trees. Note that to ﬁt tree classiﬁers in this deﬁnition their output (usually a class number) should be translated into a class codeword. As before, let f t(x) ∈RM−1 be the predictor available after t boosting iterations. At iteration t + 1 a step is given along the direction g(x) ∈H of largest decrease of the risk R[f(x)]. For this, we consider the directional functional derivative of R[f(x)] along the direction of the functional g : X →RM−1, at point f(x) = f t(x). δR[f t; g] = ∂R[f t + ϵg] ∂ϵ ϵ=0 . (29) As shown in Appendix G, −δR[f t; g] = n i=1 < g(xi), wi > (30) where wi ∈RM−1 wi = 1 2e−1 2 <f t(xi),yi> M k=1 (yi −yk)e 1 2 <f t(xi),yk>. (31) The direction of greatest risk decrease is the weak learner g∗(x) = arg max g∈H n i=1 < g(xi), wi >, (32) and the optimal step size along this direction α∗= arg min α∈R R[f t(x) + αg∗(x)]. (33) The predictor is updated to f t+1(x) = f t(x)+α∗g∗(x). This procedure is summarised in Algorithm 1-right, and denoted GD-MCBoost. Since (13) is convex, it converges to the global minimum of the risk. 5 Algorithm 1 CD-MCBoost and GD-MCBoost Input: Number of classes M, set of codewords Y = {y1, . . . , yM}, number of iterations N and dataset S = {(x1, y1), ..., (xn, yn)}, where xi are examples and yi ∈Y are their class codewords. Initialization: set t = 0, and f t = 0 ∈RM−1 CD-MCBoost GD-MCBoost while t < N do for j = 1 to M −1 do Compute wj i with (24) Find g∗ j (x), α∗ j using (25) and (26) Update f t+1 j (x) = f t j(x) + α∗ jg∗ j (x) Update f t+1 k (x) = f t k(x) ∀k ̸= j t = t + 1 end for end while while t < N do Compute wi with (31) Find g∗(x), α∗using (32) and (33) Update f t+1(x) = f t(x) + α∗g∗(x) t = t + 1 end while Output: decision rule: F(x) = arg maxk < f N(x), yk > 4 Comparison to previous methods Multi-dimensional predictors and codewords have been used implicitly, [7, 18, 16, 6], or explicitly, [12, 9], in all previous multiclass boosting methods. “one vs all”, “all pairs” and “ECOC” [1]: as shown in [1], these methods can be interpreted as assigning a codeword yk to each class, where yk ∈{+1, 0, −1}l and l = M for “one vs all”, l = M(M−1) 2 for “all pairs” and l is variable for “ECOC”, depending on the error correction code. In all these methods, binary classiﬁers are learned independently for each of the codeword components. This does not guarantee an optimal joint predictor. These methods are similar to CD-MCBoost in the sense that the predictor components are updated individually at each boosting iteration. However, in CD-MCBoost, the codewords are not restricted to {+1, 0, −1} and the predictor components are learned jointly. AdaBoost-MH [18, 9]: This method converts the M-ary classiﬁcation problem into a binary one, learned from a M times larger training set, where each example x is augmented with a feature y that identiﬁes a class. Examples such that x belongs to class y receive binary label 1, while the remaining receive the label −1 [9]. In this way, the binary classiﬁer learns if the multiclass label y is correct for x or not. AdaBoost-MH uses weak learners ht : X × {1, . . . , M} →R and the decision rule ¯F(x) = arg max j∈{1,2,..M} t ht(x, j) (34) where t is the iteration number. This is equivalent to the decision rule of (11) if f(x) is an Mdimensional predictor with jth component fj(x) = t ht(x, j), and the label codewords are deﬁned as yj = 1j. This method is comparable to CD-MCBoost in the sense that it does not require multiclass weak learners. However, there are no guarantees that the weak learners in common use are able to discriminate the complex classes of the augmented binary problem. AdaBoost-M1 [7] and AdaBoost-Cost [16]: These methods use multiclass weak learners ht : X →{1, 2, ..M} and a classiﬁcation rule of the form ¯F(x) = arg max j∈{1,2,..M} t\|ht(x)=j αtht(x), (35) where t is the boosting iteration and αt the coefﬁcient of weak learner ht(x). This is equivalent to the decision rule of (11) if f(x) is an M-dimensional predictor with jth component fj(x) = t\|ht(x)=j αtht(x) and label codewords yj = 1j. These methods are comparable to GD-MCBoost, in the sense that they update the predictor components simultaneously. However, they have not been shown to be Bayes consistent, and it is not clear that they can be interpreted as maximizing the multiclass margin. 6 −1 −0.5 0 0.5 1 1.5 −1 −0.5 0 0.5 1 f1(x) f2(x) y1 y2 y3 −2 −1 0 1 2 3 −4 −2 0 2 4 f1(x) f2(x) class 1 class 2 class 3 y1 y2 y3 −2 −1 0 1 2 3 −4 −2 0 2 4 f1(x) f2(x) Class 1 Class 2 Class 3 y1 y2 y3 t = 0 t = 10 t = 100 Figure 2: Classiﬁer predictions of CD-MCBoost, on the test set, after t = 0, 10, 100 boosting iterations. SAMME [12]: This method explicitly uses M-dimensional predictors with codewords yj = M1j −1 M −1 = −1 M −1, −1 M −1, ..., 1, −1 M −1, −1 M −1 ∈RM, (36) and decision rule ¯F(x) = arg max j∈{1,2,..M} fj(x). (37) Since, as discussed in Section 2.3, the optimal detector is not unique when the predictor is Mdimensional, this algorithm includes the additional constraint M j=1 fj(x) = 0 and solves a constrained optimization problem [12, 9]. It is comparable to GD-MCBoost in the sense that it updates the predictor components simultaneously, but uses the loss function LSAMME[yk, f(x)] = e−1 M <yk,f(x)>. Using (36), the minimization of this loss is equivalent to maximizing M′(f(x), yk) =< f(x), yk >= fk(x) − 1 M −1 j̸=k fj(x), (38) which is not a proper margin since M′(f(x), yk) > 0 does not imply correct classiﬁcation i.e. fk(x) > fj(x) ∀j ̸= k. Hence, SAMME does not guarantee a large margin solution for the multiclass problem. When compared to all these methods, MCBoost has the advantage of combining 1) a Bayes consistent and margin enforcing loss function, 2) an optimal set of codewords, 3) the ability to boost any type of weak learner, 4) guaranteed convergence to the global minimum of (21), for CD-MCBoost, or (28), for GD-MCBoost, and 5) equivalence to the classical AdaBoost algorithm for binary problems. It is worth emphasizing that MCBoost can boost any type of weak learners of non-zero directional derivative, i.e. non-zero (23) for CD-MCBoost and (30) for GD-MCBoost. This is independent of the type of weak learner output, and unlike previous multiclass boosting approaches, which can only boost weak learners of speciﬁc output types. Note that, although the weak learner selection criteria of previous approaches can have interesting interpretations, e.g. based on weighted error rates [16], these only hold for speciﬁc weak learners. Finally, MCBoost extends the deﬁnition of margin and loss function to multi-dimensional predictors. The derivation of Section 2 can easily be generalized to the design of other multiclass boosting algorithms by the use of 1) alternative φ(v) functions in (14) (e.g. those of the logistic [9] or Tangent [13] losses for increased outlier robustness, asymmetric losses for cost-sensitive classiﬁcation, etc.), and 2) alternative optimization approaches (e.g. Newton’s method [9, 17]). 5 Evaluation A number of experiments were conducted to evaluate the MCBoost algorithms2. 5.1 Synthetic data We start with a synthetic example, for which the optimal decision rule is known. This is a three class problem, with two-dimensional Gaussian classes of means [1, 2], [−1, 0], [2, −1] and covariances of 2Codes for CD-MCBoost and GD-MCBoost are available from [2]. 7 Table 1: Accuracy of multiclass boosting methods, using decision stumps, on six UCI data sets method landsat letter pendigit optdigit shuttle isolet One Vs All 84.80% 50.92% 86.56% 89.93% 87.11% 88.97% AdaBoost-MH [18] 47.70% 15.73% 24.41% 73.62% 79.16% 66.71% CD-MCBoost 85.70% 49.60% 89.51% 92.82% 88.01% 91.02% Table 2: Accuracy of multiclass boosting methods, using trees of max depth 2, on six UCI data sets method landsat letter pendigit optdigit shuttle isolet AdaBoost-M1[7] 72.85% − − − 96.45% − AdaBoost-SAMME[12] 79.80% 45.65% 83.82% 87.53% 99.70% 61.00% AdaBoost-Cost [16] 83.95% 42.00% 80.53% 86.20% 99.55% 63.69% GD-MCBoost 86.65% 59.65% 92.94% 92.32% 99.73% 84.28% [1, 0.5; 0.5, 2],[1, 0.3; 0.3, 1],[.4, 0.1; 0.1, 0.8] respectively. Training and test sets of 1, 000 examples each were randomly sampled and the Bayes rule computed in closed form [5]. The associated Bayes error rate was 11.67% in the training and 11.13% in the test set. A classiﬁer was learned with CD-MCBoost and decision stumps. Figure 2) shows predictions3 of f t(x) on the test set, for t = 0, 10, 100. Note that f 0(xi) = [0, 0] for all examples xi. However, as the iterations proceed, CD-MCBoost produces predictions that are more aligned with the true class codewords, shown as dashed lines, while maximizing the distance between examples of different classes (by increasing their distance to the origin). In this context, “alignment of f(x) with yk” implies that < f(x), yk >≥< f(x), yj >, ∀j ̸= k. This combination of alignment and distance maximization results in higher margins, leading to more accurate and robust classiﬁcation. The test error rate after 100 iterations of boosting was 11.30%, and very close to the Bayes error rate of 11.13%. 5.2 CD-MCBoost We next conducted a number of experiments to evaluate the performance of CD-MCBoost on the six UCI datasets of Table 1. Among the methods identiﬁed as comparable in the previous section, we implemented “one vs all” and AdaBoost-MH [18]. In all cases, decision stumps were used as weak learners, and we used the training/test set decomposition speciﬁed for each dataset. The “one vs all” detectors were trained with 20 iterations. The remaining methods were then allowed to include the same number of weak learners in their ﬁnal decision rules. Table 1 presents the resulting classiﬁcation accuracies. CD-MCBoost produced the most accurate classiﬁer in four of the ﬁve datasets, and was a close second in the remaining one. “One vs all” achieved the next best performance, with AdaBoost-MH producing the worst classiﬁers. 5.3 GD-MCBoost Finally, the performance of GD-MCBoost was compared to AdaBoost-M1 [7], AdaBoost-Cost [16] and AdaBoost-SAMME [12]. The experiments were based on the UCI datasets of the previous section, but the weak learners were now trees of depth 2. These were built with a greedy procedure so as to 1) minimize the weighted error rate of AdaBoost-M1 [7] and AdaBoost-SAMME[12], 2) minimize the classiﬁcation cost of AdaBoost-Cost [16], or 3) maximize (32) for GD-MCBoost. Table 2 presents the classiﬁcation accuracy of each method, for 50 training iterations. GD-MCBoost achieved the best accuracy on all datasets, reaching substantially larger classiﬁcation rate than all other methods in the most difﬁcult datasets, e.g. from a previous best of 63.69% to 84.28% in isolet, 45.65% to 59.65% in letter, and 83.82% to 92.94% in pendigit. Among the remaining methods, AdaBoost-SAMME achieved the next best performance, although this was close to that of AdaBoost-Cost. AdaBoost-M1 had the worst results, and was not able to boost the weak learners used in this experiment for four of the six datasets. It should be noted that the results of Tables 1 and 2 are not directly comparable, since the classiﬁers are based on different types of weak learners and have different complexities. 3We emphasize the fact that these are plots of f t(x) ∈R2, not x ∈R2. 8 References [1] E. L. Allwein, R. E. Schapire, and Y. Singer. Reducing multiclass to binary: a unifying approach for margin classiﬁers. J. Mach. Learn. Res., 1:113–141, September 2001. [2] N. N. Author. 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4,404	Composite Multiclass Losses Elodie Vernet ENS Cachan evernet@ens-cachan.fr Robert C. Williamson ANU and NICTA Bob.Williamson@anu.edu.au Mark D. Reid ANU and NICTA Mark.Reid@anu.edu.au Abstract We consider loss functions for multiclass prediction problems. We show when a multiclass loss can be expressed as a “proper composite loss”, which is the composition of a proper loss and a link function. We extend existing results for binary losses to multiclass losses. We determine the stationarity condition, Bregman representation, order-sensitivity, existence and uniqueness of the composite representation for multiclass losses. We subsume existing results on “classiﬁcation calibration” by relating it to properness and show that the simple integral representation for binary proper losses can not be extended to multiclass losses. 1 Introduction The motivation of this paper is to understand the intrinsic structure and properties of suitable loss functions for the problem of multiclass prediction, which includes multiclass probability estimation. Suppose we are given a data sample S := (xi,yi)i∈[m] where xi ∈X is an observation and yi ∈ {1,..,n} =: [n] is its corresponding class. We assume the sample S is drawn iid according to some distribution P = PX ,Y on X × [n]. Given a new observation x we want to predict the probability pi := P(Y = i\|X = x) of x belonging to class i, for i ∈[n]. Multiclass classiﬁcation requires the learner to predict the most likely class of x; that is to ﬁnd ˆy = argmaxi∈[n] pi. A loss measures the quality of prediction. Let ∆n := {(p1,..., pn): ∑i∈[n] pi = 1,and 0 ≤pi ≤1, ∀i ∈ [n]} denote the n-simplex. For multiclass probability estimation, ℓ: ∆n →Rn +. For classiﬁcation, the loss ℓ: [n] →Rn +. The partial losses ℓi are the components of ℓ(q) = (ℓ1(q),...,ℓn(q))′. Proper losses are particularly suitable for probability estimation. They have been studied in detail when n = 2 (the “binary case”) where there is a nice integral representation [1, 2, 3], and characterization [4] when differentiable. Classiﬁcation calibrated losses are an analog of proper losses for the problem of classiﬁcation [5]. The relationship between classiﬁcation calibration and properness was determined in [4] for n = 2. Most of these results have had no multiclass analogue until now. The design of losses for multiclass prediction has received recent attention [6, 7, 8, 9, 10, 11, 12] although none of these papers developed the connection to proper losses, and most restrict consideration to margin losses (which imply certain symmetry conditions). Glasmachers [13] has shown that certain learning algorithms can still behave well when the losses do not satisfy the conditions in these earlier papers because the requirements are actually stronger than needed. Our contributions are: We relate properness, classiﬁcation calibration, and the notion used in [8] which we rename “prediction calibrated” §3; we provide a novel characterization of multiclass properness §4; we study composite proper losses (the composition of a proper loss with an invertible link) presenting new uniqueness and existence results §5; we show how the above results can aid in the design of proper losses §6; and we present a (somewhat surprising) negative result concerning the integral representation of proper multiclass losses §7. Many of our results are characterisations. Full proofs are provided in the extended version [14]. 1 2 Formal Setup Suppose X is some set and Y = {1,...,n} = [n] is a set of labels. We suppose we are given data (xi,yi)i∈[m] such that Yi ∈Y is the label corresponding to xi ∈X . These data follow a joint distribution PX ,Y . We denote by EX ,Y and EY \|X respectively, the expectation and the conditional expectation with respect to PX ,Y . The conditional risk L associated with a loss ℓis the function L: ∆n ×∆n ∋(p,q) 7→L(p,q) = EY∼pℓY(q) = p′ ·ℓ(q) = ∑ i∈[n] piℓi(q) ∈R+, where Y ∼p means Y is drawn according to a multinomial distribution with parameter p. In a typical learning problem one will make an estimate q: X →∆n. The full risk is L(q) = EX EY \|X ℓY(q(X)). Minimizing L(q) over q: X →∆n is equivalent to minimizing L(p(x),q(x)) over q(x) ∈∆n for all x ∈X where p(x) = (p1(x),..., pn(x))′, p′ is the transpose of p, and pi(x) = P(Y = i\|X = x). Thus it sufﬁces to only consider the conditional risk; confer [3]. A loss ℓ: ∆n →Rn + is proper if L(p, p) ≤L(p,q), ∀p,q ∈∆n. It is strictly proper if the inequality is strict when p ̸= q. The conditional Bayes risk L: ∆n ∋p 7→infq∈∆n L(p,q). This function is always concave [2]. If ℓis proper, then L(p) = L(p, p) = p′ · ℓ(p). Strictly proper losses induce Fisher consistent estimators of probabilities: if ℓis strictly proper, p = argminq L(p,q). In order to differentiate the losses we project the n-simplex into a subset of Rn−1. We denote by Π∆: ∆n ∋p = (p1,..., pn)′ 7→˜p = (p1,..., pn−1)′ ∈˜∆n := {(p1,..., pn−1)′ : pi ≥0, ∀i ∈ [n], ∑n−1 i=1 pi ≤1}, the projection of the n-simplex ∆n, and Π−1 ∆ : ˜∆n ∋˜p = ( ˜p1,..., ˜pn−1) 7→p = ( ˜p1,..., ˜pn−1,1−∑n−1 i=1 ˜pi)′ ∈∆n its inverse. The losses above are deﬁned on the simplex ∆n since the argument (an estimator) represents a probability vector. However it is sometimes desirable to use another set V of predictions. One can consider losses ℓ: V →Rn +. Suppose there exists an invertible function ψ : ∆n →V . Then ℓcan be written as a composition of a loss λ deﬁned on the simplex with ψ−1. That is, ℓ(v) = λ ψ(v) := λ(ψ−1(v)). Such a function λ ψ is a composite loss. If λ is proper, we say ℓis a proper composite loss, with associated proper loss λ and link ψ. We use the following notation. The kth unit vector ek is the n vector with all components zero except the kth which is 1. The n-vector 1n := (1,...,1)′. The derivative of a function f is denoted D f and its Hessian H f. Let ˚∆n := {(p1,..., pn): ∑i∈[n] pi = 1,and 0 < pi < 1, ∀i ∈[n]} and ∂∆n := ∆n \ ˚∆n. 3 Relating Properness to Classiﬁcation Calibration Properness is an attractive property of a loss for the task of class probability estimation. However if one is merely interested in classifying (predicting ˆy ∈[n] given x ∈X ) then one requires less. We relate classiﬁcation calibration (the analog of properness for classiﬁcation problems) to properness. Suppose c ∈˚∆n. We cover ∆n with n subsets each representing one class: Ti(c) := {p ∈∆n : ∀j ̸= i picj ≥p jci}. Observe that for i ̸= j, the sets {p ∈R: picj = pjcj} are subsets of dimension n−2 through c and all ek such that k ̸= i and k ̸= j. These subsets partition ∆n into two parts, the subspace Ti is the intersection of the subspaces delimited by the precedent (n−2)-subspace and in the same side as ei. We will make use of the following properties of Ti(c). Lemma 1 Suppose c ∈˚∆n, i ∈[n]. Then the following hold: 1. For all p ∈∆n, there exists i such that p ∈Ti(c). 2. Suppose p ∈∆n. Ti(c)∩Tj(c) ⊆{p ∈∆n : picj = pjci}, a subspace of dimension n−2. 3. Suppose p ∈∆n. If p ∈Tn i=1 Ti(c) then p = c. 4. For all p,q ∈∆n, p ̸= q, there exists c ∈˚∆n, and i ∈[n] such that p ∈Ti(c) and q /∈Ti(c). 2 Classiﬁcation calibrated losses have been developed and studied under some different deﬁnitions and names [6, 5]. Below we generalise the notion of c-calibration which was proposed for n = 2 in [4] as a generalisation of the notion of classiﬁcation calibration in [5]. Deﬁnition 2 Suppose ℓ: ∆n →Rn + is a loss and c ∈˚∆n. We say ℓis c-calibrated at p ∈∆n if for all i ∈[n] such that p /∈Ti(c) then ∀q ∈Ti(c), L(p) < L(p,q). We say that ℓis c-calibrated if ∀p ∈∆n, ℓis c-calibrated at p. Deﬁnition 2 means that if the probability vector q one predicts doesn’t belong to the same subset (i.e. doesn’t predict the same class) as the real probability vector p, then the loss might be larger. Classiﬁcation calibration in the sense used in [5] corresponds to 1 2-calibrated losses when n = 2. If cmid := ( 1 n,..., 1 n)′, cmid-calibration induces Fisher-consistent estimates in the case of classiﬁcation. Furthermore “ℓis cmid-calibrated and for all i ∈[n], and ℓi is continuous and bounded below” is equivalent to “ℓis inﬁnite sample consistent as deﬁned by [6]”. This is because if ℓis continuous and Ti(c) is closed, then ∀q ∈Ti(c), L(p) < L(p,q) if and only if L(p) < infq∈Ti(c) L(p,q). The following result generalises the correspondence between binary classiﬁcation calibration and properness [4, Theorem 16] to multiclass losses (n > 2). Proposition 3 A continuous loss ℓ: ∆n →Rn + is strictly proper if and only if it is c-calibrated for all c ∈˚∆n. In particular, a continuous strictly proper loss is cmid-calibrated. Thus for any estimator ˆqn of the conditional probability vector one constructs by minimizing the empirical average of a continuous strictly proper loss, one can build an estimator of the label (corresponding to the largest probability of ˆqn) which is Fisher consistent for the problem of classiﬁcation. In the binary case, ℓis classiﬁcation calibrated if and only if the following implication holds [5]: L( fn) →min g L(g) ⇒ PX ,Y (Y ̸= fn(X)) →min g PX ,Y (Y ̸= g(X)) . (1) Tewari and Bartlett [8] have characterised when (1) holds in the multiclass case. Since there is no reason to assume the equivalence between classiﬁcation calibration and (1) still holds for n > 2, we give different names for these two notions. We keep the name of classiﬁcation calibration for the notion linked to Fisher consistency (as deﬁned before) and call prediction calibrated the notion of Tewari and Bartlett (equivalent to (1)). Deﬁnition 4 Suppose ℓ: V →Rn + is a loss. Let Cℓ= co({ℓ(v): v ∈V }), the convex hull of the image of V . ℓis said to be prediction calibrated if there exists a prediction function pred: Rn →[n] such that ∀p ∈∆n : inf z∈Cℓ,ppred(z)<maxipi p′ ·z > inf z∈Cℓ p′ ·z = L(p). Observe that the class is predicted from ℓ(p) and not directly from p (which is equivalent if the loss is invertible). Suppose that ℓ: ∆n →Rn + is such that ℓis prediction calibrated and pred(ℓ(p)) ∈ argmaxi pi. Then ℓis cmid-calibrated almost everywhere. By introducing a reference “link” ¯ψ (which corresponds to the actual link if ℓis a proper composite loss) we now show how the pred function can be canonically expressed in terms of argmaxi pi. Proposition 5 Suppose ℓ: V →Rn + is a loss. Let ¯ψ(p) ∈argminv∈V L(p,v) and λ = ℓ◦¯ψ. Then λ is proper. If ℓis prediction calibrated then pred(λ(p)) ∈argmaxi pi. 4 Characterizing Properness We ﬁrst present some simple (but new) consequences of properness. We say f : C ⊂Rn →Rn is monotone on C when for all x and y in C, ( f(x)−f(y))′ ·(x−y) ≥0; confer [15]. Proposition 6 Suppose ℓ: ∆n →Rn + is a loss. If ℓis proper, then −ℓis monotone. 3 Proposition 7 If ℓis strictly proper then it is invertible. A theme of the present paper is the extensibility of results concerning binary losses to multiclass losses. The following proposition shows how the characterisation of properness in the general (not necessarily differentiable) multiclass case can be reduced to the binary case. In the binary case, the two classes are often denoted −1 and 1 and the loss is denoted ℓ= (ℓ1,ℓ−1)′. We project the 2-simplex ∆2 into [0,1]: η ∈[0,1] is the projection of (η,1−η) ∈∆2. Proposition 8 Suppose ℓ: ∆n →Rn + is a loss. Deﬁne ˜ℓp,q : [0,1] ∋η 7→ ˜ℓp,q 1 (η) ˜ℓp,q −1(η) = q′ ·ℓ p+η(q−p) p′ ·ℓ p+η(q−p) . Then ℓis (strictly) proper if and only if ˜ℓp,q is (strictly) proper ∀p,q ∈∂∆n. This proposition shows that in order to check if a loss is proper one needs only to check the properness in each line. One could use the easy characterization of properness for differentiable binary losses (ℓ: [0,1] →R2 + is proper if and only if ∀η ∈[0,1], −ℓ′ 1(η) 1−η = ℓ′ −1(η) η ≥0, [4]). However this needs to be checked for all lines deﬁned by p,q ∈∂∆n. We now extend some characterisations of properness to the multiclass case by using Proposition 8. Lambert [16] proved that in the binary case, properness is equivalent to the fact that the further your prediction is from reality, the larger the loss (“order sensitivity”). The result relied upon on the total order of R. In the multiclass case, there does not exist such a total order. Yet, one can compare two predictions if they are in the same line as the true real class probability. The next result is a generalization of the binary case equivalence of properness and order sensitivity. Proposition 9 Suppose ℓ: ∆n →Rn + is a loss. Then ℓis (strictly) proper if and only if ∀p,q ∈∆n, ∀0 ≤h1 ≤h2, L(p, p+h1(q−p)) ≤L(p, p+h2(q−p)) (the inequality is strict if h1 ̸= h2). “Order sensitivity” tells us more about properness: the true class probability minimizes the risk and if the prediction moves away from the true class probability in a line then the risk increases. This property appears convenient for optimization purposes: if one reaches a local minimum in the second argument of the risk and the loss is strictly proper then it is a global minimum. If the loss is proper, such a local minimum is a global minimum or a constant in an open set. But observe that typically one is minimising the full risk L(q(·)) over functions q: X →∆n. Order sensitivity of ℓ does not imply this optimisation problem is well behaved; one needs convexity of q 7→L(p,q) for all p ∈∆n to ensure convexity of the functional optimisation problem. The order sensitivity along a line leads to a new characterisation of differentiable proper losses. As in the binary case, one condition comes from the fact that the derivative is zero at a minimum and the other ensures that it is really a minimum. Corollary 10 Suppose ℓ: ∆n →Rn + is a loss such that ˜ℓ= ℓ◦Π−1 ∆ is differentiable. Let M(p) = D ˜ℓ(Π∆(p))·DΠ∆(p). Then ℓis proper if and only if p′ ·M(p) = 0 (q−r)′ ·M(p)·(q−r) ≤ 0 ∀q,r ∈∆n, ∀p ∈˚∆n. (2) We know that for any loss, its Bayes risk L(p) = infq∈∆n L(p,q) = infq∈∆n p′ ·ℓ(q) is concave. If ℓis proper, L(p) = p′ ·ℓ(p). Rather than working with the loss ℓ: V →Rn + we will now work with the simpler associated conditional Bayes risk L: V →R+. We need two deﬁnitions from [15]. Suppose f : Rn →R is concave. Then limt↓0 f(x+td)−f(x) t exists, and is called the directional derivative of f at x in the direction d and is denoted D f(x,d). By analogy with the usual deﬁnition of subdifferential, the superdifferential ∂f(x) of f at x is ∂f(x) := s ∈Rn : s′ ·y ≥D f(x,y), ∀y ∈Rn = s ∈Rn : f(y) ≤f(x)+s′ ·(y−x), ∀y ∈Rn . A vector s ∈∂f(x) is called a supergradient of f at x. The next proposition is a restatement of the well known Bregman representation of proper losses; see [17] for the differentiable case, and [2, Theorem 3.2] for the general case. 4 Proposition 11 Suppose ℓ: ∆n →Rn + is a loss. Then ℓis proper if and only if there exists a concave function f and ∀q ∈∆n, there exists a supergradient A(q) ∈∂f(q) such that ∀p,q ∈∆n, p′ ·ℓ(q) = L(p,q) = f(q)+(p−q)′ ·A(q). Then f is unique and f(p) = L(p, p) = L(p). The fact that f is deﬁned on a simplex is not a problem. Indeed, the superdifferential becomes ∂f(x) = {s ∈Rn : s′ ·d ≥D f(x,d),∀d ∈∆n} = {s ∈Rn : f(y) ≤f(x)+s′ ·(y−x), ∀y ∈∆n}. If ˜f = f ◦Π−1 ∆is differentiable at ˜q ∈˜∆n, A(q) = (D ˜f(Π∆(q)),0)′+α1′ n, α ∈R. Then (p−q)′·A(q) = D ˜f(Π∆(q)) · (Π∆(p) −Π∆(q)). Hence for any concave differentiable function f, there exists an unique proper loss whose Bayes risk is equal to f (we say that f is differentiable when ˜f is differentiable). The last property gives us the form of the proper losses associated with a Bayes risk. Suppose L: ∆n →R+ is concave. The proper losses whose Bayes risk is equal to L are ℓ: ∆n ∋q 7→ L(q)+(ei −q)′ ·A(q) n i=1 ∈Rn +, ∀A(q) ∈∂L(q). (3) This result suggests that some information is lost by representing a proper loss via its Bayes risk (when the last is not differentiable). The next proposition elucidates this by showing that proper losses which have the same Bayes risk are equal almost everywhere. Proposition 12 Two proper losses ℓ1 and ℓ2 have the same conditional Bayes risk function L if and only if ℓ1 = ℓ2 almost everywhere. If L is differentiable, ℓ1 = ℓ2 everywhere. We say that L is differentiable at p if ˜L = L◦Π−1 ∆is differentiable at ˜p = Π∆(p). Proposition 13 Suppose ℓ: ∆n →Rn + is a proper loss. Then ℓis continuous in ˚∆n if and only if L is differentiable on ˚∆n; ℓis continuous at p ∈˚∆n if and only if, L is differentiable at p ∈˚∆n. 5 The Proper Composite Representation: Uniqueness and Existence It is sometimes helpful to deﬁne a loss on some set V rather than ∆n; confer [4]. Composite losses (see the deﬁnition in §2) are a way of constructing such losses: given a proper loss λ : ∆n →Rn + and an invertible link ψ : ∆n →V , one deﬁnes λ ψ : V →Rn + using λ ψ = λ ◦ψ−1. We now consider the question: given a loss ℓ: V →Rn +, when does ℓhave a proper composite representation (whereby ℓ can be written as ℓ= λ ◦ψ−1), and is this representation unique? We ﬁrst consider the binary case and study the uniqueness of the representation of a loss as a proper composite loss. Proposition 14 Suppose ℓ= λ ◦ψ−1 : V →R2 + is a proper composite loss and that the proper loss λ is differentiable and the link function ψ is differentiable and invertible. Then the proper loss λ is unique. Furthermore ψ is unique if ∀v1,v2 ∈R, ∃v ∈[v1,v2], ℓ′ 1(v) ̸= 0 or ℓ′ −1(v) ̸= 0. If there exists ¯v1, ¯v2 ∈R such that ℓ′ 1(v) = ℓ′ −1(v) = 0 ∀v ∈[¯v1, ¯v2], one can choose any ψ\|[¯v1,¯v2] such that ψ is differentiable, invertible and continuous in [¯v1, ¯v2] and obtain ℓ= λ ◦ψ−1, and ψ is uniquely deﬁned where ℓis invertible. Proposition 15 Suppose ℓ: V →R2 + is a differentiable binary loss such that ∀v ∈V , ℓ′ −1(v) ̸= 0 or ℓ′ 1(v) ̸= 0. Then ℓcan be expressed as a proper composite loss if and only if the following three conditions hold: 1) ℓ1 is decreasing (increasing); 2) ℓ−1 is increasing (decreasing); and 3) f : V ∋v 7→ℓ′ 1(v) ℓ′ −1(v) is strictly increasing (decreasing) and continuous. Observe that the last condition is alway satisﬁed if both ℓ1 and ℓ−1 are convex. Suppose ϕ : R →R+ is a function. The loss deﬁned via ℓϕ : V ∋v 7→(ℓ−1(v),ℓ1(v))′ = (ϕ(−v),ϕ(v))′ ∈R2 + is called a binary margin loss. Binary margin losses are often used for classiﬁcation problems. We will now show how the above proposition applies to them. 5 Corollary 16 Suppose ϕ : R →R+ is differentiable and ∀v ∈R, ϕ′(v) ̸= 0 or ϕ′(−v) ̸= 0. Then ℓϕ can be expressed as a proper composite loss if and only if f : R ∋v 7→−ϕ′(v) ϕ′(−v) is strictly monotonic continuous and ϕ is monotonic. If ϕ is convex or concave then f deﬁned above is monotonic. However not all binary margin losses are composite proper losses. One can even build a smooth margin loss which cannot be expressed as a proper composite loss. Consider ϕ(x) = 1−1 π arctan(x−1). Then f(v) = ϕ′(−v) ϕ′(−v)+ϕ′(v) = x2−2x+2 2x2+4 which is not invertible. We now generalize the above results to the multiclass case. Proposition 17 Suppose ℓhas two proper composite representations ℓ= λ ◦ψ−1 = µ ◦φ −1 where λ and µ are proper losses and ψ and φ are continuous invertible. Then λ = m almost everywhere. If ℓis continuous and has a composite representation, then the proper loss (in the decomposition) is unique (λ = µ everywhere). If ℓis invertible and has a composite representation, then the representation is unique. ℓ1(v) ℓ2(v) q hL(v) q = {x: x·q = L(v)} Sℓ x = ℓ(v) ℓ(V ) Given a loss ℓ: V →Rn +, we denote by Sℓ= ℓ(V )+ [0,∞)n = {λ : ∃v ∈V , ∀i ∈[n], λi ≥ℓi(v)} the superprediction set of ℓ(confer e.g. [18]). We introduce a set of hyperplanes for p ∈∆n and β ∈R, hβ p = {x ∈ Rn : x′ · p = β}. A hyperplane hβ p supports a set A at x ∈A when x ∈hβ p and for all a ∈A , a′ · p ≥β or for all a ∈A , a′ · p ≤β. We say that Sℓis strictly convex in its inner part when for all p ∈∆n, there exists an unique x ∈ℓ(V ) such that there exists a hyperplane hβ p supporting Sℓat x. Sℓis said to be smooth when for all x ∈ℓ(V ), there exists an unique hyperplane supporting Sℓat x. If ℓis invertible, we can express these two deﬁnitions in terms of v ∈V rather than x ∈ℓ(V ). If ℓ: V →Rn + is strictly convex, then Sℓwill be strictly convex in its inner part. Proposition 18 Suppose ℓ: V →Rn + is a continuous invertible loss. Then ℓhas a strictly proper composite representation if and only if Sℓis convex, smooth and strictly convex in its inner part. Proposition 19 Suppose ℓ: V →Rn + is a continuous loss. If ℓhas a proper composite representation, then Sℓis convex and smooth. If ℓis also invertible, then Sℓis strictly convex in its inner part. 6 Designing Proper Losses We now build a family of conditional Bayes risks. Suppose we are given n(n−1) 2 concave functions {Li1,i2 : ∆2 →R}1≤i1<i2≤n on ∆2, and we want to build a concave function L on ∆n which is equal to one of the given functions on each edge of the simplex (∀1 ≤i1 < i2 ≤n, L(0,.,0, pi1,0,.,0, pi2,0,.,0) = Li1,i2(pi1, pi2)). This is equivalent to choosing a binary loss function, knowing that the observation is in the class i1 or i2. The result below gives one possible construction. (There exists an inﬁnity of solutions — one can simply add any concave function equal to zero in each edge). Lemma 20 Suppose we have a family of concave functions {Li1,i2 : ∆2 →R}1≤i1<i2≤n, then L: ∆n ∋p 7→L(p1,..., pn) = ∑ 1≤i1<i2≤n (pi1 + pi2)Li1,i2 pi1 pi1 + pi2 , pi2 pi1 + pi2 is concave and ∀1 ≤i1 < i2 ≤n, L(0,.,0, pi1,0,.,0, pi2,0,.,0) = Li1,i2(pi1, pi2). 6 Using this family of Bayes risks, one can build a family of proper losses. Lemma 21 Suppose we have a family of binary proper losses ℓi1,i2 : ∆2 →R2. Then ℓ: ∆n ∋p 7→ℓ(p) = j−1 ∑ i=1 ℓi, j −1 pi pi + pj + n ∑ i=j+1 ℓi,j 1 p j pi + pj !n j=1 ∈Rn + is a proper n-class loss such that ℓi((0,.,0, pi1,0,.,0, pi2,0,.,0)) =    ℓi1,i2 1 (pi1) i = i1 ℓi1,i2 −1 (pi1) i = i2 0 otherwise . Observe that it is much easier to work at ﬁrst with the Bayes risk and then using the correspondence between Bayes risks and proper losses. 7 Integral Representations of Proper Losses Unlike the natural generalisation of the results from proper binary to proper multiclass losses above, there is one result that does not carry over: the integral representation of proper losses [1]. In the binary case there exists a family of “extremal” loss functions (cost-weighted generalisations of the 0-1 loss) each parametrised by c ∈[0,1] and deﬁned for all η ∈[0,1] by ℓc −1(η) := cJη ≥cK and ℓc 1 := (1 −c)Jη < cK. As shown in [1, 3], given these extremal functions, any proper binary loss ℓ can be expressed as the weighted integral ℓ= R 1 0 ℓc w(c)dc + constant with w(c) = −L′′(c). This representation is a special case of a representation from Choquet theory [19] which characterises when every point in some set can be expressed as a weighted combination of the “extremal points” of the set. Although there is such a representation when n > 2, the difﬁculty is that the set of extremal points is much larger and this rules out the existence of a nice small set of “primitive” proper losses when n > 2. The rest of this section makes this statement precise. A convex cone K is a set of points closed under linear combinations of positive coefﬁcients. That is, K = αK + βK for any α,β ≥0. A point f ∈K is extremal if f = 1 2(g + h) for g,h ∈K implies ∃α ∈R+ such that g = α f. That is, f cannot be represented as a non-trivial combination of other points in K . The set of extremal points for K will be denoted exK . Suppose U is a bounded closed convex set in Rd, and Kb(U) is the set of convex functions on U bounded by 1, then Kb(U) is compact with respect to the topology of uniform convergence. Theorem 2.2 of [20] shows that the extremal points of the convex cone K (U) = {α f +βg : f,g ∈Kb(U),α,β ≥0} are dense (w.r.t. the topology of uniform convergence) in K (U) when d > 1. This means for any function f ∈K (U) there is a sequence of functions (gi)i such that for all i gi ∈exK (U) and limi→∞∥f −gi∥∞= 0, where ∥f∥∞:= supu∈U \|f(u)\|. We use this result to show that the set of extremal Bayes risks is dense in the set of Bayes risks when n > 2. In order to simplify our analysis, we restrict attention to fair proper losses. A loss is fair if each partial loss is zero on its corresponding vertex of the simplex (ℓi(ei) = 0, ∀i ∈[n]). A proper loss is fair if and only if its Bayes risk is zero at each vertex of the simplex (in this case the Bayes risk is also called fair). One does not lose generality by studying fair proper losses since any proper loss is a sum of a fair proper loss and a constant vector. The set of fair proper losses deﬁned on ∆n form a closed convex cone, denoted Ln. The set of concave functions which are zero on all the vertices of the simplex ∆n is denoted Fn and is also a closed convex cone. Proposition 22 Suppose n > 2. Then for any fair proper loss ℓ∈Ln there exists a sequence (ℓi)i of extremal fair proper losses (ℓi ∈exLn) which converges almost everywhere to ℓ. The proof of Proposition 22 requires the following lemma which relies upon the correspondence between a proper loss and its Bayes risk (Proposition 11) and the fact that two continuous functions equal almost everywhere are equal everywhere. Lemma 23 If ℓ∈exLn then its corresponding Bayes risk L is extremal in Fn. Conversely, if L ∈exFn then all the proper losses ℓwith Bayes risk equal to L are extremal in Ln. 7 We also need a correspondence between the uniform convergence of a sequence of Bayes risk functions and the convergence of their associated proper losses. Lemma 24 Suppose L,Li ∈Fn for i ∈N and suppose ℓand ℓi, i ∈N are associated proper losses. Then (Li)i converges uniformly to L if and only if (ℓi)i converges almost everywhere to ℓ. Figure 1: Complexity of extremal concave functions in two dimensions (corresponds to n = 3). Graph of an extremal concave function in two dimensions. Lines are where the slope changes. The pattern of these lines can be arbitrarily complex. Bronshtein [20] and Johansen [21] showed how to construct a set of extremal convex functions which is dense in K (U). With a trivial change of sign this leads to a family of extremal proper fair Bayes risks that is dense in the set of Bayes risks in the topology of uniform convergence. This means that it is not possible to have a small set of extremal (“primitive”) losses from which one can construct any proper fair loss by linear combinations when n > 2. A convex polytope is a compact convex intersection of a ﬁnite set of half-spaces and is therefore the convex hull of its vertices. Let {ai}i be a ﬁnite family of afﬁne functions deﬁned on ∆n. Now deﬁne the convex polyhedral function f by f(x) := maxi ai(x). The set K := {Pi = {x ∈∆n : f(x) = ai(x)}} is a covering of ∆n by polytopes. Theorem 2.1 of [20] shows that for f, Pi and K so deﬁned, f is extremal if the following two conditions are satisﬁed: 1) for all polytopes Pi in K and for every face F of Pi, F ∩∆n ̸= ∅implies F has a vertex in ∆n; 2) every vertex of Pi in ∆n belongs to n distinct polytopes of K. The set of all such f is dense in K (U). Using this result it is straightforward to exhibit some sets of extremal fair Bayes risks {Lc(p): c ∈ ∆n}. Two examples are when Lc(p) = n ∑ i=1 pi ci ∏ j̸=i J pi ci ≤p j c j K or Lc(p) = ^ i∈[n] 1−pi 1−ci . 8 Conclusion We considered loss functions for multiclass prediction problems and made four main contributions: • We extended existing results for binary losses to multiclass prediction problems including several characterisations of proper losses and the relationship between properness and classiﬁcation calibration; • We related the notion of prediction calibration to classiﬁcation calibration; • We developed some new existence and uniqueness results for proper composite losses (which are new even in the binary case) which characterise when a loss has a proper composite representation in terms of the geometry of the associated superprediction set; and • We showed that the attractive (simply parametrised) integral representation for binary proper losses can not be extended to the multiclass case. Our results suggest that in order to design losses for multiclass prediction problems it is helpful to use the composite representation, and design the proper part via the Bayes risk as suggested for the binary case in [1]. The proper composite representation is used in [22]. Acknowledgements The work was performed whilst Elodie Vernet was visiting ANU and NICTA, and was supported by the Australian Research Council and NICTA, through backing Australia’s ability. 8 References [1] Andreas Buja, Werner Stuetzle and Yi Shen. Loss functions for binary class probability estimation and classiﬁcation: Structure and applications. Technical report, University of Pennsylvania, November 2005. http://www-stat.wharton.upenn.edu/˜buja/PAPERS/ paper-proper-scoring.pdf. [2] Tilmann Gneiting and Adrian E. Raftery. Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association, 102(477):359-378, March 2007. [3] Mark D. Reid and Robert C. Williamson. Information, divergence and risk for binary experiments. Journal of Machine Learning Research, 12:731-817, March 2011. [4] Mark D. Reid and Robert C. Williamson. Composite binary losses. Journal of Machine Learning Research, 11:2387-2422, 2010. [5] Peter L. Bartlett, Michael I. Jordan and Jon D. McAuliffe. Convexity, classiﬁcation, and risk bounds. Journal of the American Statistical Association, 101(473):138-156, March 2006. [6] Tong Zhang. Statistical analysis of some multi-category large margin classiﬁcation methods. Journal of Machine Learning Research, 5:1225-1251, 2004. [7] Simon I. Hill and Arnaud Doucet. A framework for kernel-based multi-category classiﬁcation. Journal of Artiﬁcial Intelligence Research, 30:525-564, 2007. [8] Ambuj Tewari and Peter L. Bartlett. On the consistency of multiclass classiﬁcation methods. Journal of Machine Learning Research, 8:1007-1025, 2007. [9] Yufeng Liu. Fisher consistency of multicategory support vector machines. Proceedings of the Eleventh International Conference on Artiﬁcial Intelligence and Statistics, side 289-296, 2007. [10] Ra´ul Santos-Rodr´ıguez, Alicia Guerrero-Curieses, Roc´ıo Alaiz-Rodriguez and Jes´us CidSueiro. Cost-sensitive learning based on Bregman divergences. Machine Learning, 76:271285, 2009. http://dx.doi.org/10.1007/s10994-009-5132-8. [11] Hui Zou, Ji Zhu and Trevor Hastie. New multicategory boosting algorithms based on multicategory Fisher-consistent losses. The Annals of Applied Statistics, 2(4):1290-1306, 2008. [12] Zhihua Zhang, Michael I. Jordan, Wu-Jun Li and Dit-Yan Yeung. Coherence functions for multicategory margin-based classiﬁcation methods. Proceedings of the Twelfth Conference on Artiﬁcial Intelligence and Statistics (AISTATS), 2009. [13] Tobias Glasmachers. Universal consistency of multi-class support vector classication. Advances in Neural Information Processing Systems (NIPS), 2010. [14] Elodie Vernet, Robert C. Williamson and Mark D. Reid. Composite multiclass losses. (with proofs). To appear in NIPS 2011, October 2011. http://users.cecs.anu.edu.au/ ˜williams/papers/P188.pdf. [15] Jean-Baptiste Hiriart-Urruty and Claude Lemar´echal. Fundamentals of Convex Analysis. Springer, Berlin, 2001. [16] Nicolas S. Lambert. Elicitation and evaluation of statistical forecasts. Technical report, Stanford University, March 2010. http://www.stanford.edu/˜nlambert/lambert_ elicitation.pdf. [17] Jes´us Cid-Sueiro and An´ıbal R. Figueiras-Vidal. On the structure of strict sense Bayesian cost functions and its applications. IEEE Transactions on Neural Networks, 12(3):445-455, May 2001. [18] Yuri Kalnishkan and Michael V. Vyugin. The weak aggregating algorithm and weak mixability. Journal of Computer and System Sciences, 74:1228-1244, 2008. [19] Robert R. Phelps. Lectures on Choquet’s Theorem, volume 1757 of Lecture Notes in Mathematics. Springer, 2nd edition, 2001. [20] Eﬁm Mikhailovich Bronshtein. Extremal convex functions. Siberian Mathematical Journal, 19:6-12, 1978. [21] Søren Johansen. The extremal convex functions. Mathematica Scandinavica, 34:61-68, 1974. [22] Tim van Erven, Mark D. Reid and Robert C. Williamson. Mixability is Bayes risk curvature relative to log loss. Proceedings of the 24th Annual Conference on Learning Theory, 2011. To appear. http://users.cecs.anu.edu.au/˜williams/papers/P186.pdf. [23] Rolf Schneider. Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, 1993. 9	2011	57
4,405	Scalable Training of Mixture Models via Coresets Dan Feldman MIT Matthew Faulkner Caltech Andreas Krause ETH Zurich Abstract How can we train a statistical mixture model on a massive data set? In this paper, we show how to construct coresets for mixtures of Gaussians and natural generalizations. A coreset is a weighted subset of the data, which guarantees that models ﬁtting the coreset will also provide a good ﬁt for the original data set. We show that, perhaps surprisingly, Gaussian mixtures admit coresets of size independent of the size of the data set. More precisely, we prove that a weighted set of O(dk3/ε2) data points sufﬁces for computing a (1 + ε)-approximation for the optimal model on the original n data points. Moreover, such coresets can be efﬁciently constructed in a map-reduce style computation, as well as in a streaming setting. Our results rely on a novel reduction of statistical estimation to problems in computational geometry, as well as new complexity results about mixtures of Gaussians. We empirically evaluate our algorithms on several real data sets, including a density estimation problem in the context of earthquake detection using accelerometers in mobile phones. 1 Introduction We consider the problem of training statistical mixture models, in particular mixtures of Gaussians and some natural generalizations, on massive data sets. Such data sets may be distributed across a cluster, or arrive in a data stream, and have to be processed with limited memory. In contrast to parameter estimation for models with compact sufﬁcient statistics, mixture models generally require inference over latent variables, which in turn depends on the full data set. In this paper, we show that Gaussian mixture models (GMMs), and some generalizations, admit small coresets: A coreset is a weighted subset of the data which guarantees that models ﬁtting the coreset will also provide a good ﬁt for the original data set. Perhaps surprisingly, we show that Gaussian mixtures admit coresets of size independent of the size of the data set. We focus on ε-semi-spherical Gaussians, where the covariance matrix Σi of each component i has eigenvalues bounded in [ε, 1/ε], but some of our results generalize even to the semi-deﬁnite case. In particular, we show that given a data set D of n points in Rd, ε > 0 and k ∈N, how one can efﬁciently construct a weighted set C of O(dk3/ε2) points, such that for any mixture of k ε-semispherical Gaussians θ = [(w1, µ1, Σ1), . . . , (wk, µk, Σk)] it holds that the log-likelihood ln P(D \| θ) of D under θ is approximated by the (properly weighted) log-likelihood ln P(C \| θ) of C under θ to arbitrary accuracy as ε →0. Thus solving the estimation problem on the coreset C (e.g., using weighted variants of the EM algorithm, see Section 3.3) is almost as good as solving the estimation problem on large data set D. Our algorithm for constructing C is based on adaptively sampling points from D and is simple to implement. Moreover, coresets can be efﬁciently constructed in a map-reduce style computation, as well as in a streaming setting (using space and update time per point of poly(dkε−1 log n log(1/δ))). Existence and construction of coresets have been investigated for a number of problems in computational geometry (such as k-means and k-median) in many recent papers (cf., surveys in [1, 2]). In this paper, we demonstrate how these techniques from computational geometry can be lifted to the realm of statistical estimation. As a by-product of our analysis, we also close an open question on the VC dimension of arbitrary mixtures of Gaussians. We evaluate our algorithms on several synthetic and real data sets. In particular, we use our approach for density estimation for acceleration data, motivated by an application in earthquake detection using mobile phones. 1 2 Background and Problem Statement Fitting mixture models by MLE. Suppose we are given a data set D = {x1, . . . , xn} ⊆Rd. We consider ﬁtting a mixture of Gaussians θ = [(w1, µ1, Σ1), . . . , (wk, µk, Σk)], i.e., the distribution P(x \| θ) = Pk i=1 wiN(x; µi, Σi), where w1, . . . , wk ≥0 are the mixture weights, P i wi = 1, and µi and Σi are mean and covariance of the i-th mixture component, which is modeled as a multivariate normal distribution N(x, µi, Σi) = 1 √ \|2πΣi\|exp −1 2(x −µi)T Σ−1 i (x −µi) . In Section 4, we will discuss extensions to more general mixture models. Assuming the data was generated i.i.d., the negative log likelihood of the data is L(D \| θ) = −P j ln P(xj \| θ), and we wish to obtain the maximum likelihood estimate (MLE) of the parameters θ∗= argminθ∈C L(D \| θ), where C is a set of constraints ensuring that degenerate solutions are avoided1. Hereby, for a symmetric matrix A, spec A is the set of all eigenvalues of A. We deﬁne C = Cε = {θ = [(w1, µ1, Σ1), . . . , (wk, µk, Σk)] \| ∀i : spec(Σi) ⊆[ε, 1/ε]} to be the set of all mixtures of k Gaussians θ, such that all the eigenvalues of the covariance matrices of θ are bounded between ε and 1/ε for some small ε > 0. Approximating the log-likelihood. Our goal is to approximate the data set D by a weighted set C = {(γ1, x′ 1), . . . , (γm, x′ m)} ⊆R × Rd, such that L(D \| θ) ≈L(C \| θ) for all θ, where we deﬁne L(C \| θ) = −P i γi ln P(x′ i \| θ). What kind of approximation accuracy may we hope to expect? Notice that there is a nontrivial issue of scale: Suppose we have a MLE θ∗for D, and let α > 0. Then straightforward linear algebra shows that we can obtain an MLE θ∗ α for a scaled data set αD = {αx : x ∈D} by simply scaling all means by α, and covariance matrices by α2. For the log-likelihood, however, it holds that L(αD \| θ∗ α) = d ln α + L(D \| θ∗). Therefore, optimal solutions on one scale can be efﬁciently transformed to optimal solutions at a different scale, while maintaining the same additive error. This means, that any algorithm which achieves absolute error ε at any scale could be used to achieve parameter estimates (for means, covariances) with arbitrarily small error, simply by applying the algorithm to a scaled data set and transforming back the obtained solution. An alternative, scaleinvariant approach may be to strive towards approximating L(D \| θ) up to multiplicative error (1 + ε). Unfortunately, this goal is also hard to achieve: Choosing a scaling parameter α such that d ln α + L(D \| θ∗) = 0 would require any algorithm that achieves any bounded multiplicative error to essentially incur no error at all when evaluating L(αD \| θ∗). The above observations hold even for the case k = 1 and Σ = I, where the mixture θ consists of a single Gaussian, and the log-likelihood is the sum of squared distances to a point µ and an additive term. Motivated by the scaling issues discussed above, we use the following error bound that was suggested in [3] (who studied the case where all Gaussians are identical spheres). We decompose the negative log-likelihood L(D \| θ) of a data set D as L(D \| θ) = − n X j=1 ln k X i=1 wi p \|2πΣi\| exp −1 2(xj −µi)T Σ−1 i (xj −µi) = −n ln Z(θ) + φ(D \| θ) where Z(θ) = P i wi √ \|2πΣi\| is a normalizer, and the function φ is deﬁned as φ(D \| θ) = − n X j=1 ln k X i=1 wi Z(θ) p \|2πΣi\| exp −1 2(xj −µi)T Σ−1 i (xj −µi) . Hereby, Z(θ) plays the role of a normalizer, which can be computed exactly, independently of the set D. φ(D \| θ) captures all dependencies of L(D \| θ) on D, and via Jensen’s inequality, it can be seen that φ(D \| θ) is always nonnegative. We can now use this term φ(D \| θ) as a reference for our error bounds. In particular, we call ˜θ a (1 + ε)-approximation for θ if (1 −ε)φ(D \| θ) ≤φ(D \| ˜θ) ≤φ(D \| θ)(1 + ε). Coresets. We call a weighted data set C a (k, ε)-coreset for another (possibly weighted) set D ⊆ Rd, if for all mixtures θ ∈C of k Gaussians it holds that (1 −ε)φ(D \| θ) ≤φ(C \| θ) ≤φ(D \| θ)(1 + ε). 1equivalently, C can be interpreted as prior thresholding. 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −4 −3 −2 −1 0 1 2 3 (a) Example data set −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −4 −3 −2 −1 0 1 2 3 (b) Iteration 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −4 −3 −2 −1 0 1 2 3 (c) Iteration 3 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −4 −3 −2 −1 0 1 2 3 (d) Final approximation B (e) Sampling distribution (f) Coreset Figure 1: Illustration of the coreset construction for example data set (a). (b,c) show two iterations of constructing the set B. Solid squares are points sampled uniformly from remaining points, hollow squares are points selected in previous iterations. Red color indicates half the points furthest away from B, which are kept for next iteration. (d) ﬁnal approximate clustering B on top of original data set. (e) Induced non-uniform sampling distribution: radius of circles indicates probability; color indicates weight, ranging from red (high weight) to yellow (low weight). (f) Coreset sampled from distribution in (e). Hereby φ(C \| θ) is generalized to weighted data sets C in the natural way (weighing the contribution of each summand x′ j ∈C by γj). Thus, as ε →0, for a sequence of (k, ε)-coresets Cε we have that supθ∈C \|L(Cε \| θ) −L(D \| θ)\| →0, i.e., L(Cε \| θ) uniformly (over θ ∈C) approximates L(D \| θ). Further, under the additional condition that all variances are sufﬁciently large (formally Q λ∈spec(Σi) λ ≥ 1 (2π)d for all components i), the log-normalizer ln Z(θ) is negative, and consequently the coreset in fact provides a multiplicative (1 + ε) approximation to the log-likelihood, i.e., (1 −ε)L(D \| θ) ≤L(C \| θ) ≤L(D \| θ)(1 + ε). More details can be found in the supplemental material. Note that if we had access to a (k, ε)-coreset C, then we could reduce the problem of ﬁtting a mixture model on D to one of ﬁtting a model on C, since the optimal solution θC is a good approximation (in terms of log-likelihood) of θ∗. While ﬁnding the optimal θC is a difﬁcult problem, one can use a (weighted) variant of the EM algorithm to ﬁnd a good solution. Moreover, if \|C\| ≪\|D\|, running EM on C may be orders of magnitude faster than solving it on D. In Section 3.3, we give more details about solving the density estimation problem on the coreset. The key question is whether small (k, ε)-coresets exist, and whether they can be efﬁciently constructed. In the following, we answer this question afﬁrmatively. We show that, perhaps surprisingly, one can efﬁciently ﬁnd coresets C of size independent of the size n of D, and with polynomial dependence on 1 ε, d and k. 3 Efﬁcient Coreset Construction via Adaptive Sampling Naive approach: uniform sampling. A naive approach towards approximating D would be to just pick a subset C uniformly at random. In particular, suppose the data set is generated from a mixture of two spherical Gaussians (Σi = I) with weights w1 = 1 √n and w2 = 1 − 1 √n. Unless m = Ω(√n) points are sampled, with constant probability no data point generated from Gaussian 2 is selected. By moving the means of the Gaussians arbitrarily far apart, L(D \| θC) can be made arbitrarily worse than L(D \| θD), where θC and θD are MLEs on C and D respectively. Thus, even for two well-separated Gaussians, uniform sampling can perform arbitrarily poorly. This example already suggests that, intuitively, in order to achieve small multiplicative error, we must devise a sampling scheme that adaptively selects representative points from all “clusters” present in the data set. However, this suggests that obtaining a coreset requires solving a chicken-and-egg problem, where we need to understand the density of the data to obtain the coreset, but simultaneously would like to use the coreset for density estimation. 3 Better approximation via adaptive sampling. The key idea behind the coreset construction is that we can break the chicken-and-egg problem by ﬁrst obtaining a rough approximation B of the clustering solution (using more than k components, but far fewer than n), and then to use this solution to bias the random sampling. Surprisingly, a simple procedure which iteratively samples a small number β of points, and removes half of the data set closest to the sampled points, provides a sufﬁciently accurate ﬁrst approximation B for this purpose. This initial clustering is then used to sample the data points comprising coreset C according to probabilities which are roughly proportional to the squared distance to the set B. This non-uniform random sampling can be understood as an importance-weighted estimate of the log-likelihood L(D \| θ), where the weights are optimized in order to reduce the variance. The same general idea has been found successful in constructing coresets for geometric clustering problems such as k-means and k-median [4]. The pseudocode for obtaining the approximation B, and for using it to obtain coreset C is given in Algorithm 1. Algorithm 1: Coreset construction Input: Data set D, ε, δ, k Output: Coreset C = (γ(x1), x1), . . . , (γ(x\|C\|), x\|C\|) D′ ←D; B ←∅; while \|D′\| > 10dk ln(1/δ) do Sample set S of β = 10dk ln(1/δ) points uniformly at random from D′; Remove ⌈\|D′\|/2⌉points x ∈D′ closest to S (i.e., minimizing dist(x, S)) from D′; Set B ←B ∪S; Set B ←B ∪D′; for each b ∈B do Db ←the points in D whose closest point in B is b. Ties broken arbitrarily; for each b ∈B and x ∈Db do m(x) ← l 5 \|Db\| + dist(x,B)2 P x′∈D dist(x′,B)2 m ; Pick a non-uniform random sample C of 10⌈dk\|B\|2 ln(1/δ)/ε2⌉points from D, where for every x′ ∈C and x ∈D, we have x′ = x with probability m(x)/ P x′∈D m(x′); for each x′ ∈C do γ(x′) ← P x∈D m(x) \|C\|·m(x′) ; We have the following result, proved in the supplemental material: Theorem 3.1. Suppose C is sampled from D using Algorithm 1 for parameters ε, δ and k. Then, with probability at least 1 −δ it holds that for all θ ∈Cε, φ(D \| θ)(1 −ε) ≤φ(C \| θ) ≤φ(D \| θ)(1 + ε). In our experiments, we compare the performance of clustering on coresets constructed via adaptive sampling, vs. clustering on a uniform sample. The size of C in Algorithm 1 depends on \|B\|2 = log2 n. By replacing B in the algorithm with a constant factor approximation B′, \|B′\| = l for the k-means problem, we can get a coreset C of size independent of n. Such a set B′ can be computed in O(ndk) time either by applying exhaustive search on the output C of the original Algorithm 1 or by using one of the existing constant-factor approximation algorithms for k-means (say, [5]). 3.1 Sketch of Analysis: Reduction to Euclidean Spaces For space limitations, the proof of Theorem 3.1 is included in the supplemental material, we only provide a sketch of the analysis, carrying the main intuition. The key insight in the proof is that the contribution log P(x \| θ) to the likelihood L(D \| θ) can be expressed in the following way: Lemma 3.2. There exist functions φ, ψ, and f such that, for any point x ∈Rd and mixture model θ, ln P(x \| θ) = −fφ(x)(ψ(θ)) + Z(θ), where f˜x(y) = −ln X i ˜wiexp −Widist(˜x −˜µi, si)2 . Hereby, φ is a function that maps a point x ∈Rd into ˜x = φ(x) ∈R2d, and ψ is a function that maps a mixture model θ into a tuple y = (s, w, ˜µ, W) where w is a k-tuple of nonnegative weights ˜w1, . . . , ˜wk summing to 1, s = s1, . . . , sk ⊆R2d is a set of k d-dimensional subspaces that are weighted by weights W1, · · · , Wk > 0, and ˜µ = ˜µ1, · · · , ˜µk ∈R2d is a set of k means. The main idea behind Lemma 3.2 is that level sets of distances between points and subspaces are quadratic forms, and can thus represent level sets of the Gaussian probability density function (see Figure 2(a) for an illustration). We recognize the “soft-min” function ∧w′(η) ≡ 4 (a) Gaussian pdf as Euclidean distances x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 C1 C2 C3 C4 C5 C6 C7 (b) Tree for coreset construction Figure 2: (a) Level sets of the distances between points on a plane (green) and (disjoint) k-dimensional subspaces are ellipses, and thus can represent contour lines of the multivariate Gaussian. (b) Tree construction for generating coresets in parallel or from data streams. Black arrows indicate “merge-and-compress” operations. The (intermediate) coresets C1, . . . , C7 are enumerated in the order in which they would be generated in the streaming case. In the parallel case, C1, C2, C4 and C5 would be constructed in parallel, followed by parallel construction of C3 and C6, ﬁnally resulting in C7. −ln P i w′ iexp (−ηi) as an approximation upper-bounding the minimum min(η) = mini ηi for ηi = Widist(˜x −˜µi, si)2 and η = [η1, . . . , ηk]. The motivation behind this transformation is that it allows expressing the likelihood P(x \| θ) of a data point x given a model θ in a purely geometric manner as soft-min over distances between points and subspaces in a transformed space. Notice that if we use the minimum min() instead of the soft-min ∧˜ w(), we recover the problem of approximating the data set D (transformed via φ) by k-subspaces. For semi-spherical Gaussians, it can be shown that the subspaces can be chosen as points while incurring a multiplicative error of at most 1/ε, and thus we recover the well-known k-means problem in the transformed space. This insight suggests using a known coreset construction for k-means, adapted to the transformation employed. The remaining challenge in the proof is to bound the additional error incurred by using the soft-min function ∧˜ w(·) instead of the minimum min(·). We tackle this challenge by proving a generalized triangle inequality adapted to the exponential transformation, and employing the framework described in [4], which provides a general method for constructing coresets for clustering problems of the form mins P i f˜x(s). As proved in [4], the key quantity that controls the size of a coreset is the pseudo-dimension of the functions Fd = {f˜x for ˜x ∈R2d}. This notion of dimension is closely related to the VC dimension of the (sub-level sets of the) functions Fd and therefore represents the complexity of this set of functions. The ﬁnal ingredient in the proof of Theorem 3.1 is a new bound on the complexity of mixtures of k Gaussians in d dimensions proved in the supplemental material. 3.2 Streaming and Parallel Computation One major advantage of coresets is that they can be constructed in parallel, as well as in a streaming setting where data points arrive one by one, and it is impossible to remember the entire data set due to memory constraints. The key insight is that coresets satisfy certain composition properties, which have previously been used by [6] for streaming and parallel construction of coresets for geometric clustering problems such as k-median and k-means. 1. Suppose C1 is a (k, ε)-coreset for D1, and C2 is a (k, ε)-coreset for D2. Then C1 ∪C2 is a (k, ε)-coreset for D1 ∪D2. 2. Suppose C is a (k, ε)-coreset for D, and C′ is a (k, δ)-coreset for C. Then C′ is a (k, (1 + ε)(1 + δ) −1)-coreset for D. In the following, we review how to exploit these properties for parallel and streaming computation. Streaming. In the streaming setting, we assume that points arrive one-by-one, but we do not have enough memory to remember the entire data set. Thus, we wish to maintain a coreset over time, while keeping only a small subset of O(log n) coresets in memory. There is a general reduction that shows that a small coreset scheme to a given problem sufﬁces to solve the corresponding problem on a streaming input [7, 6]. The idea is to construct and save in memory a coreset for every block of poly(dk/ε) consecutive points arriving in a stream. When we have two coresets in memory, we can merge them (resulting in a (k, ε)-coreset via property (1)), and compress by computing a single coreset from the merged coresets (via property (2)) to avoid increase in the coreset size. An important subtlety arises: While merging two coresets (via property (1)) does not increase the approximation error, compressing a coreset (via property (2)) does increase the error. A naive approach that merges and compresses immediately as soon as two coresets have been constructed, can incur an exponential increase in approximation error. Fortunately, it is possible to organize the merge-and-compress operations in a binary tree of height O(log n), where we need to store in memory a single coreset 5 for each level on the tree (thus requiring only poly(dkε−1 log n) memory). Figure 2(b) illustrates this tree computation. In order to construct a coreset for the union of two (weighted) coresets, we use a weighted version of Algorithm 1, where we consider a weighted point as duplicate copies of a non-weighted point (possibly with fractional weight). A more formal description can be found in [8]. We summarize our streaming result in the following theorem. Theorem 3.3. A (k, ε)-coreset for a stream of n points in Rd can be computed for the εsemi-spherical GMM problem with probability at least 1 −δ using space and update time poly(dkε−1 log n log(1/δ)). Parallel/Distributed computations. Using the same ideas from the streaming model, a (nonparallel) coreset construction can be transformed into a parallel one. We partition the data into sets, and compute coresets for each set, independently, on different computers in a cluster. We then (in parallel) merge (via property (1)) two coresets, and compute a single coreset for every pair of such coresets (via property (2)). Continuing in this manner yields a process that takes O(log n) iterations of parallel computation. This computation is also naturally suited for map-reduce [9] style computations, where the map tasks compute coresets for disjoint parts of D, and the reduce tasks perform the merge-and-compress operations. Figure 2(b) illustrates this parallel construction. Theorem 3.4. A (k, ε)-coreset for a set of n points in Rd can be computed for the ε-semispherical GMM problem with probability at least 1 −δ using m machines in time (n/m) · poly(dkε−1 log(1/δ) log n). 3.3 Fitting a GMM on the Coreset using Weighted EM One approach, which we employ in our experiments, is to use a natural generalization of the EM algorithm, which takes the coreset weights into account. We here describe the algorithm for the case of GMMs. For other mixture distributions, the E and M steps are modiﬁed appropriately. Algorithm 2: Weighted EM for Gaussian mixtures Input: Coreset C, k, TOL Output: Mixture model θC Lold = ∞; Initialize means µ1, . . . , µk by sampling k points from C with probability proportional to their weight. Initialize Σi = I and wi = 1 k for all i; repeat Lold = L(C \| θ); for j = 1 to n do for i = 1 to k do Compute ηi,j = γi wiN (x′ j;µi,Σi) P ℓwℓN (x′ j;µℓ,Σℓ); for i = 1 to k do wi ←wi/ P ℓwi; µi ←P j ηi,jx′ j/ P j ηi,j; Σi ←P j ηi,j x′ j −µi x′ j −µi T / P j ηi,j; until L(C \| θ) ≥Lold −TOL ; Using a similar analysis as for the standard EM algorithm, Algorithm 2 is guaranteed to converge, but only to a local optimum. However, since it is applied on a much smaller set, it can be initialized using multiple random restarts. 4 Extensions and Generalizations We now show how the connection between estimating the parameters for mixture models and problems in computational geometry can be leveraged further. Our observations are based on the link between mixture of Gaussians and projective clustering (multiple subspace approximation) as shown in Lemma 3.2. Generalizations to non-semi-spherical GMMs. For simplicity, we generalized the coreset construction for the k-means problem, which required assumptions that the Gaussians are ε-semispherical. However, several more complex coresets for projective clustering were suggested recently (cf., [4]). Using the tools developed in this article, each such coreset implies a corresponding coreset for GMMs and generalizations. As an example, the coresets for approximating points by lines [10] implies that we can construct small coresets for GMMs even if the smallest singular value of one of the corresponding covariance matrices is zero. Generalizations to ℓq distances and other norms. Our analysis is based on combinatorics (such as the complexity of sub-levelsets of GMMs) and probabilistic methods (non-uniform random sampling). Therefore, generalizations to other non-Euclidean distance functions, or error functions such as (non-squared) distances (mixture of Laplace distributions) is straightforward. The main property 6 10 1 10 2 10 3 10 4 10 5 44 45 46 47 48 49 50 51 52 Training Set Size Log Likelihood on Test Data Set Full Set Uniform Sample Coreset (a) MNIST 10 2 10 3 10 4 10 5 −1800 −1600 −1400 −1200 −1000 −800 −600 −400 −200 0 Training Set Size Log Likelihood on Test Data Set Uniform Sample Coreset Full Set (b) Tetrode recordings 10 1 10 2 10 3 10 4 10 5 −250 −200 −150 −100 −50 0 Training Set Size Log Likelihood on Test Data Set Uniform Sample Full Set Coreset (c) CSN data 10 1 10 2 10 3 10 4 10 5 0.55 0.6 0.65 0.7 0.75 Training Set Size Area Under ROC Curve Uniform Sample Full Set Coreset (d) CSN detection Figure 3: Experimental results for three real data sets. We compare likelihood of the best model obtained on subsets C constructed by uniform sampling, and by the adaptive coreset sampling procedure. that we need is a generalization of the triangle inequality, as proved in the supplemental material. For example, replacing the squared distances by non-squared distances yields a coreset for mixture of Laplace distributions. The double triangle inequality ∥a −c∥2 ≤2(∥a −b∥+ ∥b −c∥2) that we used in this paper is replaced by H¨older’s inequality, ∥a −c∥2 ≤2O(q) ∥a −b∥+ 2 ∥b −c∥2. Such a result is straight-forward from our analysis, and we summarize it in the following theorem. Theorem 4.1. Let q ≥1 be an integer. Consider Algorithm 1, where dist(·, ·)2 is replaced by dist(·, ·)q and ε2 is replaced by εO(q). Suppose C is sampled from D using this updated version of Algorithm 1 for parameters ε, δ and k. Then, with prob. at least 1 −δ it holds that for all θ ∈Cε, φ(D \| θ)(1 −ε) ≤φ(C \| θ) ≤φ(D \| θ)(1 + ε), where Z(θ) = P i wi g(θi) and φ(D \| θ) = −P x∈D ln Pk i=1 wi Z(θ)g(θi)exp −1 2 Σ−1/2 i (x −µi) q using the normalizer g(θi) = R exp −1 2 Σ−1/2 i (x −µi) q dx. 5 Experiments We experimentally evaluate the effectiveness of using coresets of different sizes for training mixture models. We compare against running EM on the full set, as well as on an unweighted, uniform sample from D. Results are presented for three real datasets. MNIST handwritten digits. The MNIST dataset contains 60,000 training and 10,000 testing grayscale images of handwritten digits. As in [11], we normalize each component of the data to have zero mean and unit variance, and then reduce each 784-pixel (28x28) image using PCA, retaining only the top d = 100 principal components as a feature vector. From the training set, we produce coresets and uniformly sampled subsets of sizes between 30 and 5000, using the parameters k = 10 (a cluster for each digit), β = 20 and δ = 0.1 (see Algorithm 1), and ﬁt GMMs using EM with 3 random restarts. The log likelihood (LLH) of each model on the testing data is shown in Figure 3(a). Notice that coresets signiﬁcantly outperform uniform samples of the same size, and even a coreset of 30 points performs very well. Further note how the test-log likelihood begins to ﬂatten out for \|C\| = 1000. Constructing the coreset and running EM on this size takes 7.9 seconds (Intel Xeon 2.6 GHz), over 100 times faster than running EM on the full set (15 minutes). Neural tetrode recordings. We also compare coresets and uniform sampling on a large dataset containing 319,209 records of rat hippocampal action potentials, measured by four co-located electrodes. As done by [11], we concatenate the 38-sample waveforms produced by each electrode to obtain a 152-dimensional vector. The vectors are normalized so each component has zero mean and unit variance. The 319,209 records are divided in half to obtain training and testing sets. From the training set, we produce coresets and uniformly sampled subsets of sizes between 70 and 1000, using the parameters k = 33 (as in [11]), β = 66, and δ = 0.1, and ﬁt GMMs. The log likelihood of each model on the held-out testing data is shown in Figure 3(b). Coreset GMMs obtain consistently higher LLH than uniform sample GMMs for sets of the same size, and even a coreset of 100 points performs very well. Overall, training on coresets achieves approximately the same likelihood as training on the full set about 95 times faster (1.2 minutes vs. 1.9 hours). CSN cell phone accelerometer data. Smart phones with accelerometers are being used by the Community Seismic Network (CSN) as inexpensive seismometers for earthquake detection. In [12], 7 GB of acceleration data were recorded from volunteers while carrying and operating their phone in normal conditions (walking, talking, on desk, etc.). From this data, 17-dimensional feature vectors were computed (containing frequency information, moments, etc.). The goal is to train, in an online 7 fashion, GMMs based on normal data, which then can be used to perform anomaly detection to detect possible seismic activity. Motivated by the limited storage on smart phones, we evaluate coresets on a data set of 40,000 accelerometer feature vectors, using the parameters k = 6, β = 12, and δ = 0.1. Figure 3(c) presents the results of this experiment. Notice that on this data set, coresets show an even larger improvement over uniform sampling. We hypothesize that this is due to the fact that the recorded accelerometer data is imbalanced, and contains clusters of vastly varying size, so uniform sampling does not represent smaller clusters well. Overall, the coresets obtain a speedup of approximately 35 compared to training on the full set. We also evaluate how GMMs trained on the coreset compare with the baseline GMMs in terms of anomaly detection performance. For each GMM, we compute ROC curves measuring the performance of detecting earthquake recordings from the Southern California Seismic Network (cf., [12]). Note that even very small coresets lead to performance comparable to training on the full set, drastically outperforming uniform sampling (Fig. 3(d)). 6 Related Work Theoretical results on mixtures of Gaussians. There has been a signiﬁcant amount of work on learning and applying GMMs (and more general distributions). Perhaps the most commonly used technique in practice is the EM algorithm [13], which is however only guaranteed to converge to a local optimum of the likelihood. Dasgupta [14] is the ﬁrst to show that parameters of an unknown GMM P can be estimated in polynomial time, with arbitrary accuracy ε, given i.i.d. samples from P. However, his algorithm assumes a common covariance, bounded excentricity, a (known) bound on the smallest component weight, as well as a separation (distance of the means), that scales as Ω( √ d). Subsequent works relax the assumption on separation to d1/4 [15] and k1/4 [16]. [3] is the ﬁrst to learn general GMMs, with separation d1/4. [17] provides the ﬁrst result that does not require any separation, but assumes that the Gaussians are axis-aligned. Recently, [18] and [19] provide algorithms with polynomial running time (except exponential dependence on k) and sample complexity for arbitrary GMMs. However, in contrast to our results, all the results described above crucially rely on the fact that the data set D is actually generated by a mixture of Gaussians. The problem of ﬁtting a mixture model with near-optimal log-likelihood for arbitrary data is studied by [3], who provides a PTAS for this problem. However, their result requires that the Gaussians are identical spheres, in which case the maximum likelihood problem is identical to the k-means problem. In contrast, our results make only mild assumptions about the Gaussian components. Furthermore, none of the algorithms described above applies to the streaming or parallel setting. Coresets. Approximation algorithms in computational geometry often make use of random sampling, feature extraction, and ϵ-samples [20]. Coresets can be viewed as a general concept that includes all of the above, and more. See a comprehensive survey on this topic in [4]. It is not clear that there is any commonly agreed-upon deﬁnition of a coreset, despite several inconsistent attempts to do so [6, 8]. Coresets have been the subject of many recent papers and several surveys [1, 2]. They have been used to great effect for a host of geometric and graph problems, including k-median [6], k-mean [8], k-center [21], k-line median [10] subspace approximation [10, 22], etc. Coresets also imply streaming algorithms for many of these problems [6, 1, 23, 8]. A framework that generalizes and improves several of these results has recently appeared in [4]. 7 Conclusion We have shown how to construct coresets for estimating parameters of GMMs and natural generalizations. Our construction hinges on a natural connection between statistical estimation and clustering problems in computational geometry. To our knowledge, our results provide the ﬁrst rigorous guarantees for obtaining compressed ε-approximations of the log-likelihood of mixture models for large data sets. The coreset construction relies on an intuitive adaptive sampling scheme, and can be easily implemented. By exploiting certain closure properties of coresets, it is possible to construct them in parallel, or in a single pass through a stream of data, using only poly(dkε−1 log n log(1/δ)) space and update time. Unlike most of the related work, our coresets provide guarantees for any given (possibly unstructured) data, without assumptions on the distribution or model that generated it. Lastly, we apply our construction on three real data sets, demonstrating signiﬁcant gains over no or naive subsampling. Acknowledgments This research was partially supported by ONR grant N00014-09-1-1044, NSF grants CNS-0932392, IIS-0953413 and DARPA MSEE grant FA8650-11-1-7156. 8 References [1] P. K. Agarwal, S. Har-Peled, and K. R. 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4,406	Recovering Intrinsic Images with a Global Sparsity Prior on Reﬂectance Peter Vincent Gehler Max Planck Institut for Informatics pgehler@mpii.de Carsten Rother Microsoft Research Cambridge carrot@microsoft.com Martin Kiefel, Lumin Zhang, Bernhard Sch¨olkopf Max Planck Institute for Intelligent Systems {mkiefel,lumin,bs}@tuebingen.mpg.de Abstract We address the challenging task of decoupling material properties from lighting properties given a single image. In the last two decades virtually all works have concentrated on exploiting edge information to address this problem. We take a different route by introducing a new prior on reﬂectance, that models reﬂectance values as being drawn from a sparse set of basis colors. This results in a Random Field model with global, latent variables (basis colors) and pixel-accurate output reﬂectance values. We show that without edge information high-quality results can be achieved, that are on par with methods exploiting this source of information. Finally, we are able to improve on state-of-the-art results by integrating edge information into our model. We believe that our new approach is an excellent starting point for future developments in this ﬁeld. 1 Introduction The task of recovering intrinsic images is to separate a given input image into its material-dependent properties, known as reﬂectance or albedo, and its light-dependent properties, such as shading, shadows, specular highlights, and inter-reﬂectance. A successful separation of these properties would be beneﬁcial to a number of computer vision tasks. For example, an image which solely depends on material-dependent properties is helpful for image segmentation and object recognition [11], while a clean image of shading is a valuable input to shape-from-shading algorithms. As in most previous work in this ﬁeld, we cast the intrinsic image recovery problem into the following simpliﬁed form, where each image pixel is the product of two components: I = sR . (1) Here I ∈R3 is the pixel’s color, in RGB space, R ∈R3 is its reﬂectance and s ∈R its “shading”. Note, we use “shading” as a proxy for all light-dependent properties, e.g. shadows. The fact that shading is only a 1D entity imposes some limitations. For example, shading effects stemming from multiple light sources can only be modeled if all light sources have the same color.1 The goal of this work is to estimate s and R given I. This problem is severely under-constraint, with 4 unknowns and 3 constraints for each pixel. Hence, a trivial solution to (1) is, for instance, I = R, s = 1 for all pixels. The main focus of this paper is on exploring sensible priors for both shading and reﬂectance. Despite the importance of this problem surprisingly little research has been conducted in recent years. Most of the inventions were done in the 70s and 80s. The recent comparative study [7] has shown that the simple Retinex method [9] from the 70s is still the top performing approach. Given 1This problem can be overcome by utilizing a 3D vector for s, as done in [4], which we however do not consider in this work. (a) Image I “paper1” (b) I (in RGB) (c) Reﬂectance R (d) R (in RGB) (e) Shading s Figure 1: An image (a), its color in RGB space (b), the reﬂectance image (c), its distribution in RGB space (d), and the shading image (e). Omer and Werman [12] have shown that an image of a natural scene often contains only a few different “basis colorlines”. Figure (b) shows a dominant gray-scale color-line and other color lines corresponding to the scribbles on the paper (a). These colorlines are generated by taking a small set of “basis colors” which are then linearly “smeared” out in RGB space. The basis colors are clearly visible in (d), where the cluster for white (top, right) is the dominant one. This “smearing effect” comes from properties of the scene (e.g. shading or shadows), and/or properties of the camera, e.g. motion blur. (Note, the few pixels in-between clusters are due to anti-aliasing effects). In this work we approximate the basis colors by a simple mixture of isotropic Gaussians. the progress in the last two decades on probabilistic models, inference and learning techniques, as well as the improved computational power, we believe that now is a good time to revisit this problem. This work, together with the recent papers [14, 4, 7, 15], are a ﬁrst step in this direction. The main motivation of our work is to develop a simple, yet powerful probabilistic model for shading and reﬂectance estimation. In total we use three different types of factors. The ﬁrst one is the most commonly used factor and is key ingredient of all Retinex-based methods. The idea is to extract those image edges which are (potentially) true reﬂectance edges and then to recover a new reﬂectance image that contains only these edges, using a set of Poisson equations. This term on its own is enough to recover a non-trivial decomposition, i.e. s ̸= 1. The next factor is a simple smoothness prior on shading between neighboring image pixels, and has been used by some previous work e.g. [14]. Note, there are a few works, which we discuss in more detail later, that extend these pairwise terms to become patch-based. The third prior term is the main contribution of our work and is conceptually very different from the local (pairwise or patch-based) constraints of previous works. We propose a new global (image-wide) sparsity prior on reﬂectance based on the ﬁndings of [12] and discussed in Fig 1. In the absence of other factors this already produces non-trivial results. This prior takes the form of a Mixture of Gaussians, and encodes the assumption that the reﬂectance value for each pixel is drawn from some mixing components, which in this context we refer to as “basis colors”. The complete model forms a latent variable Random Field model for which we perform MAP estimation. By combining the different terms we are able to outperform state-of-the art. If we use image optimal parameter settings we perform on par with methods that use multiple images as input. To empirically validate this we use the database introduced in the comparative study [7]. 2 Related Work There is a vast amount of literature on the problem of recovering intrinsic images. We refer the reader to detailed surveys in [8, 17, 7], and limit our attention to some few related works. Barrow and Tenenbaum [2] were the ﬁrst to deﬁne the term “intrinsic image”. Around the same time the ﬁrst solution to this problem was developed by Land and McCann [9] known as the Retinex algorithm. After that the Retinex algorithm was extended to two dimensions by Blake [3] and Horn [8], and later applied to color images [6]. The basic Retinex algorithm is a 2-step procedure: 1) detect all image gradients which are caused by changes in reﬂectance; 2) recover a reﬂectance image which preserves the detected reﬂectance gradients. The basic assumption of this approach is that small image gradients are more likely caused by a shading effect and strong gradients by a change in reﬂectance. For color images this rule can be extended by treating changes in the 1D brightness domain differently to changes in the 2D chromaticity space.2 This method, which we denote as “Color Retinex” was the top performing method in the recent comparison paper [7]. Note, 2Note, a gradient in chromaticity can only be caused by differently colored light sources, or inter-reﬂectance. 2 the only approach which could beat Retinex utilizes multiple images [19]. Surprisingly, the study [7] also shows that more sophisticated methods for training the reﬂectance edge detector, using e.g. images patches, did not perform better than the basic Retinex method. In particular the study tested two methods of Tappen et al. [17, 16]. A plausible explanation is offered, namely that these methods may have over-ﬁtted the small amount of training data. The method [17] has an additional intermediate step where a Markov Random Field (MRF) is used to “propagate” reﬂectance gradients along contour lines. The paper [15] implements the same intuition as done here, namely that there is a sparse set of reﬂectances present in the scene. However both approaches bear the following differences. In [15] a sparsity enforcing term is included, that is penalizing reﬂectance differences from some prototype references. This term encourages all reﬂectances to take on the same value, while the model we propose in this paper allows for a mixture of different material reﬂectances and thus keeps their diversity. Also, in contrast to [15], where a gradient aware wavelet transform is used as a new representation, here we work directly in the RGB domain. By doing so we directly extend previous intrinsic image models which makes evident the gains that can be attributed to a global sparse reﬂectance term alone. Recently, Shen et al. [14] introduced an interesting extension of the Retinex method, which bears some similarity with our approach. The key idea in their work is to perform a pre-processing step where the (normalized) reﬂectance image is partitioned into a few clusters. Each cluster is treated as a non-local “super-pixel”. Then a variant of the Retinex method is run on this super-pixel image. The conceptual similarity to our approach is the idea of performing an image-wide clustering step. However, the differences are that they do not formulate this idea as a joint probabilistic model over latent reﬂectance “basis colors” and shading variables. Furthermore, every pixel in a super-pixel must have the same intensity, which is not the case in our work. Also, they need a Retinex type of edge term to avoid the trivial solution of s = 1. Finally, let us brieﬂy mention techniques which use patch-based constraints, instead of pair-wise terms. The seminal work of Freeman et al. on learning low-level vision [5] formulates a probabilistic model for intrinsic images. In essence, they build a patch-based prior jointly over shading and reﬂectance. In a new test image the best explanation for reﬂectance and shading is determined. The key idea is that patches do overlap, and hence form an MRF, where long-range propagation is possible. Since no large-scale ground database was available at that time, they only train and test on computer generated images of blob-like textures. Another patch-based method was recently suggested in [4]. They introduce a new energy term which is satisﬁed when all reﬂectance values in a small, e.g. 3 × 3, patch lie on a plane in RGB space. This idea is derived from the Laplacian matrix used for image matting [10]. On its own this term gives in practice often the trivial solution s = 1. For that reason additional user scribbles are provided to achieve high-quality results.3 3 A Probabilistic Model for Intrinsic Images The model outlined here falls into the class of Conditional Random Fields, specifying a conditional probability distribution over reﬂectance R and shading S components for a given image I p(s, R \| I) ∝exp (−E(s, R \| I)) . (2) Before we describe the energy function E in detail, let us specify the notation. We will denote with subscripts i the values at location i in the image. Thus Ii is an image pixel (vector of dimension 3), Ri a reﬂectance vector (a 3-vector), si the shading (a scalar). The total number of pixels in an image is N. With boldface we denote vectors of components, e.g. s = (s1, . . . , sN). There are two ways to use the relationship (1) to formulate a model for shading and reﬂectance, corresponding to two different image likelihoods p(I \| s, R). One possible way is to relax the relation (1) and for example assume a Gaussian likelihood p(I \| s, R) ∝exp(−∥I −sR∥2) to account for some noise in the image formation process. This yields an optimization problem with 4N unknowns. The second possibility is to assume a delta-prior around sR which results in the following complexity reduction. Since Ic i = siRc i has to hold of all color channels c = {R, G, B}, the unknown variables are speciﬁed up to scalar multipliers, in other words the direction of Ri is already known. We rewrite Ri = ri ⃗Ri, with ⃗Ri = Ii/∥Ii∥, leaving r = (r1, . . . , rN) to be the 3We performed initial tests with this term. However, we found that it did not help to improve performance. 3 only unknown variable. The shading components can be computed using si = ∥Ii∥/ri. Thus the optimization problem is reduced to a search of N variables. The latter reduction is commonly exploited by intrinsic image algorithms in order to simplify the model [7, 14, 4] and in the remainder we will also make use of it. This allows us to write all model parts in terms of r. Note that there is a global scalar k by which the result s, R can be modiﬁed without effecting eq. (1), i.e. I = (sk)(1/kR). For visualization purpose k is chosen such that the results are visually closest to the known ground truth. 3.1 Model The energy function we describe here consists of three different terms that are linearly combined. We will describe the three components and their inﬂuence in greater detail below, ﬁrst we write the optimization problem that corresponds to a MAP solution in its most general form min ri,αi;i=1,...,n wsEs(r) + wrEret(r) + wclEcl(r, α). (3) Note, the global scale of the energy is not important, hence we can always ﬁx one non-zero weight ws, wr, wcl to 1. Shading Prior (Es) We expect the shading of an image to vary smoothly over the image and we encode this in the following pairwise factors Es(r) = X i∼j r−1 i ∥Ii∥−r−1 j ∥Ij∥ 2 , (4) where we use a 4-connected pixel graph to encode the neighborhood relation which we denote with i ∼j. Because of the dependency on the inverse of r, this term is not jointly convex in r. Any model that includes this smoothness prior thus has the (potential) problem of multiple local minima. Empirically we have seen that, however, this function seems to be very well behaved, a large range of different starting points for r resulted in the same minimum. Nevertheless, we use multiple restarts with different starting points, see optimization selection 3.2. Gradient Consistency (Eret) As discussed in the introduction, the main idea of the Retinex algorithm is to disambiguate between edges that are due to shading variations from those that are caused by material reﬂectance changes. This idea is then implemented as follows. Assume that we already know, or have classiﬁed, that an edge at location i, j in the input image is caused by a change in reﬂectance. Then we know the magnitude of the gradient that has to appear in the reﬂectance map by noting that log(Ii)−log(Ij) = log(ri ⃗Ri)−log(rj ⃗Rj). Using the fact log(∥Ii∥) = log(Ic i )−log(⃗Rc i) (for all channels c) and assuming a squared deviation around the log gradient magnitude, this translates into the following Gaussian MRF term on the reﬂectances Eret(r) = X i∼j (log(ri) −log(rj) −gij(I)(log(∥Ii∥) −log(∥Ij∥)))2 . (5) It remains to specify the classiﬁcation function g(I) for the image edges. In this work we adopt the Color Retinex version that has been proposed in [7]. For each pixel i and a neighbor j we compute the gradient of the intensity image and the gradient of the chromaticity change. If both gradients exceed a certain threshold (θg and θc resp.), the edge at i, j is classiﬁed as being a “reﬂectance edge” and in this case gij(I) = 1. The two parameters which are the thresholds θg, θc for the intensity and the chromaticity change are then estimated using leave-one-out-cross validation. It is worth noting that this term is qualitatively different from the smoothness prior on shading (4) even for pixels where gij(I) = 0. Here, the log-difference is penalized whereas the shading smoothness does also depend on the intensity values ∥Ii∥, ∥Ij∥. By setting wcl, ws = 0 in Eq. (2) we recover Color Retinex [7]. Global Sparse Reﬂectance Prior (Ecl ) Motivated by the ﬁndings of [12] we include a term that acts as a global potential on the reﬂectances and favors the decomposition into some few reﬂectance clusters. We assume C different reﬂectance clusters, each of which is denoted by ˜Rc, c ∈{1, . . . , C}. Every reﬂectance component ri belongs to one of the clusters and we denote its cluster membership with the variable αi ∈{1, . . . , C}. This is summarized in the following energy term 4 Figure 2: A crop from the image “panther”. Left: input image I and true decomposition (R, s). Note, the colors in reﬂectance image (True R) have been modiﬁed on purpose such that there are exactly 4 different colors. The second column shows a clustering (here from the solution with ws = 0), where each cluster has an arbitrary color. The remaining columns show results with various settings for C and ws (left reﬂectance image, right shading image). Top row is the result for C = 4 and bottom row for C = 50 clusters, columns are results for ws = 0, 10−5, and 0.1. Below the images is the corresponding LMSE score (described in Section 4.1). (Note, results are visually slightly different since the unknown overall global scaling factor k is set differently, that is I = (sk)(1/kR). Ecl(r, α) = n X i=1 ∥ri ⃗Ri −˜Rαi∥2. (6) Here, both continuous r and discrete α variables are mixed. This represents a global potential, since the cluster means depend on the assignment of all pixels in the image. For ﬁxed α, this term is convex in r and for ﬁxed r the optimum of α is a simple assignment problem. The cluster means ˜Rc are optimally determined given r and α: ˜Rc = 1 \|{i:αi=c}\| P i:αi=c ri ⃗Ri. Relationship between Ecl and Es The example in Figure 2 highlights the inﬂuence of the terms. We use a simpliﬁed model (2), namely Ecl + wsEs, and vary ws as well as the number of clusters. Let us ﬁrst consider the case where ws = 0 (third column). Independent of the clustering we get an imperfect result. This is expected since there is no constraint across clusters. Hence the shading within one cluster looks reasonable, but is not aligned across clusters. By adding a little bit of smoothing (ws = 10−5; 4’th column), this problem is cured for both clusterings. It is very important to note that too many clusters (here C=50) do not affect the result very much. The reason is that enough clustering constraints are present to recover the variation in shading. If we were to give each pixel its own cluster this would no longer be true and we would get the trivial solution of s = 1. Finally, results deteriorate when the smoothing term is too strong (last column ws = 0.1), since it prefers a constant shading. Note, that for this simple toy example the smoothness prior was not important, however for real images the best results are achieved by using a non-zero ws. 3.2 Optimization of (3) Algorithm 1 Coordinate Descent for solving (3) 1: Select r0 as described in the text 2: α0 ←K-Means clustering of {r0 i ⃗Ri, i = 1, . . . , N} 3: t ←0 4: repeat 5: rt+1 ←optimize (3) with αt ﬁxed 6: ˜Rc = P i:αi=c ri ⃗Ri/\|{i : αi = c}\| 7: αt+1 ←assign new cluster labels with rt+1 ﬁxed 8: t ←t + 1 9: until E(rt−1, αt−1) −E(rt, αt) < θ The MAP problem (3) consists of both discrete and continuous variables and we solve it using coordinate descent. The entire algorithm is summarized in Algorithm 1. 4 Given an initial value for α we have seen empirically that our function tends to yield same solutions, irrespective of the starting point r. In order to be also robust with respect to this initial choice, we choose from a range of initial r values as described next. From these starting points we choose the one with the lowest objective value (energy) and its corresponding result. 4Code available http://people.tuebingen.mpg.de/mkiefel/projects/intrinsic 5 comment Es Ecl Eret LOO-CV best single image opt. Color Retinex ✓ 29.5 29.5 25.5 no edge information ✓ ✓ 30.0 30.6 18.2 Col-Ret+ global term ✓ ✓ 27.2 24.4 18.1 full model ✓ ✓ ✓ 27.4 24.4 16.1 Table 1: Comparing the effect of including different terms. The column “best-single” is the parameter set that works best on all 16 images jointly, “image opt.” is the result when choosing the parameters optimal for each image individually, based on ground truth information. We have seen empirically that this procedure gives stable results. For instance, we virtually always achieve a lower energy compared to using the ground truth r as initial start point. Initialization of r It is reasonable to assume that the output has a ﬁxed range, i.e. 0 ≥Rc i, si ≥1 (for all c, i).5 In particular, this is true for the data in [7]. From these constraints we can derive that ∥Ii∥≥ri ≥3. Given that, we use the following three starting points for r, by varying γ ∈ {0.3, 0.5, 0.7}: ri = γ∥Ii∥+ 3(1 −γ). Additionally we choose the start point r = 1. From these four different initial settings we choose the result which corresponds to the lowest ﬁnal energy. Initialization of α Given an initial value for r we can compute the terms in Eq.(6) and use KMeans clustering to optimize it. We use the best solution from ﬁve restarts. Updating r for a given ﬁxed α this is implemented using a conjugate gradient descent solver [1]. This typically converges in some few hundred iterations for the images used in the experiments. Updating α for given r this is a simple assignment problem: αi = argminc=1,...,C∥ri ⃗Ri −˜Rc∥2. 4 Experiments For the empirical evaluation we use the intrinsic image database that has been introduced in [7]. This dataset consists of 16 different images for all of which the ground truth shading and reﬂectance components are available. We refer to [7] for details on how this data was collected. Some of the images can be seen in Figure 3. In all experiments we compare against Color Retinex which was found to be the best performing method among those that take a single image as input. The method from [19] yields better results but requires multiple input images from different light variations. 4.1 Error metric We report the performance of the algorithms using the two different error metrics that have been suggested by the creators of the database [7]. The ﬁrst metric is the average of the localized mean squared error (LMSE) between the predicted and true shading and predicted and true reﬂectance image. 6 Since the LMSE vary considerably we also use the average rank of the algorithm. 4.2 Experimental set-up and parameter learning All free parameters of the models, e.g. the weights wcl, ws, wr and the gradient thresholds θc, θg have been chosen using a leave-one-out estimate (LOO-CV). Due to the high variance of the scores for the images we used the median error to score the parameters. Thus for image i the parameter was chosen that leads to the lowest median error on all images except i. Additionally we record the best single parameter set that works well on all images, and the score that is obtained when using the optimal parameters on each image individually. Although the latter estimate involves knowing ground truth estimates we are interested in the lower bound of the performance, in an interactive scenario a user can provide additional information to achieve this, as in [4]. We select the parameters from the following ranges. Whenever used, we ﬁx wcl = 1 since it sufﬁces to specify the relative difference between the parameters. For models using both the cluster and shading smoothness terms, we select from ws ∈{0.001, 0.01, 0.1}, for models that use the cluster and Color Retinex term wr ∈{0.001, 0.01, 0.1, 1, 10}. When all three terms are non-zero, we vary ws as above paired with wr ∈×{0.1ws, ws, 10ws}. The gradient thresholds are varied in θg, θc ∈{0.075, 1} which yields four possible conﬁgurations. The reﬂectance cluster count is varied in C ∈{10, 50, 150}. 5This assumption is violated if there is no global scalar k such that 0 ≥(1/kRc i), (ksi) ≥1. 6We multiply by 1000 for easier readability 6 4.3 Comparison - Model variations In a ﬁrst set of experiments we investigate the inﬂuence of using combinations of the prior terms described in Section 3.1. The numerical results are summarized in Table 1. The ﬁrst observation is that the Color Retinex algorithm (1st row) performs about similar to the system using a shading smoothness prior together with the global factor Ecl (2nd row). Note that the latter system does not use any gradient information for estimation. This conﬁrms our intuition that the term Ecl provides strong coupling information between reﬂectance components, as also discussed in Figure 2. The lower value for the image optimal setting of 18.2 compared to 25.5 for Color Retinex indicates that one would beneﬁt from a better parameter estimate, i.e. the ﬂexibility of this algorithm is higher. Equipping Color Retinex with the global reﬂectance term improves all recorded results (3rd vs 2nd row). Again it seems that the LOO-CV parameter estimation is more stable in this case. Combining all three parts (4th row) does not improve the results over Color Retinex with the reﬂectance prior. With knowledge about the optimal image parameter it yields a lower LMSE score (16.1 vs 18.1). 4.4 Comparison to Literature LOO-CV rank best single im. opt. TAP05 [17] 56∗ TAP06 [16] 39∗ SHE [14]+ n/a n/a 56.2 n/a SHE [15]× n/a n/a (20.4) BAS [7] 72.6 5.1 60.3 36.6 Gray-Ret [7] 40.7 4.9 40.7 28.9 Col-Ret 29.5 3.7 29.5 25.5 full model 27.4 3.0 24.4 16.1 Weiss [19] 21.5 2.7 21.5 21.5 Weiss+Ret [7] 16.4 1.7 16.4 15.0 Table 2: Method comparison with other intrinsic image algorithms also compared in [7]. Refer to Tab. 1 for a description of the quantities. Note that the last two methods from [19] use multiple input images. For entries ’-’ we had no individual results (and no code), the two numbers marked ∗are estimated from Fig4.a [7]. SHE+ is our implementation. SHE× Note that in [15] results were only given for 13 of 16 images from [7]. The additional data was kindly provided by authors. In Table 2 we compare the numerical results of our method to other intrinsic image algorithms. We again include the single best parameter and image dependent optimal parameter set. Although those are positively biased and obviously decrease with model complexity we believe that they are informative, given the parameter estimation problems due to the diverse and small database. The full model using all terms Ecl, Es and Ecret improves over all the compared methods that use only a single image as input, but SHE× (see below). The difference in rank between (ColRet) and (full model) indicates that the latter model is almost always better (direct comparison: 13 out of 16 images) than Color Retinex alone. The full model is even better on 6/16 images than the Weiss algorithm [19] that uses multiple images. Regarding the results of SHE×, we could not resolve with certainty whether the reported results should be compared as “best single” or “im.opt.” (most parameters in [15] are common to all images, the strategy for setting λmax is not entirely speciﬁed). Assuming “best single” SHE× is better in terms of LMSE, in direct comparison both models are better on 8/16 images. Comparing as an “im.opt.” setting, our full model yields lower LMSE and is better on 12/16 images. 4.5 Visual Comparison Additionally to the quantitative numbers we present some visual comparison in Figure 3, since the numbers not always reﬂect a visually pleasing results. For example note that the method BAS that either attributes all variations to shading (r = 1) or to reﬂectance alone (s = 1) already yields a LMSE of 36.6, if for every image the optimal choice between the two is made. Numerically this is better than [16, 17] and “Gray-Ret” with proper model selection. However the results of those algorithms are of course visually more pleasing. We have also tested our method on various other real-world images and results are visually similar to [15, 4]. Due to missing ground truth and lack of space we do not show them. Figure 3 shows results with various models and settings. The “turtle” example (top three rows) shows the effect of the global term. Without the global term (Color Retinex with LOO-CV and image optimal) the result is imperfect. The key problem of Retinex is highlighted in the two zoomin pictures with blue border (second column, left side). The upper one shows the detected edges in black. As expected the Retinex result has discontinuities at these edges, but over-smooths otherwise (lower picture). With a global term (remaining three results) the images look visually much better. 7 Figure 3: Various results obtained with different methods and settings (more in supplementary material); For each result: left reﬂectance image, right shading image Note that the third row shows an extreme variation for the full model when switching from image optimal setting to LOO-CV setting. The example “teabag2” illustrates nicely the point that Color Retinex and our model without edge term (i.e. no Retinex term) achieve very complementary results. Our model without edges is sensitive to edge transitions, while Color Retinex has problems with ﬁne details, e.g. the small text below “TWININGS”. Combing all terms (full model) gives the best result with lowest LMSE score (16.4). Note, in this case we chose for both methods the image optimal settings to illustrate the potential of each model. 5 Discussion and Conclusion We have introduced a new probabilistic model for intrinsic images that explicitly models the reﬂectance formation process. Several extensions are conceivable, e.g. one can relax the condition I = sR to allow deviations. Another reﬁnement would be to replace the Gaussian cluster term with a color line term [12]. Building on the work of [5, 4] one can investigate various higher-order (patch-based) priors for both reﬂectance and shading. A main concern is that in order to develop more advanced methods a larger and even more diverse database than the one of [7] is needed. This is especially true to enable learning of richer models such as Fields of Experts [13] or Gaussian CRFs [18]. We acknowledge the complexity of collecting ground truth data, but do believe that the creation of a new, much enlarged dataset, is a necessity for future progress in this ﬁeld. 8 References [1] www.gatsby.ucl.ac.uk/˜edward/code/minimize. [2] H. G. Barrow and J. M. Tenenbaum. Recovering intrinsic scene characteristics from images. Computer Vision Systems, 1978. [3] A. Blake. Boundary conditions for lightness computation in mondrian world. Computer Vision, Graphics, and Image Processing, 1985. [4] A. Bousseau, S. Paris, and F. Durand. User assisted intrinsic images. SIGGRAPH Asia, 2009. [5] W. T. Freeman, E. C. Pasztor, and O. T. Carmichael. Learning low-level vision. International Journal of Computer Vision (IJCV), 2000. [6] B. V. Funt, M. S. Drew, and M. Brockington. Recovering shading from color images. In European Conference on Computer Vision (ECCV), 1992. [7] R. Grosse, M. K. Johnson, E. H. Adelson, and W. T. Freeman. Ground-truth dataset and baseline evaluations for intrinsic image algorithms. In International Conference on Computer Vision (ICCV), 2009. [8] B. K. Horn. Robot Vision. MIT press, 1986. [9] E. Land and J. McCann. Lightness and retinex theory. Journal of the Optical Society of America, 1971. [10] A. Levin, D. Lischinski, and Y. Weiss. A closed form solution to natural image matting. IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), 30(2), 2008. [11] Y.-H. W. Ming Shao. Recovering facial intrinsic images from a single input. Lecture Notes in Computer Science, 2009. [12] I. Omer and M. Werman. Color lines: Image speciﬁc color representation. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2004. [13] S. Roth and M. J. Black. Fields of experts. International Journal of Computer Vision (IJCV), 82(2):205– 229, 2009. [14] L. Shen, P. Tan, and S. Lin. Intrinsic image decomposition with non-local texture cues. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2008. [15] L. Shen and C. Yeo. Intrinsic images decomposition using a local and global sparse representation of reﬂectance. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2011. [16] M. Tappen, E. Adelson, and W. Freeman. Estimating intrinsic component images using non-linear regression. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2006. [17] M. Tappen, W. Freeman, and E. Adelson. Recovering intrinsic images from a single image. IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), 2005. [18] M. Tappen, C. Liu, E. H. Adelson, and W. T.Freeman. Learning gaussian conditional random ﬁelds for low-level vision. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2007. [19] Y. Weiss. Deriving intrinsic images from image sequences. In International Conference on Computer Vision (ICCV), 2001. 9	2011	59
4,407	The Kernel Beta Process Lu Ren∗ Electrical & Computer Engineering Dept. Duke University Durham, NC 27708 lr22@duke.edu Yingjian Wang∗ Electrical & Computer Engineering Dept. Duke University Durham, NC 27708 yw65@duke.edu David Dunson Department of Statistical Science Duke University Durham, NC 27708 dunson@stat.duke.edu Lawrence Carin Electrical & Computer Engineering Dept. Duke University Durham, NC 27708 lcarin@duke.edu Abstract A new L´evy process prior is proposed for an uncountable collection of covariatedependent feature-learning measures; the model is called the kernel beta process (KBP). Available covariates are handled efﬁciently via the kernel construction, with covariates assumed observed with each data sample (“customer”), and latent covariates learned for each feature (“dish”). Each customer selects dishes from an inﬁnite buffet, in a manner analogous to the beta process, with the added constraint that a customer ﬁrst decides probabilistically whether to “consider” a dish, based on the distance in covariate space between the customer and dish. If a customer does consider a particular dish, that dish is then selected probabilistically as in the beta process. The beta process is recovered as a limiting case of the KBP. An efﬁcient Gibbs sampler is developed for computations, and state-of-the-art results are presented for image processing and music analysis tasks. 1 Introduction Feature learning is an important problem in statistics and machine learning, characterized by the goal of (typically) inferring a low-dimensional set of features for representation of high-dimensional data. It is desirable to perform such analysis in a nonparametric manner, such that the number of features may be learned, rather than a priori set. A powerful tool for such learning is the Indian buffet process (IBP) [4], in which the data samples serve as “customers”, and the potential features serve as “dishes”. It has recently been demonstrated that the IBP corresponds to a marginalization of a beta-Bernoulli process [15]. The IBP and beta-Bernoulli constructions have found signiﬁcant utility in factor analysis [7, 17], in which one wishes to infer the number of factors needed to represent data of interest. The beta process was developed originally by Hjort [5] as a L´evy process prior for “hazard measures”, and was recently extended for use in feature learning [15], the interest of this paper; we therefore here refer to it as a “feature-learning measure.” The beta process is an example of a L´evy process [6], another example of which is the gamma process [1]; the normalized gamma process is well known as the Dirichlet process [3, 14]. A key characteristic of such models is that the data samples are assumed exchangeable, meaning that the order/indices of the data may be permuted with no change in the model. ∗The ﬁrst two authors contributed equally to this work. 1 An important line of research concerns removal of the assumption of exchangeability, allowing incorporation of covariates (e.g., spatial/temporal coordinates that may be available with the data). As an example, MacEachern introduced the dependent Dirichlet process [8]. In the context of feature learning, the phylogenetic IBP removes the assumption of sample exchangeability by imposing prior knowledge on inter-sample relationships via a tree structure [9]. The form of the tree may be constituted as a result of covariates that are available with the samples, but the tree is not necessarily unique. A dependent IBP (dIBP) model has been introduced recently, with a hierarchical Gaussian process (GP) used to account for covariate dependence [16]; however, the use of a GP may constitute challenges for large-scale problems. Recently a dependent hierarchical beta process (dHBP) has been developed, yielding encouraging results [18]. However, the dHBP has the disadvantage of assigning a kernel to each data sample, and therefore it scales unfavorably as the number of samples increases. In this paper we develop a new L´evy process prior, termed the kernel beta process (KBP), which yields an uncountable number of covariate-dependent feature-learning measures, with the beta process a special case. This model may be interpreted as inferring covariates x∗ i for each feature (dish), indexed by i. The generative process by which the nth data sample, with covariates xn, selects features may be viewed as a two-step process. First the nth customer (data sample) decides whether to “examine” dish i by drawing z(1) ni ∼Bernoulli(K(xn, x∗ i ; ψ∗ i )), where ψ∗ i are dish-dependent kernel parameters that are also inferred (the {ψ∗ i } deﬁning the meaning of proximity/locality in covariate space). The kernels are designed to satisfy K(xn, x∗ i ; ψ∗ i ) ∈(0, 1], K(x∗ i , x∗ i ; ψ∗ i ) = 1, and K(xn, x∗ i ; ψ∗ i ) →0 as ∥xn −x∗ i ∥2 →∞. In the second step, if z(1) ni = 1, customer n draws z(2) ni ∼Bernoulli(πi), and if z(2) ni = 1, the feature associated with dish i is employed by data sample n. The parameters {x∗ i , ψ∗ i , πi} are inferred by the model. After computing the posterior distribution on model parameters, the number of kernels required to represent the measures is deﬁned by the number of features employed from the buffet (typically small relative to the data size); this is a signiﬁcant computational savings relative to [18, 16], for which the complexity of the model is tied to the number of data samples, even if a small number of features are ultimately employed. In addition to introducing this new L´evy process, we examine its properties, and demonstrate how it may be efﬁciently applied in important data analysis problems. The hierarchical construction of the KBP is fully conjugate, admitting convenient Gibbs-sampling (complicated sampling methods were required for the method in [18]). To demonstrate the utility of the model we consider imageprocessing and music-analysis applications, for which state-of-the-art performance is demonstrated compared to other relevant methods. 2 Kernel Beta Process 2.1 Review of beta and Bernoulli processes A beta process B ∼BP(c, B0) is a distribution on positive random measures over the space (Ω, F). Parameter c(ω) is a positive function over ω ∈Ω, and B0 is the base measure deﬁned over Ω. The beta process is an example of a L´evy process, and the L´evy measure of BP(c, B0) is ν(dπ, dω) = c(ω)π−1(1 −π)c(ω)−1dπB0(dω) (1) To draw B, one draws a set of points (ωi, πi) ∈Ω× [0, 1] from a Poisson process with measure ν, yielding B = ∞ X i=1 πiδωi (2) where δωi is a unit point measure at ωi; B is therefore a discrete measure, with probability one. The inﬁnite sum in (2) is a consequence of drawing Poisson(λ) atoms {ωi, πi}, with λ = R Ω R [0,1] ν(dω, dπ) = ∞. Additionally, for any set A ⊂F, B(A) = P i: ωi∈A πi. If Zn ∼BeP(B) is the nth draw from a Bernoulli process, with B deﬁned as in (2), then Zn = ∞ X i=1 bniδωi , bni ∼Bernoulli(πi) (3) 2 A set of N such draws, {Zn}n=1,N, may be used to deﬁne whether feature ωi ∈Ωis utilized to represent the nth data sample, where bni = 1 if feature ωi is employed, and bni = 0 otherwise. One may marginalize out the measure B analytically, yielding conditional probabilities for the {Zn} that correspond to the Indian buffet process [15, 4]. 2.2 Covariate-dependent L´evy process In the above beta-Bernoulli construction, the same measure B ∼BP(c, B0) is employed for generation of all {Zn}, implying that each of the N samples have the same probabilities {πi} for use of the respective features {ωi}. We now assume that with each of the N samples of interest there are an associated set of covariates, denoted respectively as {xn}, with each xn ∈X. We wish to impose that if samples n and n′ have similar covariates xn and xn′, that it is probable that they will employ a similar subset of the features {ωi}; if the covariates are distinct it is less probable that feature sharing will be manifested. Generalizing (2), consider B = ∞ X i=1 γiδωi , ωi ∼B0 (4) where γi = {γi(x) : x ∈X} is a stochastic process (random function) from X →[0, 1] (drawn independently from the {ωi}). Hence, B is a dependent collection of L´evy processes with the measure speciﬁc to covariate x ∈X being Bx = P∞ i=1 γi(x)δωi. This constitutes a general speciﬁcation, with several interesting special cases. For example, one might consider γi(x) = g{µi(x)}, where g : R →[0, 1] is any monotone differentiable link function and µi(x) : X →R may be modeled as a Gaussian process [10], or related kernel-based construction. To choose g{µi(x)} one can potentially use models for the predictor-dependent breaks in probit, logistic or kernel stick-breaking processes [13, 11, 2]. In the remainder of this paper we propose a special case for design of γi(x), termed the kernel beta process (KBP). 2.3 Characteristic function of the kernel beta process Recall from Hjort [5] that B ∼BP(c(ω), B0) is a beta process on measure space (Ω, F) if its characteristic function satisﬁes E[ejuB(A)] = exp{ Z [0,1]×A (ejuπ −1)ν(dπ, dω)} (5) where here j = √−1, and A is any subset in F. The beta process is a particular class of the L´evy process, with ν(dπ, dω) deﬁned as in (1). For kernel K(x, x∗; ψ∗), let x ∈X, x∗∈X, and ψ∗∈Ψ; it is assumed that K(x, x∗; ψ∗) ∈[0, 1] for all x, x∗and ψ∗. As a speciﬁc example, for the radial basis function K(x, x∗; ψ∗) = exp[−ψ∗∥x −x∗∥2], where ψ∗∈R+. Let x∗represent random variables drawn from probability measure H, with support on X, and ψ∗is also a random variable drawn from an appropriate probability measure Q with support over Ψ (e.g., in the context of the radial basis function, ψ∗are drawn from a probability measure with support over R+). We now deﬁne a new L´evy measure νX = H(dx∗)Q(dψ∗)ν(dπ, dω) (6) where ν(dπ, dω) is the L´evy measure associated with the beta process, deﬁned in (1). Theorem 1 Assume parameters {x∗ i , ψ∗ i , πi, ωi} are drawn from measure νX in (6), and that the following measure is constituted Bx = ∞ X i=1 πiK(x, x∗ i ; ψ∗ i )δωi (7) which may be evaluated for any covariate x ∈ X. For any ﬁnite set of covariates S = {x1, . . . , x\|S\|}, we deﬁne the \|S\|-dimensional random vector K = (K(x1, x∗; ψ∗), . . . , K(x\|S\|, x∗; ψ∗))T , with random variables x∗and ψ∗drawn from H and Q, respectively. For any set A ⊂ F, the B evaluated at covariates S, on the set A, 3 yields an \|S\|-dimensional random vector B(A) = (Bx1(A), . . . , Bx\|S\|(A))T , where Bx(A) = P i: ωi∈A πiK(x, x∗ i ; ψ∗ i ). Expression (7) is a covariate-dependent L´evy process with L´evy measure (6), and characteristic function for an arbitrary set of covariates S satisfying E[ej<u,B(A)>] = exp{ Z X×Ψ×[0,1]×A (ej<u,Kπ> −1)νX (dx∗, dψ∗, dπ, dω)} (8) 2 A proof is provided in the Supplemental Material. Additionally, for notational convenience, below a draw of (7), valid for all covariates in X, is denoted B ∼KBP(c, B0, H, Q), with c and B0 deﬁning ν(dπ, dω) in (1). 2.4 Relationship to the beta-Bernoulli process If the covariate-dependent measure Bx in (7) is employed to deﬁne covariate-dependent feature usage, then Zx ∼BeP(Bx), generalizing (3). Hence, given {x∗ i , ψ∗ i , πi}, the feature-usage measure is Zx = P∞ i=1 bxiδωi, with bxi ∼Bernoulli(πiK(x, x∗ i ; ψ∗ i )). Note that it is equivalent in distribution to express bxi = z(1) xi z(2) xi , with z(1) xi ∼Bernoulli(K(x, x∗ i ; ψ∗ i )) and z(2) xi ∼Bernoulli(πi). This model therefore yields the two-step generalization of the generative process of the beta-Bernoulli process discussed in the Introduction. The condition z(1) xi = 1 only has a high probability when observed covariates x are near the (latent/inferred) covariates x∗ i . It is deemed attractive that this intuitive generative process comes as a result of a rigorous L´evy process construction, the properties of which are summarized next. 2.5 Properties of B For all Borel subsets A ∈F, if B is drawn from the KBP and for covariates x, x′ ∈X, we have E[Bx(A)] = B0(A)E(Kx) Cov(Bx(A), Bx′(A)) = E(KxKx′) Z A B0(dω)(1 −B0(dω)) c(ω) + 1 −Cov(Kx, Kx′) Z A B2 0(dω) where, E(Kx) = R X×Ψ K(x, x∗; ψ∗)H(dx∗)Q(dψ∗). If K(x, x∗; ψ∗) = 1 for all x ∈X, E(Kx) = E(KxKx′) = 1, and Cov(Kx, Kx′) = 0, and the above results reduce to the those for the original BP [15]. Assume c(ω) = c, where c ∈R+ is a constant, and let Kx = (K(x, x∗ 1; ψ∗ 1), K(x, x∗ 2; ψ∗ 2), . . . )T represent an inﬁnite-dimensional vector, then for ﬁxed kernel parameters {x∗ i , ψ∗ i }, Corr(Bx(A), Bx′(A)) = < Kx, Kx′ > ∥Kx∥2 · ∥Kx′∥2 (9) where it is assumed < Kx, Kx′ >, ∥Kx∥2, ∥Kx′∥2 are ﬁnite; the latter condition is always met when we (in practice) truncate the number of terms used in (7). The expression in (9) clearly imposes the desired property of high correlation in Bx and Bx′ when x and x′ are proximate. Proofs of the above properties are provided in the Supplemental Material. 3 Applications 3.1 Model construction We develop a covariate-dependent factor model, generalizing [7, 17], which did not consider covariates. Consider data yn ∈RM with associated covariates xn ∈RL, with n = 1, . . . , N. The factor loadings in the factor model here play the role of “dishes” in the buffet analogy, and we model the data as yn = D(wn ◦bn) + ϵn Zxn ∼BeP(Bxn), B ∼KBP(c, B0, H, Q), B0 ∼DP(α0G0) (10) wn ∼N(0, α−1 1 IT ), ϵn ∼N(0, α−1 2 IM) 4 with gamma priors placed on α0, α1 and α2, with ◦representing the pointwise (Hadamard) vector product, and with IM representing the M × M identity matrix. The Dirichlet process [3] base measure G0 = N(0, 1 M IM), and the KBP base measure B0 is a mixture of atoms (factor loadings). For the applications considered it is important that the same atoms be reused at different points {x∗ i } in covariate space, to allow for repeated structure to be manifested as a function of space or time, within the image and music applications, respectively. The columns of D are deﬁned respectively by (ω1, ω2, . . . ) in B, and the vector bn = (bn1, bn2, . . . ) with bnk = Zxn(ωk). Note that B is drawn once from the KBP, and when drawing the Zxn we evaluate B as deﬁned by the respective covariate xn. When implementing the KBP, we truncate the sum in (7) to T terms, and draw the πi ∼ Beta(1/T, 1), which corresponds to setting c = 1. We set T large, and the model infers the subset of {πi}i=1,T that have signiﬁcant amplitude, thereby estimating the number of factors needed for representation of the data. In practice we let H and Q be multinomial distributions over a discrete and ﬁnite set of, respectively, locations for {x∗ i } and kernel parameters for {ψ∗ i }, details of which are discussed in the speciﬁc examples. In (10), the ith column of D, denoted Di, is drawn from B0, with B0 drawn from a Dirichlet process (DP). There are multiple ways to perform such DP clustering, and here we apply the P´olya urn scheme [3]. Assume D1, D2, . . . , Di−1 are a series of i.i.d. random draws from B0, then the successive conditional distribution of Di is of the following form: Di\|D1, . . . , Di−1, α0, G0 ∼ Nu X l=1 n∗ l i −1 + α0 δD∗ l + α0 i −1 + α0 G0, (11) where {D∗ l }l=1,Nu are the unique dictionary elements shared by the ﬁrst i −1 columns of D, and n∗ l = Pi−1 j=1 δ(Dj = D∗ l ). For model inference, an indicator variable ci is introduced for each Di, and ci = l with a probability proportional to n∗ l , with l = 1, . . . , Nu, with ci equal to Nu + 1 with a probability controlled by α0. If ci = l for l = 1, . . . , Nu, Di takes the value D∗ l ; otherwise Di is drawn from the prior G0 = N(0, 1 M IM), and a new dish/factor loading D∗ Nu+1 is hence introduced. 3.2 Extensions It is relatively straightforward to include additional model sophistication into (10), one example of which we will consider in the context of the image-processing example. Speciﬁcally, in many applications it is inappropriate to assume a Gaussian model for the noise or residual ϵn. In Section 4.3 we consider the following augmented noise model: ϵn = λn ◦mn + ˆϵn (12) λn ∼N(0, α−1 λ IM), mnp ∼Bernoulli(˜πn), ˜πn ∼Beta(a0, b0), ˆϵn ∼N(0, α−1 3 IM) with gamma priors placed on αλ and α2, and with p = 1, . . . , M. The term λn ◦mn accounts for “spiky” noise, with potentially large amplitude, and ˆπn represents the probability of spiky noise in data sample n. This type of noise model was considered in [18], with which we compare. 3.3 Inference The model inference is performed with a Gibbs sampler. Due to the limited space, only those variables having update equations distinct from those in the BP-FA of [17] are included here. Assume T is the truncation level for the number of dictionary elements, {Di}i=1,T ; Nu is the number of unique dictionary elements values in the current Gibbs iteration, {D∗ l }l=1,Nu. For the applications considered in this paper, K(xn, x∗ i ; ψ∗ i ) is deﬁned based on the Euclidean distance: K(xn, x∗ i ; ψ∗ i ) = exp[−ψ∗ i \|\|xn −x∗ i \|\|2] for i = 1, . . . , T; both ψ∗ i and x∗ i are updated from multinomial distributions (deﬁning Q and H, respectively) over a set of discretized values with a uniform prior for each; more details on this are discussed in Sec. 4. • Update {D∗ l }l=1,L: D∗ l ∼N(µl, Σl), µl = Σl[α2 N X n=1 X i:ci=l (bniwni)y−l n ], Σl = [α2 N X n=1 X i:ci=l (bniwni)2 + M]−1IM, 5 where y−l n = yn −P i:ci̸=l Di(bniwni). • Update {ci}i=1,T : p(ci) ∼Mult(pi), p(ci = l\|−) ∝ ( n∗ l −i T −1+α0 QN n=1 exp{−α2 2 ∥y−i n −D∗ l (bniwni)∥2 2}, if l is previously used, α0 T −1+α0 QN n=1 exp{−α2 2 ∥y−i n −D∗ lnew(bniwni)∥2 2}, if l = lnew, where n∗ l −i = P j:j̸=i δ(Dj = D∗ l ), and y−i n = yn −P k:k̸=i Dk(bnkwnk); pi is realized by normalizing the above equation. • Update {Zxn}n=1,N: for Zxn, update each component p(bni) ∼Bernoulli(vni) for i = 1, . . . , K, p(bni = 1) p(bni = 0) = exp{−α2 2 DT i Diw2 ni −2wniDT i y−i n }πiK(xn, x∗ i ; ψ∗ i ) 1 −πiK(xn, x∗ i ; ψ∗ i ) . vni is calculated by normalizing p(bni) with the above constraint. • Update {πi}i=1,T : Introduce two sets of auxiliary variables {z(1) ni }i=1,T and {z(2) ni }i=1,T for each data yn. Assume z(1) ni ∼Bernoulli(πi) and z(2) ni ∼Bernoulli(K(xn, x∗ i ; ψ∗ i )). For each speciﬁc n, – If bni = 1, z(1) ni = 1 and z(2) ni = 1; – If bni = 0,          p(z(1) ni = 0, z(2) ni = 0\|bni = 0) = (1−πi) 1−K(xn,x∗ i ;ψ∗ i ) 1−πiK(xn,x∗ i ;ψ∗ i ) p(z(1) ni = 0, z(2) ni = 1\|bni = 0) = (1−πi)K(xn,x∗ i ;ψ∗ i ) 1−πiK(xn,x∗ i ;ψ∗ i ) p(z(1) ni = 1, z(2) ni = 0\|bni = 0) = πi 1−K(xn,x∗ i ;ψ∗ i ) 1−πiK(xn,x∗ i ;ψ∗ i ) From the above equations, we derive the conditional distribution for πi, πi ∼Beta 1 T + X n z(1) ni , 1 + X n (1 −z(1) ni ) . 4 Results 4.1 Hyperparameter settings For both α1 and α2 the corresponding prior was set to Gamma(10−6, 10−6); the concentration parameter α0 was given a prior Gamma(1, 0.1). For both experiments below, the number of dictionary elements T was truncated to 256, the number of unique dictionary element values was initialized to 100, and {πi}i=1,T were initialized to 0.5. All {ψ∗ i }i=1,T were initialized to 10−5 and updated from a set {10−5, 10−4, 10−3, 10−2, 10−1, 1} with a uniform prior Q. The remaining variables were initialized randomly. No parameter tuning or optimization has been performed. 4.2 Music analysis We consider the same music piece as described in [12]: “A Day in the Life” from the Beatles’ album Sgt. Pepper’s Lonely Hearts Club Band. The acoustic signal was sampled at 22.05 KHz and divided into 50 ms contiguous frames; 40-dimensional Mel frequency cepstral coefﬁcients (MFCCs) were extracted from each frame, shown in Figure 1(a). A typical goal of music analysis is to infer interrelationships within the music piece, as a function of time [12]. For the audio data, each MFCC vector yn has an associated time index, the latter used as the covariate xn. The ﬁnite set of temporal sample points (covariates) were employed to deﬁne a library for the {x∗ i }, and H is a uniform distribution over this set. After 2000 burn-in iterations, we collected samples every ﬁve iterations. Figure 1(b) shows the frequency for the number of unique dictionary elements used by the data, based on the 1600 collected samples; and Figure 1(c) shows the frequency for the number of total dictionary elements used. With the model deﬁned in (10), the sparse vector bn◦wn indicates the importance of each dictionary element from {Di}i=1,T to data yn. Each of these N vectors {bn ◦wn}n=1,N was normalized 6 observation index feature values 1000 2000 3000 4000 5000 6000 5 10 15 20 25 30 35 40 −6 −4 −2 0 2 4 (a) 25 30 35 40 45 50 55 0 100 200 300 400 500 600 The number of unique dictionary elements Frequency calculated from the collected samples (b) 165 170 175 180 185 190 195 200 205 0 50 100 150 200 250 300 The number of dictionary elements taken by the data Frequency calculated from the collected samples (c) Figure 1: (a) MFCCs features used in music analysis, where the horizontal axis corresponds to time, for “A Day in the Life”. Based on the Gibbs collection samples: (b) frequency on number of unique dictionary elements, and (c) total number of dictionary elements. within each Gibbs sample, and used to compute a correlation matrix associated with the N time points in the music. Finally, this matrix was averaged across the collection samples, to yield a correlation matrix relating one part of the music to all others. For a fair comparison between our methods and the model proposed in [12] (which used an HMM, and computed correlations over windows of time), we divided the whole piece into multiple consecutive short-time windows. Each temporal window includes 75 consecutive feature vectors, and we compute the average correlation coefﬁcients between the features within each pair of windows. There were 88 temporal windows in total (each temporal window is de noted as a sequence in Figure 2), and the dimension of the correlation matrix is accordingly 88 × 88. The computed correlation matrix for the proposed KBP model is presented in Figure 2(a). We compared KBP performance with results based on BP-FA [17] in which covariates are not employed, and with results from the dynamic clustering model in [12], in which a dynamic HMM is employed (in [12] a dynamic HDP, or dHDP, was used in concert with an HMM). The BP-FA results correspond to replacing the KBP with a BP. The correlation matrix computed from the BP-FA and the dHDP-HMM [12] are shown in Figures 2(b) and (c), respectively. The dHDP-HMM results yield a reasonably good segmentation of the music, but it is unable to infer subtle differences in the music over time (for example, all voices in the music are clustered together, even if they are different). Since the BP-FA does not capture as much localized information in the music (the probability of dictionary usage is the same for all temporal positions), it does not manifest as good a music segmentation as the dHDP-HMM. By contrast, the KBP-FA model yields a good music segmentation, while also capturing subtle differences in the music over time (e.g., in voices). Note that the use of the DP to allow repeated use of dictionary elements as a function of time (covariates) is important here, due to the repetition of structure in the piece. One may listen to the music and observe the segmentation at http://www.youtube.com/watch?v=35YhHEbIlEI. Sequence index Sequence index 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (a) Sequence index Sequence index 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 0.8 0.85 0.9 0.95 1 (b) sequence index sequence index 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (c) Figure 2: Inference of relationships in music as a function of time, as computed via a correlation of the dictionary-usage weights, for (a) and (b), and based upon state usage in an HMM, for (c). Results are shown for “A Day in the Life.” The results in (c) are from [12], as a courtesy from the authors of that paper. (a) KBP-FA, (b) BP-FA, (c) dHDP-HMM . 4.3 Image interpolation and denoising We consider image interpolation and denoising as two additional potential applications. In both of these examples each image is divided into N 8 × 8 overlapping patches, and each patch is stacked into a vector of length M = 64, constituting observation yn ∈RM. The covariate xn represents the 7 patch coordinates in the 2-D space. The probability measure H corresponds to a uniform distribution over the centers of all 8 × 8 patches. The images were recovered based on the average of the collection samples, and each pixel was averaged across all overlapping patches in which it resided. For the image-processing examples, 5000 Gibbs samples were run, with the ﬁrst 2000 discarded as burn-in. For image interpolation, we only observe a fraction of the image pixels, sampled uniformly at random. The model infers the underlying dictionary D in the presence of this missing data, as well as the weights on the dictionary elements required for representing the observed components of {yn}; using the inferred dictionary and associated weights, one may readily impute the missing pixel values. In Table 1 we present average PSNR values on the recovered pixel values, as a function of the fraction of pixels that are observed (20% in Table 1 means that 80% of the pixels are missing uniformly at random). Comparisons are made between a model based on BP and one based on the proposed KBP; the latter generally performs better, particularly when a large fraction of the pixels are missing. The proposed algorithm yields results that are comparable to those in [18], which also employed covariates within the BP construc tion. However, the proposed KBP construction has the signiﬁcant computational advantages of only requiring kernels centered at the locations of the dictionary-dependent covariates {x∗ i }, while the model in [18] has a kernel for each of the image patches, and therefore it scales unfavorably for large images. Table 1: Comparison of BP and KBP for interpolating images with pixels missing uniformly at random, using standard image-processing images. The top and bottom rows of each cell show results of BP and KBP, respectively. Results are shown when 20%, 30% and 50% of the pixels are observed, selected uniformly at random. RATIO C.MAN HOUSE PEPPERS LENA BARBARA BOATS F.PRINT MAN COUPLE HILL 20% 23.75 29.75 25.56 30.97 26.84 27.84 26.49 28.29 27.76 29.38 24.02 30.89 26.29 31.38 28.93 28.11 26.89 28.37 28.03 29.67 30% 25.59 33.09 28.64 33.30 30.13 30.20 29.23 29.89 29.97 31.19 25.75 34.02 29.29 33.33 31.46 30.24 29.37 30.12 30.33 31.25 50% 28.66 38.26 32.53 36.79 35.95 33.05 33.50 33.19 33.61 34.19 28.78 38.35 32.69 35.89 36.03 33.18 32.18 32.35 32.35 32.60 In the image-denoising example in Figure 3 the images were corrupted with both white Gaussian noise (WGN) and sparse spiky noise, as considered in [18]. The sparse spiky noise exists in particular pixels, selected uniformly at random, with amplitude distributed uniformly between −255 and 255. For the pepper image, 15% of the pixels were corrupted by spiky noise, and the standard deviation of the WGN was 15; for the house image, 10% of the pixels were corrupted by spiky noise and the standard deviation of WGN was 10. We compared with different methods on both two images: the augmented KBP-FA model (KBP-FA+) in Sec. 3.2, the BP-FA model augmented with a term for spiky noise (BP-FA+) and the original BP-FA model. The model proposed with KBP showed the best denoising result for both visual and quantitative evaluations. Again, these results are comparable to those in [18], with the signiﬁcant computational advant age discussed above. Note that here the imposition of covariates and the KBP yields marked improvements in this application, relative to BP-FA alone. Figure 3: Denoising Result: the ﬁrst column shows the noisy images (PSNR is 15.56 dB for Peppers and 17.54 dB for House); the second and third column shows the results inferred from the BP-FA model (PSNR is 16.31 dB for Peppers and 17.95 dB for House), with the dictionary elements shown in column two and the reconstruction in column three; the fourth and ﬁfth columns show results from BP-FA+ (PSNR is 23.06 dB for Peppers and 26.71 dB for House); the sixth and seventh column shows the results of the KBP-FA+ (PSNR is 27.37 dB for Peppers and 34.89 dB for House). In each case the dictionaries are ordered based on their frequency of usage, starting from top-left. 8 5 Summary A new L´evy process, the kernel beta process, has been developed for the problem of nonparametric Bayesian feature learning, with example results presented for music analysis, image denoising, and image interpolation. 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4,408	Sparse Inverse Covariance Matrix Estimation Using Quadratic Approximation Cho-Jui Hsieh, M´aty´as A. Sustik, Inderjit S. Dhillon, and Pradeep Ravikumar Department of Computer Science University of Texas at Austin Austin, TX 78712 USA {cjhsieh,sustik,inderjit,pradeepr}@cs.utexas.edu Abstract The ℓ1 regularized Gaussian maximum likelihood estimator has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix, or alternatively the underlying graph structure of a Gaussian Markov Random Field, from very limited samples. We propose a novel algorithm for solving the resulting optimization problem which is a regularized log-determinant program. In contrast to other state-of-the-art methods that largely use ﬁrst order gradient information, our algorithm is based on Newton’s method and employs a quadratic approximation, but with some modiﬁcations that leverage the structure of the sparse Gaussian MLE problem. We show that our method is superlinearly convergent, and also present experimental results using synthetic and real application data that demonstrate the considerable improvements in performance of our method when compared to other state-of-the-art methods. 1 Introduction Gaussian Markov Random Fields; Covariance Estimation. Increasingly, in modern settings statistical problems are high-dimensional, where the number of parameters is large when compared to the number of observations. An important class of such problems involves estimating the graph structure of a Gaussian Markov random ﬁeld (GMRF) in the high-dimensional setting, with applications ranging from inferring gene networks and analyzing social interactions. Speciﬁcally, given n independently drawn samples {y1, y2, . . . , yn} from a p-variate Gaussian distribution, so that yi ∼N(µ, Σ), the task is to estimate its inverse covariance matrix Σ−1, also referred to as the precision or concentration matrix. The non-zero pattern of this inverse covariance matrix Σ−1 can be shown to correspond to the underlying graph structure of the GMRF. An active line of work in high-dimensional settings where p < n is thus based on imposing some low-dimensional structure, such as sparsity or graphical model structure on the model space. Accordingly, a line of recent papers [2, 8, 20] has proposed an estimator that minimizes the Gaussian negative log-likelihood regularized by the ℓ1 norm of the entries (off-diagonal entries) of the inverse covariance matrix. The resulting optimization problem is a log-determinant program, which is convex, and can be solved in polynomial time. Existing Optimization Methods for the regularized Gaussian MLE. Due in part to its importance, there has been an active line of work on efﬁcient optimization methods for solving the ℓ1 regularized Gaussian MLE problem. In [8, 2] a block coordinate descent method has been proposed which is called the graphical lasso or GLASSO for short. Other recent algorithms proposed for this problem include PSM that uses projected subgradients [5], ALM using alternating linearization [14], IPM an inexact interior point method [11] and SINCO a greedy coordinate descent method [15]. For typical high-dimensional statistical problems, optimization methods typically suffer sub-linear rates of convergence [1]. This would be too expensive for the Gaussian MLE problem, since the 1 number of matrix entries scales quadratically with the number of nodes. Luckily, the log-determinant problem has special structure; the log-determinant function is strongly convex and one can observe linear (i.e. geometric) rates of convergence for the state-of-the-art methods listed above. However, at most linear rates in turn become infeasible when the problem size is very large, with the number of nodes in the thousands and the number of matrix entries to be estimated in the millions. Here we ask the question: can we obtain superlinear rates of convergence for the optimization problem underlying the ℓ1 regularized Gaussian MLE? One characteristic of these state-of-the-art methods is that they are ﬁrst-order iterative methods that mainly use gradient information at each step. Such ﬁrst-order methods have become increasingly popular in recent years for high-dimensional problems in part due to their ease of implementation, and because they require very little computation and memory at each step. The caveat is that they have at most linear rates of convergence [3]. For superlinear rates, one has to consider second-order methods which at least in part use the Hessian of the objective function. There are however some caveats to the use of such second-order methods in high-dimensional settings. First, a straightforward implementation of each second-order step would be very expensive for high-dimensional problems. Secondly, the log-determinant function in the Gaussian MLE objective acts as a barrier function for the positive deﬁnite cone. This barrier property would be lost under quadratic approximations so there is a danger that Newton-like updates will not yield positive-deﬁnite matrices, unless one explicitly enforces such a constraint in some manner. Our Contributions. In this paper, we present a new second-order algorithm to solve the ℓ1 regularized Gaussian MLE. We perform Newton steps that use iterative quadratic approximations of the Gaussian negative log-likelihood, but with three innovations that enable ﬁnessing the caveats detailed above. First, we provide an efﬁcient method to compute the Newton direction. As in recent methods [12, 9], we build on the observation that the Newton direction computation is a Lasso problem, and perform iterative coordinate descent to solve this Lasso problem. However, the naive approach has an update cost of O(p2) for performing each coordinate descent update in the inner loop, which makes this resume infeasible for this problem. But we show how a careful arrangement and caching of the computations can reduce this cost to O(p). Secondly, we use an Armijo-rule based step size selection rule to obtain a step-size that ensures sufﬁcient descent and positive-deﬁniteness of the next iterate. Thirdly, we use the form of the stationary condition characterizing the optimal solution to then focus the Newton direction computation on a small subset of free variables, in a manner that preserves the strong convergence guarantees of second-order descent. Here is a brief outline of the paper. In Section 3, we present our algorithm that combines quadratic approximation, Newton’s method and coordinate descent. In Section 4, we show that our algorithm is not only convergent but superlinearly so. We summarize the experimental results in Section 5, using real application data from [11] to compare the algorithms, as well as synthetic examples which reproduce experiments from [11]. We observe that our algorithm performs overwhelmingly better (quadratic instead of linear convergence) than the other solutions described in the literature. 2 Problem Setup Let y be a p-variate Gaussian random vector, with distribution N(µ, Σ). We are given n independently drawn samples {y1, . . . , yn} of this random vector, so that the sample covariance matrix can be written as S = 1 n n ! k=1 (yk −ˆµ)(yk −ˆµ)T , where ˆµ = 1 n n ! i=1 yi. (1) Given some regularization penalty λ > 0, the ℓ1 regularized Gaussian MLE for the inverse covariance matrix can be estimated by solving the following regularized log-determinant program: arg min X≻0 " −log det X + tr(SX) + λ∥X∥1 # = arg min X≻0 f(X), (2) where ∥X∥1 = $p i,j=1 \|Xij\| is the elementwise ℓ1 norm of the p × p matrix X. Our results can be also extended to allow a regularization term of the form ∥λ ◦X∥1 = $p i,j=1 λij\|Xij\|, i.e. different nonnegative weights can be assigned to different entries. This would include for instance the popular off-diagonalℓ1 regularization variant where we penalize $ i̸=j \|Xij\|, but not the diagonal entries. The addition of such ℓ1 regularization promotes sparsity in the inverse covariance matrix, and thus encourages sparse graphical model structure. For further details on the background of ℓ1 regularization in the context of GMRFs, we refer the reader to [20, 2, 8, 15]. 2 3 Quadratic Approximation Method Our approach is based on computing iterative quadratic approximations to the regularized Gaussian MLE objective f(X) in (2). This objective function f can be seen to comprise of two parts, f(X) ≡ g(X) + h(X), where g(X) = −log det X + tr(SX) and h(X) = λ∥X∥1. (3) The ﬁrst component g(X) is twice differentiable, and strictly convex, while the second part h(X) is convex but non-differentiable. Following the standard approach [17, 21] to building a quadratic approximation around any iterate Xt for such composite functions, we build the secondorder Taylor expansion of the smooth component g(X). The second-order expansion for the log-determinant function (see for instance [4, Chapter A.4.3]) is given by log det(Xt + ∆) ≈ log det Xt+tr(X−1 t ∆)−1 2 tr(X−1 t ∆X−1 t ∆). We introduce Wt = X−1 t and write the second-order approximation ¯gXt(∆) to g(X) = g(Xt + ∆) as ¯gXt(∆) = tr((S −Wt)∆) + (1/2) tr(Wt∆Wt∆) −log det Xt + tr(SXt). (4) We deﬁne the Newton direction Dt for the entire objective f(X) can then be written as the solution of the regularized quadratic program: Dt = arg min ∆¯gXt(∆) + h(Xt + ∆). (5) This Newton direction can be used to compute iterative estimates {Xt} for solving the optimization problem in (2). In the sequel, we will detail three innovations which makes this resume feasible. Firstly, we provide an efﬁcient method to compute the Newton direction. As in recent methods [12], we build on the observation that the Newton direction computation is a Lasso problem, and perform iterative coordinate descent to ﬁnd its solution. However, the naive approach has an update cost of O(p2) for performing each coordinate descent update in the inner loop, which makes this resume infeasible for this problem. We show how a careful arrangement and caching of the computations can reduce this cost to O(p). Secondly, we use an Armijo-rule based step size selection rule to obtain a step-size that ensures sufﬁcient descent and positive-deﬁniteness of the next iterate. Thirdly, we use the form of the stationary condition characterizing the optimal solution to then focus the Newton direction computation on a small subset of free variables, in a manner that preserves the strong convergence guarantees of second-order descent. We outline each of these three innovations in the following three subsections. We then detail the complete method in Section 3.4. 3.1 Computing the Newton Direction The optimization problem in (5) is an ℓ1 regularized least squares problem, also called Lasso [16]. It is straightforward to verify that for a symmetric matrix ∆we have tr(Wt∆Wt∆) = vec(∆)T (Wt ⊗ Wt) vec(∆), where ⊗denotes the Kronecker product and vec(X) is the vectorized listing of the elements of matrix X. In [7, 18] the authors show that coordinate descent methods are very efﬁcient for solving lasso type problems. However, an obvious way to update each element of ∆to solve for the Newton direction in (5) needs O(p2) ﬂoating point operations since Q := Wt⊗Wt is a p2×p2 matrix, thus yielding an O(p4) procedure for approximating the Newton direction. As we show below, our implementation reduces the cost of one variable update to O(p) by exploiting the structure of Q or in other words the speciﬁc form of the second order term tr(Wt∆Wt∆). Next, we discuss the details. For notational simplicity we will omit the Newton iteration index t in the derivations that follow. (Hence, the notation for ¯gXt is also simpliﬁed to ¯g.) Furthermore, we omit the use of a separate index for the coordinate descent updates. Thus, we simply use D to denote the current iterate approximating the Newton direction and use D′ for the updated direction. Consider the coordinate descent update for the variable Xij, with i < j that preserves symmetry: D′ = D+µ(eieT j +ejeT i ). The solution of the one-variable problem corresponding to (5) yields µ: arg min µ ¯g(D + µ(eieT j + ejeT i )) + 2λ\|Xij + Dij + µ\|. (6) As a matter of notation: we use xi to denote the i-th column of the matrix X. We expand the terms appearing in the deﬁnition of ¯g after substituting D′ = D + µ(eieT j + ejeT i ) for ∆in (4) and omit the terms not dependent on µ. The contribution of tr(SD′)−tr(WD′) yields 2µ(Sij −Wij), while 3 the regularization term contributes 2λ\|Xij + Dij + µ\|, as seen from (6). The quadratic term can be rewritten using tr(AB) = tr(BA) and the symmetry of D and W to yield: tr(WD′WD′) = tr(WDWD) + 4µwT i Dwj + 2µ2(W 2 ij + WiiWjj). (7) In order to compute the single variable update we seek the minimum of the following function of µ: 1 2(W 2 ij + WiiWjj)µ2 + (Sij −Wij + wT i Dwj)µ + λ\|Xij + Dij + µ\|. (8) Letting a = W 2 ij +WiiWjj, b = Sij −Wij +wT i Dwj, and c = Xij +Dij the minimum is achieved for: µ = −c + S(c −b/a, λ/a), (9) where S(z, r) = sign(z) max{\|z\| −r, 0} is the soft-thresholding function. The values of a and c are easy to compute. The main cost arises while computing the third term contributing to coefﬁcient b, namely wT i Dwj. Direct computation requires O(p2) time. Instead, we maintain U = DW by updating two rows of the matrix U for every variable update in D costing O(p) ﬂops, and then compute wT i uj using also O(p) ﬂops. Another way to view this arrangement is that we maintain a decomposition WDW = $p k=1 wkuT k throughout the process by storing the uk vectors, allowing O(p) computation of update (9). In order to maintain the matrix U we also need to update two coordinates of each uk when Dij is modiﬁed. We can compactly write the row updates of U as follows: ui· ←ui· + µwj· and uj· ←uj· + µwi·, where ui· refers to the i-th row vector of U. We note that the calculation of the Newton direction can be simpliﬁed if X is a diagonal matrix. For instance, if we are starting from a diagonal matrix X0, the terms wT i Dwj equal Dij/((X0)ii(X0)jj), which are independent of each other implying that we only need to update each variable according to (9) only once, and the resulting D will be the optimum of (5). Hence, the time cost of ﬁnding the ﬁrst Newton direction is reduced from O(p3) to O(p2). 3.2 Computing the Step Size Following the computation of the Newton direction Dt, we need to ﬁnd a step size α ∈(0, 1] that ensures positive deﬁniteness of the next iterate Xt + αDt and sufﬁcient decrease in the objective function. We adopt Armijo’s rule [3, 17] and try step-sizes α ∈{β0, β1, β2, . . . } with a constant decrease rate 0 < β < 1 (typically β = 0.5) until we ﬁnd the smallest k ∈N with α = βk such that Xt + αDt (a) is positive-deﬁnite, and (b) satisﬁes the following condition: f(Xt + αDt) ≤f(Xt) + ασ∆t, ∆t = tr(∇g(Xt)Dt) + λ∥Xt + Dt∥1 −λ∥Xt∥1 (10) where 0 < σ < 0.5 is a constant. To verify positive deﬁniteness, we use a Cholesky factorization costing O(p3) ﬂops during the objective function evaluation to compute log det(Xt +αDt) and this step dominates the computational cost in the step-size computations. In the Appendix in Lemma 9 we show that for any Xt and Dt, there exists a ¯αt > 0 such that (10) and the positive-deﬁniteness of Xt + αDt are satisﬁed for any α ∈(0, ¯αt], so we can always ﬁnd a step size satisfying (10) and the positive-deﬁniteness even if we do not have the exact Newton direction. Following the line search and the Newton step update Xt+1 = Xt + αDt we efﬁciently compute Wt+1 = X−1 t+1 by reusing the Cholesky decomposition of Xt+1. 3.3 Identifying which variables to update In this section, we propose a way to select which variables to update that uses the stationary condition of the Gaussian MLE problem. At the start of any outer loop computing the Newton direction, we partition the variables into free and ﬁxed sets based on the value of the gradient. Speciﬁcally, we classify the (Xt)ij variable as ﬁxed if \|∇ijg(Xt)\| < λ −ϵ and (Xt)ij = 0, where ϵ > 0 is small. (We used ϵ = 0.01 in our experiments.) The remaining variables then constitute the free set. The following lemma shows the property of the ﬁxed set: Lemma 1. For any Xt and the corresponding ﬁxed and free sets Sfixed, Sfree, the optimized update on the ﬁxed set would not change any of the coordinates. In other words, the solution of the following optimization problem is ∆= 0: arg min ∆f(Xt + ∆) such that ∆ij = 0 ∀(i, j) ∈Sfree. 4 The proof is given in Appendix 7.2.3. Based on the above observation, we perform the inner loop coordinate descent updates restricted to the free set only (to ﬁnd the Newton direction). This reduces the number of variables over which we perform the coordinate descent from O(p2) to the number of non-zeros in Xt, which in general is much smaller than p2 when λ is large and the solution is sparse. We have observed huge computational gains from this modiﬁcation, and indeed in our main theorem we show the superlinear convergence rate for the algorithm that includes this heuristic. The attractive facet of this modiﬁcation is that it leverages the sparsity of the solution and intermediate iterates in a manner that falls within a block coordinate descent framework. Speciﬁcally, suppose as detailed above at any outer loop Newton iteration, we partition the variables into the ﬁxed and free set, and then ﬁrst perform a Newton update restricted to the ﬁxed block, followed by a Newton update on the free block. According to Lemma 1 a Newton update restricted to the ﬁxed block does not result in any changes. In other words, performing the inner loop coordinate descent updates restricted to the free set is equivalent to two block Newton steps restricted to the ﬁxed and free sets consecutively. Note further, that the union of the free and ﬁxed sets is the set of all variables, which as we show in the convergence analysis in the appendix, is sufﬁcient to ensure the convergence of the block Newton descent. But would the size of free set be small? We initialize X0 to the identity matrix, which is indeed sparse. As the following lemma shows, if the limit of the iterates (the solution of the optimization problem) is sparse, then after a ﬁnite number of iterations, the iterates Xt would also have the same sparsity pattern. Lemma 2. Assume {Xt} converges to X∗. If for some index pair (i, j), \|∇ijg(X∗)\| < λ (so that X∗ ij = 0), then there exists a constant ¯t > 0 such that for all t > ¯t, the iterates Xt satisfy \|∇ijg(Xt)\| < λ and (Xt)ij = 0. (11) The proof comes directly from Lemma 11 in the Appendix. Note that \|∇ijg(X∗)\| < λ implying X∗ ij = 0 follows from the optimality condition of (2). A similar (so called shrinking) strategy is used in SVM or ℓ1-regularized logistic regression problems as mentioned in [19]. In Appendix 7.4 we show in experiments this strategy can reduce the size of variables very quickly. 3.4 The Quadratic Approximation based Method We now have the machinery for a description of our algorithm QUIC standing for QUadratic Inverse Covariance. A high level summary of the algorithm is shown in Algorithm 1, while the the full details are given in Algorithm 2 in the Appendix. Algorithm 1: Quadratic Approximation method for Sparse Inverse Covariance Learning (QUIC) Input : Empirical covariance matrix S, scalar λ, initial X0, inner stopping tolerance ϵ Output: Sequence of Xt converging to arg minX≻0 f(X), where f(X) = −log det X + tr(SX) + λ∥X∥1. 1 for t = 0, 1, . . . do 2 Compute Wt = X−1 t . 3 Form the second order approximation ¯fXt(∆) := ¯gXt(∆) + h(Xt + ∆) to f(Xt + ∆). 4 Partition the variables into free and ﬁxed sets based on the gradient, see Section 3.3. 5 Use coordinate descent to ﬁnd the Newton direction Dt = arg min∆¯fXt(Xt + ∆) over the free variable set, see (6) and (9). (A Lasso problem.) 6 Use an Armijo-rule based step-size selection to get α s.t. Xt+1 = Xt + αDt is positive deﬁnite and the objective value sufﬁciently decreases, see (10). 7 end 4 Convergence Analysis In this section, we show that our algorithm has strong convergence guarantees. Our ﬁrst main result shows that our algorithm does converge to the optimum of (2). Our second result then shows that the asymptotic convergence rate is actually superlinear, speciﬁcally quadratic. 4.1 Convergence Guarantee We build upon the convergence analysis in [17, 21] of the block coordinate gradient descent method applied to composite objectives. Speciﬁcally, [17, 21] consider iterative updates where at each 5 iteration t they update just a block of variables Jt. They then consider a Gauss-Seidel rule: % j=0,...,T −1 Jt+j ⊇N ∀t = 1, 2, . . . , (12) where N is the set of all variables and T is a ﬁxed number. Note that the condition (12) ensures that each block of variables will be updated at least once every T iterations. Our Newton steps with the free set modiﬁcation is a special case of this framework: we set J2t, J2t+1 to be the ﬁxed and free sets respectively. As outlined in Section 3.3, our selection of the ﬁxed sets ensures that a block update restricted to the ﬁxed set would not change any values since these variables in ﬁxed sets already satisfy the coordinatewise optimality condition. Thus, while our algorithm only explicitly updates the free set block, this is equivalent to updating variables in ﬁxed and free blocks consecutively. We also have J2t ∪J2t+1 = N, implying the Gauss-Seidel rule with T = 3. Further, the composite objectives in [17, 21] have the form F(x) = g(x) + h(x), where g(x) is smooth (continuously differentiable), and h(x) is non-differentiable but separable. Note that in our case, the smooth component is the log-determinant function g(X) = −log det X + tr(SX), while the non-differentiable separable component is h(x) = λ∥x∥1. However, [17, 21] impose the additional assumption that g(x) is smooth over the domain Rn. In our case g(x) is smooth over the restricted domain of the positive deﬁnite cone Sp ++ . In the appendix, we extend the analysis so that convergence still holds under our setting. In particular, we prove the following theorem in Appendix 7.2: Theorem 1. In Algorithm 1, the sequence {Xt} converges to the unique global optimum of (2). 4.2 Asymptotic Convergence Rate In addition to convergence, we further show that our algorithm has a quadratic asymptotic convergence rate. Theorem 2. Our algorithm QUIC converges quadratically, that is for some constant 0 < κ < 1: lim t→∞ ∥Xt+1 −X∗∥F ∥Xt −X∗∥2 F = κ. The proof, given in Appendix 7.3, ﬁrst shows that the step size as computed in Section 3.2 would eventually become equal to one, so that we would be eventually performing vanilla Newton updates. Further we use the fact that after a ﬁnite number of iterations, the sign pattern of the iterates converges to the sign pattern of the limit. From these two assertions, we build on the convergence rate result for constrained Newton methods in [6] to show that our method is quadratically convergent. 5 Experiments In this section, we compare our method QUIC with other state-of-the-art methods on both synthetic and real datasets. We have implemented QUIC in C++, and all the experiments were executed on 2.83 GHz Xeon X5440 machines with 32G RAM and Linux OS. We include the following algorithms in our comparisons: • ALM: the Alternating Linearization Method proposed by [14]. We use their MATLAB source code for the experiments. • GLASSO: the block coordinate descent method proposed by [8]. We used their Fortran code available from cran.r-project.org, version 1.3 released on 1/22/09. • PSM: the Projected Subgradient Method proposed by [5]. We use the MATLAB source code available at http://www.cs.ubc.ca/˜schmidtm/Software/PQN.html. • SINCO: the greedy coordinate descent method proposed by [15]. The code can be downloaded from https://projects.coin-or.org/OptiML/browser/trunk/sinco. • IPM: An inexact interior point method proposed by [11]. The source code can be downloaded from http://www.math.nus.edu.sg/˜mattohkc/Covsel-0.zip. Since some of the above implementations do not support the generalized regularization term ∥λ ◦ X∥1, our comparisons use λ∥X∥1 as the regularization term. The GLASSO algorithm description in [8] does not clearly specify the stopping criterion for the Lasso iterations. Inspection of the available Fortran implementation has revealed that a separate 6 Table 1: The comparisons on synthetic datasets. p stands for dimension, ∥Σ−1∥0 indicates the number of nonzeros in ground truth inverse covariance matrix, ∥X∗∥0 is the number of nonzeros in the solution, and ϵ is a speciﬁed relative error of objective value. ∗indicates the run time exceeds our time limit 30,000 seconds (8.3 hours). The results show that QUIC is overwhelmingly faster than other methods, and is the only one which is able to scale up to solve problem where p = 10000. Dataset setting Parameter setting Time (in seconds) pattern p ∥Σ−1∥0 λ ∥X∗∥0 ϵ QUIC ALM Glasso PSM IPM Sinco chain 1000 2998 0.4 3028 10−2 0.30 18.89 23.28 15.59 86.32 120.0 10−6 2.26 41.85 45.1 34.91 151.2 520.8 chain 4000 11998 0.4 11998 10−2 11.28 922 1068 567.9 3458 5246 10−6 53.51 1734 2119 1258 5754 * chain 10000 29998 0.4 29998 10−2 216.7 13820 * 8450 * * 10−6 986.6 28190 * 19251 * * random 1000 10758 0.12 10414 10−2 0.52 42.34 10.31 20.16 71.62 60.75 10−6 1.2 28250 20.43 59.89 116.7 683.3 0.075 55830 10−2 1.17 65.64 17.96 23.53 78.27 576.0 10−6 6.87 * 60.61 91.7 145.8 4449 random 4000 41112 0.08 41910 10−2 23.25 1429 1052 1479 4928 7375 10−6 160.2 * 2561 4232 8097 * 0.05 247444 10−2 65.57 * 3328 2963 5621 * 10−6 478.8 * 8356 9541 13650 * random 10000 91410 0.08 89652 10−2 337.7 26270 21298 * * * 10−6 1125 * * * * * 0.04 392786 10−2 803.5 * * * * * 10−6 2951 * * * * * threshold is computed and is used for these inner iterations. We found that under certain conditions the threshold computed is smaller than the machine precision and as a result the overall algorithm occasionally displayed erratic convergencebehavior and slow performance. We modiﬁed the Fortran implementation of GLASSO to correct this error. 5.1 Comparisons on synthetic datasets We ﬁrst compare the run times of the different methods on synthetic data. We generate the two following types of graph structures for the underlying Gaussian Markov Random Fields: • Chain Graphs: The ground truth inverse covariance matrix Σ−1 is set to be Σ−1 i,i−1 = −0.5 and Σ−1 i,i = 1.25. • Graphs with Random Sparsity Structures: We use the procedure mentioned in Example 1 in [11] to generate inverse covariance matrices with random non-zero patterns. Speciﬁcally, we ﬁrst generate a sparse matrix U with nonzero elements equal to ±1, set Σ−1 to be U T U and then add a diagonal term to ensure Σ−1 is positive deﬁnite. We control the number of nonzeros in U so that the resulting Σ−1 has approximately 10p nonzero elements. Given the inverse covariance matrix Σ−1, we draw a limited number, n = p/2 i.i.d. samples, to simulate the high-dimensional setting, from the corresponding GMRF distribution. We then compare the algorithms listed above when run on these samples. We can use the minimum-norm sub-gradient deﬁned in Lemma 5 in Appendix 7.2 as the stopping condition, and computing it is easy because X−1 is available in QUIC. Table 1 shows the results for timing comparisons in the synthetic datasets. We vary the dimensionality from 1000, 4000 to 10000 for each dataset. For chain graphs, we select λ so that the solution had the (approximately) correct number of nonzero elements. To test the performance of algorithms on different parameters (λ), for random sparse pattern we test the speed under two values of λ, one discovers correct number of nonzero elements, and one discovers 5 times the number of nonzero elements. We report the time for each algorithm to achieve ϵ-accurate solution deﬁned by f(Xk) −f(X∗) < ϵf(X∗). Table 1 shows the results for ϵ = 10−2 and 10−6, where ϵ = 10−2 tests the ability for an algorithm to get a 7 good initial guess (the nonzero structure), and ϵ = 10−6 tests whether an algorithm can achieve an accurate solution. Table 1 shows that QUIC is consistently and overwhelmingly faster than other methods, both initially with ϵ = 10−2, and at ϵ = 10−6. Moreover, for p = 10000 random pattern, there are p2 = 100 million variables, the selection of ﬁxed/free sets helps QUIC to focus only on very small part of variables, and can achieve an accurate solution in about 15 minutes, while other methods fails to even have an initial guess within 8 hours. Notice that our λ setting is smaller than [14] because here we focus on the λ which discovers true structure, therefore the comparison between ALM and PSM are different from [14]. 5.2 Experiments on real datasets We use the real world biology datasets preprocessed by [11] to compare the performance of our method with other state-of-the-art methods. The regularization parameter λ is set to 0.5 according to the experimental setting in [11]. Results on the following datasets are shown in Figure 1: Estrogen (p = 692), Arabidopsis (p = 834), Leukemia (p = 1, 225), Hereditary (p = 1, 869). We plot the relative error (f(Xt) −f(X∗))/f(X∗) (on a log scale) against time in seconds. On these real datasets, QUIC can be seen to achieve super-linear convergence, while other methods have at most a linear convergence rate. Overall QUIC can be ten times faster than other methods, and even more faster when higher accuracy is desired. 6 Acknowledgements We would like to thank Professor Kim-Chuan Toh for providing the data set and the IPM code. We would also like to thank Professor Katya Scheinberg and Shiqian Ma for providing the ALM implementation. This research was supported by NSF grant IIS-1018426 and CCF-0728879. ISD acknowledges support from the Moncrief Grand Challenge Award. 0 10 20 30 40 50 60 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 Time (sec) Relative error (log scale) ALM Sinco PSM Glasso IPM QUIC (a) Time for Estrogen, p = 692 0 10 20 30 40 50 60 70 80 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 Time (sec) Relative error (log scale) ALM Sinco PSM Glasso IPM QUIC (b) Time for Arabidopsis, p = 834 0 50 100 150 200 250 300 350 400 450 500 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 Time (sec) Relative error (log scale) ALM Sinco PSM Glasso IPM QUIC (c) Time for Leukemia, p = 1, 255 0 200 400 600 800 1000 1200 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 Time (sec) Relative error (log scale) ALM Sinco PSM Glasso IPM QUIC (d) Time for hereditarybc, p = 1, 869 Figure 1: Comparison of algorithms on real datasets. The results show QUIC converges faster than other methods. 8 References [1] A. Agarwal, S. Negahban, and M. Wainwright. Convergence rates of gradient methods for high-dimensional statistical recovery. 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Sebag, editors, Machine Learning and Knowledge Discovery in Databases, volume 6323 of Lecture Notes in Computer Science, pages 196–212. Springer Berlin / Heidelberg, 2010. [16] R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, 58:267–288, 1996. [17] P. Tseng and S. Yun. A coordinate gradient descent method for nonsmooth separable minimization. Mathematical Programming, 117:387–423, 2007. [18] T. T. Wu and K. Lange. Coordinate descent algorithms for lasso penalized regression. The Annals of Applied Statistics, 2(1):224–244, 2008. [19] G.-X. Yuan, K.-W. Chang, C.-J. Hsieh, and C.-J. Lin. A comparison of optimization methods and software for large-scale l1-regularized linear classiﬁcation. Journal of Machine Learning Research, 11:3183–3234, 2010. [20] M. Yuan and Y. Lin. Model selection and estimation in the gaussian graphical model. Biometrika, 94:19–35, 2007. [21] S. Yun and K.-C. Toh. 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4,409	Testing a Bayesian Measure of Representativeness Using a Large Image Database Joshua T. Abbott Department of Psychology University of California, Berkeley Berkeley, CA 94720 joshua.abbott@berkeley.edu Katherine A. Heller Department of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 kheller@mit.edu Zoubin Ghahramani Department of Engineering University of Cambridge Cambridge, CB2 1PZ, U.K. zoubin@eng.cam.ac.uk Thomas L. Grifﬁths Department of Psychology University of California, Berkeley Berkeley, CA 94720 tom griffiths@berkeley.edu Abstract How do people determine which elements of a set are most representative of that set? We extend an existing Bayesian measure of representativeness, which indicates the representativeness of a sample from a distribution, to deﬁne a measure of the representativeness of an item to a set. We show that this measure is formally related to a machine learning method known as Bayesian Sets. Building on this connection, we derive an analytic expression for the representativeness of objects described by a sparse vector of binary features. We then apply this measure to a large database of images, using it to determine which images are the most representative members of different sets. Comparing the resulting predictions to human judgments of representativeness provides a test of this measure with naturalistic stimuli, and illustrates how databases that are more commonly used in computer vision and machine learning can be used to evaluate psychological theories. 1 Introduction The notion of “representativeness” appeared in cognitive psychology as a proposal for a heuristic that people might use in the place of performing a probabilistic computation [1, 2]. For example, we might explain why people believe that the sequence of heads and tails HHTHT is more likely than HHHHH to be produced by a fair coin by saying that the former is more representative of the output of a fair coin than the latter. This proposal seems intuitive, but raises a new problem: How is representativeness itself deﬁned? Various proposals have been made, connecting representativeness to existing quantities such as similarity [1] (itself an ill-deﬁned concept [3]), or likelihood [2]. Tenenbaum and Grifﬁths [4] took a different approach to this question, providing a “rational analysis” of representativeness by trying to identify the problem that such a quantity solves. They proposed that one sense of representativeness is being a good example of a concept, and then showed how this could be quantiﬁed via Bayesian inference. The resulting model outperformed similarity and likelihood in predicting human representativeness judgments for two kinds of simple stimuli. In this paper, we extend this deﬁnition of representativeness, and provide a more comprehensive test of this account using naturalistic stimuli. The question of what makes a good example of a concept is of direct relevance to computer scientists as well as cognitive scientists, providing a way to build better systems for retrieving images or documents relevant to a user’s query. However, the 1 model presented by Tenenbaum and Grifﬁths [4] is overly restrictive in requiring the concept to be pre-deﬁned, and has not been tested in the context of a large-scale information retrieval system. We extend the Bayesian measure of representativeness to apply to the problem of deciding which objects are good examples of a set of objects, show that the resulting model is closely mathematically related to an existing machine learning method known as Bayesian Sets [5], and compare this model to similarity and likelihood as an account of people’s judgments of the extent to which images drawn from a large database are representative of different concepts. In addition, we show how measuring the representativeness of items in sets can also provide a novel method of ﬁnding outliers in sets. By extending the Bayesian measure of representativeness to apply to sets of objects and testing it with a large image database, we are taking the ﬁrst steps towards a closer integration of the methods of cognitive science and machine learning. Cognitive science experiments typically use a small set of artiﬁcial stimuli, and evaluate different models by comparing them to human judgments about those stimuli. Machine learning makes use of large datasets, but relies on secondary sources of “cognitive” input, such as the labels people have applied to images. We combine these methods by soliciting human judgments to test cognitive models with a large set of naturalistic stimuli. This provides the ﬁrst experimental comparison of the Bayesian Sets algorithm to human judgments, and the ﬁrst evaluation of the Bayesian measure of representativeness in a realistic applied setting. The plan of the paper is as follows. Section 2 provides relevant background information, including psychological theories of representativeness and the deﬁnition of Bayesian Sets. Section 3 then introduces our extended measure of representativeness, and shows how it relates to Bayesian Sets. Section 4 describes the dataset derived from a large image database that we use for evaluating this measure, together with the other psychological models we use for comparison. Section 5 presents the results of an experiment soliciting human judgments about the representativeness of different images. Section 6 provides a second form of evaluation, focusing on identifying outliers from sets. Finally, Section 7 concludes the paper. 2 Background To approach our main question of which elements of a set are most representative of that set, we ﬁrst review previous psychological models of representativeness with a particular focus on the rational model proposed by Tenenbaum and Grifﬁths [4]. We then introduce Bayesian Sets [5]. 2.1 Representativeness While the notion of representativeness has been most prominent in the literature on judgment and decision-making, having been introduced by Kahneman and Tversky [1], similar ideas have been explored in accounts of human categorization and inductive inference [6, 7]. In these accounts, representativeness is typically viewed as a form of similarity between an outcome and a process or an object and a concept. Assume some data d has been observed, and we want to evaluate its representativeness of a hypothesized process or concept h. Then d is representative of h if it is similar to the observations h typically generates. Computing similarity requires deﬁning a similarity metric. In the case where we want to evaluate the representativeness of an outcome to a set, we might use metrics of the kind that are common in categorization models: an exemplar model deﬁnes similarity in terms of the sum of the similarities to the other objects in the set (e.g., [8, 9]), while a prototype model deﬁnes similarity in terms of the similarity to a prototype that captures the characteristics of the set (e.g., [10]). An alternative to similarity is the idea that representativeness might track the likelihood function P(d\|h) [11]. The main argument for this proposed equivalence is that the more frequently h leads to observing d, the more representative d should be of h. However, people’s judgments from the coin ﬂip example with which we started the paper go against this idea of equivalence, since both ﬂips have equal likelihood yet people tend to judge HHTHT as more representative of a fair coin. Analyses of typicality have also argued against the adequacy of frequency for capturing people’s judgments about what makes a good example of a category [6]. Tenenbaum and Grifﬁths [4] took a different approach to this question, asking what problem representativeness might be solving, and then deriving an optimal solution to that problem. This approach is similar to that taken in Shepard’s [12] analysis of generalization, and to Anderson’s [13] idea of 2 rational analysis. The resulting rational model of representativeness takes the problem to be one of selecting a good example, where the best example is the one that best provides evidence for the target process or concept relative to possible alternatives. Given some observed data d and a set of of hypothetical sources, H, we assume that a learner uses Bayesian inference to infer which h ∈H generated d. Tenenbaum and Grifﬁths [4] deﬁned the representativeness of d for h to be the evidence that d provides in favor of a speciﬁc h relative to its alternatives, R(d, h) = log P(d\|h) P h′̸=h P(d\|h′)P(h′), (1) where P(h′) in the denominator is the prior distribution on hypotheses, re-normalized over h′ ̸= h. 2.2 Bayesian Sets If given a small set of items such as “ketchup”, “mustard”, and “mayonnaise” and asked to produce other examples that ﬁt into this set, one might give examples such as “barbecue sauce”, or “honey”. This task is an example of clustering on-demand, in which the original set of items represents some concept or cluster such as “condiment” and we are to ﬁnd other items that would ﬁt appropriately into this set. Bayesian Sets is a formalization of this process in which items are ranked by a modelbased probabilistic scoring criterion, measuring how well they ﬁt into the original cluster [5]. More formally, given a data collection D, and a subset of items Ds = {x1, . . . , xN} ⊂D representing a concept, the Bayesian Sets algorithm ranks an item x∗∈{D \ Ds} by the following scoring criterion Bscore(x∗) = p(x∗, Ds) p(x∗)p(Ds) (2) This ratio intuitively compares the probability that x∗and Ds were generated by some statistical model with the same, though unknown, model parameters θ, versus the probability that x∗and Ds were generated by some statistical model with different model parameters θ1 and θ2. Each of the three terms in Equation 2 are marginal likelihoods and can be expressed as the following integrals over θ since the model parameter is assumed to be unknown: p(x∗) = R p(x∗\|θ)p(θ)dθ, p(Ds) = R hQN n=1 p(xn\|θ) i p(θ)dθ, and p(x∗, Ds) = R hQN n=1 p(xn\|θ) i p(x∗\|θ)p(θ)dθ. For computational efﬁciency reasons, Bayesian Sets is typically run on binary data. Thus, each item in the data collection, xi ∈D, is represented as a binary feature vector xi = (xi1, . . . , xiJ) where xij ∈{0, 1}, and deﬁned under a model in which each element of xi has an independent Bernoulli distribution p(xi\|θ) = Q j θxij j (1 −θj)1−xij and conjugate Beta prior p(θ\|α, β) = Q j Γ(αj+βj) Γ(αj)Γ(βj) θαj−1 j (1−θj)βj−1. Under these assumptions, the scoring criterion for Bayesian Sets reduces to Bscore(x∗) = p(x∗, Ds) p(x∗)p(Ds) = Y j αj + βj αj + βj + N ˜αj αj x∗j ˜βj βj !1−x∗j (3) where ˜αj = αj + PN n=1 xnj and ˜βj = βj + N −PN n=1 xnj. The logarithm of this score is linear in x and can be computed efﬁciently as log Bscore(x∗) = c + X j sjx∗j (4) where c = P j log(αj +βj)−log(αj +βj +N)+log ˜βj −log βj, sj = log ˜αj −log αj −log ˜βj + log βj, and x∗j is the jth component of x∗. The Bayesian Sets method has been tested with success on numerous datasets, over various applications including content-based image retrieval [14] and analogical reasoning with relational data [15]. Motivated by this method, we now turn to extending the previous measure of representativeness for a sample from a distribution, to deﬁne a measure of representativeness for an item to a set. 3 3 A Bayesian Measure of Representativeness for Sets of Objects The Bayesian measure of representativeness introduced by Tenenbaum and Grifﬁths [4] indicated the representativeness of data d for a hypothesis h. However, in many cases we might not know what statistical hypothesis best describes the concept that we want to illustrate through an example. For instance, in an image retrieval problem, we might just have a set of images that are all assigned to the same category, without a clear idea of the distribution that characterizes that category. In this section, we show how to extend the Bayesian measure of representativeness to indicate the representativeness of an element of a set, and how this relates to the Bayesian Sets method summarized above. Formally, we have a set of data Ds and we want to know how representative an element d of that set is of the whole set. We can perform an analysis similar to that given for the representativeness of d to a hypothesis, and obtain the expression R(d, Ds) = P(d\|Ds) P D′̸=Ds P(d\|D′)P(D′) (5) which is simply Equation 1 with hypotheses replaced by datasets. The quantities that we need to compute to apply this measure, P(d\|Ds) and P(D′), we obtain by marginalizing over all hypotheses. For example, P(d\|Ds) = P h P(d\|h)P(h\|Ds), being the posterior predictive distribution associated with Ds. If the hypotheses correspond to the continuous parameters of a generative model, then this is better expressed as P(d\|Ds) = R P(d\|θ)P(θ\|Ds). In the case where the set of possible datasets that is summed over in the denominator is large, this denominator will approximate P D′ P(d\|D′)P(D′), which is just P(d). This allows us to observe that this measure of representativeness will actually closely approximate the logarithm of the quantity Bscore produced by Bayesian Sets for the dataset Ds, with R(d, Ds) = log P(d\|Ds) P D′̸=Ds P(d\|D′)P(D′) ≈log P(d\|Ds) P(d) = log P(d, Ds) P(d)P(Ds) = log Bscore(d) This relationship provides a link between the cognitive science literature on representativeness and the machine learning literature on information retrieval, and a new way to evaluate psychological models of representativeness. 4 Evaluating Models of Representativeness Using Image Databases Having developed a measure of the representativeness of an item in a set of objects, we now focus on the problem of evaluating this measure. The evaluation of psychological theories has historically tended to use simple artiﬁcial stimuli, which provide precision at the cost of ecological validity. In the case of representativeness, the stimuli previously used by Tenenbaum and Grifﬁths [4] to evaluate different representativeness models consisted of 4 coin ﬂip sequences and 45 arguments based on predicates applied to a set of 10 mammals. One of the aims of this paper is to break the general trend of using such restricted kinds of stimuli, and the formal relationship between our rational model and Bayesian Sets allows us to do so. Any dataset that can be represented as a sparse binary matrix can be used to test the predictions of our measure. We formulate our evaluation problem as one of determining how representative an image is of a labeled set of images. Using an existing image database of naturalistic scenes, we can better test the predictions of different representativeness theories with stimuli much more in common with the environment humans naturally confront. In the rest of this section, we present the dataset used for evaluation and outline the implementations of existing models of representativeness we compare our rational Bayesian model against. 4.1 Dataset We use the dataset presented in [14], a subset of images taken from the Corel database commonly used in content-based image retrieval systems. The images in the dataset are partitioned into 50 labeled sets depicting unique categories, with varying numbers of images in each set (the mean is 264). The dataset is of particular interest for testing models of representativeness as each image 4 Algorithm 1 Representativeness Framework input: a set of items, Dw, for a particular category label w for each item xi ∈Dw do let Dwi = {Dw \ xi} compute score(xi, Dwi) end for rank items in Dw by this score output: ranked list of items in Dw (a) (b) Figure 1: Results of the Bayesian model applied to the set labelled coast. (a) The top nine ranked images. (b) The bottom nine ranked images. from the Corel database comes with multiple labels given by human judges. The labels have been criticized for not always being of high quality [16], which provides an additional (realistic) challenge for the models of representativeness that we aim to evaluate. The images in this dataset are represented as 240-dimensional feature vectors, composed of 48 Gabor texture features, 27 Tamura texture features, and 165 color histogram features. The images were additionally preprocessed through a binarization stage, transforming the entire dataset into a sparse binary matrix that represents the features which most distinguish each image from the rest of the dataset. Details of the construction of this feature representation are presented in [14]. 4.2 Models of Representativeness We compare our Bayesian model against a likelihood model and two similarity models: a prototype model and an exemplar model. We build upon a simple leave-one-out framework to allow a fair comparison of these different representativeness models. Given a set of images with a particular category label, we iterate through each image in the set and compute a score for how well this image represents the rest of the set (see Algorithm 1). In this framework, only score(xi, Dwi) varies across the different models. We present the different ways to compute this score below. Bayesian model. Since we have already shown the relationship between our rational measure and Bayesian Sets, the score in this model is computed efﬁciently via Equation 2. The hyperparameters α and β are set empirically from the entire dataset, α = κm, β = κ(1 −m), where m is the mean of x over all images, and κ is a scaling factor. An example of using this measure on the set of 299 images for category label coast is presented in Figure 1. Panels (a) and (b) of this ﬁgure show the top nine and bottom nine ranked images, respectively, where it is quite apparent that the top ranked images depict a better set of coast examples than the bottom rankings. It also becomes clear how poorly this label applies to some of the images in the bottom rankings, which is an important issue if using the labels provided with the Corel database as part of a training set for learning algorithms. 5 Likelihood model. This model treats representative judgments of an item x∗as p(x∗\|Ds) for a set Ds = {x1, . . . , xN}. Since this probability can also be expressed as p(x∗,Ds) p(Ds) , we can derive an efﬁcient scheme for computing the score similar to the Bayesian Sets scoring criterion by making the same model assumptions. The likelihood model scoring criterion is Lscore(x∗) = p(x∗, Ds) p(Ds) = Y j 1 αj + βj + N (˜αj)x∗j ˜βj 1−x∗j (6) where ˜αj = αj + PN n=1 xnj and ˜βj = βj + N −PN n=1 xnj. The logarithm of this score is also linear in x and can be computed efﬁciently as log Lscore(x∗) = c + X j wjx∗j (7) where c = P j log βj −log(αj + βj + N) and wj = log ˜αj −log ˜βj. The hyperparameters α and β are initialized to the same values used in the Bayesian model. Prototype model. In this model we deﬁne a prototype vector xproto to be the modal features for a set of items Ds. The similarity measure then becomes Pscore(x∗) = exp{−λ dist(x∗, xproto)} (8) where dist(·, ·) is the Hamming distance between the two vectors and λ is a free parameter. Since we are primarily concerned with ranking images, λ does not need to be optimized as it plays the role of a scaling constant. Exemplar model. We deﬁne the exemplar model using a similar scoring metric to the prototype model, except rather than computing the distance of x∗to a single prototype, we compute a distance for each item in the set Ds. Our similarity measure is thus computed as Escore(x∗) = X xj∈Ds exp{−λ dist(x∗, xj)} (9) where dist(·, ·) is the Hamming distance between two vectors and λ is a free parameter. In this case, λ does need to be optimized as the sum means that different values for λ can result in different overall similarity scores. 5 Modeling Human Ratings of Representativeness Given a set of images provided with a category label, how do people determine which images are good or bad examples of that category? In this section we present an experiment which evaluates our models through comparison with human judgments of the representativeness of images. 5.1 Methods A total of 500 participants (10 per category) were recruited via Amazon Mechanical Turk and compensated $0.25. The stimuli were created by identifying the top 10 and bottom 10 ranked images for each of the 50 categories for the Bayesian, likelihood, and prototype models and then taking the union of these sets for each category. The exemplar model was excluded in this process as it required optimization of its λ parameter, meaning that the best and worst images could not be determined in advance. The result was a set of 1809 images, corresponding to an average of 36 images per category. Participants were shown a series of images and asked to rate how good an example each image was of the assigned category label. The order of images presented was randomized across subjects. Image quality ratings were made on a scale of 1-7, with a rating of 1 meaning the image is a very bad example and a rating of 7 meaning the image is a very good example. 5.2 Results Once the human ratings were collected, we computed the mean ratings for each image and the mean of the top 10 and bottom 10 results for each algorithm used to create the stimuli. We also computed 6 Top 10 Rankings Bottom 10 Rankings 3 3.5 4 4.5 5 5.5 6 Mean quality ratings Bayes Likelihood Prototype Figure 2: Mean quality ratings of the top 10 and bottom 10 rankings of the different representativeness models over 50 categories. Error bars show one standard error. The vertical axis is bounded by the best possible top 10 ratings and the worst possible bottom 10 ratings across categories. bounds for the ratings based on the optimal set of top 10 and bottom 10 images per category. These are the images which participants rated highest and lowest, regardless of which algorithm was used to create the stimuli. The mean ratings for the optimal top 10 images was slightly less than the highest possible rating allowed (m = 6.018, se = 0.074), while the mean ratings for the optimal bottom 10 images was signiﬁcantly higher than the lowest possible rating allowed (m = 2.933, se = 0.151). The results are presented in Figure 2. The Bayesian model had the overall highest ratings for its top 10 rankings (m = 5.231, se = 0.026) and the overall lowest ratings for its bottom 10 rankings (m = 3.956, se = 0.031). The other models performed signiﬁcantly worse, with likelihood giving the next highest top 10 (m = 4.886, se = 0.028), and next lowest bottom 10 (m = 4.170, se = 0.031), and prototype having the lowest top 10 (m = 4.756, se = 0.028), and highest bottom 10 (m = 4.249, se = 0.031). We tested for statistical signiﬁcance via pairwise t-tests on the mean differences of the top and bottom 10 ratings over all 50 categories, for each pair of models. The Bayesian model outperformed both other algorithms (p < .001). As a second analysis, we ran a Spearman rank-order correlation to examine how well the actual scores from the models ﬁt with the entire set of human judgments. Although we did not explicitly ask participants to rank images, their quality ratings implicitly provide an ordering on the images that can be compared against the models. This also gives us an opportunity to evaluate the exemplar model, optimizing its λ parameter to maximize the ﬁt to the human data. To perform this correlation we recorded the model scores over all images for each category, and then computed the correlation of each model with the human judgments within that category. Correlations were then averaged across categories. The Bayesian model had the best mean correlation (ρ = 0.352), while likelihood (ρ = 0.220), prototype (ρ = 0.160), and the best exemplar model (λ = 2.0, ρ = 0.212) all performed less well. Paired t-tests showed that the Bayesian model produced statistically signiﬁcantly better performance than the other three models (all p < .01). 5.3 Discussion Overall, the Bayesian model of representativeness provided the best account of people’s judgments of which images were good and bad examples of the different categories. The mean ratings over the entire dataset were best predicted by our model, indicating that on average, the model predictions for images in the top 10 results were deemed of high quality and the predictions for images in the bottom 10 results were deemed of low quality. Since the images from the Corel database come with labels given by human judges, few images are actually very bad examples of their prescribed labels. This explains why the ratings for the bottom 10 images are not much lower. Additionally, there was some variance as to which images the Mechanical Turk workers considered to be “most representative”. This explains why the ratings for the top 10 images are not much higher, and thus why the difference between top and bottom 10 on average is not larger. When comparing the actual 7 Table 1: Model comparisons for the outlier experiment Model Average Outlier Position S.E. Bayesian Sets 0.805 ± 0.014 Likelihood 0.779 ± 0.013 Prototype 0.734 ± 0.015 Exemplar 0.734 ± 0.016 scores from the different models against the ranked order of human quality ratings, the Bayesian account was also signiﬁcantly more accurate than the other models. While the actual correlation value was less than 1, the dataset was rather varied in terms of quality for each category and thus it was not expected to be a perfect correlation. The methods of the experiment were also not explicitly testing for this effect, providing another source of variation in the results. 6 Finding Outliers in Sets Measuring the representativeness of items in sets can also provide a novel method of ﬁnding outliers in sets. Outliers are deﬁned as an observation that appears to deviate markedly from other members of the sample in which it occurs [17]. Since models of representativeness can be used to rank items in a set by how good an example they are of the entire set, outliers should receive low rankings. The performance of these different measures in detecting outliers provides another indirect means of assessing their quality as measures of representativeness. To empirically test this idea we can take an image from a particular category and inject it into all other categories, and see whether the different measures can identify it as an outlier. To ﬁnd a good candidate image we used the top ranking image per category as ranked by the Bayesian model. We justify this method because the Bayesian model had the best performance in predicting human quality judgments. Thus, the top ranked image for a particular category is assumed to be a bad example of the other categories. We evaluated how low this outlier was ranked by each of the representativeness measures 50 times, testing the models with a single injected outlier from each category to get a more robust measure. The ﬁnal evaluation was based on the normalized outlier ranking for each category (position of outlier divided by total number of images in the category), averaged over the 50 injections. The closer this quantity is to 1, the lower the ranking of outliers. The results of this analysis are depicted in Table 1, where it can be seen that the Bayesian model outperforms the other models. It is interesting to note that these measures are all quite distant from 1. We interpret this as another indication of the noisiness of the original image labels in the dataset since there were a number of images in each category that were ranked lower than the outlier. 7 Conclusions We have extended an existing Bayesian model of representativeness to handle sets of items and showed how it closely approximates a method of clustering on-demand – Bayesian Sets – that had been developed in machine learning. We exploited this relationship to allow us to evaluate a set of psychological models of representativeness using a large database of naturalistic images. Our Bayesian measure of representativeness signiﬁcantly outperformed other proposed accounts in predicting human judgments of how representative images were of different categories. These results provide strong evidence for this characterization of representativeness, and a new source of validation for the Bayesian Sets algorithm. We also introduced a novel method of detecting outliers in sets of data using our representativeness measure, and showed that it outperformed other measures. We hope that the combination of methods from cognitive science and computer science that we used to obtain these results is the ﬁrst step towards closer integration between these disciplines, linking psychological theories and behavioral methods to sophisticated algorithms and large databases. 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 to TLG and a National Science Foundation Postdoctoctoral Fellowship to KAH. 8 References [1] D. Kahneman and A. Tversky. Subjective probability: A judgment of representativeness. Cognitive Psychology, 3:430–454, 1972. [2] G. Gigerenzer. On narrow norms and vague heuristics: A reply to Kahneman and Tversky (1996). Psychological Review, 103:592, 1996. [3] G. L. Murphy and D. L. Medin. The role of theories in conceptual coherence. Psychological Review, 92:289–316, 1985. [4] J. B. Tenenbaum and T. L. Grifﬁths. The rational basis of representativeness. In Proc. 23rd Annu. Conf. Cogn. Sci. Soc., pages 1036–1041, 2001. [5] Z. Ghahramani and K. A. Heller. Bayesian sets. In Advances in Neural Information Processing Systems, volume 18, 2005. [6] C.B. Mervis and E. Rosch. Categorization of natural objects. Annual Review of Psychology, 32:89–115, 1981. [7] D.N. Osherson, E.E. Smith, O. Wilkie, A. Lopez, and E. Shaﬁr. Category-based induction. Psychological Review, 97:185, 1990. [8] D. L. Medin and M. M. Schaffer. Context theory of classiﬁcation learning. Psychological Review, 85:207– 238, 1978. [9] R. M. Nosofsky. Attention and learning processes in the identiﬁcation and categorization of integral stimuli. Journal of Experimental Psychology: Learning, Memory, and Cognition, 13:87–108, 1987. [10] S. K. Reed. Pattern recognition and categorization. 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Technometrics, 11:1–21, 1969. 9	2011	61
4,410	Dynamical segmentation of single trials from population neural data Biljana Petreska Gatsby Computational Neuroscience Unit University College London biljana@gatsby.ucl.ac.uk Byron M. Yu ECE and BME Carnegie Mellon University byronyu@cmu.edu John P. Cunningham Dept of Engineering University of Cambridge jpc74@cam.ac.uk Gopal Santhanam, Stephen I. Ryu†, Krishna V. Shenoy‡ Electrical Engineering ‡Bioengineering, Neurobiology and Neurosciences Program Stanford University †Dept of Neurosurgery, Palo Alto Medical Foundation {gopals,seoulman,shenoy}@stanford.edu Maneesh Sahani Gatsby Computational Neuroscience Unit University College London maneesh@gatsby.ucl.ac.uk Abstract Simultaneous recordings of many neurons embedded within a recurrentlyconnected cortical network may provide concurrent views into the dynamical processes of that network, and thus its computational function. In principle, these dynamics might be identiﬁed by purely unsupervised, statistical means. Here, we show that a Hidden Switching Linear Dynamical Systems (HSLDS) model— in which multiple linear dynamical laws approximate a nonlinear and potentially non-stationary dynamical process—is able to distinguish different dynamical regimes within single-trial motor cortical activity associated with the preparation and initiation of hand movements. The regimes are identiﬁed without reference to behavioural or experimental epochs, but nonetheless transitions between them correlate strongly with external events whose timing may vary from trial to trial. The HSLDS model also performs better than recent comparable models in predicting the ﬁring rate of an isolated neuron based on the ﬁring rates of others, suggesting that it captures more of the “shared variance” of the data. Thus, the method is able to trace the dynamical processes underlying the coordinated evolution of network activity in a way that appears to reﬂect its computational role. 1 Introduction We are now able to record from hundreds—and very likely soon from thousands—of neurons in vivo. By studying the activity of these neurons in concert we may hope to gain insight not only into the computations performed by speciﬁc neurons, but also into the computations performed by the population as a whole. The dynamics of such collective computations can be seen in the coordinated activity of all of the neurons within the local network; although each individual such neuron may reﬂect this coordinated component only noisily. Thus, we hope to identify the computationallyrelevant network dynamics by purely statistical, unsupervised means—capturing the shared evolu1 tion through latent-variable state-space models [1, 2, 3, 4, 5, 6, 7, 8]. The situation is similar to that of a camera operating at the extreme of its light sensitivity. A single pixel conveys very little information about an object in the scene, both due to thermal and shot noise and due to the ambiguity of the single-channel signal. However, by looking at all of the noisy pixels simultaneously and exploiting knowledge about the structure of natural scenes, the task of extracting the object becomes feasible. In a similar way, noisy data from many neurons participating in a local network computation needs to be combined with the learned structure of that computation—embodied by a suitable statistical model—to reveal the progression of the computation. Neural spiking activity is usually analysed by averaging across multiple experimental trials, to obtain a smooth estimate of the underlying ﬁring rates [2, 3, 4, 5]. However, even under carefully controlled experimental conditions, the animal’s behavior may vary from trial-to-trial. Reaction time in motor or decision-making tasks for example, reﬂects internal processes that can last for measurably different periods of time. In these cases traditional methods are challenging to apply, as there is no obvious way of aligning the data from different trials. It is thus essential to develop methods for the analysis of neural data that can account for the timecourse of a neural computation during a single trial. Single-trial methods are also attractive for analysing speciﬁc trials in which the subject exhibits erroneous behavior. In the case of a surprisingly long movement preparation time or a wrong decision, it becomes possible to identify the sources of error at the neural level. Furthermore, single-trial methods allow the use of more complex experimental paradigms where the external stimuli can arise at variable times (e.g. variable time delays). Here, we study a method for the unsupervised identiﬁcation of the evolution of the network computational state on single trials. Our approach is based on a Hidden Switching Linear Dynamical System (HSLDS) model, in which the coordinated network inﬂuence on the population is captured by a low-dimensional latent variable which evolves at each time step according to one of a set of available linear dynamical laws. Similar models have a long history in tracking, speech and, indeed, neural decoding applications [9, 10, 11] where they are variously known as Switching Linear Dynamical System models, Jump Markov models or processes, switching Kalman Filters or Switching Linear Gaussian State Space models [12]. We add the preﬁx “Hidden” to stress that in our application neither the switching process nor the latent dynamical variable are ever directly observed, and so learning of the parameters of the model is entirely unsupervised—and again, learning in such models has a long history [13]. The details of the HSLDS model, inference and learning are reviewed in Section 2. In our models, the transitions between linear dynamical laws may serve two purposes. First, they may provide a piecewise-linear approximation to a more accurate non-linear dynamical model [14]. Second, they may reﬂect genuine changes in the dynamics of the local network, perhaps due to changes in the goals of the underlying computation under the control of signals external to the local area. This second role leads to a computational segmentation of individual trials, as we will see below. We compare the performance of the HSLDS model to Gaussian Processes Factor Analysis (GPFA), a method introduced by [8] which analyses multi-neuron data on a single-trial basis with similar motivation to our own. Instead of explicitly modeling the network computation as a dynamical process, GPFA assumes that the computation evolves smoothly in time. In this sense, GPFA is less restrictive and would perform better if the HSLDS provided a bad model of the real network dynamics. However GPFA assumes that the latent dimensions evolve independently, making GPFA more restrictive than HSLDS in which the latent dimensions can be coupled. Coupling the latent dynamics introduces complex interactions between the latent dimensions, which allows a richer set of behaviors. To validate our HSLDS model against GPFA and a single LDS we will use the cross-prediction measure introduced with GPFA [8] in which the ﬁring rate of each neuron is predicted using only the ﬁring rates of the rest of the neurons; thus the metric measures how well each model captures the shared components of the data. GPFA and cross-prediction are reviewed brieﬂy in Section 3, which also introduces the dataset used; and the cross-prediction performance of the models is compared in Section 4. Having validated the HSLDS approach, we go on to study the dynamical segmentation identiﬁed by the model in the rest of Section 4, leading to the conclusions of Section 5. 2 2 Hidden Switching Linear Dynamical Systems Our goal is to extract the structure of computational dynamics in a cortical network from the recorded ﬁring rates of a subset of neurons in that network. We use a Hidden Switching Linear Dynamical Systems (HSLDS) model to capture the component of those ﬁring rates which is shared by multiple cells, thus exploiting the intuition that network computations should be reﬂected in coordinated activity across a local population. This will yield a latent low-dimensional subspace of dynamical states embedded within the space of noisy measured ﬁring rates, along with a model of the dynamics within that latent space. The dynamics of the HSLDS model combines a number of linear dynamical systems (LDS), each of which capture linear Markovian dynamics using a ﬁrst-order linear autoregressive (AR) rule [9, 15]. By combining multiple such rules, the HSLDS model can provide a piecewise linear approximation to nonlinear dynamics, and also capture changes in the dynamics of the local network driven by external inﬂuences that presumably reﬂect task demands. In the model implemented here, transitions between LDS rules themselves form a Markov chain. Let x:,t ∈IRp×1 be the low-dimensional computational state that we wish to estimate. This latent computational state reﬂects the network-level computation performed at timepoint t that gives rise to the observed spiking activity y:,t ∈IRq×1. Note that the dimensionality of the computational state p is lower than the dimensionality of the recorded neural data q which corresponds to the number of recorded neurons. The evolution of the computational state x:,t is given by x:,t\|x:,t−1, st ∼N(Astx:,t−1, Kst) (1) where N(µ, Σ) denotes a Gaussian distribution with mean µ and covariance Σ. The linear dynamical matrices Ast ∈IRp×p and innovations covariance matrices Kst ∈IRp×p are parameters of the model and need to be learned. These matrices are indexed by a switch variable st ∈{1, ..., S} such that different Ast and Kst need to be learned for each of the S possible linear dynamical systems. If the dependencies on st are removed, Eq. 1 deﬁnes a single LDS. The switch variable st speciﬁes which linear dynamical law guides the evolution of the latent state x:,t at timepoint t and as such provides a piecewise approximation to the nonlinear dynamics with which x:,t may evolve. The variable st itself is drawn from a Markov transition matrix M learned from the data: st ∼Discrete(M:,st−1) As mentioned above, the observed neural activity y:,t ∈IRq×1 is generated by the latent dynamics and denotes the spike counts (Gaussianised as described below) of q simultaneously recorded neurons at timepoints t ∈{1, ..., T}. The observations y:,t are related to the latent computational states x:,t through a linear-Gaussian relationship: y:,t\|x:,t ∼N(Cx:,t + d, R). where the observation matrix C ∈IRq×p, offset d ∈IRq×1, and covariance matrix R ∈IRq×q are model parameters that need to be learned. We force R to be diagonal and to keep track only of the independent noise variances. This means that the ﬁring rates of different neurons are independent conditioned on the latent dynamics, compelling the shared variance to live only in the latent space. Note that different neurons can have different independent noise variances. We use a Gaussian relationship instead of a point-process likelihood model for computational tractability. Finally, the observation dynamics do not depend on which linear dynamical system is used (i.e., are independent of st). A graphical model of the particular HSLDS instance we have used is shown in Figure 2. Inference and learning in the model are performed by approximate Expectation Maximisation (EM). Inference (or the E-step) requires ﬁnding appropriate expected sufﬁcient statistics under the distributions of the computational latent state and switch variable at each point in time given the observed neural data p(x1:T , s1:T \|y1:T ). Inference in the HSLDS is computationally intractable because of the following exponential complexity. At the initial timepoint, s0 can take one of S discrete values. At the next timepoint, each of the S possible latent states can again evolve according to S different linear dynamical laws, such that at timepoint t we need to keep track of St possible solutions. To avoid 3 Figure 1: Graphical model of the HSLDS. The ﬁrst layer corresponds to the discrete switch variable that dictates which of the S available linear dynamical systems (LDSs) will guide the latent dynamics shown in the second layer. The latent dynamics evolves as a linear dynamical system at timepoint t and presumably captures relevant aspects of the computation performed at the level of the recorded neural network. The relationship between the latent dynamics and neural data (third layer) is again linear-Gaussian, such that each computational state is associated to a speciﬁc denoised ﬁring pattern. The dimensionality of the latent dynamics x is lower than that of the observations y (equivalent to the number of recorded neurons), meaning that x extracts relevant features reﬂected in the shared variance of y. Note that there are no connections between xt−1 and st, nor st and y. this exponential scaling, we use an approximate inference algorithm based on Assumed Density Filtering [16, 17, 18] and Assumed Density Smoothing [19]. The algorithm comprises a single forward pass that estimates the ﬁltered posterior distribution p(xt, st\|y1:t) and a single backward pass that estimates the smoothed posterior distribution p(xt, st\|y1:T ). The key idea is to approximate these posterior distributions by a simple tractable form such as a single Gaussian. The approximated distribution is then propagated through time conditioned on the new observation. The smoothing step requires an additional simplifying assumption where p(xt+1\|st, st+1, y1:T ) ≈p(xt+1\|st+1, y1:T ) as proposed in [19]. It is also possible to use a mixture of a ﬁxed number of Gaussians as the approximating distribution, at the cost of greater computational time. We found that this approach yielded similar results in pilot runs, and thus retained the single-Gaussian approximation. Learning the model parameters (or the M-step) can be performed using the standard procedure of maximizing the expected joint log-likelihood: N X n=1 ⟨log p(xn 1:T , yn 1:T )⟩pold(xn\|yn) with respect to the parameters Ast, Kst, M, C, d and R, where the superscript n indexes data from each of N different trials. In practice, the estimated individual variance of particularly low-ﬁring neurons was very low and likely to be incorrectly estimated. Therefore we assumed a Wishart prior on the observation covariance matrix R, which resulted in an update rule that adds a ﬁxed parameter ψ ∈IR to all of the values at the diagonal. In the analyses below ψ was ﬁxed to the value that gave the best cross-prediction results (see Section 3.2). Finally, the most likely state of the switch variable s∗ 1:T = arg maxs1:T p(s1:T \|y1:T ) was estimated using the standard Viterbi algorithm [20], which ensures that the most likely switch variable path is in fact possible in terms of the transitions allowed by M. 3 Model Comparison and Experimental Data 3.1 Gaussian Process Factor Analysis Below, we compare the performance of the HSLDS model to Gaussian Process Factor Analysis (GPFA), another method for estimating the functional computation of a set of neurons. GPFA is an extension of Factor Analysis that leverages time-label information, introduced in [8]. In this model, the latent dynamics evolve as a Gaussian Process (GP), with a smooth correlation structure between the latent states at different points in time. This combination of FA and the GP prior work together to identify smooth low-dimensional latent trajectories. 4 Formally, each dimension of the low-dimensional latent states x:,t is indexed by i ∈{1, ..., p} and deﬁnes a separate GP: xi,: ∼N(0, Ki) where xi,: ∈IR1×T is the trajectory in time of the ith latent dimension and Ki ∈IRT ×T is the ith GP smoothing covariance matrix. Ki is set to the commonly-used squared exponential (SE) covariance function as deﬁned in [8]. Whereas HSLDS explicitly models the dynamics of the network computation, GPFA only assumes that the evolution of the computational state is smooth. Thus GPFA is a less restrictive model than HSLDS, but being model-free makes it also less informative of the dynamical rules that underlie the computation. A major advantage of GPFA over HSLDS is that the solution is approximation-free and faster to run. 3.2 Cross-prediction performance measure To compare model goodness-of-ﬁt we adopt the cross-prediction metric of [8]. All of these models attempt to capture the shared variance in the data, and so performance may be measured by how well the activity of one neuron can be predicted using the activity of the rest of the neurons. It is important to measure the cross-prediction error on trials that have not been used for learning the parameters of the model. We arrange the observed neural data in a matrix Y = [y:,1, ..., y:,T ] ∈IRq×T where each row yj,: represents the activity of neuron j in time. The model cross-prediction for this neuron j is ˆyj,: = E[yj,:\|Y−j,:] where Y−j,: ∈IR(q−1)×T represents all but the jth row of Y . We ﬁrst estimate the trajectories in the latent space using all but the jth neuron P(x1:p,:\|Y−j,:) in a set of testing trials. We then project this estimate back to the high-dimensional space to obtain the model cross-prediction ˆyj,: using ˆyj,t = Cj,: · E[x(:, t)\|Y−j,:] + dj. The error is computed as the sumof-squared errors between the model cross-prediction and the observed Gaussianised spike counts across all neurons and timepoints; and we plot the difference between this error (per time bin) and the average temporal variance of the corresponding neuron in the corresponding trial (denoted as Var-MSE). Note that the performance of difference models can be evaluated as a function of the dimensionality of the latent state. The HSLDS model has two futher free parameters which inﬂuence crossprediction peformance: the number of available LDSs S and the concentration of the Wishart prior ψ. 3.3 Data We applied the model to data recorded in the premotor and motor cortices of a rhesus macaque while it performed a delayed center-out reach task. A trial began with the animal touching and looking at an illuminated point at the center of a vertically oriented screen. A target was then illuminated at a distance of 10cm and in one of seven directions (0, 45, 90, 135, 180, 225, 315) away from this central starting point. The target remained visible while the animal prepared but withheld a movement to touch it. After a random delay of between 200 and 700ms, the illumination of the starting point was extinguished, which was the animal’s cue (the “go cue”) to reach to the target to obtain a reward. Neural activity was recorded from 105 single and multi-units, using a 96-electrode array (Blackrock, Salt Lake City, UT). All active units were included in the analysis without selection based on tuning. The spike-counts were binned at a relatively ﬁne time-scale of 10ms (non-overlapping bins). As in [8], the observations were taken to be the square-roots of these spike counts, a transformation that helps to Gaussianise and stabilise the variance of count data [21]. 4 Results We ﬁrst compare the cross-prediction-derived goodness-of-ﬁt of the HSLDS model to that of the single LDS and GPFA models in section 4.1. We ﬁnd that HSLDS provides a better model of the shared component of the recorded data than do the two other methods. We then study the dynamical segmentation found by the HSLDS model, ﬁrst by looking at a typical example (section 4.2) and then by correlating dynamical switches to behavioural events (section 4.3). We show that the latent 5 4 5 6 7 8 9 10 11 12 13 14 15 6 6.2 6.4 6.6 6.8 7 7.2 x 10 −3 Latent state dimensionality p Var-MSE HSLDS, S=7 LDS GPFA Figure 2: Performance of the HSLDS (green solid line), LDS (blue dashed) and GPFA (red dash-dotted) models. Analyses are based on one movement type with the target in the 45◦direction. Crossprediction error was computed using 4fold cross-validation. HSLDS with different values of S also outperformed the LDS case (which is equivalent to S = 1). Performance was more sensitive to the strength ψ of the Wishart prior, and the best performing model is shown. trajectories and dynamical transitions estimated by the model predict reaction time, a behavioral covariate that varies from trial-to-trial. Finally we argue that these behavioral correlates are difﬁcult to obtain using a standard neural analysis method. 4.1 Cross-prediction To validate the HSLDS model we compared it to the GPFA model described in section 3.1 and a single LDS model. Since all of these models attempt to capture the shared variance of the data across neurons and multiple trials, we used cross-prediction to measure their performance. Crossprediction looks at how well the spiking activity of one neuron is predicted just by looking at the spiking activity of all of the other neurons (described in detail in Section 3.2). We found that both the single LDS and HSLDS models that allow for coupled latent dynamics do better than GPFA, shown in Figure 2, which could be attributed to the fact that GPFA constrains the different dimensions of the latent computational state to evolve independently. The HSLDS model also outperforms a single LDS yielding the lowest prediction error for all of the latent dimensions we have looked at, arguing that a nonlinear model of the latent dynamics is better than a linear model. Note that the minimum prediction error asymptotes after 10 latent dimensions. It is tempting to suggest that for this particular task the effective dimensionality of the spiking activity is much lower than that of the 105 recorded neurons, thereby justifying the use of a low-dimensional manifold to describe the underlying computation. This could be interpreted as evidence that neurons may carry redundant information and that the (nonlinear) computational function of the network is better reﬂected at the level of the population of neurons, rather than in single neurons. 4.2 Data segmentation By deﬁnition, the HSLDS model partitions the latent dynamics underlying the observed data into time-labeled segments that may evolve linearly. The segments found by HSLDS correspond to periods of time in which the latent dynamics seem to evolve according to different linear dynamical laws, suggesting that the observed ﬁring pattern of the network has changed as a whole. Thus, by construction, the HSLDS model can subdivide the network activity into different ﬁring regimes for each trial speciﬁcally. For the purpose of visualization, we have applied an additional orthonormalization post-processing step (as in [8]) that helps us order the latent dimensions according to the amount of covariance explained. The orthonormalization consists of ﬁnding the singular-value decomposition of C, allowing us to write the product Cx:,t as UC(DCV ′ Cx:,t), where UC ∈IRq×p is a matrix with orthonormal columns. We will refer to ˜x:,t = DCV ′ Cx:,t as the orthonormalised latent state at time t. The ﬁrst dimension of the orthonormalised latent state in time ˜x1,: corresponds then to the latent trajectory which explains the most covariance. Since the columns of UC are orthonormal, the relationship between the orthonormalised latent trajectories and observed data can be interpreted in an intuitive way, similarly to Principal Components Analysis (PCA). The results presented here were obtained by setting the number of switching LDSs S, latent space dimensionality p and Wishart prior ψ to values that yielded a reasonably low cross-prediction error. Figure 3 shows a typical example of the HSLDS model applied to data in one movement direction, where the different trials are fanned out vertically for illustration purposes. The ﬁrst orthonormalized 6 Figure 3: HSLDS applied to neural data from the 45◦direction movement (S = 7, p = 7, ψ = 0.05). The ﬁrst dimension of the orthonormalised latent trajectory is shown. The colors denote the different linear dynamical systems used by the model. Each line is a different trial, aligned to the target onset (left) and go cue (right), and sorted by reaction time. Switches reliably follow the target onset and precede the movement onset, with a time lag that is correlated with reaction time. latent dimension indicates a transient in the recorded population activity shortly after target onset (which is marked by the red dots) and a sustained change of activity after the go cue (marked by the green dots). The colours of the lines indicate the most likely setting of the switching variable at each time. It is evident that the learned solution segments each trial into a broadly reproducible sequence of dynamical epochs. Some transitions appear to reliably follow or precede external events (even though these events were not used to train the segmentation) and may reﬂect actual changes in dynamics due to external inﬂuences. Others seem to follow each other in quick succession, and may instead reﬂect linear approximations to non-linear dynamical processes—evident particularly during transiently rapid changes in the latent state. Unfortunately, the current model does not allow us to distinguish quantitatively between these two types of transition. Note that the delays (time from target onset to go cue) used in the experiment varied from 200 to 700ms, such that the model systematically detected a change in the neural ﬁring rates shortly after the go cue appeared on each individual trial. The model succeeds at detecting these changes in a purely unsupervised fashion as it was not given any time information about the external experimental inputs. 4.3 Behavioral correlates during single trials It is not surprising that the ﬁring rates of the recorded neurons change during different behavioral periods. For example, neural activity is often observed to be higher during movement execution than during movement preparation. However, the HSLDS method reliably detects the behaviourallycorrelated changes in the pattern of neural activity across many neurons on single trials. In order to ensure that HSLDS captures trial-speciﬁc information we have looked at whether the time post-go-cue at which the model estimates a ﬁrst switch in the neural dynamics could predict the subsequent onset of movement and thus the trial reaction time (RT). We found that the ﬁltered model (which does not incorporate spiking data from future times into its estimate of the switching variable) could explain 52% of the reaction time variance on average, across the 7 reach directions (Figure 4). Could a more conventional approach do better? We attempted to use a combination of the “population vector” (PV) method and the “rise-to-threshold” hypothesis. The PV sums the preferred directions of a population of neurons, weighted by the respective spike counts in order to decode the represented direction of movement [22]. The rise-to-threshold hypothesis asserts that neural ﬁring rates rise during a preparatory period and movement is initiated when the population rate crosses a threshold [23]. The neural data used for this analysis were smoothed with a Gaussian window and sampled at 1 ms. We ﬁrst estimated the preferred direction ˆpq of the neuron indexed by q as the 7 Figure 4: Correlation (R2 = 0.52) between the reaction time and ﬁrst ﬁltered HSLDS switch following the go cue, on a trial-bytrial basis and averaged across directions. Symbols correspond to movements in different directions. Note that in two catch trials the model did not switch following the go cue, so we considered the last switch before the cue. unit vector in the direction of ⃗pq = P7 d=1 rd i ⃗vd where d indexes the instructed movement direction ⃗vd and rd q is the mean ﬁring rate of neuron q during all movements in direction d. The preferred direction of a given neuron often differed between plan and movement activity, so we used data from movement onset until the movement end to estimate rd q as this gave us better results when trying to estimate a threshold in the rising movement-related activity. We then estimated the instanteneous amplitude of the network PV at time t as sd t = \|\| PQ q=1 yq,t⃗pq\|\|, where yq,t is the smoothed spike count of neuron q at time t, Q is the number of neurons and \|\|⃗w\|\| denotes the norm of the vector ⃗w. Finally, we searched for a threshold length (one per direction), such that the time at which the PV exceeded this length on each trial was best correlated with RT. Note that this approach uses considerable supervision that was denied to the HSLDS model. First, the movement epoch of each trial was identiﬁed to deﬁne the PV. Second, the thresholds were selected so as to maximize the RT correlation—a direct form of supervision. Finally, this selection was based on the same data as were used to evaluate the correlation score, thus leading to potential overﬁtting in the explained variance. The HSLDS model was also trained on the same trials, which could lead to some overﬁtting in terms of likelihood, but should not introduce overﬁtting in the correlation between switch times and RT, which is not directly optimised. Despite these considerable advantages, the PV approach did not predict RT as well as did the HSLDS, yielding an average variance explained across conditions of 48%. 5 Conclusion It appears that the Hidden Switching Linear Dynamical System (HSLDS) model is able to appropriately extract relevant aspects of the computation reﬂected in a network of ﬁring neurons. HSLDS explicitly models the nonlinear dynamics of the computation as a piecewise linear process that captures the shared variance in the neural data across neurons and multiple trials. One limitation of HSLDS is the approximate EM algorithm used for inference and learning of the model parameters. We have traded off computational tractability with accuracy, such that the model may settle into a solution that is simpler than the optimum. A second limitation of HSLDS is the slow training time of EM, enforcing an ofﬂine learning of the model parameters. Despite these simplications, HSLDS can be used to dynamically segment the neural activity at the level of the whole population of neurons into periods of different ﬁring regimes. We showed that in a delayed-reach task the ﬁring regimes found correlate well with the experimental behavioral periods. The computational trajectories found by HSLDS are trial-speciﬁc and with a dimensionality that is more suitable for visualization than the high-dimensional spiking activity. Overall, HSLDS are attractive models for uncovering behavioral correlates in neural data on a single-trial basis. Acknowledgments. This work was supported by DARPA REPAIR (N66001-10-C-2010), the Swiss National Science Foundation Fellowship PBELP3-130908, the Gatsby Charitable Foundation, UK EPSRC EP/H019472/1 and NIH-NINDS-CRCNS-R01, NDSEG and NSF Graduate Fellowships, Christopher and Dana Reeve Foundation. We are very grateful to Jacob Macke, Lars Buesing and Alexander Lerchner for discussion. 8 References [1] A. C. Smith and E. N. Brown. Estimating a state-space model from point process observations. Neural Computation, 15(5):965–991, 2003. [2] M. Stopfer, V. Jayaraman, and G. Laurent. Intensity versus identity coding in an olfactory system. Neuron, 39:991–1004, 2003. [3] S. L. Brown, J. Joseph, and M. Stopfer. Encoding a temporally structured stimulus with a temporally structured neural representation. Nature Neuroscience, 8(11):1568–1576, 2005. [4] R. Levi, R. Varona, Y. I. Arshavsky, M. I. Rabinovich, and A. I. Selverston. The role of sensory network dynamics in generating a motor program. 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4,411	Approximating Semideﬁnite Programs in Sublinear Time Dan Garber Technion - Israel Institute of Technology Haifa 32000 Israel dangar@cs.technion.ac.il Elad Hazan Technion - Israel Institute of Technology Haifa 32000 Israel ehazan@ie.technion.ac.il Abstract In recent years semideﬁnite optimization has become a tool of major importance in various optimization and machine learning problems. In many of these problems the amount of data in practice is so large that there is a constant need for faster algorithms. In this work we present the ﬁrst sublinear time approximation algorithm for semideﬁnite programs which we believe may be useful for such problems in which the size of data may cause even linear time algorithms to have prohibitive running times in practice. We present the algorithm and its analysis alongside with some theoretical lower bounds and an improved algorithm for the special problem of supervised learning of a distance metric. 1 Introduction Semideﬁnite programming (SDP) has become a tool of great importance in optimization in the past years. In the ﬁeld of combinatorial optimization for example, numerous approximation algorithms have been discovered starting with Goemans and Williamson [1] and [2, 3, 4]. In the ﬁeld of machine learning solving semideﬁnite programs is at the heart of many learning tasks such as learning a distance metric [5], sparse PCA [6], multiple kernel learning [7], matrix completion [8], and more. It is often the case in machine learning that the data is assumed no be noisy and thus when considering the underlying optimization problem, one can settle for an approximated solution rather then an exact one. Moreover it is also common in such problems that the amounts of data are so large that fast approximation algorithms are preferable to exact generic solvers, such as interior-point methods, which have impractical running times and memory demands and are not scalable. In the problem of learning a distance metric [5] one is given a set of points in Rn and similarity information in the form of pairs of points and a label indicating weather the two points are in the same class or not. The goal is to learn a distance metric over Rn which respects this similarity information. That is it assigns small distances to points in the same class and bigger distances to points in different classes. Learning such a metric is important for other learning tasks which rely on having a good metric over the input space, such as K-means, nearest-neighbours and kernel-based algorithms. In this work we present the ﬁrst approximation algorithm for general semideﬁnite programming which runs in time that is sublinear in the size of the input. For the special case of learning a pseudo-distance metric, we present an even faster sublinear time algorithm. Our algorithms are the fastest possible in terms of the number of constraints and the dimensionality, although slower than other methods in terms of the approximation guarantee. 1.1 Related Work Semideﬁnite programming is a notoriously difﬁcult optimization formulation, and has attracted a host of attempts at fast approximation methods. Klein and Lu [9] gave a fast approximate solver for 1 the MAX-CUT semideﬁnite relaxation of [1]. Various faster and more sophisticated approximate solvers followed [10, 11, 12], which feature near-linear running time albeit polynomial dependence on the approximation accuracy. For the special case of covering an packing SDP problems, [13] and [14] respectively give approximation algorithms with a smaller dependency on the approximation parameter ϵ. Our algorithms are based on the recent work of [15] which described sublinear algorithms for various machine learning optimization problems such has linear classiﬁcation and minimum enclosing ball. We describe here how such methods, coupled with techniques, may be used for semideﬁnite optimization. 2 Preliminaries In this paper we denote vectors in Rn by a lower case letter (e.g. v) and matrices in Rn×n by upper case letters (e.g. A). We denote by ∥v∥the standard euclidean norm of the vector v and by ∥A∥the frobenius norm norm of the matrix A, that is ∥A∥= qP i,j A(i, j)2. We denote by ∥v∥1 the l1-norm of v. The notation X ⪰0 states that the matrix X is positive semi deﬁnite, i.e. it is symmetric and all of its eigenvalues are non negative. The notation X ⪰B states that X −B ⪰0. The notation C ◦X is just the dot product between matrices, that is C ◦X = P i,j C(i, j)X(i, j). We denote by ∆m the m-dimensional simplex, that is ∆m = {p\| Pm i=1 pi = 1, ∀i : pi ≥0}. We denote by 1n the all ones n-dimensional vector and by 0n×n the all zeros n × n matrix. We denote by I the identity matrix when its size is obvious from context. Throughout the paper we will use the complexity notation ˜O(·) which is the same as the notation O(·) with the difference that it suppresses poly-logarithmic factors that depend on n, m, ϵ−1. We consider the following general SDP problem Maximise C ◦X (1) subject to Ai ◦X ≥ 0 i = 1, ..., m X ⪰0 Where C, A1, ..., Am ∈Rn×n. For reasons that will be made clearer in the analysis, we will assume that for all i ∈[m], ∥Ai∥≤1 The optimization problem (1) can be reduced to a feasibility problem by a standard reduction of performing a binary search over the value of the objective C◦X and adding an appropriate constraint. Thus we will only consider the feasibility problem of ﬁnding a solution that satisﬁes all constraints. The feasibility problem can be rewritten using the following min-max formulation max X⪰0 min i∈[m] Ai ◦X (2) Clearly if the optimum value of (2) is non-negative, then a feasible solution exists and vice versa. Denoting the optimum of (2) by σ, an ϵ additive approximation algorithm to (2) is an algorithm that produces a solution X such that X ⪰0 and for all i ∈[m], Ai ◦X ≥σ −ϵ. For the simplicity of the presentation we will only consider constraints of the form A ◦X ≥0 but we mention in passing that SDPs with other linear constraints can be easily rewritten in the form of (1). We will be interested in a solution to (2) which lies in the bounded semideﬁnite cone K = {X\|X ⪰0, Tr(X) ≤1}. The demand on a solution to (2) to have bounded trace is due to the observation that in case σ > 0, any solution needs to be bounded or else the products Ai ◦X could be made to be arbitrarily large. Learning distance pseudo metrics In the problem of learning a distance metric from examples, we are given a set triplets S = {{xi, x′ i, yi}}m i=1 such that xi, x′ i ∈Rn and yi ∈{−1, 1}. A value yi = 1 indicates that the vectors xi, x′ i are in the same class and a value yi = −1 indicates that they are from different classes. Our goal is to learn a pseudo-metric over Rn which respects the example set. A pseudo-metric is a function d : R×R →R, which satisﬁes three conditions: (i) d(x, x′) ≥0, (ii) d(x, x′) = d(x′, x) , and (iii) d(x1, x2) + d(x2, x3) ≥d(x1, x3). We consider pseudo-metrics of the form dA(x, x′) ≡ p (x −x′)⊤A(x −x′). Its easily veriﬁed that if A ⪰0 then dA is indeed a pseudo-metric. A reasonable demand from a ”good” pseudo metric is that it separates the examples 2 (assuming such a separation exists). That is we would like to have a matrix A ⪰0 and a threshold value b ∈R such that for all {xi, x′ i, yi} ∈S it will hold that, (dA(xi −x′ i))2 = (xi −x′ i)⊤A(xi −x′ i) ≤ b −σ/2 yi = 1 (3) (dA(xi −x′ i))2 = (xi −x′ i)⊤A(xi −x′ i) ≥ b + σ/2 yi = −1 where σ is the margin of separation which we would like to maximize. Denoting by vi = (xi −x′ i) for all i ∈[m], (3) can be summarized into the following formalism: yi b −v⊤ i Avi ≥σ Without loss of generality we can assume that b = 1 and derive the following optimization problem max A⪰0 min i∈[m] yi 1 −v⊤ i Avi (4) 3 Algorithm for General SDP Our algorithm for general SDPs is based on the generic framework for constrained optimization problems that ﬁt a max-min formulation, such as (2), presented in [15]. Noticing that mini∈[m] Ai ◦ X = minp∈∆m P i∈[m] p(i)Ai ◦X, we can rewrite (2) in the following way max x∈K min p∈∆m p(i)A⊤ i x (5) Building on [15], we use an iterative primal-dual algorithm that simulates a repeated game between two online algorithms: one that wishes to maximize P i∈[m] p(i)Ai ◦X as a function of X and the other that wishes to minimize P i∈[m] p(i)Ai ◦X as a function of p. If both algorithms achieve sublinear regret, then this framework is known to approximate max-min problems such as (5), in case a feasible solution exists [16]. The primal algorithm which controls X is a gradient ascent algorithm that given p adds to the current solution a vector in the direction of the gradient P i∈[m] p(i)Ai. Instead of adding the exact gradient we actually only sample from it by adding Ai with probability p(i) (lines 5-6). The dual algorithm which controls p is a variant of the well known multiplicative (or exponential) update rule for online optimization over the simplex which updates the weight p(i) according to the product Ai ◦X (line 11). Here we replace the exact computation of Ai ◦X by employing the l2-sampling technique used in [15] in order to estimate this quantity by viewing only a single entry of the matrix Ai (line 9). An important property of this sampling procedure is that if ∥Ai∥≤1, then E[˜vt(i)2] ≤1. Thus, we can estimate the product Ai ◦X with constant variance, which is important for our analysis. A problem that arises with this estimation procedure is that it might yield unbounded values which do not ﬁt well with the multiplicative weights analysis. Thus we use a clipping procedure clip(z, V ) ≡min{V, max{−V, Z}} to bound these estimations in a certain range (line 10). Clipping the samples yields unbiased estimators of the products Ai ◦X but the analysis shows that this bias is not harmful. The algorithm is required to generate a solution X ∈K. This constraint is enforced by performing a projection step onto the convex set K after each gradient improvement step of the primal online algorithm. A projection of a matrix Y ∈Rn×n onto K is given by Yp = arg minX∈K ∥Y −X∥. Unlike the algorithms in [15] that perform optimization over simple sets such as the euclidean unit ball which is trivial to project onto, projecting onto the bounded semideﬁnite cone is more complicated and usually requires to diagonalize the projected matrix (assuming it is symmetric). Instead, we show that one can settle for an approximated projection which is faster to compute (line 4). Such approximated projections could be computed by Hazan’s algorithm for ofﬂine optimization over the bounded semideﬁnite cone, presented in [12]. Hazan’s algorithm gives the following guarantee Lemma 3.1. Given a matrix Y ∈Rn×n, ϵ > 0, let f(X) = −∥Y −X∥2 and denote X∗= arg maxX∈K f(X). Then Hazan’s algorithm produces a solution ˜X ∈K of rank at most ϵ−1 such that ∥Y −˜X∥2 −∥Y −X∗∥2 ≤ϵ in O n2 ϵ1.5 time. We can now state the running time of our algorithm. Lemma 3.2. Algorithm SublinearSDP has running time ˜O m ϵ2 + n2 ϵ5 . 3 Algorithm 1 SublinearSDP 1: Input: ϵ > 0, Ai ∈Rn×n for i ∈[m]. 2: Let T ←602ϵ−2 log m, Y1 ←0n×n, w1 ←1m, η ← q log m T , ϵP ←ϵ/2. 3: for t = 1 to T do 4: pt ← wt ∥wt∥1 , Xt ←ApproxProject(Yt, ϵ2 P ). 5: Choose it ∈[m] by it ←i w.p. pt(i). 6: Yt+1 ←Yt + 1 √ 2T Ait 7: Choose (jt, lt) ∈[n] × [n] by (jt, lt) ←(j, l) w.p. Xt(j, l)2/∥Xt∥2. 8: for i ∈[m] do 9: ˜vt ←Ai(jt, lt)∥Xt∥2/Xt(jt, lt) 10: vt(i) ←clip(˜vt(i), 1/η) 11: wt+1(i) ←wt(i)(1 −ηvt(i) + η2vt(i)2) 12: end for 13: end for 14: return ¯X = 1 T P t Xt We also have the following lower bound. Theorem 3.3. Any algorithm which computes an ϵ-approximation with probability at least 2 3 to (2) has running time Ω m ϵ2 + n2 ϵ2 . We note that while the dependency of our algorithm on the number of constraints m is close to optimal (up to poly-logarithmic factors), there is a gap of ˜O(ϵ−3) between the dependency of our algorithm on the size of the constraint matrices n2 and the above lower bound. Here it is important to note that our lower bound does not reﬂect the computational effort in computing a general solution that is positive semideﬁnite which is in fact the computational bottleneck of our algorithm (due to the use of the projection procedure). 4 Analysis We begin with the presentation of the Multiplicative Weights algorithm used in our algorithm. Deﬁnition 4.1. Consider a sequence of vectors q1, ..., qT ∈Rm. The Multiplicative Weights (MW) algorithm is as follows. Let 0 < η ∈R, w1 ←1m, and for t ≥1, pt ←wt/∥wt∥1, wt+1 ←wt(i)(1 −ηqt(i) + η2qt(i)2) The following lemma gives a bound on the regret of the MW algorithm, suitable for the case in which the losses are random variables with bounded variance. Lemma 4.2. The MW algorithm satisﬁes X t∈[T ] p⊤ t qt ≤min i∈[m] X t∈[T ] max{qt(i), −1 η } + log m η + η X t∈[t] p⊤ t q2 t The following lemma gives concentration bounds on our random variables from their expectations. Lemma 4.3. For 1/4 ≥η ≥ q log m T , with probability at least 1 −O(1/m), it holds that (i) maxi∈[m] P t∈[T ][vt(i) −Ai ◦Xt] ≤4ηT (ii) X t∈[T ] Ait ◦Xt − X t∈[T ] p⊤ t vt ≤8ηT The following Lemma gives a regret bound on the lazy gradient ascent algorithm used in our algorithm (line 6). For a proof see Lemma A.2 in [17]. 4 Lemma 4.4. Consider matrices A1, ..., AT ∈Rn×n such that for all i ∈[m] ∥Ai∥≤1. Let X0 = 0n×n and for all t ≥1 let Xt+1 = arg minX∈K 1 √ 2T Pt τ=1 Aτ −X Then max X∈K X t∈[T ] At ◦X − X t∈[T ] At ◦Xt ≤2 √ 2T We are now ready to state the main theorem and prove it. Theorem 4.5 (Main Theorem). With probability 1/2, the SublinearSDP algorithm returns an ϵadditive approximation to (5). Proof. At ﬁrst assume that the projection onto the set K in line 4 is an exact projection and not an approximation and denote by ˜Xt the exact projection of Yt. In this case, by lemma 4.4 we have max x∈K X t∈[T ] Ait ◦X − X t∈[T ] Ait ◦˜Xt ≤2 √ 2T (6) By the law of cosines and lemma 3.1 we have for every t ∈[T] ∥Xt −˜Xt∥2 ≤∥Yt −Xt∥2 −∥Yt −˜Xt∥2 ≤ϵ2 P (7) Rewriting (6) we have max x∈K X t∈[T ] Ait ◦X − X t∈[T ] Ait ◦Xt − X t∈[T ] Ait ◦( ˜Xt −Xt) ≤2 √ 2T Using the Cauchy-Schwarz inequality, ∥Ait∥≤1 and (7) we get max x∈K X t∈[T ] Ait ◦X − X t∈[T ] Ait ◦Xt ≤2 √ 2T + X t∈[T ] ∥Ait∥∥˜Xt −Xt∥≤2 √ 2T + TϵP Rearranging and plugging maxx∈K mini∈[m] Ai ◦X = σ we get X t∈[T ] Ait ◦Xt ≥Tσ −2 √ 2T −TϵP (8) Turning to the MW part of the algorithm, by the MW Regret Lemma 4.2, and using the clipping of vt(i) we have X t∈[T ] p⊤ t vt ≤min i∈[i] X t∈[t] vt(i) + (log m)/η + η X t∈[T ] p⊤ t v2 t By Lemma 4.3, with high probability and for any i ∈[n], X t∈[T ] vt(i) ≤ X t∈[T ] Ai ◦Xt + 4ηT Thus with high probability it holds that X t∈[T ] p⊤ t vt ≤min i∈[i] X t∈[t] Ai ◦Xt + (log m)/η + η X t∈[T ] p⊤ t v2 t + 4ηT (9) Combining (8) and (9) we get min i∈[i] X t∈[t] Ai ◦Xt ≥−(log m)/η −η X t∈[T ] p⊤ t v2 t −4ηT + Tσ −2 √ 2T − X t∈[T ] p⊤ t vt − X t∈[T ] Ait ◦Xt −TϵP By a simple Markov inequality argument it holds that w.p. at least 3/4, X t∈[T ] p⊤ t v2 t ≤8T 5 Combined with lemma 4.3, we have w.p. at least 3 4 −O( 1 n) ≥1 2 min i∈[i] X t∈[t] Ai ◦Xt ≥ −(log m)/η −8ηT −4ηT + Tσ −2 √ 2T −8ηT −TϵP ≥ Tσ −log m η −20ηT −2 √ 2T −TϵP Dividing through by T and plugging in our choice for η and ϵP , we have mini∈[m] Ai ◦¯X ≥σ −ϵ w.p. at least 1/2. 5 Application to Learning Pseudo-Metrics As in the problem of general SDP, we can also rewrite (4) by replacing the mini∈[m] objective with minp∈∆m and arrive at the following formalism, max A⪰0 min p∈∆m yi 1 −v⊤ i Avi (10) As we demanded a solution to general SDP to have bounded trace, here we demand that ∥A∥≤1. Letting v′ i = vi 1 and deﬁning the set of matrices P = A 0 0 −1 \|A ⪰0, ∥A∥≤1 , we can rewrite (10) in the following form. max A∈P min p∈∆m −yiv′ iv′⊤ i ◦A (11) In what comes next, we use the notation Ai = −yiv′ iv′ i. Since projecting a matrix onto the set P is as easy as projecting a matrix onto the set {A ⪰0, ∥A∥≤1}, we assume for the simplicity of the presentation that the set on which we optimize is indeed P = {A ⪰0, ∥A∥≤1}. We proceed with presenting a simpler algorithm for this problem than the one given for general SDP. The gradient of yiv′ iv′⊤ i ◦A with respect to A is a symmetric rank one matrix and here we have the following useful fact that was previously stated in [18]. Theorem 5.1. If A ∈Rn×n is positive semi deﬁnite, v ∈Rn and α ∈R then the matrix B = A + αvv⊤has at most one negative eigenvalue. The proof is due to the eigenvalue Interlacing Theorem (see [19] pp. 94-97 and [20] page 412). Thus after performing a gradient step improvement of the form Yt+1 = Xt + ηyiviv⊤ i , projecting Yt+1 onto to the feasible set P comes down to the removal of at most one eigenvalue in case we subtracted a rank one matrix (yit = −1) or normalizing the l2 norm in case we added a rank one matrix (yit = 1). Since in practice computing eigenvalues fast, using the Power or Lanczos methods, can be done only up to a desired approximation, in fact the resulting projection Xt+1 might not be positive semideﬁnite. Nevertheless, we show by care-full analysis that we can still settle for a single eigenvector computation in order to compute an approximated projection with the price that Xt+1 ⪰−ϵ3I. That is Xt+1 might be slightly outside of the positive semideﬁnite cone. The beneﬁt is an algorithm with improved performance over the general SDP algorithm since far less eigenvalue computations are required than in Hazan’s algorithm. The projection to the set P is carried out in lines 7-11. In line 7 we check if Yt+1 has a negative eigenvalue and if so, we compute the corresponding eigenvector in line 8 and remove it in line 9. In line 11 we normalize the l2 norm of the solution. The procedure Sample(Ai, Xt) will be detailed later on when we discuss the running time. The following Lemma is a variant of Zinkevich’s Online Gradient Ascent algorithm [21] suitable for the use of approximated projections when Xt is not necessarily inside the set P. Lemma 5.2. Consider a set of matrices A1, ..., AT ∈Rn×n such that ∥Ai∥≤1. Let X0 = 0n×n and for all t ≥0 let Yt+1 = Xt + ηAt, ˜Xt+1 = arg min X∈P ∥Yt+1 −X∥ 6 Algorithm 2 SublinearPseudoMetric 1: Input: ϵ > 0, Ai = yiviv⊤ i ∈Rn×n for i ∈[m]. 2: Let T ←602ϵ−2 log m, X1 =←0n×n, w1 ←1m, η ← q log m T . 3: for t = 1 to T do 4: pt ← wt ∥wt∥1 . 5: Choose it ∈[m] by it ←i w.p. pt(i). 6: Yt+1 ←Xt + q 2 T yitvitv⊤ it 7: if yi < 0 and λmin(Yt+1) < 0 then 8: u ←arg minz:∥z∥=1 z⊤Yt+1z 9: Yt+1 = Yt+1 −λuu⊤ 10: end if 11: Xt+1 ← Yt+1 max {1,∥Yt+1∥} 12: for i ∈[m] do 13: vt(i) ←clip(Sample(Ai, Xt), 1/η) 14: wt+1(i) ←wt(i)(1 −ηvt(i) + η2vt(i)2) 15: end for 16: end for 17: return ¯X = 1 T P t Xt and let Xt+1 be such that ˜Xt+1 −Xt+1 ≤ϵd. Then, for a proper choice of η it holds that, max X∈P X t∈[T ] At ◦X − X t∈[T ] At ◦Xt ≤ √ 2T + 3 2ϵdT 3/2 The following lemma states the connection between the precision used in eigenvalues approximation in lines 7-8, and the quality of the approximated projection. Lemma 5.3. Assume that on each iteration t of the algorithm, the eigenvalue computation in line 7 is a δ = ϵd 4T 1.5 additive approximation of the smallest eigenvalue of Yt+1 and let ˜Xt = arg minX∈P ∥Yt −X∥. It holds that ∥˜Xt −Xt∥≤ϵd Theorem 5.4. Algorithm SublinearPseudoMetric computes an ϵ additive approximation to (11) w.p. 1/2. Proof. Combining lemmas 5.2, 5.3 we have, max X∈P X t∈[T ] At ◦X − X t∈[T ] At ◦Xt ≤ √ 2T + 3 2ϵdT 3/2 Setting ϵd = 2ϵP 3 √ T where ϵP is the same as in theorem 4.5 yields, arg max X∈P X t∈[T ] At ◦X − X t∈[T ] At ◦Xt ≤ √ 2T + ϵP T The rest of the proof follows the same lines as theorem 4.5. We move on to discus the time complexity of the algorithm. It is easily observed from the algorithm that for all t ∈[T], the matrix Xt can be represented as the sum of kt ≤2T symmetric rank-one matrices. That is Xt is of the form Xt = P i∈[kt] αiziz⊤ i , ∥zi∥= 1 for all i. Thus instead of computing Xt explicitly, we may represent it by the vectors zi and scalars αi. Denote by α the vector of length kt in which the ith entry is just αi, for some iteration t ∈[T]. Since ∥Xt∥≤ 1 it holds that ∥α∥≤1. The sampling procedure Sample(Ai, Xt) in line 13, returns the value Ai(j,l)∥α∥2 zk(j)zk(l)αk with probability α2 k ∥α∥2 · (zk(j)zk(l))2. That is we ﬁrst sample a vector zi according to 7 α and then we sample an entry (j, l) according to the chosen vector zi. It is easily observed that ˜vt(i) = Sample(Ai, Xt) is an unbiased estimator of Ai ◦Xt. It also holds that: E[˜vt(i)2] = X j∈[n],l∈[n],k∈[kt] α2 k ∥α∥2 (zk(j)zk(l))2 · Ai(j, l)2∥α∥4 (zk(j)zk(l))2α2 k = kt∥α∥2∥Ai∥2 = ˜O(ϵ−2) Thus taking ˜vt(i) to be the average of ˜O(ϵ−2) i.i.d samples as described above yields an unbiased estimator of Ai · Xt with variance at most 1 as required for the analysis of our algorithm. We can now state the running time of the algorithm. Lemma 5.5. Algorithm SublinearPseudoMetric can be implemented to run in time ˜O m ϵ4 + n ϵ6.5 . Proof. According the lemmas 5.3, 5.4, the required precision in eigenvalue approximation is ϵ O(1)T 2 . Using the Lanczos method for eigenvalue approximation and the sparse representation of Xt described above, a single eigenvalue computation takes ˜O(nϵ−4.5) time per iteration. Estimating the products Ai ◦Xt on each iteration takes by the discussion above ˜O(mϵ−2). Overall the running time on all iteration is as stated in the lemma. 6 Conclusions We have presented the ﬁrst sublinear time algorithm for approximate semi-deﬁnite programming, a widely used optimization framework in machine learning. The algorithm’s running time is optimal up to poly-logarithmic factors and its dependence on ε - the approximation guarantee. The algorithm is based on the primal-dual approach of [15], and incorporates methods from previous SDP solvers [12]. For the problem of learning peudo-metrics, we have presented further improvements to the basic method which entail an algorithm that performs O( log n ε2 ) iterations, each encompassing at most one approximate eigenvector computation. Acknowledgements This work was supported in part by the IST Programme of the European Community, under the PASCAL2 Network of Excellence, IST-2007-216886. This publication only reﬂects the authors’ views. References [1] Michel. X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisﬁability problems using semideﬁnite programming. In Journal of the ACM, volume 42, pages 1115–1145, 1995. [2] Sanjeev Arora, Satish Rao, and Umesh Vazirani. Expander ﬂows, geometric embeddings and graph partitioning. In Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, STOC ’04, pages 222–231, 2004. [3] Amit Agarwal, Moses Charikar, Konstantin Makarychev, and Yury Makarychev. O(sqrt(log n)) approximation algorithms for min uncut, min 2cnf deletion, and directed cut problems. In Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, STOC ’05, pages 573–581, 2005. [4] Sanjeev Arora, James R. Lee, and Assaf Naor. Euclidean distortion and the sparsest cut. In Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, STOC ’05, pages 553–562, 2005. [5] Eric P. Xing, Andrew Y. Ng, Michael I. Jordan, and Stuart Russell. Distance metric learning, with application to clustering with side-information. In Advances in Neural Information Processing Systems 15, pages 505–512, 2002. 8 [6] Alexandre d’Aspremont, Laurent El Ghaoui, Michael I. Jordan, and Gert R. G. Lanckriet. A direct formulation of sparse PCA using semideﬁnite programming. In SIAM Review, volume 49, pages 41–48, 2004. [7] Gert R. G. Lanckriet, Nello Cristianini, Laurent El Ghaoui, Peter Bartlett, and Michael I. Jordan. 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Sparse approximate solutions to semideﬁnite programs. In Proceedings of the 8th Latin American conference on Theoretical informatics, LATIN’08, pages 306–316, 2008. [13] Garud Iyengar, David J. Phillips, and Clifford Stein. Feasible and accurate algorithms for covering semideﬁnite programs. In SWAT, pages 150–162, 2010. [14] Garud Iyengar, David J. Phillips, and Clifford Stein. Approximating semideﬁnite packing programs. In SIAM Journal on Optimization, volume 21, pages 231–268, 2011. [15] Kenneth L. Clarkson, Elad Hazan, and David P. Woodruff. Sublinear optimization for machine learning. In Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS ’10, pages 449–457, 2010. [16] Elad Hazan. Approximate convex optimization by online game playing. CoRR, abs/cs/0610119, 2006. [17] Kenneth L. Clarkson, Elad Hazan, and David P. Woodruff. Sublinear optimization for machine learning. CoRR, abs/1010.4408, 2010. [18] Shai Shalev-shwartz, Yoram Singer, and Andrew Y. Ng. Online and batch learning of pseudometrics. In ICML, pages 743–750, 2004. [19] James Hardy Wilkinson. The algebric eigenvalue problem. Claderon Press, Oxford, 1965. [20] Gene H. Golub and Charles F. Van Loan. Matrix computations. John Hopkins University Press, 1989. [21] Martin Zinkevich. Online convex programming and generalized inﬁnitesimal gradient ascent. In ICML, pages 928–936, 2003. 9	2011	63
4,412	Active Classiﬁcation based on Value of Classiﬁer Tianshi Gao Department of Electrical Engineering Stanford University Stanford, CA 94305 tianshig@stanford.edu Daphne Koller Department of Computer Science Stanford University Stanford, CA 94305 koller@cs.stanford.edu Abstract Modern classiﬁcation tasks usually involve many class labels and can be informed by a broad range of features. Many of these tasks are tackled by constructing a set of classiﬁers, which are then applied at test time and then pieced together in a ﬁxed procedure determined in advance or at training time. We present an active classiﬁcation process at the test time, where each classiﬁer in a large ensemble is viewed as a potential observation that might inform our classiﬁcation process. Observations are then selected dynamically based on previous observations, using a value-theoretic computation that balances an estimate of the expected classiﬁcation gain from each observation as well as its computational cost. The expected classiﬁcation gain is computed using a probabilistic model that uses the outcome from previous observations. This active classiﬁcation process is applied at test time for each individual test instance, resulting in an efﬁcient instance-speciﬁc decision path. We demonstrate the beneﬁt of the active scheme on various real-world datasets, and show that it can achieve comparable or even higher classiﬁcation accuracy at a fraction of the computational costs of traditional methods. 1 Introduction As the scope of machine learning applications has increased, the complexity of the classiﬁcation tasks that are commonly tackled has grown dramatically. On one dimension, many classiﬁcation problems involve hundreds or even thousands of possible classes [8]. On another dimension, researchers have spent considerable effort developing new feature sets for particular applications, or new types of kernels. For example, in an image labeling task, we have the option of using GIST feature [26], SIFT feature [23], spatial HOG feature [33], Object Bank [21] and more. The beneﬁts of combining information from different types of features can be very signiﬁcant [12, 33]. To solve a complex classiﬁcation problem, many researchers have resorted to ensemble methods, in which multiple classiﬁers are combined to achieve an accurate classiﬁcation decision. For example, the Viola-Jones classiﬁer [32] uses a cascade of classiﬁers, each of which focuses on different spatial and appearance patterns. Boosting [10] constructs a committee of weak classiﬁers, each of which focuses on different input distributions. Multiclass classiﬁcation problems are very often reduced to a set of simpler (often binary) decisions, including one-vs-one [11], one-vs-all, error-correcting output codes [9, 1], or tree-based approaches [27, 13, 3]. Intuitively, different classiﬁers provide different “expertise” in making certain distinctions that can inform the classiﬁcation task. However, as we discuss in Section 2, most of these methods use a ﬁxed procedure determined at training time to apply the classiﬁers without adapting to each individual test instance. In this paper, we take an active and adaptive approach to combine multiple classiﬁers/features at test time, based on the idea of value of information [16, 17, 24, 22]. At training time, we construct a rich family of classiﬁers, which may vary in the features that they use or the set of distinctions that they make (i.e., the subset of classes that they try to distinguish). Each of these classiﬁers is trained on all of the relevant training data. At test time, we dynamically select an instance-speciﬁc 1 subset of classiﬁers. We view each our pre-trained classiﬁer as a possible observation we can make about an instance; each one adds a potential value towards our ability to classify the instance, but also has a cost. Starting from an empty set of observations, at each stage, we use a myopic value-ofinformation computation to select the next classiﬁer to apply to the instance in a way that attempts to increase the accuracy of our classiﬁcation state (e.g., decrease the uncertainty about the class label) at a low computational cost. This process stops when one of the suitable criteria is met (e.g., if we are sufﬁciently conﬁdent about the prediction). We provide an efﬁcient probabilistic method for estimating the uncertainty of the class variable and about the expected gain from each classiﬁer. We show that this approach provides a natural trajectory, in which simple, cheap classiﬁers are applied initially, and used to provide guidance on which of our more expensive classiﬁers is likely to be more informative. In particular, we show that we can get comparable (or even better) performance to a method that uses a large range of expensive classiﬁers, at a fraction of the computational cost. 2 Related Work Our classiﬁcation model is based on multiple classiﬁers, so it resembles ensemble methods like boosting [10], random forests [4] and output-coding based multiclass classiﬁcation [9, 1, 29, 14]. However, these methods use a static decision process, where all classiﬁers have to be evaluated before any decision can be made. Moreover, they often consider a homogeneous set of classiﬁers, but we consider a variety of heterogeneous classiﬁers with different features and function forms. Some existing methods can make classiﬁcation decisions based on partial observations. One example is a cascade of classiﬁers [32, 28], where an instance goes through a chain of classiﬁers and the decision can be made at any point if the classiﬁer response passes some threshold. Another type of method focuses on designing the stopping criteria. Schwing et al. [30] proposed a stopping criterion for random forests such that decisions can be made based on a subset of the trees. However, these methods have a ﬁxed evaluation sequence for any instance, so there is no adaptive selection of which classiﬁers to use based on what we have already observed. Instance-speciﬁc decision paths based on previous observations can be found in decision tree style models, e.g., DAGSVM [27] and tree-based methods [15, 13, 3]. Instead of making hard decisions based on individual observations like these methods, we use a probabilistic model to fuse information from multiple observations and only make decisions when it is sufﬁciently conﬁdent. When observations are associated with different features, our method also performs feature selection. Instead of selecting a ﬁxed set of features in the learning stage [34], we actively select instancespeciﬁc features in the test stage. Furthermore, our method also considers computational properties of the observations. Our selection criterion trades off between the statistical gain and the computational cost of the classiﬁer, resulting in a computationally efﬁcient cheap-to-expensive evaluation process. Similar ideas are hard-coded by Vedaldi et al. [31] without adaptive decisions about when to switch to which classiﬁer with what cost. Angelova et al. [2] performed feature selection to achieve certain accuracy under some computational budget, but the selection is at training time without adaptation to individual test instances. Chai et al. [5] considered test-time feature value acquisition with a strong assumption that observations are conditionally independent given the class variable. Finally, our work is inspired by decision-making under uncertainty based on value of information [16, 17, 24, 22]. For classiﬁcation, Krause and Guestrin [19] used it to compute a conditional plan for asking the expert, trying to optimize classiﬁcation accuracy while requiring as little expert interaction as possible. In machine learning, Cohn et al. [7] used active learning to select training instances to reduce the labeling cost and speedup the learning, while our work focuses on inference. 3 Model We denote the instance and label pair as (X, Y ). Furthermore, we assume that we have been provided a set of trained classiﬁers H, where hi ∈H : X →R can be any real-valued classiﬁers (functions) from existing methods. For example, for multiclass classiﬁcation, hi can be one-vs-all classiﬁers, one-vs-one classiﬁers and weak learners from the boosting algorithms. Note that hi’s do not have to be homogeneous meaning that they can have different function forms, e.g., linear or nonlinear, and more importantly they can be trained on different types of features with various computational costs. Given an instance x, our goal is to infer Y by sequentially selecting one hi to evaluate at a time, based on what has already been observed, until we are sufﬁciently conﬁdent about 2 Y or some other stopping criterion is met, e.g., the computational constraint. The key in this process is how valuable we think a classiﬁer hi is, so we introduce the value of a classiﬁer as follows. Value of Classiﬁer. Let O be the set of classiﬁers that have already been evaluated (empty at the beginning). Denote the random variable Mi = hi(X) as the response/margin of the i-th classiﬁer in H and denote the random vector for the observed classiﬁers as MO = [Mo1, Mo2, . . . , Mo\|O\|]T , where ∀oi ∈O. Given the actual observed values mO of MO, we have a posterior P(Y \|mO) over Y . For now, suppose we are given a reward R : P →R which takes in a distribution P and returns a real value indicating how preferable P is. Furthermore, we use C(hi\|O) to denote the computational cost of evaluating classiﬁer hi conditioned on the set of evaluated classiﬁers O. This is because if hi shares the same feature with some oi ∈O, we do not need to compute the feature again. With some chosen reward R and a computational model C(hi\|O), we deﬁne the value of an unobserved classiﬁer as follows. Deﬁnition 1 The value of classiﬁer V (hi\|mO) for a classiﬁer hi given the observed classiﬁer responses mO is the combination of the expected reward of the state informed by hi and the computational cost of hi. Formally, V (hi\|mO) ∆= Z P(mi\|mO)R(P(Y \|mi, mO))dmi −1 τ C(hi\|O) =Emi∼P (Mi\|mO) R(P(Y \|mi, mO)) −1 τ C(hi\|O) (1) The value of classiﬁer has two parts corresponding to the statistical and computational properties of the classiﬁer respectively. The ﬁrst part VR(hi\|mO) ∆= E R(P(Y \|mi, mO)) is the expected reward of P(Y \|mi, mO), where the expectation is with respect to the posterior of Mi given mO. The second part VC(hi\|mO) ∆= −1 τ C(hi\|O) is a computational penalty incurred by evaluating the classiﬁer hi. The constant τ controls the tradeoff between the reward and the cost. Given the deﬁnition of the value of classiﬁer, at each step of our sequential evaluations, our goal is to pick hi with the highest value: h∗= argmax hi∈H\O V (hi\|mO) = argmax hi∈H\O VR(hi\|mO) + VC(hi\|mO) (2) We introduce the building blocks of the value of classiﬁer, i.e., the reward, the cost and the probabilistic model in the following, and then explain how to compute it. Reward Deﬁnition. We propose two ways to deﬁne the reward R : P →R. Residual Entropy. From the information-theoretical point of view, we want to reduce the uncertainty of the class variable Y by observing classiﬁer responses. Therefore, a natural way to deﬁne the reward is to consider the negative residual entropy, that is the lower the entropy the higher the reward. Formally, given some posterior distribution P(Y \|mO) , we deﬁne R(P(Y \|mO)) = −H(Y \|mO) = X y P(y\|mO) log P(y\|mO) (3) The value of classiﬁer under this reward deﬁnition is closely related to information gain. Speciﬁcally, VR(hi\|mO) =Emi∼P (Mi\|mO) −H(Y \|mi, mO) + H(Y \|mO) −H(Y \|mO) =I(Y ; Mi\|mO) −H(Y \|mO) (4) Since H(Y \|mO) is a constant w.r.t. hi, we have h∗= argmax hi∈H/O VR(hi\|mO) + VC(hi\|mO) = argmax hi∈H/O I(Y ; Mi\|mO) + VC(hi\|mO) (5) Therefore, at each step, we want to pick the classiﬁer with the highest mutual information with the class variable Y given the observed classiﬁer responses mO with a computational constraint. Classiﬁcation Loss. From the classiﬁcation loss point of view, we want to minimize the expected loss when choosing classiﬁers to evaluate. Therefore, given a loss function ∆(y, y′) specifying the 3 penalty of classifying an instance of class y to y′, we can deﬁne the reward as the negative of the minimum expected loss: R(P(Y \|mO)) = −min y′ X y P(y\|mO)∆(y, y′) = −min y′ Ey∼P (Y \|mO) ∆(y, y′) (6) To gain some intuition about this deﬁnition, consider a 0-1 loss function, i.e., ∆(y, y′) = 1{y ̸= y′}, then R(P(Y \|mO)) = −1 + maxy′ P(y′\|mO). To maximize R, we want the peak of P(Y \|mO) to be as high as possible. In our experiment, these two reward deﬁnitions give similar results. Classiﬁcation Cost. The cost of evaluating a classiﬁer h on an instance x can be broken down into two parts. The ﬁrst part is the cost of computing the feature φ : X →Rn on which h is built, and the second is the cost of computing the function value of h given the input φ(x). If h shares the same feature as some evaluated classiﬁers in O, then C(h\|O) only consists of the cost of evaluating the function h, otherwise it will also include the cost of computing the feature input φ. Note that computing φ is usually much more expensive than evaluating the function value of h. Probabilistic Model. Given a test instance x, we construct an instance-speciﬁc joint distribution over Y and the selected observations MO. Our probabilistic model is a mixture model, where each component corresponds to a class Y = y, and we use a uniform prior P(Y ). Starting from an empty O, we model P(Mi, Y ) as a mixture of Gaussian distributions. At each step, given the selected MO, we model the new joint distribution P(Mi, MO, Y ) = P(Mi\|MO, Y )P(MO, Y ) by modeling the new P(Mi\|MO, Y = y) as a linear Gaussian, i.e., P(Mi\|MO, Y = y) = N(θT y MO, σ2 y). As we show in Section 5, this choice of probabilistic model works well empirically. We discuss how to learn the distribution and do inference in the next section. 4 Learning and Inference Learning P(Mi\|mO, y). Given the subset of the training set {(x(j), y(j) = y)}Ny j=1 corresponding to the instances from class y, we denote m(j) i = hi(x(j)), then our goal is to learn P(Mi\|mO, y) from {(m(j), y(j) = y)}Ny j=1. If O = ∅, then P(Mi\|mO, y) reduces to the marginal distribution P(Mi\|y) = N(µy, σ2 y), and based on maximum likelihood estimation, we have µy = 1 Ny P j m(j) i , and σ2 y = 1 Ny P j(m(j) i −µy)2. If O ̸= ∅, we assume that P(Mi\|mO, y) is a linear Gaussian, i.e., µy = θT y mO. Note that we also append a constant 1 to mO as the bias term. Since we know mO at test time, we estimate θy and σ2 y by maximizing the local likelihood with a Gaussian prior on θy. Speciﬁcally, for each training instance j from class y, let wj = e− ∥mO−m(j) O ∥2 β , where β is a bandwidth parameter, then the regularized local log likelihood is L(θy, σy; mO) = −λ ∥θy ∥2 2 + Ny X j=1 wj log N(m(j) i ; θT y m(j) O , σ2 y) (7) where we overload the notation N(x; µy, σ2 y) to mean the value of a Gaussian PDF with mean µy and variance σ2 y evaluated at x. Note that maximizing (7) is equivalent to locally weighted regression [6] with ℓ2 regularization. Maximizing (7) results in: ˆθy = argmin θy λ ∥θy ∥2 2 + Ny X j=1 wj ∥m(j) i −θT y m(j) O ∥2 2= ( ¯MT OW ¯ MO + λI)−1 ¯MT OW ¯ Mi (8) where ¯MO is a matrix whose j-th row is m(j)T O , W is a diagonal matrix whose diagonal entries are wj’s , ¯ Mi is an column vector whose j-th element is m(j) i , and I is an identity matrix. It is worth noting that ( ¯ MT OW ¯ MO + λI)−1W in (8) does not depend on i, so it can be computed once and shared for different classiﬁers hi’s. Finally, the estimated σ2 y is ˆσy 2 = 1 PNy j=1 wj Ny X j=1 wj ∥m(j) i −ˆθy T m(j) O ∥2 (9) 4 Computing V (fi\|mO). Given the learned distribution, we can easily compute the two CPDs in (1), i.e., P(Mi\|mO) and P(Y \|mi, mO). P(Mi\|mO) can be obtained as P(Mi\|mO) = P y P(Mi\|mO, y)P(y\|mO), where P(Y \|mO) is the posterior over Y given some observation mO which is tracked over iterations. Speciﬁcally, P(Y \|mi, mO) ∝P(mi, mO\|Y )P(Y ) = P(mi\|mO, Y )P(mO\|Y )P(Y ), where all terms are available by caching previous computations. Finally, to compute V (fi\|mO), the computational part VC(fi\|mO) is just a lookup in a cost table, and the expected reward part VR(fi\|mO) can be rewritten as: VR(hi\|mO) = X y P(y\|mO)Emi∼P (Mi\|mO,y) R(P(Y \|mi, mO)) (10) Therefore, each component Emi∼P (Mi\|mO,y) R(P(Y \|mi, mO)) is the expectation of a function of a scalar Gaussian variable. We use Gaussian quadrature [18] 1 to approximate each component expectation, and then do the weighted average to get VR(hi\|mO). Dynamic Inference. Given the building blocks introduced before, one can execute the classiﬁcation process in \|H\| steps, where at each step, the values of all the remaining classiﬁers are computed. However, this will incur a large scheduling cost. This is due to the fact that usually \|H\| is large. For example, in multiclass classiﬁcation, if we include all one-vs-one classiﬁers into H, \|H\| is quadratic in the number of classes. Since we are maintaining a belief over Y as observations are accumulated, we can use it to make the inference process more adaptive resulting in small scheduling cost. Early Stopping. Based on the posterior P(Y \|mO), we can make dynamic and adaptive decision about whether to continue observing new classiﬁers or stop the process. We propose two stopping criteria. We stop the inference process whenever either of them is met, and use the posterior over Y at that point to make classiﬁcation decision. The ﬁrst criterion is based on the information-theoretic point of view. Given the current posterior estimation P(Y \|mi, mO) and the previous posterior estimation P(Y \|mO), the relative entropy (KL-divergence) between them is D P(Y \|mO) ∥P(Y \|mi, mO) . We stop the inference procedure when this divergence is below some threshold t. The second criterion is based on the classiﬁcation point of view. We consider the gap between the probability of the current best class and that of the runner-up. Speciﬁcally, we deﬁne the margin given a posterior P(Y \|mO) as δm(P(Y \|mO)) = P(y∗\|mO) −maxy̸=y∗P(Y \|mO), where y∗= argmaxy P(y\|mO). If δm(P(Y \|mO)) ≥t′, then the inference stops. Dynamic Pruning of Class Space. In many cases, a class is mainly confused with a small number of other classes (the confusion matrix is often close to sparse). This implies that after observing a few classiﬁers, the posterior P(Y \|mO) is very likely to be dominated by a few modes leaving the rest with very small probability. For those classes y with very small P(y\|mO), their contributions to the value of classiﬁer (10) are negligible. Therefore, when computing (10), we ignore the components whose P(y\|mO) is below some small threshold (equivalent to setting the contribution from this component to 0). Furthermore, when P(y\|mO) falls below some very small threshold for a class y, we will not estimate the likelihood related to y, i.e., P(Mi\|mO, y), but use a small constant. Dynamic Classiﬁer Space. To avoid computing the values of all the remaining classiﬁers, we can dynamically restrict the search space of classiﬁers to those having high expected mutual information with Y with respect to the current posterior P(Y \|mO). Speciﬁcally, during the training, for each classiﬁer hi we can compute the mutual information I(Mi; By) between its response Mi and a class y, where By is a binary variable indicating whether an instance is from class y or not. Given our current posterior P(Y \|mO), we tried two ways to rank the unobserved classiﬁers. First, we simply select the top L classiﬁers with the highest I(Mi; Bˆy), where ˆy is the most probable class based on current posterior. Since we can sort classiﬁers in the training stage, this step is constant time. Another way is that for each classiﬁer, we can compute a weighted mutual information score, i.e., P y P(y\|mO)I(Mi; By), and we restrict the classiﬁer space to those with the top L scores. Note that computing the scores is very efﬁcient, since it is just an inner product between two vectors, where I(Y ; By)’s have been computed and cached before testing. Our experiments showed that these two scores have similar performances, and we used the ﬁrst method to report the results. Analysis of Time Complexity. At each iteration t, the scheduling overhead includes selecting the top L candidate observations, and for each candidate i, learning P(Mi\|mO, y) and computing 1We found that 3 or 5 points provide an accurate approximation. 5 0 5 10 15 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 number of evaluated classifiers test classification accuracy results on satimage dataset selection by value of classifier random selection one−vs−all dagsvm one−vs−one tree 0 5 10 15 20 25 30 35 40 45 0.4 0.5 0.6 0.7 0.8 0.9 1 number of evaluated classifiers test classification accuracy results on pendigits dataset selection by value of classifier random selection one−vs−all dagsvm one−vs−one tree 0 5 10 15 20 25 30 35 40 45 50 55 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 number of evaluated classifiers test classification accuracy results on vowel dataset selection by value of classifier random selection one−vs−all dagsvm one−vs−one tree 10 0 10 1 10 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 number of evaluated classifiers test classification accuracy results on letter dataset selection by value of classifier random selection one−vs−all dagsvm one−vs−one tree Figure 1: (Best viewed magniﬁed and in colors) Performance comparisons on UCI datasets. From the left to right are the results on satimage, pendigits, vowel and letter (in log-scale) datasets. Note that the error bars for pendigits and letter datasets are very small (around 0.5% on average). V (fi\|mO). First, selecting the top L candidate observations is a constant time, since we can sort the observations based on I(Mi; By) before the test process. Second, estimating P(Mi\|mO, y) requires computing (8) and (9) for different y’s. Given our dynamic pruning of class space, suppose there are only Nt,Y promising classes to consider instead of the total number of classes K. Since ( ¯ MT OW ¯ MO + λI)−1W in (8) does not depend on i, we compute it for each promising class, which takes O(tN 2 y + t2Ny + t3) ﬂoating point operations, and share it for different i’s. After computing this shared component, for each pair of i and a promising class, computing (8) and (9) both take O(tNy). Finally, computing (10) takes O(N 2 t,Y ). Putting everything together, the overall cost at iteration t is O(Nt,Y (tN 2 y + t2Ny + t3) + LNt,Y tNy + LN 2 t,Y ). The key to have a low cost is to effectively prune the class space (small Nt,Y ) and reach a decision quickly (small t). 5 Experimental Results We performed experiments on a collection of four UCI datasets [25] and on a scene recognition dataset [20]. All tasks are multiclass classiﬁcation problems. The ﬁrst set of experiments focuses on a single feature type and aims to show that (i) our probabilistic model is able to combine multiple binary classiﬁers to achieve comparable or higher classiﬁcation accuracy than traditional methods; (ii) our active evaluation strategy successfully selects a signiﬁcantly fewer number of classiﬁers. The second set of experiments considers multiple features, with varying computational complexities. This experiment shows the real power of our active scheme. Speciﬁcally, it dynamically selects an instance-speciﬁc subset of features, resulting in higher classiﬁcation accuracy of using all features but with a signiﬁcant reduction in the computational cost. Basic Setup. Given a feature φ, our set of classiﬁers Hφ consists of all one-vs-one classiﬁers, all one-vs-all classiﬁers, and all node classiﬁers from a tree-based method [13], where a node classiﬁer can be trained to distinguish two arbitrary clusters of classes. Therefore, for a K-class problem, the number of classiﬁers given a single feature is \|Hφ\| = (K−1)K 2 + K + Nφ,tree, where Nφ,tree is the number of nodes in the tree model. If there are multiple features {φi}F i=1, our pool of classiﬁers is H = ∪F i=1Hφi. The form of all classiﬁers is linear SVM for the ﬁrst set of experiments and nonlinear SVM with various kernels for the second set of experiments. During training, in addition to learning the classiﬁers, we also need to compute the response m(j) i of each classiﬁer hi ∈H for each training instance x(j). In order to make the training distribution of the classiﬁer responses better match the test distribution, when evaluating classiﬁer hi on x(j), we do not want hi to be trained on x(j). To achieve this, we use a procedure similar to cross validation. Speciﬁcally, we split the training set into 10 folds, and for each fold, instances from this fold are tested using the classiﬁers trained on the other 9 folds. After this procedure, each training instance x(j) will be evaluated by all hi’s. Note that the classiﬁers used in the test stage are trained on the entire training set. Although for different training instances x(j) and x(k) from different folds and a test instance x, m(j) i , m(k) i and mi are obtained using different hi’s, our experimental results conﬁrmed that their empirical distributions are close enough to achieve good performance. Standard Multiclass Problems from UCI Repository. The ﬁrst set of experiments are done on four standard multiclass problems from the UCI machine learning repository [25]: vowel (speech recognition, 11 classes), letter (optical character recognition, 26 classes), satimage (pixel-based classiﬁcation/segmentation on satellite images, 6 classes) and pendigits (hand written digits recognition, 6 10 classes). We used the same training/test split as speciﬁed in the UCI repository. For each dataset, there is only one type of feature, so it will be computed at the ﬁrst step no matter which classiﬁer is selected. After that, all classiﬁers have the same complexity, so the results will be independent of the τ parameter in the deﬁnition of value of classiﬁer (1). For the baselines, we have one-vs-one with max win, one-vs-all, DAGSVM [27] and a tree-based method [13]. These methods vary both in terms of what set of classiﬁers they use and how those classiﬁers are evaluated and combined. To evaluate the effectiveness of our classiﬁer selection scheme, we introduce another baseline that selects classiﬁers randomly, for which we repeated the experiments for 10 times and the average and one standard deviation are reported. We compare different methods in terms of both the classiﬁcation accuracy and the number of evaluated classiﬁers. For our algorithm and the random selection baseline, we show the accuracy over iterations as well. Since in our framework the number of iterations (classiﬁers) needed varies over instances due to early stopping, the maximum number of iterations shown is deﬁned as the mean plus one standard derivation of the number of classiﬁer evaluations of all test instances. In addition, for the tree-based method, the number of evaluated classiﬁers is the mean over all test instances. Figure 1 shows a set of results. As can be seen, our method can achieve comparable or higher accuracy than traditional methods. In fact, we achieved the best accuracy on three datasets and the gains over the runner-up methods are 0.2%, 5.2%, 8.2% for satimage, vowel, and letter datasets respectively. We think the statistical gain might come from two facts: (i) we are performing instancespeciﬁc “feature selection” to only consider those most informative classiﬁers; (ii) another layer of probabilistic model is used to combine the classiﬁers instead of the uniform voting of classiﬁers used by many traditional methods. In terms of the number of evaluated classiﬁers, our active scheme is very effective: the mean number of classiﬁer evaluations for 6-class, 10-class, 11-class and 26-class problems are 4.50, 3.22, 6.15 and 7.72. Although the tree-based method can also use a few number of classiﬁers, sometimes it suffers from a signiﬁcant drop in accuracy like on the vowel and letter datasets. Furthermore, compared to the random selection scheme, our method can effectively select more informative classiﬁers resulting in faster convergence to a certain classiﬁcation accuracy. The performance gain of our method is not free. To maintain a belief over the class variable Y and to dynamically select classiﬁers with high value, we have introduced additional computational costs, i.e., estimating conditional distributions and computing the value of classiﬁers. For example, this additional cost is around 10ms for satimage, however, evaluating a linear classiﬁer only takes less than 1ms due to very low feature dimension, so the actual running time of the active scheme is higher than one-vs-one. Therefore, our method will have a real computational advantage only if the cost of evaluating the classiﬁers is higher than the cost of our probabilistic inference. We demonstrate such beneﬁt of our method in the context of multiple high dimensional features below. Scene Recognition. We test our active classiﬁcation on a benchmark scene recognition dataset Scene15 [20]. It has 15 scene classes and 4485 images in total. Following the protocol used in [20, 21], 100 images per class are randomly sampled for training and the remaining 2985 for test. model accuracy feature cost classiﬁer scheduling total (# of features) cost cost running time all features 86.40% 52.645s (184) 0.426s 0 53.071s best feature OB [21] 83.38% 6.20s 0.024s 0 6.224s fastest feature GIST [26] 72.70% 0.399s 0.0002s 0 0.3992s ours τ = 25 86.26% 1.718s (5.62) 0.010s 0.141s 1.869s (28.4x) ours τ = 100 86.77% 6.573s (4.71) 0.014s 0.116s 6.703s (7.9x) ours τ = 600 88.11% 19.821s (4.46) 0.031s 0.094s 19.946s (2.7x) Table 1: Detailed performance comparisons on Scene15 dataset with various feature types. For our methods, we show the speedup factors with respective to using all the features in a static way. We consider various types of features, since as shown in [33], the classiﬁcation accuracy can be signiﬁcantly improved by combining multiple features but at a high computational cost. Our feature set includes 7 features from [33], including GIST, spatial HOG, dense SIFT, Local Binary Pattern, self-similarity, texton histogram, geometry speciﬁc histograms (please refer to [33] for details), and another recently proposed high-level image feature Object Bank [21]. The basic idea of Object Bank is to use the responses of various object detectors as the feature. The current release of the code from the authors selected 177 object detectors, each of which outputs a feature vector φi with 7 dimension 252. These individual vectors are concatenated together to form the ﬁnal feature vector Φ = [φ1; φ2; . . . ; φ177] ∈R44,604. Instead of treating Φ as an undecomposable single feature vector, we can think of it as a collection of 177 different features {φi}177 i=1. Therefore, our feature pool consists of 184 features in total. Their computational costs vary from 0.035 to 13.796 seconds, with the accuracy from 54% to 83%. One traditional way to combine these features is through multiple kernel learning. Speciﬁcally, we take the average of individual kernels constructed based on individual features, and train a one-vs-all SVM using the joint average kernel. Surprisingly, this simple average kernel performs comparably with learning the weights to combine them [12]. For our active classiﬁcation, we will not compute all features at the beginning of the evaluation process, but will only compute a component φi when a classiﬁer h based on it is selected. We will cache all evaluated φi’s, so different classiﬁers sharing the same φi will not induce repeated computation of the common φi. We decompose the computational costs per instance into three parts: (1) the feature cost, which is the time spent on computing the features; (2) the classiﬁer cost, which is the time spent on evaluating the function value of the classiﬁers; (3) the scheduling cost, which is the time spent on selecting the classiﬁers using our method. To demonstrate the trade-off between the accuracy and computational cost in the deﬁnition of value of classiﬁer, we run multiple experiments with various τ’s. 0 5 10 15 20 25 30 35 40 45 50 0.65 0.7 0.75 0.8 0.85 0.9 running time (seconds) test classification accuracy results on scene15 dataset sequentially adding features active classification GIST LBP spatial HOG Object Bank dense SIFT Figure 2: Classiﬁcation accuracy versus running time for the baseline, active classiﬁcation, and various individual features. The results are shown in Table 1. We also report comparisons to the best individual features in terms of either accuracy or speed (the reported accuracy is the best of one-vs-one and one-vs-all). As can be seen, combining all features using the traditional method indeed improves the accuracy signiﬁcantly over those individual features, but at an expensive computational cost. However, using active classiﬁcation, to achieve similar accuracy as the baseline of all features, we can get 28.4x speedup (τ = 25). Note that at this conﬁguration, our method is faster than the state-of-the-art individual feature [21], and is also 2.8% better in accuracy. Furthermore, if we put more emphasis on the accuracy, we can get the best accuracy 88.11% when τ = 600. To further test the effectiveness of our active selection scheme, we compare with another baseline that sequentially adds one feature at a time from a ﬁltered pool of features. Speciﬁcally, we ﬁrst rank the individual features based on their classiﬁcation accuracy, and only consider the top 80 features (using 80 features achieves essentially the same accuracy as using 184 features). Given this selected pool, we arrange the features in order of increasing computational complexity, and then train a classiﬁer based on the top N features for all values of N from 1 to 80. As shown in Figure 2, our active scheme is one order of magnitude faster than the baseline given the same level of accuracy. 6 Conclusion and Future Work In this paper, we presented an active classiﬁcation process based on the value of classiﬁer. We applied this active scheme in the context of multiclass classiﬁcation, and achieved comparable and even higher classiﬁcation accuracy with signiﬁcant computational savings compared to traditional static methods. One interesting future direction is to estimate the value of features instead of individual classiﬁers. This is particularly important when computing the feature is much more expensive than evaluating the function value of classiﬁers, which is often the case. Once a feature has been computed, a set of classiﬁers that are built on it will be cheap to evaluate. Therefore, predicting the value of the feature (equivalent to the joint value of multiple classiﬁers sharing the same feature) can potentially lead to more computationally efﬁcient classiﬁcation process. Acknowledgment. This work was supported by the NSF under grant No. RI-0917151, the Ofﬁce of Naval Research MURI grant N00014-10-10933, and the Boeing company. We thank Pawan Kumar and the reviewers for helpful feedbacks. 8 References [1] E. L. Allwein, R. E. Schapire, and Y. Singer. Reducing multiclass to binary: a unifying approach for margin classiﬁers. J. Mach. Learn. Res., 1:113–141, 2001. [2] A. Angelova, L. Matthies, D. Helmick, and P. Perona. 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4,413	Efﬁcient Learning of Generalized Linear and Single Index Models with Isotonic Regression Sham M. Kakade Microsoft Research and Wharton, U Penn skakade@microsoft.com Adam Tauman Kalai Microsoft Research adum@microsoft.com Varun Kanade SEAS, Harvard University vkanade@fas.harvard.edu Ohad Shamir Microsoft Research ohadsh@microsoft.com Abstract Generalized Linear Models (GLMs) and Single Index Models (SIMs) provide powerful generalizations of linear regression, where the target variable is assumed to be a (possibly unknown) 1-dimensional function of a linear predictor. In general, these problems entail non-convex estimation procedures, and, in practice, iterative local search heuristics are often used. Kalai and Sastry (2009) provided the ﬁrst provably efﬁcient method, the Isotron algorithm, for learning SIMs and GLMs, under the assumption that the data is in fact generated under a GLM and under certain monotonicity and Lipschitz (bounded slope) constraints. The Isotron algorithm interleaves steps of perceptron-like updates with isotonic regression (ﬁtting a one-dimensional non-decreasing function). However, to obtain provable performance, the method requires a fresh sample every iteration. In this paper, we provide algorithms for learning GLMs and SIMs, which are both computationally and statistically efﬁcient. We modify the isotonic regression step in Isotron to ﬁt a Lipschitz monotonic function, and also provide an efﬁcient O(n log(n)) algorithm for this step, improving upon the previous O(n2) algorithm. We provide a brief empirical study, demonstrating the feasibility of our algorithms in practice. 1 Introduction The oft used linear regression paradigm models a dependent variable Y as a linear function of a vector-valued independent variable X. Namely, for some vector w, we assume that E[Y \|X] = w·X. Generalized linear models (GLMs) provide a ﬂexible extension of linear regression, by assuming that the dependent variable Y is of the form, E[Y \|X] = u(w · X); u is referred to as the inverse link function or transfer function (see [1] for a review). Generalized linear models include commonly used regression techniques such as logistic regression, where u(z) = 1/(1 + e−z) is the logistic function. The class of perceptrons also falls in this category, where u is a simple piecewise linear function of the form /¯, with the slope of the middle piece being the inverse of the margin. In the case of linear regression, the least-squares method is an highly efﬁcient procedure for parameter estimation. Unfortunately, in the case of GLMs, even in the setting when u is known, the problem of ﬁtting a model that minimizes squared error is typically not convex. We are not aware of any classical estimation procedure for GLMs which is both computationally and statistically efﬁcient, and with provable guarantees. The standard procedure is iteratively reweighted least squares, based on Newton-Raphson (see [1]). The case when both u and w are unknown (sometimes referred to as Single Index Models (SIMs)), involves the more challenging (and practically relevant) question of jointly estimating u and w, 1 where u may come from a large non-parametric family such as all monotonic functions. There are two questions here: 1) What statistical rate is achievable for simultaneous estimation of u and w? 2) Is there a computationally efﬁcient algorithm for this joint estimation? With regards to the former, under mild Lipschitz-continuity restrictions on u, it is possible to characterize the effectiveness of an (appropriately constrained) joint empirical risk minimization procedure. This suggests that, from a purely statistical viewpoint, it may be worthwhile to attempt jointly optimizing u and w on empirical data. However, the issue of computationally efﬁciently estimating both u and w (and still achieving a good statistical rate) is more delicate, and is the focus of this work. We note that this is not a trivial problem: in general, the joint estimation problem is highly non-convex, and despite a signiﬁcant body of literature on the problem, existing methods are usually based on heuristics, which are not guaranteed to converge to a global optimum (see for instance [2, 3, 4, 5, 6]). The Isotron algorithm of Kalai and Sastry [7] provides the ﬁrst provably efﬁcient method for learning GLMs and SIMs, under the common assumption that u is monotonic and Lipschitz, and assuming that the data corresponds to the model.1 The sample and computational complexity of this algorithm is polynomial, and the sample complexity does not explicitly depend on the dimension. The algorithm is a variant of the “gradient-like” perceptron algorithm, where apart from the perceptronlike updates, an isotonic regression procedure is performed on the linear predictions using the Pool Adjacent Violators (PAV) algorithm, on every iteration. While the Isotron algorithm is appealing due to its ease of implementation (it has no parameters other than the number of iterations to run) and theoretical guarantees (it works for any u, w), there is one principal drawback. It is a batch algorithm, but the analysis given requires the algorithm to be run on fresh samples each batch. In fact, as we show in experiments, this is not just an artifact of the analysis – if the algorithm loops over the same data in each update step, it really does overﬁt in very high dimensions (such as when the number of dimensions exceeds the number of examples). Our Contributions: We show that the overﬁtting problem in Isotron stems from the fact that although it uses a slope (Lipschitz) condition as an assumption in the analysis, it does not constrain the output hypothesis to be of this form. To address this issue, we introduce the SLISOTRON algorithm (pronounced slice-o-tron, combining slope and Isotron). The algorithm replaces the isotonic regression step of the Isotron by ﬁnding the best non-decreasing function with a bounded Lipschitz parameter - this constraint plays here a similar role as the margin in classiﬁcation algorithms. We also note SLISOTRON (like Isotron) has a signiﬁcant advantage over standard regression techniques, since it does not require knowing the transfer function. Our two main contributions are: 1. We show that the new algorithm, like Isotron, has theoretical guarantees, and signiﬁcant new analysis is required for this step. 2. We provide an efﬁcient O(n log(n)) time algorithm for ﬁnding the best non-decreasing function with a bounded Lipschitz parameter, improving on the previous O(n2) algorithm [10]. This makes SLISOTRON practical even on large datasets. We begin with a simple perceptron-like algorithm for ﬁtting GLMs, with a known transfer function u which is monotone and Lipschitz. Somewhat surprisingly, prior to this work (and Isotron [7]) a computationally efﬁcient procedure that guarantees to learn GLMs was not known. Section 4 contains the more challenging SLISOTRON algorithm and also the efﬁcient O(n log(n)) algorithm for Lipschitz isotonic regression. We conclude with a brief empirical analysis. 2 Setting We assume the data (x, y) are sampled i.i.d. from a distribution supported on Bd × [0, 1], where Bd = {x ∈Rd : ∥x∥≤1} is the unit ball in d-dimensional Euclidean space. Our algorithms and 1In the more challenging agnostic setting, the data is not required to be distributed according to a true u and w, but it is required to ﬁnd the best u, w which minimize the empirical squared error. Similar to observations of Kalai et al. [8], it is straightforward to show that this problem is likely to be computationally intractable in the agnostic setting. In particular, it is at least as hard as the problem of “learning parity with noise,” whose hardness has been used as the basis for designing multiple cryptographic systems. Shalev-Shwartz et al. [9] present a kernel-based algorithm for learning certain types of GLMs and SIMs in the agnostic setting. However, their worst-case guarantees are exponential in the norm of w (or equivalently the Lipschitz parameter). 2 Algorithm 1 GLM-TRON Input: data ⟨(xi, yi)⟩m i=1 ∈Rd × [0, 1], u : R →[0, 1], held-out data ⟨(xm+j, ym+j)⟩s j=1 w1 := 0; for t = 1, 2, . . . do ht(x) := u(wt · x); wt+1 := wt + 1 m m X i=1 (yi −u(wt · xi))xi; end for Output: arg minht Ps j=1(ht(xm+j) −ym+j)2 analysis also apply to the case where Bd is the unit ball in some high (or inﬁnite)-dimensional kernel feature space. We assume there is a ﬁxed vector w, such that ∥w∥≤W, and a non-decreasing 1-Lipschitz function u : R →[0, 1], such that E[y\|x] = u(w · x) for all x. The restriction that u is 1-Lipschitz is without loss of generality, since the norm of w is arbitrary (an equivalent restriction is that ∥w∥= 1 and that u is W-Lipschitz for an arbitrary W). Our focus is on approximating the regression function well, as measured by the squared loss. For a real valued function h : Bd →[0, 1], deﬁne err(h) = E(x,y) (h(x) −y)2 ε(h) = err(h) −err(E[y\|x]) = E(x,y) (h(x) −u(w · x))2 err(h) measures the error of h, and ε(h) measures the excess error of h compared to the Bayesoptimal predictor x 7→u(w · x). Our goal is to ﬁnd h such that ε(h) (equivalently, err(h)) is as small as possible. In addition, we deﬁne the empirical counterparts c err(h), ˆε(h), based on a sample (x1, y1), . . . , (xm, ym), to be c err(h) = 1 m m X i=1 (h(xi) −yi)2; ˆε(h) = 1 m m X i=1 (h(xi) −u(w · xi))2. Note that ˆε is the standard ﬁxed design error (as this error conditions on the observed x’s). Our algorithms work by iteratively constructing hypotheses ht of the form ht(x) = ut(wt·x), where ut is a non-decreasing, 1-Lipschitz function, and wt is a linear predictor. The algorithmic analysis provides conditions under which ˆε(ht) is small, and using statistical arguments, one can guarantee that ε(ht) would be small as well. 3 The GLM-TRON algorithm We begin with the simpler case, where the transfer function u is assumed to be known (e.g. a sigmoid), and the problem is estimating w properly. We present a simple, parameter-free, perceptronlike algorithm, GLM-TRON (Alg. 1), which efﬁciently ﬁnds a close-to-optimal predictor. We note that the algorithm works for arbitrary non-decreasing, Lipschitz functions u, and thus covers most generalized linear models. We refer the reader to the pseudo-code in Algorithm 1 for some of the notation used in this section. To analyze the performance of the algorithm, we show that if we run the algorithm for sufﬁciently many iterations, one of the predictors ht obtained must be nearly-optimal, compared to the Bayesoptimal predictor. Theorem 1. Suppose (x1, y1), . . . , (xm, ym) are drawn independently from a distribution supported on Bd × [0, 1], such that E[y\|x] = u(w · x), where ∥w∥≤W, and u : R →[0, 1] is a known nondecreasing 1-Lipschitz function. Then for any δ ∈(0, 1), the following holds with probability at least 1 −δ: there exists some iteration t < O(W p m/ log(1/δ)) of GLM-TRON such that the hypothesis ht(x) = u(wt · x) satisﬁes max{ˆε(ht), ε(ht)} ≤O r W 2 log(m/δ) m ! . 3 Algorithm 2 SLISOTRON Input: data ⟨(xi, yi)⟩m i=1 ∈Rd × [0, 1], held-out data ⟨(xm+j, ym+j)⟩s j=1 w1 := 0; for t = 1, 2, . . . do ut := LIR ((wt · x1, y1), . . . , (wt · xm, ym)) // Fit 1-d function along wt wt+1 := wt + 1 m m X i=1 (yi −ut(wt · xi))xi end for Output: arg minht Ps j=1(ht(xm+j) −ym+j)2 In particular, the theorem implies that some ht has small enough ε(ht). Since ε(ht) equals err(ht) up to a constant, we can easily ﬁnd an appropriate ht by picking the one that has least c err(ht) on a held-out set. The main idea of the proof is showing that at each iteration, if ˆε(ht) is not small, then the squared distance wt+1 −w 2 is substantially smaller than ∥wt −w∥2. Since the squared distance is bounded below by 0, and w0 −w 2 ≤W 2, there is an iteration (arrived at within reasonable time) such that the hypothesis ht at that iteration is highly accurate. Although the algorithm minimizes empirical squared error, we can bound the true error using a uniform convergence argument. The complete proofs are provided in the full version of the paper ([11] Appendix A). 4 The SLISOTRON algorithm In this section, we present SLISOTRON (Alg. 2), which is applicable to the harder setting where the transfer function u is unknown, except for it being non-decreasing and 1-Lipschitz. SLISOTRON does have one parameter, the Lipschitz constant; however, in theory we show that this can simply be set to 1. The main difference between SLISOTRON and GLM-TRON is that now the transfer function must also be learned, and the algorithm keeps track of a transfer function ut which changes from iteration to iteration. The algorithm is inspired by the Isotron algorithm [7], with the main difference being that at each iteration, instead of applying the PAV procedure to ﬁt an arbitrary monotonic function along the direction wt, we use a different procedure, (Lipschitz Isotonic Regression) LIR, to ﬁt a Lipschitz monotonic function, ut, along wt. This key difference allows for an analysis that does not require a fresh sample each iteration. We also provide an efﬁcient O(m log(m)) time algorithm for LIR (see Section 4.1), making SLISOTRON an extremely efﬁcient algorithm. We now turn to the formal theorem about our algorithm. The formal guarantees parallel those of the GLM-TRON algorithm. However, the rates achieved are somewhat worse, due to the additional difﬁculty of simultaneously estimating both u and w. Theorem 2. Suppose (x1, y1), . . . , (xm, ym) are drawn independently from a distribution supported on Bd × [0, 1], such that E[y\|x] = u(w · x), where ∥w∥≤W, and u : R →[0, 1] is an unknown non-decreasing 1-Lipschitz function. Then the following two bounds hold: 1. (Dimension-dependent) With probability at least 1 −δ, there exists some iteration t < O W m d log(W m/δ) 1/3 of SLISOTRON such that max{ˆε(ht), ε(ht)} ≤O dW 2 log(Wm/δ) m 1/3! . 2. (Dimension-independent) With probability at least 1 −δ, there exists some iteration t < O W m log(m/δ) 1/4 of SLISOTRON such that max{ˆε(ht), ε(ht)} ≤O W 2 log(m/δ) m 1/4! 4 As in the case of Thm. 1, one can easily ﬁnd ht which satisﬁes the theorem’s conditions, by running the SLISOTRON algorithm for sufﬁciently many iterations, and choosing the hypothesis ht which minimizes c err(ht) on a held-out set. The algorithm minimizes empirical error and generalization bounds are obtained using a uniform convergence argument. The proofs are somewhat involved and appear in the full paper ([11] Appendix B). 4.1 Lipschitz isotonic regression The SLISOTRON algorithm (Alg. 2) performs Lipschitz Isotonic Regression (LIR) at each iteration. The goal is to ﬁnd the best ﬁt (least squared error) non-decreasing 1-Lipschitz function that ﬁts the data in one dimension. Let (z1, y1), . . . (zm, ym) be such that zi ∈R, yi ∈[0, 1] and z1 ≤z2 ≤ · · · ≤zm. The Lipschitz Isotonic Regression (LIR) problem is deﬁned as the following quadratic program: Minimize w.r.t ˆyi :1 2 m X i=1 (yi −ˆyi)2 (1) subject to: ˆyi ≤ˆyi+1 1 ≤i ≤m −1 (Monotonicity) (2) ˆyi+1 −ˆyi ≤(zi+1 −zi) 1 ≤i ≤m −1 (Lipschitz) (3) Once the values ˆyi are obtained at the data points, the actual function can be constructed by interpolating linearly between the data points. Prior to this work, the best known algorithm for this problem wass due to Yeganova and Wilbur [10] and required O(m2) time for m points. In this work, we present an algorithm that performs the task in O(m log(m)) time. The actual algorithm is fairly complex and relies on designing a clever data structure. We provide a high-level view here; the details are provided in the full version ([11] Appendix D). Algorithm Sketch: We deﬁne functions Gi(·), where Gi(s) is the minimum squared loss that can be attained if ˆyi is ﬁxed to be s, and ˆyi+1, . . . ˆym are then chosen to be the best ﬁt 1-Lipschitz non-decreasing function to the points (zi, yi), . . . , (zm, ym). Formally, for i = 1, . . . , m, deﬁne the functions, Gi(s) = min ˆyi+1,...,ˆym 1 2(s −yi)2 + 1 2 m X j=i+1 (ˆyj −yj)2 (4) subject to the constraints (where s = ˆyi), ˆyj ≤ˆyj+1 i ≤j ≤m −1 (Monotonic) ˆyj+1 −ˆyj ≤zj+1 −zj i ≤j ≤m −1 (Lipschitz) Furthermore, deﬁne: s∗ i = mins Gi(s). The functions Gi are piecewise quadratic, differentiable everywhere and strictly convex, a fact we prove in full paper [11]. Thus, Gi is minimized at s∗ i and it is strictly increasing on both sides of s∗ i . Note that Gm(s) = (1/2)(s −ym)2 and hence is piecewise quadratic, differentiable everywhere and strictly convex. Let δi = zi+1 −zi. The remaining Gi obey the following recursive relation. Gi−1(s) = 1 2(s −yi−1)2 + ( Gi(s + δi−1) If s ≤s∗ i −δi−1 Gi(s∗ i ) If s∗ i −δi−1 < s ≤s∗ i Gi(s) If s∗ i < s (5) As intuition for the above relation, note that Gi−1(s) is obtained ﬁxing ˆyi−1 = s and then by choosing ˆyi as close to s∗ i (since Gi is strictly increasing on both sides of s∗ i ) as possible without violating either the monotonicity or Lipschitz constraints. The above argument can be immediately translated into an algorithm, if the values s∗ i are known. Since s∗ 1 minimizes G1(s), which is the same as the objective of (1), start with ˆy1 = s∗ 1, and then successively chose values for ˆyi to be as close to s∗ i as possible without violating the Lipschitz or monotonicity constraints. This will produce an assignment for ˆyi which achieves loss equal to G1(s∗ 1) and hence is optimal. 5 b (a) (b) Figure 1: (a) Finding the zero of G′ i. (b) Update step to transform representation of G′ i to G′ i−1 The harder part of the algorithm is ﬁnding the values s∗ i . Notice that G′ i are all piecewise linear, continuous and strictly increasing, and obey a similar recursive relation (G′ m(s) = s −ym): G′ i−1(s) = (s −yi−1) + ( G′ i(s + δi−1) If s ≤s∗ i −δi−1 0 If s∗ i −δi−1 < s ≤s∗ i G′ i(s) If s∗ i < s (6) The algorithm then ﬁnds s∗ i by ﬁnding zeros of G′ i. Starting from m, G′ m = s −ym, and s∗ m = ym. We design a special data structure, called notable red-black trees, for representing piecewise linear, continuous, strictly increasing functions. We initialize such a tree T to represent G′ m(s) = s −ym. Assuming that at some time it represents G′ i, we need to support two operations: 1. Find the zero of G′ i to get s∗ i . Such an operation can be done efﬁciently O(log(m)) time using a tree-like structure (Fig. 1 (a)). 2. Update T to represent G′ i−1. This operation is more complicated, but using the relation (6), we do the following: Split the interval containing s′ i. Move the left half of the piecewise linear function G′ i by δi−1 (Fig. 1(b)), adding the constant zero function in between. Finally, we add the linear function s −yi−1 to every interval, to get G′ i−1, which is again piecewise linear, continuous and strictly increasing. To perform the operations in step (2) above, we cannot na¨ıvely apply the transformations, shift-by(δi−1) and add(s −yi−1) to every node in the tree, as it may take O(m) operations. Instead, we simply leave a note (hence the name notable red-black trees) that such a transformation should be applied before the function is evaluated at that node or at any of its descendants. To prevent a large number of such notes accumulating at any given node we show that these notes satisfy certain commutative and additive relations, thus requiring us to keep track of no more than 2 notes at any given node. This lazy evaluation of notes allows us to perform all of the above operations in O(log(m)) time. The details of the construction are provided in the full paper ([11] Appendix D). 5 Experiments In this section, we present an empirical study of the SLISOTRON and GLM-TRON algorithms. We perform two evaluations using synthetic data. The ﬁrst one compares SLISOTRON and Isotron [7] and illustrates the importance of imposing a Lipschitz constraint. The second one demonstrates the advantage of using SLISOTRON over standard regression techniques, in the sense that SLISOTRON can learn any monotonic Lipschitz function. We also report results of an evaluation of SLISOTRON, GLM-TRON and several competing approaches on 5 UCI[12] datasets. All errors are reported in terms of average root mean squared error (RMSE) using 10 fold cross validation along with the standard deviation. 5.1 Synthetic Experiments Although, the theoretical guarantees for Isotron are under the assumption that we get a fresh sample each round, one may still attempt to run Isotron on the same sample each iteration and evaluate the 6 −1 −0.8 −0.6 −0.4 −0.2 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 Slisotron Isotron SLISOTRON Isotron ∆ 0.289 ± 0.014 0.334 ± 0.026 0.045 ± 0.018 (a) Synthetic Experiment 1 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Slisotron SLISOTRON Logistic ∆ 0.058 ± 0.003 0.073 ± 0.006 0.015 ± 0.004 (b) Synthetic Experiment 2 Figure 2: (a) The ﬁgure shows the transfer functions as predicted by SLISOTRON and Isotron. The table shows the average RMSE using 10 fold cross validation. The ∆column shows the average difference between the RMSE values of the two algorithms across the folds. (b) The ﬁgure shows the transfer function as predicted by SLISOTRON. Table shows the average RMSE using 10 fold cross validation for SLISOTRON and Logistic Regression. The ∆column shows the average difference between the RMSE values of the two algorithms across folds. empirical performance. Then, the main difference between SLISOTRON and Isotron is that while SLISOTRON ﬁts the best Lipschitz monotonic function using LIR each iteration, Isotron merely ﬁnds the best monotonic ﬁt using PAV. This difference is analogous to ﬁnding a large margin classiﬁer vs. just a consistent one. We believe this difference will be particularly relevant when the data is sparse and lies in a high dimensional space. Our ﬁrst synthetic dataset is the following: The dataset is of size m = 1500 in d = 500 dimensions. The ﬁrst co-ordinate of each point is chosen uniformly at random from {−1, 0, 1}. The remaining co-ordinates are all 0, except that for each data point one of the remaining co-ordinates is randomly set to 1. The true direction is w = (1, 0, . . . , 0) and the transfer function is u(z) = (1 + z)/2. Both SLISOTRON and Isotron put weight on the ﬁrst co-ordinate (the true direction). However, Isotron overﬁts the data using the remaining (irrelevant) co-ordinates, which SLISOTRON is prevented from doing because of the Lipschitz constraint. Figure 2(a) shows the transfer functions as predicted by the two algorithms, and the table below the plot shows the average RMSE using 10 fold cross validation. The ∆column shows the average difference between the RMSE values of the two algorithms across the folds. A principle advantage of SLISOTRON over standard regression techniques is that it is not necessary to know the transfer function in advance. The second synthetic experiment is designed as a sanity check to verify this claim. The dataset is of size m = 1000 in d = 4 dimensions. We chose a random direction as the “true” w and used a piecewise linear function as the “true” u. We then added random noise (σ = 0.1) to the y values. We compared SLISOTRON to Logistic Regression on this dataset. SLISOTRON correctly recovers the true function (up to some scaling). Fig. 2(b) shows the actual transfer function as predicted by SLISOTRON, which is essentially the function we used. The table below the ﬁgure shows the performance comparison between SLISOTRON and logistic regression. 5.2 Real World Datasets We now turn to describe the results of experiments performed on the following 5 UCI datasets: communities, concrete, housing, parkinsons, and wine-quality. We compared the performance of SLISOTRON (Sl-Iso) and GLM-TRON with logistic transfer function (GLM-t) against Isotron (Iso), as well as standard logistic regression (Log-R), linear regression (Lin-R) and a simple heuristic algorithm (SIM) for single index models, along the lines of standard iterative maximum-likelihood procedures for these types of problems (e.g., [13]). The SIM algorithm works by iteratively ﬁxing the direction w and ﬁnding the best transfer function u, and then ﬁxing u and 7 optimizing w via gradient descent. For each of the algorithms we performed 10-fold cross validation, using 1 fold each time as the test set, and we report averaged results across the folds. Table 1 shows average RMSE values of all the algorithms across 10 folds. The ﬁrst column shows the mean Y value (with standard deviation) of the dataset for comparison. Table 2 shows the average difference between RMSE values of SLISOTRON and the other algorithms across the folds. Negative values indicate that the algorithm performed better than SLISOTRON. The results suggest that the performance of SLISOTRON (and even Isotron) is comparable to other regression techniques and in many cases also slightly better. The performance of GLM-TRON is similar to standard implementations of logistic regression on these datasets. This suggests that these algorithms should work well in practice, while providing non-trivial theoretical guarantees. It is also illustrative to see how the transfer functions found by SLISOTRON and Isotron compare. In Figure 3, we plot the transfer functions for concrete and communities. We see that the ﬁts found by SLISOTRON tend to be smoother because of the Lipschitz constraint. We also observe that concrete is the only dataset where SLISOTRON performs noticeably better than logistic regression, and the transfer function is indeed somewhat far from the logistic function. Table 1: Average RMSE values using 10 fold cross validation. The ¯Y column shows the mean Y value and standard deviation. dataset ¯Y Sl-Iso GLM-t Iso Lin-R Log-R SIM communities 0.24 ± 0.23 0.13 ± 0.01 0.14 ± 0.01 0.14 ± 0.01 0.14 ± 0.01 0.14 ± 0.01 0.14 ± 0.01 concrete 35.8 ± 16.7 9.9 ± 0.9 10.5 ± 1.0 9.9 ± 0.8 10.4 ± 1.1 10.4 ± 1.0 9.9 ± 0.9 housing 22.5 ± 9.2 4.65 ± 1.00 4.85 ± 0.95 4.68 ± 0.98 4.81 ± 0.99 4.70 ± 0.98 4.63 ± 0.78 parkinsons 29 ± 10.7 10.1 ± 0.2 10.3 ± 0.2 10.1 ± 0.2 10.2 ± 0.2 10.2 ± 0.2 10.3 ± 0.2 winequality 5.9 ± 0.9 0.78 ± 0.04 0.79 ± 0.04 0.78 ± 0.04 0.75 ± 0.04 0.75 ± 0.04 0.78 ± 0.03 Table 2: Performance comparison of SLISOTRON with the other algorithms. The values reported are the average difference between RMSE values of the algorithm and SLISOTRON across the folds. Negative values indicate better performance than SLISOTRON. dataset GLM-t Iso Lin-R Log-R SIM communities 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 concrete 0.56 ± 0.35 0.04 ± 0.17 0.52 ± 0.35 0.55 ± 0.32 -0.03 ± 0.26 housing 0.20 ± 0.48 0.03 ± 0.55 0.16 ± 0.49 0.05 ± 0.43 -0.02 ± 0.53 parkinsons 0.19 ± 0.09 0.01 ± 0.03 0.11 ± 0.07 0.09 ± 0.07 0.21 ± 0.20 winequality 0.01 ± 0.01 0.00 ± 0.00 -0.03 ± 0.02 -0.03 ± 0.02 0.01 ± 0.01 −1 −0.8 −0.6 −0.4 −0.2 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 Slisotron Isotron −1 −0.8 −0.6 −0.4 −0.2 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 Slisotron Isotron (a) concrete (b) communities Figure 3: The transfer function u as predicted by SLISOTRON (blue) and Isotron (red) for the concrete and communities datasets. The domain of both functions was normalized to [−1, 1]. 8 References [1] P. McCullagh and J. A. Nelder. Generalized Linear Models (2nd ed.). Chapman and Hall, 1989. [2] P. Hall W. 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4,414	Co-regularized Multi-view Spectral Clustering Abhishek Kumar∗ Dept. of Computer Science University of Maryland, College Park, MD abhishek@cs.umd.edu Piyush Rai∗ Dept. of Computer Science University of Utah, Salt Lake City, UT piyush@cs.utah.edu Hal Daum´e III Dept. of Computer Science University of Maryland, College Park, MD hal@umiacs.umd.edu Abstract In many clustering problems, we have access to multiple views of the data each of which could be individually used for clustering. Exploiting information from multiple views, one can hope to ﬁnd a clustering that is more accurate than the ones obtained using the individual views. Often these different views admit same underlying clustering of the data, so we can approach this problem by looking for clusterings that are consistent across the views, i.e., corresponding data points in each view should have same cluster membership. We propose a spectral clustering framework that achieves this goal by co-regularizing the clustering hypotheses, and propose two co-regularization schemes to accomplish this. Experimental comparisons with a number of baselines on two synthetic and three real-world datasets establish the efﬁcacy of our proposed approaches. 1 Introduction Many real-world datasets have representations in the form of multiple views [1, 2]. For example, webpages usually consist of both the page-text and hyperlink information; images on the web have captions associated with them; in multi-lingual information retrieval, the same document has multiple representations in different languages, and so on. Although these individual views might be sufﬁcient on their own for a given learning task, they can often provide complementary information to each other which can lead to improved performance on the learning task at hand. In the context of data clustering, we seek a partition of the data based on some similarity measure between the examples. Our of the numerous clustering algorithms, Spectral Clustering has gained considerable attention in the recent past due to its strong performance on arbitrary shaped clusters, and due to its well-deﬁned mathematical framework [3]. Spectral clustering is accomplished by constructing a graph from the data points with edges between them representing the similarities, and solving a relaxation of the normalized min-cut problem on this graph [4]. For the multi-view clustering problem, we work with the assumption that the true underlying clustering would assign corresponding points in each view to the same cluster. Given this assumption, we can approach the multi-view clustering problem by limiting our search to clusterings that are compatible across the graphs deﬁned over each of the views: corresponding nodes in each graph should have the same cluster membership. In this paper, we propose two spectral clustering algorithms that achieve this goal by co-regularizing the clustering hypotheses across views. Co-regularization is a well-known technique in semisupervised literature; however, not much is known on using it for unsupervised learning problems. We propose novel spectral clustering objective functions that implicitly combine graphs from multiple views of the data to achieve a better clustering. Our proposed methods give us a way to combine multiple kernels (or similarity matrices) for the clustering problem. Moreover, we would like to note here that although multiple kernel learning has met with considerable success on supervised learning problems, similar investigations for unsupervised learning have been found lacking so far, which is one of the motivations behind this work. ∗Authors contributed equally 1 2 Co-regularized Spectral Clustering We assume that we are given data having multiple representations (i.e., views). Let X = {x(v) 1 , x(v) 2 , . . . , x(v) n } denote the examples in view v and K(v) denote the similarity or kernel matrix of X in this view. We write the normalized graph Laplacian for this view as: L(v) = D(v)−1/2K(v)D(v)−1/2. The single view spectral clustering algorithm of [5] solves the following optimization problem for the normalized graph Laplacian L(v): max U(v)∈Rn×k tr U(v)T L(v)U(v) , s.t. U(v)T U(v) = I (1) where tr denotes the matrix trace. The rows of matrix U(v) are the embeddings of the data points that can be given to the k-means algorithm to obtain cluster memberships. For a detailed introduction to both theoretical and practical aspects of spectral clustering, the reader is referred to [3]. Our multi-view spectral clustering framework builds on the standard spectral clustering with a single view, by appealing to the co-regularization framework typically used in the semi-supervised learning literature [1]. Co-regularization in semi-supervised learning essentially works by making the hypotheses learned from different views of the data agree with each other on unlabeled data [6]. The framework employs two main assumptions for its success: (a) the true target functions in each view should agree on the labels for the unlabeled data (compatibility), and (b) the views are independent given the class label (conditional independence). The compatibility assumption allows us to shrink the space of possible target hypotheses by searching only over the compatible functions. Standard PAC-style analysis [1] shows that this also leads to reductions in the number of examples needed to learn the target function, since this number depends on the size of the hypothesis class. The independence assumption makes it unlikely for compatible classiﬁers to agree on wrong labels. In the case of clustering, this would mean that a data point in both views would be assigned to the correct cluster with high probability. Here, we propose two co-regularization based approaches to make the clustering hypotheses on different graphs (i.e., views) agree with each other. The effectiveness of spectral clustering hinges crucially on the construction of the graph Laplacian and the resulting eigenvectors that reﬂect the cluster structure in the data. Therefore, we construct an objective function that consists of the graph Laplacians from all the views of the data and regularize on the eigenvectors of the Laplacians such that the cluster structures resulting from each Laplacian look consistent across all the views. Our ﬁrst co-regularization scheme (Section 2.1) enforces that the eigenvectors U(v) and U(w)of a view pair (v, w) should have high pairwise similarity (using a pair-wise co-regularization criteria we will deﬁne in Section 2.1). Our second co-regularization scheme (Section 2.3) enforces the view-speciﬁc eigenvectors to look similar by regularizing them towards a common consensus (centroid based co-regularization). The idea is different from previously proposed consensus clustering approaches [7] that commit to individual clusterings in the ﬁrst step and then combine them to a consensus in the second step. We optimize for individual clusterings as well as the consensus using a joint cost function. 2.1 Pairwise Co-regularization In standard spectral clustering, the eigenvector matrix U(v) is the data representation for subsequent k-means clustering step (with i’th row mapping to the original i’th sample). In our proposed objective function, we encourage the pairwise similarities of examples under the new representation (in terms of rows of U(·)’s) to be similar across all the views. This amounts to enforcing the spectral clustering hypotheses (which are based on the U(·)’s) to be the same across all the views. We will work with two-view case for the ease of exposition. This will later be extended to more than two views. We propose the following cost function as a measure of disagreement between clusterings of two views: D(U(v), U(w)) = KU(v) \|\|KU(v)\|\|2 F − KU(w) \|\|KU(w)\|\|2 F 2 F . (2) KU(v) is the similarity matrix for U(v), and \|\| · \|\|F denotes the Frobenius norm of the matrix. The similarity matrices are normalized by their Frobenius norms to make them comparable across 2 views. We choose linear kernel, i.e., k(xi, xj) = xT i xj as our similarity measure in Equation 2. This implies that we have KU(v) = U(v)U(v)T . The reason for choosing linear kernel to measure similarity of U(·) is twofold. First, the similarity measure (or kernel) used in the Laplacian for spectral clustering has already taken care of the non-linearities present in the data (if any), and the embedding U(·) being real-valued cluster indicators, can be considered to obey linear similarities. Secondly, we get a nice optimization problem by using linear kernel for U(·). We also note that \|\|KU(v)\|\|2 F = k, where k is the number of clusters. Substituting this in Equation 2 and ignoring the constant additive and scaling terms that depend on the number of clusters, we get D(U(v), U(w)) = −tr U(v)U(v)T U(w)U(w)T We want to minimize the above disagreement between the clusterings of views v and w. Combining this with the spectral clustering objectives of individual views, we get the following joint maximization problem for two graphs: max U(v)∈Rn×k U(w)∈Rn×k tr U(v)T L(v)U(v) + tr U(w)T L(w)U(w) + λ tr U(v)U(v)T U(w)U(w)T s.t. U(v)T U(v) = I, U(w)T U(w) = I (3) The hyperparameter λ trades-off the spectral clustering objectives and the spectral embedding (dis)agreement term. The joint optimization problem given by Equation 3 can be solved using alternating maximization w.r.t. U(v) and U(w). For a given U(w), we get the following optimization problem in U(v): max U(v)∈Rn×k tr n U(v)T L(v) + λU(w)U(w)T U(v)o , s.t. U(v)T U(v) = I. (4) This is a standard spectral clustering objective on view v with graph Laplacian L(v) +λU(w)U(w)T . This can be seen as a way of combining kernels or Laplacians. The difference from standard kernel combination (kernel addition, for example) is that the combination is adaptive since U(w) keeps getting updated at each step, as guided by the clustering algorithm. The solution U(v) is given by the top-k eigenvectors of this modiﬁed Laplacian. Since the alternating maximization can make the algorithm stuck in a local maximum [8], it is important to have a sensible initialization. If there is no prior information on which view is more informative about the clustering, we can start with any of the views. However, if we have some a priori knowledge on this, we can start with the graph Laplacian L(w) of the more informative view and initialize U(w). The alternating maximization is carried out after this until convergence. Note that one possibility could be to regularize directly on the eigenvectors U(v)’s and make them close to each other (e.g., in the sense of the Frobenious norm of the difference between U(v) and U(w)). However, this type of regularization could be too restrictive and could end up shrinking the hypothesis space of feasible clusterings too much, thus ruling out many valid clusterings. For ﬁxed λ and n, the joint objective of Eq. 3 can be shown to be bounded from above by a constant. Since the objective is non-decreasing with the iterations, the algorithm is guaranteed to converge. In practice, we monitor the convergence by the difference in the value of the objective between consecutive iterations, and stop when the difference falls below a minimum threshold of ǫ = 10−4. In all our experiments, we converge within less than 10 iterations. Note that we can use either U(v) or U(w) in the ﬁnal k-means step of the spectral clustering algorithm. In our experiments, we note a marginal difference in the clustering performance depending on which U(·) is used in the ﬁnal step of k-means clustering. 2.2 Extension to Multiple Views We can extend the co-regularized spectral clustering proposed in the previous section for more than two views. This can be done by employing pair-wise co-regularizers in the objective function of Eq. 3. For m number of views, we have max U(1),U(2),...,U(m)∈Rn×k m X v=1 tr U(v)T L(v)U(v) + λ X 1≤v,w≤m v̸=w tr U(v)U(v)T U(w)U(w)T , s.t. U(v)T U(v) = I, ∀1 ≤v ≤V (5) 3 We use a common λ for all pair-wise co-regularizers for simplicity of exposition, however different λ’s can be used for different pairs of views. Similar to the two-view case, we can optimize it by alternating maximization cycling over the views. With all but one U(v) ﬁxed, we have the following optimization problem: max U(v) tr n U(v)T L(v) + λ X 1≤w≤m, w̸=v U(w)U(w)T U(v)o , s.t. U(v)T U(v) = I (6) We initialize all U(v), 2 ≤v ≤m by solving the spectral clustering problem for single views. We solve the objective of Eq. 6 for U(1) given all other U(v), 2 ≤v ≤m. The optimization is then cycled over all views while keeping the previously obtained U(·)’s ﬁxed. 2.3 Centroid-Based Co-regularization In this section, we present an alternative regularization scheme that regularizes each view-speciﬁc set of eigenvectors U(v) towards a common centroid U∗(akin to a consensus set of eigenvectors) . In contrast with the pairwise regularization approach which has m 2 pairwise regularization terms, where m is the number of views, the centroid based regularization scheme has m pairwise regularization terms. The objective function can be written as: max U(1),U(2),...,U(m),U∗∈Rn×k m X v=1 tr U(v)T L(v)U(v) + X v λvtr U(v)U(v)T U∗U∗T , s.t. U(v)T U(v) = I, ∀1 ≤v ≤V, U∗T U∗= I (7) This objective tries to balance a trade-off between the individual spectral clustering objectives and the agreement of each of the view-speciﬁc eigenvectors U(v) with the consensus eigenvectors U∗. Each regularization term is weighted by a parameter λv speciﬁc to that view, where λv can be set to reﬂect the importance of view v. Just like for Equation 6, the objective in Equation 7 can be solved in an alternating fashion optimizing each of the U(v)’s one at a time, keeping all other variables ﬁxed, followed by optimizing the consensus U∗, keeping all the U(v)’s ﬁxed. It is easy to see that with all other view-speciﬁc eigenvectors and the consensus U∗ﬁxed, optimizing U(v) for view v amounts to solving the following: max U(v)∈Rn×k tr U(v)T L(v)U(v) + λvtr U(v)U(v)T U∗U∗T , s.t. U(v)T U(v) = I (8) which is nothing but equivalent to solving the standard spectral clustering objective for U(v) with a modiﬁed Laplacian L(v) + λvU∗U∗T . Solving for the consensus U∗requires solving the following objective: max U∗∈Rn×k X v λvtr U(v)U(v)T U∗U∗T , s.t. U∗T U∗= I (9) Using the circular property of matrix traces, Equation 9 can be rewritten as: max U∗∈Rn×k tr ( U∗T X v λv U(v)U(v)T ! U∗ ) , s.t. U∗T U∗= I (10) which is equivalent to solving the standard spectral clustering objective for U∗with a modiﬁed Laplacian P v λv U(v)U(v)T . In contrast with the pairwise co-regularization approach of Section 2.1 which computes optimal view speciﬁc eigenvectors U(v)’s, which ﬁnally need to be combined (e.g., via column-wise concatenation) before running the k-means step, the centroid-based co-regularization approach directly ﬁnds an optimal U∗to be used in the k-means step. One possible downside of the centroid-based co-regularization approach is that noisy views could potentially affect the optimal U∗as it depends on all the views. To deal with this, careful selection of the weighing parameter λv is required. If it is a priori known that some views are noisy, then it is advisable to use a small value of λv for such views, so as to prevent them from adversely affecting U∗. 4 3 Experiments We compare both of our co-regularization based multi-view spectral clustering approaches with a number of baselines. In particular, we compare with: • Single View: Using the most informative view, i.e., one that achieves the best spectral clustering performance using a single view of the data. • Feature Concatenation: Concatenating the features of each view, and then running standard spectral clustering using the graph Laplacian derived from the joint view representation of the data. • Kernel Addition: Combining different kernels by adding them, and then running standard spectral clustering on the corresponding Laplacian. As suggested in earlier ﬁndings [9], even this seemingly simple approach often leads to near optimal results as compared to more sophisticated approaches for classiﬁcation. It can be noted that kernel addition reduces to feature concatenation for the special case of linear kernel. In general, kernel addition is same as concatenation of features in the Reproducing Kernel Hilbert Space. • Kernel Product (element-wise): Multiplying the corresponding entries of kernels and applying standard spectral clustering on the resultant Laplacian. For the special case of Gaussian kernel, element-wise kernel product would be same as simple feature concatenation if both kernels use same width parameter σ. However, in our experiments, we use different width parameters for different views so the performances of kernel product may not be directly comparable to feature concatenation. • CCA based Feature Extraction: Applying CCA for feature fusion from multiple views of the data [10], and then running spectral clustering using these extracted features. We apply both standard CCA and kernel CCA for feature extraction and report the clustering results for whichever gives the best performance. • Minimizing-Disagreement Spectral Clustering: Our last baseline is the minimizingdisagreement approach to spectral clustering [11], and is perhaps most closely related to our coregularization based approach to spectral clustering. This algorithm is discussed more in Sec. 4. To distinguish between the results of our two co-regularization based approaches, in the tables containing the results, we use symbol “P” to denote the pairwise co-regularization method and symbol “C” to denote the centroid based co-regularization method. For datasets with more than 2 views, we have also explicitly mentioned the number of views in parentheses. We report experimental results on two synthetic and three real-world datasets. We give a brief description of each dataset here. • Synthetic data 1: Our ﬁrst synthetic dataset consists of two views and is generated in a manner akin to [12] which ﬁrst chooses the cluster ci each sample belongs to, and then generates each of the views x(1) i and x(2) i from a two-component Gaussian mixture model. These views are combined to form the sample (x(1) i , x(2) i , ci). We sample 1000 points from each view. The cluster means in view 1 are µ(1) 1 = (1 1) , µ(1) 2 = (2 2), and in view 2 are µ(2) 1 = (2 2) , µ(2) 2 = (1 1). The covariances for the two views are given below. Σ(1) 1 = 1 0.5 0.5 1.5 , Σ(2) 1 = 0.3 0 0 0.6 , Σ(1) 2 = 0.3 0 0 0.6 , Σ(2) 2 = 1 0.5 0.5 1.5 • Synthetic data 2: Our second synthetic dataset consists of three views. Moreover, the features are correlated. Each view still has two clusters. Each view is generated by a two component Gaussian mixture model. The cluster means in view 1 are µ(1) 1 = (1 1) , µ(1) 2 = (3 4); in view 2 are µ(2) 1 = (1 2) , µ(2) 2 = (2 2); and in view 3 are µ(3) 1 = (1 1) , µ(3) 2 = (3 3). The covariances for the three views are given below. The notation Σ(v) c denotes the parameter for c’th cluster in v’th view. Σ(1) 1 = 1 0.5 0.5 1.5 , Σ(2) 1 = 1 −0.2 −0.2 1 , Σ(3) 1 = 1.2 0.2 0.2 1 Σ(1) 2 = 0.3 0.2 0.2 0.6 , Σ(2) 2 = 0.6 0.1 0.1 0.5 , Σ(3) 2 = 1 0.4 0.4 0.7 5 • Reuters Multilingual data: The test collection contains feature characteristics of documents originally written in ﬁve different languages (English, French, German, Spanish and Italian), and their translations, over a common set of 6 categories [13]. This corpus is built by sampling parts of the Reuters RCV1 and RCV2 collections [14, 15]. We use documents originally in English as the ﬁrst view and their French translations as the second view. We randomly sample 1200 documents from this collection in a balanced manner, with each of the 6 clusters having 200 documents. The documents are in bag-of-words representation which implies that the features are extremely sparse and high-dimensional. The standard similarity measures (like Gaussian kernel) in very high dimensions are often unreliable. Since spectral clustering essentially works with similarities of the data, we ﬁrst project the data using Latent Semantic Analysis (LSA) [16] to a 100-dimensional space and compute similarities in this lower dimensional space. This is akin to a computing topic based similarity of documents [17]. • UCI Handwritten digits data: Our second real-world dataset is taken from the handwritten digits (0-9) data from the UCI repository. The dataset consists of 2000 examples, with view-1 being the 76 Fourier coefﬁcients, and view-2 being the 216 proﬁle correlations of each example image. • Caltech-101 data: Our third real-world dataset is a subset of the Caltech-101 data from the Multiple Kernel Learning repository from which we chose 450 examples having 30 underlying clusters. We experiment with 4 kernels from this dataset. In particular, we chose the “pixel features”, the “Pyramid Histogram Of Gradients”, bio-inspired “Sparse Localized Features”, and SIFT descriptors as our four views. We report results on our co-regularized spectral clustering for two, three and four views cases. We use normalized mutual information (NMI) as the clustering quality evaluation measure, which gives the mutual information between obtained clustering and the true clustering normalized by the cluster entropies. NMI ranges between 0 and 1 with higher value indicating closer match to the true clustering. We use Gaussian kernel for computing the graph similarities in all the experiments, unless mentioned otherwise. The standard deviation of the kernel is taken equal to the median of the pair-wise Euclidean distances between the data points. In our experiments, the co-regularization parameter λ is varied from 0.01 to 0.05 and the best result is reported (we keep λ the same for all views; one can however also choose different λ’s based on the importance of individual views). We experiment with λ values more exhaustively later in this Section where we show that our approach outperforms other baselines for a wide range of λ. In the results table, the numbers in the parentheses are the standard deviations of the performance measures obtained with 20 different runs of k-means with random initializations. 3.1 Results The results for all datasets are shown in Table 1. For two-view synthetic data (Synthetic Data 1), both the co-regularized spectral clustering approaches outperform all the baselines by a signiﬁcant margin, with the pairwise approach doing marginally better than the centroid-based approach. The closest performing approaches are kernel addition and CCA. For synthetic data, order-2 polynomial kernel based kernel-CCA gives best performance among all CCA variants, while Gaussian kernel based kernel-CCA performs poorly. We do not report results for Gaussian kernel CCA here. All the multi-view baselines outperform the single view case for the synthetic data. For three-view synthetic data (Synthetic Data 2), we can see that simple feature concatenation does not help much. In fact, it reduces the performance when the third view is added, so we report the performance with only two views for feature concatenation. Kernel addition with three views gives a good improvement over single view case. As compared to other baselines (with two views), both our co-regularized spectral clustering approaches with two views perform better. For both approaches, addition of third view also results in improving the performance beyond the two view case. For the document clustering results on Reuters multilingual data, English and French languages are used as the two views. On this dataset too, both our approaches outperform all the baselines by a signiﬁcant margin. The next best performance is attained by minimum-disagreement spectral clustering [11] approach. It should be noted that CCA and element-wise kernel product performances are worse than that of single view. For UCI Handwritten digits dataset, quite a few approaches including kernel addition, element-wise kernel multiplication, and minimum-disagreement are close to both of our co-regularized spectral 6 Method Synth data 1 Synth data 2 Reuters Handwritten Caltech Best Single View 0.267 (0.0) 0.898 (0.0) 0.287 (0.019) 0.641 (0.008) 0.510 (0.008) Feature Concat 0.294 (0.0) 0.923 (0.0) 0.298 (0.020) 0.619 (0.015) – Kernel Addition 0.339 (0.0) 0.973 (0.0) 0.323 (0.021) 0.744 (0.030) 0.383 (0.008) Kernel Product 0.277 (0.0) 0.959 (0.0) 0.123 (0.010) 0.754 (0.026) 0.429 (0.007) CCA 0.330 (0.0) 0.932 (0.0) 0.147 (0.003) 0.682 (0.019) 0.466 (0.007) Min-Disagreement 0.313 (0.0) 0.936 (0.0) 0.342 (0.024) 0.745 (0.024) 0.389 (0.008) Co-regularized (P) (2) 0.378 (0.0) 0.981 (0.0) 0.375 (0.002) 0.759 (0.031) 0.527 (0.007) Co-regularized (P) (3) – 0.989 (0.0) – – 0.533 (0.008) Co-regularized (P) (4) – – – – 0.564 (0.007) Co-regularized (C) (2) 0.367 (0.0) 0.955 (0.0) 0.360 (0.025) 0.768 (0.025) 0.522 (0.004) Co-regularized (C) (3) – 0.989 (0.0) – – 0.512 (0.007) Co-regularized (C) (4) – – – – 0.561 (0.005) Table 1: NMI results on various datasets for different baselines and the proposed approaches. Numbers in parentheses are the std. deviations. The numbers (2), (3) and (4) indicate the number of views used in our co-regularized spectral clustering approach. Other multi-view baselines were run with maximum number of views available (or maximum number of views they can handle). Letters (P) and (C) indicate pairwise and centroid based regularizations respectively. clustering approaches. It can be also be noted that feature concatenation actually performs worse than single view on this dataset. For Caltech-101 data, we cannot do feature concatenation since only kernels are available. Surprisingly, on this dataset, all the baselines perform worse than the single view case. On the other hand, both of our co-regularized spectral clustering approaches with two views outperform the single view case. As we added more views that were available for the Caltech-101 datasets, we found that the performance of the pairwise approach consistently went up as we added the third and the fourth view. On the other hand, the performance of the centroid-based approach slightly got worse upon adding the third view (possibly due to the view being noisy which affected the learned U∗); however addition of the fourth view brought the performance almost close to that of the pairwise case. 0 0.02 0.04 0.06 0.08 0.1 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 Co−regularization parameter λ NMI Score Co−regularization approach Closest performing baseline (a) 0 0.02 0.04 0.06 0.08 0.1 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 Co−regularization Parameter λ NMI Score Co−regularization approach Closed performing baseline (b) Figure 1: NMI scores of Co-regularized Spectral Clustering as a function of λ for (a) Reuters multilingual data and (b) Caltech-101 data We also experiment with various values of co-regularization parameter λ and observe its effect on the clustering performance. Our reported results are for the pairwise co-regularization approach. Similar trends were observed for the centroid-based co-regularization approach and therefore we do not report them here. Fig. 1(a) shows the plot for Reuters multilingual data. The NMI score shoots up right after λ starts increasing from 0 and reaches a peak at λ = 0.01. After reaching a second peak at about 0.025, it starts decreasing and hovers around the second best baseline (Minimizingdisagreement in this case) for a while. The NMI becomes worse than the second best baseline after λ = 0.075. The plot for Caltech-101 data is shown in Fig. 1(b). The normalized mutual information (NMI) starts increasing as the value of lambda is increased away from 0, and reaches a peak at λ = 0.01. It starts to decrease after that with local ups and downs. For the range of λ shown in the plot, the NMI for co-regularized spectral clustering is greater than the closest baseline for most of 7 the λ values. These results indicate that although the performance of our algorithms depends on the weighing parameter λ, it is reasonably stable across a wide range of λ. 4 Related Work A number of clustering algorithms have been proposed in the past to learn with multiple views of the data. Some of them ﬁrst extract a set of shared features from the multiple views and then apply any off-the-shelf clustering algorithm such as k-means on these features. The Canonical Correlation Analysis (CCA) [2, 10] based approach is an example of this. Alternatively, some other approaches exploit the multiple views of the data as part of the clustering algorithm itself. For example, [19] proposed an Co-EM based framework for multi-view clustering in mixture models. Co-EM approach computes expected values of hidden variables in one view and uses these in the M-step for other view, and vice versa. This process is repeated until a suitable stopping criteria is met. The algorithm often does not converge. Multi-view clustering algorithms have also been proposed in the framework of spectral clustering [11, 20, 21]. In [20], the authors obtain a graph cut which is good on average over the multiple graphs but may not be the best for a single graph. They give a random walk based formulation for the problem. [11] approaches the problem of two-view clustering by constructing a bipartite graph from nodes of both views. Edges of the bipartite graph connect nodes from one view to those in the other view. Subsequently, they solve standard spectral clustering problem on this bipartite graph. In [21], a co-training based framework is proposed where the similarity matrix of one view is constrained by the eigenvectors of the Laplacian in the other view. In [22], the information from multiple graphs are fused using Linked Matrix Factorization. Consensus clustering approaches can also be applied to the problem of multi-view clustering [7]. These approaches do not generally work with original features. Instead, they take different clusterings of a dataset coming from different sources as input and reconcile them to ﬁnd a ﬁnal clustering. 5 Discussion We proposed a multi-view clustering approach in the framework of spectral clustering. The approach uses the philosophy of co-regularization to make the clusterings in different views agree with each other. Co-regularization idea has been used in the past for semi-supervised learning problems. To the best of our knowledge, this is the ﬁrst work to apply the idea to the problem of unsupervised learning, in particular to spectral clustering. The co-regularized spectral clustering has a joint optimization function for spectral embeddings of all the views. An alternating maximization framework reduces the problem to the standard spectral clustering objective which is efﬁciently solvable using state-ofthe-art eigensolvers. It is possible to extend the proposed framework to the case where some of the views have missing data. For missing data points, the corresponding entries in the similarity matrices would be unavailable. We can estimate these missing similarities by the corresponding similarities in other views. One possible approach to estimate the missing entry could be to simply average the similarities from views in which the data point is available. Proper normalization of similarities (possibly by Frobenius norm of the whole matrix) might be needed before averaging to make them comparable. Other methods for missing kernel entries estimation can also be used. It is also possible to assign weights to different views in the proposed objective function as done in [20], if we have some a priori knowledge about the informativeness of the views. Our co-regularization based framework can also be applied to other unsupervised problems such as spectral methods for dimensionality reduction. For example, the Kernel PCA algorithm [23] can be extended to work with multiple views by deﬁning each view as having its own Kernel PCA objective function and having a regularizer which enforces the embeddings to look similar across all views (e.g., by enforcing the similarity matrices deﬁned on embeddings of each view to be close to each other). Theoretical analysis of the proposed approach can also be pursued as a separate line of work. There has been very little prior work analyzing spectral clustering methods. For instance, there has been some work on consistency analysis of single view spectral clustering [24], which provides results about the rate of convergence as the sample size increases, using tools from theory of linear operators and empirical processes. Similar convergence properties could be studied for multi-view spectral clustering. We can expect the convergence to be faster for multi-view case. Coregularization reduces the size of hypothesis space and hence less number of examples should be needed to converge to a solution. 8 References [1] A. Blum and T. Mitchell. Combining labeled and unlabeled data with co-training. In Conference on Learning Theory, 1998. [2] Kamalika Chaudhuri, Sham M. Kakade, Karen Livescu, and Karthik Sridharan. Multi-view Clustering via Canonical Correlation Analysis. In International Conference on Machine Learning, 2009. [3] Ulrike von Luxburg. A Tutorial on Spectral Clustering. Statistics and Computing, 2007. [4] J. Shi and J. Malik. Normalized cuts and Image Segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22:888–905, 1997. [5] A. Ng, M. Jordan, and Y. Weiss. On spectral clustering: analysis and an algorithm. In Advances in Neural Information Processing Systems, 2002. [6] Vikas Sindhwani, Partha Niyogi, and Mikhail Belkin. A Co-regularization approach to semisupervised learning with multiple views. 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4,415	A concave regularization technique for sparse mixture models Martin Larsson School of Operations Research and Information Engineering Cornell University mol23@cornell.edu Johan Ugander Center for Applied Mathematics Cornell University jhu5@cornell.edu Abstract Latent variable mixture models are a powerful tool for exploring the structure in large datasets. A common challenge for interpreting such models is a desire to impose sparsity, the natural assumption that each data point only contains few latent features. Since mixture distributions are constrained in their L1 norm, typical sparsity techniques based on L1 regularization become toothless, and concave regularization becomes necessary. Unfortunately concave regularization typically results in EM algorithms that must perform problematic non-concave M-step maximizations. In this work, we introduce a technique for circumventing this difﬁculty, using the so-called Mountain Pass Theorem to provide easily veriﬁable conditions under which the M-step is well-behaved despite the lacking concavity. We also develop a correspondence between logarithmic regularization and what we term the pseudo-Dirichlet distribution, a generalization of the ordinary Dirichlet distribution well-suited for inducing sparsity. We demonstrate our approach on a text corpus, inferring a sparse topic mixture model for 2,406 weblogs. 1 Introduction The current trend towards ‘big data’ has created a strong demand for techniques to efﬁciently extract structure from ever-accumulating unstructured datasets. Speciﬁc contexts for this demand include latent semantic models for organizing text corpora, image feature extraction models for navigating large photo datasets, and community detection in social networks for optimizing content delivery. Mixture models identify such latent structure, helping to categorize unstructured data. Mixture models approach datasets as a set D of element d ∈D, for example images or text documents. Each element consists of a collection of words w ∈W drawn with replacement from a vocabulary W. Each element-word pair observation is further assumed to be associated with an unobserved class z ∈Z, where Z is the set of classes. Ordinarily it is assumed that \|Z\| ≪\|D\|, namely that the number of classes is much less than the number of elements. In this work we explore an additional sparsity assumption, namely that individual elements only incorporate a small subset of the \|Z\| classes, so that each element arises as a mixture of only ℓ≪\|Z\| classes. We develop a framework to overcome mathematical difﬁculties in how this assumption can be harnessed to improve the performance of mixture models. Our primary context for mixture modeling in this work will be latent semantic models of text data, where elements d are documents, words w are literal words, and classes z are vocabulary topics. We apply our framework to models based on Probabilistic Latent Semantic Analysis (PLSA) [1]. While PLSA is often outperformed within text applications by techniques such as Latent Dirichlet Allocation (LDA) [2], it forms the foundation of many mixture model techniques, from computer vision [3] to network community detection [4], and we emphasize that our contribution is an optimization technique intended for broad application outside merely topic models for text corpora. The 1 near-equivalence between PLSA and Nonnegative Matrix Factorization (NMF) [5, 6] implies that our technique is equally applicable to NMF problems as well. Sparse inference as a rule targets point estimation, which makes PLSA-style models appropriate since they are inherently frequentist, deriving point-estimated models via likelihood maximization. In contrast, fully Bayesian frameworks such as Latent Dirichlet Allocation (LDA) output a posterior distribution across the model space. Sparse inference is commonly achieved through two largely equivalent techniques: regularization or MAP inference. Regularization modiﬁes ordinary likelihood maximization with a penalty on the magnitudes of the parameters. Maximum a posteriori (MAP) inference employs priors concentrated towards small parameter values. MAP PLSA is an established technique [7], but earlier work has been limited to log-concave prior distributions (corresponding to convex regularization functions) that make a concave contribution to the posterior log-likelihood. While such priors allow for tractable EM algorithms, they have the effect of promoting smoothing rather than sparsity. In contrast, sparsity-inducing priors are invariably convex in their contribution. In this work we resolve this difﬁculty by showing how, even though concavity fails in general, we are able to derive simple checkable conditions that guarantee a unique stationary point to the M-step objective function that serves as the unique global maximum. This rather surprising result, using the so-called Mountain Pass Theorem, is a noteworthy contribution to the theory of learning algorithms which we expect has many applications outside merely PLSA. Section 2 brieﬂy outlines the structure of MAP inference for PLSA. Section 3 discusses priors appropriate for inducing sparsity, and introduces a generalization of the Dirichlet distribution which we term the pseudo-Dirichlet distribution. Section 4 contains our main result, a tractable EM algorithm for PLSA under sparse pseudo-Dirichlet priors using the Mountain Pass Theorem. Section 5 presents empirical results for a corpus of 2,406 weblogs, and section 6 concludes with a discussion. 2 Background and preliminaries 2.1 Standard PLSA Within the PLSA framework, word-document-topic triplets (w, d, z) are assumed to be i.i.d. draws from a joint distribution on W × D × Z of the form P(w, d, z \| θ) = P(w \| z)P(z \| d)P(d), (1) where θ consists of the model parameters P(w \| z), P(z \| d) and P(d) for (w, d, z) ranging over W × D × Z. Following [1], the corresponding data log-likelihood can be written ℓ0(θ) = X w,d n(w, d) log h X z P(w \| z)P(z \| d) i + X d n(d) log P(d), (2) where n(w, d) is the number of occurrences of word w in document d, and n(d) = P w n(w, d) is the total number of words in d. The goal is to maximize the likelihood over the set of admissible θ. This is accomplished using the EM algorithm, iterating between the following two steps: E-step: Find P(z \| w, d, θ′), the posterior distribution of the latent variable z, given (w, d) and a current parameter estimate θ′. M-step: Maximize Q0(θ \| θ′) over θ, where Q0(θ \| θ′) = X d n(d) log P(d) + X w,d,z n(w, d)P(z \| w, d, θ′) log h P(w \| z)P(z \| d) i . We refer to [1] for details on the derivations, as well as extensions using so-called tempered EM. The resulting updates corresponding to the E-step and M-step are, respectively, P(z \| w, d, θ) = P(w \| z)P(z \| d) P z′ P(w \| z′)P(z′ \| d) (3) and P(w \| z) = P d P(z \| w, d, θ′)n(w, d) P w′,d P(z \| w′, d, θ′)n(w′, d), P(z \| d) = P w P(z \| w, d, θ′)n(w, d) n(d) , P(d) = n(d) P d′ n(d′). (4) 2 Note that PLSA has an alternative parameterization, where (1) is replaced by P(w, d, z \| θ) = P(w\|z)P(d\|z)P(z). This formulation is less interesting in our context, since our sparsity assumption is intended as a statement about the vectors (P(z \| d) : z ∈Z), d ∈D. 2.2 MAP PLSA The standard MAP extension of PLSA is to introduce a prior density P(θ) on the parameter vector, and then maximize the posterior data log-likelihood ℓ(θ) = ℓ0(θ) + log P(θ) via the EM algorithm. In order to simplify the optimization problem, we impose the reasonable restriction that the vectors (P(w \| z) : w ∈W) for z ∈Z, (P(z \| d) : z ∈Z) for d ∈D, and (P(d) : d ∈D) be mutually independent under the prior P(θ). That is, P(θ) = Y z∈Z fz(P(w \| z) : w ∈W) × Y d∈D gd(P(z \| d) : z ∈Z) × h(P(d) : d ∈D), for densities fz, gd and h on the simplexes in R\|W\|, R\|Z\| and R\|D\|, respectively. With this structure on P(θ) one readily veriﬁes that the M-step objective function for the MAP likelihood problem, Q(θ \| θ′) = Q0(θ \| θ′) + log P(θ), is given by Q(θ \| θ′) = X z Fz(θ \| θ′) + X d Gd(θ \| θ′) + H(θ \| θ′), where Fz(θ \| θ′) = X w,d P(z \| w, d, θ′)n(w, d) log P(w \| z) + log fz(P(w \| z) : w ∈W), Gd(θ \| θ′) = X w,z P(z \| w, d, θ′)n(w, d) log P(z \| d) + log gd(P(z \| d) : z ∈Z), H(θ \| θ′) = X d n(d) log P(d) + log h(P(d) : d ∈D). As a comment, notice that if the densities fz, gd, or h are log-concave then Fz, Gd, and H are concave in θ. Furthermore, the functions Fz, Gd, and H can be maximized independently, since the corresponding non-negativity and normalization constraints are decoupled. In particular, the \|Z\| + \|D\| + 1 optimization problems can be solved in parallel. 3 The pseudo-Dirichlet prior The parameters for PLSA models consist of \|Z\| + \|D\| + 1 probability distributions taking their values on \|Z\| + \|D\| + 1 simplexes. The most well-known family of distributions on the simplex is the Dirichlet family, which has many properties that make it useful in Bayesian statistics [8]. Unfortunately the Dirichlet distribution is not a suitable prior for modeling sparsity for PLSA, as we shall see, and to address this we introduce a generalization of the Dirichlet distribution which we call the pseudo-Dirichlet distribution. To illustrate why the Dirichlet distribution is unsuitable in the present context, consider placing a symmetric Dirichlet prior on (P(z \| d) : z ∈Z) for each document d. That is, for each d ∈D, gd(P(z \| d) : z ∈Z) ∝ Y z∈Z P(z \| d)α−1, where α > 0 is the concentration parameter. Let fz and h be constant. The relevant case for sparsity is when α < 1, which concentrates the density toward the (relative) boundary of the simplex. It is easy to see that the distribution is in this case log-convex, which means that the contribution to the log-likelihood and M-step objective function Gd(θ \| θ′) will be convex. We address this problem in Section 4. A bigger problem, however, is that for α < 1 the density of the symmetric Dirichlet distribution is unbounded and the MAP likelihood problem does not have a well-deﬁned solution, as the following result shows. 3 Proposition 1 Under the above assumptions on fz, gd and h there are inﬁnitely many sequences (θm)m≥1, converging to distinct limits, such that limm→∞Q(θm \| θm) = ∞. As a consequence, ℓ(θm) tends to inﬁnity as well. Proof. Choose θm as follows: P(d) = \|D\|−1 and P(w \| z) = \|W\|−1 for all w, d and z. Fix d0 ∈D and z0 ∈Z, and set P(z0 \| d0) = m−1, P(z \| d0) = 1−m−1 \|Z\|−1 for z ̸= z0, and P(z \| d) = \|Z\|−1 for all z and d ̸= d0. It is then straightforward to verify that Q(θm \| θm) tends to inﬁnity. The choice of d0 and z0 was arbitrary, so by choosing two other points we get a different sequence with a different limit. Taking convex combinations yields the claimed inﬁnity of sequences. The second statement follows from the well-known fact that Q(θ \| θ′) ≤ℓ(θ) for all θ and θ′. □ This proposition is a formal statement of the observation that when the Dirichlet prior is unbounded, any single zero element in P(z\|d) leads to an inﬁnite posterior likelihood, and so the optimization problem is not well-posed. To overcome these unbounded Dirichlet priors while retaining their sparsity-inducing properties, we introduce the following class of distributions on the simplex. Deﬁnition 1 A random vector conﬁned to the simplex in Rp is said to follow a pseudo-Dirichlet distribution with concentration parameter α = (α1, . . . , αp) ∈Rp and perturbation parameter ϵ = (ϵ1, . . . , ϵp) ∈Rp + if it has a density on the simplex given by P(x1, . . . , xp \| α, ϵ) = C p Y i=1 (ϵi + xi)αi−1 (5) for a normalizing constant C depending on α and ϵ. If αi = α and ϵi = ϵ for all i and some ﬁxed α ∈R, ϵ ≥0, we call the resulting distribution symmetric pseudo-Dirichlet. Notice that if ϵi > 0 for all i, the pseudo-Dirichlet density is indeed bounded for all α. If ϵi = 0 and αi > 0 for all i, we recover the standard Dirichlet distribution. If ϵi = 0 and αi ≤0 for some i then the density is not integrable, but can still be used as an improper prior. Like the Dirichlet distribution, when α < 1 the pseudo-Dirichlet distribution is log-convex, and it will make a convex contribution to the M-step objective function of any EM algorithm. The psuedo-Dirichlet distribution can be viewed as a bounded perturbation of the Dirichlet distribution, and for small values of the perturbation parameter ϵ, many of the properties of the original Dirichlet distribution hold approximately. In our discussion section we offer a justiﬁcation for allowing α ≤0, framed within a regularization approach. 4 EM under pseudo-Dirichlet priors We now derive an EM algorithm for PLSA under sparse pseudo-Dirichlet priors. The E-step is the same as for standard PLSA, and is given by (3). The M-step consists in optimizing each Fz, Gd and H individually. While our M-step will not offer a closed-form maximization, we are able to derive simple checkable conditions under which the M-step has a stationary point that is also the global maximum. Once the conditions are satisﬁed, the M-step optimum can be found via a practitioner’s favorite root-ﬁnding algorithm. For consideration, we propose an iteration scheme that in practice we ﬁnd converges rapidly and well. Because our sparsity assumption focuses on the parameters P(z\|d), we perform our main analysis on Gd, but for completeness we state the corresponding result for Fz. The less applicable treatment of H is omitted. Consider the problem of maximizing Gd(θ \| θ′) over (P(z \| d) : z ∈Z) subject to P z P(z \| d) = 1 and P(z \| d) ≥0 for all z. We use symmetric pseudo-Dirichlet priors with parameters αd = (αd, . . . , αd) and ϵd = (ϵd, . . . , ϵd) for αd ∈R and ϵd > 0. Since each Gd is treated separately, let us ﬁx d and write xz = P(z \| d), cz = X w P(z \| w, d, θ′)n(w, d), where the dependence on d is suppressed in the notation. For x = (xz : z ∈Z) and a ﬁxed θ′, we write Gd(x) = Gd(θ \| θ′), which yields, up to an additive constant, Gd(x) = X z h (αd −1) log(ϵd + xz) + cz log xz i . 4 The task is to maximize Gd, subject to P z xz = 1 and xz ≥0 for all z. Assuming that every word w is observed in at least one document d and that all components of θ′ are strictly positive, Lemma 1 below implies that any M-step optimizer must have strictly positive components. The non-negativity constraint is therefore never binding, so the appropriate Lagrangian for this problem is Ld(x; λ) = Gd(x) + λ h 1 − X z xz i , and it sufﬁces to consider its stationary points. Lemma 1 Assume that every word w has been observed in at least one document d, and that P(z \| w, d; θ′) > 0 for all (w, d, z). If xz →0 for some z, and the nonnegativity and normalization constraints are maintained, then Gd(x) →−∞. Proof. The assumption implies that cz > 0, ∀z. Therefore, since log(ϵd + xz) and log xz are bounded from above, ∀z, when θ stays in the feasible region, xz →0 leads to Gd(x) →−∞. □ The next lemma establishes a property of the stationary points of the Lagrangian Ld which will be the key to proving our main result. Lemma 2 Let (x, λ) be any stationary point of Ld such that xz > 0 for all z. Then λ ≥n(d) − (1 −αd)\|Z\|. If in addition to the assumptions of Lemma 1 we have n(d) ≥(1 −αd)\|Z\|, then ∂2Gd ∂x2z (x, λ) < 0 for all z ∈Z. Proof. We have ∂Ld ∂xz = cz xz −1−αd ϵd+xz −λ. Since ∂Ld ∂xz (x, λ) = 0 at the stationary point, we get λxz = cz −(1−αd) xz ϵd+xz ≥cz −(1−αd), which, after summing over z and using that P z xz = 1, yields λ ≥ X z cz −(1 −αd)\|Z\|. Furthermore, P z cz = P w n(w, d) P z P(z \| w, d, θ′) = n(d), so λ ≥n(d) −(1 −αd)\|Z\|. For the second assertion, using once again that ∂Ld ∂xz (x, λ) = 0 at the stationary point, a calculation shows that ∂2Gd ∂x2z (x, λ) = − 1 x2z(ϵd + xz) h x2 zλ + czϵd i . The assumptions imply that cz > 0, so it sufﬁces to prove that λ ≥0. This follows from our hypothesis and the ﬁrst part of the lemma. □ This allows us to obtain our main result result concerning the structure of the optimization problem associated with the M-step. Theorem 1 Assume that (i) every word w has been observed in at least one document d, (ii) P(z \| w, d, θ′) > 0 for all (w, d, z), and (iii) n(d) > (1 −αd)\|Z\| for each d. Then each Lagrangian Ld has a unique stationary point, which is the global maximum of the corresponding optimization problem, and whose components are strictly positive. The proof relies on the following version of the so-called Mountain Pass Theorem. Lemma 3 (Mountain Pass Theorem) Let O ⊂Rn be open, and consider a continuously differentiable function φ : O →R s.t. φ(x) →−∞whenever x tends to the boundary of O. If φ has two distinct strict local maxima, it must have a third stationary point that is not a strict local maximum. Proof. See p. 223 in [9], or Theorem 5.2 in [10]. □ Proof of Theorem 1. Consider a ﬁxed d. We ﬁrst prove that the corresponding Lagrangian Ld can have at most one stationary point. To simplify notation, assume without loss of generality that Z = {1, . . . , K}, and deﬁne eGd(x1, . . . , xK−1) = Gd x1, . . . , xK−1, 1 − K−1 X k=1 xk . 5 The constrained maximization of Gd is then equivalent to maximizing eGd over the open set O = {(x1, . . . , xK−1) ∈RK−1 ++ : P k xk < 1}. The following facts are readily veriﬁed: (i) If (x, λ) is a stationary point of Ld, then (x1, . . . , xK−1) is a stationary point of eGd. (ii) If (x1, . . . , xK−1) is a stationary point of eGd, then (x, λ) is a stationary point of Ld, where xK = 1 −PK−1 k=1 xk and λ = cK xK − 1−αd ϵd+xK . (iii) For any y = (y1, . . . , yK−1, PK−1 k=1 yk), we have yT ∇2 eGdy = PK k=1 y2 k ∂2Gd ∂x2 k . Now, suppose that (x, λ) is a stationary point of Ld. Property (i) and property (iii) in conjunction with Lemma 2 imply that (x1, . . . , xK−1) is a stationary point of eGd and that ∇2 eGd is negative deﬁnite there. Hence it is a strict local maximum. Next, suppose for contradiction that there are two distinct such points. By Lemma 1, eGd tends to −∞near the boundary of O, so we may apply the mountain pass theorem to get the existence of a third point (˜x1, . . . , ˜xK−1), stationary for eGd, that is not a strict local maximum. But by (ii), this yields a corresponding stationary point (˜x, ˜λ) for Ld. The same reasoning as above then shows that (˜x1, . . . , ˜xK−1) has to be a strict local max for eGd, which is a contradiction. We deduce that Ld has at most one stationary point. Finally, the continuity of Gd together with its boundary behavior (Lemma 1) implies that a maximizer exists and has strictly positive components. But the maximizer must be a stationary point of Ld, so together with the previously established uniqueness, the result follows. □ Condition (i) in Theorem 1 is not a real restriction, since a word that does not appear in any document typically will be removed from the vocabulary. Moreover, if the EM algorithm is initialized such that P(z \| w, d; θ′) > 0 for all (w, d, z), Theorem 1 ensures that this will be the case for all future iterates as well. The critical assumption is Condition (iii). It can be thought of as ensuring that the prior does not drown the data. Indeed, sufﬁciently large negative values of αd, corresponding to strong prior beliefs, will cause the condition to fail. While there are various methods available for ﬁnding the stationary point of Ld, we have found that the following ﬁxed-point type iterative scheme produces satisfactory results. xz ← cz n(d) + (1 −αd) h 1 ϵd+xz −P z′ xz′ ϵd+xz′ i, xz ← xz P z xz . (6) To motivate this particular update rule, recall that ∂Ld ∂xz = cz xz −1 −αd ϵd + xz −λ, ∂Ld ∂λ = 1 − X z xz. At the stationary point, λxz = cz −1−αd ϵd+xz xz, so by summing over z and using that P z xz = 1 and P z cz = n(d), we get λ = n(d) −(1 −αd) P z xz ϵd+xz . Substituting this for λ in ∂Ld ∂xz = 0 and rearranging terms yields the ﬁrst part of (6). Notice that Lemma 2 ensures that the denominator stays strictly positive. Further, the normalization is a classic technique to restrict x to the simplex. Note that (6) reduces to the standard PLSA update (4) if αd = 1. For completeness we also consider the topic-vocabulary distribution (P(w\|z) : w ∈W). We impose a symmetric pseudo-Dirichlet prior on the vector (P(w \| z) : w ∈W) for each z ∈Z. The corresponding parameters are denoted by αz and ϵz. Each Fz is optimized individually, so we ﬁx z ∈Z and write yw = P(w \| z). The objective function Fz(y) = Fz(θ \| θ′) is then given by Fz(y) = X w h (αz −1) log(ϵz + yw) + bw log yz i , bw = X d P(z \| w, d, θ′)n(w, d). (7) The following is an analog of Theorem 1, whose proof is essentially the same and therefore omitted. Theorem 2 Assume condition (i) and (ii) of Theorem 1 are satisﬁed, and that for each z ∈Z, P w bw ≥(1 −αz)\|W\|. Then each Fz has a unique local optimum on the simplex, which is also a global maximum and whose components are strictly positive. 6 Unfortunately there is no simple expression for P w bw in terms of the inputs to the problem. On the other hand, the sum can be evaluated at the beginning of each M-step, which makes it possible to verify that αz is not too negative. 5 Empirical evaluation To evaluate our framework for sparse mixture model inference, we develop a MAP PLSA topic model for a corpus of 2,406 blogger.com blogs, a dataset originally analyzed by Schler et al. [11] for the role of gender in language. Unigram frequencies for the blogs were built using the python NLTK toolkit [12]. Inference was run on the document-word distribution of 2,406 blogs and 2,000 most common words, as determined by the aggregate frequencies across the entire corpus. The implications of Section 4 is that in order to adapt PLSA for sparse MAP inference, we simply need to replace equation (4) from PLSA’s ordinary M-step with an iteration of (6). The corpus also contains a user-provided ‘category’ for each blog, indicating one of 28 categories. We focused our analysis on 8 varied but representative topics, while the complete corpus contained over 19,000 blogs. The user-provided topic labels are quite noisy, and so in order to have cleaner ground truth data for evaluating our model we chose to also construct a synthetic, sparse dataset. This synthetic dataset is employed to evaluate parameter choices within the model. To generate our synthetic data, we ran PLSA on our text corpus and extracted the inferred P(w\|z) and P(d) distributions, while creating 2,406 synthetic P(z\|d) distributions where each synthetic blog was a uniform mixture of between 1 and 4 topics. These distributions were then used to construct a ground-truth word-document distribution Q(w, d), which we then sampled N times, where N is the total number of words in our true corpus. In this way we were able to generate a realistic synthetic dataset with a sparse and known document-topic distribution. We evaluate the quality of each model by calculating the model perplexity of the reconstructed worddocument distribution as compared to the underlying ground truth distribution used to generate the synthetic data. Here model perplexity is given by P(P(w, d)) = 2−P w,d Q(w,d) log2 P (w,d), where Q(w, d) is the true document-word distribution used to generate the synthetic dataset and P(w, d) is the reconstructed matrix inferred by the model. Using this synthetic dataset we are able to evaluate the roles of α and ϵ in our algorithm, as seen in Figure 1. From Figure 1 we can conclude that α should in practice be chosen close the algorithm’s feasible lower bound, and ϵ can be almost arbitrarily small. Choosing α = ⌈1−maxd n(d)/k⌉and ϵ = 10−6, we return to our blog data with its user-provided labels. In Figure 2 we see that sparse inference indeed results in P(z\|d) distributions with signiﬁcantly sparser support. Furthermore, we can more easily see how certain categories of blogs cluster in their usage of certain topics. For example, a majority of the blogs self-categorized as pertaining to ‘religion’ employ almost exclusively the second topic vocabulary of the model. The ﬁve most exceptional unigrams for this topic are ‘prayer’, ‘christ’, ‘jesus’, ‘god’, and ‘church’. 6 Discussion We have shown how certain latent variable mixture models can be tractably extended with sparsityinducing priors using what we call the pseudo-Dirichlet distribution. Our main theoretical result shows that the resulting M-step maximization problem is well-behaved despite the lack of concavity, and empirical ﬁndings indicate that the approach is indeed effective. Our use of the Mountain Pass Theorem to prove that all local optima coincide is to the best of our knowledge new in the literature, and we ﬁnd it intriguing and surprising that the global properties of maximizers, which are very rarely susceptible to analysis in the absence of concavity, can be studied using this tool. The use of log-convex priors (equivalently, concave regularization functions) to encourage sparsity is particularly relevant when the parameters of the model correspond to probability distributions. Since each distribution has a ﬁxed L1 norm equal to one, the use of L1-regularization, which otherwise would be the natural choice for inducing sparsity, becomes toothless. The pseudo-Dirichlet prior we introduce corresponds to a concave regularization of the form P i log(xi + ϵ). We mention in 7 −50 −40 −30 −20 −10 0 2.85 2.9 2.95 x 10 5 model perplexity α ε=10−6 ε=0.1 ε=1 PLSA α Figure 1: Model perplexity for inferred models with k = 8 topics as a function of the concentration parameter α of the pseudo-Dirichlet prior, shown from the algorithm’s lower bound α = 1−n(d)/k to the uniform prior case of α = 1. Three different choices of ϵ are shown, as well as the baseline PLSA perplexity corresponding to a uniform prior. The dashed line indicates the perplexity P(Q(w, d)), which should be interpreted as a lower bound. P(z\|d), PLSA, Religion P(z\|d), Ps-Dir MAP, Religion P(z\|d), PLSA, Internet P(z\|d), Ps-Dir MAP, Internet P(z\|d), PLSA, Engineering P(z\|d), Ps-Dir MAP, Engineering P(z\|d), PLSA, Technology P(z\|d), Ps-Dir MAP, Technology P(z\|d), PLSA, Fashion P(z\|d), Ps-Dir MAP, Fashion P(z\|d), PLSA, Media P(z\|d), Ps-Dir MAP, Media P(z\|d), PLSA, Tourism P(z\|d), Ps-Dir MAP, Tourism P(z\|d), PLSA, Law P(z\|d), Ps-Dir MAP, Law Figure 2: Document-topic distributions P(z\|d) for the 8 different categories of blogs studied. All distributions share the same color scale. passing that the same sum-log regularization has also been used for sparse signal recovery in [13]. It should be emphasized that the notion of sparsity we discuss in this work is not in the formal sense of a small L0 norm. Indeed, Theorem 1 shows that, no different from ordinary PLSA, the estimated parameters for MAP PLSA will all be strictly positive. Instead, we seek sparsity in the sense that most parameters should be almost zero. Next, let us comment on the possibility to allow the concentration parameter αd to be negative, assuming for simplicity that fz and h are constant. Consider the normalized likelihood, where clearly ℓ(θ) may be replaced by ℓ(θ)/N, ℓ(θ) N = ℓ0(θ) N − X d 1 −αd N X z log(ϵd + P(z \| d)), which by (2) we deduce only depends on the data through the normalized quantities n(w, d)/N. This indicates that the quantity (1 −αd)/N, which plays the role of a regularization ‘gain’ in the normalized problem, must be non-negligible in order for the regularization to have an effect. For realistic sizes of N, allowing αd < 0 therefore becomes crucial. Finally, while we have chosen to present our methodology as applied to topic models, we expect the same techniques to be useful in a notably broader context. In particular, our methodology is directly applicable to problems solved through Nonnegative Matrix Factorization (NMF), a close relative of PLSA where matrix columns or rows are often similarly constrained in their L1 norm. Acknowledgments: This work is supported in part by NSF grant IIS-0910664. 8 References [1] T. Hofmann. Unsupervised learning by probabilistic latent semantic analysis. Machine Learning, 42:177–196, 2001. [2] D.M. Blei, A.Y. Ng, and M.I. Jordan. Latent Dirichlet allocation. Journal of Machine Learning Research, 3:993–1022, 2003. [3] A. Bosch, A. Zisserman, and X. Munoz. Scene Classiﬁcation via pLSA. In European Conference on Computer Vision, 2006. [4] I. Psorakis and B. Sheldon. Soft Partitioning in Networks via Baysian Non-negative Matrix Factorization. In NIPS, 2010. [5] C. Ding, T. Li, and W. Peng. Nonnegative matrix factorization and probabilistic latent semantic indexing: Equivalence chi-square statistic, and a hybrid method. In Proceedings of AAAI ’06, volume 21, page 342, 2006. [6] E. Gaussier and C. Goutte. Relation between PLSA and NMF and implications. In Proceedings of ACM SIGIR, pages 601–602. ACM, 2005. [7] A. Asuncion, M. Welling, P. Smyth, and Y.W. Teh. On smoothing and inference for topic models. In Proc. of the 25th Conference on Uncertainty in Artiﬁcial Intelligence, pages 27–34, 2009. [8] A. Gelman. Bayesian data analysis. CRC Press, 2004. [9] R. Courant. Dirichlet’s principle, conformal mapping, and minimal surfaces. Interscience, New York, 1950. [10] Y. Jabri. The Mountain Pass Theorem: Variants, Generalizations and Some Applications. Cambridge University Press, 2003. [11] J. Schler, M. Koppel, S. Argamon, and J. Pennebaker. Effects of age and gender on blogging. In Proc. of the AAAI Spring Symposium on Computational Approaches for Analyzing Weblogs, pages 191–197, 2006. [12] S. Bird, E. Klein, and Loper E. Natural language processing with Python. O’Reilly Media, 2009. [13] E.J. Cand`es, M.B. Wakin, and S.P. Boyd. Enhancing sparsity by reweighted ℓ1 minimization. Journal of Fourier Analysis and Applications, 14:877–905, 2008. 9	2011	67
4,416	Image Parsing via Stochastic Scene Grammar Yibiao Zhao∗ Department of Statistics University of California, Los Angeles Los Angeles, CA 90095 ybzhao@ucla.edu Song-Chun Zhu Department of Statistics and Computer Science University of California, Los Angeles Los Angeles, CA 90095 sczhu@stat.ucla.edu Abstract This paper proposes a parsing algorithm for scene understanding which includes four aspects: computing 3D scene layout, detecting 3D objects (e.g. furniture), detecting 2D faces (windows, doors etc.), and segmenting background. In contrast to previous scene labeling work that applied discriminative classiﬁers to pixels (or super-pixels), we use a generative Stochastic Scene Grammar (SSG). This grammar represents the compositional structures of visual entities from scene categories, 3D foreground/background, 2D faces, to 1D lines. The grammar includes three types of production rules and two types of contextual relations. Production rules: (i) AND rules represent the decomposition of an entity into sub-parts; (ii) OR rules represent the switching among sub-types of an entity; (iii) SET rules represent an ensemble of visual entities. Contextual relations: (i) Cooperative “+” relations represent positive links between binding entities, such as hinged faces of a object or aligned boxes; (ii) Competitive “-” relations represents negative links between competing entities, such as mutually exclusive boxes. We design an efﬁcient MCMC inference algorithm, namely Hierarchical cluster sampling, to search in the large solution space of scene conﬁgurations. The algorithm has two stages: (i) Clustering: It forms all possible higher-level structures (clusters) from lower-level entities by production rules and contextual relations. (ii) Sampling: It jumps between alternative structures (clusters) in each layer of the hierarchy to ﬁnd the most probable conﬁguration (represented by a parse tree). In our experiment, we demonstrate the superiority of our algorithm over existing methods on public dataset. In addition, our approach achieves richer structures in the parse tree. 1 Introduction Scene understanding is an important task in neural information processing systems. By analogy to natural language parsing, we pose the scene understanding problem as parsing an image into a hierarchical structure of visual entities (in Fig.1(i)) using the Stochastic Scene Grammar (SSG). The literature of scene parsing can be categorized into two categories: discriminative approaches and generative approaches. Discriminative approaches focus on classifying each pixel (or superpixel) to a semantic label (building, sheep, road, boat etc.) by discriminative Conditional Random Fields (CRFs) model [5][7]. Without an understanding of the scene structure, the pixel-level labeling is insufﬁcient to represent the knowledge of object occlusions, 3D relationships, functional space etc. To address this problem, geometric descriptions were added to the scene interpretation. Hoiem et al. [1] and Saxena et al. [8] generated the surface orientation labels and the depth labels by exploring rich geometric ∗http://www.stat.ucla.edu/˜ybzhao/research/sceneparsing 1 (i) a parse tree 3D background 2D faces 1D line segments scene 3D foregrounds (ii) input image and line detection (iii) geometric parsing result (iv) reconstructed via line segments Figure 1: A parse tree of geometric parsing result. Figure 2: 3D synthesis of novel views based on the parse tree. features and context information. Gupta et al. [9] posed the 3D objects as blocks and infers its 3D properties such as occlusion, exclusion and stableness in addition to surface orientation labels. They showed the global 3D prior does help the 2D surface labeling. For the indoor scene, Hedau et al. [2], Wang et al. [3] and Lee et al. [4] adopted different approaches to model the geometric layout of the background and/or foreground objects, and ﬁt their models into Structured SVM (or Latent SVM) settings [10]. The Structured SVM uses features extracted jointly from input-output pairs and maximizes the margin over the structured output space. These algorithms involve hidden variables or structured labels in discriminative training. However, these discriminative approaches lack a general representation of visual vocabulary and a principled approach for exploring the compositional structure. Generative approaches make efforts to model the reconﬁgurable graph structures in generative probabilistic models. The stochastic grammar were used to parse natural languages [11]. Compositional models for the hierarchical structure and sharing parts were studied in visual object recognition [12]-[15]. Zhu and Mumford [16] proposed an AND/OR Graph Model to represent the compositional structures in vision. However, the expressive power of conﬁgurable graph structures comes at the cost of high computational complexity of searching in a large conﬁguration space. In order to accelerate the inference, the Adaptor Grammars [17] applied an idea of ”adaptor” (re-using subtree) that induce dependencies among successive uses. Han and Zhu [18] applied grammar rules, in a greedy manner, to detect rectangular structures in man-made scenes. Porway et al. [19] [20]allowed the Markov chain jumping between competing solutions by a C4 algorithm. 2 Overview of the approach. In this paper, we parse an image into a hierarchical structure, namely a parse tree as shown in Fig.1. The parse tree covers a wide spectrum of visual entities, including scene categories, 3D foreground/background, 2D faces, and 1D line segments. With the low-level information of the parse tree, we reconstruct the original image by the appearance of line segments, as shown in Fig.1(iv). With the high-level information of the parse tree, we further recover the 3D scene by the geometry of 3D background and foreground objects, as shown in Fig.2. This paper has two major contributions to the scene parsing problems: (I) A Stochastic Scene Grammar (SSG) is introduced to represent the hierarchical structure of visual entities. The grammar starts with a single root node (the scene) and ends with a set of terminal nodes (line segments). In between, we generate all intermediate 3D/2D sub-structures by three types of production rules and two types of contextual relations, as illustrated in Fig.3. Production rules: AND, OR, and SET. (i) The AND rule encodes how sub-parts are composed into a larger structure. For example, three hinged faces form a 3D box, four linked line segments form a rectangle, a background and inside objects form a scene in Fig.3(i); (ii) The SET rule represents an ensemble of entities, e.g. a set of 3D boxes or a set of 2D regions as in Fig.3(ii); (iii)The OR rule represents a switch between different sub-types, e.g. a 3D foreground and 3D background have several switchable types in Fig.3(iii). Contextual relations: Cooperative “+” and Competitive “-”. (i) If the visual entities satisfy a cooperative “+” relation, they tend to bind together, e.g. hinged faces of a foreground box showed in Fig.3(a). (ii) If entities satisfy a competitive “-” relation, they compete with each other for presence, e.g. two exclusive foreground boxes competing for a same space in Fig.3(b). (II) A hierarchical cluster sampling algorithm is proposed to perform inference efﬁciently in SSG model. The algorithm accelerates a Markov chain search by exploring contextual relations. It has two stages: (i) Clustering. Based on the detected line segments in Fig.1(ii), we form all possible larger structures (clusters). In each layer, the entities are ﬁrst ﬁltered by the Cooperative “+” constraints, they then form a cluster only if they satisfy the “+” constraints, e.g. several faces form a cluster of a box when their edges are hinged tightly. (ii) Sampling. The sampling process makes a big reversible jumps by switching among competing sub-structures (e.g. two exclusive boxes). In summary, the Stochastic Scene Grammar is a general framework to parse a scene with a large number of geometric conﬁgurations. We demonstrate the superiority of our algorithm over existing methods in the experiment. 2 Stochastic Scene Grammar The Stochastic Scene Grammar (SSG) is deﬁned as a four-tuple G = (S, V, R, P), where S is a start symbol at the root (scene); V = V N ∪V T , V N is a ﬁnite set of non-terminal nodes (structures or sub-structures), V T is a ﬁnite set of terminal nodes (line segments); R = {r : α →β} is a set of production rules, each of which represents a generating process from a parent node α to its child nodes β = Chα. P(r) = P(β\|α) is an expansion probability for each production rule (r : α →β). A set of all valid conﬁgurations C derived from production rules is called a language: L(G) = {C : S {ri} −−−→C, {ri} ⊂R, C ⊂V T , P({ri}) > 0}. Production rules. We deﬁne three types of stochastic production rules RAND,ROR,RSET to represent the structural regularity and ﬂexibility of visual entities. The regularity is enforced by the AND rule and the ﬂexibility is expressed by the OR rule. The SET rule is a mixture of OR and AND rules. (i) An AND rule (rAND : A →a · b · c) represents the decomposition of a parent node A into three sub-parts a, b, and c. The probability P(a, b, c\|A) measures the compatibility (contextual relations) among sub-structures a, b, c. As seen Fig.3(i), the grammar outputs a high probability if the three faces of a 3D box are well hinged, and a low probability if the foreground box lays out of the background. (ii) An OR rule (rOR : A →a \| b) represents the switching between two sub-types a and b of a parent node A. The probability P(a\|A) indicates the preference for one subtype over others. For 3D foreground in Fig.3(iii), the three sub-types in the third row represent objects below the horizon. These objects appear with high probabilities. Similarly, for the 3D background in Fig.3(iii), the camera rarely faces the ceiling or the ground, hence, the three sub-types in the middle row have 3 3D foreground types 3D background types (i) AND rules (ii) SET rules (a) "+" relations (b) "-" relations (iii) OR rules hinged faces linked lines aligned faces aligned boxes nested faces stacked boxes exclusive faces invalid scene layout exclusive boxes Figure 3: Three types of production rules: AND (i), SET (ii) OR (iii), and two types of contextual relations: cooperative “+” relations (a), competitive “-” relations (b). higher probabilities (the higher the darker). Moreover, OR rules also model the discrete size of entities, which is useful to rule out the extreme large or small entities. (iii) An SET rule (rSET : A →{a}k, k ≥0) represents an ensemble of k visual entities. The SET rule is equivalent to a mixture of OR and AND rules (rSET : A →∅\| a \| a · a \| a · a · a \| · · · ). It ﬁrst chooses a set size k by ORing, and forms an ensemble of k entities by ANDing. It is worth noting that the OR rule essentially changes the graph topology of the output parse tree by changing its node size k. In this way, as seen in Fig.3(ii), the SET rule generates a set of 3D/2D entities which satisfy some contextual relations. Contextual relations. There are two kinds of contextual relations, Cooperative “+” relations and Competitive “-” relations, which involve in the AND and SET rules. (i) The cooperative “+” relations specify the concurrent patterns in a scene, e.g. hinged faces, nested rectangle, aligned windows in Fig.3(a). The visual entities satisfying a cooperative “+” relation tend to bind together. (i) The competitive “-” relations specify the exclusive patterns in a scene. If entities satisfy competitive “-” relations, they compete with each other for the presence. As shown in Fig.3(b), if a 3D box is not contained by its background, or two 2D/3D objects are exclusive with one another, these cases will rarely be in a solution simultaneously. The tight structures vs. the loose structure: If several visual entities satisfy a cooperative “+” relation, they tend to bind together, and we call them tight structures. These tight structures are grouped into clusters in the early stage of inference (Sect.4). If the entities neither satisfy any cooperative “+” relations nor violate a competitive “-” relation, they may be loosely combined. We call them loose structures, whose combinations are sampled in a later stage of inference (Sect.4). With the three production rules and two contextual relations, SSG is able to handle an enormous number of conﬁgurations and large geometric variations, which are the major difﬁculties in our task. 3 Bayesian formulation of the grammar We deﬁne a posterior distribution for a solution (a parse tree) pt conditioned on an input image I. This distribution is speciﬁed in terms of the statistics deﬁned over the derivation of production rules. P(pt\|I) ∝P(pt)P(I\|pt) = P(S) Y v∈V N P(Chv\|v) Y v∈V T P(I\|v) (1) where I is the input image, pt is the parse tree. The probability derivation represents a generating process of the production rules {r : v →Chv} from the start symbol S to the nonterminal nodes v ∈V N, and to the children of non-terminal nodes Chv. The generating process stops at the terminal nodes v ∈V T and generates the image I. We use a probabilistic graphical model of AND/OR graph [12, 17] to formulate our grammar. The graph structure G = (V, E) consists of a set of nodes V and a set of edges E. The edge deﬁne a 4 (i) initial distribution (iii) with competitive(-) relations (iv) with both (+/-) relations (ii) with cooperative(+) relations Figure 4: Learning to synthesize. (a)-(d) Some typical samples drawn from Stochastic Scene Grammar model with/without contextual relations. parent-child conditional dependency for each production rule. The posterior distribution of a parse graph pt is given by a family of Gibbs distributions: P(pt\|I; λ) = 1/Z(I; λ) exp{−E(pt\|I)}, where Z(I; λ) = P pt∈Ωexp{−E(pt\|I)} is a partition function summation over the solution space Ω. The energy is decomposed into three potential terms: E(pt\|I) = X v∈V OR EOR(AT (Chv)) + X v∈V AND EAND(AG(Chv)) + X Λv∈ΛI,v∈V T ET (I(Λv)) (2) (i) The energy for OR nodes is deﬁned over ”type” attributes AT (Chv) of ORing child nodes. The potential captures the prior statistics on each switching branch. EOR(AT (v)) = −log P(v → AT (v)) = −log{ #(v→AT (v)) P u∈Ch(v) #(v→u) }. The switching probability of foreground objects and the background layout is shown in Fig.3(iii). (ii) The energy for AND nodes is deﬁned over ”geometry” attribute AG(Chv) of ANDing child nodes. They are Markov Random Fields (MRFs) inside a tree-structure. We deﬁne both “+” relations and “-” relations as EAND = λ+h+(AG(Chv)) + λ−h−(AG(Chv)), where h(∗) are sufﬁcient statistics in the exponential model, λ are their parameters. For 2D faces as an example, the “+” relation speciﬁes a quadratic distance between their connected joints h+(AG(Chv)) = P a,b∈Chv(X(a) −X(b))2, and the “-” relation speciﬁes an overlap rate between their occupied image area h−(AG(Chv)) = (Λa ∩Λb)/(Λa ∪Λb), a, b ∈Chv. (iii) The energy for Terminal nodes is deﬁned over bottom-up image features I(Λv) on the image area Λv. The features used in this paper include: (a) surface labels of geometric context [1], (b) a 3D orientation map [21], (c) the MDL coding length of line segments [20]. This term only captures the features from their dominant image area Λv, and avoids the double counting of the shared edges and the occluded areas. We learn the context-sensitive grammar model of SSG from a context-free grammar. Under the learning framework of minimax entropy [25], we enforce the contextual relations by adding statistical constraints sequentially. The learning process matches the statistics between the current distribution p and a targeted distribution f by adding the most violated constraint in each iteration. Fig.4 shows the typical samples drawn from the learned SSG model. With more contextual relations being added, the sampled conﬁgurations become more similar to a real scene, and the statistics of the learned distribution become closer to that of target distribution. 4 Inference with hierarchical cluster sampling We design a hierarchical cluster sampling algorithm to infer the optimal parse tree for the SSG model. A parse tree speciﬁes a conﬁguration of visual entities. The combination of conﬁgurations makes the solution space expand exponentially, and it is NP-hard to enumerate all parse trees in such a large space. 5 50 100 150 200 250 300 700 600 500 400 300 200 0 iterations energy iteration 100 iteration 300 iteration 250 iteration 200 iteration 150 iteration 50 iteration 0 Figure 5: The hierarchical cluster sampling process. In order to detecting scene components, neither sliding window (top-down) nor binding (bottom-up) approaches can handle the large geometric variations and an enormous number of conﬁgurations. In this paper we combine the bottom-up and top-down process by exploring the contextual relations deﬁned on the grammar model. The algorithm ﬁrst perform a bottom-up clustering stage and follow by a top-down sampling stage. In the clustering stage, we group visual entities into clusters (tight structures) by ﬁltering the entities based on cooperative “+” relations. With the low-level line segments as illustrated in Fig.1.(iv), we detect substructures, such as 2D faces, aligned and nested 2D faces, 3D boxes, aligned and stacked 3D boxes (in Fig.3(a)) layer by layer. The clusters Cl are formed only if the cooperative “+” constraints are satisﬁed. The proposal probability for each cluster Cl is deﬁned as P+(Cl\|I) = Y v∈ClOR P OR(AT (v)) Y u,v∈ClAND P AND + (AG(u), AG(v)) Y v∈ClT P T (I(Λv)). (3) Clusters with marginal probabilities below threshold are pruned. The threshold is learned by a probably approximately admissible (PAA) bound [23]. The clusters so deﬁned are enumerable. In the sampling stage, we performs an efﬁcient MCMC inference to search in the combinational space. In each step, the Markov chain jumps over a cluster (a big set of nodes) given information of ”what goes together” from clustering. The algorithm proposes a new parse tree: pt∗= pt+Cl∗with the cluster Cl∗conditioning on the current parse tree pt. To avoid heavy computation, the proposal probability is deﬁned as Q(pt∗\|pt, I) = P+(Cl∗\|I) Y u∈ClAND,v∈ptAND P AND − (AG(u)\|AG(v)). (4) The algorithm gives more weights to the proposals with strong bottom-up support and tight “+” relations by P+(Cl\|I), and simultaneously avoids the exclusive proposals with “-” relations by P AND − (AG(u)\|AG(v)). All of these probabilities are pre-computed before sampling. The marginal probability of each cluster P+(Cl\|I) is computed during the clustering stage, and the probability for each pair-wise negative “-” relations P AND − (AG(u)\|AG(v)) is then calculated and stored in a look-up table. The algorithm also proposes a new parse tree by pruning current parse tree randomly. By applying the Metropolis-Hastings acceptance probability α(pt →pt∗) = min{1, Q(pt\|pt∗,I) Q(pt∗\|pt,I) · P (pt∗\|I) P (pt\|I) }, the Markov chain search satisﬁes the detailed balance principle, which implies that the Markov chain search will converge to the global optimum in Fig.5. 5 Experiments We evaluate our algorithm on both the UIUC indoor dataset [2] and our own dataset. The UIUC dataset contains 314 cluttered indoor images, of which the ground-truth is two label maps of background layout with/without foreground objects. Our dataset contains 220 images which cover six 6 2D face detection 0 0.2 0.4 0.6 0.8 1 False nagative rate True positive rate cluster proposals after inference 0 0.2 0.4 0.6 0.8 1 3D foreground detection 0 0.2 0.4 0.6 0.8 1 False nagative rate True positive rate cluster proposals after inference 0 0.2 0.4 0.6 0.8 1 (a) (b) (c) Figure 6: Quantitative performance of 2D face detection (a) and 3D foreground detection (b) in our dataset. (c) An example of the top proposals and the result after inference. indoor scene categories: bedroom, living room, kitchen, classroom, ofﬁce room, and corridor. The dataset is available on the project webpage1. The ground-truths are hand labeled segments for scene components for each image. Our algorithm usually takes 20s in clustering, 40s in sampling, and 1m in preparing input features. Qualitative evaluation: The experimental results in Fig.7 is obtained by applying different production rules to images in our dataset. With the AND rules only, the algorithm obtains reasonable results and successfully recovers some salient 3D foreground objects and 2D faces. With both the AND and SET rules, the cooperative “+” relations help detect some weak visual entities. Fig.8 lists more experimental results of the UIUC dataset. The proposed algorithm recovers most of the indoor components. In the last row, we show some challenging images with missing detections and false positives. Weak line information, ambiguous overlapping objects, salient patterns and clustered structures would confuse our algorithm. Quantitative evaluation: We ﬁrst evaluate the detection of 2D faces, 3D foreground objects in our dataset. The detection error is measured on the pixel level, it indicates how many pixels are correctly labelled. In Fig.6, the red curves show the ROC of 2D faces / 3D objects detection in clustering stage. They are computed by thresholding cluster probabilities given by Eq.3. The blue curves show the ROC of ﬁnal detection given a partial parse tree after MCMC inference. They are computed by thresholding the marginal probability given Eq.2. Using the UIUC dataset, we compare our algorithm to four other state-of-the-art indoor scene parsing algorithms, Hoiem et al. [1], Hedau et al. [2], Wang et al. [3] and Lee et al. [4]. All of these four algorithms used discriminative learning of Structure-SVM (or Latent-SVM). By applying the production rules and the contextual relations, our generative grammar model outperforms others as shown in Table.1. 6 Conclusion In this paper, we propose a framework of geometric image parsing using Stochastic Scene Grammar (SSG). The grammar model is used to represent the compositional structure of visual entities. It is beyond the traditional probabilistic context-free grammars (PCFGs) in a few aspects: spatial context, production rules for multiple occurrences of objects, richer image appearance and geometric properties. We also design a hierarchical cluster sampling algorithm that uses contextual relations to accelerate the Markov chain search. The SSG model is ﬂexible to model other compositional structures by applying different production rules and contextual relations. An interesting extension of our work can be adding semantic labels, such as chair, desk, shelf etc., to 3D objects. This will be interesting to discover new relations between TV and sofa, desk and chair, bed and night table as demonstrated in [26]. Acknowledgments The work is supported by grants from NSF IIS-1018751, NSF CNS-1028381 and ONR MURI N00014-10-1-0933. 1http://www.stat.ucla.edu/˜ybzhao/research/sceneparsing 7 Figure 7: Experimental results by applying the AND/OR rules (the ﬁrst row) and applying all AND/OR/SET rules (the second row) in our dataset Figure 8: Experimental results of more complex indoor images in UIUC dataset [2]. The last row shows some challenging images with missing detections and false positives of proposed algorithm. Table 1: Segmentation precision compared with Hoiem et al. 2007 [1], Hedau et al. 2009 [2], Wang et al. 2010 [3] and Lee et al. 2010 [4] in the UIUC dataset [2]. Segmentation precision [1] [2] [3] [4] Our method Without rules 73.5% 78.8% 79.9% 81.4% 80.5% With 3D “-” constraints 83.8% 84.4% With AND, OR rules 85.1% With AND, OR, SET rules 85.5% 8 References [1] Hoiem, D., Efors, A., & Hebert, M. (2007) Recovering Surface Layout from an Image IJCV 75(1). [2] Hedau, V., Hoiem, D., & Forsyth, D. (2009) Recovering the spatial layout of cluttered rooms. In ICCV. [3] Wang, H., Gould, S. & Koller, D. (2010) Discriminative Learning with Latent Variables for Cluttered Indoor Scene Understanding. ECCV. [4] Lee, D., Gupta, A. Hebert, M., & Kanade, T. 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4,417	Dynamic Pooling and Unfolding Recursive Autoencoders for Paraphrase Detection Richard Socher, Eric H. Huang, Jeffrey Pennington∗, Andrew Y. Ng, Christopher D. Manning Computer Science Department, Stanford University, Stanford, CA 94305, USA ∗SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA richard@socher.org, {ehhuang,jpennin,ang,manning}@stanford.edu Abstract Paraphrase detection is the task of examining two sentences and determining whether they have the same meaning. In order to obtain high accuracy on this task, thorough syntactic and semantic analysis of the two statements is needed. We introduce a method for paraphrase detection based on recursive autoencoders (RAE). Our unsupervised RAEs are based on a novel unfolding objective and learn feature vectors for phrases in syntactic trees. These features are used to measure the word- and phrase-wise similarity between two sentences. Since sentences may be of arbitrary length, the resulting matrix of similarity measures is of variable size. We introduce a novel dynamic pooling layer which computes a ﬁxed-sized representation from the variable-sized matrices. The pooled representation is then used as input to a classiﬁer. Our method outperforms other state-of-the-art approaches on the challenging MSRP paraphrase corpus. 1 Introduction Paraphrase detection determines whether two phrases of arbitrary length and form capture the same meaning. Identifying paraphrases is an important task that is used in information retrieval, question answering [1], text summarization, plagiarism detection [2] and evaluation of machine translation [3], among others. For instance, in order to avoid adding redundant information to a summary one would like to detect that the following two sentences are paraphrases: S1 The judge also refused to postpone the trial date of Sept. 29. S2 Obus also denied a defense motion to postpone the September trial date. We present a joint model that incorporates the similarities between both single word features as well as multi-word phrases extracted from the nodes of parse trees. Our model is based on two novel components as outlined in Fig. 1. The ﬁrst component is an unfolding recursive autoencoder (RAE) for unsupervised feature learning from unlabeled parse trees. The RAE is a recursive neural network. It learns feature representations for each node in the tree such that the word vectors underneath each node can be recursively reconstructed. These feature representations are used to compute a similarity matrix that compares both the single words as well as all nonterminal node features in both sentences. In order to keep as much of the resulting global information of this comparison as possible and deal with the arbitrary length of the two sentences, we then introduce our second component: a new dynamic pooling layer which outputs a ﬁxed-size representation. Any classiﬁer such as a softmax classiﬁer can then be used to classify whether the two sentences are paraphrases or not. We ﬁrst describe the unsupervised feature learning with RAEs followed by a description of pooling and classiﬁcation. In experiments we show qualitative comparisons of different RAE models and describe our state-of-the-art results on the Microsoft Research Paraphrase (MSRP) Corpus introduced by Dolan et al. [4]. Lastly, we discuss related work. 1 Recursive Autoencoder 1 2 3 4 5 6 7 3 4 5 1 2 n Dynamic Pooling and Classification 12345 1 2 3 4 5 6 7 cats catch mice The Cats eat mice Softmax Classifier Paraphrase Dynamic Pooling Layer Fixed-Sized Matrix Variable-Sized Similarity Matrix Figure 1: An overview of our paraphrase model. The recursive autoencoder learns phrase features for each node in a parse tree. The distances between all nodes then ﬁll a similarity matrix whose size depends on the length of the sentences. Using a novel dynamic pooling layer we can compare the variable-sized sentences and classify pairs as being paraphrases or not. 2 Recursive Autoencoders In this section we describe two variants of unsupervised recursive autoencoders which can be used to learn features from parse trees. The RAE aims to ﬁnd vector representations for variable-sized phrases spanned by each node of a parse tree. These representations can then be used for subsequent supervised tasks. Before describing the RAE, we brieﬂy review neural language models which compute word representations that we give as input to our algorithm. 2.1 Neural Language Models The idea of neural language models as introduced by Bengio et al. [5] is to jointly learn an embedding of words into an n-dimensional vector space and to use these vectors to predict how likely a word is given its context. Collobert and Weston [6] introduced a new neural network model to compute such an embedding. When these networks are optimized via gradient ascent the derivatives modify the word embedding matrix L ∈Rn×\|V \|, where \|V \| is the size of the vocabulary. The word vectors inside the embedding matrix capture distributional syntactic and semantic information via the word’s co-occurrence statistics. For further details and evaluations of these embeddings, see [5, 6, 7, 8]. Once this matrix is learned on an unlabeled corpus, we can use it for subsequent tasks by using each word’s vector (a column in L) to represent that word. In the remainder of this paper, we represent a sentence (or any n-gram) as an ordered list of these vectors (x1, . . . , xm). This word representation is better suited for autoencoders than the binary number representations used in previous related autoencoder models such as the recursive autoassociative memory (RAAM) model of Pollack [9, 10] or recurrent neural networks [11] since the activations are inherently continuous. 2.2 Recursive Autoencoder Fig. 2 (left) shows an instance of a recursive autoencoder (RAE) applied to a given parse tree as introduced by [12]. Unlike in that work, here we assume that such a tree is given for each sentence by a parser. Initial experiments showed that having a syntactically plausible tree structure is important for paraphrase detection. Assume we are given a list of word vectors x = (x1, . . . , xm) as described in the previous section. The binary parse tree for this input is in the form of branching triplets of parents with children: (p →c1c2). The trees are given by a syntactic parser. Each child can be either an input word vector xi or a nonterminal node in the tree. For both examples in Fig. 2, we have the following triplets: ((y1 →x2x3), (y2 →x1y1)), ∀x, y ∈Rn. Given this tree structure, we can now compute the parent representations. The ﬁrst parent vector p = y1 is computed from the children (c1, c2) = (x2, x3) by one standard neural network layer: p = f(We[c1; c2] + b), (1) where [c1; c2] is simply the concatenation of the two children, f an element-wise activation function such as tanh and We ∈Rn×2n the encoding matrix that we want to learn. One way of assessing how well this n-dimensional vector represents its direct children is to decode their vectors in a 2 Recursive Autoencoder Unfolding Recursive Autoencoder x1' x2' x3' x2 x3 x1 y2 y1 x1' y1' We x2 x3 x1 y2 y1 y1' Wd We We We Wd Wd Figure 2: Two autoencoder models with details of the reconstruction at node y2. For simplicity we left out the reconstruction layer at the ﬁrst node y1 which is the same standard autoencoder for both models. Left: A standard autoencoder that tries to reconstruct only its direct children. Right: The unfolding autoencoder which tries to reconstruct all leaf nodes underneath each node. reconstruction layer and then to compute the Euclidean distance between the original input and its reconstruction: [c′ 1; c′ 2] = f(Wdp + bd) Erec(p) = \|\|[c1; c2] −[c′ 1; c′ 2]\|\|2 . (2) In order to apply the autoencoder recursively, the same steps repeat. Now that y1 is given, we can use Eq. 1 to compute y2 by setting the children to be (c1, c2) = (x1, y1). Again, after computing the intermediate parent vector p = y2, we can assess how well this vector captures the content of the children by computing the reconstruction error as in Eq. 2. The process repeats until the full tree is constructed and each node has an associated reconstruction error. During training, the goal is to minimize the reconstruction error of all input pairs at nonterminal nodes p in a given parse tree T : Erec(T ) = X p∈T Erec(p) (3) For the example in Fig. 2 (left), we minimize Erec(T ) = Erec(y1) + Erec(y2). Since the RAE computes the hidden representations it then tries to reconstruct, it could potentially lower reconstruction error by shrinking the norms of the hidden layers. In order to prevent this, we add a length normalization layer p = p/\|\|p\|\| to this RAE model (referred to as the standard RAE). Another more principled solution is to use a model in which each node tries to reconstruct its entire subtree and then measure the reconstruction of the original leaf nodes. Such a model is described in the next section. 2.3 Unfolding Recursive Autoencoder The unfolding RAE has the same encoding scheme as the standard RAE. The difference is in the decoding step which tries to reconstruct the entire spanned subtree underneath each node as shown in Fig. 2 (right). For instance, at node y2, the reconstruction error is the difference between the leaf nodes underneath that node [x1; x2; x3] and their reconstructed counterparts [x′ 1; x′ 2; x′ 3]. The unfolding produces the reconstructed leaves by starting at y2 and computing [x′ 1; y′ 1] = f(Wdy2 + bd). (4) Then it recursively splits y′ 1 again to produce vectors [x′ 2; x′ 3] = f(Wdy′ 1 + bd). (5) In general, we repeatedly use the decoding matrix Wd to unfold each node with the same tree structure as during encoding. The reconstruction error is then computed from a concatenation of the word vectors in that node’s span. For a node y that spans words i to j: Erec(y(i,j)) = [xi; . . . ; xj] − x′ i; . . . ; x′ j 2 . (6) The unfolding autoencoder essentially tries to encode each hidden layer such that it best reconstructs its entire subtree to the leaf nodes. Hence, it will not have the problem of hidden layers shrinking in norm. Another potential problem of the standard RAE is that it gives equal weight to the last merged phrases even if one is only a single word (in Fig. 2, x1 and y1 have similar weight in the last merge). In contrast, the unfolding RAE captures the increased importance of a child when the child represents a larger subtree. 3 2.4 Deep Recursive Autoencoder Both types of RAE can be extended to have multiple encoding layers at each node in the tree. Instead of transforming both children directly into parent p, we can have another hidden layer h in between. While the top layer at each node has to have the same dimensionality as each child (in order for the same network to be recursively compatible), the hidden layer may have arbitrary dimensionality. For the two-layer encoding network, we would replace Eq. 1 with the following: h = f(W (1) e [c1; c2] + b(1) e ) (7) p = f(W (2) e h + b(2) e ). (8) 2.5 RAE Training For training we use a set of parse trees and then minimize the sum of all nodes’ reconstruction errors. We compute the gradient efﬁciently via backpropagation through structure [13]. Even though the objective is not convex, we found that L-BFGS run with mini-batch training works well in practice. Convergence is smooth and the algorithm typically ﬁnds a good locally optimal solution. After the unsupervised training of the RAE, we demonstrate that the learned feature representations capture syntactic and semantic similarities and can be used for paraphrase detection. 3 An Architecture for Variable-Sized Similarity Matrices Now that we have described the unsupervised feature learning, we explain how to use these features to classify sentence pairs as being in a paraphrase relationship or not. 0.7 0.2 0.6 0.1 0.1 Figure 3: Example of the dynamic min-pooling layer ﬁnding the smallest number in a pooling window region of the original similarity matrix S. 3.1 Computing Sentence Similarity Matrices Our method incorporates both single word and phrase similarities in one framework. First, the RAE computes phrase vectors for the nodes in a given parse tree. We then compute Euclidean distances between all word and phrase vectors of the two sentences. These distances ﬁll a similarity matrix S as shown in Fig. 1. For computing the similarity matrix, the rows and columns are ﬁrst ﬁlled by the words in their original sentence order. We then add to each row and column the nonterminal nodes in a depth-ﬁrst, right-to-left order. Simply extracting aggregate statistics of this table such as the average distance or a histogram of distances cannot accurately capture the global structure of the similarity comparison. For instance, paraphrases often have low or zero Euclidean distances in elements close to the diagonal of the similarity matrix. This happens when similar words align well between the two sentences. However, since the matrix dimensions vary based on the sentence lengths one cannot simply feed the similarity matrix into a standard neural network or classiﬁer. 3.2 Dynamic Pooling Consider a similarity matrix S generated by sentences of lengths n and m. Since the parse trees are binary and we also compare all nonterminal nodes, S ∈R(2n−1)×(2m−1). We would like to map S into a matrix Spooled of ﬁxed size, np × np. Our ﬁrst step in constructing such a map is to partition the rows and columns of S into np roughly equal parts, producing an np × np grid.1 We then deﬁne Spooled to be the matrix of minimum values of each rectangular region within this grid, as shown in Fig. 3. The matrix Spooled loses some of the information contained in the original similarity matrix but it still captures much of its global structure. Since elements of S with small Euclidean distances show that 1The partitions will only be of equal size if 2n −1 and 2m −1 are divisible by np. We account for this in the following way, although many alternatives are possible. Let the number of rows of S be R = 2n −1. Each pooling window then has ⌊R/np⌋many rows. Let M = R mod np, be the number of remaining rows. We then evenly distribute these extra rows to the last M window regions which will have ⌊R/np⌋+ 1 rows. The same procedure applies to the number of columns for the windows. This procedure will have a slightly ﬁner granularity for the single word similarities which is desired for our task since word overlap is a good indicator for paraphrases. In the rare cases when np > R, the pooling layer needs to ﬁrst up-sample. We achieve this by simply duplicating pixels row-wise until R ≥np. 4 Center Phrase Recursive Average RAE Unfolding RAE the U.S. the U.S. and German the Swiss the former U.S. suffering low morale suffering a 1.9 billion baht UNK 76 million suffering due to no fault of my own suffering heavy casualties to watch hockey to watch one Jordanian border policeman stamp the Israeli passports to watch television to watch a video advance to the next round advance to ﬁnal qualifying round in Argentina advance to the ﬁnal of the UNK 1.1 million Kremlin Cup advance to the semis a prominent political ﬁgure such a high-proﬁle ﬁgure the second high-proﬁle opposition ﬁgure a powerful business ﬁgure Seventeen people were killed ”Seventeen people were killed, including a prominent politician ” Fourteen people were killed Fourteen people were killed conditions of his release ”conditions of peace, social stability and political harmony ” conditions of peace, social stability and political harmony negotiations for their release Table 1: Nearest neighbors of randomly chosen phrases. Recursive averaging and the standard RAE focus mostly on the last merged words and incorrectly add extra information. The unfolding RAE captures most closely both syntactic and semantic similarities. there are similar words or phrases in both sentences, we keep this information by applying a min function to the pooling regions. Other functions, like averaging, are also possible, but might obscure the presence of similar phrases. This dynamic pooling layer could make use of overlapping pooling regions, but for simplicity, we consider only non-overlapping pooling regions. After pooling, we normalize each entry to have 0 mean and variance 1. 4 Experiments For unsupervised RAE training we used a subset of 150,000 sentences from the NYT and AP sections of the Gigaword corpus. We used the Stanford parser [14] to create the parse trees for all sentences. For initial word embeddings we used the 100-dimensional vectors computed via the unsupervised method of Collobert and Weston [6] and provided by Turian et al. [8]. For all paraphrase experiments we used the Microsoft Research paraphrase corpus (MSRP) introduced by Dolan et al. [4]. The dataset consists of 5,801 sentence pairs. The average sentence length is 21, the shortest sentence has 7 words and the longest 36. 3,900 are labeled as being in the paraphrase relationship (technically deﬁned as “mostly bidirectional entailment”). We use the standard split of 4,076 training pairs (67.5% of which are paraphrases) and 1,725 test pairs (66.5% paraphrases). All sentences were labeled by two annotators who agreed in 83% of the cases. A third annotator resolved conﬂicts. During dataset collection, negative examples were selected to have high lexical overlap to prevent trivial examples. For more information see [4, 15]. As described in Sec. 2.4, we can have deep RAE networks with two encoding or decoding layers. The hidden RAE layer (see h in Eq. 8) was set to have 200 units for both standard and unfolding RAEs. 4.1 Qualitative Evaluation of Nearest Neighbors In order to show that the learned feature representations capture important semantic and syntactic information even for higher nodes in the tree, we visualize nearest neighbor phrases of varying length. After embedding sentences from the Gigaword corpus, we compute nearest neighbors for all nodes in all trees. In Table 1 the ﬁrst phrase is a randomly chosen phrase and the remaining phrases are the closest phrases in the dataset that are not in the same sentence. We use Euclidean distance between the vector representations. Note that we do not constrain the neighbors to have the same word length. We compare the two autoencoder models above: RAE and unfolding RAE without hidden layers, as well as a recursive averaging baseline (R.Avg). R.Avg recursively takes the average of both child vectors in the syntactic tree. We only report results of RAEs without hidden layers between the children and parent vectors. Even though the deep RAE networks have more parameters to learn complex encodings they do not perform as well in this and the next task. This is likely due to the fact that they get stuck in local optima during training. 5 Encoding Input Generated Text from Unfolded Reconstruction a December summit a December summit the ﬁrst qualifying session the ﬁrst qualifying session English premier division club Irish presidency division club the safety of a ﬂight the safety of a ﬂight the signing of the accord the signing of the accord the U.S. House of Representatives the U.S. House of Representatives enforcement of the economic embargo enforcement of the national embargo visit and discuss investment possibilities visit and postpone ﬁnancial possibilities the agreement it made with Malaysia the agreement it made with Malaysia the full bloom of their young lives the lower bloom of their democratic lives the organization for which the men work the organization for Romania the reform work a pocket knife was found in his suitcase in the plane’s cargo hold a bomb corpse was found in the mission in the Irish car language case Table 2: Original inputs and generated output from unfolding and reconstruction. Words are the nearest neighbors to the reconstructed leaf node vectors. The unfolding RAE can reconstruct perfectly almost all phrases of 2 and 3 words and many with up to 5 words. Longer phrases start to get incorrect nearest neighbor words. For the standard RAE good reconstructions are only possible for two words. Recursive averaging cannot recover any words. Table 1 shows several interesting phenomena. Recursive averaging is almost entirely focused on an exact string match of the last merged words of the current phrase in the tree. This leads the nearest neighbors to incorrectly add various extra information which would break the paraphrase relationship if we only considered the top node vectors and ignores syntactic similarity. The standard RAE does well though it is also somewhat focused on the last merges in the tree. Finally, the unfolding RAE captures most closely the underlying syntactic and semantic structure. 4.2 Reconstructing Phrases via Recursive Decoding In this section we analyze the information captured by the unfolding RAE’s 100-dimensional phrase vectors. We show that these 100-dimensional vector representations can not only capture and memorize single words but also longer, unseen phrases. In order to show how much of the information can be recovered we recursively reconstruct sentences after encoding them. The process is similar to unfolding during training. It starts from a phrase vector of a nonterminal node in the parse tree. We then unfold the tree as given during encoding and ﬁnd the nearest neighbor word to each of the reconstructed leaf node vectors. Table 2 shows that the unfolding RAE can very well reconstruct phrases of up to length ﬁve. No other method that we compared had such reconstruction capabilities. Longer phrases retain some correct words and usually the correct part of speech but the semantics of the words get merged. The results are from the unfolding RAE that directly computes the parent representation as in Eq. 1. 4.3 Evaluation on Full-Sentence Paraphrasing We now turn to evaluating the unsupervised features and our dynamic pooling architecture in our main task of paraphrase detection. Methods which are based purely on vector representations invariably lose some information. For instance, numbers often have very similar representations, but even small differences are crucial to reject the paraphrase relation in the MSRP dataset. Hence, we add three number features. The ﬁrst is 1 if two sentences contain exactly the same numbers or no number and 0 otherwise, the second is 1 if both sentences contain the same numbers and the third is 1 if the set of numbers in one sentence is a strict subset of the numbers in the other sentence. Since our pooling-layer cannot capture sentence length or the number of exact string matches, we also add the difference in sentence length and the percentage of words and phrases in one sentence that are in the other sentence and vice-versa. We also report performance without these three features (only S). For all of our models and training setups, we perform 10-fold cross-validation on the training set to choose the best regularization parameters and np, the size of the pooling matrix S ∈Rnp×np. In our best model, the regularization for the RAE was 10−5 and 0.05 for the softmax classiﬁer. The best pooling size was consistently np = 15, slightly less than the average sentence length. For all sentence pairs (S1, S2) in the training data, we also added (S2, S1) to the training set in order to make the most use of the training data. This improved performance by 0.2%. 6 Model Acc. F1 All Paraphrase Baseline 66.5 79.9 Rus et al. (2008) [16] 70.6 80.5 Mihalcea et al. (2006) [17] 70.3 81.3 Islam and Inkpen (2007) [18] 72.6 81.3 Qiu et al. (2006) [19] 72.0 81.6 Fernando and Stevenson (2008) [20] 74.1 82.4 Wan et al. (2006) [21] 75.6 83.0 Das and Smith (2009) [15] 73.9 82.3 Das and Smith (2009) + 18 Features 76.1 82.7 Unfolding RAE + Dynamic Pooling 76.8 83.6 Table 3: Test results on the MSRP paraphrase corpus. Comparisons of unsupervised feature learning methods (left), similarity feature extraction and supervised classiﬁcation methods (center) and other approaches (right). In our ﬁrst set of experiments we compare several unsupervised feature learning methods: Recursive averaging as deﬁned in Sec. 4.1, standard RAEs and unfolding RAEs. For each of the three methods, we cross-validate on the training data over all possible hyperparameters and report the best performance. We observe that the dynamic pooling layer is very powerful because it captures the global structure of the similarity matrix which in turn captures the syntactic and semantic similarities of the two sentences. With the help of this powerful dynamic pooling layer and good initial word vectors even the standard RAE and recursive averaging perform well on this dataset with an accuracy of 75.5% and 75.9% respectively. We obtain the best accuracy of 76.8% with the unfolding RAE without hidden layers. We tried adding 1 and 2 hidden encoding and decoding layers but performance only decreased by 0.2% and training became slower. Next, we compare the dynamic pooling to simpler feature extraction methods. Our comparison shows that the dynamic pooling architecture is important for achieving high accuracy. For every setting we again exhaustively cross-validate on the training data and report the best performance. The settings and their accuracies are: (i) S-Hist: 73.0%. A histogram of values in the matrix S. The low performance shows that our dynamic pooling layer better captures the global similarity information than aggregate statistics. (ii) Only Feat: 73.2%. Only the three features described above. This shows that simple binary string and number matching can detect many of the simple paraphrases but fails to detect complex cases. (iii) Only Spooled: 72.6%. Without the three features mentioned above. This shows that some information still gets lost in Spooled and that a better treatment of numbers is needed. In order to better recover exact string matches it may be necessary to explore overlapping pooling regions. (iv) Top Unfolding RAE Node: 74.2%. Instead of Spooled, use Euclidean distance between the two top sentence vectors. The performance shows that while the unfolding RAE is by itself very powerful, the dynamic pooling layer is needed to extract all information from its trees. Table 3 shows our results compared to previous approaches (see next section). Our unfolding RAE and dynamic similarity pooling architecture achieves state-of-the-art performance without handdesigned semantic taxonomies and features such as WordNet. Note that the effective range of the accuracy lies between 66% (most frequent class baseline) and 83% (interannotator agreement). In Table 4 we show several examples of correctly classiﬁed paraphrase candidate pairs together with their similarity matrix after dynamic min-pooling. The ﬁrst and last pair are simple cases of paraphrase and not paraphrase. The second example shows a pooled similarity matrix when large chunks are swapped in both sentences. Our model is very robust to such transformations and gives a high probability to this pair. Even more complex examples such as the third with very few direct string matches (few blue squares) are correctly classiﬁed. The second to last example is highly interesting. Even though there is a clear diagonal with good string matches, the gap in the center shows that the ﬁrst sentence contains much extra information. This is also captured by our model. 5 Related Work The ﬁeld of paraphrase detection has progressed immensely in recent years. Early approaches were based purely on lexical matching techniques [22, 23, 19, 24]. Since these methods are often based on exact string matches of n-grams, they fail to detect similar meaning that is conveyed by synonymous words. Several approaches [17, 18] overcome this problem by using Wordnet- and corpus-based semantic similarity measures. In their approach they choose for each open-class word the single most similar word in the other sentence. Fernando and Stevenson [20] improved upon this idea by computing a similarity matrix that captures all pair-wise similarities of single words in the two sentences. They then threshold the elements of the resulting similarity matrix and compute the mean 7 L Pr Sentences Sim.Mat. P 0.95 (1) LLEYTON Hewitt yesterday traded his tennis racquet for his ﬁrst sporting passion Australian football - as the world champion relaxed before his Wimbledon title defence (2) LLEYTON Hewitt yesterday traded his tennis racquet for his ﬁrst sporting passionAustralian rules football-as the world champion relaxed ahead of his Wimbledon defence P 0.82 (1) The lies and deceptions from Saddam have been well documented over 12 years (2) It has been well documented over 12 years of lies and deception from Saddam P 0.67 (1) Pollack said the plaintiffs failed to show that Merrill and Blodget directly caused their losses (2) Basically, the plaintiffs did not show that omissions in Merrill’s research caused the claimed losses N 0.49 (1) Prof Sally Baldwin, 63, from York, fell into a cavity which opened up when the structure collapsed at Tiburtina station, Italian railway ofﬁcials said (2) Sally Baldwin, from York, was killed instantly when a walkway collapsed and she fell into the machinery at Tiburtina station N 0.44 (1) Bremer, 61, is a onetime assistant to former Secretaries of State William P. Rogers and Henry Kissinger and was ambassador-at-large for counterterrorism from 1986 to 1989 (2) Bremer, 61, is a former assistant to former Secretaries of State William P. Rogers and Henry Kissinger N 0.11 (1) The initial report was made to Modesto Police December 28 (2) It stems from a Modesto police report Table 4: Examples of sentence pairs with: ground truth labels L (P - Paraphrase, N - Not Paraphrase), the probabilities our model assigns to them (Pr(S1, S2) > 0.5 is assigned the label Paraphrase) and their similarity matrices after dynamic min-pooling. Simple paraphrase pairs have clear diagonal structure due to perfect word matches with Euclidean distance 0 (dark blue). That structure is preserved by our min-pooling layer. Best viewed in color. See text for details. of the remaining entries. There are two shortcomings of such methods: They ignore (i) the syntactic structure of the sentences (by comparing only single words) and (ii) the global structure of such a similarity matrix (by computing only the mean). Instead of comparing only single words [21] adds features from dependency parses. Most recently, Das and Smith [15] adopted the idea that paraphrases have related syntactic structure. Their quasisynchronous grammar formalism incorporates a variety of features from WordNet, a named entity recognizer, a part-of-speech tagger, and the dependency labels from the aligned trees. In order to obtain high performance they combine their parsing-based model with a logistic regression model that uses 18 hand-designed surface features. We merge these word-based models and syntactic models in one joint framework: Our matrix consists of phrase similarities and instead of just taking the mean of the similarities we can capture the global layout of the matrix via our min-pooling layer. The idea of applying an autoencoder in a recursive setting was introduced by Pollack [9] and extended recently by [10]. Pollack’s recursive auto-associative memories are similar to ours in that they are a connectionist, feedforward model. One of the major shortcomings of previous applications of recursive autoencoders to natural language sentences was their binary word representation as discussed in Sec. 2.1. Recently, Bottou discussed related ideas of recursive autoencoders [25] and recursive image and text understanding but without experimental results. Larochelle [26] investigated autoencoders with an unfolded “deep objective”. Supervised recursive neural networks have been used for parsing images and natural language sentences by Socher et al. [27, 28]. Lastly, [12] introduced the standard recursive autoencoder as mentioned in Sect. 2.2. 6 Conclusion We introduced an unsupervised feature learning algorithm based on unfolding, recursive autoencoders. The RAE captures syntactic and semantic information as shown qualitatively with nearest neighbor embeddings and quantitatively on a paraphrase detection task. Our RAE phrase features allow us to compare both single word vectors as well as phrases and complete syntactic trees. In order to make use of the global comparison of variable length sentences in a similarity matrix we introduce a new dynamic pooling architecture that produces a ﬁxed-sized representation. 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4,418	Lower Bounds for Passive and Active Learning Maxim Raginsky∗ Coordinated Science Laboratory University of Illinois at Urbana-Champaign Alexander Rakhlin Department of Statistics University of Pennsylvania Abstract We develop uniﬁed information-theoretic machinery for deriving lower bounds for passive and active learning schemes. Our bounds involve the so-called Alexander’s capacity function. The supremum of this function has been recently rediscovered by Hanneke in the context of active learning under the name of “disagreement coefﬁcient.” For passive learning, our lower bounds match the upper bounds of Gin´e and Koltchinskii up to constants and generalize analogous results of Massart and N´ed´elec. For active learning, we provide ﬁrst known lower bounds based on the capacity function rather than the disagreement coefﬁcient. 1 Introduction Not all Vapnik-Chervonenkis classes are created equal. This was observed by Massart and N´ed´elec [24], who showed that, when it comes to binary classiﬁcation rates on a sample of size n under a margin condition, some classes admit rates of the order 1/n while others only (log n)/n. The latter classes were called “rich” in [24]. As noted by Gin´e and Koltchinskii [15], the ﬁne complexity notion that deﬁnes this “richness” is in fact embodied in Alexander’s capacity function.1 Somewhat surprisingly, the supremum of this function (called the disagreement coefﬁcient by Hanneke [19]) plays a key role in risk bounds for active learning. The contribution of this paper is twofold. First, we prove lower bounds for passive learning based on Alexander’s capacity function, matching the upper bounds of [15] up to constants. Second, we prove lower bounds for the number of label requests in active learning in terms of the capacity function. Our proof techniques are information-theoretic in nature and provide a uniﬁed tool to study active and passive learning within the same framework. Active and passive learning. Let (X, A) be an arbitrary measurable space. Let (X, Y ) be a random variable taking values in X × {0, 1} according to an unknown distribution P = Π ⊗PY \|X, where Π denotes the marginal distribution of X. Here, X is an instance (or a feature, a predictor variable) and Y is a binary response (or a label). Classical results in statistical learning assume availability of an i.i.d. sample {(Xi, Yi)}n i=1 from P. In this framework, the learner is passive and has no control on how this sample is chosen. The classical setting is well studied, and the following question has recently received attention: do we gain anything if data are obtained sequentially, and the learner is allowed to modify the design distribution Π of the predictor variable before receiving the next pair (Xi, Yi)? That is, can the learner actively use the information obtained so far to facilitate faster learning? Two paradigms often appear in the literature: (i) the design distribution is a Dirac delta function at some xi that depends on (xi−1, Y i−1), or (ii) the design distribution is a restriction of the original distribution to some measurable set. There is rich literature on both approaches, and we only mention a few results here. The paradigm (i) is closely related to learning with membership queries [21], generalized binary search [25], and coding with noiseless feedback [6]. The goal is to actively choose the next xi so that the observed Yi ∼PY \|X=xi is sufﬁciently “informative” for the classiﬁcation task. In this paradigm, the sample no longer provides information about the distribution ∗Afﬁliation until January, 2012: Department of Electrical and Computer Engineering, Duke University. 1To be precise, the capacity function depends on the underlying probability distribution. 1 Π (see [7] for further discussion and references). The setting (ii) is often called selective sampling [9, 13, 8], although the term active learning is also used. In this paradigm, the aim is to sequentially choose subsets Di ⊆X based on the observations prior to the ith example, such that the label Yi is requested only if Xi ∈Di. The sequence {Xi}n i=1 is assumed to be i.i.d., and so, form the view point of the learner, the Xi is sampled from the conditional distribution Π(·\|Di). In recent years, several interesting algorithms for active learning and selective sampling have appeared in the literature, most notably: the A2 algorithm of Balcan et al. [4], which explicitly maintains Di as a “disagreement” set of a “version space”; the empirical risk minimization (ERM) based algorithm of Dasgupta et al. [11], which maintains the set Di implicitly through synthetic and real examples; and the importance-weighted active learning algorithm of Beygelzimer et al. [5], which constructs the design distribution through careful reweighting in the feature space. An insightful analysis has been carried out by Hanneke [20, 19], who distilled the role of the so-called disagreement coefﬁcient in governing the performance of several of these active learning algorithms. Finally, Koltchinskii [23] analyzed active learning procedures using localized Rademacher complexities and Alexander’s capacity function, which we discuss next. Alexander’s capacity function. Let F denote a class of candidate classiﬁers, where a classiﬁer is a measurable function f : X →{0, 1}. Suppose the VC dimension of F is ﬁnite: VC-dim(F) = d. The loss (or risk) of f is its probability of error, RP (f) ≜EP [1{f(X)̸=Y }] = P(f(X) ̸= Y ). It is well known that the risk is globally minimized by the Bayes classiﬁer f ∗= f ∗ P , deﬁned by f ∗(x) ≜1{2η(x)≥1}, where η(x) ≜E[Y \|X = x] is the regression function. Deﬁne the margin as h ≜infx∈X \|2η(x) −1\|. If h > 0, we say the problem satisﬁes Massart’s noise condition. We deﬁne the excess risk of a classiﬁer f by EP (f) ≜RP (f) −RP (f ∗), so that EP (f) ≥0, with equality if and only if f = f ∗Π-a.s. Given ε ∈(0, 1], deﬁne Fε(f ∗) ≜{f ∈F : Π(f(X) ̸= f ∗(X)) ≤ε} , Dε(f ∗) ≜{x ∈X : ∃f ∈Fε(f ∗) s.t. f(x) ̸= f ∗(x)} The set Fε consists of all classiﬁers f ∈F that are ε-close to f ∗in the L1(Π) sense, while the set Dε consists of all points x ∈X, for which there exists a classiﬁer f ∈Fε that disagrees with the Bayes classiﬁer f ∗at x. The Alexander’s capacity function [15] is deﬁned as τ(ε) ≜Π(Dε(f ∗))/ε, (1) that is, τ(ε) measures the relative size (in terms of Π) of the disagreement region Dε compared to ε. Clearly, τ(ε) is always bounded above by 1/ε; however, in some cases τ(ε) ≤τ0 with τ0 < ∞. The function τ was originally introduced by Alexander [1, 2] in the context of exponential inequalities for empirical processes indexed by VC classes of functions, and Gin´e and Koltchinskii [15] generalized Alexander’s results. In particular, they proved (see [15, p. 1213]) that, for a VC-class of binary-valued functions with VC-dim(F) = d, the ERM solution bfn = arg minf∈F 1 n Pn i=1 1{f(Xi)̸=Yi} under Massart’s noise condition satisﬁes EP ( bfn) ≤C d nh log τ d nh2 + s nh (2) with probability at least 1 −Ks−1e−s/K for some constants C, K and any s > 0. The upper bound (2) suggests the importance of the Alexander’s capacity function for passive learning, leaving open the question of necessity. Our ﬁrst contribution is a lower bound which matches the upper bound (2) up to constant, showing that, in fact, dependence on the capacity is unavoidable. Recently, Koltchinskii [23] made an important connection between Hanneke’s disagreement coefﬁcient and Alexander’s capacity function. Under Massart’s noise condition, Koltchinskii showed (see [23, Corollary 1]) that, for achieving an excess loss of ε with conﬁdence 1−δ, the number of queries issued by his active learning algorithm is bounded above by C τ0 log(1/ε) h2 [d log τ0 + log(1/δ) + log log(1/ε) + log log(1/h)] , (3) where τ0 = supε∈(0,1] τ(ε) is Hanneke’s disagreement coefﬁcient. Similar bounds based on the disagreement coefﬁcient have appeared in [19, 20, 11]. The second contribution of this paper is a lower bound on the expected number of queries based on Alexander’s capacity τ(ε). 2 Comparison to known lower bounds. For passive learning, Massart and N´ed´elec [24] proved two lower bounds which, in fact, correspond to τ(ε) = 1/ε and τ(ε) = τ0, the two endpoints on the complexity scale for the capacity function. Without the capacity function at hand, the authors emphasize that “rich” VC classes yield a larger lower bound. Our Theorem 1 below gives a uniﬁed construction for all possible complexities τ(ε). In the PAC framework, the lower bound Ω(d/ε + (1/ε) log(1/δ)) goes back to [12]. It follows from our results that in the noisy version of the problem (h ̸= 1), the lower bound is in fact Ω((d/ε) log(1/ε) + (1/ε) log(1/δ)) for classes with τ(ε) = Ω(1/ε). For active learning, Castro and Nowak [7] derived lower bounds, but without the disagreement coefﬁcient and under a Tsybakov-type noise condition. This setting is out of the scope of this paper. Hanneke [19] proved a lower bound on the number of label requests speciﬁcally for the A2 algorithm in terms of the disagreement coefﬁcient. In contrast, lower bounds of Theorem 2 are valid for any algorithm and are in terms of Alexander’s capacity function. Finally, a result by K¨a¨ari¨ainen [22] (strengthened by [5]) gives a lower bound of Ω(ν2/ε2) where ν = inff∈F EP (f). A closer look at the construction of the lower bound reveals that it is achieved by considering a speciﬁc margin h = ε/ν. Such an analysis is somewhat unsatisfying, as we would like to keep h as a free parameter, not necessarily coupled with the desired accuracy ε. This point of view is put forth by Massart and N´ed´elec [24, p. 2329], who argue for a non-asymptotic analysis where all the parameters of the problem are made explicit. We also feel that this gives a better understanding of the problem. 2 Setup and main results We suppose that the instance space X is a countably inﬁnite set. Also, log(·) ≡loge(·) throughout. Deﬁnition 1. Given a VC function class F and a margin parameter h ∈[0, 1], let C(F, h) denote the class of all conditional probability distributions PY \|X of Y ∈{0, 1} given X ∈X, such that: (a) the Bayes classiﬁer f ∗∈F, and (b) the corresponding regression function satisﬁes the Massart condition with margin h > 0. Let P(X) denote the space of all probability measures on X. We now introduce Alexander’s capacity function (1) into the picture. Whenever we need to specify explicitly the dependence of τ(ε) on f ∗and Π, we will write τ(ε; f ∗, Π). We also denote by T the set of all admissible capacity functions τ : (0, 1] →R+, i.e., τ ∈T if and only if there exist some f ∗∈F and Π ∈P(X), such that τ(ε) = τ(ε; f ∗, Π) for all ε ∈(0, 1]. Without loss of generality, we assume τ(ε) ≥2. Deﬁnition 2. Given some Π ∈P(X) and a pair (F, h) as in Def. 1, we let P(Π, F, h) denote the set of all joint distributions of (X, Y ) ∈X × {0, 1} of the form Π ⊗PY \|X, such that PY \|X ∈C(F, h). Moreover, given an admissible function τ ∈T and some ε ∈(0, 1], we let P(Π, F, h, τ, ε) denote the subset of P(Π, F, h), such that τ(ε; f ∗, Π) = τ(ε). Finally, we specify the type of learning schemes we will be dealing with. Deﬁnition 3. An n-step learning scheme S consists of the following objects: n conditional probability distributions Π(t) Xt\|Xt−1,Y t−1, t = 1, . . . , n, and a mapping ψ : X n × {0, 1}n →F. This deﬁnition covers the passive case if we let Π(t) Xt\|Xt−1,Y t−1(·\|xt−1, yt−1) = Π(·), ∀(xt−1, yt−1) ∈X t−1 × {0, 1}t−1 as well as the active case, in which Π(t) Xt\|Xt−1,Y t−1 is the user-controlled design distribution for the feature at time t given all currently available information. The learning process takes place sequentially as follows: At each time step t = 1, . . . , n, a random feature Xt is drawn according to Π(t) Xt−1,Y t−1(·\|Xt−1, Y t−1), and then a label Yt is drawn given Xt. After the n samples {(Xt, Yt)}n t=1 are collected, the learner computes the candidate classiﬁer bfn = ψ(Xn, Y n). To quantify the performance of such a scheme, we need the concept of an induced measure, which generalizes the set-up of [14]. Speciﬁcally, given some P = Π ⊗PY \|X ∈P(Π, F, h), deﬁne the 3 following probability measure on X n × {0, 1}n: PS(xn, yn) = n Y t=1 PY \|X(yt\|xt)Π(t) Xt\|Xt−1,Y t−1(xt\|xt−1, yt−1). Deﬁnition 4. Let Q be a subset of P(Π, F, h). Given an accuracy parameter ε ∈(0, 1) and a conﬁdence parameter δ ∈(0, 1), an n-step learning scheme S is said to (ε, δ)-learn Q if sup P ∈Q PS EP ( bfn) ≥εh ≤δ. (4) Remark 1. Leaving the precision as εh makes the exposition a bit cleaner in light of the fact that, under Massart’s noise condition with margin h, EP (f) ≥h∥f −f ∗ P ∥L1(Π) = hΠ(f(X) ̸= f ∗ P (X)) (cf. Massart and N´ed´elec [24, p. 2352]). With these preliminaries out of the way, we can state the main results of this paper: Theorem 1 (Lower bounds for passive learning). Given any τ ∈T , any sufﬁciently large d ∈N and any ε ∈(0, 1], there exist a probability measure Π ∈P(X) and a VC class F with VC-dim(F) = d with the following properties: (1) Fix any K > 1 and δ ∈(0, 1/2). If there exists an n-step passive learning scheme that (ε/2, δ)learns P(Π, F, h, τ, ε) for some h ∈(0, 1 −K−1], then n = Ω (1 −δ)d log τ(ε) Kεh2 + log 1 δ Kεh2 . (5) (2) If there exists an n-step passive learning scheme that (ε/2, δ)-learns P(Π, F, 1, τ, ε), then n = Ω (1 −δ)d ε . (6) Theorem 2 (Lower bounds for active learning). Given any τ ∈T , any sufﬁciently large d ∈N and any ε ∈(0, 1], there exist a probability measure Π ∈P(X) and a VC class F with VC-dim(F) = d with the following property: Fix any K > 1 and any δ ∈(0, 1/2). If there exists an n-step active learning scheme that (ε/2, δ)-learns P(Π, F, h, τ, ε) for some h ∈(0, 1 −K−1], then n = Ω (1 −δ)d log τ(ε) Kh2 + τ(ε) log 1 δ Kh2 . (7) Remark 2. The lower bound in (6) is well-known and goes back to [12]. We mention it because it naturally arises from our construction. In fact, there is a smooth transition between (5) and (6), with the extra log τ(ε) factor disappearing as h approaches 1. As for the active learning lower bound, we conjecture that d log τ(ε) is, in fact, optimal, and the extra factor of τ0 in dτ0 log τ0 log(1/ε) in (3) arises from the use of a passive learning algorithm as a black box. The remainder of the paper is organized as follows: Section 3 describes the required informationtheoretic tools, which are then used in Section 4 to prove Theorems 1 and 2. The proofs of a number of technical lemmas can be found in the Supplementary Material. 3 Information-theoretic framework Let P and Q be two probability distributions on a common measurable space W. Given a convex function φ : [0, ∞) →R such that φ(1) = 0, the φ-divergence2 between P and Q [3, 10] is given by Dφ(P∥Q) ≜ Z W dQ dµ φ dP/dµ dQ/dµ dµ, (8) where µ is an arbitrary σ-ﬁnite measure that dominates both P and Q.3 For the special case of W = {0, 1}, when P and Q are the distributions of a Bernoulli(p) and a Bernoulli(q) random 2We deviate from the standard term “f-divergence” since f is already reserved for a generic classiﬁer. 3For instance, one can always take µ = P + Q. It it easy to show that the value of Dφ(P∥Q) in (8) does not depend on the choice of the dominating measure. 4 variable, we will denote their φ-divergence by dφ(p∥q) = q · φ p q + (1 −q) · φ 1 −p 1 −q . (9) Two particular choices of φ are of interest: φ(u) = u log u, which gives the ordinary Kullback– Leibler (KL) divergence D(P∥Q), and φ(u) = −log u, which gives the reverse KL divergence D(Q∥P), which we will denote by Dre(P∥Q). We will write d(·∥·) for the binary KL divergence. Our approach makes fundamental use of the data processing inequality that holds for any φdivergence [10]: if P and Q are two possible probability distributions for a random variable W ∈W and if PZ\|W is a conditional probability distribution of some other random variable Z given W, then Dφ(PZ∥QZ) ≤Dφ(P∥Q), (10) where PZ (resp., QZ) is the marginal distribution of Z when W has distribution P (resp., Q). Consider now an arbitrary n-step learning scheme S. Let us ﬁx a ﬁnite set {f1, . . . , fN} ⊂F and assume that to each m ∈[N] we can associate a probability measure P m = Π⊗P m Y \|X ∈P(Π, F, h) with the Bayes classiﬁer f ∗ Pm = fm. For each m ∈[N], let us deﬁne the induced measure PS,m(xn, yn) ≜ n Y t=1 P m Y \|X(yt\|xt)Π(t) Xt\|Xt−1,Y t−1(xt\|xt−1, yt−1). (11) Moreover, given any probability distribution π over [N], let PS,π(m, xn, yn) ≜π(m)PS,m(xn, yn). In other words, PS,π is the joint distribution of (M, Xn, Y n) ∈[N] × X n × {0, 1}n, under which M ∼π and P(Xn, Y n\|M = m) = PS,m(Xn, Y n). The ﬁrst ingredient in our approach is standard [27, 14, 24]. Let {f1, . . . , fN} be an arbitrary 2εpacking subset of F (that is, ∥fi −fj∥L1(Π) > 2ε for all i ̸= j). Suppose that S satisﬁes (4) on some Q that contains {P 1, . . . , P N}. Now consider c M ≡c M(Xn, Y n) ≜arg min 1≤m≤N ∥bfn −fm∥L1(Π). (12) Then the following lemma is easily proved using triangle inequality: Lemma 1. With the above deﬁnitions, PS,π(c M ̸= M) ≤δ. The second ingredient of our approach is an application of the data processing inequality (10) with a judicious choice of φ. Let W ≜(M, Xn, Y n), let M be uniformly distributed over [N], π(m) = 1 N for all m ∈[N], and let P be the induced measure PS,π. Then we have the following lemma (see also [17, 16]): Lemma 2. Consider any probability measure Q for W, under which M is distributed according to π and independent of (Xn, Y n). Let the divergence-generating function φ be such that the mapping p 7→dφ(p∥q) is nondecreasing on the interval [q, 1]. Then, assuming that δ ≤1 −1 N , Dφ(P∥Q) ≥1 N · φ (N(1 −δ)) + 1 −1 N · φ Nδ N −1 . (13) Proof. Deﬁne the indicator random variable Z = 1{c M=M}. Then P(Z = 1) ≥1 −δ by Lemma 1. On the other hand, since Q can be factored as Q(m, xn, yn) = 1 N QXn,Y n(xn, yn), we have Q(Z = 1) = N X m=1 Q(M = m, c M = m) = 1 N N X m=1 X xn,yn QXn,Y n(xn, yn)1{c M(xn,yn)=m} = 1 N . Therefore, Dφ(P∥Q) ≥Dφ(PZ∥QZ) = dφ(P(Z = 1)∥Q(Z = 1)) ≥dφ(1 −δ∥1/N), where the ﬁrst step is by the data processing inequality (10), the second is due to the fact that Z is binary, and the third is by the assumed monotonicity property of φ. Using (9), we arrive at (13). Next, we need to choose the divergence-generating function φ and the auxiliary distribution Q. 5 Choice of φ. Inspection of the right-hand side of (13) suggests that the usual Ω(log N) lower bounds [14, 27, 24] can be obtained if φ(u) behaves like u log u for large u. On the other hand, if φ(u) behaves like −log u for small u, then the lower bounds will be of the form Ω log 1 δ . These observations naturally lead to the respective choices φ(u) = u log u and φ(u) = −log u, corresponding to the KL divergence D(P∥Q) and the reverse KL divergence Dre(P∥Q) = D(Q∥P). Choice of Q. One obvious choice of Q satisfying the conditions of the lemma is the product of the marginals PM ≡π and PXn,Y n ≡N −1 PN m=1 PS,m: Q = PM ⊗PXn,Y n. With this Q and φ(u) = u log u, the left-hand side of (13) is given by D(P∥Q) = D(PM,Xn,Y n∥PM ⊗PXn,Y n) = I(M; Xn, Y n), (14) where I(M; Xn, Y n) is the mutual information between M and (Xn, Y n) with joint distribution P. On the other hand, it is not hard to show that the right-hand side of (13) can be lower-bounded by (1 −δ) log N −log 2. Combining with (14), we get I(M; Xn, Y n) ≥(1 −δ) log N −log 2, which is (a commonly used variant of) the well-known Fano’s inequality [14, Lemma 4.1], [18, p. 1250], [27, p. 1571]. The same steps, but with φ(u) = −log u, lead to the bound L(M; Xn, Y n) ≥ 1 −1 N log 1 δ −log 2 ≥1 2 log 1 δ −log 2, where L(M; Xn, Y n) ≜Dre(PM,Xn,Y n∥PM ⊗PXn,Y n) is the so-called lautum information between M and (Xn, Y n) [26], and the second inequality holds whenever N ≥2. However, it is often more convenient to choose Q as follows. Fix an arbitrary conditional distribution QY \|X of Y ∈{0, 1} given X ∈X. Given a learning scheme S, deﬁne the probability measure QS(xn, yn) ≜ n Y t=1 QY \|X(yt\|xt)Π(t) Xt\|Xt−1,Y t−1(xt\|xt−1, yt−1) (15) and let Q(m, xn, yn) = 1 N QS(xn, yn) for all m ∈[N]. Lemma 3. For each xn ∈X n and y ∈X, let N(y\|xn) ≜\|{1 ≤t ≤n : xt = y}\|. Then D(P∥Q) = 1 N N X m=1 X x∈X D(P m Y \|X(·\|x)∥QY \|X(·\|x))EPS,m [N(x\|Xn)] ; (16) Dre(P∥Q) = 1 N N X m=1 X x∈X Dre(P m Y \|X(·\|x)∥QY \|X(·\|x))EQ [N(x\|Xn)] . (17) Moreover, if the scheme S is passive, then Eq. (17) becomes Dre(P∥Q) = n · EXEM h Dre(P M Y \|X(·\|X)∥QY \|X(·\|X)) i , (18) and the same holds for Dre replaced by D. 4 Proofs of Theorems 1 and 2 Combinatorial preliminaries. Given k ∈N, onsider the k-dimensional Boolean cube {0, 1}k = {β = (β1, . . . , βk) : βi ∈{0, 1}, i ∈[k]}. For any two β, β′ ∈{0, 1}k, deﬁne their Hamming distance dH(β, β′) ≜Pk i=1 1{βi̸=β′ i}. The Hamming weight of any β ∈{0, 1}k is the number of its nonzero coordinates. For k > d, let {0, 1}k d denote the subset of {0, 1}k consisting of all binary strings with Hamming weight d. We are interested in large separated and well-balanced subsets of {0, 1}k d. To that end, we will use the following lemma: Lemma 4. Suppose that d is even and k > 2d. Then, for d sufﬁciently large, there exists a set Mk,d ⊂{0, 1}k d with the following properties: (i) log \|Mk,d\| ≥d 4 log k 6d; (ii) dH(β, β′) > d for any two distinct β, β′ ∈M(2) k,d ; (iii) for any j ∈[k], d 2k ≤ 1 \|Mk,d\| X β∈Mk,d βj ≤3d 2k (19) 6 Proof of Theorem 1. Without loss of generality, we take X = N. Let k = dτ(ε) (we increase ε if necessary to ensure that k ∈N), and consider the probability measure Π that puts mass ε/d on each x = 1 through x = k and the remaining mass 1 −ετ(ε) on x = k + 1. (Recall that τ(ε) ≤1/ε.) Let F be the class of indicator functions of all subsets of X with cardinality d. Then VC-dim(F) = d. We will focus on a particular subclass F′ of F. For each β ∈{0, 1}k d, deﬁne fβ : X →{0, 1} by fβ(x) = βx if x ∈[k] and 0 otherwise, and take F′ = {fβ : β ∈{0, 1}k d}. For p ∈[0, 1], let νp denote the probability distribution of a Bernoulli(p) random variable. Now, to each fβ ∈F′ let us associate the following conditional probability measure P β Y \|X: P β Y \|X(y\|x) = ν(1+h)/2(y)βx + ν(1−h)/2(y)(1 −βx) 1{x∈[k]} + 1{y=0}1{x̸∈[k]} It is easy to see that each P β Y \|X belongs to C(F, h). Moreover, for any two fβ, fβ′ ∈F we have ∥fβ −fβ′∥L1(Π) = Π(fβ(X) ̸= fβ′(X)) = ε d k X i=1 1{βi̸=β′ i} ≡ε ddH(β, β′). Hence, for each choice of f ∗= fβ∗∈F we have Fε(fβ∗) = {fβ : dH(β, β∗) ≤d}. This implies that Dε(fβ∗) = [k], and therefore τ(ε; fβ∗, Π) = Π([k])/ε = τ(ε). We have thus established that, for each β ∈{0, 1}k d, the probability measure P β = Π ⊗P β Y \|X is an element of P(Π, F, h, τ, ε). Finally, let Mk,d ⊂{0, 1}k d be the set described in Lemma 4, and let G ≜{fβ : β ∈Mk,d}. Then for any two distinct β, β′ ∈Mk,d we have ∥fβ −fβ′∥L1(Π) = ε ddH(β, β′) > ε. Hence, G is a ε-packing of F′ in the L1(Π)-norm. Now we are in a position to apply the lemmas of Section 3. Let {β(1), . . . , β(N)}, N = \|Mk,d\|, be a ﬁxed enumeration of the elements of Mk,d. For each m ∈[N], let us denote by P m Y \|X the conditional probability measure P β(m) Y \|X , by P m the measure Π ⊗P m Y \|X on X × {0, 1}, and by fm ∈G the corresponding Bayes classiﬁer. Now consider any n-step passive learning scheme that (ε/2, δ)-learns P(Π, F, h, τ, ε), and deﬁne the probability measure P on [N] × X n × {0, 1}n by P(m, xn, yn) = 1 N PS,m(xn, yn), where PS,m is constructed according to (11). In addition, for every γ ∈(0, 1) deﬁne the auxiliary measure Qγ on [N] × X n × {0, 1}n by Qγ(m, xn, yn) = 1 N QS γ(xn, yn), where QS γ is constructed according to (15) with Qγ Y \|X(y\|x) ≜νγ(y)1{x∈[k]} + 1{y=0}1{x̸∈[k]}. Applying Lemma 2 with φ(u) = u log u, we can write D(P∥Qγ) ≥(1 −δ) log N −log 2 ≥(1 −δ)d 4 log k 6d −log 2 (20) Next we apply Lemma 3. Deﬁning η = 1+h 2 and using the easily proved fact that D(P m Y \|X(·\|x)∥Qγ Y \|X(·\|x)) = [d(η∥γ) −d(1 −η∥γ)] fm(x) + d(1 −η∥γ)1{x∈[k]}, we get D(P∥Qγ) = nε [d(η∥γ) + (τ(ε) −1)d(1 −η∥γ)] . (21) Therefore, combining Eqs. (20) and (21) and using the fact that k = dτ(ε), we obtain n ≥ (1 −δ)d log τ(ε) 6 −log 16 4ε [d(η∥γ) + (τ(ε) −1)d(1 −η∥γ)], ∀γ ∈(0, 1) (22) This bound is valid for all h ∈(0, 1], and the optimal choice of γ for a given h can be calculated in closed form: γ∗(h) = 1−h 2 + h τ(ε). We now turn to the reverse KL divergence. First, suppose that h ̸= 1. Lemma 2 gives Dre(P∥Q1−η) ≥(1/2) log(1/δ) −log 2. On the other hand, using the fact that Dre(P m Y \|X(·\|x)∥Q1−η Y \|X(·\|x)) = d(η∥1 −η)fm(x) (23) 7 and applying Eq. (18), we can write Dre(P∥Q1−η) = nε · d(η∥1 −η) = nε · h log 1 + h 1 −h. (24) We conclude that n ≥ 1 2 log 1 δ −log 2 εh log 1+h 1−h . (25) For h = 1, we get the vacuous bound n ≥0. Now we consider the two cases of Theorem 1. (1) For a ﬁxed K > 1, it follows from the inequality log u ≤u −1 that h log 1+h 1−h ≤Kh2 for all h ∈(0, 1 −K−1]. Choosing γ = 1−h 2 and using Eqs. (22) and (25), we obtain (5). (2) For h = 1, we use (22) with the optimal setting γ∗(1) = 1/τ(ε), which gives (6). The transition between h = 1 and h ̸= 1 is smooth and determined by γ∗(h) = 1−h 2 + h τ(ε). Proof of Theorem 2. We work with the same construction as in the proof of Theorem 1. First, let QXn,Y n ≜ 1 N PN m=1 PS,m. and Q = π ⊗QXn,Y n, where π is the uniform distribution on [N]. Then, by convexity, D(P∥Q) ≤ 1 N 2 N X m,m′=1 EP " n X t=1 log P m Y \|X(Yt\|Xt) P m′ Y \|X(Yt\|Xt) # ≤n max m,m′∈[N] max x∈[k] D(P m Y \|X(·\|x)∥P m′ Y \|X(·\|x)) which is upper bounded by nh log 1+h 1−h. Applying Lemma 2 with φ(u) = u log u, we therefore obtain n ≥(1 −δ)d log k 6d −log 16 4h log 1+h 1−h . (26) Next, consider the auxiliary measure Q1−η with η = 1+h 2 . Then Dre(P∥Q1−η) (a)= 1 N N X M=1 k X x=1 Dre(P m Y \|X(·\|x)∥Q1−η Y \|X(·\|x))EQ1−η[N(x\|Xn)] (b)= d(η∥1 −η) N N X m=1 k X x=1 fm(x)EQ1−η[N(x\|Xn)] = d(η∥1 −η) k X x=1 1 N N X m=1 fm(x) ! EQ1−η[N(x\|Xn)] (c)= d(η∥1 −η) k X x=1 1 N N X m=1 β(m) x ! EQ1−η[N(x\|Xn)] (d) ≤ 3 2τ(ε)h log 1 + h 1 −hEQ1−η " k X x=1 N(x\|Xn) # (e) ≤ 3n 2τ(ε)h log 1 + h 1 −h, where (a) is by Lemma 3, (b) is by (23), (c) is by deﬁnition of {fm}, (d) is by the balance condition (19) satisﬁed by Mk,d, and (e) is by the fact that Pk x=1 N(x\|Xn) ≤P x∈X N(x\|Xn) = n. Applying Lemma 2 with φ(u) = −log u, we get n ≥τ(ε) log 1 δ −log 4 3h log 1+h 1−h (27) Combining (26) and (27) and using the bound h log 1+h 1−h ≤Kh2 for h ∈(0, 1 −K−1], we get (7). 8 References [1] K.S. Alexander. 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4,419	Trace Lasso: a trace norm regularization for correlated designs ´Edouard Grave INRIA, Sierra Project-team ´Ecole Normale Sup´erieure, Paris edouard.grave@inria.fr Guillaume Obozinski INRIA, Sierra Project-team ´Ecole Normale Sup´erieure, Paris guillaume.obozinski@inria.fr Francis Bach INRIA, Sierra Project-team ´Ecole Normale Sup´erieure, Paris francis.bach@inria.fr Abstract Using the ℓ1-norm to regularize the estimation of the parameter vector of a linear model leads to an unstable estimator when covariates are highly correlated. In this paper, we introduce a new penalty function which takes into account the correlation of the design matrix to stabilize the estimation. This norm, called the trace Lasso, uses the trace norm of the selected covariates, which is a convex surrogate of their rank, as the criterion of model complexity. We analyze the properties of our norm, describe an optimization algorithm based on reweighted least-squares, and illustrate the behavior of this norm on synthetic data, showing that it is more adapted to strong correlations than competing methods such as the elastic net. 1 Introduction The concept of parsimony is central in many scientiﬁc domains. In the context of statistics, signal processing or machine learning, it takes the form of variable or feature selection problems, and is commonly used in two situations: ﬁrst, to make the model or the prediction more interpretable or cheaper to use, i.e., even if the underlying problem does not admit sparse solutions, one looks for the best sparse approximation. Second, sparsity can also be used given prior knowledge that the model should be sparse. Many methods have been designed to learn sparse models, namely methods based on greedy algorithms [1, 2], Bayesian inference [3] or convex optimization [4, 5]. In this paper, we focus on the regularization by sparsity-inducing norms. The simplest example of such norms is the ℓ1-norm, leading to the Lasso, when used within a least-squares framework. In recent years, a large body of work has shown that the Lasso was performing optimally in highdimensional low-correlation settings, both in terms of prediction [6], estimation of parameters or estimation of supports [7, 8]. However, most data exhibit strong correlations, with various correlation structures, such as clusters (i.e., close to block-diagonal covariance matrices) or sparse graphs, such as for example problems involving sequences (in which case, the covariance matrix is close to a Toeplitz matrix [9]). In these situations, the Lasso is known to have stability problems: although its predictive performance is not disastrous, the selected predictor may vary a lot (typically, given two correlated variables, the Lasso will only select one of the two, at random). Several remedies have been proposed to this instability. First, the elastic net [10] adds a strongly convex penalty term (the squared ℓ2-norm) that will stabilize selection (typically, given two correlated variables, the elastic net will select the two variables). However, it is blind to the exact 1 correlation structure, and while strong convexity is required for some variables, it is not for other variables. Another solution is to consider the group Lasso, which will divide the predictors into groups and penalize the sum of the ℓ2-norm of these groups [11]. This is known to accomodate strong correlations within groups [12]; however it requires to know the groups in advance, which is not always possible. A third line of research has focused on sampling-based techniques [13, 14, 15]. An ideal regularizer should thus take into account the design (like the group Lasso, with oracle groups), but without requiring human intervention (like the elastic net); it should thus add strong convexity only where needed, and not modifying variables where things behave correctly. In this paper, we propose a new norm towards this end. More precisely we make the following contributions: • We propose in Section 2 a new norm based on the trace norm (a.k.a. nuclear norm) that interpolates between the ℓ1-norm and the ℓ2-norm depending on correlations. • We show that there is a unique minimum when penalizing with this norm in Section 2.2. • We provide optimization algorithms based on reweighted least-squares in Section 3. • We study the second-order expansion around independence and relate it to existing work on including correlations in Section 4. • We perform synthetic experiments in Section 5, where we show that the trace Lasso outperforms existing norms in strong-correlation regimes. Notations. Let M ∈Rn×p. We use superscripts for the columns of M, i.e., M(i) denotes the i-th column, and subscripts for the rows, i.e., Mi denotes the i-th row. For M ∈Rp×p, diag(M) ∈Rp is the diagonal of the matrix M, while for u ∈Rp, Diag(u) ∈Rp×p is the diagonal matrix whose diagonal elements are the ui. Let S be a subset of {1, ..., p}, then uS is the vector u restricted to the support S, with 0 outside the support S. We denote by Sp the set of symmetric matrices of size p. We will use various matrix norms, here are the notations we use: ∥M∥∗is the trace norm, i.e., the sum of the singular values of the matrix M, ∥M∥op is the operator norm, i.e., the maximum singular value of the matrix M, ∥M∥F is the Frobenius norm, i.e., the ℓ2-norm of the singular values, which is also equal to p tr(M⊤M) and ∥M∥2,1 is the sum of the ℓ2-norm of the columns of M: ∥M∥2,1 = p X i=1 ∥M(i)∥2. 2 Deﬁnition and properties of the trace Lasso We consider the problem of predicting y ∈R, given a vector x ∈Rp, assuming a linear model y = w⊤x + ε, where ε is an additive (typically Gaussian) noise with mean 0 and variance σ2. Given a training set X = (x1, ..., xn)⊤∈Rn×p and y = (y1, ..., yn)⊤∈Rn, a widely used method to estimate the parameter vector w is penalized empirical risk minimization ˆw ∈argmin w 1 n n X i=1 ℓ(yi, w⊤xi) + λf(w), (1) where ℓis a loss function used to measure the error we make by predicting w⊤xi instead of yi, while f is a regularization term used to penalize complex models. This second term helps avoiding overﬁtting, especially in the case where we have many more parameters than observation, i.e., n ≪p. 2.1 Related work We will now present some classical penalty functions for linear models which are widely used in the machine learning and statistics community. The ﬁrst one, known as Tikhonov regularization [16] or ridge [17], is the squared ℓ2-norm. When used with the square loss, estimating the parameter vector w is done by solving a linear system. One of the main drawbacks of this penalty function is the fact 2 that it does not perform variable selection and thus does not behave well in sparse high-dimensional settings. Hence, it is natural to penalize linear models by the number of variables used by the model. Unfortunately, this criterion, sometimes denoted by ∥· ∥0 (ℓ0-penalty), is not convex and solving the problem in Eq. (1) is generally NP-Hard [18]. Thus, a convex relaxation for this problem was introduced, replacing the size of the selected subset by the ℓ1-norm of w. This estimator is known as the Lasso [4] in the statistics community and basis pursuit [5] in signal processing. Under some assumptions, the two problems are in fact equivalent (see for example [19] and references therein). When two predictors are highly correlated, the Lasso has a very unstable behavior: it often only selects the variable that is the most correlated with the residual. On the other hand, Tikhonov regularization tends to shrink coefﬁcients of correlated variables together, leading to a very stable behavior. In order to get the best of both worlds, stability and variable selection, Zou and Hastie introduced the elastic net [10], which is the sum of the ℓ1-norm and squared ℓ2-norm. Unfortunately, this estimator needs two regularization parameters and is not adaptive to the precise correlation structure of the data. Some authors also proposed to use pairwise correlations between predictors to interpolate more adaptively between the ℓ1-norm and squared ℓ2-norm, by introducing a method called pairwise elastic net [20] (see comparisons with our approach in Section 5). Finally, when one has more knowledge about the data, for example clusters of variables that should be selected together, one can use the group Lasso [11]. Given a partition (Si) of the set of variables, it is deﬁned as the sum of the ℓ2-norms of the restricted vectors wSi: ∥w∥GL = k X i=1 ∥wSi∥2. The effect of this penalty function is to introduce sparsity at the group level: variables in a group are selected all together. One of the main drawback of this method, which is also sometimes one of its quality, is the fact that one needs to know the partition of the variables, and so one needs to have a good knowledge of the data. 2.2 The ridge, the Lasso and the trace Lasso In this section, we show that Tikhonov regularization and the Lasso penalty can be viewed as norms of the matrix X Diag(w). We then introduce a new norm involving this matrix. The solution of empirical risk minimization penalized by the ℓ1-norm or ℓ2-norm is not equivariant to rescaling of the predictors X(i), so it is common to normalize the predictors. When normalizing the predictors X(i), and penalizing by Tikhonov regularization or by the Lasso, people are implicitly using a regularization term that depends on the data or design matrix X. In fact, there is an equivalence between normalizing the predictors and not normalizing them, using the two following reweighted ℓ2 and ℓ1-norms instead of Tikhonov regularization and the Lasso: ∥w∥2 2 = p X i=1 ∥X(i)∥2 2 w2 i and ∥w∥1 = p X i=1 ∥X(i)∥2 \|wi\|. (2) These two norms can be expressed using the matrix X Diag(w): ∥w∥2 = ∥X Diag(w)∥F and ∥w∥1 = ∥X Diag(w)∥2,1, and a natural question arises: are there other relevant choices of functions or matrix norms? A classical measure of the complexity of a model is the number of predictors used by this model, which is equal to the size of the support of w. This penalty being non-convex, people use its convex relaxation, which is the ℓ1-norm, leading to the Lasso. Here, we propose a different measure of complexity which can be shown to be more adapted in model selection settings [21]: the dimension of the subspace spanned by the selected predictors. This is equal to the rank of the selected predictors, or also to the rank of the matrix X Diag(w). Like the size of the support, this function is non-convex, and we propose to replace it by a convex surrogate, the trace norm, leading to the following penalty that we call “trace Lasso”: Ω(w) = ∥X Diag(w)∥∗. 3 The trace Lasso has some interesting properties: if all the predictors are orthogonal, then, it is equal to the ℓ1-norm. Indeed, we have the decomposition: X Diag(w) = p X i=1 ∥X(i)∥2wi X(i) ∥X(i)∥2 e⊤ i , where ei are the vectors of the canonical basis. Since the predictors are orthogonal and the ei are orthogonal too, this gives the singular value decomposition of X Diag(w) and we get ∥X Diag(w)∥∗= p X i=1 ∥X(i)∥2\|wi\| = ∥X Diag(w)∥2,1. On the other hand, if all the predictors are equal to X(1), then X Diag(w) = X(1)w⊤, and we get ∥X Diag(w)∥∗= ∥X(1)∥2∥w∥2 = ∥X Diag(w)∥F , which is equivalent to Tikhonov regularization. Thus when two predictors are strongly correlated, our norm will behave like Tikhonov regularization, while for almost uncorrelated predictors, it will behave like the Lasso. Always having a unique minimum is an important property for a statistical estimator, as it is a ﬁrst step towards stability. The trace Lasso, by adding strong convexity exactly in the direction of highly correlated covariates, always has a unique minimum, and is thus much more stable than the Lasso. Proposition 1. If the loss function ℓis strongly convex with respect to its second argument, then the solution of the empirical risk minimization penalized by the trace Lasso, i.e., Eq. (1), is unique. The technical proof of this proposition can be found in [22], and consists in showing that in the ﬂat directions of the loss function, the trace Lasso is strongly convex. 2.3 A new family of penalty functions In this section, we introduce a new family of penalties, inspired by the trace Lasso, allowing us to write the ℓ1-norm, the ℓ2-norm and the newly introduced trace Lasso as special cases. In fact, we note that ∥Diag(w)∥∗= ∥w∥1 and ∥p−1/21⊤Diag(w)∥∗= ∥w⊤∥∗= ∥w∥2. In other words, we can express the ℓ1 and ℓ2-norms of w using the trace norm of a given matrix times the matrix Diag(w). A natural question to ask is: what happens when using a matrix P other than the identity or the line vector p−1/21⊤, and what are good choices of such matrices? Therefore, we introduce the following family of penalty functions: Deﬁnition 1. Let P ∈Rk×p, all of its columns having unit norm. We introduce the norm ΩP as ΩP(w) = ∥P Diag(w)∥∗. Proof. The positive homogeneity and triangle inequality are direct consequences of the linearity of w 7→P Diag(w) and the fact that ∥· ∥∗is a norm. Since all the columns of P are not equal to zero, we have P Diag(w) = 0 ⇔w = 0, and so, ΩP separates points and thus is a norm. As stated before, the ℓ1 and ℓ2-norms are special cases of the family of norms we just introduced. Another important penalty that can be expressed as a special case is the group Lasso, with nonoverlapping groups. Given a partition (Sj) of the set {1, ..., p}, the group Lasso is deﬁned by ∥w∥GL = X Sj ∥wSj∥2. We deﬁne the matrix PGL by PGL ij = 1/ p \|Sk\| if i and j are in the same group Sk, 0 otherwise. 4 Figure 1: Unit balls for various value of P⊤P. See the text for the value of P⊤P. (Best seen in color). Then, PGL Diag(w) = X Sj 1Sj p \|Sj\| w⊤ Sj. (3) Using the fact that (Sj) is a partition of {1, ..., p}, the vectors 1Sj are orthogonal and so are the vectors wSj. Hence, after normalizing the vectors, Eq. (3) gives a singular value decomposition of PGL Diag(w) and so the group Lasso penalty can be expressed as a special case of our family of norms: ∥PGL Diag(w)∥∗= X Sj ∥wSj∥2 = ∥w∥GL. In the following proposition, we show that our norm only depends on the value of P⊤P. This is an important property for the trace Lasso, where P = X, since it underlies the fact that this penalty only depends on the correlation matrix X⊤X of the covariates. Proposition 2. Let P ∈Rk×p, all of its columns having unit norm. We have ΩP(w) = ∥(P⊤P)1/2 Diag(w)∥∗. We plot the unit ball of our norm for the following value of P⊤P (see ﬁgure 1): 1 0.9 0.1 0.9 1 0.1 0.1 0.1 1 ! 1 0.7 0.49 0.7 1 0.7 0.49 0.7 1 ! 1 1 0 1 1 0 0 0 1 ! We can lower bound and upper bound our norms by the ℓ2-norm and ℓ1-norm respectively. This shows that, as for the elastic net, our norms interpolate between the ℓ1-norm and the ℓ2-norm. But the main difference between the elastic net and our norms is the fact that our norms are adaptive, and require a single regularization parameter to tune. In particular for the trace Lasso, when two covariates are strongly correlated, it will be close to the ℓ2-norm, while when two covariates are almost uncorrelated, it will behave like the ℓ1-norm. This is a behavior close to the one of the pairwise elastic net [20]. Proposition 3. Let P ∈Rk×p, all of its columns having unit norm. We have ∥w∥2 ≤ΩP(w) ≤∥w∥1. 2.4 Dual norm The dual norm is an important quantity for both optimization and theoretical analysis of the estimator. Unfortunately, we are not able in general to obtain a closed form expression of the dual norm for the family of norms we just introduced. However we can obtain a bound, which is exact for some special cases: Proposition 4. The dual norm, deﬁned by Ω∗ P(u) = max ΩP(v)≤1 u⊤v, can be bounded by: Ω∗ P(u) ≤∥P Diag(u)∥op. 5 Proof. Using the fact that diag(P⊤P) = 1, we have u⊤v = tr Diag(u)P⊤P Diag(v) ≤∥P Diag(u)∥op∥P Diag(v)∥∗, where the inequality comes from the fact that the operator norm ∥· ∥op is the dual norm of the trace norm. The deﬁnition of the dual norm then gives the result. As a corollary, we can bound the dual norm by a constant times the ℓ∞-norm: Ω∗ P(u) ≤∥P Diag(u)∥op ≤∥P∥op∥Diag(u)∥op = ∥P∥op∥u∥∞. Using proposition (3), we also have the inequality Ω∗ P(u) ≥∥u∥∞. 3 Optimization algorithm In this section, we introduce an algorithm to estimate the parameter vector w when the loss function is equal to the square loss: ℓ(y, w⊤x) = 1 2(y −w⊤x)2 and the penalty is the trace Lasso. It is straightforward to extend this algorithm to the family of norms indexed by P. The problem we consider is thus min w 1 2∥y −Xw∥2 2 + λ∥X Diag(w)∥∗. We could optimize this cost function by subgradient descent, but this is quite inefﬁcient: computing the subgradient of the trace Lasso is expensive and the rate of convergence of subgradient descent is quite slow. Instead, we consider an iteratively reweighted least-squares method. First, we need to introduce a well-known variational formulation for the trace norm [23]: Proposition 5. Let M ∈Rn×p. The trace norm of M is equal to: ∥M∥∗= 1 2 inf S⪰0 tr M⊤S−1M + tr (S) , and the inﬁmum is attained for S = MM⊤1/2 . Using this proposition, we can reformulate the previous optimization problem as min w inf S⪰0 1 2∥y −Xw∥2 2 + λ 2 w⊤Diag diag(X⊤S−1X) w + λ 2 tr(S). This problem is jointly convex in (w, S) [24]. In order to optimize this objective function by alternating the minimization over w and S, we need to add a term λµi 2 tr(S−1). Otherwise, the inﬁmum over S could be attained at a non invertible S, leading to a non convergent algorithm. The inﬁmum over S is then attained for S = X Diag(w)2X⊤+ µiI 1/2. Optimizing over w is a least-squares problem penalized by a reweighted ℓ2-norm equal to w⊤Dw, where D = Diag diag(X⊤S−1X) . It is equivalent to solving the linear system (X⊤X + λD)w = X⊤y. This can be done efﬁciently by using a conjugate gradient method. Since the cost of multiplying (X⊤X+λD) by a vector is O(np), solving the system has a complexity of O(knp), where k ≤n+1 is the number of iterations needed to converge (see theorem 10.2.5 of [9]). Using warm restarts, k can be even smaller than n, since the linear system we are solving does not change a lot from an iteration to another. Below we summarize the algorithm: ITERATIVE ALGORITHM FOR ESTIMATING w Input: the design matrix X, the initial guess w0, number of iteration N, sequence µi. For i = 1...N: • Compute the eigenvalue decomposition U Diag(sk)U⊤of X Diag(wi−1)2X⊤. • Set D = Diag(diag(X⊤S−1X)), where S−1 = U Diag(1/√sk + µi)U⊤. • Set wi by solving the system (X⊤X + λD)w = X⊤y. For the sequence µi, we use a decreasing sequence converging to ten times the machine precision. 6 3.1 Choice of λ We now give a method to choose the initial parameter λ of the regularization path. In fact, we know that the vector 0 is solution if and only if λ ≥Ω∗(X⊤y) [25]. Thus, we need to start the path at λ = Ω∗(X⊤y), corresponding to the empty solution 0, and then decrease λ. Using the inequalities on the dual norm we obtained in the previous section, we get ∥X⊤y∥∞≤Ω∗(X⊤y) ≤∥X∥op∥X⊤y∥∞. Therefore, starting the path at λ = ∥X∥op∥X⊤y∥∞is a good choice. 4 Approximation around the Lasso We recall that when P = I ∈Rp×p, our norm is equal to the ℓ1-norm, and we want to understand its behavior when P departs from the identity. Thus, we compute a second order approximation of our norm around the Lasso: we add a small perturbation ∆∈Sp to the identity matrix, and using Prop. 6 of [22], we obtain the following second order approximation: ∥(I + ∆) Diag(w)∥∗= ∥w∥1 + diag(∆)⊤\|w\|+ X \|wi\|>0 X \|wj\|>0 (∆ji\|wi\| −∆ij\|wj\|)2 4(\|wi\| + \|wj\|) + X \|wi\|=0 X \|wj\|>0 (∆ij\|wj\|)2 2\|wj\| + o(∥∆∥2). We can rewrite this approximation as ∥(I + ∆) Diag(w)∥∗= ∥w∥1 + diag(∆)⊤\|w\| + X i,j ∆2 ij(\|wi\| −\|wj\|)2 4(\|wi\| + \|wj\|) + o(∥∆∥2), using a slight abuse of notation, considering that the last term is equal to 0 when wi = wj = 0. The second order term is quite interesting: it shows that when two covariates are correlated, the effect of the trace Lasso is to shrink the corresponding coefﬁcients toward each other. We also note that this term is very similar to pairwise elastic net penalties, which are of the form \|w\|⊤P\|w\|, where Pij is a decreasing function of ∆ij. 5 Experiments In this section, we perform experiments on synthetic data to illustrate the behavior of the trace Lasso and other classical penalties when there are highly correlated covariates in the design matrix. The support S of w is equal to {1, ..., k}, where k is the size of the support. For i in the support of w, wi is independently drawn from a uniform distribution over [−1, 1]. The observations xi are drawn from a multivariate Gaussian with mean 0 and covariance matrix Σ. For the ﬁrst setting, Σ is set to the identity, for the second setting, Σ is block diagonal with blocks equal to 0.2I + 0.811⊤ corresponding to clusters of four variables, ﬁnally for the third setting, we set Σij = 0.95\|i−j\|, corresponding to a Toeplitz design. For each method, we choose the best λ. We perform a ﬁrst series of experiments (p = 1024, n = 256) for which we report the estimation error. For the second series of experiments (p = 512, n = 128), we report the Hamming distance between the estimated support and the true support. In all six graphs of Figure 2, we observe behaviors that are typical of Lasso, ridge and elastic net: the Lasso performs very well on very sparse models but its performance degrades for denser models. The elastic net performs better than the Lasso for settings where there are strongly correlated covariates, thanks to its strongly convex ℓ2 term. In setting 1, since the variables are uncorrelated, there is no reason to couple their selection. This suggests that the Lasso should be the most appropriate convex regularization. The trace Lasso approaches the Lasso when n is much larger than p, but the weak coupling induced by empirical correlations is sufﬁcient to slightly decrease its performance compared to that of the Lasso. By contrast, in settings 2 and 3, the trace Lasso outperforms other methods (including the pairwise elastic net) since variables that should be selected together are indeed correlated. As for the penalized elastic net, since it takes into account the correlations between variables, it is not surprising that in experiments 2 and 3 it performs better than methods that do not. We do not have a compelling explanation for its superior performance in experiment 1. 7 Figure 2: Left: estimation error (p = 1024, n = 256), right: support recovery (p = 512, n = 128). (Best seen in color. e-net stands for elastic net, pen stands for pairwise elastic net and trace stands for trace Lasso. Error bars are obtained over 20 runs.) 6 Conclusion We introduce a new penalty function, the trace Lasso, which takes advantage of the correlation between covariates to add strong convexity exactly in the directions where needed, unlike the elastic net for example, which blindly adds a squared ℓ2-norm term in every directions. We show on synthetic data that this adaptive behavior leads to better estimation performance. In the future, we want to show that if a dedicated norm using prior knowledge such as the group Lasso can be used, the trace Lasso will behave similarly and its performance will not degrade too much, providing theoretical guarantees to such adaptivity. Finally, we will seek applications of this estimator in inverse problems such as deblurring, where the design matrix exhibits strong correlation structure. Acknowledgments Guillaume Obozinski and Francis Bach are supported in part by the European Research Council (SIERRA ERC-239993). 8 References [1] S.G. Mallat and Z. 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4,420	Linear Submodular Bandits and their Application to Diversiﬁed Retrieval Yisong Yue iLab, Heinz College Carnegie Mellon University yisongyue@cmu.edu Carlos Guestrin Machine Learning Department Carnegie Mellon University guestrin@cs.cmu.edu Abstract Diversiﬁed retrieval and online learning are two core research areas in the design of modern information retrieval systems. In this paper, we propose the linear submodular bandits problem, which is an online learning setting for optimizing a general class of feature-rich submodular utility models for diversiﬁed retrieval. We present an algorithm, called LSBGREEDY, and prove that it efﬁciently converges to a near-optimal model. As a case study, we applied our approach to the setting of personalized news recommendation, where the system must recommend small sets of news articles selected from tens of thousands of available articles each day. In a live user study, we found that LSBGREEDY signiﬁcantly outperforms existing online learning approaches. 1 Introduction User feedback has become an invaluable source of training data for optimizing information retrieval systems in a rapidly expanding range of domains, most notably content recommendation (e.g., news, movies, ads). When designing retrieval systems that adapt to user feedback, two important challenges arise. First, the system should recommend optimally diversiﬁed content that maximizes coverage of the information the user ﬁnds interesting (to maximize positive feedback). Second, the system should make exploratory recommendations in order to learn a reliable model from feedback. Challenge 1: diversiﬁcation. In most retrieval settings, the retrieval system must recommend sets of articles, rather than individual articles. Furthermore, the recommended articles should be well diversiﬁed. This is motivated by the principle that recommending redundant articles leads to diminishing returns on utility, since users need to consume redundant information only once. This notion of diminishing returns is well-captured by submodular utility models, which have become an increasingly popular approach to modeling diversiﬁed retrieval tasks in recent years [24, 25, 18, 3, 21, 9, 16]. Challenge 2: feature-based exploration. In most retrieval settings, users typically only provide feedback on the articles recommended to them. This partial feedback issue leads to an inherent tension between exploration and exploitation when deciding which articles to recommend to the user. Furthermore, it is typically desirable to learn a feature-based model that can generalize to new or previously unseen articles and users; this is often called the contextual bandits problem [13, 15, 7]. Although there exist approaches that have addressed these challenges individually, to our knowledge there is no single approach which solves both simultaneously and is also practical to implement. For instance, existing online approaches for optimizing submodular functions typically assume a featurefree model, and thus cannot generalize easily [18, 22, 23]. Such approaches measure performance relative to the single best set (e.g., of articles). Thus, they are not suitable for many retrieval settings since the set of available articles can change frequently (e.g., news recommendation). In this paper, we address both challenges in a uniﬁed framework. We propose the linear submodular bandits problem, which is an online learning setting for optimizing a general class of feature-based 1 submodular utility models. To make learning practical, we represent the beneﬁt of adding an article to an existing set of selected articles as a linear model with respect to the user’s preferences. This class of models encompasses several existing information coverage utility models for diversiﬁed retrieval [24, 25, 9], and allows us to learn ﬂexible models that can generalize to new predictions. Similar to the contextual bandits setting considered in [15], our setting can be characterized as a feature-based exploration-exploitation problem, where the uncertainty lies in how best to model user interests using the available features. In contrast to [15], we aim to recommend optimally diversiﬁed sets of articles rather than just single articles. From that standpoint, modeling this additional layer of complexity in the bandit setting is our main technical contribution. We present an algorithm, called LSBGREEDY, to optimize this exploration-exploitation trade-off. When learning a d-dimensional model to recommend sets of L articles for T time steps, we prove that LSBGREEDY incurs regret that grows as O(d p LT) (ignoring log factors). This regret matches the convergence rates of analogous algorithms for the conventional linear bandits setting [1, 20, 8]. As a case study, we applied our approach to the setting of personalized news recommendation [9, 15, 16]. In addition to simulation experiments, we conducted a live user study over a period of ten rounds, where in each round the retrieval system must recommend a small set of news articles selected from tens of thousands of available articles for that round. We compared against existing online learning approaches that either employ no exploration [9], or learn to recommend only single articles (and thus do not model diversity) [15]. Compared to previous approaches, we ﬁnd that LSBGREEDY can signiﬁcantly improve the performance of the retrieval system even when learning for a limited number of rounds. Our empirical results demonstrate the advantage of jointly tackling the challenges of diversiﬁcation and feature-based exploration, as well as showcase the practicality of our approach. 2 Submodular Information Coverage Models Before presenting our online learning setting, we ﬁrst describe the class of utility functions that we optimize over. Throughout this paper, we use personalized news recommendation as our motivating example. In this setting, utility corresponds to the amount of interesting information covered by the set of recommended articles. Suppose that news articles are represented using a set of d “topics” or “concepts” that we wish to cover (e.g., the Middle East or the weather).1 Intuitively, recommending two articles that cover highly overlapping topics might not be more beneﬁcial than recommending just one of the articles – this is the notion of diminishing returns we wish to capture in our information coverage model. Two key properties we will exploit are that our utility functions are monotone and submodular. A set function F mapping sets of recommended articles A to real values (e.g., the total information covered by A) is monotone and submodular if and only if F(A [ {a}) ≥F(A) and F(A [ {a}) −F(A) ≥F(B [ {a}) −F(B), respectively, for all articles a and sets A ✓B. In other words, since A is smaller than B, the beneﬁt of adding a to A is larger than the beneﬁt of adding a to B. Submodularity provides a natural framework for characterizing diminishing returns in information coverage, since the gain of adding a second (redundant) article on a topic will be smaller than the gain of adding the ﬁrst. For each topic i, let Fi(A) be a monotone submodular function corresponding to how well the recommended articles A cover topic i. We write the total utility of recommending A as F(A\|w) = w>hF1(A), . . . , Fd(A)i, (1) where w 2 <d + is a parameter vector indicating the user’s interest level in each topic. Thus, F(A\|w) corresponds to the weighted information coverage of A, and depends on the preferences of the particular user. Since sums of monotone submodular functions are themselves monotone submodular, this implies that F(A\|w) is also monotone submodular (this would not hold if w has negative components). When making recommendations, the goal then is to select the A that maximizes F(A\|w). This class of information utility models encompasses several existing models of information coverage for diversiﬁed retrieval [24, 25, 9]. 1In general, these features can represent any “nugget of information”, such as a single word [24, 25, 9]. 2 Example: Probabilistic Coverage. As an illustrative example, we now describe the probabilistic coverage model proposed in [9]. This will also be the coverage model used in our case study (see Section 5). Each article a has some probability P(i\|a) of covering topic i.2 Assuming each article a 2 A has an independent probability of covering each topic, then we can write Fi(A) as Fi(A) = 1 − Y a2A (1 −P(i\|a)), (2) which corresponds to the probability that topic i is covered by at least one article in A. It is straightforward to check that Fi in (2) is monotone submodular [9]. Local Linearity. One attractive property of F(A\|w) in (1) is that the incremental gains are locally linear. In particular, the incremental gain of adding a to A can be written as w>∆(a\|A), where ∆(a\|A) = h F1(A [ {a}) −F1(A) , . . . , Fd(A [ {a}) −Fd(A) i. (3) In other words, the i-th component of ∆(a\|A) corresponds to the incremental coverage (i.e., submodular advantage) of topic i by article a, conditioned on articles A having already been selected. This property will be exploited by our online learning algorithm presented in Section 4. Optimization. Another attractive property of monotone submodular functions is that the myopic greedy algorithm is guaranteed to produce a near-optimal solution [17]. For any budget L (e.g., L = 10 articles), the constrained optimization problem, argmaxA:\|A\|L F(A\|w), can be solved greedily to produce a solution that is within a factor (1 −1/e) ⇡0.63 of optimal. Achieving better than (1 −1/e)OPT is known to be intractable unless P = NP [10]. In practice, the greedy algorithm can often perform much better than this worst case guarantee (cf. [14]), and will be a central component in our online learning algorithm. 3 Problem Formulation We propose the linear submodular bandits problem which is described in the following. At each time step t = 1, . . . , T, our algorithm interacts with the user in the following way: • A set of articles At is made available to the algorithm. Each article a 2 At is represented using a set of d basis coverage functions F1, . . . , Fd, deﬁned as in Section 2, which is known to the algorithm. • The algorithm chooses a ranked set of L articles, denoted At = (a(1) t , . . . , a(L) t ), using the basis coverage functions of the articles and the outcomes of previous time steps. • The user provides feedback (e.g., clicks on or ignores each article), and the rewards for each recommended articles rt(At) (4) are observed. In order to develop our algorithm, we require a model of user behavior. We assume the user scans the recommended articles A = (a(1), . . . , a(L)) one by one in top-down fashion. For each article a(`), the user considers the new information covered by a(`) and not covered by the above articles A(1:`−1) (A(1:`) denotes the articles in the ﬁrst ` slots). In our representation, this new information is ∆(a(`)\|A(1:`−1)) as in (3). The user then clicks on (or likes) a(`) with independent probability (w⇤)> ∆(a(`)\|A(1:`−1)), where w⇤is the hidden preferences of the user. Formally, for any set of articles A chosen at time t, the rewards rt(A) can be written as the sum of rewards at each slot, rt(A) = L X `=1 r(`) t (A). (4) We assume each r(`) t is an independent random variable bounded in [0, 1] and satisﬁes E h r(`) t (A) i = (w⇤)> ∆(a(`)\|A(1:`−1)), (5) where w⇤is a weight vector unknown to the algorithm with kw⇤k S. In other words, the expected reward in each slot is realizable, linear in ∆(a(`)\|A(1:`−1)), and independent of the other slots. We call this independence property conditional submodular independence, which we will leverage in 2E.g., the topics and coverage probabilities can be derived from a topic model such as LDA [4]. 3 Algorithm 1 LSBGREEDY 1: input: λ, ↵t 2: for t = 1, . . . , T do 3: Mt λId + Pt−1 ⌧=1 PL `=1 ∆(`) ⌧ ⇣ ∆(`) ⌧ ⌘> //covariance matrix 4: bt Pt−1 ⌧=1 PL `=1 ˆr(`) ⌧∆(`) ⌧ //aggregate feedback so far 5: wt M −1 t bt //linear regression using previous feedback as training data 6: At ; 7: for ` = 1, . . . , L do 8: 8a 2 At \ A(t) : µa w> t ∆(a\|At) //compute mean estimate of utility gain 9: 8i 2 At \ A(t) : ca ↵t q ∆(a\|At)> M −1 t ∆(a\|At) //compute conﬁdence interval 10: set a(`) t argmaxa(µa + ca) //select article with highest upper conﬁdence bound 11: store ∆(`) t ∆ ⇣ a(`) t %%% A(1:`−1) t ⌘ , At At [ n a(`) t o 12: end for 13: recommend articles At in the order selected, and observe rewards ˆr(1) t , . . . , ˆr(L) t for each slot 14: end for our analysis. While conditional submodular independence may seem ideal, we will show in our user study experiments that it is not required for our proposed algorithm to achieve good performance. Equations (4) and (5) imply that E[rt(A)] = F(A\|w⇤) for F deﬁned as in (1). Thus, E[rt] is monotone submodular, and a clairvoyant system with perfect knowledge of w⇤can greedily select articles to achieve (expected) reward at least (1−1/e)OPT, where OPT denotes the total expected reward of the optimal recommendations for t = 1, . . . , T. Let A⇤ t denote the optimal set of articles at time t. We quantify performance using the following notion of regret which we call greedy regret, RegG(T) = ✓ 1 −1 e ◆ T X t=1 E [rt(A⇤ t )] − T X t=1 rt(At) ⌘ ✓ 1 −1 e ◆ OPT − T X t=1 rt(At). (6) 4 Algorithm and Main Results A central question in the study of bandit problems is how best to balance the trade-off between exploration and exploitation (cf. [15]). To minimize regret (6), an algorithm must exploit its past experience to recommend sets of articles that appear to maximize information coverage. However, topics that appear good (i.e., interesting to the user) may actually be suboptimal due to imprecision in the algorithm’s knowledge. In order to avoid this situation, the algorithm must explore by recommending articles about seemingly poor topics in order to gather more information about them. In this section, we present an algorithm, called LSBGREEDY, which automatically trades off between exploration and exploitation (Algorithm 1). LSBGREEDY balances exploration and exploitation using upper conﬁdence bounds on the estimated gain in utility, and builds upon upper conﬁdence bound style algorithms for the conventional linear bandits setting [8, 20, 15, 7, 1]. Intuitively, the algorithm can be decomposed into the following components. Training a Model. Since we employ a linear model, at each time t, we can ﬁt an estimate wt of the true w⇤via linear regression on the previous feedback. Lines 3–5 in Algorithm 1 describe this step, where ∆(`) ⌧ denotes the incremental coverage features of the article selected at time ⌧and slot `, and ˆr(`) ⌧ denotes the associated reward. Note that λ in Line 3 is the standard regularization parameter. Estimating Incremental Coverage. Given wt, we can now estimate the incremental gain of adding any article a to an existing set of results A. As discussed in Section 3, the true (expected) incremental gain is (w⇤)> ∆(a\|A). Our algorithm’s estimate is w> t ∆(a\|A) (Line 8). If our algorithm were to purely exploit prior knowledge, then it would greedily choose articles that maximize w> t ∆(a\|A).3 Computing Conﬁdence Intervals. Of course, each wt is an imprecise estimate of the true w⇤. Given such uncertainty, a natural approach is to use conﬁdence intervals which contain the true w⇤ 3Note that wt may have negative components, which would make F(·\|wt) not monotone submodular. However, regret is measured by F(·\|w⇤), which is monotone submodular. We show in our analysis that having negative components in wt does not hinder our ability to converge efﬁciently to w⇤in a regret sense. 4         t A(1) t r(1) t A(2) t r(2) t 1 a1 1 a2 0 2 b1 1 b3 1 Figure 1: Illustrative example of LSBGREEDY for L = 2 and 2 days. Each day comprises 3 articles covering 4 topics, which are depicted in the two plots. Each row in the table describes the choices of LSBGREEDY and the resulting feedback. In day 1, LSBGREEDY recommends articles to explore topics 1, 2, and 3, and the user indicates liking a1 and disliking a2. In day 2, LSBGREEDY recommends b1 to exploitatively cover topic 1, and b3 to both cover topic 1 and explore topic 4. with some target conﬁdence (e.g., 95%). Our algorithm’s uncertainty in the gain of article a given set A depends directly to how much feedback we have collected regarding prominent topics in ∆(a\|A). In our linear setting, uncertainty is measured using the inverse covariance matrix M −1 t of the submodular features of the previously selected articles (Line 9). If our algorithm were to purely explore, then it would greedily select articles that have maximal uncertainty q ∆(a\|A)>M −1 t ∆(a\|A). Balancing Exploration and Exploitation. In order to achieve low regret, LSBGREEDY greedily selects articles that maximize a compromise between estimated gain and uncertainty (Line 10), with ↵t controlling the tradeoff. For any δ 2 (0, 1), Lemma 3 in Appendix A.2 provides sufﬁcient conditions on ↵t for constructing conﬁdence intervals, w> t ∆(a\|A) ± ↵t q ∆(a\|A)>M −1 t ∆(a\|A) ⌘w> t ∆(a\|A) ± ↵tk∆(a\|A)kM −1 t , (7) that contain the true value, (w⇤)>∆(a\|A), with probability at least 1 −δ. In this sense, Line 10 maximizes the upper conﬁdence bound on the true expected reward. Figure 1 provides an illustrative example of the behavior of LSBGREEDY. We now state our main result, which essentially bounds the greedy regret (6) of LSBGREEDY as O(d p TL) (ignoring log factors). This means that the average loss incurred per slot and per day by LSBGREEDY relative to (1 −1/e)OPT decreases at a rate of O(d/ p TL). Theorem 1. For L d, λ = L, and ↵t deﬁned as ↵t = q 2 log & 2 det(Mt)1/2 det(λId)−1/2/δ ' + S p λ, (8) with probability at least 1 −δ, LSBGREEDY achieves greedy regret (6) bounded by RegG(T) ↵T p 8TL log det(MT +1) + s 2(1 + TL) log ✓p 1 + TL δ/2 ◆ = O ✓ Sd p TL log ✓TL δ ◆◆ . The proof of Theorem 1 is presented in Appendix A in the supplementary material. In practice, the choice of ↵t in (8) may be overly conservative. As we show in our experiments, more aggressive choices of ↵t can often lead to faster convergence. 5 Empirical Analysis: Case Study in News Recommendation We applied LSBGREEDY to the setting of personalized news recommendation (cf. [9, 15, 16]), where the system is tasked with recommending sets of articles that maximally cover the interesting information of the available articles. The user provides feedback (e.g., by indicating that she likes or dislikes each article), and the goal is to maximize the total positive feedback by personalizing to the user. We conducted both simulation experiments as well as a live user study. Since real users are unlikely to behave exactly according to our modeling assumptions (e.g., obey conditional submodular independence), our user study tests the effectiveness of our approach in settings beyond those considered in our theoretical analysis. 5.1 Simulations Data. We ran simulations using both synthetic datasets as well as the blog dataset from [9]. For each setting, we generated a hidden true preference vector w⇤. For the synthetic data, all articles 5                                                                                 Figure 2: Simulation results comparing LSBGREEDY (red), RankLinUCB (black thick), Multiplicative Weighting (black thin), and ✏-Greedy (dashed thin). The middle column computes regret versus the clairvoyant greedy solution, and not (1 −1/e)OPT. Unless speciﬁed, results are for L = 5. were randomly generated using d = 25 topics, and w⇤was randomly generated and re-scaled so the most likely articles were liked with probability ⇡75%. For the blog dataset, articles are represented using d = 100 topics generated using Latent Dirichlet Allocation [4], and w⇤was derived from a preliminary version of our user study. Our simulated user behaves according to the user model described in Section 3. We use probabilistic coverage (2) as the submodular basis functions. Competing Methods. We compared LSBGREEDY against the following online learning algorithms. Note that all learning algorithms use the same underlying submodular utility model. • Multiplicative Weighting (MW) as proposed in [9], which does not employ exploration. • RankLinUCB, which combines the LinUCB algorithm [8, 20, 15, 7, 1] with Ranked Bandits [18, 22]. RankLinUCB is similar to LSBGREEDY except that it maintains a separate weight vector per slot since it employs a reduction to L separate linear bandits (one per slot). In a sense, this is the natural application of existing approaches to our setting.4 • ✏-Greedy, which randomly explores with probability ✏, and exploits otherwise [15]. Results. Figure 2 shows a representative sample of our simulation results.5 We see that both ✏Greedy and Multiplicative Weighting achieve signiﬁcantly worse results than LSBGREEDY. We also observe the performance of Multiplcative Weigthing diverge in the synthetic dataset, which is due to the fact that it does not employ exploration. RankLinUCB is more competitive, and achieves matching performance in the synthetic dataset. We also see that RankLinUCB is more sensitive to the choice of ↵. Interestingly, both LSBGREEDY and RankLinUCB approach the same performance when recommending L = 10 articles. This can be explained by the user’s interests being saturated by 10 articles, and suggests that the bound in Theorem 1 could potentially be further reﬁned. Additional details can be found in Appendix B in the supplementary material. 5.2 User Studies Design. The design of our study is similar to the personalization study conducted in [9]. We presented each user with ten articles per day over ten days from January 18, 2009 to January 27, 2009. Each day, the articles are selected using an interleaving of two policies (described below). The articles are displayed as a title with its contents viewable via a preview pane. The user is instructed 4One can show that RankLinUCB achieves greedy regret (6) that grows as O(dL p T) (ignoring log factors), which is a factor p L worse than the regret guarantee of LSBGREEDY. 5For all methods, we ﬁnd performance to be relatively stable w.r.t. the tuning parameters (e.g., ↵t for LSBGREEDY). Unless speciﬁed, we set all parameters to values that achieve good results for their respective algorithms. In particular we set ↵t = 1 for LSBGREEDY, ↵t = 0.6 for RankLinUCB, β = 0.9 for MW, and ✏= 0.1 for ✏-Greedy. LSBGREEDY, RankLinUCB, and ✏-Greedy train linear models with regularization parameter λ, which we kept constant at λ = 1. 6                 Figure 3: Displaying normalized learned preferences of LSBGREEDY (dark) and MW (light) for two user study sessions. In the left session, MW overﬁts to the “world” topic. In the right session, the user likes very few articles, and MW does not discover any topics that interest the user. COMPARISON #SESSIONS WIN/TIE/LOSE GAIN PER DAY % OF LIKES LSBGREEDY vs Static Baseline 24 24 / 0 / 0 1.07 63% (67%) LSBGREEDY vs Mult. Weighting 26 24 / 1 / 1 0.54 57% (63%) LSBGREEDY vs RankLinUCB 27 21 / 2 / 4 0.58 57% (61%) Table 1: User study comparing LSBGREEDY with competing algorithms. The parenthetical values in the last column are computed ignoring clicks on articles jointly recommended by both algorithms (see Section 5.2). All results are statistically signiﬁcant with 95% conﬁdence. to brieﬂy skim each article to get a sense of its content and, one by one, mark each article as “interested in reading in detail” (like), or “not interested” (dislike). As in [9], for each decision, the user is told to take into account the articles shown above in the current day, so as to capture the notion of incremental coverage. For example, a user might be interested in reading an article regarding the Middle East appearing at the top slot, and would mark it as “interested.” However, if several very similar articles appear below it, the user may mark the subsequent articles as “not interested.” Evaluation. For each day, we generate an interleaving of recommendations from two algorithms. Interleaving allows us to make paired comparisons such that we simultaneously control for the particular user and particular day (certain days may contain more or less interesting content to the user than other days). Like other interleaving approaches [19], our approach maintains a notion of fairness so that both competing algorithms recommend the same amount of content. After each day, the user’s feedback is collected and given to the two competing algorithms. Additional details of our experimental setup can be found in Appendix C in the supplementary material. Data. In order to distinguish the gains of the algorithms from other effects (such as imperfections in the features, or having too high a dimension to converge), we performed dimensionality reduction. We created 18 genres (examples shown in Figure 3), labeled relevant articles and trained a model via linear regression for each genre. Note that many articles are relevant to multiple genres. We compared LSBGREEDY against the static baseline (i.e., no personalization), Multiplicative Weighting (MW) from [9], and RankLinUCB. We evaluated each comparison setting using approximately twenty ﬁve participants, most of whom are graduate students or young professionals. Results. Table 1 describes our results. We ﬁrst aggregated per user, and then aggregated over all users. For each user, we computed three statistics: (1) whether LSBGREEDY won, tied, or lost in terms of total number of liked articles, (2) the difference in liked articles per day, and (3) the fraction of liked articles recommended by LSBGREEDY. Jointly recommended articles can be either counted as half to each algorithm or ignored (these results are shown in parentheticals in Table 1). Overall, about 90% of users preferred recommendations by LSBGREEDY over the competing algorithms. On average, LSBGREEDY obtains about one additional liked article per day and 63% of all liked articles versus the static baseline, and about half an additional liked article per day and 57% of all liked articles versus the two competing learning algorithms. The gains we observe are all statistically signiﬁcant with 95% conﬁdence, and show that LSBGREEDY can be effective even when the assumptions in our theoretical analysis may not be satisﬁed. Figure 3 shows the learned preferences by LSBGREEDY and MW on two sessions. Since MW does not employ exploration, it can either overﬁt to its previous experience and not ﬁnd new topics that interest the user (left plot), or fail to discover any good topics (right plot). We do not include a comparison with RankLinUCB since it learns L preference vectors, which are difﬁcult to visualize. 7 6 Related Work Diversiﬁed Retrieval. We are chieﬂy interested in training ﬂexible submodular utility models, since such models yield practical algorithmic approaches. At one extreme are feature-free models that do not require training. However, such models are limited to unpersonalized settings that ignore context, such as recommending a global set of blogs to monitor [14]. On the other hand, methods that use feature-rich models typically either employ unsupervised training [24] or require ﬁne-grained subtopic labels [25]. Such learning approaches cannot easily adapt to new domains. One exception is [9], whose proposed online learning approach does not incorporate exploration. As shown in our experiments, this signiﬁcantly inhibits the learning ability of their approach. Beyond submodular models of information coverage, other approaches include methods that balance relevance and novelty [5, 26, 6] and graph-based methods [27]. For such models, it remains a challenge to design provably efﬁcient online learning algorithms. Bandit Learning. From the perspective of our work, existing bandit approaches can be categorized along two dimensions: single-prediction versus set-prediction, and feature-based versus feature-free. Most feature-based settings are designed to predict single results, rather than sets of results. Of such settings, the most relevant to ours is the linear stochastic bandits setting [8, 20, 15, 7, 1], which we build upon in our approach. One limitation here is the assumption of realizability – that the “true” user model lies within our class. It may be possible to develop more robust algorithms for our submodular bandits setting by building upon algorithms with more general guarantees (e.g., [2]). Most set-based settings, such as bandit submodular optimization or the general bandit slate problem, assume a feature-free model [18, 22, 23, 12]. As such, performance is quantiﬁed relative to a ﬁxed set of articles, which is not appropriate for many retrieval settings (e.g., news recommendation). One exception is [21], which assumes that document and user models lie within a metric space. However, it is unclear how to incorporate our submodular features into their setting. 7 Discussion of Limitations and Future Work Submodular Basis Features. Our approach requires access to submodular basis functions as features. In practice these basis features are often derived using various topic modeling or dimensionality reduction techniques. However, the resulting features are almost always noisy or biased. Furthermore, one expects that different users will be better modeled using different basis features. As such, one important direction for future work is to learn the appropriate basis features from user feedback, which is similar to the setting of interactive topic modeling [11]. Moreover, user behavior is likely to be inﬂuenced by many factors beyond those well-modeled by submodular basis features. For example, the probability of the user liking a certain article could be inﬂuenced by the time of day, or day of the week. A more uniﬁed approach would be to incorporate both these standard features as well as submodular basis features in a joint model. Curse of Dimensionality. The convergence rate of LSBGREEDY depends linearly on the number of features d (which appears unavoidable without further assumptions). Thus, our approach may not be practical for settings that use a very large number of features. One possible extension is to jointly learn from multiple users simultaneously. If users tend to have similar preferences, then learning jointly from multiple users may yield convergence rates that are sub-linear in d. 8 Conclusion We proposed an online learning setting for optimizing a general class of submodular functions. This setting is well-suited for modeling diversiﬁed retrieval systems that interactively learn from user feedback. We presented an algorithm, LSBGREEDY, and proved that it efﬁciently converges to a near-optimal model. We conducted simulations as well as user studies in the setting of news recommendation, and found that LSBGREEDY outperforms competing online learning approaches. Acknowledgements. This work was funded in part by ONR (PECASE) N000141010672 and ONR Young Investigator Program N00014-08-1-0752. The authors also thank Khalid El-Arini, Joey Gonzalez, Sue Ann Hong, Jing Xiang, and the anonymous reviewers for their helpful comments. 8 References [1] Y. Abbasi-Yadkori, D. Pal, and C. Szepesvari. Online least squares estimation with self-normalized processes: An application to bandit problems, 2011. http://arxiv.org/abs/1102.2670. [2] J. Abernathy, E. Hazan, and A. Rakhlin. Competing in the dark: An efﬁcient algorithm for bandit linear optimization. In Conference on Learning Theory (COLT), 2008. [3] R. Agrawal, S. Gollapudi, A. Halverson, and S. Ieong. Diversifying search results. In ACM Conference on Web Search and Data Mining (WSDM), 2009. [4] D. Blei, A. Ng, and M. Jordan. Latent dirichlet allocation. Journal of Machine Learning Research (JMLR), 3:993–1022, 2003. [5] J. Carbonell and J. Goldstein. 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4,421	Learning Eigenvectors for Free Wouter M. Koolen Royal Holloway and CWI wouter@cs.rhul.ac.uk Wojtek Kotłowski Centrum Wiskunde & Informatica kotlowsk@cwi.nl Manfred K. Warmuth UC Santa Cruz manfred@cse.ucsc.edu Abstract We extend the classical problem of predicting a sequence of outcomes from a ﬁnite alphabet to the matrix domain. In this extension, the alphabet of n outcomes is replaced by the set of all dyads, i.e. outer products uu⊤where u is a vector in Rn of unit length. Whereas in the classical case the goal is to learn (i.e. sequentially predict as well as) the best multinomial distribution, in the matrix case we desire to learn the density matrix that best explains the observed sequence of dyads. We show how popular online algorithms for learning a multinomial distribution can be extended to learn density matrices. Intuitively, learning the n2 parameters of a density matrix is much harder than learning the n parameters of a multinomial distribution. Completely surprisingly, we prove that the worst-case regrets of certain classical algorithms and their matrix generalizations are identical. The reason is that the worst-case sequence of dyads share a common eigensystem, i.e. the worst case regret is achieved in the classical case. So these matrix algorithms learn the eigenvectors without any regret. 1 Introduction We consider the extension of the classical online problem of predicting outcomes from a ﬁnite alphabet to the matrix domain. In this extension, the alphabet of n outcomes is replaced by a set of all dyads, i.e. outer products uu⊤where u is a unit vector in Rn. Whereas classically the goal is to learn as well as the best multinomial distribution over outcomes, in the matrix case we desire to learn the distribution over dyads that best explains the sequence of dyads seen so far. A distribution on dyads is summarized as a density matrix, i.e. a symmetric positive-deﬁnite1 matrix of unit trace. Such matrices are heavily used in quantum physics, where dyads represent states. We will show how popular online algorithms for learning multinomials can be extended to learn density matrices. Considerable attention has been placed recently on generalizing algorithms for learning and optimization problems from probability vector parameters to density matrices [17, 19]. Efﬁcient semideﬁnite programming algorithms have been devised [1] and better approximation algorithms for NP-hard problems have been obtained [2] by employing on-line algorithms that update a density matrix parameter. Also two important quantum complexity classes were shown to collapse based on these algorithms [8]. Even though the matrix generalization led to progress in many contexts, in the original domain of on-line learning, the regret bounds proven for the algorithms in the matrix case are often the same as those provable for the original classical ﬁnite alphabet case [17, 19]. Therefore it was posed as an open problem to determine whether this is just a case of loose classical bound or whether there truly exists a “free matrix lunch” for some of these algorithms [18]. Such algorithms essentially would learn the eigensystem of the data for free without incurring any additional regret. This is non-intuitive, since one would expect a matrix to have n2 parameters and be much harder to learn than an n dimensional parameter vector. 1We use positive in the non-strict sense, and omit ‘symmetric’ and ‘deﬁnite’. Our matrices are real-valued. 1 for trial t = 1, 2, . . . , T do Algorithm predicts with probability vector ωt−1 Nature responds with outcome xt. Algorithm incurs loss −log ωt−1,xt. end for Probability vector prediction for trial t = 1, 2, . . . , T do Algorithm predicts with density matrix Wt−1 Nature responds with density matrix Xt. Algorithm incurs loss −tr Xt log(Wt−1) . end for Density matrix prediction Figure 1: Protocols In this paper we investigate this frivolously named but deep “free matrix lunch” question in arguably the simplest context: learning a multinomial distribution. In the classical case, there are n ≥2 outcomes and a distribution is parametrized by an n-dimensional probability vector ω, where ωi is the probability of outcome i. One can view the base vectors ei as the elementary events and the probability vector as a mixture of these events: ω = P i ωiei. We deﬁne a “matrix generalization” of a multinomial which is parametrized by a density matrix W (positive matrix of unit trace). Now the elementary events are dyads of the form uu⊤, where u is a unit vector in Rn. Dyads are the representations of states used in quantum physics [20]. A density matrix is a mixture of dyads. Whereas probability vectors represent uncertainty over n basis vectors, density matrices can be viewed as representing uncertainty over inﬁnitely many dyads in Rn. In the classical case, the algorithm predicts at trial t with multinomial ωt−1. Nature produces an outcome xt ∈{1, . . . , n}, and the algorithm incurs loss −log(ωt−1,xt). The most common heuristic (a.k.a. the Laplace estimator) chooses ωt−1,i proportional to 1 plus the number of previous trials in which outcome i was observed. The on-line algorithms are evaluated by their worst-case regret over data sequences, where the regret is the additional loss of the algorithm over the total loss of the best probability vector chosen in hindsight. In this paper we develop the corresponding matrix setting, where the algorithm predicts with a density matrix Wt−1, Nature produces a dyad xtx⊤ t , and the algorithm incurs loss −x⊤ t log(Wt−1)xt. Here log denotes the matrix logarithm. We are particularly interested in how the regret changes when the algorithms are generalized to the matrix case. Surprisingly we can show that for the Laplace as well as the Krichevsky-Troﬁmov [10] estimators the worst-case regret is the same in the matrix case as it is in the classical case. For the Last-Step Minimax algorithm [16], we can prove the same regret bound for the matrix case that was proven for the classical case. Why are we doing this? Most machine learning algorithms deal with vector parameters. The goal of this line of research is to develop methods for handling matrix parameters. We are used to dealing with probability vectors. Recently a probability calculus was developed for density matrices [20] including various Bayes rules for updating generalized conditionals. The vector problems are typically retained as special cases of the matrix problems, where the eigensystem is ﬁxed and only the vectors of eigenvalues has to be learned. We exhibit for the ﬁrst time a basic fundamental problem, for which the regret achievable in the matrix case is no higher than the regret achievable in the original vector setting. Paper outline Deﬁnitions and notation are given in the next section, followed by proofs of the free matrix lunch for the three discussed algorithms in Section 3. At the core of our proofs is a new technical lemma for mixing quantum entropies. We also discuss the minimax algorithm for multinomials due to Shtarkov, and corresponding minimax algorithm for density matrices. We provide strong experimental evidence that the free matrix lunch holds for this algorithm as well. To put the results into context, we motivate and discuss our choice of the loss function, and compare it to several alternatives in Section 4. More discussion and perspective is provided in the Section 5. 2 Setup The protocol for the classical probability vector prediction problem and the new density matrix prediction problem are displayed side-by-side in Figure 1. We explain the latter problem. Learning proceeds in trials. During trial t the algorithm predicts with a density matrix Wt−1. We use index t−1 to indicate that is based on the t−1 previous outcomes. Then nature responds with an outcome 2 density matrix Xt. The discrepancy between prediction and outcome is measured by the matrix entropic loss ℓ(Wt−1, Xt) := −tr Xt log(Wt−1) , (1) where log denotes matrix logarithm2. When the outcome density matrix Xt is a dyad xtx⊤ t , then this loss becomes −x⊤ t log(Wt−1)xt, which is the simpliﬁed form of the entropic loss discussed in the introduction. Also if the prediction density matrix is diagonal, i.e. it has the form Wt−1 = P i ωt−1,i eie⊤ i for some probability vector ωt−1, and the outcome Xt is an eigendyad eje⊤ j of the same eigensystem, then this loss simpliﬁes to the classical log loss: ℓ(Wt−1, Xt) = −log(ωt−1,j). The above deﬁnition is not the only way to promote the log loss to the matrix domain. Yet, in Section 4 we justify this choice. We aim to design algorithms with low regret compared to the best ﬁxed density matrix in hindsight. The loss of the best ﬁxed density matrix can be expressed succinctly in terms of the von Neumann entropy, which is deﬁned for any density matrix D as H(D) := −tr(D log D), and the sufﬁcient statistic ST = PT t=1 Xt as follows: infW PT t=1 ℓ(W , Xt) = TH ST T . For ﬁxed data X1, . . . , XT , the regret of a strategy that issues prediction Wt after observing X1, . . . , Xt is T X t=1 ℓ(Wt−1, Xt) −TH ST T , (2) and the worst-case regret on T trials is obtained by taking supX1,...,XT over (2). Our aim is to design strategies for density matrix prediction that have low worst-case regret. 3 Free Matrix Lunches In this section, we will show how four popular online algorithms for learning multinomials can be extended to learning density matrices. We start with the simple Laplace estimator, continue with its improved version known as the Krichevsky-Troﬁmov estimator, and also extend the less known Last Step Minimax strategy which has even less regret. We will prove a version of the free matrix lunch (FML) for all three algorithms. Finally we discuss the minimax algorithm for which we have experimental evidence that the free matrix lunch holds as well. 3.1 Laplace After observing classical data with sufﬁcient statistic vector σt = Pt q=1 exq, classical Laplace predicts with the probability vector ωt := σt+1 t+n consisting of the normalized smoothed counts. By analogy, after observing matrix data with sufﬁcient statistic St = Pt q=1 Xt, matrix Laplace predicts with the correspondingly smoothed matrix Wt := St+I t+n . Classical Laplace is commonly motivated as either the Bayes predictive distribution w.r.t. the uniform prior or as a loss minimization with virtual outcomes [3]. The latter motivation can be “lifted” to the matrix domain by adding n virtual outcomes at I/n: Wt = argmin W dens. mat. ( n ℓ(W , I/n) + t X q=1 ℓ(W , Xq) ) = St + I t + n . (3) The worst-case regret of classical Laplace after T iterations equals log T +n−1 n−1 ≤(n−1) log(T +1) (see e.g. [6]). We now show that in the matrix case, no additional regret is incurred. Theorem 1 (Laplace FML). The worst-case regrets of classical and matrix Laplace coincide. Proof. Let W ∗ t denote the best density matrix for the ﬁrst t outcomes. The regret (2) of matrix Laplace can be bounded as follows: T X t=1 ℓ(Wt−1, Xt) − T X t=1 ℓ(W ∗ T , Xt) ≤ T X t=1 ℓ(Wt−1, Xt) −ℓ(W ∗ t , Xt) . (4) 2For any positive matrix with eigendecomposition A = P i αi aia⊤ i , log(A) := P i log(αi) aia⊤ i . 3 Now consider each term in the right-hand sum separately. The tth term equals −tr Xt log St−1 + I t −1 + n −log St t = log t −1 + n t −tr Xt log(St−1 + I) −log St . Note that the ﬁrst term constitutes the “classical” part of the per-round regret, while the second term is the “matrix” part. The matrix part is non-positive since St−1 + I ⪰St, and the logarithm is a matrix monotone operation (i.e. A ⪰B implies log A ⪰log B). By omitting it, we obtain an upper bound on the regret of matrix Laplace, that is tight: for any sequence of identical dyads the matrix part is zero and (4) holds with equality since W ∗ t = W ∗ T for all t ≤T. The same upper bound is also met by classical Laplace on any sequence of identical outcomes [6]. We just showed that matrix Laplace has the same worst-case regret as classical Laplace, albeit matrix Laplace learns a matrix of n2 parameters whereas classical Laplace only learns n probabilities. No additional regret is incurred for learning the eigenvectors. Matrix Laplace can update Wt in O(n2) time per trial. The same will be true for our next algorithm. 3.2 Krichevsky-Troﬁmov (KT) Classical and matrix KT smooth by adding 1 2 to each count, i.e. ωt := σt+1/2 t+n/2 and Wt := St+I/2 t+n/2 . The former can again be obtained as the Bayes predictive distribution w.r.t. Jeffreys’ prior, the latter as the solution to the matrix entropic loss minimization problem (3) with n/2 virtual outcomes instead of n for Laplace. The leading term in the worst-case regret for classical KT is the optimal 1 2 log(T) rate per parameter instead of the log(T) rate for Laplace. More precisely, classical KT’s worst-case regret after T iterations is known to be log Γ(T +n/2) Γ(T +1/2) + log Γ(1/2) Γ(n/2) ≤n−1 2 log(T + 1) + log(π) (see e.g. [6]). Again we show that no additional regret is incurred in the matrix case. Theorem 2 (KT FML). The worst-case regrets of classical and matrix KT coincide. The proof uses the following key entropy decomposition lemma (proven in Appendix A): Lemma 1. For positive matrices A, B with A = P i αi aia⊤ i the eigendecomposition of A: H(A + B) ≥ n X i=1 a⊤ i Bai tr(B) H A + tr(B) aia⊤ i , Proof of Theorem 2. We start by telescoping the regret (2) of matrix KT as follows T X t=1 −tr Xt log(Wt−1) −tH St−1 + Xt t + (t −1)H St−1 t −1 . (5) We bound each term separately. Let us denote the eigendecomposition of St−1 by St−1 = Pn i=1 σi sis⊤ i . Notice that since Wt−1 plays in the eigensystem of St−1, we have: −tr Xt log(Wt−1) = −tr Xt n X i=1 log(ωt−1,i) sis⊤ i = − n X i=1 s⊤ i Xtsi log(ωt−1,i). Moreover, it follows from Lemma 1 that: H St−1 + Xt t ≥ n X i=1 s⊤ i XtsiH St−1 + sis⊤ i t . Taking this equality and inequality into account, the tth term in (5) is bounded above by: δt := n X i=1 s⊤ i Xtsi −log(ωt−1,i) −tH St−1 + sis⊤ i t + (t −1)H St−1 t −1 , (6) which, in turn, is at most: δt ≤sup i −log(ωt−1,i) −tH St−1 + sis⊤ i t + (t −1)H St−1 t −1 . 4 In other words the per-round regret increase is largest for one of the eigenvectors of the sufﬁcient statistic St−1, i.e. for classical data. To get an upper bound, maximize over S0, . . . , ST −1 independently, each with the constraint that tr(St) = t. A particular maximizer is St = t e1e⊤ 1 , which is the sufﬁcient statistic of the sequence of outcomes all equal to Xt = e1e⊤ 1 . For this sequence all bounding steps hold with equality. Hence the matrix KT regret is below the classical KT regret. The reverse is obvious. 3.3 Last Step Minimax The bounding technique, developed using Lemma 1 and applied to KT can be used to prove bounds for a much broader class of prediction strategies. The crucial part of the KT proof was showing that each term in the telescoped regret (5) can be bounded above by δt as deﬁned in (6), in which all matrices share the same eigensystem, and which is hence equivalent to the corresponding classical expression. The only property of the prediction strategy that we used was that it plays in the eigensystem of the past sufﬁcient statistic. Therefore, using the same line of argument, we can show that if for some classical prediction strategy we can obtain a meaningful regret bound by bounding each term in the regret δt independently, we can obtain the same bound for the corresponding matrix strategy, i.e. its spectral promotion. In particular, we can push this argument to its limit by considering the algorithm designed to minimize δt in each iteration. This algorithm is known as Last Step Minimax. In fact, the Last Step Minimax (LSM) principle is a general recipe for online prediction, which states that the algorithm should minimize the worst-case regret with respect to the next outcome [16]. In other words, it should act as the minimax algorithm given that the time horizon is one iteration ahead. In the classical case for the multinomial distribution, after observing data with sufﬁcient statistic σt−1, classical LSM predicts with ωt−1 := argmin ω max xt ( ℓ(ω, xt) \| {z } −log(ωt−1,xt) − t X q=1 ℓ(ω∗ t , xq) \| {z } tH( σt t ) ) = n X i=1 exp −tH( σt−1+ei t ) P j exp −tH( σt−1+ej t ) ei. (7) Classical LSM is analyzed in [16] for the Bernoulli (n = 2) case. For our straightforward generalization to the classical multinomial case, the regret is bounded by n−1 2 ln(T + 1) + 1. LSM is therefore slightly better than KT. Applying the Last Step Minimax principle to density prediction, we obtain matrix LSM which issues prediction: Wt−1 := argmin W max Xt −tr Xt log(W ) −tH St t . We show that matrix LSM learns the eigenvectors without additional regret. Theorem 3 (LSM FML). The regrets of classical and matrix LSM are at most n−1 2 ln(T + 1) + 1. Proof. We determine the form of Wt−1. By Sion’s minimax theorem [15]: min W max Xt −tr Xt log(W ) −tH St t = max P min W EP −tr Xt log(W ) −tH St t , where P ranges over probability distribution on density matrices Xt. Plugging in the minimizer W = EP [Xt], the right hand side becomes: max P H EP [Xt] −EP tH St t . (8) Now decompose St−1 as Pn i=1 σi sis⊤ i . Using Lemma 1, we can bound the second expression inside the maximum: EP tH St t ≥EP " t n X i=1 s⊤ i XtsiH St−1 + sis⊤ i t # = t n X i=1 s⊤ i EP [Xt] siH St−1 + sis⊤ i t . 5 On the other hand, we know that the entropy does not decrease when we replace the argument EP [Xt] by its pinching (a.k.a. projective measurement) Pn i=1(u⊤ i EP [Xt]ui) uiu⊤ i w.r.t. any eigensystem ui [12]. Therefore, we have: H EP [Xt] ≤H n X i=1 (s⊤ i EP [Xt]si) sis⊤ i ! = H(p), where the last entropy is a classical entropy and p is a vector such that pi = s⊤ i EP [Xt]si. Combining those two results together, we have: H EP [Xt] −EP tH St t ≤H(p) −t n X i=1 piH σt−1 + ei t . Note that we have equality only when the distribution P puts nonzero mass only on the eigenvectors s1, . . . , sn. This means that when p is ﬁxed, we will maximize (8) by using a distribution with such a property, i.e. P is restricted to the eigensystem of St−1. This, in turn, means that Wt−1 = EP [Xt] will play in the eigensystem of St−1 as well. It follows that Wt−1 is the classical LSM strategy in the eigensystem of St−1, i.e. Wt−1 = P i ωt−1,i sis⊤ i , where ωt−1 are taken as in (7). The proof of the classical LSM guarantee is based on bounding the per-round regret increase: δt := −log(ωt−1,xt) −tH σt−1 + ext t + (t −1)H σt−1 t −1 , by choosing the worst case w.r.t. xt and σt−1. Since, for matrices, the worst case for the corresponding matrix version of δt, see (6), is the diagonal case, the whole analysis immediately goes through and we get the same bound as for classical LSM. Note that the bound for LSM is not tight, i.e. there exists no data sequence for which the bound is achieved. Therefore, the bound for matrix LSM is also not tight. This theorem is a weaker FML because it only relates worst-case regret bounds. We have veriﬁed experimentally that the actual regrets coincide in dimension n = 2 for up to T = 5 outcomes, using a grid of 30 dyads per trial, with uniformly spaced (x⊤e1)2. So we believe that in fact Conjecture 1 (LSM FML). The worst-case regrets of classical and matrix LSM coincide. To execute the LSM matrix strategy, we need to have the eigendecomposition of the sufﬁcient statistic. For density matrix data Xt, we may need to recompute it each trial in Ω(n3) time. For dyadic data xtx⊤ t it can be incrementally updated in O(n2) per trial with methods along the lines of [11]. 3.4 Shtarkov Fix horizon T. The minimax algorithm for multinomials, due to Shtarkov [14], minimizes the worstcase regret inf ω0 sup x1 . . . inf ωT −1 sup xT T X t=1 ℓ(ωt−1, xt) −TH σT T . (9) After observing data with sufﬁcient statistic σt and hence with r := T −t rounds remaining, classical Shtarkov predicts with ωt := n X i=1 φr−1(σt + ei) φr(σt) ei where φr(σ) := X c1,...,cn Pn i=1 ci=r r c1, . . . , cn ! exp −TH σ + c T . (10) The so-called Shtarkov sum φr can be evaluated in time O n r log(r) using a straightforward extension of the method described in [9] for computing φT (0), which is based on dynamic programming and Fast Fourier Transforms. The regret of classical Shtarkov equals log φT (0) ≈n−1 2 log(T) −log(n −2) + 1 [6]. This is again better than Last Step Minimax, which is in turn better than KT which dominates Laplace. 6 The minimax algorithm for density matrices, called matrix Shtarkov, optimizes the worst-case regret inf W0 sup X1 . . . inf WT −1 sup XT T X t=1 ℓ(Wt−1, Xt) −TH ST T . (11) To this end, after observing data with sufﬁcient statistic St, with r rounds remaining, it predicts with Wt := argmin W sup X ℓ(W , X) + Rr−1(St + X), where Rr is the tail sequence of inf/sups of (11) of length r. We now argue that the FML holds for matrix Shtarkov. Matrix Shtarkov is surprisingly difﬁcult to analyze. However, we provide a simplifying conjecture that we veriﬁed experimentally. A rigorous proof remains an open problem. Our conjecture is that Lemma 1 holds with the entropy H replaced by the minimax regret tail Rr: Conjecture 2. For each integer r, for each pair of positive matrices A and B Rr(A + B) ≥ X i a⊤ i Bai tr(B) Rr A + tr(B) aia⊤ i . Note that this conjecture generalizes Lemma 1, which is retained as the case r = 0. It follows from this conjecture, using the same argument as for LSM, that matrix Shtarkov predicts in the eigensystem of St, i.e. with Wt = P i ωt,i sis⊤ i , where ωt as in (10), and furthermore that Conjecture 3 (Shtarkov FML). The worst-case regrets of classical and matrix Shtarkov coincide. We have veriﬁed Conjecture 3 for the matrix Bernoulli case (n = 2) up to T = 5 outcomes, using a grid of 30 dyads per trial, with uniformly spaced (x⊤e1)2. Then assuming that Rr(S) = log(φ(σ)), where σ are the eigenvalues of S, for each n from 2 to 5 we drew 105 trace pairs uniformly from [0, 10], then drew matrix pairs A and B uniformly at random with those traces. Conjecture 2 always held. Obtaining the FML for the minimax algorithm is mathematically challenging and of academic interest but of minor practical relevance. First, the time horizon T must be speciﬁed in advance, so the minimax algorithm can not be used in a purely online fashion. Secondly, the running time is superlinear in the number of rounds remaining, while it is constant for the previous three algorithms. 4 Motivation and Discussion of the Loss Function The matrix entropic loss (1) that we choose as our loss function has a coding interpretation and it is a proper scoring rule. The latter seems to be a necessary condition for the free matrix lunch. Quantum coding Classical log-loss forecasting can be motivated from the point of view of data compression and variable-length coding [7]. In information theory, the Kraft-McMillan inequality states that, ignoring rounding issues, for every uniquely decodable code with a code length function λ, there is a probability distribution ω such that λi = −log ωi for all symbols i = 1, . . . , n, and vice versa. Therefore, the log loss can be interpreted as the code length assigned to the observed outcome. Quantum information theory[13, 5] generalizes variable length coding to the quantum/density matrix case. Instead of messages composed of bits, the sender and the receiver exchange messages described by density matrices, and the role analogous to the message length is now played by the dimension of the density matrix. Variable-length quantum coding requires the deﬁnition of a code length operator L, which is a positive matrix such that for any density matrix X, tr(XL) gives the expected dimension (“length”) of the message assigned to X. The quantum version of Kraft’s inequality states that, ignoring rounding issues, for every variable-length quantum code with codelength operator L, there exists a density matrix W such that L = −log W . Therefore, the matrix entropic loss can be interpreted as the (expected) code length of the observed outcome. Proper score function In decision theory, the loss function ℓ(ω, x) assessing the quality of predictions is also referred to as a score function. A score function is said to be proper, if for any distribution p on outcomes, the expected loss is minimized by predicting with p itself, i.e. argminω Ex∼p[ℓ(ω, x)] = p. Minimization of a proper score function leads to well-calibrated forecasting. The log loss is known to be a proper score function [4]. 7 We will say that a matrix loss function ℓ(W , X) is proper if for any distribution P on density matrix outcomes, the expected loss with respect to P is minimized by predicting with the mean outcome of P, i.e. argminW EX∼P [ℓ(W , X)] = EX∼P [X]. The matrix entropic loss (1) is proper, for EX∼P [−tr(X log W )] = −tr EX∼P [X] log W is minimized at W = EX∼P [X] [12]. Therefore, minimization of the matrix entropic loss leads to well-calibrated forecasting, as in the classical case. A second generalization of the log loss to the matrix domain used in quantum physics [12] is the log trace loss ℓ(W , X) := −log tr(XW ) . Note that here the trace and the logarithm are exchanged compared to (1). The expression tr(XW ) plays an important role in quantum physics as the expected value of a measurement outcome, and for X = xx⊤, tr(xx⊤W ) is interpreted as a probability. However, log trace loss is not proper. The counterexample is straightforward: if we take P uniform on {x1x⊤ 1 , x2x⊤ 2 }, then the minimizer of the expected log trace loss is W ∝(x1 + x2)(x1 + x2)⊤, which differs from EX∼P [X] = 1 2(x1x⊤ 1 + x2x⊤ 2 ). Also for log trace loss we found an example (not presented) against the FML for the minimax algorithm. A third generalization of the loss is ℓ(W , X) := −log tr(X ⊙W ) , where ⊙denotes the commutative “product” between matrices that underlies the probability calculus of [20].3 This loss upper bounds the log trace loss. We don’t know whether it is a proper scoring function. However, it equals the matrix entropic loss when X is a dyad. Finally, another loss explored in the on-line learning community is the trace loss ℓ(W , X) := tr(W X). This loss is not a proper scoring function (it behaves like the absolute loss in the vector case) and we have an example that shows that there is no FML for the minimax algorithm in this case (not presented). In summary, for there to exist a FML, properness of the loss function seems to be required. 5 Conclusion We showed that the free matrix lunch holds for the matrix version of the KT estimator. Thus the conjectured free matrix lunch [18] is realized. Our paper raises many open questions. Perhaps the main one is whether the free matrix lunch holds for the matrix minimax algorithm. Also we would like to know what properties of the loss function and algorithm cause the free matrix lunch to occur. From the examples given in this paper it is tempting to believe that you always get a free matrix lunch when upgrading any classical sufﬁcient-statistics-based predictor to a matrix version by just playing this predictor in the eigensystem of the current matrix sufﬁcient statistics. However the following counter example shows that a general reduction must be more subtle: Consider ﬂoored KT, which predicts with ωt,i ∝⌊σt,i⌋+ 1/2. For T = 5 trials in dimension n = 2, the worst-case regret is 1.297 for the classical log loss and 1.992 for matrix entropic loss. A Proof of Lemma 1 We prove the following slightly stronger inequality for all γ ≥0. The lemma is the case γ = 1. f(γ) := H(A + γB) − n X i=1 a⊤ i Bai tr(B) H(A + γ tr(B)aia⊤ i ) ≥0. Since f(0) = 0, it sufﬁces to show that f ′(γ) ≥0. Since ∂H(D) ∂D = −log(D) −I, f ′(γ) = −tr B log(A + γB) + n X i=1 a⊤ i Bai tr aia⊤ i log A + γ tr(B) aia⊤ i = tr B log A + γ tr(B)I −tr B log(A + γB) . 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4,422	Neural Reconstruction with Approximate Message Passing (NeuRAMP) Alyson K. Fletcher University of California, Berkeley alyson@eecs.berkeley.edu Sundeep Rangan Polytechnic Institute of New York University srangan@poly.edu Lav R. Varshney IBM Thomas J. Watson Research Center lrvarshn@us.ibm.com Aniruddha Bhargava University of Wisconsin Madison aniruddha@wisc.edu Abstract Many functional descriptions of spiking neurons assume a cascade structure where inputs are passed through an initial linear ﬁltering stage that produces a lowdimensional signal that drives subsequent nonlinear stages. This paper presents a novel and systematic parameter estimation procedure for such models and applies the method to two neural estimation problems: (i) compressed-sensing based neural mapping from multi-neuron excitation, and (ii) estimation of neural receptive ﬁelds in sensory neurons. The proposed estimation algorithm models the neurons via a graphical model and then estimates the parameters in the model using a recently-developed generalized approximate message passing (GAMP) method. The GAMP method is based on Gaussian approximations of loopy belief propagation. In the neural connectivity problem, the GAMP-based method is shown to be computational efﬁcient, provides a more exact modeling of the sparsity, can incorporate nonlinearities in the output and signiﬁcantly outperforms previous compressed-sensing methods. For the receptive ﬁeld estimation, the GAMP method can also exploit inherent structured sparsity in the linear weights. The method is validated on estimation of linear nonlinear Poisson (LNP) cascade models for receptive ﬁelds of salamander retinal ganglion cells. 1 Introduction Fundamental to describing the behavior of neurons in response to sensory stimuli or to inputs from other neurons is the need for succinct models that can be estimated and validated with limited data. Towards this end, many functional models assume a cascade structure where an initial linear stage combines inputs to produce a low-dimensional output for subsequent nonlinear stages. For example, in the widely-used linear nonlinear Poisson (LNP) model for retinal ganglion cells (RGCs) [1,2], the time-varying input stimulus vector is ﬁrst linearly ﬁltered and summed to produce a low (typically one or two) dimensional output, which is then passed through a memoryless nonlinear function that outputs the neuron’s instantaneous Poisson spike rate. An initial linear ﬁltering stage also appears in the well-known integrate-and-ﬁre model [3]. The linear ﬁltering stage in these models reduces the dimensionality of the parameter estimation problem and provides a simple characterization of a neuron’s receptive ﬁeld or connectivity. However, even with the dimensionality reduction from assuming such linear stages, parameter estimation may be difﬁcult when the stimulus is high-dimensional or the ﬁlter lengths are large. Compressed sensing methods have been recently proposed [4] to reduce the dimensionality further. The key insight is that although most experiments for mapping, say visual receptive ﬁelds, expose the 1 Linear filtering Stimulus (eg. n pixel image) [ ] 1 u t ( * )[ ] 1 1 u w t ( * )[ ] u w t n n [ ] u t n Poisson spike process [ ] z t [ ]t  Nonlinearity [ ] d t Gaussian noise Spike count [ ] y t Figure 1: Linear nonlinear Poisson (LNP) model for a neuron with n stimuli. neural system under investigation to a large number of stimulus components, the overwhelming majority of the components do not affect the instantaneous spiking rate of any one particular neuron due to anatomical sparsity [5,6]. As a result, the linear weights that model the response to these stimulus components will be sparse; most of the coefﬁcients will be zero. For the retina, the stimulus is typically a large image, whereas the receptive ﬁeld of any individual neuron is usually only a small portion of that image. Similarly, for mapping cortical connectivity to determine the connectome, each neuron is typically only connected to a small fraction of the neurons under test [7]. Due to the sparsity of the weights, estimation can be performed via sparse reconstruction techniques similar to those used in compressed sensing (CS) [8–10]. This paper presents a CS-based estimation of linear neuronal weights via a recently-developed generalized approximate message passing (GAMP) methods from [11] and [12]. GAMP, which builds upon earlier work in [13, 14], is a Gaussian approximation of loopy belief propagation. The beneﬁts of the GAMP method for neural mapping are that it is computationally tractable with large sums of data, can incorporate very general graphical model descriptions of the neuron and provides a method for simultaneously estimating the parameters in the linear and nonlinear stages. In contrast, methods such as the common spike-triggered average (STA) perform separate estimation of the linear and nonlinear components. Following the simulation methodology in [4], we show that the GAMP method offers signiﬁcantly improved reconstruction of cortical wiring diagrams over other state-of-the-art CS techniques. We also validate the GAMP-based sparse estimation methodology in the problem of ﬁtting LNP models of salamander RGCs. LNP models have been widely-used in systems modeling of the retina, and they have provided insights into how ganglion cells communicate to the lateral geniculate nucleus, and further upstream to the visual cortex [15]. Such understanding has also helped clarify the computational purpose of cell connectivity in the retina. The ﬁlter shapes estimated by the GAMP algorithm agree with other ﬁndings on RGC cells using STA methods, such as [16]. What is important here is that the ﬁlter coefﬁcients can be estimated accurately with a much smaller number of measurements. This feature suggests that GAMP-based sparse modeling may be useful in the future for other neurons and more complex models. 2 Linear Nonlinear Poisson Model 2.1 Mathematical Model We consider the following simple LNP model for the spiking output of a single neuron under n stimulus components shown in Fig. 1, cf. [1, 2]. Inputs and outputs are measured in uniform time intervals t = 0, 1, . . . , T −1, and we let uj[t] denote the jth stimulus input in the tth time interval, j = 1, . . . , n. For example, if the stimulus is a sequence of images, n would be the number of pixels in each image and uj[t] would be the value of the jth pixel over time. We let y[t] denote the number of spikes in the tth time interval, and the general problem is to ﬁnd a model that explains the relation between the stimuli uj[t] and spike outputs y[t]. As the name suggests, the LNP model is a cascade of three stages: linear, nonlinear and Poisson. In the ﬁrst (linear) stage, the input stimulus is passed through a set of n linear ﬁlters and then summed 2 to produce the scalar output z[t] given by z[t] = n X j=1 (wj ∗uj)[t] = n X j=1 L−1 X ℓ=0 wj[ℓ]uj[t −ℓ], (1) where wj[·] is the linear ﬁlter applied to the jth stimulus component and (wj ∗uj)[t] is the convolution of the ﬁlter with the input. We assume the ﬁlters have ﬁnite impulse response (FIR) with L taps, wj[ℓ], ℓ= 0, 1, . . . , L −1. In the second (nonlinear) stage of the LNP model, the scalar linear output z[t] passes through a memoryless nonlinear random function to produce a spike rate λ[t]. We assume a nonlinear mapping of the form λ[t] = f(v[t]) = log h 1 + exp φ(v[t]; α) i , (2a) v[t] = z[t] + d[t], d[t] ∼N(0, σ2 d), (2b) where d[t] is Gaussian noise to account for randomness in the spike rate and φ(v; α) is the ν-th order polynomial, φ(v; α) = α0 + α1v + · · · + ανvν. (3) The form of the function in (2b) ensures that the spike rate λ[t] is always positive. In the third and ﬁnal stage of the LNP model, the number of spikes is modeled as a Poisson process with mean λ[t]. That is, Pr y[t] = k λ[t] = e−λ[t]λ[t]k/k!, k = 0, 1, 2, . . . (4) This LNP model is sometimes called a one-dimensional model since z[t] is a scalar. 2.2 Conventional Estimation Methods The parameters in the neural model can be written as the vector θ = (w, α, σ2 d), where w is the nLdimensional vector of the ﬁlter coefﬁcients, the vector α contains the ν + 1 polynomial coefﬁcients in (3) and σ2 d is the noise variance. The basic problem is to estimate the parameters θ from the input/output data uj[t] and y[t]. We brieﬂy summarize three conventional methods: spike-triggered average (STA), reverse correlation (RC) and maximum likelihood (ML), all described in several texts including [1]. The STA and RC methods are based on simple linear regression. The vector z of linear ﬁlter outputs z[t] in (1) can be written as z = Aw, where A is a known block Toeplitz matrix with the input data uj[t]. The STA and RC methods then both attempt to ﬁnd a w such that output z has high linear correlation with measured spikes y. The RC method ﬁnds this solution with the least squares estimate bwRC = (A∗A + σ2I)−1A∗y, (5) for some parameter σ2, and the STA is an approximation given by bwSTA = 1 T A∗y. (6) The statistical properties of the estimates are discussed in [17,18]. Once the estimate bw = bwSTA or bwRC has been computed, one can compute an estimate bz = Abw for the linear output z and then use any scalar estimation method to ﬁnd a nonlinear mapping from z[t] to λ[t] based on the outputs y[t]. A shortcoming of the STA and RC methods is that the ﬁlter coefﬁcients w are selected to maximize the linear correlation and may not work well when there is a strong nonlinearity. A maximum likelihood (ML) estimate may overcome this problem by jointly optimizing over nonlinear and linear parameters. To describe the ML estimate, ﬁrst ﬁx parameters α and σ2 d in the nonlinear stage. Then, given the vector output z from the linear stage, the spike count components y[t] are independent: Pr y z, α, σ2 d = T−1 Y t=0 Pr y[t] z[t], α, σ2 d (7) 3 where the component distributions are given by P y[t] z[t], α, σ2 d = Z ∞ 0 Pr y[t] λ[t] p λ[t] z[t], α, σ2 d dλ[t], (8) and p λ[t] z[t], α, σ2 d can be computed from the relation (2b) and Pr y[t] λ[t] is the Poisson distribution (4). The ML estimate is then given by the solution to the optimization bθML := arg max (w,α,σ2 d) T−1 Y t=0 Pr y[t] z[t], α, σ2 d , z = Aw. (9) In this way, the ML estimate attempts to maximize the goodness of ﬁt by simultaneously searching over the linear and nonlinear parameters. 3 Estimation via Compressed Sensing 3.1 Bayesian Model with Group Sparsity A difﬁculty in the above methods is that the number, Ln, of ﬁlter coefﬁcients in w may be large and require an excessive number of measurements to estimate accurately. As discussed above, the key idea in this work is that most stimulus components have little effect on the spiking output. Most of the ﬁlter coefﬁcients wj[ℓ] will be zero and exploiting this sparsity may be able to reduce the number of measurements while maintaining the same estimation accuracy. The sparse nature of the ﬁlter coefﬁcients can be modeled with the following group sparsity structure: Let ξj be a binary random variable with ξj = 1 when stimulus j is in the receptive ﬁeld of the neuron and ξj = 0 when it is not. We call the variables ξj the receptive ﬁeld indicators, and model these indicators as i.i.d. Bernoulli variables with Pr(ξj = 1) = 1 −Pr(ξj = 0) = ρ, (10) where ρ ∈[0, 1] is the average fraction of stimuli in the receptive ﬁeld. We then assume that, given the vector ξ of receptive ﬁeld indicators, the ﬁlter weight coefﬁcients are independent with distribution p wj[ℓ] ξ = p wj[ℓ] ξj = 0 if ξj = 0 N(0, σ2 x) if ξj = 1. (11) That is, the linear weight coefﬁcients are zero outside the receptive ﬁeld and Gaussian within the receptive ﬁeld. Since our algorithms are general, other distributions can also be used—we use the Gaussian for illustration. The distribution on w deﬁned by (10) and (11) is often called a group sparse model, since the components of the vector w are zero in groups. Estimation with this sparse structure leads naturally to a compressed sensing problem. Speciﬁcally, we are estimating a sparse vector w through a noisy version y of a linear transform z = Aw, which is precisely the problem of compressed sensing [8–10]. With a group structure, one can employ a variety of methods including the group Lasso [19–21] and group orthogonal matching pursuit [22]. However, these methods are designed for either AWGN or logistic outputs. In the neural model, the spike count y[t] is a nonlinear, random function of the linear output z[t] described by the probability distribution in (8). 3.2 GAMP-Based Sparse Estimation To address the nonlinearities in the outputs, we use the generalized approximate message passing (GAMP) algorithm [11] with extensions in [12]. The GAMP algorithm is a general approximate inference method for graphical models with linear mixing. To place the neural estimation problem in the GAMP framework, ﬁrst ﬁx the stimulus input vector u, nonlinear output parameters α and σ2 d. Then, the conditional joint distribution of the outputs y, linear ﬁlter weights w and receptive ﬁeld indicators ξ factor as p y, ξ, w u, α, σ2 d = n Y j=1 " Pr(ξj) L−1 Y ℓ=0 p wj[ℓ] ξj # T−1 Y t=0 Pr y[t] z[t], α, σ2 d , z = Aw. (12) 4 Data matrix with input stimuli uj[t] A Receptive field indicators ξ   [ ] p wj j   ( ) P j  2 [ ] [ ], , p y t z t d        α Filter weights w Filter outputs z Observed spike counts y Nonlinear parameters 2 , d  α Figure 2: The neural estimation problem represented as a graphical model with linear mixing. Solid circles are unknown variables, dashed circles are observed variables (in this case, spike counts) and squares are factors in the probability distribution. The linear mixing component of the graph indicates the constraints that z = Aw. Similar to standard graphical model estimation [23], GAMP is based on the ﬁrst representing the distribution in (12) via a factor graph as shown in Fig. 2. In the factor graph, the solid circles represent the components of the unknown vectors w, ξ, . . ., and the dashed circles the components of the observed or measured variables y. Each square corresponds to one factor in the distribution (12). What is new for the GAMP methodology, is that the factor graph also contains a component to indicate the linear constraints that z = Aw, which would normally be represented by a set of additional factor nodes. Inference on graphical models is often performed by some variant of loopy belief propagation (BP). Loopy BP attempts to reduce the joint estimation of all the variables to a sequence of lower dimensional estimation problems associated with each of the factors in the graph. Estimation at the factor nodes is performed iteratively, where after each iteration, “beliefs” of the variables are passed to the factors to improve the estimates in the subsequent iterations. Details can be found in [23]. However, exact implementation of loopy BP is intractable for the neural estimation problem: The linear constraints z = Aw create factor nodes that connect each of the variables z[t] to all the variables wj[ℓ] where uj[t −ℓ] is non-zero. In the RGC experiments below, the pixels value uj[t] are non-zero 50% of the time, so each variable z[t] will be connected to, on average, half of the Ln ﬁlter weight coefﬁcients through these factor nodes. Since exact implementation of loopy BP grows exponentially in the degree of the factor nodes, loopy BP would be infeasible for the neural problem, even for moderate values of Ln. The GAMP method reduces the complexity of loopy BP by exploiting the linear nature of the relations between the variables w and z. Speciﬁcally, it is shown that when each term z[t] is a linear combination of a large number of terms wj[ℓ], the belief messages across the factor node for the linear constraints can be approximated as Gaussians and the factor node updates can be computed with a central limit theorem approximation. Details are in [11] and [12]. 4 Receptive Fields of Salamander Retinal Ganglion Cells The sparse LNP model with GAMP-based estimation was evaluated on data from recordings of neural spike trains from salamander retinal ganglion cells exposed to random checkerboard images, following the basic methods of [24].1 In the experiment, spikes from individual neurons were measured over an approximately 1900s period at a sampling interval of 10ms. During the recordings, the salamander was exposed to 80 × 60 pixel random black-white binary images that changed every 3 to 4 sampling intervals. The pixels of each image were i.i.d. with a 50-50 black-white probability. We compared three methods for ﬁtting an L = 30 tap one-dimensional LNP model for the RGC neural responses: (i) truncated STA, (ii) approximate ML, and (iii) GAMP estimation with the sparse LNP model. Methods (i) and (ii) do not exploit sparsity, while method (iii) does. The truncated STA method was performed by ﬁrst computing a linear ﬁlter estimate as in (6) for the entire 80 × 60 image and then setting all coefﬁcients outside an 11 × 11 pixel subarea around the pixel with the largest estimated response to zero. The 11 × 11 size was chosen since it is sufﬁciently large to contain these neurons’ entire receptive ﬁelds. This truncation signiﬁcantly improves the STA estimate by removing spurious estimates that anatomically cannot have relation to the neural 1Data from the Leonardo Laboratory at the Janelia Farm Research Campus. 5 Non−sparse LNP w/ STA 400 s Training Sparse LNP w/ GAMP 600 s Training 0 100 200 300 Delay (ms) 1000 s Training 0 100 200 300 Delay (ms) (a) Filter responses over time Non−sparse LNP w/ STA Sparse LNP w/ GAMP (b) Spatial receptive ﬁeld Figure 3: Estimated ﬁlter responses and visual receptive ﬁeld for salamander RGCs using a nonsparse LNP model with STA estimation and a sparse LNP model with GAMP estimation. responses; this provides a better comparison to test other methods. From the estimate bwSTA of the linear ﬁlter coefﬁcients, we compute an estimate bz = Abw of the linear ﬁlter output. The output parameters α and σ2 d are then ﬁt by numerical maximization of likelihood P(y\|bz, α, σ2 d) in (7). We used a (ν = 1)-order polynomial, since higher orders did not improve the prediction. The fact that only a linear polynomial was needed in the output is likely due to the fact that random checkerboard images rarely align with the neuron’s ﬁlters and therefore do not excite the neural spiking into a nonlinear regime. An interesting future experiment would be to re-run the estimation with swatches of natural images as in [25]. We believe that under such experimental conditions, the advantages of the GAMP-based nonlinear estimation would be even larger. The RC estimate (5) was also computed, but showed no appreciable difference from the STA estimate for this matrix A. As a result, we discuss only STA results below. The GAMP-based sparse estimation used the STA estimate for initialization to select the 11 × 11 pixel subarea and the variances σ2 x in (11). As in the STA case, we used only a (ν = 1)-order linear polynomial in (3). The linear coefﬁcient α1 was set to 1 since other scalings could be absorbed into the ﬁlter weights w. The constant term α0 was incorporated as another linear regression coefﬁcient. For a third algorithm, we approximately computed the ML estimate (9) by running the GAMP algorithm, but with all the factors for the priors on the weights w removed. To illustrate the qualitative differences between the estimates, Fig. 3 shows the estimated responses for the STA and GAMP-based sparse LNP estimates for one neuron using three different lengths of training data: 400, 600 and 1000 seconds of the total 1900 second training data. For brevity, the approximate ML estimate is omitted, but is similar to the STA estimate. The estimated responses in Fig. 3(a) are displayed as 11 × 11 = 121 curves, each curve representing the linear ﬁlter response with L = 30 taps over the 30×10 = 300ms response. Fig. 3(b) shows the estimated spatial receptive ﬁelds plotted as the total magnitude of the 11 × 11 ﬁlters. One can immediately see that the GAMP based sparse estimate is signiﬁcantly less noisy than the STA estimate, as the smaller, unreliable responses are zeroed out in the GAMP-based sparse LNP estimate. The improved accuracy of the GAMP-estimation with the sparse LNP model was veriﬁed in the cross validation, as shown in Fig. 4. In this plot, the length of the training data was varied from 200 to 1000 seconds, with the remaining portion of the 1900 second data used for cross-validation. At each training length, each of the three methods—STA, GAMP-based sparse LNP and approximate ML—were used to produce an estimate bθ = (bw, bα, bσ2 d). Fig. 4 shows, for each of these methods, the cross-validation scores P(y\|bz, bα, bσ2 d)1/T , which is the geometric mean of the likelihood in (7). It can be seen that the GAMP-based sparse LNP estimate signiﬁcantly outperforms the STA and 6 200 400 600 800 1000 0.895 0.9 0.905 0.91 0.915 0.92 0.925 Train time (sec) Cross−valid score Sparse LNP w/ GAMP Non−sparse LNP w/ STA Non−sparse LNP w/ approx ML Figure 4: Prediction accuracy of sparse and non-sparse LNP estimates for data from salamander RGC cells. Based on cross-validation scores, the GAMP-based sparse LNP estimation provides a signiﬁcantly better estimate for the same amount of training. 10 −4 10 −3 10 −2 10 −1 10 0 0 0.2 0.4 0.6 0.8 1 False alarm prob, pFA Missed detect prob, pMD RC CoSaMP GAMP Figure 5: Comparison of reconstruction methods on cortical connectome mapping with multi-neuron excitation based on simulation model in [4]. In this case, connectivity from n = 500 potential pre-synaptic neurons are estimated from m = 300 measurements with 40 neurons excited in each measurement. In the simulation, only 6% of the n potential neurons are actually connected to the postsynaptic neuron under test. approximate ML estimates that do not assume any sparse structure. Indeed, by the measure of the cross-validation score, the sparse LNP estimate with GAMP after only 400 seconds of data was as accurate as the STA estimate with 1000 seconds of data. Interestingly, the approximate ML estimate is actually worse than the STA estimate, presumably since it overﬁts the model. 5 Neural Mapping via Multi-Neuron Excitation The GAMP methodology was also applied to neural mapping from multi-neuron excitation, originally proposed in [4]. A single post-synaptic neuron has connections to n potential pre-synaptic neurons. The standard method to determine which of the n neurons are connected to the postsynaptic neurons is to excite one neuron at a time. This process is wasteful, since only a small fraction of the neurons are typically connected. In the method of [4], multiple neurons are excited in each measurement. Then, exploiting the sparsity in the connectivity, compressed sensing techniques can be used to recover the mapping from m < n measurements. Unfortunately, the output stage of spiking neurons is often nonlinear and most CS methods cannot directly incorporate such nonlinearities into the estimation. The GAMP methodology thus offers the possibility of improved performance for reconstruction. To validate the methodology, we compared the performance of GAMP to various reconstruction methods following a simulation of mapping of cortical neurons with multi-neuron excitation in [4]. The simulation assumes an LNP model of Section 2.1, where the inputs uj[t] are 1 or 0 depending on whether the jth pre-synaptic input is excited in tth measurement. The ﬁlters have a single tap (i.e. L=1), which are modeled as a Bernoulli-Weibull distribution with a probability ρ = 0.06 of being on (the neuron is connected) or 1 −ρ of being zero (the neuron is not connected). The output has a strong nonlinearity including a thresholding and saturation – the levels of which must be estimated. Connectivity detection amounts to determining which of the n pre-synaptic neurons have non-zero weights. Fig. 5 plots the missed detection vs. false alarm rate of the various detectors. It can be seen that the GAMP-based connectivity detection signiﬁcantly outperforms both non-sparse RC reconstruction as well as a state-of-the-art greedy sparse method CoSaMP [26,27]. 7 6 Conclusions and Future Work A general method for parameter estimation in neural models based on generalized approximate message passing was presented. The GAMP methodology is computationally tractable for large data sets, can exploit sparsity in the linear coefﬁcients and can incorporate a wide range of nonlinear modeling complexities in a systematic manner. Experimental validation of the GAMP-based estimation of a sparse LNP model for salamander RGC cells shows signiﬁcantly improved prediction in cross-validation over simple non-sparse estimation methods such as STA. Beneﬁts over state-of-theart sparse reconstruction methods are also apparent in simulated models of cortical mapping with multi-neuron excitation. Going forward, the generality offered by the GAMP model will enable accurate parameter estimation for other complex neural models. For example, the GAMP model can incorporate other prior information such as a correlation between responses in neighboring pixels. Future work may also include experiments with integrate-and-ﬁre models [3]. An exciting future possibility for cortical mapping is to decode memories, which are thought to be stored as the connectome [7,28]. Throughout this paper, we have presented GAMP as an experimental data analysis method. One might wonder, however, whether the brain itself might use compressive representations and message-passing algorithms to make sense of the world. There have been several previous suggestions that visual and general cortical regions of the brain may use belief propagation-like algorithms [29, 30]. There have also been recent suggestions that the visual system uses compressive representations [31]. As such, we assert the biologically plausibility of the brain itself using the algorithms presented herein for receptive ﬁeld and memory decoding. 7 Acknowledgements We thank D. B. Chklovskii and T. Hu for formulative discussions on the problem, A. Leonardo for providing experimental data and further discussions, and B. Olshausen for discussions. 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4,423	Bayesian Partitioning of Large-Scale Distance Data David Adametz Volker Roth Department of Computer Science & Mathematics University of Basel Basel, Switzerland {david.adametz,volker.roth}@unibas.ch Abstract A Bayesian approach to partitioning distance matrices is presented. It is inspired by the Translation-invariant Wishart-Dirichlet process (TIWD) in [1] and shares a number of advantageous properties like the fully probabilistic nature of the inference model, automatic selection of the number of clusters and applicability in semi-supervised settings. In addition, our method (which we call fastTIWD) overcomes the main shortcoming of the original TIWD, namely its high computational costs. The fastTIWD reduces the workload in each iteration of a Gibbs sampler from O(n3) in the TIWD to O(n2). Our experiments show that the cost reduction does not compromise the quality of the inferred partitions. With this new method it is now possible to ‘mine’ large relational datasets with a probabilistic model, thereby automatically detecting new and potentially interesting clusters. 1 Introduction In cluster analysis we are concerned with identifying subsets of n objects that share some similarity and therefore potentially belong to the same sub-population. Many practical applications leave us without direct access to vectorial representations and instead only supply pairwise distance measures collected in a matrix D. This poses a serious challenge, because great parts of geometric information are hereby lost that could otherwise help to discover hidden structures. One approach to deal with this is to encode geometric invariances in the probabilistic model, as proposed in [1]. The most important properties that distinguish this Translation-invariant Wishart-Dirichlet Process (TIWD) from other approaches working on pairwise data are its fully probabilistic model, automatic selection of the number of clusters, and its applicability in semi-supervised settings in which not all classes are known in advance. Its main drawback, however, is the high computational cost of order O(n3) per sweep of a Gibbs sampler, limiting its applicability to relatively small data sets. In this work we present an alternative method which shares all the positive properties of the TIWD while reducing the computational workload to O(n2) per Gibbs sweep. In analogy to [1] we call this new approach fastTIWD. The main idea is to solve the problem of missing geometric information by a normalisation procedure, which chooses one particular geometric embedding of the distance data and allows us to use a simple probabilistic model for inferring the unknown underlying partition. The construction we use is guaranteed to give the optimal such geometric embedding if the true partition was known. Of course, this is only a hypothetical precondition, but we show that even rough prior estimates of the true partition signiﬁcantly outperform ‘naive’ embedding strategies. Using a simple hierarchical clustering model to produce such prior estimates leads to clusterings being at least of the same quality as those obtained by the original TIWD. The algorithmic contribution here is an efﬁcient algorithm for performing this normalisation procedure in O(n2) time, which makes the whole pipeline from distance matrix to inferred partition an O(n2) process (assuming a constant number of Gibbs sweeps). Detailed complexity analysis shows not only a worst-case complexity reduction from O(n3) to O(n2), but also a drastic speed improvement. We demonstrate 1 this performance gain for a dataset containing ≈350 clusters, which now can be analysed in 6 hours instead of ≈50 days with the original TIWD. It should be noted that both the TIWD and our fastTIWD model expect (squared) Euclidean distances on input. While this might be seen as a severe limitation, we argue that (i) a ‘zoo’ of Mercer kernels has been published in the last decade, e.g. kernels on graphs, sequences, probability distributions etc. All these kernels allow the construction of squared Euclidean distances; (ii) efﬁcient preprocessing methods like randomised versions of kernel PCA have been proposed, which can be used to transform an initial matrix into one of squared Euclidean type; (iii) one might even use an arbitrary distance matrix hoping that the resulting model mismatch can be tolerated. In the next section we introduce a probabilistic model for partitioning inner product matrices, which is generalised in section 3 to distance matrices using a preprocessing step that breaks the geometric symmetry inherent in distance representations. Experiments in section 4 demonstrate the high quality of clusterings found by our method and its superior computational efﬁciency over the TIWD. 2 A Wishart Model for Partitioning Inner Product Matrices Suppose there is a matrix X ∈Rn×d representing n objects in Rd that belong to one of k subpopulations. For identifying the underlying cluster structure, we formulate a generative model by assuming the columns xi ∈Rn, i = 1 . . . d are i.i.d. according to a normal distribution with zero mean and covariance Σn×n, i.e. xi ∼N(0n, Σ), or in matrix notation: X ∼N(0n×d, Σ ⊗I). Then, S = 1 dXXt ∈Rn×n is central Wishart distributed, S ∼Wd(Σ). For convenience we deﬁne the generalised central Wishart distribution which also allows rank-deﬁcient S and/or Σ as p(S\|Ψ, d) ∝det(S) 1 2 (d−n−1) det(Ψ) d 2 exp −d 2 tr(ΨS) , (1) where det(•) is the product of non-zero eigenvalues and Ψ denotes the (generalised) inverse of Σ. The likelihood as a function in Ψ is L(Ψ) = det(Ψ) d 2 exp −d 2 tr(ΨS) . (2) Consider now the case where we observe S without direct access to X. Then, an orthogonal transformation X ←OX cannot be retrieved anymore, but it is reasonable to assume such rotations are irrelevant for ﬁnding the partition. Following the Bayesian inference principle, we complement the likelihood with a prior over Ψ. Since by assumption there is an underlying joint normal distribution, a zero entry in Ψ encodes conditional independence between two objects, which means that block diagonal Ψ matrices deﬁne a suitable partitioning model in which the joint normal is decomposed into independent cluster-wise normals. Note that the inverse of a block diagonal matrix is also block diagonal, so we can formulate the prior in terms of Σ, which is easier to parametrise. For this purpose we adapt the method in [2] using a Multinomial-Dirichlet process model [3, 4, 5] to deﬁne a ﬂexible prior distribution over block matrices without specifying the exact number of blocks. We only brieﬂy sketch this construction and refer the reader to [1, 2] for further details. Let Bn be the set of partitions of the index set [n]. A partition B ∈Bn can be represented in matrix form as B(i, j) = 1 if y(i) = y(j) and B(i, j) = 0 otherwise, with y being a function that maps [n] to some label set L. Alternatively, B may be represented as a set of disjoint non-empty subsets called ‘blocks’ b. A partition process is a series of distributions Pn on the set Bn in which Pn is the marginal of Pn+1. Using a multinomial model for the labels and a Dirichlet prior with rate parameter ξ on the mixing proportions, we may integrate out the latter and derive a Dirichlet-Multinomial prior over labels. Finally, after using a ‘label forgetting’ transformation, the prior over B is: p(B\|ξ, k) = k! (k −kB)! Γ(ξ) Q b∈B Γ(nb + ξ/k) [Γ(ξ/k)]kBΓ(n + ξ) . (3) In this setting, k is the number of blocks in the population (k can be inﬁnite, which leads to the Ewens Process [6], a.k.a. Chinese Restaurant Process), nb is the number of objects in block b and kB ≤k is the total number of blocks in B. The prior is exchangeable meaning rows and columns can be (jointly) permuted arbitrarily and therefore partition matrices can always be brought to block diagonal form. To specify the variances of the normal distributions, the models in [1, 2] use two global parameters, α, β, for the within- and between-class scatter. This model can be easily extended to include block-wise scatter parameters, but for the sake of simplicity we will stay with the simple parametrisation here. The ﬁnal block diagonal covariance matrix used in (2) has the form Σ = Ψ−1 = α(In + θB), with θ := β/α. (4) 2 Inference by way of Gibbs sampling. Multiplying the Wishart likelihood (2), the prior over partitions (3) and suitable priors over α, θ gives the joint posterior. Inference for B, α and θ can then be carried out via a Gibbs sampler. Each Gibbs sweep can be efﬁciently implemented since both trace and determinant in (2) can be computed analytically, see [1]: tr(ΨS) = P b∈B 1 α tr(Sbb) − θ 1+nbθ ¯Sbb = 1 α tr(S) −P b∈B θ 1+nbθ ¯Sbb , (5) where Sbb denotes the block submatrix corresponding to the bth diagonal block in B, and ¯Sbb = 1t bSbb1b. 1b is the indicator function mapping block b to a {0, 1}n vector, whose elements are 1 if a sample is contained in b, or 0 otherwise. For the determinant one derives det(Ψ) = α−n Q b∈B(1 + θnb)−1. (6) The conditional likelihood for α is Inv-Gamma(r, s) with shape parameter r = n·d/2−1 and scale s = d 2 tr(S) −P b∈B θ 1+nbθ ¯Sbb . Using the prior α ∼Inv-Gamma(r0 · d/2, s0 · d/2), the posterior is of the same functional form, and we can integrate out α analytically: Pn(B\|•) ∝Pn(B\|ξ, k) det(Ψ)d/2 (α=1) d 2 tr(ΨS)(α=1) + s0 −(n+r0)d/2 , (7) where det(Ψ)(α=1) = Q b∈B(1 + θnb)−1 and tr(ΨS)(α=1) = tr(S) −P b∈B θ 1+nbθ ¯Sbb. Note that the (usually unknown) degree of freedom d has the formal role of an annealing parameter, and it can indeed be used to ‘cool’ the Markov chain by increasing d, if desired, until a partition is ‘frozen’. Complexity analysis. In one sweep of the Gibbs sampler, we have to iteratively compute the membership probability of one object indexed by i to the kB currently existing blocks in partition B (plus one new block), given the assignments for the n−1 remaining ones denoted by the superscript (−i) [7, 8]. In every step of this inner loop over kB existing blocks we have to evaluate the Wishart likelihood, i.e. trace (5) and determinant (6). Given trace tr(−i), we update ¯Sbb for kB blocks b ∈B which in total needs O(n) operations. Given det(−i), the computation of all kB updated determinants induces costs of O(kB). In total, there are n objects, so a full sweep requires O(n2 + nkB) operations, which is equal to O(n2) since the maximum number of blocks is n, i.e. kB ≤n. Following [1], we update θ on a discretised grid of values which adds O(kB) to the workload, thus not changing the overall complexity of O(n2). Compared to the original TIWD, the worst case complexity in the Dirichlet process model with an inﬁnite number of blocks in the population, k = ∞, is reduced from O(n3) to O(n2) . 3 The fastTIWD Model for Partitioning Distance Matrices Consider now the case where S is not accessible, but only squared pairwise distances D ∈Rn×n: D(i, j) = S(i, i) + S(j, j) −2 S(i, j). (8) Observing one speciﬁc D does not imply a unique corresponding S, since there is a surjective mapping from a set of S-matrices to D, S(D) 7→D. Hereby, not only do we lose information about orthogonal transformations of X, but also information about the origin of the coordinate system. If S∗is one (any) matrix that fulﬁlls (8) for a speciﬁc D, the set S(D) is formally deﬁned as S = {S\|S = S∗+ 1vt + v1t, S ⪰0, v ∈Rn} [9]. The Wishart distribution, however, is not invariant against the choice of S ∈S. In fact, if S∗∼W(Σ), the distribution of a general S ∈S is non-central Wishart, which can be easily seen as follows: S is exactly the set of inner product matrices that can be constructed by varying c ∈Rd in a modiﬁed matrix normal model X ∼N(M, Σ ⊗Id) with mean matrix M = 1nct. Note that now the d columns in X are still independent, but no longer identically distributed. Note further that ‘shifts’ ci do not affect pairwise distances between rows in X. The modiﬁed matrix normal distribution implies that S = 1 dXXt is non-central Wishart, S ∼W(Σ, Θ), with non-centrality matrix Θ := Σ−1MM t. The practical use, however, is limited by its complicated form and the fundamental problem of estimating Θ based on only one single observation S. It is thus desirable to work with a simpler probabilistic model. In principle, there are two possibilities: either the likelihood is reformulated as being constant over all S ∈S (the approach taken in [1], called the translation-invariant Wishart distribution), or one tries to ﬁnd a ‘good’ candidate matrix S′ ∗that is ‘close’ to the underlying S∗and uses the much 3 simpler central Wishart model. Both approaches have their pros and cons: encoding the translation invariance directly in the likelihood is methodologically elegant and seems to work well in a couple of experiments (cf. [1]), but it induces high computational cost. The alternative route of searching for a good candidate S′ ∗close to S∗is complicated, because S∗is unknown and it is not immediately clear what ‘close’ means. The positive aspect of this approach is the heavily reduced computational cost due to the formal simplicity of the central Wishart model. It is important to discuss the ‘naive’ way of ﬁnding a good candidate S′ ∗by subtracting the empirical column means in X, thus removing the shifts ci. This normalisation procedure can be implemented solely based on S, leading to the well-known centering procedure in kernel PCA, [10]: Sc = QIS Qt I, with projection QI = I −(1/n)11t. (9) Contrary to the PCA setting, however, this column normalisation induced by QI does not work well here, because the elements of a column vector in X are not independent. Rather, they are coupled via the Σ component in the covariance tensor Σ ⊗Id. Hereby, we not only remove the shifts ci, but also alter the distribution: the non-centrality matrix does not vanish in general and as a result, Sc is no longer central Wishart distributed. In the following we present a solution to the problem of ﬁnding a candidate matrix S′ ∗that recasts inference based on the translation-invariant Wishart distribution as a method to reconstruct the optimal S∗. Our proposal is guided by a particular analogy between trees and partition matrices and aims at exploiting a tree-structure to guarantee low computational costs. The construction has the same functional form as (9), but uses a different projection matrix Q. The translation-invariant Wishart distribution. Let S∗induce pairwise distances D. Assuming that S∗∼Wd(Σ), the distribution of an arbitrary member S ∈S(D) can be derived analytically as a generalised central Wishart distribution with a rank-deﬁcient covariance, see [2]. Its likelihood in the rank-deﬁcient inverse covariance matrix eΨ is L(eΨ) ∝det(eΨ) d 2 exp −d 2tr(eΨS∗) = det(eΨ) d 2 exp d 4tr(eΨD) , (10) with eΨ = Ψ−(1tΨ1)−1Ψ11tΨ. Note that although S∗appears in the ﬁrst term in (10), the density is constant on all S ∈S(D), meaning it can be replaced by any other member of S(D). Note further that S also contains rank-deﬁcient matrices (like, e.g. the column normalised Sc). By multiplying (10) with the product of nonzero eigenvalues of such a matrix raised to the power of (d −n −1)/2, a valid generalised central Wishart distribution is obtained (see (1)), which is normalised on the manifold of positive semi-deﬁnite matrices of rank r = n −1 with r distinct positive eigenvalues [11, 12, 13]. Unfortunately, (10) has a simple form only in eΨ, but not in the original Ψ, which ﬁnally leads to the O(n3) complexity of the TIWD model. Selecting an optimal candidate S∗. Introducing the projection matrix Q = I − 1 1tΨ111tΨ, (11) one can rewrite eΨ in (10) as ΨQ or, equivalently, as QtΨQ, see [2] for details. Assume now S ∼Wd(Σ) induces distances D and consider the transformed S∗= QSQt. Note that this transformation does not change the distances, i.e. S ∈S(D) ⇔S∗∈S(D), and that QSQt has rank r = n −1 (because Q is a projection with kernel 1). Plugging our speciﬁc S∗= QSQt into (10), extending the likelihood to a generalised central Wishart (1) with rank-deﬁcient inverse covariance eΨ, exploiting the identity QQ = Q and using the the cyclic property of the trace, we arrive at p(QSQt\|eΨ, d) ∝det(QSQt) 1 2 (d−n−1) det(eΨ) d 2 exp −d 2 tr(ΨQSQt) . (12) By treating Q as a ﬁxed matrix, this expression can also be seen as a central Wishart in the transformed matrix S∗= QSQt, parametrised by the full-rank matrix Ψ if det(eΨ) is substituted by the appropriate normalisation term det(Ψ). From this viewpoint, inference using the translation-invariant Wishart distribution can be interpreted as ﬁnding a (rank-deﬁcient) representative S∗= QSQt ∈S(D) which follows a generalised central Wishart distribution with full-rank inverse covariance matrix Ψ. For inferring Ψ, the rank deﬁciency of S∗is not relevant, since only the likelihood is needed. Thus S∗can be seen as an optimal candidate inner-product matrix in the set S(D) for a central Wishart model parametrised by Ψ. 4 Approximating S∗with trees. The above selection of S∗∈S(D) cannot be directly used in a constructive way, since Q in (11) depends on unknown Ψ. If, on the other hand, we had some initial estimate of Ψ, we could ﬁnd a reasonable transformation Q′ and hereby a reasonable candidate S′ ∗. Note that even if the estimate of Ψ is far away from the true inverse covariance, the pairwise distances are at least guaranteed not to change under Q′S(Q′)t. One particular estimate would be to assume that every object forms a singleton cluster, which means that our estimate of Ψ is an identity matrix. After substitution into (11) it is easily seen that this assumption results in the column-normalisation projection QI deﬁned in (9). However, if we assume that there is some non-trivial cluster structure in the data, this would be a very poor approximation. The main difﬁculty in ﬁnding a better estimate is to not specify the number of blocks. Our construction is guided by an analogy between binary trees and weighted sums of cut matrices, which are binary complements of partition matrices with two blocks. We use a binary tree with n leaves representing n objects. It encodes a path distance matrix Dtree between those n objects, and for an optimal tree Dtree = D. Such an optimal tree exists only if D is additive, and the task of ﬁnding an approximation is a well-studied problem. We will not discuss the various tree reconstruction algorithms, but only mention that there exist algorithms for reconstructing the closest ultrametric tree (in the ℓ∞norm) in O(n2) time, [14]. Figure 1: From left to right: Unknown samples X, pairwise distances collected in D, closest tree structure and an exemplary building block. A tree metric induced by Dtree is composed of elementary cut (pseudo-)metrics. Any such metric lies in the metric space L1 and is also a member of (L2)2, which is the metric part of the space of squared Euclidean distance matrices D. Thus, there exists a positive (semi-)deﬁnite Stree such that (Dtree)ij = (Stree)ii + (Stree)jj −2(Stree)ij. In fact, any matrix Stree has a canonical decomposition into a weighted sum of 2-block partition matrices, which is constructed by cutting all edges (2n −2 for a rooted tree) and observing the resulting classiﬁcation of leaf nodes. Suppose, we keep track of such an assignment with indicator 1j induced by a single cut j, then the inner product matrix is Stree = P2n−2 j=1 λj(1j1t j + ¯1j¯1t j), (13) where λj is the weight of edge j to be cut and ¯1j 7→{0, 1}n is the complementary assignment, i.e. 1j ﬂipped. Each term (1j1t j + ¯1j¯1t j) is a 2-block partition matrix. We demonstrate the construction of Stree in Fig. 2 for a small dataset of n = 25 objects sampled from S ∼Wd(Σ) with d = 25 and Σ = α(In + θB) as deﬁned in (4) with α = 2 and θ = 1. B contains 3 blocks and is depicted in the ﬁrst panel. The remaining panels show the single-linkage clustering tree, all 2n −2 = 48 weighted 2-block partition matrices, and the ﬁnal Stree (= sum of all individual 2-block matrices, rescaled to full gray-value range). Note that single-linkage fails to identify the clusters in the three branches closest to root, but still the structure of B is clearly visible in Stree. Figure 2: Inner product matrix of a tree. Left to right: Partition matrix B for n = 25 objects in 3 clusters, single-linkage tree, all weighted 2-block partition matrices, ﬁnal Stree. The idea is now to have Stree as an estimate of Σ, and use its inverse Ψtree to construct Qtree in (11), which, however, naively would involve an O(n3) Cholesky decomposition of Stree. 5 Theorem 1. The n × n matrix S∗= QtreeSQt tree can be computed in O(n2) time. For the proof we need the following lemma: Lemma 1. The product of Stree ∈Rn×n and a vector y ∈Rn can be computed in O(n) time. Proof. (of lemma 1) Restating (13) and deﬁning m := 2n −2, we have Streey = Xm j=1 λj 1j1t j + ¯1j¯1t j y = Xm j=1 λj 1j Xn l=1 1jl yl + ¯1j Xn l=1 ¯1jl yl = Xn l=1 yl Xm j=1 λj¯1j + Xm j=1 λj1j Xn l=1 1jlyl − Xm j=1 λj¯1j Xn l=1 1jlyl. (14) In the next step, let us focus speciﬁcally on the ith element of the resulting vector. Furthermore, assume Ri is the set of all nodes on the branch starting from node i and leading to the tree’s root: (Streey)i = Xn l=1 yl X j /∈Ri λj + X j∈Ri λj X l∈Rj yl − X j /∈Ri λj X l∈Rj yl = Xn l=1 yl Xm j=1 λj − X j∈Ri λj + 2 X j∈Ri λjyj − Xm j=1 λjyj. (15) Note that Pn l=1 yl, Pm j=1 λj and Pm j=1 λjyj are constants and computed in O(n) time. For each element i, we are now left to ﬁnd Ri in order to determine the remaining two terms. This can be done directly on the tree structure in two separate traversals: 1. Bottom up: Starting from the leaf nodes, store the sum of both childrens’ y values in their parent node j (see Fig. 1, rightmost), then ascend. Do the same for λj and compute λjyj. 2. Top down: Starting from the root node, recursively descend into the child nodes j and sum up λj and λjyj until reaching the leafs. This implicitly determines Ri. It is important to stress that the above two tree traversals fully describe the complete algorithm. Proof. (of theorem 1) First, note that only the matrix-vector product a := Ψtree1 is needed in QtreeSQt tree = I − 1 1tΨtree111t Ψtree S I −Ψtree 1 1tΨtree111t = S −(1/1ta) 1atS −(1/1ta) S a1t + (1/1ta)2 1atS a1t. (16) One way of computing a = Ψtree1 is to employ conjugate gradients (CG) and iteratively minimise \|\|Streea −1\|\|2. Theoretically, CG is guaranteed to ﬁnd the true a in O(n) iterations, each evaluating one matrix-vector product (Streey), y ∈Rn. Due to lemma 1, a can be computed in O(n2) time and is used in (16) to compute S∗= QtreeSQt tree (only matrix-vector products, so O(n2) complexity is maintained). 4 Experiments Synthetic examples: normal clusters. In a ﬁrst experiment we investigate the performance of our method on artiﬁcial datasets generated in accordance with underlying model assumptions. A partition matrix B of size n = 200 containing k = 3 blocks is sampled from which we construct ΣB = α(I + θB). Then, X is drawn from N(M = 40 · 1n1t d, Σ = ΣB ⊗Id) with d = 300 to generate S = 1 dXXt and D. The covariance parameters are set to α = 2 and θ = 15/d, which deﬁnes a rather difﬁcult clustering problem with a hardly visible structure in D as can be seen in the left part of Fig. 3. We compared the method to three different hierarchical clustering strategies (single-linkage, complete-linkage, Ward’s method), to the standard central Wishart model using two different normalisations of S (‘WD C’: column normalisation using Sc = QISQt I and ‘WD R’: additional row normalisation after embedding Sc using kernel PCA) and to the original TIWD model. The experiment was repeated 200 times and the quality of the inferred clusters was measured by the adjusted Rand index w.r.t. the true labels. For the hierarchical methods we report two different performance values: splitting the tree such that the ‘true’ number k = 3 of clusters is obtained and computing the best value among all possible splits into [2, n] clusters (‘*.best’ in the boxplot). The reader should notice that both values are in favour of the hierarchical algorithms, since neither the true k nor the true labels are used for inferring the clusters in the Wishart-type methods. From the right part of Fig. 3 we conclude that (i) both ‘naive’ normalisation strategies WD C and WD R are clearly outperformed by TIWD and fastTIWD (‘fTIWD’ in the boxplot). Signiﬁcance of pairwise performance differences is measured with a nonparametric Kruskal-Wallis test with a 6 Bonferroni-corrected post-test of Dunn’s type, see the rightmost panel; (ii) the hierarchical methods have severe problems with high dimensionality and low class separation, and optimising the tree cutting does not help much. Even Ward’s method (being perfectly suited for spherical clusters) has problems; (iii) there is no signiﬁcant difference between TIWD and fastTIWD. Figure 3: Normal distributed toy data. Left half: Partition matrix (top), distance matrix (bottom) and 2D-PCA embedding of a dataset drawn from the generative model. Right half: Agreement with ‘true’ labels measured by the adjusted Rand index (left) and outcome of a Kruskal-Wallis/Dunn test (right). Black squares mean two methods are different at a ‘family’ p-value ≤0.05. Synthetic examples: log-normal clusters. In a second toy example we explicitly violate underlying model assumptions. For this purpose we sample again 3 clusters in d = 300 dimensions, but now use a log-normal distribution that tends to produce a high number of ‘atypical’ samples. Note that such a distribution should not induce severe problems for hierarchical methods when optimising the Rand index over all possible tree cuttings, since the ‘atypical’ samples are likely to form singleton clusters while the main structure is still visible in other branches of the tree. This should be particularly true for Ward’s method, since we still have spherically shaped clusters. As for the fastTIWD model, we want to test if the prior over partitions is ﬂexible enough to introduce additional singleton clusters: In the experiment, it performed at least as well as Ward’s method, and clearly outperformed single- and complete-linkage. We also compared it to the afﬁnity-propagation method (AP), which, however, has severe problems on this dataset, even when optimising the input preference parameter that affects the number of clusters in the partition. Figure 4: Log-normal distributed toy data. Left: Agreement with ‘true’ labels measured by the adjusted Rand index. Right: Outcome of a Kruskal-Wallis/Dunn test, analogous to Fig. 3. Semi-supervised clustering of protein sequences. As large-scale application we present a semisupervised clustering example which is an upscaled version of an experiment with protein sequences presented in [1]. While traditional semi-supervised classiﬁers assume at least one labelled object per class, our model is ﬂexible enough to allow additional new clusters that have no counterpart in the subset of labelled objects. We apply this idea on two different databases, one being high quality due to manual annotation with a stringent review process (SwissProt) while the other contains automatically annotated proteins and is not reviewed (TrEMBL). The annotations in SwissProt are used as supervision information resulting in a set of class labels, whereas the proteins in TrEMBL are treated as unlabelled objects, potentially forming new clusters. In contrast to a relatively small set of globin sequences in [1], we extract a total number of 12,290 (manually or automatically) annotated proteins to have some role in oxygen transport or binding. This set contains a richer class including, for instance, hemocyanins, hemerythrins, chlorocruorins and erythrocruorins. The proteins are represented as a matrix of pairwise alignment scores. A subset of 1731 annotated sequences is from SwissProt, resulting in 356 protein classes. Among the 10,559 TrEMBL sequences 7 we could identify 23 new clusters which are dissimilar to any SwissProt proteins, see Fig. 5. Most of the newly identiﬁed clusters contain sequences sharing some rare and speciﬁc properties. In accordance with the results in [1], we ﬁnd a large new cluster containing ﬂavohemoglobins from speciﬁc species of funghi and bacteria that share a certain domain architecture composed of a globin domain fused with ferredoxin reductase-like FAD- and NAD-binding modules. An additional example is a cluster of proteins with chemotaxis methyl-accepting receptor domain from a very special class of magnetic bacteria to orient themselves according to earth’s magnetic ﬁeld. The domain architecture of these proteins involving 6 domains is unique among all sequences in our dataset. Another cluster contains iron-sulfur cluster repair di-iron proteins that build on a polymetallic system, the di-iron center, constituted by two iron ions bridged by two sulﬁde ions. Such di-iron centers occur only in this new cluster. Figure 5: Partition of all 12,290 proteins into 379 clusters: 356 predeﬁned by sequences from SwissProt and 23 new formed by sequences from TrEMBL (red box). In order to gain the above results, 5000 Gibbs sweeps were conducted in a total runtime of ≈6 hours. Although section 2 highlighted the worst-case complexity of the original TIWD, it is also important to experimentally compare both models in a real world scenario: we ran 100 sweeps with each fastTIWD and TIWD and hereby observed an average improvement of factor 192, which would lead to an estimated runtime of 1152 hours (≈50 days) for the latter model. On a side note, automatic cluster identiﬁcation is a nice example for beneﬁts of large-scale data mining: clearly, one could theoretically also identify special sequences by digging into various protein domain databases, but without precise prior knowledge, this would hardly be feasible for ≈12,000 proteins. 5 Conclusion We have presented a new model for partitioning pairwise distance data, which is motivated by the great success of the TIWD model, shares all its positive properties, and additionally reduces the computational workload from O(n3) to O(n2) per sweep of the Gibbs sampler. Compared to vectorial representations, pairwise distances do not convey information about translations and rotations of the underlying coordinate system. While in the TIWD model this lack of information is handled by making the likelihood invariant against such geometric transformations, here we break this symmetry by choosing one particular inner-product representation S∗and thus, one particular coordinate system. The advantage is being able to use a standard (i.e. central) Wishart distribution for which we present an efﬁcient Gibbs sampling algorithm. We show that our construction principle for selecting S∗among all inner product matrices corresponding to an observed distance matrix D and ﬁnds an optimal candidate if the true covariance was known. Although it is a pure theoretical guarantee, it is successfully exploited by a simple hierarchical cluster method to produce an initial covariance estimate—all without specifying the number of clusters, which is one of the model’s key properties. On the algorithmic side, we prove that S∗ can be computed in O(n2) time using tree traversals. Assuming the number of Gibbs sweeps necessary is independent of n (which, of course, depends on the problem), we now have a probabilistic algorithm for partitioning distance matrices running in O(n2) time. Experiments on simulated data show that the quality of partitions found is at least comparable to that of the original TIWD. It is now possible for the ﬁrst time to use the Wishart-Dirichlet process model for large matrices. Our experiment containing ≈12,000 proteins shows that fastTIWD can be successfully used to mine large relational datasets and leads to automatic identiﬁcation of protein clusters sharing rare structural properties. Assuming that in most clustering problems it is acceptable to obtain a solution within some hours, any further size increase of the input matrix will become more and more a problem of memory capacity rather than computation time. Acknowledgments This work has been partially supported by the FP7 EU project SIMBAD. 8 References [1] J. Vogt, S. 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4,424	Dimensionality Reduction Using the Sparse Linear Model Ioannis Gkioulekas Harvard SEAS Cambridge, MA 02138 igkiou@seas.harvard.edu Todd Zickler Harvard SEAS Cambridge, MA 02138 zickler@seas.harvard.edu Abstract We propose an approach for linear unsupervised dimensionality reduction, based on the sparse linear model that has been used to probabilistically interpret sparse coding. We formulate an optimization problem for learning a linear projection from the original signal domain to a lower-dimensional one in a way that approximately preserves, in expectation, pairwise inner products in the sparse domain. We derive solutions to the problem, present nonlinear extensions, and discuss relations to compressed sensing. Our experiments using facial images, texture patches, and images of object categories suggest that the approach can improve our ability to recover meaningful structure in many classes of signals. 1 Introduction Dimensionality reduction methods are important for data analysis and processing, with their use motivated mainly from two considerations: (1) the impracticality of working with high-dimensional spaces along with the deterioration of performance due to the curse of dimensionality; and (2) the realization that many classes of signals reside on manifolds of much lower dimension than that of their ambient space. Linear methods in particular are a useful sub-class, for both the reasons mentioned above, and their potential utility in resource-constrained applications like low-power sensing [1, 2]. Principal component analysis (PCA) [3], locality preserving projections (LPP) [4], and neighborhood preserving embedding (NPE) [5] are some common approaches. They seek to reveal underlying structure using the global geometry, local distances, and local linear structure, respectively, of the signals in their original domain; and have been extended in many ways [6–8].1 On the other hand, it is commonly observed that geometric relations between signals in their original domain are only weakly linked to useful underlying structure. To deal with this, various feature transforms have been proposed to map signals to different (typically higher-dimensional) domains, with the hope that geometric relations in these alternative domains will reveal additional structure, for example by distinguishing image variations due to changes in pose, illumination, object class, and so on. These ideas have been incorporated into methods for dimensionality reduction by ﬁrst mapping the input signals to an alternative (higher-dimensional) domain and then performing dimensionality reduction there, for example by treating signals as tensors instead of vectors [9,10] or using kernels [11]. In the latter case, however, it can be difﬁcult to design a kernel that is beneﬁcial for a particular signal class, and ad hoc selections are not always appropriate. In this paper, we also address dimensionality reduction through an intermediate higher-dimensional space: we consider the case in which input signals are samples from an underlying dictionary model. This generative model naturally suggests using the hidden covariate vectors as intermediate features, and learning a linear projection (of the original domain) to approximately preserve the Euclidean geometry of these vectors. Throughout the paper, we emphasize a particular instance of this model that is related to sparse coding, motivated by studies suggesting that data-adaptive sparse representations 1Other linear methods, most notably linear discriminant analysis (LDA), exploit class labels to learn projections. In this paper, we focus on the unsupervised setting. 1 are appropriate for signals such as natural images and facial images [12, 13], and enable state-ofthe-art performance for denoising, deblurring, and classiﬁcation tasks [14–19]. Formally, we assume our input signal to be well-represented by a sparse linear model [20], previously used for probabilistic sparse coding. Based on this generative model, we formulate learning a linear projection as an optimization problem with the objective of preservation, in expectation, of pairwise inner products between sparse codes, without having to explicitly obtain the sparse representation for each new sample. We study the solutions of this optimization problem, and we discuss how they are related to techniques proposed for compressed sensing. We discuss applicability of our results to general dictionary models, and nonlinear extensions. Finally, by applying our method to the visualization, clustering, and classiﬁcation of facial images, texture patches, and general images, we show experimentally that it improves our ability to uncover useful structure. Omitted proofs and additional results can be found in the accompanying supplementary material. 2 The sparse linear model We use RN to denote the ambient space of the input signals, and assume that each signal x ∈RN is generated as the sum of a noise term ε ∈RN and a linear combination of the columns, or atoms, of a N × K dictionary matrix D = [d1, . . . , dK], with the coefﬁcients arranged as a vector a ∈RK, x = Da + ε. (1) We assume the noise to be white Gaussian, ε ∼N(0N×1, σ2IN×N). We are interested in the sparse linear model [20], according to which the elements of a are a-priori independent from ε and are identically and independently drawn from a Laplace distribution, p (a) = K Y i=1 p (ai) , p (ai) = 1 2τ exp −\|ai\| τ . (2) In the context of this model, D is usually overcomplete (K > N), and in practice often learned in an unsupervised manner from training data. Several efﬁcient algorithms exist for dictionary learning [21–23], and we assume in our analysis that a dictionary D adapted to the signals of interest is given. Our adoption of the sparse linear model is motivated by signiﬁcant empirical evidence that it is accurate for certain signals of interest, such as natural and facial images [12, 13], as well as the fact that it enables high performance for such diverse tasks as denoising and inpainting [14, 24], deblurring [15], and classiﬁcation and clustering [13, 16–19]. Typically, the model (1) with an appropriate dictionary D is employed as a means for feature extraction, in which input signals x in RN are mapped to higher-dimensional feature vectors a ∈RK. When inferring features a (termed sparse codes) through maximum-a-posteriori (MAP) estimation, they are solutions to min a 1 σ2 ∥x −Da∥2 2 + 1 τ ∥a∥1 . (3) This problem, known as the lasso [25], is a convex relaxation of the more general problem of sparse coding [26] (in the rest of the paper we use both terms interchangeably). A number of efﬁcient algorithms for computing a exist, with both MAP [21,27] and fully Bayesian [20] procedures. 3 Preserving inner products Linear dimensionality reduction from RN to RM, M < N, is completely speciﬁed by a projection matrix L that maps each x ∈RN to y = Lx, y ∈RM, and different algorithms for linear dimensionality reduction correspond to different methods for ﬁnding this matrix. Typically, we are interested in projections that reveal useful structure in a given set of input signals. As mentioned in the introduction, structure is often better revealed in a higher-dimensional space of features, say a ∈RK. When a suitable feature transform can be found, this structure may exist as simple Euclidean geometry and be encoded in pairwise Euclidean distances or inner products between feature vectors. This is used, for example, in support vector machines and nearest-neighbor classiﬁers based on Euclidean distance, as well as k-means and spectral clustering based on pairwise inner products. For the problem of dimensionality reduction, this motivates learning a projection matrix L such that, for any two input samples, the inner product between their resulting low-dimensional representations is close to that of their corresponding high-dimensional features. 2 More formally, for two samples xk, k = 1, 2 with corresponding low-dimensional representations yk = Lxk and feature vectors ak, we deﬁne δp = yT 1 y2 −aT 1 a2 as a quantity whose magnitude we want on average to be small. Assuming that an accurate probabilistic generative model for the samples x and features a is available, we propose learning L by solving the optimization problem (E denoting expectation with respect to subscripted variables) min LM×N E x1,x2,a1,a2 δp2 . (4) Solving (4) may in general be a hard optimization problem, depending on the model used for ak and xk. Here we solve it for the case of the sparse linear model of Section 2, under which the feature vectors are the sparse codes. Using (1) and denoting S = LT L, (4) becomes min LM×N E a1,a2,ε1,ε2 h aT 1 DT SD −I a2 + εT 1 SDa2 + εT 2 SDa1 + εT 1 Sε2 2i . (5) Assuming that x1 and x2 are drawn independently, we prove that (5) is equivalent to problem min LM×N 4τ 4 DT SD −I 2 F + 4τ 2σ2 ∥SD∥2 F + σ4 ∥S∥2 F , (6) where ∥·∥F is the Frobenius norm, which has the closed-form solution (up to an arbitrary rotation): L = diag (f (λM)) V T M. (7) Here, λM = (λ1, . . . , λM) is a M × 1 vector composed of the M largest eigenvalues of the N × N matrix DDT , and V M is the N × M matrix with the corresponding eigenvectors as columns. The function f (·) is applied element-wise to the vector λM such that f (λi) = s 4τ 4λi σ4 + 4τ 2σ2λi + 4τ 4λ2 i , (8) and diag (f (λM)) is the M × M diagonal matrix formed from f (λM). This solution assumes that DDT has full rank N, which in practice is almost always true as D is overcomplete. Through comparison with (5), we observe that (6) is a trade-off between bringing inner products of sparse codes and their projections close (ﬁrst term), and suppressing noise (second and third terms). Their relative inﬂuence is controlled by the variance of ε and a, through the constants σ and τ respectively. It is interesting to compare their roles in (3) and (6): as σ increases relative to τ, data ﬁtting in (3) becomes less important, and (7) emphasizes noise suppression. As τ increases, l1-regularization in (3) is weighted less, and the ﬁrst term in (6) more. In the extreme case of σ = 0, the data term in (3) becomes a hard constraint, whereas (6) and (7) simplify, respectively, to min LM×N DT SD −I 2 F , and L = diag (λM)−1 2 V T M. (9) Interestingly, in this noiseless case, an ambiguity arises in the solution of (9), as a minimizer is obtained for any subset of M eigenpairs and not necessarily the M largest ones. The solution to (7) is similar—and in the noiseless case identical—to the whitening transform of the atoms of D. When the atoms are centered at the origin, this essentially means that solving (4) for the sparse linear model amounts to performing PCA on dictionary atoms learned from training samples instead of the training samples themselves. The above result can also be interpreted in the setting of [28]: dimensionality reduction in the case of the sparse linear model with the objective of (4) corresponds to kernel PCA using the kernel DDT , modulo centering and the normalization. 3.1 Other dictionary models Even though we have presented our results using the sparse linear model described in Section 2, it is important to realize that our analysis is not limited to this model. The assumptions required for deriving (5) are that signals are generated by a linear dictionary model such as (1), where the coefﬁcients of each of the noise and code vectors are independent and identically distributed according to some zero-mean distribution, with the two vectors also independent from each other. The above assumptions apply for several other popular dictionary models. Examples include the models used implicitly by ridge and bridge regression [29] and elastic-net [30], where the Laplace 3 prior on the code coefﬁcients is replaced by a Gaussian, and priors of the form exp(−λ ∥a∥q q) and exp(−λ ∥a∥1−γ ∥a∥2 2), respectively. In the context of sparse coding, other sparsity-inducing priors that have been proposed in the literature, such as Student’s t-distribution [31], also fall into the same framework. We choose to emphasize the sparse linear model, however, due to the apparent structure present in dictionaries learned using this model, and its empirical success in diverse applications. It is possible to derive similar results for a more general model. Speciﬁcally, we make the same assumptions as above, except that we only require that elements of a be zero-mean and not necessarily identically distributed, and similarly for ε. Then, we prove that (4) becomes min LM×N DT SD −I ⊙ p W 1 2 F + (SD) ⊙ p W 2 2 F + S ⊙ p W 3 2 F , (10) where ⊙denotes the Hadamard product and √ W ij = q (W )ij. The elements of the weight matrices W 1, W 2 and W 3 in (10), of sizes K × K, N × K, and N × N respectively, are (W 1)ij = E a2 1ia2 2j , (W 2)ij = E ε2 1ia2 2j + E ε2 2ia2 1j , (W 3)ij = E ε2 1iε2 2j . (11) Problem (10) can still be solved efﬁciently, see for example [32]. 3.2 Extension to the nonlinear case We consider a nonlinear extension of the above analysis through the use of kernels. We denote by Φ : RN →H a mapping from the signal domain to a reproducing kernel Hilbert space H associated with a kernel function k : RN × RN →R [33]. Using a set D = { ˜di ∈H, i = 1, . . . , K} as dictionary, we extend the sparse linear model of Section 2 by replacing (1) for each x ∈RN with Φ (x) = Da + ˜ε, (12) where Da ≡PK i=1 ai ˜di. For a ∈RK we make the same assumptions as in the sparse linear model. The term ˜ε denotes a Gaussian process over the domain RN whose sample paths are functions in H and with covariance operator C˜ε = σ2I, where I is the identity operator on H [33,34]. This nonlinear extension of the sparse linear model is valid only in ﬁnite dimensional spaces H. In the inﬁnite dimensional case, constructing a Gaussian process with both sample paths in H and identity covariance operator is not possible, as that would imply that the identity operator in H has ﬁnite Hilbert-Schmidt norm [33, 34]. Related problems arise in the construction of cylindrical Gaussian measures on inﬁnite dimensional spaces [35]. We deﬁne ˜ε this way to obtain a probabilistic model for which MAP inference of a corresponds to the kernel extension of the lasso (3) [36], min a∈RK 1 2σ2 ∥Φ (x) −Da∥2 H + 1 τ ∥a∥1 , (13) where ∥·∥H is the norm H deﬁned through k. In the supplementary material, we discuss an alternative to (12) that resolves these problems by requiring that all Φ (x) be in the subspace spanned by the atoms of D. Our results can be extended to this alternative, however in the following we adopt (12) and limit ourselves to ﬁnite dimensional spaces H, unless mentioned otherwise. In the kernel case, the equivalent of the projection matrix L (transposed) is a compact, linear operator V : H →RM, that maps an element x ∈RN to y = VΦ (x) ∈RM. We denote by V∗: RM →H the adjoint of V, and by S : H →H the self-adjoint positive semi-deﬁnite linear operator of rank M from their synthesis, S = V∗V. If we consider optimizing over S, we prove that (4) reduces to min S 4τ 4 K X i=1 K X i=1 D ˜di, S ˜dj E H −δij 2 + 4τ 2σ2 K X i=1 D S ˜di, S ˜di E H + ∥S∥2 HS , (14) where ∥·∥HS is the Hilbert-Schmidt norm. Assuming that KDD has full rank (which is almost always true in practice due to the very large dimension of the Hilbert spaces used) we extend the representer theorem of [37] to prove that all solutions of (14) can be written in the form S = (DB) ⊗(DB) , (15) where ⊗denotes the tensor product between all pairs of elements of its operands, and B is a K ×M matrix. Then, denoting Q = BBT , problem (14) becomes min BK×M 4τ 4 ∥KDDQKDD −I∥2 F + 4τ 2σ2 KDDQK 1 2 DD 2 F + σ4 K 1 2 DDQK 1 2 DD 2 F , (16) 4 Figure 1: Two-dimensional projection of CMU PIE dataset, colored by identity. Shown at high resolution and at their respective projections are identity-averaged faces across the dataset for various illuminations, poses, and expressions. Insets show projections of samples from only two distinct identities. (Best viewed in color.) where KDD (i, j) = ⟨˜di, ˜dj⟩H, i, j = 1, . . . , K. We can replace ˜L = BT K 1 2 DD to turn (16) into an equivalent problem over ˜L of the form (6), with K 1 2 DD instead of D, and thus use (8) to obtain B = V M diag (g (λM)) (17) where, similar to the linear case, λM and V M are the M largest eigenpairs of the matrix KDD, and g (λi) = 1 √λi f (λi) = s 4τ 4 σ4 + 4τ 2σ2λi + 4τ 4λ2 i . (18) Using the derived solution, a vector x ∈RN is mapped to y = BT KD (x), where KD (x) = [⟨˜d1, Φ (x)⟩H, . . . , ⟨˜dM, Φ (x)⟩H]T . As in the linear case, this is similar to the result of applying kernel PCA on the dictionary D instead of the training samples. Note that, in the noiseless case, σ = 0, the above analysis is also valid for inﬁnite dimensional spaces H. Expression (17) simpliﬁes to B = V M diag (λM)−1 where, as in the linear case, any subset of M eigenvalues may be selected. Even though in the inﬁnite dimensional case selecting the M largest eigenvalues cannot be justiﬁed probabilistically, it is a reasonable heuristic given the analysis in the ﬁnite dimensional case. 3.3 Computational considerations It is interesting to compare the proposed method in the nonlinear case with kernel PCA, in terms of computational and memory requirements. If we require dictionary atoms to have pre-images in RN, that is D = Φ (di) , di ∈RN, i = 1, . . . , K [36], then the proposed algorithm requires calculating and decomposing the K × K kernel matrix KDD when learning V, and performing K kernel evaluations for projecting a new sample x. For kernel PCA on the other hand, the S×S matrix KXX and S kernel evaluations are needed respectively, where X = Φ (xi) , xi ∈RN, i = 1, . . . , S and xi are the representations of the training samples in H, with S ≫K. If the pre-image constraint is dropped and the usual alternating procedure [21] is used for learning D, then the representer theorem of [38] implies that D = XF , where F is an S × K matrix. In this case, the proposed method also requires calculating KXX during learning and S kernel evaluations for out-of-sample projections, but only the eigendecomposition of the K × K matrix F T K2 XX F is required. On the other hand, we have assumed so far, in both the linear and nonlinear cases, that a dictionary is given. When this is not true, we need to take into account the cost of learning a dictionary, which greatly outweights the computational savings described above, despite advances in dictionary learning algorithms [21, 22]. In the kernel case, whereas imposing the pre-image constraint has the advantages we mentioned, it also makes dictionary learning a harder nonlinear optimization problem, due to the need for evaluation of kernel derivatives. In the linear case, the computational savings from applying (linear) PCA to the dictionary instead of the training samples are usually negligible, and therefore the difference in required computation becomes even more severe. 5 Figure 2: Classiﬁcation accuracy results. From left to right: CMU PIE (varying value of M); CMU PIE (varying number of training samples); brodatz texture patches; Caltech-101. (Best viewed in color.) 4 Experimental validation In order to evaluate our proposed method, we compare it with other unsupervised dimensionality reduction methods on visualization, clustering, and classiﬁcation tasks. We use facial images in the linear case, and texture patches and images of object categories in the kernel case. Facial images: We use the CMU PIE [39] benchmark dataset of faces under pose, illumination and expression changes, and speciﬁcally the subset used in [8].2 We visualize the dataset by projecting all face samples to M = 2 dimensions using LPP and the proposed method, as shown in Figure 1. Also shown are identity-averaged faces over the dataset, for various illumination, pose, and expression combinations, at the location of their projection. We observe that our method recovers a very clear geometric structure, with changes in illumination corresponding to an ellipsoid, changes in pose to moving towards its interior, and changes in expression accounting for the density on the horizontal axis. We separately show the projections of samples from two distinct indviduals, and see that different identities are mapped to parallely shifted ellipsoids, easily separated by a nearestneighbor classiﬁer. On the other hand, such structure is not apparent when using LPP. A larger version of Figure 1 and the corresponding for PCA are provided in the supplementary material. To assess how well identity structure is recovered for increasing values of the target dimension M, we also perform face recognition experiments. We compare against three baseline methods, PCA, NPE, and LPP, linear extensions (spectral regression “SRLPP” [7], spatially smooth LPP “SmoothLPP” [8]), and random projections (see Section 5). We produce 20 random splits into training and testing sets, learn a dictionary and projection matrices from the training set, and use the obtained low-dimensional representations with a k-nearest neighbor classiﬁer (k = 4) to classify the test samples, as is common in the literature. In Figure 2, we show the average recognition accuracy for the various methods as the number of projections is varied, when using 100 training samples for each of the 68 individuals in the dataset. Also, we compare the proposed method with the best performing alternative, when the number of training samples per individual is varied from 40 to 120. We observe that the proposed method outperforms all other by a wide margin, in many cases even when trained with fewer samples. However, it can only be used when there are enough training samples to learn a dictionary, a limitation that does not apply to the other methods. For this reason, we do not experiment with cases of 5-20 samples per individual, as commonly done in the literature. Texture patches: We perform classiﬁcation experiments on texture patches, using the Brodatz dataset [40], and speciﬁcally classes 4, 5, 8, 12, 17, 84, and 92 from the 2-texture images. We extract 12 × 12 patches and use those from the training images to learn dictionaries and projections for the Gaussian kernel.3 We classify the low-dimensional representations using an one-versus-all linear SVM. In Figure 2, we compare the classiﬁcation accuracy of the proposed method (“ker.dict”) with the kernel variants of PCA and LPP (“KPCA” and “KLPP” respectively), for varying M. KLPP and the proposed method both outperform KPCA. Our method achieves much higher accuracy at small values of M, and KLPP is better for large values; otherwise they perform similarly. This dataset provides an illustrative example for the discussion in Section 3.3. For 20000 training samples, KPCA and KLPP require storing and processing a 20000×20000 kernel matrix, as opposed to 512 × 512 for our method. On the other hand, training a dictionary with K = 512 for this dataset takes approximately 2 hours, on an 8 core machine and using a C++ implementation of the learning algorithm, as opposed to the few minutes required for the eigendecompositions in KPCA and KLPP. 2Images are pre-normalized to unit length. We use the algorithm of [21] to learn dictionaries, with K equal to the number of pixels N = 1024, due to the limited amount of training data, and λ = σ2 τ = 0.05 as in [19]. 3Following [36], we set the kernel parameter γ = 8, and use their method for dictionary learning with K = 512 and λ = 0.30, but with a conjugate gradient optimizer for the dictionary update step. 6 Method Accuracy NMI Rand Index KPCA (k-means) 0.6217 0.6380 0.4279 KLPP (spectral clustering) 0.6900 0.6788 0.5143 ker.dict (k-means) 0.7233 0.7188 0.5275 Table 1: Clustering results on Caltech-101. Images of object categories: We use the Caltech-101 [41] object recognition dataset, with the average of the 39 kernels used in [42]. Firstly, we use 30 training samples from each class to learn a dictionary4 and projections using KPCA, KLPP, and the proposed method. In Figure 2, we plot the classiﬁcation accuracy achieved using a linear SVM for each method and varying M. We see that the proposed method and KPCA perform similarly and outperform KLPP. Our algorithm performs consistently well in both the datasets we experiment with in the kernel case. We also perform unsupervised clustering experiments, where we randomly select 30 samples from each of the 20 classes used in [43] to learn projections with the three methods, over a range of values for M between 10 and 150. We combine each with three clustering algorithms, k-means, spectral clustering [44], and afﬁnity propagation [43] (using negative Euclidean distances of the low-dimensional representations as similarities). In Table 1, we report for each method the best overall result in terms of accuracy, normalized mutual information, and rand index [45], along with the clustering algorithm for which these are achieved. We observe that the low-dimensional representations from the proposed method produce the best quality clusterings, for all three measures. 5 Discussion and future directions As we remarked in Section 3, the proposed method uses available training samples to learn D and ignores them afterwards, relying exclusively on the assumed generative model and the correlation information in D. To see how this approach could fail, consider the degenerate case when D is the identity matrix, that is the signal and sparse domains coincide. Then, to discover structure we need to directly examine the training samples. Better use of the training samples within our framework can be made by adopting a richer probabilistic model, using available data to train it, naturally with appropriate regularization to avoid overﬁtting, and then minimizing (4) for the learned model. For example, we can use the more general model of Section 3.1, and assume that each ai follows a Laplace distribution with a different τi. Doing so agrees with empirical observations that, when D is learned, the average magnitude of coefﬁcients ai varies signiﬁcantly with i. An orthogonal approach is to forgo adopting a generative model, and learn a projection matrix directly from training samples using an appropriate empirical loss function. One possibility is minimizing ∥AT A−XT LT LX∥2 F , where the columns of X and A are the training samples and corresponding sparse code estimates, which is an instance of multidimensional scaling [46] (as modiﬁed to achieve linear induction). For the sparse linear model case, objective function (4) is related to the Restricted Isometry Property (RIP) [47], used in the compressed sensing literature as a condition enabling reconstruction of a sparse vector a ∈RK from linear measurements y ∈RM when M ≪K. The RIP is a worstcase condition, requiring approximate preservation, in the low-dimensional domain, of pairwise Euclidean distances of all a, and therefore stronger than the expectation condition (4). Verifying the RIP for an arbitrary matrix is a hard problem, but it is known to hold for the equivalent dictionary ˜D = LD with high probability, if L is drawn from certain random distributions, and M is of the order of only O k log K k [48]. Despite this property, our experiments demonstrate that a learned matrix L is in practice more useful than random projections (see left of Figure 2). The formal guarantees that preservation of Euclidean geometry of sparse codes is possible with few linear projections are unique for the sparse linear model, thus further justifying our choice to emphasize this model throughout the paper. Another quantity used in compressed sensing is the mutual coherence of ˜D [49], and its approximate minimization has been proposed as a way for learning L for signal reconstruction [50,51]. One of the optimization problems arrived at in this context [51] is the same as problem (9) we derived in the noiseless case, the solution of which as we mentioned in Section 3 is not unique. This ambiguity has been addressed heuristically by weighting the objective function with appropriate multiplicative terms, so that it becomes ∥Λ−ΛV T LT LV Λ∥2 F , where Λ and V are eigenpairs of DDT [51]. This 4We use a kernel extension of the algorithm of [21] without pre-image constraints. We select K = 300 and λ = 0.1 from a range of values, to achieve about 10% non-zero coefﬁcients in the sparse codes and small reconstruction error for the training samples. Using K = 150 or 600 affected accuracy by less than 1.5%. 7 problem admits as only minimizer the one corresponding to the M largest eigenvalues. Our analysis addresses the above issue naturally by incorporating noise, thus providing formal justiﬁcation for the heuristic. Also, the closed-form solution of (9) is not shown in [51], though its existence is mentioned, and the (weighted) problem is instead solved through an iterative procedure. In Section 3, we motivated preserving inner products in the sparse domain by considering existing algorithms that employ sparse codes. As our understanding of sparse coding continues to improve [52], there is motivation for considering other structure in RK. Possibilities include preservation of linear subspace (as determined by the support of the sparse codes) or local group relations in the sparse domain. Extending our analysis to also incorporate supervision is another important future direction. Linear dimensionality reduction has traditionally been used for data preprocessing and visualization, but we are also beginning to see its utility for low-power sensors. A sensor can be designed to record linear projections of an input signal, instead of the signal itself, with projections implemented through a low-power physical process like optical ﬁltering. In these cases, methods like the ones proposed in this paper can be used to obtain a small number of informative projections, thereby reducing the power and size of the sensor while maintaining its effectiveness for tasks like recognition. An example for visual sensing is described in [2], where a heuristically-modiﬁed version of our linear approach is employed to select projections for face detection. Rigorously extending our analysis to this domain will require accounting for noise and constraints on the projections (for example non-negativity, limited resolution) induced by fabrication processes. We view this as a research direction worth pursuing. 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4,425	RTRMC: A Riemannian trust-region method for low-rank matrix completion Nicolas Boumal∗ ICTEAM Institute Universit´e catholique de Louvain B-1348 Louvain-la-Neuve nicolas.boumal@uclouvain.be P.-A. Absil ICTEAM Institute Universit´e catholique de Louvain B-1348 Louvain-la-Neuve absil@inma.ucl.ac.be Abstract We consider large matrices of low rank. We address the problem of recovering such matrices when most of the entries are unknown. Matrix completion ﬁnds applications in recommender systems. In this setting, the rows of the matrix may correspond to items and the columns may correspond to users. The known entries are the ratings given by users to some items. The aim is to predict the unobserved ratings. This problem is commonly stated in a constrained optimization framework. We follow an approach that exploits the geometry of the low-rank constraint to recast the problem as an unconstrained optimization problem on the Grassmann manifold. We then apply ﬁrst- and second-order Riemannian trust-region methods to solve it. The cost of each iteration is linear in the number of known entries. Our methods, RTRMC 1 and 2, outperform state-of-the-art algorithms on a wide range of problem instances. 1 Introduction We address the problem of recovering a low-rank m-by-n matrix X of which a few entries are observed, possibly with noise. Throughout, we assume that r = rank(X) ≪m ≤n and note Ω⊂{1 . . . m} × {1 . . . n} the set of indices of the observed entries of X, i.e., Xij is known iff (i, j) ∈Ω. Solving this problem is namely useful in recommender systems, where one tries to predict the ratings users would give to items they have not purchased. 1.1 Related work In the noiseless case, one could state the minimum rank matrix recovery problem as follows: min ˆ X∈Rm×n rank ˆX, such that ˆXij = Xij ∀(i, j) ∈Ω. (1) This problem, however, is NP hard [CR09]. A possible convex relaxation of (1) introduced by Cand`es and Recht [CR09] is to use the nuclear norm of ˆX as objective function, i.e., the sum of its singular values, noted ∥ˆX∥∗. The SVT method [CCS08] attempts to solve such a convex problem using tools from compressed sensing and the ADMiRA method [LB10] does so using matching pursuit-like techniques. Alternatively, one may minimize the discrepancy between ˆX and X at entries Ωunder the constraint that rank( ˆX) ≤r for some small constant r. Since any matrix ˆX of rank at most r may be written in the form UW with U ∈Rm×r and W ∈Rr×n, a reasonable formulation of the problem reads: min U∈Rm×r min W ∈Rr×n X (i,j)∈Ω (UW)ij −Xij 2. (2) ∗Web: http://perso.uclouvain.be/nicolas.boumal/ 1 The LMaFit method [WYZ10] does a good job at solving this problem by alternatively ﬁxing either of the variables and solving the resulting least-squares problem efﬁciently. One drawback of the latter formulation is that the factorization of a matrix ˆX into the product UW is not unique. Indeed, for any r-by-r invertible matrix M, we have UW = (UM)(M −1W). All the matrices UM share the same column space. Hence, the optimal value of the inner optimization problem in (2) is a function of col(U)— the column space of U—rather than U speciﬁcally. Dai et al. [DMK11, DKM10] exploit this to recast (2) on the Grassmann manifold G(m, r), i.e., the set of r-dimensional vector subspaces of Rm (see Section 2): min U ∈G(m,r) min W ∈Rr×n X (i,j)∈Ω (UW)ij −Xij 2, (3) where U ∈Rm×r is any matrix such that col(U) = U and is often chosen to be orthonormal. Unfortunately, the objective function of the outer minimization in (3) may be discontinuous at points U for which the leastsquares problem in W does not have a unique solution. Dai et al. proposed ingenious ways to deal with the discontinuity. Their focus, though, was on deriving theoretical performance guarantees rather than developing fast algorithms. Keshavan et al. [KO09, KM10] state the problem on the Grassmannian too, but propose to simultaneously optimize on the row and column spaces, yielding a smaller least-squares problem which is unlikely to not have a unique solution, resulting in a smooth objective function. In one of their recent papers [KM10], they solve: min U ∈G(m,r),V ∈G(n,r) min S∈Rr×r X (i,j)∈Ω (USV ⊤)ij −Xij 2 + λ2 USV ⊤ 2 F , (4) where U and V are any orthonormal bases of U and V , respectively, and λ is a regularization parameter. The authors propose an efﬁcient SVD-based initial guess for U and V which they reﬁne using a steepest descent method, along with strong theoretical guarantees. Meyer et al. [MBS11] proposed a Riemannian approach to linear regression on ﬁxed-rank matrices. Their regression framework encompasses matrix completion problems. Likewise, Balzano et al. [BNR10] introduced GROUSE for subspace identiﬁcation on the Grassmannian, applicable to matrix completion. Finally, in the preprint [Van11] which became public while we were preparing the camera-ready version of this paper, Vandereycken proposes an approach based on the submanifold geometry of the sets of ﬁxed-rank matrices. 1.2 Our contribution and outline of the paper Dai et al.’s initial formulation (3) has a discontinuous objective function on the Grassmannian. The OptSpace formulation (4) on the other hand has a continuous objective and comes with a smart initial guess, but optimizes on a higher-dimensional search space, while it is arguably preferable to keep the dimension of the manifold search space low, even at the expense of a larger least-squares problem. Furthermore, the OptSpace regularization term is efﬁciently computable since USV ⊤ F = ∥S∥F, but it penalizes all entries instead of just the entries (i, j) /∈Ω. In an effort to combine the best of both worlds, we equip (3) with a regularization term weighted by λ > 0, which yields a smooth objective function deﬁned over an appropriate search space: min U ∈G(m,r) min W ∈Rr×n 1 2 X (i,j)∈Ω C2 ij (UW)ij −Xij 2 + λ2 2 X (i,j)/∈Ω (UW)2 ij. (5) Here, we introduced a conﬁdence index Cij > 0 for each observation Xij, which may be useful in applications. As we will see, introducing a regularization term is essential to ensure smoothness of the objective and hence obtain good convergence properties. It may not be critical for practical problem instances though. We further innovate on previous works by using a Riemannian trust-region method, GenRTR [ABG07], as optimization algorithm to minimize (5) on the Grassmannian. GenRTR is readily available as a free Matlab package and comes with strong convergence results that are naturally inherited by our algorithms. In Section 2, we rapidly cover the essential useful tools on the Grassmann manifold. In Section 3, we derive expressions for the gradient and the Hessian of our objective function while paying special attention to complexity. Section 4 sums up the main properties of the Riemannian trust-region method. Section 5 shows a few results of numerical experiments demonstrating the effectiveness of our approach. 2 2 Geometry of the Grassmann manifold Our objective function f (10) is deﬁned over the Grassmann manifold G(m, r), i.e., the set of r-dimensional vector subspaces of Rm. Absil et al. [AMS08] give a computation-oriented description of the geometry of this manifold. Here, we only give a summary of the important tools we use. Each point U ∈G(m, r) is a vector subspace we may represent numerically as the column space of a fullrank matrix U ∈Rm×r: U = col(U). For numerical reasons, we will only use orthonormal matrices U ∈ U(m, r) = {U ∈Rm×r : U ⊤U = Ir}. The set U(m, r) is the Stiefel manifold. The Grassmannian is a Riemannian manifold, and as such we can deﬁne a tangent space to G(m, r) at each point U , noted TU G(m, r). The latter is a vector space of dimension dim G(m, r) = r(m −r). A tangent vector H ∈TU G(m, r), where we represent U as the orthonormal matrix U, is represented by a unique matrix H ∈Rm×r verifying U ⊤H = 0 and d dt col(U + tH) t=0 = H . For practical purposes we may, with a slight abuse of notation we often commit hereafter, write H = H—assuming U is known from the context— and TUG(m, r) = {H ∈Rm×r : U ⊤H = 0}. Each tangent space is endowed with an inner product, the Riemannian metric, that varies smoothly from point to point. It is inherited from the embedding space of the matrix representation of tangent vectors Rm×r: ∀H1, H2 ∈TUG(m, r) : ⟨H1, H2⟩U = Trace(H⊤ 2 H1). The orthogonal projector from Rm×r onto the tangent space TUG(m, r) is given by: PU : Rm×r →TUG(m, r) : H 7→PUH = (I −UU ⊤)H. One can also project a vector onto the tangent space of the Stiefel manifold: P St U : Rm×r →TUU(m, r) : H 7→P St U H = (I −UU ⊤)H + Uskew(U ⊤H), where skew(X) = (X −X⊤)/2 extracts the skew-symmetric part of X. This is useful for the computation of gradf(U) ∈TUG(m, r). Indeed, according to [AMS08, eqs. (3.37) and (3.39)], considering ¯f : Rm×r →R, its restriction ¯f U(m,r) to the Stiefel manifold and f : G(m, r) →R such that f(col(U)) = ¯f U(m,r) (U) is well-deﬁned, as will be the case in Section 3, we have (with a slight abuse of notation): gradf(U) = grad ¯f U(m,r) (U) = P St U grad ¯f(U). (6) Similarly, since PU ◦P St U = PU, the Hessian of f at U along H is given by [AMS08, eqs. (5.14) and (5.18)]: Hessf(U)[H] = PU(D(U 7→P St U grad ¯f(U))(U)[H]), (7) where Dg(X)[H] is the directional derivative of g at X along H, in the classical sense. For our optimization algorithms, it is important to be able to move along the manifold from some initial point U in some prescribed direction speciﬁed by a tangent vector H. To this end, we use the retraction: RU(H) = qf(U + H), (8) where qf(X) ∈U(m, r) designates the m-by-r Q-factor of the QR decomposition of X ∈Rm×r. 3 Computation of the objective function and its derivatives We seek an m-by-n matrix ˆX of rank not more than r such that ˆX is as close as possible to a given matrix X at the entries in the observation set Ω. Furthermore, we are given a weight matrix C ∈Rm×n indicating the conﬁdence we have in each observed entry of X. The matrix C is positive at entries in Ωand zero elsewhere. To this end, we consider the following function, where (XΩ)ij equals Xij if (i, j) ∈Ωand is zero otherwise: ˆf : Rm×r × Rr×n →R : (U, W) 7→ˆf(U, W) = 1 2 ∥C ⊙(UW −XΩ)∥2 Ω+ λ2 2 ∥UW∥2 ¯Ω, (9) where ⊙is the entry-wise product, λ > 0 is a regularization parameter, ¯Ωis the complement of the set Ωand ∥M∥2 Ω≜P (i,j)∈ΩM 2 ij. Picking a small but positive λ will ensure that the objective function f (10) is smooth. For a ﬁxed U, computing the matrix W that minimizes ˆf is a least-squares problem. The mapping between U and this (unique) optimal W, noted WU, U 7→WU = argmin W ∈Rr×n ˆf(U, W), 3 is smooth and easily computable—see Section 3.3. By virtue of the discussion in Section 1, we know that the mapping U 7→ˆf(U, WU), with U ∈Rm×r, is constant over sets of full-rank matrices U spanning the same column space. Hence, considering these sets as equivalence classes U , the following function f over the Grassmann manifold is well-deﬁned: f : G(m, r) →R : U 7→f(U ) = ˆf(U, WU), (10) with any full-rank U ∈Rm×r such that col(U) = U . The interpretation is as follows: we are looking for an optimal matrix ˆX = UW of rank at most r; we have conﬁdence Cij that ˆXij should equal Xij for (i, j) ∈Ω and (very small) conﬁdence λ that ˆXij should equal 0 for (i, j) /∈Ω. 3.1 Rearranging the objective Considering (9), it looks like evaluating ˆf(U, W) will require the computation of the product UW at the entries in Ωand ¯Ω, i.e., we would need to compute the whole matrix UW, which cannot cost much less than O(mnr). Since applications typically involve very large values of the product mn, this is not acceptable. Alternatively, if we restrict ourselves—without loss of generality—to orthonormal matrices U, we observe that ∥UW∥2 Ω+ ∥UW∥2 ¯Ω= ∥UW∥2 F = ∥W∥2 F . Consequently, for all U in U(m, r), we have ˆf(U, WU) = ˇf(U, WU), where ˇf(U, W) = 1 2 ∥C ⊙(UW −XΩ)∥2 Ω+ λ2 2 ∥W∥2 F −λ2 2 ∥UW∥2 Ω. (11) This only requires the computation of UW at entries in Ω, at a cost of O(\|Ω\|r). Finally, let ¯f : Rm×r →R : U 7→ˇf(U, WU), and observe that f(col(U)) = ¯f U(m,r) (U) for all U in U(m, r), as in the setting of Section 2. 3.2 Gradient and Hessian of the objective We now derive formulas for the ﬁrst and second order derivatives of f. In deriving these formulas, it is useful to note that, for a suitably smooth mapping g, grad X 7→1/2 ∥g(X)∥2 F (X) = Dg(X) ∗[g(X)], (12) where Dg(X) ∗is the adjoint of the differential of g at X. For ease of notation, let us deﬁne the following m-by-n matrix with the sparsity structure induced by Ω: ˆCij = C2 ij −λ2 if (i, j) ∈Ω, 0 otherwise. (13) We also introduce a sparse residue matrix RU that will come up in various formulas: RU = ˆC ⊙(UWU −XΩ) −λ2XΩ. (14) Successively using the chain rule, the optimality of WU and (12), we obtain: grad ¯f(U) = d dU ˇf(U, WU) = ∂ ∂U ˇf(U, WU) + ∂ ∂W ˇf(U, WU) · d dU WU = ∂ ∂U ˇf(U, WU) = RUW ⊤ U. Indeed, since WU is optimal, ∂ ∂W ˇf(U, WU) = U ⊤RU + λ2WU = 0. Then, according to the identity (6) and since U ⊤RU = −λ2WU, the gradient of f at U = col(U) on the Grassmannian is given by: gradf(U) = grad ¯f U(m,r) (U) = P St U grad ¯f(U) = (I −UU ⊤)RUW ⊤ U + Uskew(U ⊤RUW ⊤ U) = (I −UU ⊤)RUW ⊤ U −λ2Uskew(WUW ⊤ U) = RUW ⊤ U + λ2U(WUW ⊤ U), (15) We now differentiate (15) according to the identity (7) to get a matrix representation of the Hessian of f at U along H . We note H a matrix representation of the tangent vector H chosen in accordance with U and WU,H ≜D(U 7→WU)(U)[H] the derivative of the mapping U 7→WU at U along the tangent direction H. Then: Hessf(U)[H] = (I −UU ⊤)Dgradf(U)[H] = (I −UU ⊤) h ˆC ⊙(HWU + UWU,H) i W ⊤ U + RUW ⊤ U,H + λ2H(WUW ⊤ U) + λ2U(WUW ⊤ U,H). (16) 4 3.3 WU and its derivative WU,H We still need to provide an explicit formula for WU and WU,H. We assume U ∈U(m, r) since we use orthonormal matrices to represent points on the Grassmannian and U ⊤H = 0 since H is a tangent vector at U . We use the vectorization operator, vec, that transforms matrices into vectors by stacking their columns—in Matlab notation, vec(A) = A(:). Denoting the Kronecker product of two matrices by ⊗, we will use the well-known identity for matrices A, Y, B of appropriate sizes [Bro05]: vec(AY B) = (B⊤⊗A)vec(Y ). We also write IΩfor the orthonormal \|Ω\|-by-mn matrix such that vecΩ(M) = IΩvec(M) is a vector of length \|Ω\| corresponding to the entries Mij for (i, j) ∈Ω, taken in order from vec(M). Computing WU comes down to minimizing the least-squares objective ˇf(U, W) (11) with respect to W. We ﬁrst manipulate ˇf to reach a standard form for least-squares, with S = IΩdiag(vec(C)): ˇf(U, W) = 1 2 ∥C ⊙(UW −XΩ)∥2 Ω+ λ2 2 ∥W∥2 F −λ2 2 ∥UW∥2 Ω = 1 2 ∥Svec(UW) −vecΩ(C ⊙XΩ)∥2 2 + λ2 2 ∥vec(W)∥2 2 −λ2 2 ∥vecΩ(UW)∥2 2 = 1 2 ∥S(In ⊗U)vec(W) −vecΩ(C ⊙XΩ)∥2 2 + 1 2 ∥λIrnvec(W)∥2 2 −1 2 ∥λIΩ(In ⊗U)vec(W)∥2 2 = 1 2 S(In ⊗U) λIrn vec(W) − vecΩ(C ⊙XΩ) 0rn 2 2 −1 2 ∥[λIΩ(In ⊗U)] vec(W)∥2 2 = 1 2 ∥A1w −b1∥2 2 −1 2 ∥A2w∥2 2 , where w = vec(W) ∈Rrn, 0rn ∈Rrn is the zero-vector and the deﬁnitions for A1, A2 and b1 are obvious. If A⊤ 1A1 −A⊤ 2A2 is positive deﬁnite there is a unique minimizing vector vec(WU), given by: vec(WU) = (A⊤ 1A1 −A⊤ 2A2)−1A⊤ 1b1. It is easy to compute the following: A⊤ 1A1 = (In ⊗U ⊤)(S⊤S)(In ⊗U) + λ2Irn, A⊤ 2A2 = (In ⊗U ⊤)(λ2I⊤ ΩIΩ)(In ⊗U), A⊤ 1b1 = (In ⊗U ⊤)S⊤vecΩ(C ⊙XΩ) = (In ⊗U ⊤)vec(C(2) ⊙XΩ). Throughout the text, we use the notation M (n) for entry-wise exponentiation, i.e., (M (n))ij = (Mij)n. Note that S⊤S −λ2I⊤ ΩIΩ= diag(vec( ˆC)). We then deﬁne A ∈Rrn×rn as: A ≜A⊤ 1A1 −A⊤ 2A2 = (In ⊗U ⊤) diag(vec( ˆC)) (In ⊗U) + λ2Irn. (17) Observe that the matrix A is block-diagonal, with n symmetric blocks of size r. Each block is indeed positivedeﬁnite provided λ > 0 (making A positive-deﬁnite too). Thanks to the sparsity of ˆC, we can compute these n blocks with O(\|Ω\|r2) ﬂops. To solve systems in A, we compute the Cholesky factorization of each block, at a total cost of O(nr3). Once these factorizations are computed, each system only costs O(nr2) to solve [TB97]. Collecting all equations in this subsection, we obtain a closed-form formula for WU: vec(WU) = A−1vec U ⊤[C(2) ⊙XΩ] , (18) where A is a function of U. We would like to differentiate WU with respect to U. Using bilinearity and associativity of ⊗as well as the formula D(Y 7→Y −1)(X)[H] = −X−1HX−1 [Bro05], some algebra yields: vec(WU,H) = −A−1vec H⊤RU + U ⊤ ˆC ⊙(HWU) . (19) 5 The most expensive operation involved in computing WU,H ought to be the resolution of a linear system in A. Fortunately, we already factored the n small diagonal blocks of A in Cholesky form to compute WU. Consequently, after computing WU, computing WU,H is cheaper than computing WU ′ for a new U ′. This means that we can beneﬁt from computing this information before we move on to a new candidate on the Grassmannian, i.e., it is worth trying second order methods. We summarize the complexities in the next subsection. 3.4 Numerical complexities By exploiting the sparsity of many of the matrices involved and the special structure of the matrix A appearing in the computation of WU and WU,H, it is possible to compute the objective f as well as its gradient and its Hessian on the Grassmannian in time essentially linear in the size of the data \|Ω\|. Memory complexities are also linear in \|Ω\|. We summarize the computational complexities in Table 1. Please note that most computations are easily parallelizable, but we do not take advantage of it here. Table 1: All complexities are essentially linear in \|Ω\|, the number of observed entries. Computation Complexity By-products Formulas WU and f(U) O(\|Ω\|r2 + nr3) Cholesky form of A (9), (10), (17), (18) gradf(U) O(\|Ω\|r + (m + n)r2) RU and WUW ⊤ U (13), (14), (15) WU,H and Hessf(U)[H] O(\|Ω\|r + (m + n)r2) (16), (19) 4 Riemannian trust-region method We use a Riemannian trust-region (RTR) method [ABG07] to minimize (10), via the freely available Matlab package GenRTR (version 0.3.0) with its default parameter values. The package is available at this address: http://www.math.fsu.edu/˜cbaker/GenRTR/?page=download. At the current iterate U = col(U), the RTR method uses the retraction RU (8) to build a quadratic model mU : TUG(m, r) →R of the lifted objective function f ◦RU (lift). It then classically minimizes the model inside a trust region on this vector space (solve), and retracts the resulting tangent vector H to a candidate U + = RU(H) on the Grassmannian (retract). The quality of U + = col(U +) is assessed using f and the step is accepted or rejected accordingly. Likewise, the radius of the trust region is adapted based on the observed quality of the model. The model mU of f ◦RU has the form: mU(H) = f(U) + ⟨gradf(U), H⟩U + 1 2 ⟨A(U)[H], H⟩U , where A(U) is some symmetric linear operator on TUG(m, r). Typically, the faster one can compute A(U)[H], the faster one can minimize mU(H) in the trust region. A powerful property of the RTR method is that global convergence of the algorithm toward critical points—local minimizers in practice since it is a descent method—is guaranteed independently of A(U) [ABG07, Thm 4.24, Cor. 4.6]. We take advantage of this and ﬁrst set it to the identity. This yields a steepest-descent algorithm we later refer to as RTRMC 1. Additionally, if we take A(U) to be the Hessian of f at U (16), we get a quadratic convergence rate, even if we only approximately minimize mU within the trust region using a few steps of a well chosen iterative method [ABG07, Thm 4.14]. This means that the RTR method only requires a few computations of the Hessian along speciﬁc directions. We call our method using the Hessian RTRMC 2. 5 Numerical experiments We test our algorithms on both synthetic and real data and compare their performances against OptSpace, ADMiRA, SVT, LMaFit and Balanced Factorization in terms of accuracy and computation time. All algorithms are run sequentially by Matlab on the same personal computer1. Table 2 speciﬁes a few implementation details. 1Intel Core i5 670 @ 3.60GHz (4), 8Go RAM, Matlab 7.10 (R2010a). 6 Table 2: All Matlab implementations call subroutines in non-Matlab code to efﬁciently deal with the sparsity of the matrices involved. PROPACK [Lar05] is a free package for large and sparse SVD computations. Method Environment Comment RTRMC 1 Matlab + some C-Mex Our method with “approximate Hessian” set to identity, i.e., no second order information. λ = 10−6. For the initial guess U0, we use the OptSpace trimmed SVD. RTRMC 2 Matlab + some C-Mex Same as RTRMC 1 but with exact Hessian. OptSpace C code [KO09] with λ = 0. Trimmed SVD + descent on Grass. ADMiRA Matlab with PROPACK [LB10] Matching pursuit based. SVT Matlab with PROPACK [CCS08] default τ and δ. Nuclear norm minimization. LMaFit Matlab + some C-Mex [WYZ10] Alternating minimization. Balanced Factorization Matlab + some C-Mex [MBS11] One of their Riemannian regression methods. Our methods (RTRMC 1 and 2) and Balanced Factorization require knowledge of the target rank r. OptSpace, ADMiRA and LMaFit include a mechanism to guess the rank, but beneﬁt from knowing it, hence we provide the true rank to these methods too. As is, the SVT code does not permit the user to specify the rank. We use the root mean square error (RMSE) criterion to assess the quality of reconstruction of X with ˆX: RMSE(X, ˆX) = ∥X −ˆX∥F/√mn. Scenario 1. We ﬁrst compare convergence behavior of the different methods on synthetic data. We pick m = n = 10 000 and r = 10. The dimension of the manifold of m-by-n matrices of rank r is d = r(m+n−r). We generate A ∈Rm×r and B ∈Rr×n with i.i.d. normal entries of zero mean and unit variance. The target matrix is X = AB. We sample 2.5d entries uniformly at random, which yields a sampling ratio of 0.5%. Figure 1 is typical and shows the evolution of the RMSE as a function of time (left) and iteration count (right). For ˆX = UV with U ∈Rm×r, V ∈Rr×n, we compute the RMSE in O((m + n)r2) ﬂops using: (mn)RMSE(AB, UV )2 = Trace((A⊤A)(BB⊤)) + Trace((U ⊤U)(V V ⊤)) −2Trace((U ⊤A)(BV ⊤)). Be wary though that this formula is numerically inaccurate when the RMSE is much smaller than the norm of either AB or UV , owing to the computation of the difference of close large numbers. Scenario 2. In this second test, we repeat the previous experience with rectangular matrices: m = 1 000, n = 30 000, r = 5 and a sampling ratio of 2.6% (5d known entries). We expect RTRMC to perform well on rectangular matrices since the dimension of the Grassmann manifold we optimize on only grows linearly with min(m, n), whereas it is the (simple) least-squares problem dimension that grows linearly in max(m, n). Figure 2 is typical and shows indeed that RTRMC is the fastest tested algorithm on this test. Scenario 3. Following the protocol in [KMO09], we test our method on the Jester dataset 1 [GRGP01] of ratings of a hundred jokes by 24 983 users. We randomly select 4 000 users and the corresponding continuous ratings in the range [−10, 10]. For each user, we extract two ratings at random as test data. We run the different matrix completion algorithms with a prescribed rank on the remaining training data, N = 100 times for each rank. Table 3 reports the average Normalized Mean Absolute Error (NMAE) on the test data along with a conﬁdence interval computed as the standard deviation of the NMAE’s obtained for the different runs divided by √ N. All methods but ADMiRA minimize a similar cost function and consequently perform the same. 6 Conclusion Our contribution is an efﬁcient numerical method to solve large low-rank matrix completion problems. RTRMC competes with the state-of-the-art and enjoys proven global and local convergence to local optima, with a quadratic convergence rate for RTRMC 2. Our methods are particularly efﬁcient on rectangular matrices. To obtain such results, we exploited the geometry of the low-rank constraint and applied techniques from the ﬁeld of optimization on manifolds. Matlab code for RTRMC 1 and 2 is available at: http://www.inma.ucl.ac.be/˜absil/RTRMC/. 7 Table 3: NMAE’s on the Jester dataset 1 (Scenario 3). All algorithms solve the problem in well under a minute for rank 7. All but ADMiRA reach similar results. As a reference, consider that a random guesser would obtain a score of 0.33. Goldberg et al. [GRGP01] report a score of 0.187 but use a different protocol. rank RTRMC 2 OptSpace LMaFit Bal. Fac. ADMiRA 1 0.1799 ± 2 · 10−4 0.1799 ± 2 · 10−4 0.1799 ± 2 · 10−4 0.1799 ± 2 · 10−4 0.1836 ± 2 · 10−4 3 0.1624 ± 2 · 10−4 0.1625 ± 2 · 10−4 0.1624 ± 2 · 10−4 0.1626 ± 2 · 10−4 0.1681 ± 2 · 10−4 5 0.1584 ± 2 · 10−4 0.1584 ± 2 · 10−4 0.1584 ± 2 · 10−4 0.1584 ± 2 · 10−4 0.1635 ± 2 · 10−4 7 0.1578 ± 2 · 10−4 0.1581 ± 2 · 10−4 0.1578 ± 2 · 10−4 0.1580 ± 2 · 10−4 0.1618 ± 2 · 10−4 Time [s] RMSE RTRMC 2 RTRMC 1 ADMiRA OptSpace LMaFit SVT Bal. Fac. Iteration count 0 50 100 0 25 50 10−8 10−5 10−2 101 10−8 10−5 10−2 101 Figure 1: Evolution of the RMSE for the six methods under Scenario 1 (m = n = 10 000, r = 10, \|Ω\|/(mn) = 0.5%, i.e., 99.5% of the entries are unknown). For RTRMC 2, we count the number of inner iterations, i.e., the number of parallelizable steps. ADMiRA stagnates and SVT diverges. All other methods eventually ﬁnd the exact solution. Time [s] RMSE RTRMC 2 RTRMC 1 ADMiRA OptSpace LMaFit SVT Bal. Fac. Iteration count 0 50 100 0 25 50 10−8 10−5 10−2 101 10−8 10−5 10−2 101 Figure 2: Evolution of the RMSE for the six methods under Scenario 2 (m = 1 000, n = 30 000, r = 5, \|Ω\|/(mn) = 2.6%). For rectangular matrices, RTRMC is especially efﬁcient owing to the linear growth of the dimension of the search space in min(m, n), whereas for most methods the growth is linear in m + n. Acknowledgments This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Ofﬁce. NB is an FNRS research fellow (Aspirant). The scientiﬁc responsibility rests with its authors. 8 References [ABG07] P.-A. Absil, C. G. Baker, and K. A. Gallivan. Trust-region methods on Riemannian manifolds. Found. Comput. Math., 7(3):303–330, July 2007. [AMS08] P.-A. Absil, R. Mahony, and R. Sepulchre. Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ, 2008. [BNR10] L. Balzano, R. Nowak, and B. Recht. Online identiﬁcation and tracking of subspaces from highly incomplete information. In Communication, Control, and Computing (Allerton), 2010 48th Annual Allerton Conference on, pages 704–711. IEEE, 2010. [Bro05] M. Brookes. The matrix reference manual. Imperial College London, 2005. [CCS08] J.F. Cai, E.J. 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4,426	Complexity of Inference in Latent Dirichlet Allocation David Sontag New York University⇤ Daniel M. Roy University of Cambridge Abstract We consider the computational complexity of probabilistic inference in Latent Dirichlet Allocation (LDA). First, we study the problem of ﬁnding the maximum a posteriori (MAP) assignment of topics to words, where the document’s topic distribution is integrated out. We show that, when the e↵ective number of topics per document is small, exact inference takes polynomial time. In contrast, we show that, when a document has a large number of topics, ﬁnding the MAP assignment of topics to words in LDA is NP-hard. Next, we consider the problem of ﬁnding the MAP topic distribution for a document, where the topic-word assignments are integrated out. We show that this problem is also NP-hard. Finally, we brieﬂy discuss the problem of sampling from the posterior, showing that this is NP-hard in one restricted setting, but leaving open the general question. 1 Introduction Probabilistic models of text and topics, known as topic models, are powerful tools for exploring large data sets and for making inferences about the content of documents. Topic models are frequently used for deriving low-dimensional representations of documents that are then used for information retrieval, document summarization, and classiﬁcation [Blei and McAuli↵e, 2008; Lacoste-Julien et al., 2009]. In this paper, we consider the computational complexity of inference in topic models, beginning with one of the simplest and most popular models, Latent Dirichlet Allocation (LDA) [Blei et al., 2003]. The LDA model is arguably one of the most important probabilistic models in widespread use today. Almost all uses of topic models require probabilistic inference. For example, unsupervised learning of topic models using Expectation Maximization requires the repeated computation of marginal probabilities of what topics are present in the documents. For applications in information retrieval and classiﬁcation, each new document necessitates inference to determine what topics are present. Although there is a wealth of literature on approximate inference algorithms for topic models, such Gibbs sampling and variational inference [Blei et al., 2003; Griﬃths and Steyvers, 2004; Mukherjee and Blei, 2009; Porteous et al., 2008; Teh et al., 2007], little is known about the computational complexity of exact inference. Furthermore, the existing inference algorithms, although well-motivated, do not provide guarantees of optimality. We choose to study LDA because we believe that it captures the essence of what makes inference easy or hard in topic models. We believe that a careful analysis of the complexity of popular probabilistic models like LDA will ultimately help us build a methodology for spanning the gap between theory and practice in probabilistic AI. Our hope is that our results will motivate discussion of the following questions, guiding research of both new topic models and the design of new approximate inference and learning ⇤This work was partially carried out while D.S. was at Microsoft Research New England. 1 algorithms. First, what is the structure of real-world LDA inference problems? Might there be structure in “natural” problem instances that makes them di↵erent from hard instances (e.g., those used in our reductions)? Second, how strongly does the prior distribution bias the results of inference? How do the hyperparameters a↵ect the structure of the posterior and the hardness of inference? We study the complexity of ﬁnding assignments of topics to words with high posterior probability and the complexity of summarizing the posterior distributions on topics in a document by either its expectation or points with high posterior density. In the former case, we show that the number of topics in the maximum a posteriori assignment determines the hardness. In the latter case, we quantify the sense in which the Dirichlet prior can be seen to enforce sparsity and use this result to show hardness via a reduction from set cover. 2 MAP inference of word assignments We will consider the inference problem for a single document. The LDA model states that the document, represented as a collection of words w = (w1, w2, . . . , wN), is generated as follows: a distribution over the T topics is sampled from a Dirichlet distribution, ✓⇠Dir(↵); then, for i 2 [N] := {1, . . . , N}, we sample a topic zi ⇠Multinomial(✓) and word wi ⇠ zi, where t, t 2 [T] are distributions on a dictionary of words. Assume that the word distributions t are ﬁxed (e.g., they have been previously estimated), and let lit = log Pr(wi\|zi = t) be the log probability of the ith word being generated from topic t. After integrating out the topic distribution vector, the joint distribution of the topic assignments conditioned on the words w is given by Pr(z1, . . . , zN\|w) / Γ(P t ↵t) Q t Γ(↵t) Q t Γ(nt + ↵t) Γ(P t ↵t + N) N Y i=1 Pr(wi\|zi), (1) where nt is the total number of words assigned to topic t. In this section, we focus on the inference problem of ﬁnding the most likely assignment of topics to words, i.e. the maximum a posteriori (MAP) assignment. This has many possible applications. For example, it can be used to cluster the words of a document, or as part of a larger system such as part-of-speech tagging [Li and McCallum, 2005]. More broadly, for many classiﬁcation tasks involving topic models it may be useful to have word-level features for whether a particular word was assigned to a given topic. From both an algorithm design and complexity analysis point of view, this MAP problem has the additional advantage of involving only discrete random variables. Taking the logarithm of Eq. 1 and ignoring constants, ﬁnding the MAP assignment is seen to be equivalent to the following combinatorial optimization problem: Φ = max xit2{0,1},nt X t logΓ( nt + ↵t) + X i,t xitlit (2) subject to X t xit = 1, X i xit = nt, where the indicator variable xit = I[zi = t] denotes the assignment of word i to topic t. 2.1 Exact maximization for small number of topics Suppose a document only uses ⌧⌧T topics. That is, T could be large, but we are guaranteed that the MAP assignment for a document uses at most ⌧di↵erent topics. In this section, we show how we can use this knowledge to eﬃciently ﬁnd a maximizing assignment of words to topics. It is important to note that we only restrict the maximum number of topics per document, letting the Dirichlet prior and the likelihood guide the choice of the actual number of topics present. We ﬁrst observe that, if we knew the number of words assigned to each topic, ﬁnding the MAP assignment is easy. For t 2 [T], let n⇤ t be the number of words assigned to topic t 2 w1 w2 w3 w4 w5 w6 t1 t2 t3 t4 t5 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 2 1 1ê2 1ê4 1 2 3 4 5 5 10 15 Figure 1: (Left) A LDA instance derived from a k-set packing instance. (Center) Plot of F(nt) = log Γ (nt + ↵) for various values of ↵. The x-axis varies nt, the number of words assigned to topic t, and the y-axis shows F(nt). (Right) Behavior of log Γ(nt + ↵) as ↵! 0. The function is stable everywhere but at zero, where the reward for sparsity increases without bound. in the MAP assignment. Then, the MAP assignment x is found by solving the following optimization problem: max xit2{0,1} X i,t xitlit (3) subject to X t xit = 1, X i xit = n⇤ t , which is equivalent to weighted b-matching in a bipartite graph (the words are on one side, the topics on the other) and can be optimally solved in time O(bm3), where b = maxt n⇤ t = O(N) and m = N + T [Schrijver, 2003]. We call (n1, . . . , nT ) a valid partition when ni ≥0 and P t nt = N. Using weighted bmatching, we can ﬁnd a MAP assignment of words to topics by trying all %T ⌧ & = ⇥(T ⌧) choices of ⌧topics and all possible valid partitions with at most ⌧non-zeros. for all subsets A ✓[T] such that \|A\| = ⌧do for all valid partitions n = (n1, n2, . . . , nT ) such that nt = 0 for t 62 A do ΦA,n Weighted-B-Matching(A, n, l) + P t logΓ( nt + ↵t) end for end for return arg maxA,n ΦA,n There are at most N ⌧−1 valid partitions with ⌧non-zero counts. For each of these, we solve the b-matching problem to ﬁnd the most likely assignment of words to topics that satisﬁes the cardinality constraints. Thus, the total running time is O((NT)⌧(N + ⌧)3). This is polynomial when the number of topics ⌧appearing in a document is a constant. 2.2 Inference is NP-hard for large numbers of topics In this section, we show that probabilistic inference is NP-hard in the general setting where a document may have a large number of topics in its MAP assignment. Let WORD-LDA(↵) denote the decision problem of whetherΦ > V (see Eq. 2) for some V 2 R, where the hyperparameters ↵t = ↵for all topics. We consider both ↵< 1 and ↵≥1 because, as shown in Figure 1, the optimization problem is qualitatively di↵erent in these two cases. Theorem 1. WORD-LDA(↵) is NP-hard for all ↵> 0. Proof. Our proof is a straightforward generalization of the approach used by Halperin and Karp [2005] to show that the minimum entropy set cover problem is hard to approximate. The proof is done by reduction from k-set packing (k-SP), for k ≥3. In k-SP, we are given a collection of k-element sets over some universe of elements ⌃with \|⌃\| = n. The goal is to ﬁnd the largest collection of disjoint sets. There exists a constant c < 1 such that it is NP-hard to decide whether a k-SP instance has (i) a solution with n/k disjoint sets 3 covering all elements (called a perfect matching), or (ii) at most cn/k disjoint sets (called a (cn/k)-matching). We now describe how to construct a LDA inference problem from a k-SP instance. This requires specifying the words in the document, the number of topics, and the word log probabilities lit. Let each element i 2 ⌃correspond to a word wi, and let each set correspond to one topic. The document consists of all of the words (i.e., ⌃). We assign uniform probability to the words in each topic, so that Pr(wi\|zi = t) = 1 k for i 2 t, and 0 otherwise. Figure 1 illustrates the resulting LDA model. The topics are on the top, and the words from the document are on the bottom. An edge is drawn between a topic (set) and a word (element) if the corresponding set contains that element. What remains is to show that we can solve some k-SP problem by using this reduction and solving a WORD-LDA(↵) problem. For technical reasons involving ↵> 1, we require that k is suﬃciently large. We will use the following result (we omit the proof due to space limitations). Lemma 2. Let P be a k-SP instance for k > (1 + ↵)2, and let P 0 be the derived WORDLDA(↵) instance. There exist constants CU and CL < CU such that, if there is a perfect matching in P, then Φ ≥CU. If, on the other hand, there is at most a (cn/k)-matching in P, then Φ < CL. Let P be a k-SP instance for k > (3 + ↵)2, P 0 be the derived WORD-LDA(↵) instance, and CU and CL < CU be as in Lemma 2. Then, by testing Φ < CL andΦ > CU we can decide whether P is a perfect matching or at best a (cn/k)-matching. Hence k-SP reduces to WORD-LDA(↵). The bold lines in Figure 1 indicate the MAP assignment, which for this example corresponds to a perfect matching for the original k-set packing instance. More realistic documents would have signiﬁcantly more words than topics used. Although this is not possible while keeping k = 3, since the MAP assignment always has ⌧≥N/k, we can instead reduce from a k-set packing problem with k ≫3. Lemma 2 shows that this is hard as well. 3 MAP inference of the topic distribution In this section we consider the task of ﬁnding the mode of Pr(✓\|w). This MAP problem involves integrating out the topic assignments, zi, as opposed to the previously considered MAP problem of integrating out the topic distribution ✓. We will see that the MAP topic distribution is not always well-deﬁned, which will lead us to deﬁne and study alternative formulations. In particular, we give a precise characterization of the MAP problem as one of ﬁnding sparse topic distributions, and use this fact to give hardness results for several settings. We also show settings for which MAP inference is tractable. There are many potential applications of MAP inference of the document’s topic distribution. For example, the distribution may be used for topic-based information retrieval or as the feature vector for classiﬁcation. As we will make clear later, this type of inference results in sparse solutions. Thus, the MAP topic distribution provides a compact summary of the document that could be useful for document summarization. Let ✓= (✓1, . . . , ✓T ). A straightforward application of Bayes’ rule allows us to write the posterior density of ✓given w as Pr(✓\|w) / T Y t=1 ✓↵t−1 t ! N Y i=1 T X t=1 ✓t it ! , (4) where it = Pr(wi\|zi = t). Taking the logarithm of the posterior and ignoring constants, we obtain Φ(✓) = T X t=1 (↵t −1) log(✓t) + N X i=1 log T X t=1 ✓t it ! (5) 4 We will use the shorthand Φ(✓) = P(✓) + L(✓), where P(✓) = PT t=1(↵t −1) log(✓t) and L(✓) = PN i=1 log(PT t=1 it✓t). To ﬁnd the MAP ✓, we maximize (5) subject to the constraint that PT t=1 ✓t = 1 and ✓t ≥0. Unfortunately, this maximization problem can be degenerate. In particular, note that if ✓t = 0 for ↵t < 1, then the corresponding term in P(✓) will take the value 1, overwhelming the likelihood term. Thus, any feasible solution with the above property could be considered ‘optimal’. A similar problem arises during the maximum-likelihood estimation of a normal mixture model, where the likelihood diverges to inﬁnity as the variance of a mixture component with a single data point approaches zero [Biernacki and Chr´etien, 2003; Kiefer and Wolfowitz, 1956]. In practice, one can enforce a lower bound on the variance or penalize such conﬁgurations. Here we consider a similar tactic. For ✏> 0, let TOPIC-LDA(✏) denote the optimization problem max ✓ Φ(✓) subject to X t ✓t = 1, ✏✓t 1. (6) For ✏= 0, we will denote the corresponding optimization problem by TOPIC-LDA. When ↵t = ↵, i.e. the prior distribution on the topic distribution is a symmetric Dirichlet, we write TOPIC-LDA(✏,↵) for the corresponding optimization problem. In the following sections we will study the structure and hardness of TOPIC-LDA, TOPIC-LDA(✏) and TOPIC-LDA(✏,↵). 3.1 Polynomial-time inference for large hyperparameters (↵t ≥1) When ↵t ≥1, Eq. 5 is a concave function of ✓. As a result, we can eﬃciently ﬁnd ✓⇤using a number of techniques from convex optimization. Note that this is in contrast to the MAP inference problem discussed in Section 2, which we showed was hard for all choices of ↵. Since we are optimizing over the simplex (✓must be non-negative and sum to 1), we can apply the exponentiated gradient method [Kivinen and Warmuth, 1995]. Initializing ✓0 to be the uniform vector, the update for time s is given by ✓s+1 t = ✓s t exp(⌘5s t) P ˆt ✓s ˆt exp(⌘5s ˆt), 5s t = ↵t −1 ✓s t + N X i=1 it PT ˆt=1 ✓s ˆt iˆt , (7) where ⌘is the step size and 5s is the gradient. When ↵= 1 the prior disappears altogether and this algorithm simply corresponds to optimizing the likelihood term. When ↵≫1, the prior corresponds to a bias toward a particular ✓topic distribution. 3.2 Small hyperparameters encourage sparsity (↵< 1) On the other hand, when ↵t < 1, the ﬁrst term in Eq. 5 is convex whereas the second term is concave. This setting, of ↵much smaller than 1, occurs frequently in practice. For example, learning a LDA model on a large corpus of NIPS abstracts with T = 200 topics, we ﬁnd that the hyperparameters found range from ↵t = 0.0009 to 0.135, with the median being 0.01. Although in this setting it is diﬃcult to ﬁnd the global optimum (we will make this precise in Theorem 6), one possibility for ﬁnding a local maximum is the Concave-Convex Procedure [Yuille and Rangarajan, 2003]. In this section we prove structural results about the TOPIC-LDA(✏,↵) solution space for when ↵< 1. These results illustrate that the Dirichlet prior encourages sparse MAP solutions: the topic distribution will be large on as few topics as necessary to explain every word of the document, and otherwise will be close to zero. The following lemma shows that in any optimal solution to TOPIC-LDA(✏,↵), for every word, there is at least one topic that both has large probability and gives non-trivial probability to this word. We use K(↵, T, N) = e−3/↵N −1T −1/↵to refer to the lower bound on the topic’s probability. 5 Lemma 3. Let ↵< 1. All optimal solutions ✓⇤to TOPIC-LDA(✏,↵) have the following property: for every word i, ✓⇤ ˆt ≥K(↵, T, N) where ˆt = arg maxt it✓⇤ t . Proof sketch. If ✏≥K(↵, T, N) the claim trivially holds. Assume for the purpose of contradiction that there exists a word ˆi such that ✓⇤ ˆt < K(↵, T, N), where ˆt = arg maxt ˆit✓⇤ t . Let Y denote the set of topics t 6= ˆt such that ✓⇤ t ≥2✏. Let β1 = P t2Y ✓⇤ t and β2 = P t62Y,t6=ˆt ✓⇤ t . Note that β2 < 2T✏. Consider ˆ✓ˆt = 1 n, ˆ✓t = ✓1 −β2 −1 n β1 ◆ ✓⇤ t for t 2 Y, ˆ✓t = ✓⇤ t for t 62 Y, t 6= ˆt. (8) It is easy to show that 8t, ˆ✓t ≥✏, and P t ˆ✓t = 1. Finally, we show that Φ(ˆ✓) > Φ(✓⇤), contradicting the optimality of ✓⇤. The full proof is given in the supplementary material. Next, we show that if a topic is not suﬃciently “used” then it will be given a probability very close to zero. By used, we mean that for at least one word, the topic is close in probability to that of the largest contributor to the likelihood of the word. To do this, we need to deﬁne the notion of the dynamic range of a word, given as i = maxt,t0: it>0, it0>0 it it0 . We let the maximum dynamic range be = maxi i. Note that ≥1 and, for most applications, it is reasonable to expect to be small (e.g., less than 1000). Lemma 4. Let ↵< 1, and let ✓⇤be any optimal solution to TOPIC-LDA(✏,↵). Suppose topic ˆt has ✓⇤ ˆt < (N)−1K(↵, T, N). Then, ✓⇤ ˆt e 1 1−↵+2✏. Proof. Suppose for the purpose of contradiction that ✓⇤ ˆt > e 1 1−↵+2✏. Consider ˆ✓deﬁned as follows: ˆ✓ˆt = ✏, and ˆ✓t = ⇣ 1−✏ 1−✓⇤ ˆt ⌘ ✓⇤ t for t 6= ˆt. We have: Φ(ˆ✓) −Φ(✓⇤) = (1 −↵) log ✓✓⇤ ˆt ✏ ◆ + (T −1)(1 −↵) log ✓1 −✓⇤ ˆt 1 −✏ ◆ + N X i=1 log P t ˆ✓t it P t ✓⇤ t it ! . Using the fact that log(1 −z) ≥−2z for z 2 [0, 1 2], it follows that (T −1)(1 −↵) log ✓1 −✓⇤ ˆt 1 −✏ ◆ ≥(T −1)(1 −↵) log % 1 −✓⇤ ˆt & ≥2(T −1)(↵−1)✓⇤ ˆt (9) ≥2(T −1)(↵−1)(N)−1K(↵, T, N) ≥2(↵−1). (10) We have ˆ✓t ≥✓⇤ t for t 6= ˆt, and so P t ˆ✓t it P t ✓⇤ t it = P t6=ˆt ˆ✓t it + ✏ iˆt P t6=ˆt ✓⇤ t it + ✓⇤ ˆt iˆt ≥ P t6=ˆt ✓⇤ t it P t6=ˆt ✓⇤ t it + ✓⇤ ˆt iˆt . (11) Recall from Lemma 3 that, for each word i and ˜t = arg maxt it✓⇤ t , we have ✓˜t > K(↵, T, N). Necessarily ˜t 6= ˆt. Therefore, using the fact that log 1 1+z ≥−z, log P t6=ˆt ✓⇤ t it P t6=ˆt ✓⇤ t it + ✓⇤ ˆt iˆt ! ≥− ✓⇤ ˆt iˆt P t6=ˆt ✓⇤ t it ≥−(N)−1K(↵, T, N) iˆt K(↵, T, N) i˜t ≥−1 n. (12) Thus,Φ( ˆ✓) −Φ(✓⇤) > (1 −↵) log e 1 1−↵+2 + 2(↵−1) −1 = 0, completing the proof. Finally, putting together what we showed in the previous two lemmas, we conclude that all optimal solutions to TOPIC-LDA(✏,↵) either have ✓t large or small, but not in between (that is, we have demonstrated a gap). We have the immediate corollary: Theorem 5. For ↵< 1, all optimal solutions to TOPIC-LDA(✏,↵) have ✓t  ⇣ e 1 1−↵+2⌘ ✏ or ✓t ≥−1e−3/↵N −2T −1/↵. 6 3.3 Inference is NP-hard for small hyperparameters (↵< 1) The previous results characterize optimal solutions to TOPIC-LDA(✏,↵) and highlight the fact that optimal solutions are sparse. In this section we show that these same properties can be the source of computational hardness during inference. In particular, it is possible to encode set cover instances as TOPIC-LDA(✏,↵) instances, where the set cover corresponds to those topics assigned appreciable probability. Theorem 6. TOPIC-LDA(✏,↵) is NP-hard for ✏K(↵, T, N)T/(1−↵)T N/(1−↵) and ↵< 1. Proof. Consider a set cover instance consisting of a universe of elements and a family of sets, where we assume for convenience that the minimum cover is neither a singleton, all but one of the family of sets, nor the entire family of sets, and that there are at least two elements in the universe. As with our previous reduction, we have one topic per set and one word in the document for each element. We let Pr(wi\|zi = t) = 0 when element wi is not in set t, and a constant otherwise (we make every topic have the uniform distribution over the same number of words, some of which may be dummy words not appearing in the document). Let Si ✓[T] denote the set of topics to which word i belongs. Then, up to additive constants, we have P(✓) = −(1 −↵) PT t=1 log(✓t) and L(✓) = PN i=1 log(P t2Si ✓t). Let C✓⇤✓[T] be those topics t 2 [T] such that ✓⇤ t ≥K(↵, T, n), where ✓⇤is an optimal solution to TOPIC-LDA(✏,↵). It immediately follows from Lemma 3 that C✓⇤is a cover. Suppose for the purpose of contradiction that C✓⇤is not a minimal cover. Let ˜C be a minimal cover, and let ˜✓t = ✏for t 62 ˜C and ˜✓t = 1−✏(T −\| ˜ C\|) \| ˜ C\| > 1 T otherwise. We will show that Φ(˜✓) > Φ(✓⇤), contradicting the optimality of ✓⇤, and thus proving that C✓⇤is in fact minimal. This suﬃces to show that TOPIC-LDA(✏,↵) is NP-hard in this regime. For all ✓in the simplex, we have P i log(maxt2Si ✓t) L(✓) 0. Thus it follows that L(✓⇤) −L(˜✓) N log T. Likewise, using the assumption that T ≥\| ˜C\| + 1, we have P(˜✓) −P(✓⇤) (1 −↵) ≥−(T −\| ˜C\|) log ✏−(\| ˜C\| + 1) log K(↵, T, N) + (T −\| ˜C\| −1) log ✏ (13) ≥log 1 ✏−T log K(↵, T, N), (14) where we have conservatively only included the terms t 62 ˜C for P(˜✓) and taken ✓⇤2 {✏, K(↵, T, N)} with \| ˜C\| + 1 terms taking the latter value. It follows that % P(˜✓) + L(˜✓) & − % P(✓⇤) + L(✓⇤) & > (1 −↵) log 1 ✏−(T log K(↵, T, N) + N log T). (15) This is greater than 0 precisely when (1 −↵) log 1 ✏> log T NK(↵, T, N)T . Note that although ✏is exponentially small in N and T, the size of its representation in binary is polynomial in N and T, and thus polynomial in the size of the set cover instance. It can be shown that as ✏! 0, the solutions to TOPIC-LDA(✏,↵) become degenerate, concentrating their support on the minimal set of topics C ✓[T] such that 8i, 9t 2 C s.t. it > 0. A generalization of this result holds for TOPIC-LDA(✏) and suggests that, while it may be possible to give a more sensible deﬁnition of TOPIC-LDA as the set of solutions for TOPIC-LDA(✏) as ✏! 0, these solutions are unlikely to be of any practical use. 4 Sampling from the posterior The previous sections of the paper focused on MAP inference problems. In this section, we study the problem of marginal inference in LDA. Theorem 7. For ↵> 1, one can approximately sample from Pr(✓\| w) in polynomial time. Proof sketch. The density given in Eq. 4 is log-concave when ↵≥1. The algorithm given in Lovasz and Vempala [2006] can be used to approximately sample from the posterior. 7 Although polynomial, it is not clear whether the algorithm given in Lovasz and Vempala [2006], based on random walks, is of practical interest (e.g., the running time bound has a constant of 1030). However, we believe our observation provides insight into the complexity of sampling when ↵is not too small, and may be a starting point towards explaining the empirical success of using Markov chain Monte Carlo to do inference in LDA. Next, we show that when ↵is extremely small, it is NP-hard to sample from the posterior. We again reduce from set cover. The intuition behind the proof is that, when ↵is small enough, an appreciable amount of the probability mass corresponds to the sparsest possible ✓vectors where the supported topics together cover all of the words. As a result, we could directly read o↵the minimal set cover from the posterior marginals E[✓t \| w]. Theorem 8. When ↵< % (4N + 4)T NΓ(N) &−1, it is NP-hard to approximately sample from Pr(✓\| w), under randomized reductions. The full proof can be found in the supplementary material. Note that it is likely that one would need an extremely large and unusual corpus to learn an ↵so small. Our results illustrate a large gap in our knowledge about the complexity of sampling as a function of ↵. We feel that tightening this gap is a particularly exciting open problem. 5 Discussion In this paper, we have shown that the complexity of MAP inference in LDA strongly depends on the e↵ective number of topics per document. When a document is generated from a small number of topics (regardless of the number of topics in the model), WORD-LDA can be solved in polynomial time. We believe this is representative of many real-world applications. On the other hand, if a document can use an arbitrary number of topics, WORD-LDA is NP-hard. The choice of hyperparameters for the Dirichlet does not a↵ect these results. We have also studied the problem of computing MAP estimates and expectations of the topic distribution. In the former case, the Dirichlet prior enforces sparsity in a sense that we make precise. In the latter case, we show that extreme parameterizations can similarly cause the posterior to concentrate on sparse solutions. In both cases, this sparsity is shown to be a source of computational hardness. In related work, Sepp¨anen et al. [2003] suggest a heuristic for inference that is also applicable to LDA: if there exists a word that can only be generated with high probability from one of the topics, then the corresponding topic must appear in the MAP assignment whenever that word appears in a document. Miettinen et al. [2008] give a hardness reduction and greedy algorithm for learning topic models. Although the models they consider are very di↵erent from LDA, some of the ideas may still be applicable. More broadly, it would be interesting to consider the complexity of learning the per-topic word distributions t. Our paper suggests a number of directions for future study. First, our exact algorithms can be used to evaluate the accuracy of approximate inference algorithms, for example by comparing to the MAP of the variational posterior. On the algorithmic side, it would be interesting to improve the running time of the exact algorithm from Section 2.1. Also, note that we did not give an analogous exact algorithm for the MAP topic distribution when the posterior has support on only a small number of topics. In this setting, it may be possible to ﬁnd this set of topics by trying all S ✓[T] of small cardinality and then doing a (non-uniform) grid search over the topic distribution restricted to support S. Finally, our structural results on the sparsity induced by the Dirichlet prior draws connections between inference in topic models and sparse signal recovery. We proved that the MAP topic distribution has, for each topic t, either ✓t ⇡✏or ✓t bounded below by some value (much larger than ✏). Because of this gap, we can approximately view the MAP problem as searching for a set corresponding to the support of ✓. Our work motivates the study of greedy algorithms for MAP inference in topic models, analogous to those used for set cover. One could even consider learning algorithms that use this greedy algorithm within the inner loop [Krause and Cevher, 2010]. 8 Acknowledgments D.M.R. is supported by a Newton International Fellowship. We thank Tommi Jaakkola and anonymous reviewers for helpful comments. References C. Biernacki and S. Chr´etien. Degeneracy in the maximum likelihood estimation of univariate Gaussian mixtures with EM. Statist. Probab. Lett., 61(4):373–382, 2003. ISSN 0167-7152. D. Blei and J. McAuli↵e. Supervised topic models. In J. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Adv. in Neural Inform. 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4,427	A Denoising View of Matrix Completion Weiran Wang Miguel ´A. Carreira-Perpi˜n´an EECS, University of California, Merced http://eecs.ucmerced.edu Zhengdong Lu Microsoft Research Asia, Beijing zhengdol@microsoft.com Abstract In matrix completion, we are given a matrix where the values of only some of the entries are present, and we want to reconstruct the missing ones. Much work has focused on the assumption that the data matrix has low rank. We propose a more general assumption based on denoising, so that we expect that the value of a missing entry can be predicted from the values of neighboring points. We propose a nonparametric version of denoising based on local, iterated averaging with meanshift, possibly constrained to preserve local low-rank manifold structure. The few user parameters required (the denoising scale, number of neighbors and local dimensionality) and the number of iterations can be estimated by cross-validating the reconstruction error. Using our algorithms as a postprocessing step on an initial reconstruction (provided by e.g. a low-rank method), we show consistent improvements with synthetic, image and motion-capture data. Completing a matrix from a few given entries is a fundamental problem with many applications in machine learning, computer vision, network engineering, and data mining. Much interest in matrix completion has been caused by recent theoretical breakthroughs in compressed sensing [1, 2] as well as by the now celebrated Netﬂix challenge on practical prediction problems [3, 4]. Since completion of arbitrary matrices is not a well-posed problem, it is often assumed that the underlying matrix comes from a restricted class. Matrix completion models almost always assume a low-rank structure of the matrix, which is partially justiﬁed through factor models [4] and fast convex relaxation [2], and often works quite well when the observations are sparse and/or noisy. The low-rank structure of the matrix essentially asserts that all the column vectors (or the row vectors) live on a low-dimensional subspace. This assumption is arguably too restrictive for problems with richer structure, e.g. when each column of the matrix represents a snapshot of a seriously corrupted motion capture sequence (see section 3), for which a more ﬂexible model, namely a curved manifold, is more appropriate. In this paper, we present a novel view of matrix completion based on manifold denoising, which conceptually generalizes the low-rank assumption to curved manifolds. Traditional manifold denoising is performed on fully observed data [5, 6], aiming to send the data corrupted by noise back to the correct surface (deﬁned in some way). However, with a large proportion of missing entries, we may not have a good estimate of the manifold. Instead, we start with a poor estimate and improve it iteratively. Therefore the “noise” may be due not just to intrinsic noise, but mostly to inaccurately estimated missing entries. We show that our algorithm can be motivated from an objective purely based on denoising, and prove its convergence under some conditions. We then consider a more general case with a nonlinear low-dimensional manifold and use a stopping criterion that works successfully in practice. Our model reduces to a low-rank model when we require the manifold to be ﬂat, showing a relation with a recent thread of matrix completion models based on alternating projection [7]. In our experiments, we show that our denoising-based matrix completion model can make better use of the latent manifold structure on both artiﬁcial and real-world data sets, and yields superior recovery of the missing entries. The paper is organized as follows: section 1 reviews nonparametric denoising methods based on mean-shift updates, section 2 extends this to matrix completion by using denoising with constraints, section 3 gives experimental results, and section 4 discusses related work. 1 1 Denoising with (manifold) blurring mean-shift algorithms (GBMS/MBMS) In Gaussian blurring mean-shift (GBMS), denoising is performed in a nonparametric way by local averaging: each data point moves to the average of its neighbors (to a certain scale), and the process is repeated. We follow the derivation in [8]. Consider a dataset {xn}N n=1 ⊂RD and deﬁne a Gaussian kernel density estimate p(x) = 1 N N X n=1 Gσ(x, xn) (1) with bandwidth σ > 0 and kernel Gσ(x, xn) ∝exp −1 2(∥x −xn∥/σ)2 (other kernels may be used, such as the Epanechnikov kernel, which results in sparse afﬁnities). The (non-blurring) mean-shift algorithm rearranges the stationary point equation ∇p(x) = 0 into the iterative scheme x(τ+1) = f(x(τ)) with x(τ+1) = f(x(τ)) = N X n=1 p(n\|x(τ))xn p(n\|x(τ)) = exp −1 2 (x(τ) −xn)/σ 2 PN n′=1 exp −1 2 (x(τ) −xn′)/σ 2. (2) This converges to a mode of p from almost every initial x ∈RD, and can be seen as taking selfadapting step sizes along the gradient (since the mean shift f(x) −x is parallel to ∇p(x)). This iterative scheme was originally proposed by [9] and it or variations of it have found widespread application in clustering [8, 10–12] and denoising of 3D point sets (surface fairing; [13, 14]) and manifolds in general [5, 6]. The blurring mean-shift algorithm applies one step of the previous scheme, initialized from every point, in parallel for all points. That is, given the dataset X = {x1, . . . , xN}, for each xn ∈X we obtain a new point ˜xn = f(xn) by applying one step of the mean-shift algorithm, and then we replace X with the new dataset ˜X, which is a blurred (shrunk) version of X. By iterating this process we obtain a sequence of datasets X(0), X(1), . . . (and a corresponding sequence of kernel density estimates p(0)(x), p(1)(x), . . .) where X(0) is the original dataset and X(τ) is obtained by blurring X(τ−1) with one mean-shift step. We can see this process as maximizing the following objective function [10] by taking parallel steps of the form (2) for each point: E(X) = N X n=1 p(xn) = 1 N N X n,m=1 Gσ(xn, xm) ∝ N X n,m=1 e−1 2∥ xn−xm σ ∥ 2 . (3) This process eventually converges to a dataset X(∞) where all points are coincident: a completely denoised dataset where all structure has been erased. As shown by [8], this process can be stopped early to return clusters (= locally denoised subsets of points); the number of clusters obtained is controlled by the bandwidth σ. However, here we are interested in the denoising behavior of GBMS. The GBMS step can be formulated in a matrix form reminiscent of spectral clustering [8] as ˜X = X P where X = (x1, . . . , xN) is a D×N matrix of data points; W is the N ×N matrix of Gaussian afﬁnities wnm = Gσ(xn, xm); D = diag (PN n=1 wnm) is the degree matrix; and P = WD−1 is an N × N stochastic matrix: pnm = p(n\|xm) ∈(0, 1) and PN n=1 pnm = 1. P (or rather its transpose) is the stochastic matrix of the random walk in a graph [15], which in GBMS represents the posterior probabilities of each point under the kernel density estimate (1). P is similar to the matrix N = D−1 2 WD−1 2 derived from the normalized graph Laplacian commonly used in spectral clustering, e.g. in the normalized cut [16]. Since, by the Perron-Frobenius theorem [17, ch. 8], all left eigenvalues of P(X) have magnitude less than 1 except for one that equals 1 and is associated with an eigenvector of constant entries, iterating ˜X = X P(X) converges to the stationary distribution of each P(X), where all points coincide. From this point of view, the product ˜X = X P(X) can be seen as ﬁltering the dataset X with a datadependent low-pass ﬁlter P(X), which makes clear the denoising behavior. This also suggests using other ﬁlters [12] ˜X = X φ(P(X)) as long as φ(1) = 1 and \|φ(r)\| < 1 for r ∈[0, 1), such as explicit schemes φ(P) = (1 −η)I + ηP for η ∈(0, 2], power schemes φ(P) = Pn for n = 1, 2, 3 . . . or implicit schemes φ(P) = ((1 + η)I −ηP)−1 for η > 0. One important problem with GBMS is that it denoises equally in all directions. When the data lies on a low-dimensional manifold, denoising orthogonally to it removes out-of-manifold noise, but 2 denoising tangentially to it perturbs intrinsic degrees of freedom of the data and causes shrinkage of the entire manifold (most strongly near its boundary). To prevent this, the manifold blurring meanshift algorithm (MBMS) [5] ﬁrst computes a predictor averaging step with GBMS, and then for each point xn a corrector projective step removes the step direction that lies in the local tangent space of xn (obtained from local PCA run on its k nearest neighbors). In practice, both GBMS and MBMS must be stopped early to prevent excessive denoising and manifold distortions. 2 Blurring mean-shift denoising algorithms for matrix completion We consider the natural extension of GBMS to the matrix completion case by adding the constraints given by the present values. We use the subindex notation XM and XP to indicate selection of the missing or present values of the matrix XD×N, where P ⊂U, M = U \ P and U = {(d, n): d = 1, . . . , D, n = 1, . . . , N}. The indices P and values XP of the present matrix entries are the data of the problem. Then we have the following constrained optimization problem: max X E(X) = N X n,m=1 Gσ(xn, xm) s.t. XP = XP. (4) This is similar to low-rank formulations for matrix completion that have the same constraints but use as objective function the reconstruction error with a low-rank assumption, e.g. ∥X −ABX∥2 with AD×L, BL×D and L < D. We initialize XM to the output of some other method for matrix completion, such as singular value projection (SVP; [7]). For simple constraints such as ours, gradient projection algorithms are attractive. The gradient of E wrt X is a matrix of D × N whose nth column is: ∇xnE(X) = 2 σ2 N X m=1 e−1 2∥ xn−xm σ ∥ 2 (xm −xn) ∝2 σ2 p(xn) −xn + N X m=1 p(m\|xn)xm ! (5) and its projection on the constraint space is given by zeroing its entries having indices in P; call ΠP this projection operator. Then, we have the following step of length α ≥0 along the projected gradient: X(τ+1) = X(τ) + αΠP(∇XE(X(τ))) ⇐⇒X(τ+1) M = X(τ) M + α ΠP(∇XE(X(τ))) M (6) which updates only the missing entries XM. Since our search direction is ascent and makes an angle with the gradient that is bounded away from π/2, and E is lower bounded, continuously differentiable and has bounded Hessian (thus a Lipschitz continuous gradient) in RNL, by carrying out a line search that satisﬁes the Wolfe conditions, we are guaranteed convergence to a local stationary point, typically a maximizer [18, th. 3.2]. However, as reasoned later, we do not perform a line search at all, instead we ﬁx the step size to the GBMS self-adapting step size, which results in a simple and faster algorithm consisting of carrying out a GBMS step on X (i.e., X(τ+1) = X(τ) P(X(τ))) and then reﬁlling XP to the present values. While we describe the algorithm in this way for ease of explanation, in practice we do not actually compute the GBMS step for all xdn values, but only for the missing ones, which is all we need. Thus, our algorithm carries out GBMS denoising steps within the missing-data subspace. We can derive this result in a different way by starting from the unconstrained optimization problem maxXP E(X) = PN n,m=1 Gσ(xn, xm) (equivalent to (4)), computing its gradient wrt XP, equating it to zero and rearranging (in the same way the mean-shift algorithm is derived) to obtain a ﬁxed-point iteration identical to our update above. Fig. 1 shows the pseudocode for our denoising-based matrix completion algorithms (using three nonparametric denoising algorithms: GBMS, MBMS and LTP). Convergence and stopping criterion As noted above, we have guaranteed convergence by simply satisfying standard line search conditions, but a line search is costly. At present we do not have a proof that the GBMS step size satisﬁes such conditions, or indeed that the new iterate X(τ+1) M increases or leaves unchanged the objective, although we have never encountered a counterexample. In fact, it turns out that none of the work about GBMS that we know about proves that either: [10] proves that ∅(X(τ+1)) ≤∅(X(τ)) for 0 < ρ < 1, where ∅(·) is the set diameter, while [8, 12] 3 notes that P(X) has a single eigenvalue of value 1 and all others of magnitued less than 1. While this shows that all points converge to the same location, which indeed is the global maximum of (3), it does not necessarily follow that each step decreases E. GBMS (k, σ) with full or k-nn graph: given XD×N, M repeat for n = 1, . . . , N Nn ←{1, . . . , N} (full graph) or k nearest neighbors of xn (k-nn graph) ∂xn ←−xn + P m∈Nn Gσ(xn,xm) P m′∈Nn Gσ(xn,xm′)xm mean-shift step end XM ←XM + (∂X)M move points’ missing entries until validation error increases return X MBMS (L, k, σ) with full or k-nn graph: given XD×N, M repeat for n = 1, . . . , N Nn ←{1, . . . , N} (full graph) or k nearest neighbors of xn (k-nn graph) ∂xn ←−xn + P m∈Nn Gσ(xn,xm) P m′∈Nn Gσ(xn,xm′)xm mean-shift step Xn ←k nearest neighbors of xn (µn, Un) ←PCA(Xn, L) estimate L-dim tangent space at xn ∂xn ←(I −UnUT n)∂xn subtract parallel motion end XM ←XM + (∂X)M move points’ missing entries until validation error increases return X LTP (L, k) with k-nn graph: given XD×N, M repeat for n = 1, . . . , N Xn ←k nearest neighbors of xn (µn, Un) ←PCA(Xn, L) estimate L-dim tangent space at xn ∂xn ←(I −UnUT n)(µn −xn) project point onto tangent space end XM ←XM + (∂X)M move points’ missing entries until validation error increases return X Figure 1: Our denoising matrix completion algorithms, based on Manifold Blurring Mean Shift (MBMS) and its particular cases Local Tangent Projection (LTP, k-nn graph, σ = ∞) and Gaussian Blurring Mean Shift (GBMS, L = 0); see [5] for details. Nn contains all N points (full graph) or only xn’s nearest neighbors (k-nn graph). The index M selects the components of its input corresponding to missing values. Parameters: denoising scale σ, number of neighbors k, local dimensionality L. However, the question of convergence as τ →∞has no practical interest in a denoising setting, because achieving a total denoising almost never yields a good matrix completion. What we want is to achieve just enough denoising and stop the algorithm, as was the case with GBMS clustering, and as is the case in algorithms for image denoising. We propose to determine the optimal number of iterations, as well as the bandwidth σ and any other parameters, by cross-validation. Specifically, we select a held-out set by picking a random subset of the present entries and considering them as missing; this allows us to evaluate an error between our completion for them and the ground truth. We stop iterating when this error increases. This argument justiﬁes an algorithmic, as opposed to an optimization, view of denoisingbased matrix completion: apply a denoising step, reﬁll the present values, iterate until the validation error increases. This allows very general deﬁnitions of denoising, and indeed a lowrank projection is a form of denoising where points are not allowed outside the linear manifold. Our formulation using the objective function (4) is still useful in that it connects our denoising assumption with the more usual low-rank assumption that has been used in much matrix completion work, and justiﬁes the reﬁlling step as resulting from the present-data constraints under a gradientprojection optimization. MBMS denoising for matrix completion Following our algorithmic-based approach to denoising, we could consider generalized GBMS steps of the form ˜X = X φ(P(X)). For clustering, Carreira-Perpi˜n´an [12] found an overrelaxed explicit step φ(P) = (1 −η)I + ηP with η ≈1.25 to achieve similar clusterings but faster. Here, we focus instead on the MBMS variant of GBMS that allows only for orthogonal, not tangential, point motions (deﬁned wrt their local tangent space as estimated by local PCA), with the goal of preserving low-dimensional manifold structure. MBMS has 3 user parameters: the bandwidth σ (for denoising), and the latent dimensionality L and the 4 number of neighbors k (for the local tangent space and the neighborhood graph). A special case of MBMS called local tangent projection (LTP) results by using a neighborhood graph and setting σ = ∞(so only two user parameters are needed: L and k). LTP can be seen as doing a low-rank matrix completion locally. LTP was found in [5] to have nearly as good performance as the best σ in several problems. MBMS also includes as particular cases GBMS (L = 0), PCA (k = N, σ = ∞), and no denoising (σ = 0 or L = D). Note that if we apply MBMS to a dataset that lies on a linear manifold of dimensionality d using L ≥d then no denoising occurs whatsoever because the GBMS updates lie on the d-dimensional manifold and are removed by the corrector step. In practice, even if the data are assumed noiseless, the reconstruction from a low-rank method will lie close to but not exactly on the d-dimensional manifold. However, this suggests using largish ranks for the low-rank method used to reconstruct X and lower L values in the subsequent MBMS run. In summary, this yields a matrix completion algorithm where we apply an MBMS step, reﬁll the present values, and iterate until the validation error increases. Again, in an actual implementation we compute the MBMS step only for the missing entries of X. The shrinking problem of GBMS is less pronounced in our matrix completion setting, because we constrain some values not to change. Still, in agreement with [5], we ﬁnd MBMS to be generally superior to GBMS. Computational cost With a full graph, the cost per iteration of GBMS and MBMS is O(N 2D) and O(N 2D + N(D + k) min(D, k)2), respectively. In practice with high-dimensional data, best denoising results are obtained using a neighborhood graph [5], so that the sums over points in eqs. (3) or (4) extend only to the neighbors. With a k-nearest-neighbor graph and if we do not update the neighbors at each iteration (which affects the result little), the respective cost per iteration is O(NkD) and O(NkD +N(D +k) min(D, k)2), thus linear in N. The graph is constructed on the initial X we use, consisting of the present values and an imputation for the missing ones achieved with a standard matrix completion method, and has a one-off cost of O(N 2D). The cost when we have a fraction µ = \|M\| ND ∈[0, 1] of missing data is simply the above times µ. Hence the run time of our mean-shift-based matrix completion algorithms is faster the more present data we have, and thus faster than the usual GBMS or MBMS case, where all data are effectively missing. 3 Experimental results We compare with representative methods of several approaches: a low-rank matrix completion method, singular value projection (SVP [7], whose performance we found similar to that of alternating least squares, ALS [3, 4]); ﬁtting a D-dimensional Gaussian model with EM and imputing the missing values of each xn as the conditional mean E {xn,Mn\|xn,Pn} (we use the implementation of [19]); and the nonlinear method of [20] (nlPCA). We initialize GBMS and MBMS from some or all of these algorithms. For methods with user parameters, we set them by cross-validation in the following way: we randomly select 10% of the present entries and pretend they are missing as well, we run the algorithm on the remaining 90% of the present values, and we evaluate the reconstruction at the 10% entries we kept earlier. We repeat this over different parameters’ values and pick the one with lowest reconstruction error. We then run the algorithm with these parameters values on the entire present data and report the (test) error with the ground truth for the missing values. 100D Swissroll We created a 3D swissroll data set with 3 000 points and lifted it to 100D with a random orthonormal mapping, and added a little noise (spherical Gaussian with stdev 0.1). We selected uniformly at random 6.76% of the entries to be present. We use the Gaussian model and SVP (ﬁxed rank = 3) as initialization for our algorithm. We typically ﬁnd that these initial X are very noisy (ﬁg. 3), with some reconstructed points lying between different branches of the manifold and causing a big reconstruction error. We ﬁxed L = 2 (the known dimensionality) for MBMS and cross-validated the other parameters: σ and k for MBMS and GBMS (both using k-nn graph), and the number of iterations τ to be used. Table 1 gives the performance of MBMS and GBMS for testing, along with their optimal parameters. Fig. 3 shows the results of different methods at a few iterations. MBMS initialized from the Gaussian model gives the most remarkable denoising effect. To show that there is a wide range of σ and number of iterations τ that give good performance with GBMS and MBMS, we ﬁx k = 50 and run the algorithm with varying σ values and plot the reconstruction error for missing entries over iterations in ﬁg. 2. Both GBMS can achieve good 5 Methods RSSE mean stdev Gaussian 168.1 2.63 1.59 + GBMS (∞, 10, 0, 1) 165.8 2.57 1.61 + MBMS (1, 20, 2, 25) 157.2 2.36 1.63 SVP 156.8 1.94 2.10 + GBMS (3, 50, 0, 1) 151.4 1.89 2.02 + MBMS (3, 50, 2, 2) 151.8 1.87 2.05 Table 1: Swissroll data set: reconstruction errors obtained by different algorithms along with their optimal parameters (σ, k, L, no. iterations τ). The three columns show the root sum of squared errors on missing entries, the mean, and the standard deviation of the pointwise reconstruction error, resp. Methods RSSE mean stdev nlPCA 7.77 26.1 42.6 SVP 6.99 21.8 39.3 + GBMS (400,140,0,1) 6.54 18.8 37.7 + MBMS (500,140,9,5) 6.03 17.0 34.9 Table 2: MNIST-7 data set: errors of the different algorithms and their optimal parameters (σ, k, L, no. iterations τ). The three columns show the root sum of squared errors on missing entries (×10−4), the mean, and the standard deviation of pixel errors, respectively. SVP + GBMS SVP + MBMS Gaussian + GBMS Gaussian + MBMS 0 1 2 3 4 5 6 7 8 910 12 14 16 18 20 150 160 170 180 iteration τ error (RSSE) 1 2 3 5 8 10 15 25 0.3 0.5 ∞ 0 1 2 3 4 5 6 7 8 910 12 14 16 18 20 150 160 170 180 iteration τ 0 1 2 3 4 5 6 7 8 910 12 14 16 18 20 150 160 170 180 iteration τ 0 1 2 3 4 5 6 7 8 910 12 14 16 18 20 150 160 170 180 iteration τ Figure 2: Reconstruction error of GBMS/MBMS over iterations (each curve is a different σ value). denoising (and reconstruction), but MBMS is more robust, with good results occurring for a wide range of iterations, indicating it is able to preserve the manifold structure better. Mocap data We use the running-motion sequence 09 01 from the CMU mocap database with 148 samples (≈1.7 cycles) with 150 sensor readings (3D positions of 50 joints on a human body). The motion is intrinsically 1D, tracing a loop in 150D. We compare nlPCA, SVP, the Gaussian model, and MBMS initialized from the ﬁrst three algorithms. For nlPCA, we do a grid search for the weight decay coefﬁcient while ﬁxing its structure to be 2 × 10 × 150 units, and use an early stopping criterion. For SVP, we do grid search on {1, 2, 3, 5, 7, 10} for the rank. For MBMS (L = 1) and GBMS (L = 0), we do grid search for σ and k. We report the reconstruction error as a function of the proportion of missing entries from 50% to 95%. For each missing-data proportion, we randomly select 5 different sets of present values and run all algorithms for them. Fig. 4 gives the mean errors of all algorithms. All methods perform well when missing-data proportion is small. nlPCA, being prone to local optima, is less stable than SVP and the Gaussian model, especially when the missing-data proportion is large. The Gaussian model gives the best and most stable initialization. At 95%, all methods fail to give an acceptable reconstruction, but up to 90% missing entries, MBMS and GBMS always beat the other algorithms. Fig. 4 shows selected reconstructions from all algorithms. MNIST digit ‘7’ The MNIST digit ‘7’ data set contains 6 265 greyscale (0–255) images of size 28 × 28. We create missing entries in a way reminiscent of run-length errors in transmission. We generate 16 to 26 rectangular boxes of an area approximately 25 pixels at random locations in each image and use them to black out pixels. In this way, we create a high dimensional data set (784 dimensions) with about 50% entries missing on average. Because of the loss of spatial correlations within the blocks, this missing data pattern is harder than random. The Gaussian model cannot handle such a big data set because it involves inverting large covariance matrices. nlPCA is also very slow and we cannot afford cross-validating its structure or the weight decay coefﬁcient, so we picked a reasonable structure (10×30×784 units), used the default weight decay parameter in the code (10−3), and allowed up to 500 iterations. We only use SVP as initialization for our algorithm. Since the intrinsic dimension of MNIST is suspected to be not very high, 6 SVP τ = 0 SVP + GBMS τ = 1 SVP + MBMS τ = 2 Gaussian τ = 0 Gaussian + GBMS τ = 1 Gaussian + MBMS τ = 25 −15 −10 −5 0 5 10 15 20 −15 −10 −5 0 5 10 15 20 −15 −10 −5 0 5 10 15 20 −15 −10 −5 0 5 10 15 20 −15 −10 −5 0 5 10 15 20 −15 −10 −5 0 5 10 15 20 −15 −10 −5 0 5 10 15 20 −15 −10 −5 0 5 10 15 20 −15 −10 −5 0 5 10 15 20 −15 −10 −5 0 5 10 15 20 −15 −10 −5 0 5 10 15 20 −15 −10 −5 0 5 10 15 20 Figure 3: Denoising effect of the different algorithms. For visualization, we project the 100D data to 3D with the projection matrix used for creating the data. Present values are reﬁlled for all plots. 50 60 70 80 85 90 95 0 1000 2000 3000 4000 5000 6000 7000 nlPCA nlPCA + GBMS nlPCA + MBMS SVP SVP + GBMS SVP + MBMS Gaussian Gaussian + GBMS Gaussian + MBMS % of missing data error frame 2 (leg distance) frame 10 (foot pose) frame 147 (leg pose) Figure 4: Left: mean of errors (RSSE) of 5 runs obtained by different algorithms for varying percentage of missing values. Errorbars shown only for Gaussian + MBMS to avoid clutter. Right: sample reconstructions when 85% percent data is missing. Row 1: initialization. Row 2: init+GBMS. Row 3: init+MBMS. Color indicates different initialization: black, original data; red, nlPCA; blue, SVP; green, Gaussian. we used rank 10 for SVP and L = 9 for MBMS. We also use the same k = 140 as in [5]. So we only had to choose σ and the number of iterations via cross-validation. Table 2 shows the methods and their corresponding error. Fig. 5 shows some representative reconstructions from different algorithms, with present values reﬁlled. The mean-shift averaging among closeby neighbors (a soft form of majority voting) helps to eliminate noise, unusual strokes and other artifacts created by SVP, which by their nature tend to occur in different image locations over the neighborhood of images. 4 Related work Matrix completion is widely studied in theoretical compressed sensing [1, 2] as well as practical recommender systems [3, 4]. Most matrix completion models rely on a low-rank assumption, and cannot fully exploit a more complex structure of the problem, such as curved manifolds. Related work is on multi-task learning in a broad sense, which extracts the common structure shared by multiple related objects and achieves simultaneous learning on them. This includes applications such as alignment of noise-corrupted images [21], recovery of images with occlusion [22], and even learning of multiple related regressors or classiﬁers [23]. Again, all these works are essentially based on a subspace assumption, and do not generalize to more complex situations. A line of work based on a nonlinear low-rank assumption (with a latent variable z of dimensionality L < D) involves setting up a least-squares error function minf,Z PN n=1 ∥xn −f(zn)∥2 = PN,D n,d=1 (xdn −fd(zn))2 where one ignores the terms for which xdn is missing, and estimates the function f and the low-dimensional data projections Z by alternating optimization. Linear functions f have been used in the homogeneity analysis literature [24], where this approach is called “missing data deleted”. Nonlinear functions f have been used recently (neural nets [20]; Gaussian processes for collaborative ﬁltering [25]). Better results are obtained if adding a projection term PN n=1 ∥zn −F(xn)∥2 and optimizing over the missing data as well [26]. 7 Orig Missing nlPCA SVP GBMS MBMS Orig Missing nlPCA SVP GBMS MBMS Figure 5: Selected reconstructions of MNIST block-occluded digits ‘7’ with different methods. Prior to our denoising-based work there have been efforts to extend the low-rank models to smooth manifolds, mostly in the context of compressed sensing. Baraniuk and Wakin [27] show that certain random measurements, e.g. random projection to a low-dimensional subspace, can preserve the metric of the manifold fairly well, if the intrinsic dimension and the curvature of the manifold are both small enough. However, these observations are not suitable for matrix completion and no algorithm is given for recovering the signal. Chen et al. [28] explicitly model a pre-determined manifold, and use this to regularize the signal when recovering the missing values. They estimate the manifold given complete data, while no complete data is assumed in our matrix completion setting. Another related work is [29], where the manifold modeled with Isomap is used in estimating the positions of satellite cameras in an iterative manner. Finally, our expectation that the value of a missing entry can be predicted from the values of neighboring points is similar to one category of collaborative ﬁltering methods that essentially use similar users/items to predict missing values [3, 4]. 5 Conclusion We have proposed a new paradigm for matrix completion, denoising, which generalizes the commonly used assumption of low rank. Assuming low-rank implies a restrictive form of denoising where the data is forced to have zero variance away from a linear manifold. More general definitions of denoising can potentially handle data that lives in a low-dimensional manifold that is nonlinear, or whose dimensionality varies (e.g. a set of manifolds), or that does not have low rank at all, and naturally they handle noise in the data. Denoising works because of the fundamental fact that a missing value can be predicted by averaging nearby present values. Although we motivate our framework from a constrained optimization point of view (denoise subject to respecting the present data), we argue for an algorithmic view of denoising-based matrix completion: apply a denoising step, reﬁll the present values, iterate until the validation error increases. In turn, this allows different forms of denoising, such as based on low-rank projection (earlier work) or local averaging with blurring mean-shift (this paper). Our nonparametric choice of mean-shift averaging further relaxes assumptions about the data and results in a simple algorithm with very few user parameters that afford user control (denoising scale, local dimensionality) but can be set automatically by cross-validation. Our algorithms are intended to be used as a postprocessing step over a user-provided initialization of the missing values, and we show they consistently improve upon existing algorithms. The MBMS-based algorithm bridges the gap between pure denoising (GBMS) and local low rank. Other deﬁnitions of denoising should be possible, for example using temporal as well as spatial neighborhoods, and even applicable to discrete data if we consider denoising as a majority voting among the neighbours of a vector (with suitable deﬁnitions of votes and neighborhood). Acknowledgments Work supported by NSF CAREER award IIS–0754089. 8 References [1] Emmanuel J. Cand`es and Benjamin Recht. Exact matrix completion via convex optimization. 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Compressive sensing on manifolds using a nonparametric mixture of factor analyzers: Algorithm and performance bounds. IEEE Trans. Signal Processing, 58(12):6140–6155, December 2010. [29] Michael B. Wakin. A manifold lifting algorithm for multi-view compressive imaging. In Proc. 27th Conference on Picture Coding Symposium (PCS’09), pages 381–384, 2009. 9	2011	78
4,428	Generalization Bounds and Consistency for Latent Structural Probit and Ramp Loss David McAllester TTI-Chicago mcallester@ttic.edu Joseph Keshet TTI-Chicago jkeshet@ttic.edu Abstract We consider latent structural versions of probit loss and ramp loss. We show that these surrogate loss functions are consistent in the strong sense that for any feature map (ﬁnite or inﬁnite dimensional) they yield predictors approaching the inﬁmum task loss achievable by any linear predictor over the given features. We also give ﬁnite sample generalization bounds (convergence rates) for these loss functions. These bounds suggest that probit loss converges more rapidly. However, ramp loss is more easily optimized on a given sample. 1 Introduction Machine learning has become a central tool in areas such as speech recognition, natural language translation, machine question answering, and visual object detection. In modern approaches to these applications systems are evaluated with quantitative performance metrics. In speech recognition one typically measures performance by the word error rate. In machine translation one typically uses the BLEU score. Recently the IBM deep question answering system was trained to optimize the Jeopardy game show score. The PASCAL visual object detection challenge is scored by average precision in recovering object bounding boxes. No metric is perfect and any metric is controversial, but quantitative metrics provide a basis for quantitative experimentation and quantitative experimentation has lead to real progress. Here we adopt the convention that a performance metric is given as a task loss — a measure of a quantity of error or cost such as the word error rate in speech recognition. We consider general methods for minimizing task loss at evaluation time. Although the goal is to minimize task loss, most systems are trained by minimizing a surrogate loss different from task loss. A surrogate loss is necessary when using scale-sensitive regularization in training a linear classiﬁer. A linear classiﬁer selects the output that maximizes an inner product of a feature vector and a weight vector. The output of a linear classiﬁer does not change when the weight vector is scaled down. But for most regularizers of interest, such as a norm of the weight vector, scaling down the weight vector drives the regularizer to zero. So directly regularizing the task loss of a linear classiﬁer is meaningless. For binary classiﬁcation standard surrogate loss functions include log loss, hinge loss, probit loss, and ramp loss. Unlike binary classiﬁcation, however, the applications mentioned above involve complex (or structured) outputs. The standard surrogate loss functions for binary classiﬁcation have generalizations to the structured output setting. Structural log loss is used in conditional random ﬁelds (CRFs) [7]. Structural hinge loss is used in structural SVMs [13, 14]. Structural probit loss is deﬁned and empirically evaluated in [6]. A version of structural ramp loss is deﬁned and empirically evaluated in [3] (but see also [12] for a treatment of the fundamental motivation for ramp loss). All four of these structural surrogate loss functions are deﬁned formally in section 2.1 1The deﬁnition of ramp loss used here is slightly different from that in [3]. 1 This paper is concerned with developing a better theoretical understanding of the relationship between surrogate loss training and task loss testing for structured labels. Structural ramp loss is justiﬁed in [3] as being a tight upper bound on task loss. But of course the tightest upper bound on task loss is the task loss itself. Here we focus on generalization bounds and consistency. A ﬁnite sample generalization bound for probit loss was stated implicitly in [9] and an explicit probit loss bound is given in [6]. Here we review the ﬁnite sample bounds for probit loss and prove a ﬁnite sample bound for ramp loss. Using these bounds we show that probit loss and ramp loss are both consistent in the sense that for any arbitrary feature map (possibly inﬁnite dimensional) optimizing these surrogate loss functions with appropriately weighted regularization approaches, in the limit of inﬁnite training data, the minimum loss achievable by a linear predictor over the given features. No convex surrogate loss function, such as log loss or hinge loss, can be consistent in this sense — for any nontrivial convex surrogate loss function one can give examples (a single feature sufﬁces) where the learned weight vector is perturbed by outliers but where the outliers do not actually inﬂuence the optimal task loss. Both probit loss and ramp loss can be optimized in practice by stochastic gradient descent. Ramp loss is simpler and easier to implement. The subgradient update for ramp loss is similar to a perceptron update — the update is a difference between a “good” feature vector and a “bad” feature vector. Ramp loss updates are closely related to updates derived from n-best lists in training machine translaiton systems [8, 2]. Ramp loss updates regularized by early stopping have been shown to be effective in phoneme alignment [10]. It is also shown in [10] that in the limit of large weight vectors the expected ramp loss update converges to the true gradient of task loss. This result suggests consistency for ramp loss, a suggestion conﬁrmed here. A practical stochastic gradient descent algorithm for structural probit loss is given in [6] where it is also shown that probit loss can be effective for phoneme recognition. Although the generalization bounds suggest that probit loss converges faster than ramp loss, ramp loss seems easier to optimize. We formulate all the notions of loss in the presence of latent structure as well as structured labels. Latent structure is information that is not given in the labeled data but is constructed by the prediction algorithm. For example, in natural language translation the alignment between the words in the source and the words in the target is not explicitly given in a translation pair. Grammatical structures are also not given in a translation pair but may be constructed as part of the translation process. In visual object detection the position of object parts is not typically annotated in the labeled data but part position estimates may be used as part of the recognition algorithm. Although the presence of latent structure makes log loss and hinge loss non-convex, latent strucure seems essential in many applications. Latent structural log loss, and the notion of a hidden CRF, is formulated in [11]. Latent structural hinge loss, and the notion of a latent structural SVM, is formulated in [15]. 2 Formal Setting and Review We consider an arbitrary input space X and a ﬁnite label space Y. We assume a source probability distribution over labeled data, i.e., a distribution over pairs (x, y), where we write Ex,y [f(x, y)] for the expectation of f(x, y). We assume a loss function L such that for any two labels y and ˆy we have that L(y, ˆy) ∈[0, 1] is the loss (or cost) when the true label is y and we predict ˆy. We will work with inﬁnite dimensional feature vectors. We let ℓ2 be the set of ﬁnite-norm inﬁnite-dimensional vectors — the set of all square-summable inﬁnite sequences of real numbers. We will be interested in linear predictors involving latent structure. We assume a ﬁnite set Z of “latent labels”. For example, we might take Z to be the set of all parse trees of source and target sentences in a machien translation system. In machine translation the label y is typically a sentence with no parse tree speciﬁed. We can recover the pure structural case, with no latent information, by taking Z to be a singleton set. It will be convenient to deﬁne S to be the set of pairs of a label and a latent label. An element s of S will be called an augmented label and we deﬁne L(y, s) by L(y, (ˆy, z)) = L(y, ˆy). We assume a feature map φ such that for an input x and augmented label s we have φ(x, s) ∈ℓ2 with \|\|φ(x, s)\|\| ≤1.2 Given an input x and a weight vector w ∈ℓ2 we deﬁne the prediction ˆsw(x) as follows. ˆsw(x) = argmax s w⊤φ(x, s) 2We note that this setting covers the ﬁnite dimensional case because the range of the feature map can be taken to be a ﬁnite dimensional subset of ℓ2 — we are not assuming a universal feature map. 2 Our goal is to use the training data to learn a weight vector w so as to minimize the expected loss on newly drawn labeled data Ex,y [L(y, ˆsw(x))]. We will assume an inﬁnite sequence of training data (x1, y1), (x2, y2), (x3, y3), . . . drawn IID from the source distribution and use the following notations. L(w, x, y) = L(y, ˆsw(x)) L(w) = Ex,y [L(w, x, y)] L∗= infw∈ℓ2 L(w) ˆLn(w) = 1 n Pn i=1 L(w, xi, yi) We adopt the convention that in the deﬁnition of L(w, x, y) we break ties in deﬁnition of ˆsw(x) in favor of augmented labels of larger loss. We will refer to this as pessimistic tie breaking. Here we deﬁne latent structural log loss, hinge loss, ramp loss and probit loss as follows. Llog(w, x, y) = ln 1 Pw(y\|x) = ln Zw(x) −ln Zw(x, y) Zw(x) = X s exp(w⊤Φ(x, s)) Zw(x, y) = X z exp(w⊤φ(x, (y, z))) Lhinge(w, x, y) = “ max s w⊤φ(x, s) + L(y, s) ” − “ max z w⊤Φ(x, (y, z)) ” Lramp(w, x, y) = “ max s w⊤φ(x, s) + L(y, s) ” − “ max s w⊤Φ(x, s) ” = “ max s w⊤φ(x, s) + L(y, s) ” −w⊤Φ(x, ˆsw(x)) Lprobit(w, x, y) = Eϵ [L(y, ˆsw+ϵ(x))] In the deﬁnition of probit loss we take ϵ to be zero-mean unit-variance isotropic Gaussian noise — for each feature dimension j we have that ϵj is an independent zero-mean unit-variance Gaussian variable.3 More generally we will write Eϵ [f(ϵ)] for the expectation of f(ϵ) where ϵ is Gaussian noise. It is interesting to note that Llog, Lhinge, and Lramp are all naturally differences of convex functions and hence can be optimized by CCCP. In the case of binary classiﬁcation we have S = Y = {−1, 1}, φ(x, y) = 1 2yφ(x), L(y, y′) = 1y̸=y′ and we deﬁne the margin m = yw⊤φ(x). We then have the following where the expression for Lprobit(w, x, y) assumes \|\|Φ(x)\|\| = 1. Llog(w, x, y) = ln (1 + e−m) Lhinge(w, x, y) = max(0, 1 −m) Lramp(w, x, y) = min(1, max(0, 1 −m)) Lprobit(w, x, y) = Pϵ∼N(0,1)[ϵ ≥m] Returning to the general case we consider the relationship between hinge and ramp loss. First we consider the case where Z is a singleton set — the case of no latent structure. In this case hinge loss is convex in w — the hinge loss becomes a maximum of linear functions. Ramp loss, however, remains a difference of nonlinear convex functions even for Z singleton. Also, in the case where Z is singleton one can easily see that hinge loss is unbounded — wrong labels may score arbitrarily better than the given label. Hinge loss remains unbounded in case of non-singleton Z. Ramp loss, on the other hand, is bounded by 1 as follows. Lramp(w, x, y) = max s w⊤Φ(x, s) + L(y, s) −w⊤Φ(x, ˆsw(x)) ≤ max s w⊤Φ(x, s) + 1 −w⊤Φ(x, ˆsw(x)) = 1 Next, as is emphasized in [3], we note that ramp loss is a tighter upper bound on task loss than is hinge loss. To see this we ﬁrst note that it is immediate that Lhinge(w, x, y) ≥Lramp(w, x, y). 3In inﬁnite dimension we have that with probability one \|\|ϵ\|\| = ∞and hence w+ϵ is not in ℓ2. The measure underling Eϵ [f(ϵ)] is a Gaussian process. However, we still have that for any unit-norm feature vector Φ the inner product ϵ⊤Φ is distributed as a zero-mean unit-norm scalar Gaussian and Lprobit(w, x, y) is therefore well deﬁned. 3 Furthermore, the following derivation shows Lramp(w, x, y) ≥L(w, x, y) where we assume pessimistic tie breaking in the deﬁnition of ˆsw(x). Lramp(w, x, y) = max s w⊤Φ(x, s) + L(y, s) −w⊤Φ(x, ˆsw(x)) ≥ w⊤Φ(x, ˆsw(x)) + L(y, ˆsw(x)) −w⊤Φ(x, ˆsw(x)) = L(y, ˆsw(x)) But perhaps the most important property of ramp loss is the following. lim α→∞Lramp(αw, x, y) = L(w, x, y) (1) This can be veriﬁed by noting that as α goes to inﬁnity the maximum of the ﬁrst term in ramp loss must occur at s = ˆsw(x). Next we note that Optimizing Lramp through subgradient descent (rather than CCCP) yields the following update rule (here we ignore regularization). ∆w ∝ φ(x, ˆsw(x)) −φ(x, ˆs+ w(x, y)) (2) ˆs+ w(x, y) = argmax s w⊤φ(x, s) + L(y, s) We will refer to (2) as the ramp loss update rule. The following is proved in [10] under mild conditions on the probability distribution over pairs (x, y). ∇wL(w) = lim α→∞αEx,y φ(x, ˆs+ αw(x, y)) −φ(x, ˆsw(x)) (3) Equation (3) expresses a relationship between the expected ramp loss update and the gradient of generalization loss. Signiﬁcant empirical success has been achieved with the ramp loss update rule using early stopping regularization [10]. But both (1) and (3) suggests that regularized ramp loss should be consistent as is conﬁrmed here. Finally it is worth noting that Lramp and Lprobit are meaningful for an arbitrary prediction space S, label space Y, and loss function L(y, s) between a label and a prediction. Log loss and hinge loss can be generalized to arbitrary prediction and label spaces provided that we assume a compatibility relation between predictions and labels. The framework of independent prediction and label spaces is explored more fully in [5] where a notion of weak-label SVM is deﬁned subsuming both ramp and hinge loss as special cases. 3 Consistency of Probit Loss We start with the consistency of probit loss which is easier to prove. We consider the following learning rule where the regularization parameter λn is some given function of n. ˆwn = argmin w ˆLn probit(w) + λn 2n \|\|w\|\|2 (4) We now prove the following fairly straightforward consequence of a generalization bound appearing in [6]. Theorem 1 (Consistency of Probit loss). For ˆwn deﬁned by (4), if the sequence λn increases without bound, and λn ln n/n converges to zero, then with probability one over the draw of the inﬁnite sample we have limn→∞Lprobit( ˆwn) = L∗. Unfortunately, and in contrast to simple binary SVMs, for a latent binary SVM (an LSVM) there exists an inﬁnite sequence w1, w2, w3, . . . such that Lprobit(wn) approaches L∗but L(wn) remains bounded away from L∗(we omit the example here). However, the learning algorithm (4) achieves consistency in the sense that the stochastic predictor deﬁned by ˆwn + ϵ where ϵ is Gaussian noise has a loss which converges to L∗. To prove theorem 1 we start by reviewing the generalization bound of [6]. The departure point for this generalization bound is the following PAC-Bayesian theorem where P and Q range over probability measures on a given space of predictors and L(Q) and ˆLn(Q) are deﬁned as expectations over selecting a predictor from Q. 4 Theorem 2 (from [1], see also [4]). For any ﬁxed prior distribution P and ﬁxed λ > 1/2 we have that with probability at least 1 −δ over the draw of the training data the following holds simultaneously for all Q. L(Q) ≤ 1 1 − 1 2λ ˆLn(Q) + λ KL(Q, P) + ln 1 δ n (5) For the space of linear predictors we take the prior P to be the zero-mean unit-variance Gaussian distribution and for w ∈ℓ2 we deﬁne the distribution Qw to be the unit-variance Gaussian centered at w. This gives the following corollary of (5). Corollary 1 (from [6]). For ﬁxed λn > 1/2 we have that with probability at least 1 −δ over the draw of the training data the following holds simultaneously for all w ∈ℓ2. Lprobit(w) ≤ 1 1 − 1 2λn ˆLn probit(w) + λn 1 2\|\|w\|\|2 + ln 1 δ n (6) To prove theorem 1 from (6) we consider an arbitrary unit-norm weight vector w∗and an arbitrary scalar α > 0. Setting δ to 1/n2, and noting that ˆwn is the minimizer of the right hand side of (6), we have the following with probability at least 1 −1/n2 over the draw of the sample. Lprobit( ˆwn) ≤ 1 1 − 1 2λn ˆLn probit(αw∗) + λn 1 2α2 + 2 ln n n (7) A standard Chernoff bound argument yields that for w∗and α > 0 selected prior to drawing the sample, we have the following with probability at least 1 −1/n2 over the choice of the sample. ˆLn probit(αw∗) ≤Lprobit(αw∗) + r ln n n (8) Combining (7) and (8) with a union bound yields that with probability at least 1 −2/n2 we have the following. Lprobit( ˆwn) ≤ 1 1 − 1 2λn Lprobit(αw∗) + r ln n n + λn 1 2α2 + 2 ln n n ! Because the probability that the above inequality is violated goes as 1/n2, with probability one over the draw of the sample we have the following. lim n→∞Lprobit( ˆwn) ≤lim n→∞ 1 1 − 1 2λn Lprobit(αw∗) + r ln n n + λn 1 2α2 + 2 ln n n ! Under the conditions on λn given in the statement of theorem 1 we then have lim n→∞Lprobit( ˆwn) ≤Lprobit(αw∗). Because this holds with probability one for any α, the following must also hold with probability one. lim n→∞Lprobit( ˆwn) ≤lim α→∞Lprobit(αw∗) (9) Now consider lim α→∞Lprobit(αw, x, y) = lim α→∞Eϵ [L(αw + ϵ, x, y)] = lim σ→0 Eϵ [L(w + σϵ, x, y)] . We have that limσ→0 Eϵ [L(w + σϵ, x, y)] is determined by the augmented labels s that are tied for the maximum value of w⊤Φ(x, s). There is some probability distribution over these tied values that occurs in the limit of small σ. Under the pessimistic tie breaking in the deﬁnition of L(w, x, y) we then get limα→∞Lprobit(αw, x, y) ≤L(w, x, y). This in turn gives the following. lim α→∞Lprobit(αw) = Ex,y h lim α→∞Lprobit(αw, x, y) i ≤Ex,y [L(w, x, y)] = L(w) (10) Combining (9) and (10) yields limn→∞Lprobit( ˆwn) ≤L(w∗). Since for any w∗this holds with probability one, with probability one we also have limn→∞Lprobit( ˆwn) ≤L∗. Finally we note Lprobit(w) = Eϵ [L(w + ϵ)] ≥L∗which then gives theorem 1. 5 4 Consistency of Ramp Loss Now we consider the following ramp loss training equation. ˆwn = argmin w ˆLn ramp(w) + γn 2n \|\|w\|\|2 (11) The main result of this paper is the following. Theorem 3 (Consistency of Ramp Loss). For ˆwn deﬁned by (11), if the sequence γn/ ln2 n increases without bound, and the sequence γn/(n ln n) converges to zero, then with probability one over the draw of the inﬁnite sample we have limn→∞Lprobit((ln n) ˆwn) = L∗. As with theorem 1, theorem 3 is derived from a ﬁnite sample generalization bound. The bound is derived from (6) by upper bounding ˆLn probit(w/σ) in terms of ˆLn ramp(w). From section 3 we have that limσ→0 Lprobit(w/σ, x, y) ≤L(w, x, y) ≤Lramp(w, x, y). This can be converted to the following lemma for ﬁnite σ where we recall that S is the set of augmented labels s = (y, z). Lemma 1. Lprobit w σ , x, y ≤Lramp(w, x, y) + σ + σ r 8 ln \|S\| σ Proof. We ﬁrst prove that for any σ > 0 we have Lprobit w σ , x, y ≤σ + max s: m(s)≤M L(y, s) (12) where m(s) = w⊤∆φ(s) ∆φ(s) = φ(x, ˆsw(x)) −φ(x, s) M = σ r 8 ln \|S\| σ . To prove (12) we note that for m(s) > M we have the following where Pϵ[Φ(ϵ)] abbreviates Eϵ 1Φ(ϵ) . Pϵ[ˆsw+σϵ(x) = s] ≤ Pϵ[(w + σϵ)⊤∆φ(s) ≤0] = Pϵ −ϵ⊤∆φ(s) ≥m(s)/σ ≤ Pϵ∼N(0,1) ϵ ≥M 2σ ≤exp −M 2 8σ2 = σ \|S\| Eϵ [L(y, ˆsw+σϵ(x))] ≤ Pϵ [∃s : m(s) > M ˆsw+ϵσ(x) = s] + max s:m(s)≤M L(y, s) ≤ σ + max s:m(s)≤M L(y, s) The following calculation shows that (12) implies the lemma. Lprobit w σ , x, y ≤ σ + max s: m(s)≤M L(y, s) ≤ σ + max s: m(s)≤M L(y, s) −m(s) + M ≤ σ + max s L(y, s) −m(s) + M = σ + Lramp(w, x, y) + M Inserting lemma 1 into (6) we get the following. Theorem 4. For λn > 1/2 we have that with probability at least 1−δ over the draw of the training data the following holds simultaneously for all w and σ > 0. Lprobit w σ ≤ 1 1 − 1 2λn ˆLn ramp(w) + σ + σ r 8 ln \|S\| σ + λn \|\|w\|\|2 2σ2 + ln 1 δ n !! (13) 6 To prove theorem 3 we now take σn = 1/ ln n and λn = γn/ ln2 n. We then have that ˆwn is the minimizer of the right hand side of (13). This observation yields the following for any unit-norm vector w∗and scalar α > 0 where we have set δ = 1/n2. Lprobit((ln n) ˆwn) ≤ 1 1 −ln2 n 2γn ˆLramp(αw∗) + 1 + p 8 ln(\|S\| ln n) ln n + γnα2 2n + 2γn n ln n ! (14) As in section 3, we use a Chernoff bound for the single vector w∗and scalar α to bound ˆLramp(αw∗) in terms of Lramp(αw∗) and then take the limit as n →∞to get the following with probability one. lim n→∞Lprobit((ln n) ˆwn) ≤Lramp(αw∗) The remainder of the proof is the same in section 3 but where we now use limα→∞Lramp(αw∗) = L(w∗) whose proof we omit. 5 A Comparison of Convergence Rates To compare the convergence rates implicit in (6) and (13) we note that in (13) we can optimize σ as a function of other quantities in the bound.4 An approximately optimal value for σ is λn\|\|w\|\|2/n 1/3 which gives the following. Lprobit w σ ≤ 1 1 − 1 2λn ˆLn ramp(w) + λn\|\|w\|\|2 n 1/3 3 2 + r 8 ln \|S\| σ ! + λn ln 1 δ n ! (15) We have that (15) gives ˜O \|\| ˆwn\|\|2/n 1/3 as opposed to (6) which gives O \|\| ˆwn\|\|2/n . This suggests that while probit loss and ramp loss are both consistent, ramp loss may converge more slowly. 6 Discussion and Open Problems The contributions of this paper are a consistency theorem for latent structural probit loss and both a generalization bound and a consistency theorem for latent structural ramp loss. These bounds suggest that probit loss converges more rapidly. However, we have only proved upper bounds on generalization loss and it remains possible that these upper bounds, while sufﬁcient to show consistency, are not accurate characterizations of the actual generalization loss. Finding more deﬁnitive statements, such as matching lower bounds, remains an open problem. The deﬁnition of ramp loss used here is not the only one possible. In particular we can consider the following variant. L′ ramp(w, x, y) = max s w⊤Φ(x, s) − max s w⊤φ(x, s) −L(y, s) Relations (1) and (3) both hold for L′ ramp as well as Lramp. Experiments indicate that L′ ramp performs somewhat better than Lramp under early stopping of subgradient descent. However it seems that it is not possible to prove a bound of the form of (15) for L′ ramp. A frustrating observation is that L′ ramp(0, x, y) = 0. Finding a meaningful ﬁnite-sample statement for L′ ramp remains an open problem. The isotropic Gaussian noise distribution used in the deﬁnition of Lprobit is not optimal. A uniformly tighter upper bound on generalization loss is achieved by optimizing the posterior in the PAC-Bayesian theorem. Finding a practical more optimal use of the PAC-Bayesian theorem also remains an open problem. 4In the consistency proof it was more convenient to set σ = 1/ln n which is plausibly nearly optimal anyway. 7 References [1] Olivier Catoni. PAC-Bayesian Supervised Classiﬁcation: The Thermodynamics of Statistical Learning. Institute of Mathematical Statistics LECTURE NOTES MONOGRAPH SERIES, 2007. [2] D. Chiang, K. Knight, and W. 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An end-to-end discriminative approach to machine translation. In International Conference on Computational Linguistics and Association for Computational Linguistics (COLING/ACL), 2006. [9] David McAllester. Generalization bounds and consistency for structured labeling. In G. Bakir nd T. Hofmann, B. Scholkopf, A. Smola, B. Taskar, and S. V. N. Vishwanathan, editors, Predicting Structured Data. MIT Press, 2007. [10] David A. McAllester, Tamir Hazan, and Joseph Keshet. Direct loss minimization for structured prediction. In Advances in Neural Information Processing Systems 24, 2010. [11] A. Quattoni, S. Wang, L.P. Morency, M Collins, and T Darrell. Hidden conditional random ﬁelds. PAMI, 29, 2007. [12] R.Collobert, F.H.Sinz, J.Weston, and L.Bottou. Trading convexity for scalability. In ICML, 2006. [13] B. Taskar, C. Guestrin, and D. Koller. Max-margin markov networks. In Advances in Neural Information Processing Systems 17, 2003. [14] I. Tsochantaridis, T. Hofmann, T. Joachims, and Y. Altun. Support vector machine learning for interdependent and structured output spaces. In Proceedings of the Twenty-First International Conference on Machine Learning, 2004. [15] Chun-Nam John Yu and T. Joachims. Learning structural svms with latent variables. In International Conference on Machine Learning (ICML), 2009. 8	2011	79
4,429	k-NN Regression Adapts to Local Intrinsic Dimension Samory Kpotufe Max Planck Institute for Intelligent Systems samory@tuebingen.mpg.de Abstract Many nonparametric regressors were recently shown to converge at rates that depend only on the intrinsic dimension of data. These regressors thus escape the curse of dimension when high-dimensional data has low intrinsic dimension (e.g. a manifold). We show that k-NN regression is also adaptive to intrinsic dimension. In particular our rates are local to a query x and depend only on the way masses of balls centered at x vary with radius. Furthermore, we show a simple way to choose k = k(x) locally at any x so as to nearly achieve the minimax rate at x in terms of the unknown intrinsic dimension in the vicinity of x. We also establish that the minimax rate does not depend on a particular choice of metric space or distribution, but rather that this minimax rate holds for any metric space and doubling measure. 1 Introduction We derive new rates of convergence in terms of dimension for the popular approach of Nearest Neighbor (k-NN) regression. Our motivation is that, for good performance, k-NN regression can require a number of samples exponential in the dimension of the input space X. This is the so-called “curse of dimension”. Formally stated, the curse of dimension is the fact that, for any nonparametric regressor there exists a distribution in RD such that, given a training size n, the regressor converges at a rate no better than n−1/O(D) (see e.g. [1, 2]). Fortunately it often occurs that high-dimensional data has low intrinsic dimension: typical examples are data lying near low-dimensional manifolds [3, 4, 5]. We would hope that in these cases nonparametric regressors can escape the curse of dimension, i.e. their performance should only depend on the intrinsic dimension of the data, appropriately formalized. In other words, if the data in RD has intrinsic dimension d << D, we would hope for a better convergence rate of the form n−1/O(d) instead of n−1/O(D). This has recently been shown to indeed be the case for methods such as kernel regression [6], tree-based regression [7] and variants of these methods [8]. In the case of k-NN regression however, it is only known that 1-NN regression (where k = 1) converges at a rate that depends on intrinsic dimension [9]. Unfortunately 1-NN regression is not consistent. For consistency, it is well known that we need k to grow as a function of the sample size n [10] . Our contributions are the following. We assume throughout that the target function f is Lipschitz. First we show that, for a wide range of values of k ensuring consistency, k-NN regression converges at a rate that only depends on the intrinsic dimension in a neighborhood of a query x. Our local notion of dimension in a neighborhood of a point x relies on the well-studied notion of doubling measure (see Section 2.3). In particular our dimension quantiﬁes how the mass of balls vary with radius, and this captures standard examples of data with low intrinsic dimension. Our second, and perhaps most important contribution, is a simple procedure for choosing k = k(x) so as to nearly achieve the minimax rate of O n−2/(2+d) in terms of the unknown dimension d in a neighborhood of x. Our ﬁnal contribution is in showing that this minimax rate holds for any metric space and doubling measure. In other words the hardness of the regression problem is not tied to a particular 1 choice of metric space X or doubling measure µ, but depends only on how the doubling measure µ expands on a metric space X. Thus, for any marginal µ on X with expansion constant Θ 2d , the minimax rate for the measure space (X, µ) is Ω n−2/(2+d) . 1.1 Discussion It is desirable to express regression rates in terms of a local notion of dimension rather than a global one because the complexity of data can vary considerably over regions of space. Consider for example a dataset made up of a collection of manifolds of various dimensions. The global complexity is necessarily of a worst case nature, i.e. is affected by the most complex regions of the space while we might happen to query x from a less complex region. Worse, it can be the case that the data is not complex locally anywhere, but globally the data is more complex. A simple example of this is a so-called space ﬁlling curve where a low-dimensional manifold curves enough that globally it seems to ﬁll up space. We will see that the global complexity does not affect the behavior of k-NN regression, provided k/n is sufﬁciently small. The behavior of k-NN regression is rather controlled by the often smaller local dimension in a neighborhood B(x, r) of x, where the neighborhood size r shrinks with k/n. Given such a neighborhood B(x, r) of x, how does one choose k = k(x) optimally relative to the unknown local dimension in B(x, r)? This is nontrivial as standard methods of (global) parameter selection do not easily apply. For instance, it is unclear how to choose k by cross-validation over possible settings: we do not know reliable surrogates for the true errors at x of the various estimators {fn,k(x)}k∈[n]. Another possibility is to estimate the dimension of the data in the vicinity of x, and use this estimate to set k. However, for optimal rates, we have to estimate the dimension exactly and we know of no ﬁnite sample result that guarantees the exact estimate of intrinsic dimension. Our method consists of ﬁnding a value of k that balances quantities which control estimator variance and bias at x, namely 1/k and distances to x’s k nearest neighbors. The method guarantees, uniformly over all x ∈X, a near optimal rate of eO n−2/(2+d) where d = d(x) is exactly the unknown local dimension on a neighborhood B(x, r) of x, where r →0 as n →∞. 2 Setup We are given n i.i.d samples (X, Y) = {(Xi, Yi)}n i=1 from some unknown distribution where the input variable X belongs to a metric space (X, ρ), and the output Y is a real number. We assume that the class B of balls on (X, ρ) has ﬁnite VC dimension VB. This is true for instance for any subset X of a Euclidean space, e.g. the low-dimensional spaces discussed in Section 2.3. The VC assumption is however irrelevant to the minimax result of Theorem 3. We denote the marginal distribution on X by µ and the empirical distribution on X by µn. 2.1 Regression function and noise The regression function f(x) = E [Y \|X = x] is assumed to be λ-Lipschitz, i.e. there exists λ > 0 such that ∀x, x′ ∈X, \|f(x) −f(x′)\| ≤λρ (x, x′). We assume a simple but general noise model: the distributions of the noise at points x ∈X have uniformly bounded tails and variance. In particular, Y is allowed to be unbounded. Formally: ∀δ > 0 there exists t > 0 such that sup x∈X PY \|X=x (\|Y −f(x)\| > t) ≤δ. We denote by tY (δ) the inﬁmum over all such t. Also, we assume that the variance of (Y \|X = x) is upper-bounded by a constant σ2 Y uniformly over all x ∈X. To illustrate our noise assumptions, consider for instance the standard assumption of bounded noise, i.e. \|Y −f(x)\| is uniformly bounded by some M > 0; then ∀δ > 0, tY (δ) ≤M, and can thus be replaced by M in all our results. Another standard assumption is that where the noise distribution has exponentially decreasing tail; in this case ∀δ > 0, tY (δ) ≤O(ln 1/δ). As a last example, in the case of Gaussian (or sub-Gaussian) noise, it’s not hard to see that ∀δ > 0, tY (δ) ≤O( p ln 1/δ). 2 2.2 Weighted k-NN regression estimate We assume a kernel function K : R+ 7→R+, non-increasing, such that K(1) > 0, and K(ρ) = 0 for ρ > 1. For x ∈X, let rk,n(x) denote the distance to its k’th nearest neighbor in the sample X. The regression estimate at x given the n-sample (X, Y) is then deﬁned as fn,k(x) = X i K (ρ(x, xi)/rk,n(x)) P j K (ρ(x, xj)/rk,n(x))Yi = X i wi,k(x)Yi. 2.3 Notion of dimension We start with the following deﬁnition of doubling measure which will lead to the notion of local dimension used in this work. We stay informal in developing the motivation and refer the reader to [?, 11, 12] for thorough overviews of the topic of metric space dimension and doubling measures. Deﬁnition 1. The marginal µ is a doubling measure if there exist Cdb > 0 such that for any x ∈X and r ≥0, we have µ(B(x, r)) ≤Cdbµ(B(x, r/2)). The quantity Cdb is called an expansion constant of µ. An equivalent deﬁnition is that, µ is doubling if there exist C and d such that for any x ∈X, for any r ≥0 and any 0 < ϵ < 1, we have µ(B(x, r)) ≤Cϵ−dµ(B(x, ϵr)). Here d acts as a dimension. It is not hard to show that d can be chosen as log2 Cdb and C as Cdb (see e.g. [?]). A simple example of a doubling measure is the Lebesgue volume in the Euclidean space Rd. For any x ∈Rd and r > 0, vol (B(x, r)) = vol (B(x, 1)) rd. Thus vol (B(x, r)) / vol (B(x, ϵr)) = ϵ−d for any x ∈Rd, r > 0 and 0 < ϵ < 1. Building upon the doubling behavior of volumes in Rd, we can construct various examples of doubling probability measures. The following ingredients are sufﬁcient. Let X ⊂RD be a subset of a d-dimensional hyperplane, and let X satisfy for all balls B(x, r) with x ∈X, vol (B(x, r) ∩X) = Θ(rd), the volume being with respect to the containing hyperplane. Now let µ be approximately uniform, that is µ satisﬁes for all such balls B(x, r), µ(B(x, r) ∩X) = Θ(vol (B(x, r) ∩X)). We then have µ(B(x, r))/µ(B(x, ϵr)) = Θ(ϵ−d). Unfortunately a global notion of dimension such as the above deﬁnition of d is rather restrictive as it requires the same complexity globally and locally. However a data space can be complex globally and have small complexity locally. Consider for instance a d-dimensional submanifold X of RD, and let µ have an upper and lower bounded density on X. The manifold might be globally complex but the restriction of µ to a ball B(x, r), x ∈X, is doubling with local dimension d, provided r is sufﬁciently small and certain conditions on curvature hold. This is because, under such conditions (see e.g. the Bishop-Gromov theorem [13]), the volume (in X) of B(x, r) ∩X is Θ(rd). The above example motivates the following deﬁnition of local dimension d. Deﬁnition 2. Fix x ∈X, and r > 0. Let C ≥1 and d ≥1. The marginal µ is (C, d)-homogeneous on B(x, r) if we have µ(B(x, r′)) ≤Cϵ−dµ(B(x, ϵr′)) for all r′ ≤r and 0 < ϵ < 1. The above deﬁnition covers cases other than manifolds. In particular, another space with small local dimension is a sparse data space X ⊂RD where each vector x has at most d non-zero coordinates, i.e. X is a collection of ﬁnitely many hyperplanes of dimension at most d. More generally suppose the data distribution µ is a mixture P i πiµi of ﬁnitely many distributions µi with potentially different low-dimensional supports. Then if all µi supported on a ball B are (Ci, d)-homogeneous on B, i.e. all have local dimension d on B, then µ is also (C, d)-homogeneous on B for some C. We want rates of convergence which hold uniformly over all regions where µ is doubling. We therefore also require (Deﬁnition 3) that C and d from Deﬁnition 2 are uniformly upper bounded. This will be the case in many situations including the above examples. Deﬁnition 3. The marginal µ is (C0, d0)-maximally-homogeneous for some C0 ≥1 and d0 ≥1, if the following holds for all x ∈X and r > 0: suppose there exists C ≥1 and d ≥1 such that µ is (C, d)-homogeneous on B(x, r), then µ is (C0, d0)-homogeneous on B(x, r). We note that, rather than assuming as in Deﬁnition 3 that all local dimensions are at most d0, we can express our results in terms of the subset of X where local dimensions are at most d0. In this case d0 would be allowed to grow with n. The less general assumption of Deﬁnition 3 allows for a clearer presentation which still captures the local behavior of k-NN regression. 3 3 Overview of results 3.1 Local rates for ﬁxed k The ﬁrst result below establishes the rates of convergence for any k ≳ln n in terms of the (unknown) complexity on B(x, r) where r is any r satisfying µ(B(x, r)) > Ω(k/n) (we need at least Ω(k) samples in the relevant neighborhoods of x). Theorem 1. Suppose µ is (C0, d0)-maximally-homogeneous, and B has ﬁnite VC dimension VB. Let 0 < δ < 1. With probability at least 1 −2δ over the choice of (X, Y), the following holds simultaneously for all x ∈X and k satisfying n > k ≥VB ln 2n + ln(8/δ). Pick any x ∈X. Let r > 0 satisfy µ(B(x, r)) > 3C0k/n. Suppose µ is (C, d)-homogeneous on B(x, r), with 1 ≤C ≤C0 and 1 ≤d ≤d0. We have \|fn,k(x) −f(x)\|2 ≤2K(0) K(1) · VB · t2 Y (δ/2n) · ln(2n/δ) + σ2 Y k + 2λ2r2 3Ck nµ(B(x, r)) 2/d . Note that the above rates hold uniformly over x, k ≳ln n, and any r where µ(B(x, r)) ≥Ω(k/n). The rate also depends on µ(B(x, r)) and suggests that the best scenario is that where x has a small neighborhood of large mass and small dimension d. 3.2 Minimax rates for a doubling measure Our next result shows that the hardness of the regression problem is not tied to a particular choice of the metric X or the doubling measure µ. The result relies mainly on the fact that µ is doubling on X. We however assume that µ has the same expansion constant everywhere and that this constant is tight. This does not however make the lower-bound less expressive, as it still tells us which rates to expect locally. Thus if µ is (C, d)-homogeneous near x, we cannot expect a better rate than O n−2/(2+d) (assuming a Lipschitz regression function f). Theorem 2. Let µ be a doubling measure on a metric space (X, ρ) of diameter 1, and suppose µ satisﬁes, for all x ∈X, for all r > 0 and 0 < ϵ < 1, C1ϵ−dµ(B(x, ϵr)) ≤µ(B(x, r)) ≤C2ϵ−dµ(B(x, ϵr)), where C1, C2 and d are positive constants independent of x, r, and ϵ. Let Y be a subset of R and let λ > 0. Deﬁne Dµ,λ as the class of distributions on X × Y such that X ∼µ and the output Y = f(X)+N(0, 1) where f is any λ-Lipschitz function from X to Y. Fix a sample size n > 0 and let fn denote any regressor on samples (X, Y) of size n, i.e. fn maps any such sample to a function fn\|(X,Y)(·) : X 7→Y in L2(µ). There exists a constant C independent of n and λ such that inf {fn} sup Dµ,λ E X,Y,x fn\|(X,Y)(x) −f(x) 2 λ2d/(2+d)n−2/(2+d) ≥C. 3.3 Choosing k for near-optimal rates at x Our last result shows a practical and simple way to choose k locally so as to nearly achieve the minimax rate at x, i.e. a rate that depends on the unknown local dimension in a neighborhood B(x, r) of x, where again, r satisﬁes µ(B(x, r)) > Ω(k/n) for good choices of k. It turns out that we just need µ(B(x, r)) > Ω(n−1/3). As we will see, the choice of k simply consists of monitoring the distances from x to its nearest neighbors. The intuition, similar to that of a method for tree-pruning in [7], is to look for a k that balances the variance (roughly 1/k) and the square bias (roughly r2 k,n(x)) of the estimate. The procedure is as follows: Choosing k at x: Pick ∆≥maxi ρ (x, Xi), and pick θn,δ ≥ln n/δ. Let k1 be the highest integer in [n] such that ∆2 · θn,δ/k1 ≥r2 k1,n(x). Deﬁne k2 = k1 + 1 and choose k as arg minki,i∈[2] θn,δ/ki + r2 ki,n(x) . 4 The parameter θn,δ guesses how the noise in Y affects the risk. This will soon be clearer. Performance guarantees for the above procedure are given in the following theorem. Theorem 3. Suppose µ is (C0, d0)-maximally-homogeneous, and B has ﬁnite VC dimension VB. Assume k is chosen for each x ∈X using the above procedure, and let fn,k(x) be the corresponding estimate. Let 0 < δ < 1 and suppose n4/(6+3d0) > (VB ln 2n + ln(8/δ)) /θn,δ. With probability at least 1 −2δ over the choice of (X, Y), the following holds simultaneously for all x ∈X. Pick any x ∈X. Let 0 < r < ∆satisfy µ(B(x, r)) > 6C0n−1/3. Suppose µ is (C, d)-homogeneous on B(x, r), with 1 ≤C ≤C0 and 1 ≤d ≤d0. We have \|fn,k(x) −f(x)\|2 ≤ 2Cn,δ θn,δ + 2λ2 1 + 4∆2 3Cθn,δ nµ(B(x, r)) 2/(2+d) , where Cn,δ = VB · t2 Y (δ/2n) · ln(2n/δ) + σ2 Y K(0)/K(1). Suppose we set θn,δ = ln2 n/δ. Then, as per the discussion in Section 2.1, if the noise in Y is Gaussian, we have t2 Y (δ/2n) = O(ln n/δ), and therefore the factor Cn,δ/θn,δ = O(1). Thus ideally we want to set θn,δ to the order of (t2 Y (δ/2n) · ln n/δ). Just as in Theorem 1, the rates of Theorem 3 hold uniformly for all x ∈X, and all 0 < r < ∆ where µ(B(x, r)) > Ω(n−1/3). For any such r, let us call B(x, r) an admissible neighborhood. It is clear that, as n grows to inﬁnity, w.h.p. any neighborhood B(x, r) of x, 0 < r < supx′∈X ρ (x, x′), becomes admissible. Once a neighborhood B(x, r) is admissible for some n, our procedure nearly attains the minimax rates in terms of the local dimension on B(x, r), provided µ is doubling on B(x, r). Again, the mass of an admissible neighborhood affects the rate, and the bound in Theorem 3 is best for an admissible neighborhood with large mass µ(B(x, r)) and small dimension d. 4 Analysis Deﬁne efn,k(x) = EY\|X fn,k(x) = P i wi,k(x)f(Xi). We will bound the error of the estimate at a point x in a standard way as \|fn,k(x) −f(x)\|2 ≤2 fn,k(x) −efn,k(x) 2 + 2 efn,k(x) −f(x) 2 . (1) Theorem 1 is therefore obtained by combining bounds on the above two r.h.s terms (variance and bias). These terms are bounded separately in Lemma 2 and Lemma 3 below. 4.1 Local rates for ﬁxed k: bias and variance at x In this section we bound the bias and variance terms of equation (1) with high probability, uniformly over x ∈X. We will need the following lemma which follows easily from standard VC theory [14] results. The proof is given in the long version [15]. Lemma 1. Let B denote the class of balls on X, with VC-dimension VB. Let 0 < δ < 1, and deﬁne αn = (VB ln 2n + ln(8/δ)) /n. The following holds with probability at least 1 −δ for all balls in B. Pick any a ≥αn. Then µ(B) ≥3a =⇒µn(B) ≥a and µn(B) ≥3a =⇒µ(B) ≥a. We start with the bias which is simpler to handle: it is easy to show that the bias of the estimate at x depends on the radius rk,n(x). This radius can then be bounded, ﬁrst in expectation using the doubling assumption on µ, then by calling on the above lemma to relate this expected bound to rk,n(x) with high probability. Lemma 2 (Bias). Suppose µ is (C0, d0)-maximally-homogeneous. Let 0 < δ < 1. With probability at least 1−δ over the choice of X, the following holds simultaneously for all x ∈X and k satisfying n > k ≥VB ln 2n + ln(8/δ). Pick any x ∈X. Let r > 0 satisfy µ(B(x, r)) > 3C0k/n. Suppose µ is (C, d)-homogeneous on B(x, r), with 1 ≤C ≤C0 and 1 ≤d ≤d0. We have: efn,k(x) −f(x) 2 ≤λ2r2 3Ck nµ(B(x, r)) 2/d . 5 Proof. First ﬁx X, x ∈X and k ∈[n]. We have: efn,k(x) −f(x) = X i wi,k(x) (f(Xi) −f(x)) ≤ X i wi,k(x) \|f(Xi) −f(x)\| ≤ X i wi,k(x)λρ (Xi, x) ≤λrk,n(x). (2) We therefore just need to bound rk,n(x). We proceed as follows. Fix x ∈X and k and pick any r > 0 such that µ(B(x, r)) > 3C0k/n. Suppose µ is (C, d)homogeneous on B(x, r), with 1 ≤C ≤C0 and 1 ≤d ≤d0. Deﬁne ϵ .= 3Ck nµ(B(x, r)) 1/d , so that ϵ < 1 by the bound on µ(B(x, r)); then by the local doubling assumption on B(x, r), we have µ(B(x, ϵr)) ≥C−1ϵdµ(B(x, r)) ≥3k/n. Let αn as deﬁned in Lemma 1, and assume k/n ≥αn (this is exactly the assumption on k in the lemma statement). By Lemma 1, it follows that with probability at least 1 −δ uniform over x, r and k thus chosen, we have µn((B(x, ϵr)) ≥k/n implying that rk,n(x) ≤ϵr. We then conclude with the lemma statement by using equation (2). Lemma 3 (Variance). Let 0 < δ < 1. With probability at least 1 −2δ over the choice of (X, Y), the following then holds simultaneously for all x ∈X and k ∈[n]: fn,k(x) −efn,k(x) 2 ≤K(0) K(1) · VB · t2 Y (δ/2n) · ln(2n/δ) + σ2 Y k . Proof. First, condition on X ﬁxed. For any x ∈X, k ∈[k], let Yx,k denote the subset of Y corresponding to points from X falling in B(x, rk,n(x)). For X ﬁxed, the number of such subsets Yx,k is therefore at most the number of ways we can intersect balls in B with the sample X; this is in turn upper-bounded by nVB as is well-known in VC theory. Let ψ(Yx,k) .= fn,k(x) −efn,k(x) . We’ll proceed by showing that with high probability, for all x ∈X, ψ(Yx,k) is close to its expectation, then we bound this expectation. Let δ0 ≤1/2n. We further condition on the event Yδ0 that for all n samples Yi, \|Yi −f(Xi)\| ≤ tY (δ0). By deﬁnition of tY (δ0), the event Yδ0 happens with probability at least 1 −nδ0 ≥1/2 . It follows that for any x ∈X E ψ(Yx,k) ≥P (Yδ0) · E Yδ0 ψ(Yx,k) ≥1 2 E Yδ0 ψ(Yx,k), where EYδ0 [·] denote conditional expectation under the event. Let ϵ > 0, we in turn have P (∃x, k, ψ(Yx,k) > 2E ψ(Yx,k) + ϵ) ≤P ∃x, k, ψ(Yx,k) > E Yδ0 ψ(Yx,k) + ϵ ≤PYδ0 ∃x, k, ψ(Yx,k) > E Yδ0 ψ(Yx,k) + ϵ + nδ0. This last probability can be bounded by applying McDiarmid’s inequality: changing any Yi value changes ψ(Yx,k) by at most wi,k · tY (δ0) when we condition on the event Yδ0. This, followed by a union-bound yields PYδ0 ∃x, k, ψ(Yx,k) > E Yδ0 ψ(Yx,k) + ϵ ≤nVB exp ( −2ϵ2/t2 Y (δ0) X i w2 i,k ) . 6 Combining with the above we get P (∃x ∈X, ψ(Yx,k) > 2E ψ(Yx,k) + ϵ) ≤nVB exp ( −2ϵ2/t2 Y (δ0) X i w2 i,k ) + nδ0. In other words, let δ0 = δ/2n, with probability at least 1 −δ, for all x ∈X and k ∈[n] fn,k(x) −efn,k(x) 2 ≤8 E Y\|X fn,k(x) −efn,k(x) 2 + t2 Y (δ/2n) VB ln(2n/δ) X i w2 i,k ! ≤8 E Y\|X fn,k(x) −efn,k(x) 2 + t2 Y (δ/2n) VB ln(2n/δ) X i w2 i,k ! , where the second inequality is an application of Jensen’s. We bound the above expectation on the r.h.s. next. In what follows (second equality below) we use the fact that for i.i.d random variables zi with zero mean, E \|P i zi\|2 = P i E \|zi\|2. We have E Y\|X fn,k(x) −efn,k(x) 2 = E Y\|X X i wi,k(x) (Yi −f(Xi)) 2 = X i w2 i,k(x) E Y\|X \|Yi −f(Xi)\|2 ≤ X i w2 i,k(x)σ2 Y . Combining with the previous bound we get that, with probability at least 1 −δ, for all x and k, fn,k(x) −efn,k(x) 2 ≤ VB · t2 Y (δ/2n) · ln(2n/δ) + σ2 Y · X i w2 i,k(x). (3) We can now bound P i w2 i,k(x) as follows: X i w2 i,k(x) ≤max i∈[n] wi,k(x) = max i∈[n] K (ρ(x, xi)/rk,n(x)) P j K (ρ(x, xj)/rk,n(x)) ≤ K(0) P j K (ρ(x, xj)/rk,n(x)) ≤ K(0) P xj∈B(x,rk,n(x)) K (ρ(x, xj)/rk,n(x)) ≤K(0) K(1)k . Plug this back into equation 3 and conclude. 4.2 Minimax rates for a doubling measure The minimax rates of theorem 2 (proved in the long version [15]) are obtained as is commonly done by constructing a regression problem that reduces to the problem of binary classiﬁcation (see e.g. [1, 2, 10]). Intuitively the problem of classiﬁcation is hard in those instances where labels (say −1, +1) vary wildly over the space X, i.e. close points can have different labels. We make the regression problem similarly hard. We will consider a class of candidate regression functions such that each function f alternates between positive and negative in neighboring regions (f is depicted as the dashed line below). + − The reduction relies on the simple observation that for a regressor fn to approximate the right f from data it needs to at least identify the sign of f in the various regions of space. The more we can make each such f change between positive and negative, the harder the problem. We are however constrained in how much f changes since we also have to ensure that each f is Lipchitz continuous. 7 4.3 Choosing k for near-optimal rates at x Proof of Theorem 3. Fix x and let r, d, C as deﬁned in the theorem statement. Deﬁne κ .= θd/(2+d) n,δ · nµ(B(x, r)) 3C 2/(2+d) and ϵ .= 3Cκ nµ(B(x, r)) 1/d . Note that, by our assumptions, µ(B(x, r)) > 6Cθn,δn−1/3 ≥6Cθn,δn−d/(2+d) = 6Cθn,δ n2/(2+d) n ≥6C κ n. (4) The above equation (4) implies ϵ < 1. Thus, by the homogeneity assumption on B(x, r), µ(B(x, ϵr)) ≥C−1ϵdµ(B(x, r)) ≥3κ/n. Now by the ﬁrst inequality of (4) we also have κ n ≥θn,δ n n4/(6+3d) ≥θn,δ n n4/(6+3d0) ≥αn, where αn = (VB ln 2n + ln(8/δ)) /n is as deﬁned in Lemma 1. We can thus apply Lemma 1 to have that, with probability at least 1 −δ, µn(B(x, ϵr)) ≥κ/n. In other words, for any k ≤κ, rk,n(x) ≤ϵr. It follows that if k ≤κ, ∆2 · θn,δ k ≥∆2 · θn,δ κ = ∆2 3Cκ nµ(B(x, r)) 2/d ≥(ϵr)2 ≥r2 k,n(x). Remember that the above inequality is exactly the condition on the choice of k1 in the theorem statement. Therefore, suppose k1 ≤κ, it must be that k2 > κ otherwise k2 is the highest integer satisfying the condition, contradicting our choice of k1. Thus we have (i) θn,δ/k2 < θn,δ/κ = ϵ2. We also have (ii) rk2,n(x) ≤21/dϵr. To see this, notice that since k1 ≤κ < k2 = k1 + 1 we have k2 ≤2κ; now by repeating the sort of argument above, we have µ(B(x, 21/dϵr)) ≥6κ/n which by Lemma 1 implies that µn(B(x, 21/dϵr)) ≥2κ/n ≥k2/n. Now suppose instead that k1 > κ, then by deﬁnition of k1, we have (iii) rk1,n(x)2 ≤∆2 · θn,δ k1 ≤∆2 · θn,δ κ = (∆ϵ)2. The following holds by (i), (ii), and (iii). Let k be chosen as in the theorem statement. Then, whether k1 > κ or not, it is true that θn,δ k + r2 k,n(x) ≤ 1 + 4∆2 ϵ2 = 1 + 4∆2 3Cθn,δ nµ(B(x, r)) 2/(2+d) . Now combine Lemma 3 with equation (2) and we have that with probability at least 1−2δ (accounting for all events discussed) \|fn,k(x) −f(x)\|2 ≤2Cn,δ θn,δ θn,δ k + 2λ2r2 k,n(x) ≤ 2Cn,δ θn,δ + 2λ2 θn,δ k + r2 k,n(x) ≤ 2Cn,δ θn,δ + 2λ2 1 + 4∆2 3Cθn,δ nµ(B(x, r)) 2/(2+d) . 5 Final remark The problem of choosing k = k(x) optimally at x is similar to the problem of local bandwidth selection for kernel-based methods (see e.g. [16, 17]), and our method for choosing k might yield insights into bandwidth selection, since k-NN and kernel regression methods only differ in their notion of neighborhood of a query x. Acknowledgments I am grateful to David Balduzzi for many useful discussions. 8 References [1] C. J. Stone. Optimal rates of convergence for non-parametric estimators. Ann. Statist., 8:1348– 1360, 1980. [2] C. J. Stone. Optimal global rates of convergence for non-parametric estimators. Ann. Statist., 10:1340–1353, 1982. [3] S. Roweis and L. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290, 2000. [4] J. Tenebaum, V. de Silva, and J. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290, 2000. [5] M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(6):1373–1396, 2003. [6] P. Bickel and B. Li. Local polynomial regression on unknown manifolds. Tech. Re. Dep. of Stats. UC Berkley, 2006. [7] S. Kpotufe. Escaping the curse of dimensionality with a tree-based regressor. Conference On Learning Theory, 2009. [8] S. Kpotufe. Fast, smooth, and adaptive regression in metric spaces. Neural Information Processing Systems, 2009. [9] S. Kulkarni and S. Posner. Rates of convergence of nearest neighbor estimation under arbitrary sampling. IEEE Transactions on Information Theory, 41, 1995. [10] L. Gyorﬁ, M. Kohler, A. Krzyzak, and H. Walk. A Distribution Free Theory of Nonparametric Regression. Springer, New York, NY, 2002. [11] C. Cutler. A review of the theory and estimation of fractal dimension. Nonlinear Time Series and Chaos, Vol. I: Dimension Estimation and Models, 1993. [12] K. Clarkson. Nearest-neighbor searching and metric space dimensions. Nearest-Neighbor Methods for Learning and Vision: Theory and Practice, 2005. [13] M. do Carmo. Riemannian Geometry. Birkhauser, 1992. [14] V. Vapnik and A. Chervonenkis. On the uniform convergence of relative frequencies of events to their expectation. Theory of probability and its applications, 16:264–280, 1971. [15] S. 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4,430	Budgeted Optimization with Concurrent Stochastic-Duration Experiments Javad Azimi, Alan Fern, Xiaoli Z. Fern School of EECS, Oregon State University {azimi, afern, xfern}@eecs.oregonstate.edu Abstract Budgeted optimization involves optimizing an unknown function that is costly to evaluate by requesting a limited number of function evaluations at intelligently selected inputs. Typical problem formulations assume that experiments are selected one at a time with a limited total number of experiments, which fail to capture important aspects of many real-world problems. This paper deﬁnes a novel problem formulation with the following important extensions: 1) allowing for concurrent experiments; 2) allowing for stochastic experiment durations; and 3) placing constraints on both the total number of experiments and the total experimental time. We develop both ofﬂine and online algorithms for selecting concurrent experiments in this new setting and provide experimental results on a number of optimization benchmarks. The results show that our algorithms produce highly effective schedules compared to natural baselines. 1 Introduction We study the optimization of an unknown function f by requesting n experiments, each specifying an input x and producing a noisy observation of f(x). In practice, the function f might be the performance of a device parameterized by x. We consider the setting where running experiments is costly (e.g. in terms of time), which renders methods that rely on many function evaluations, such as stochastic search or empirical gradient methods, impractical. Bayesian optimization (BO) [8, 4] addresses this issue by leveraging Bayesian modeling to maintain a posterior over the unknown function based on previous experiments. The posterior is then used to intelligently select new experiments to trade-off exploring new parts of the experimental space and exploiting promising parts. Traditional BO follows a sequential approach where only one experiment is selected and run at a time. However, it is often desirable to select more than one experiment at a time so that multiple experiments can be run simultaneously to leverage parallel facilities. Recently, Azimi et al. (2010) proposed a batch BO algorithm that selects a batch of k ≥1 experiments at a time. While this broadens the applicability of BO, it is still limited to selecting a ﬁxed number of experiments at each step. As such, prior work on BO, both batch and sequential, completely ignores the problem of how to schedule experiments under ﬁxed experimental budget and time constraints. Furthermore, existing work assumes that the durations of experiments are identical and deterministic, whereas in practice they are often stochastic. Consider one of our motivating applications of optimizing the power output of nano-enhanced Microbial Fuel Cells (MFCs). MFCs [3] use micro-organisms to generate electricity. Their performance depends 1 strongly on the surface properties of the anode [10]. Our problem involves optimizing nano-enhanced anodes, where various types of nano-structures, e.g. carbon nano-wire, are grown directly on the anode surface. Because there is little understanding of how different nano-enhancements impact power output, optimizing anode design is largely guess work. Our original goal was to develop BO algorithms for aiding this process. However, many aspects of this domain complicate the application of BO. First, there is a ﬁxed budget on the number of experiments that can be run due to limited funds and a ﬁxed time period for the project. Second, we can run multiple concurrent experiments, limited by the number of experimental apparatus. Third, the time required to run each experiment is variable because each experiment requires the construction of a nano-structure with speciﬁc properties. Nano-fabrication is highly unpredictable and the amount of time to successfully produce a structure is quite variable. Clearly prior BO models fail to capture critical aspects of the experimental process in this domain. In this paper, we consider the following extensions. First, we have l available labs (which may correspond to experimental stations at one location or to physically distinct laboratories), allowing up to l concurrent experiments. Second, experiments have stochastic durations, independently and identically distributed according to a known density function pd. Finally, we are constrained by a budget of n total experiments and a time horizon h by which point we must ﬁnish. The goal is to maximize the unknown function f by selecting experiments and when to start them while satisfying the constraints. We propose ofﬂine (Section 4) and online (Section 5) scheduling approaches for this problem, which aim to balance two competing factors. First, a scheduler should ensure that all n experiments complete within the horizon h, which encourages high concurrency. Second, we wish to select new experiments given as many previously completed experiments as possible to make more intelligent experiment selections, which encourages low concurrency. We introduce a novel measure of the second factor, cumulative prior experiments (CPE) (Section 3), which our approaches aim to optimize. Our experimental results indicate that these approaches signiﬁcantly outperform a set of baselines across a range of benchmark optimization problems. 2 Problem Setup Let X ⊆ℜd be a d-dimensional compact input space, where each dimension i is bounded in [ai, bi]. An element of X is called an experiment. An unknown real-valued function f : X →ℜrepresents the expected value of the dependent variable after running an experiment. For example, f(x) might be the result of a wetlab experiment described by x. Conducting an experiment x produces a noisy outcome y = f(x) + ϵ, where ϵ is a random noise term. Bayesian Optimization (BO) aims to ﬁnd an experiment x ∈X that approximately maximizes f by requesting a limited number of experiments and observing their outcomes. We extend traditional BO algorithms and study the experiment scheduling problem. Assuming a known density function pd for the experiment durations, the inputs to our problem include the total number of available labs l, the total number of experiments n, and the time horizon h by which we must ﬁnish. The goal is to design a policy π for selecting when to start experiments and which ones to start to optimize f. Speciﬁcally, the inputs to π are the set of completed experiments and their outcomes, the set of currently running experiments with their elapsed running time, the number of free labs, and the remaining time till the horizon. Given this information, π must select a set of experiments (possibly empty) to start that is no larger than the number of free labs. Any run of the policy ends when either n experiments are completed or the time horizon is reached, resulting in a set X of n or fewer completed experiments. The objective is to obtain a policy with small regret, which is the expected difference between the optimal value of f and the value of f for the predicted best experiment in X. In theory, the optimal policy can be found by solving a POMDP with hidden state corresponding to the unknown function f. However, this POMDP is beyond the reach of any existing solvers. Thus, we focus on deﬁning and comparing several principled policies that work well in practice, but without optimality guarantees. Note that this problem has not been studied in the literature to the best of our knowledge. 2 3 Overview of General Approach A policy for our problem must make two types of decisions: 1) scheduling when to start new experiments, and 2) selecting the speciﬁc experiments to start. In this work, we factor the problem based on these decisions and focus on approaches for scheduling experiments. We assume a black box function SelectBatch for intelligently selecting the k ≥1 experiments based on both completed and currently running experiments. The implementation of SelectBatch is described in Section 6. Optimal scheduling to minimize regret appears to be computationally hard for non-trivial instances of SelectBatch. Further, we desire scheduling approaches that do not depend on the details of SelectBatch, but work well for any reasonable implementation. Thus, rather than directly optimizing regret for a speciﬁc SelectBatch, we consider the following surrogate criteria. First, we want to ﬁnish all n experiments within the horizon h with high probability. Second, we would like to select each experiment based on as much information as possible, measured by the number of previously completed experiments. These two goals are at odds, since maximizing the completion probability requires maximizing concurrency of the experiments, which minimizes the second criterion. Our ofﬂine and online scheduling approaches provide different ways for managing this trade-off. 0 20 40 60 80 100 120 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 CPE Regret Cosines 0 20 40 60 80 100 120 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 CPE Regret Hydrogen Figure 1: The correlation between CPE and regret for 30 different schedulers on two BO benchmarks. To quantify the second criterion, consider a complete execution E of a scheduler. For any experiment e in E, let priorE(e) denote the number of experiments in E that completed before starting e. We deﬁne the cumulative prior experiments (CPE) of E as: P e∈E priorE(e). Intuitively, a scheduler with a high expected CPE is desirable, since CPE measures the total amount of information SelectBatch uses to make its decisions. CPE agrees with intuition when considering extreme policies. A poor scheduler that starts all n experiments at the same time (assuming enough labs) will have a minimum CPE of zero. Further, CPE is maximized by a scheduler that sequentially executes all experiments (assuming enough time). However, in between these extremes, CPE fails to capture certain intuitive properties. For example, CPE increases linearly in the number of prior experiments, while one might expect diminishing returns as the number of prior experiments becomes large. Similarly, as the number of experiments started together (the batch size) increases, we might also expect diminishing returns since SelectBatch must choose the experiments based on the same prior experiments. Unfortunately, quantifying these intuitions in a general way is still an open problem. Despite its potential shortcomings, we have found CPE to be a robust measure in practice. To empirically examine the utility of CPE, we conducted experiments on a number of BO benchmarks. For each domain, we used 30 manually designed diverse schedulers, some started more experiments early on than later, and vice-versa, while others included random and uniform schedules. We measured the average regret achieved for each scheduler given the same inputs and the expected CPE of the executions. Figure 1 shows the results for two of the domains (other results are highly similar), where each point corresponds to the average regret and CPE of a particular scheduler. We observe a clear and non-trivial correlation between regret and CPE, which provides empirical evidence that CPE is a useful measure to optimize. Further, as we will see in our experiments, the performance of our methods is also highly correlated with CPE. 4 Ofﬂine Scheduling We now consider ofﬂine schedules, which assign start times to all n experiments before the experimental process begins. Note that while the schedules are ofﬂine, the overall BO policy has online characteristics, since the exact experiments to run are only speciﬁed when they need to be started by SelectBatch, based 3 on the most recent information. This ofﬂine scheduling approach is often convenient in real experimental domains where it is useful to plan out a static equipment/personnel schedule for the duration of a project. Below we ﬁrst consider a restricted class of schedules, called staged schedules, for which we present a solution that optimizes CPE. Next, we describe an approach for a more general class of schedules. 4.1 Staged Schedules A staged schedule deﬁnes a consecutive sequence of N experimental stages, denoted by a sequence of tuples ⟨(ni, di)⟩N i=1, where 0 < ni ≤l, P i di ≤h, and P i ni ≤n. Stage i begins by starting up ni new experiments selected by SelectBatch using the most recent information, and ends after a duration of di, upon which stage i+1 starts. In some applications, staged schedules are preferable as they allow project planning to focus on a relatively small number of time points (the beginning of each stage). While our approach tries to ensure that experiments ﬁnish within their stage, experiments are never terminated and hence might run longer than their speciﬁed duration. If, because of this, at the beginning of stage i there are not ni free labs, the experiments will wait till labs free up. We say that an execution E of a staged schedule S is safe if each experiment is completed within its speciﬁed duration in S. We say that a staged schedule S is p-safe if with probability at least p an execution of S is safe which provides a probabilistic guarantee that all n experiments complete within the horizon h. Further, it ensures with probability p that the maximum number of concurrent experiments when executing S is maxi ni (since experiments from two stages will not overlap with probability p). As such, we are interested in ﬁnding staged schedules that are p-safe for a user speciﬁed p, e.g. 95%. Meanwhile, we want to maximize CPE. The CPE of any safe execution of S (slightly abusing notation) is: CPE(S) = PN i=2 ni Pi−1 j=1 nj. Typical applications will use relative high values of p, since otherwise experimental resources would be wasted, and thus with high probability we expect the CPE of an execution of S to equal CPE(S). Our goal is thus to maximize CPE(S) while ensuring p-safeness. It turns out that for any ﬁxed number of stages N, the schedules that maximize CPE(S) must be uniform. A staged schedule is deﬁned to be uniform if ∀i, j, \|ni −nj\| ≤1, i.e., the batch sizes across stages may differ by at most a single experiment. Proposition 1. For any number of experiments n and labs l, let SN be the set of corresponding N stage schedules, where N ≥⌈n/l⌉. For any S ∈SN, CPE(S) = maxS′∈SN CPE(S′) if and only if S is uniform. Algorithm 1 Algorithm for computing a p-safe uniform schedule with maximum number of stages. Input:number of experiments (n), number of labs (l), horizon (h), safety probability (p) Output:A p-safe uniform schedule with maximum number of stages N = ⌈n/l⌉, S ←null loop S′ ←MaxProbUniform(N, n, l, h) if S′ is not p-safe then return S end if S ←S′, N ←N + 1 end loop It is easy to verify that for a given n and l, an N stage uniform schedule achieves a strictly higher CPE than any N −1 stage schedule. This implies that we should prefer uniform schedules with maximum number of stages allowed by the p-safeness restriction. This motivates us to solve the following problem: Find a p-safe uniform schedule with maximum number of stages. Our approach, outlined in Algorithm 1, considers N stage schedules in order of increasing N, starting at the minimum possible number of stages N = ⌈n/l⌉for running all experiments. For each value of N, the call to MaxProbUniform computes a uniform schedule S with the highest probability of a safe execution, among all N stage uniform schedules. If the resulting schedule is p-safe then we consider N + 1 stages. Otherwise, there is no uniform N stage schedule that is p-safe and we return a uniform N −1 stage schedule, which was computed in the previous iteration. 4 It remains to describe the MaxProbUniform function, which computes a uniform N stage schedule S = ⟨(ni, di)⟩N i=1 that maximizes the probability of a safe execution. First, any N stage uniform schedule must have N ′ = (n mod N) stages with n′ = ⌊n/N⌋+1 experiments and N−N ′ stages with n′−1 experiments. Furthermore, the probability of a safe execution is invariant to the ordering of the stages, since we assume i.i.d. distribution on the experiment durations. The MaxProbUniform problem is now reduced to computing the durations di of S that maximize the probability of safeness for each given ni. For this we will assume that the distribution of the experiment duration pd is log-concave, which allows us to characterize the solution using the following lemma. Lemma 1. For any duration distribution pd that is log-concave, if an N stage schedule S = ⟨(ni, di)⟩N i=1 is p-safe, then there is a p-safe N stage schedule S′ = ⟨(ni, d′ i)⟩N i=1 such that if ni = nj then d′ i = d′ j. This lemma suggests that any stages with equal ni’s should have equal di’s to maximize the probability of safe execution. For a uniform schedule, ni is either n′ or n′ −1. Thus we only need to consider schedules with two durations, d′ for stages with ni = n′ and d′′ for stages with ni = n′ −1. Since all durations must sum to h, d′ and d′′ are deterministically related by: d′′ = h−d′·N ′ N−N ′ . Based on this, for any value of d′ the probability of the uniform schedule using durations d′ and d′′ is as follows, where Pd is the CDF of pd. Pd(d′) N′·n′ Pd h −d′ · N ′ N −N ′ (N−N′)·(n′−1) (1) We compute MaxProbUniform by maximizing Equation 1 with respect to d′ and using the corresponding duration for d′′. Putting everything together we get the following result. Theorem 1. For any log-concave pd, computing MaxProbUniform by maximizing Equation 1 over d′, if a p-safe uniform schedule exists, Algorithm 1 returns a maximum-stage p-safe uniform schedule. 4.2 Independent Lab Schedules We now consider a more general class of ofﬂine schedules and a heuristic algorithm for computing them. This class allows the start times of different labs to be decoupled, desirable in settings where labs are run by independent experimenters. Further, our online scheduling approach is based on repeatedly calling an ofﬂine scheduler, which requires the ﬂexibility to make schedules for labs in different stages of execution. An independent lab (IL) schedule S speciﬁes a number of labs k < l and for each lab i, a number of experiments mi such that P i mi = n. Further, for each lab i a sequence of mi durations Di = ⟨d1 i , . . . , dmi i ⟩ is given. The execution of S runs each lab independently, by having each lab start up experiments whenever they move to the next stage. Stage j of lab i ends after a duration of dj i, or after the experiment ﬁnishes when it runs longer than dj i (i.e. we do not terminate experiments). Each experiment is selected according to SelectBatch, given information about all completed and running experiments across all labs. We say that an execution of an IL schedule is safe if all experiments ﬁnish within their speciﬁed durations, which also yields a notion of p-safeness. We are again interested in computing p-safe schedules that maximizes the CPE. Intuitively, CPE will be maximized if the amount of concurrency during an execution is minimized, suggesting the use of as few labs as possible. This motivates the problem of ﬁnding a p-safe IL schedule that use the minimum number of labs. Below we describe our heuristic approach to this problem. Algorithm Description. Starting with k = 1, we compute a k labs IL schedule with the goal of maximizing the probability of safe execution. If this probability is less than p, we increment k, and otherwise output the schedule for k labs. To compute a schedule for each value of k, we ﬁrst allocate the number of experiments mi across k labs as uniformly as possible. In particular, (n mod k) labs will have ⌊n/k⌋+ 1 experiments and k −(n mod k) labs will have ⌊n/k⌋experiments. This choice is motivated by the intuition that the best way to maximize the probability of a safe execution is to distribute the work across labs as uniformly as possible. Given mi for each lab, we assign all durations of lab i to be h/mi, which can be shown to be optimal for log-concave pd. In this way, for each value of k the schedule we compute has just two possible values of mi and labs with the same mi have the same stage durations. 5 5 Online Scheduling Approaches We now consider online scheduling, which selects the start time of experiments online. The ﬂexibility of the online approaches offers the potential to outperform ofﬂine schedules by adapting to speciﬁc stochastic outcomes observed during experimental runs. Below we ﬁrst describe two baseline online approaches, followed by our main approach, policy switching, which aims to directly optimize CPE. Online Fastest Completion Policy (OnFCP). This baseline policy simply tries to ﬁnish all of the n experiments as quickly as possible. As such, it keeps all l labs busy as long as there are experiments left to run. Speciﬁcally whenever a lab (or labs) becomes free the policy immediately uses SelectBatch with the latest information to select new experiments to start right away. This policy will achieve a low value of expected CPE since it maximizes concurrency. Online Minimum Eager Lab Policy (OnMEL). One problem with OnFCP is that it does not attempt to use the full time horizon. The OnMEL policy simply restricts OnFCP to use only k labs, where k is the minimum number of labs required to guarantee with probability at least p that all n experiments complete within the horizon. Monte-Carlo simulation is used to estimate p for each k. Policy Switching (PS). Our policy switching approach decides the number of new experiments to start at each decision epoch. Decision epochs are assumed to occur every ∆units of time, where ∆is a small constant relative to the expected experiment durations. The motivation behind policy switching is to exploit the availability of a policy generator that can produce multiple policies at any decision epoch, where at least one of them is expected to be good. Given such a generator, the goal is to deﬁne a new (switching) policy that performs as well or better than the best of the generated policies in any state. In our case, the objective is to improve CPE, though other objectives can also be used. This is motivated by prior work on policy switching [6] over a ﬁxed policy library, and generalize that work to handle arbitrary policy generators instead of static policy libraries. Below we describe the general approach and then the speciﬁc policy generator that we use. Let t denote the number of remaining decision epochs (stages-to-go), which is originally equal to ⌊h/∆⌋and decremented by one each epoch. We use s to denote the experimental state of the scheduling problem, which encodes the number of completed experiments and ongoing experiments with their elapsed running time. We assume access to a policy generator Π(s, t) which returns a set of base scheduling policies (possibly nonstationary) given inputs s and t. Prior work on policy switching [6] corresponds to the case where Π(s, t) returns a ﬁxed set of policies regardless of s and t. Given Π(s, t), ¯π(s, t, π) denotes the resulting switching policy based on s, t, and the base policy π selected in the previous epoch. The decision returned by ¯π is computed by ﬁrst conducting N simulations of each policy returned by Π(s, t) along with π to estimate their CPEs. The base policy with the highest estimated CPE is then selected and its decision is returned by ¯π. The need to compare to the previous policy π is due to the use of a dynamic policy generator, rather than a ﬁxed library. The base policy passed into policy switching for the ﬁrst decision epoch can be arbitrary. Despite its simplicity, we can make guarantees about the quality of ¯π assuming a bound on the CPE estimation error. In particular, the CPE of the switching policy will not be much worse than the best of the policies produced by our generator given accurate simulations. We say that a CPE estimator is ϵ-accurate if it can estimate the CPE Cπ t (s) of any base policy π for any s and t within an accuracy bound of ϵ. Below we denote the expected CPE of ¯π for s, t, and π to be C ¯π t (s, π). Theorem 2. Let Π(s, t) be a policy generator and ¯π be the switching policy computed with ϵ-accurate estimates. For any state s, stages-to-go t, and base policy π, C ¯π t (s, π) ≥maxπ′∈Π(s,t)∪{π} Cπ′ t (s) −2tϵ. We use a simple policy generator Π(s, t) that makes multiple calls to the ofﬂine IL scheduler described earlier. The intuition is to notice that the produced p-safe schedules are fairly pessimistic in terms of the experiment runtimes. In reality many experiments will ﬁnish early and we can adaptively exploit such situations. Speciﬁcally, rather than follow the ﬁxed ofﬂine schedule we may choose to use fewer labs and hence improve CPE. Similarly if experiments run too long, we will increase the number of labs. 6 Table 1: Benchmark Functions Cosines(2)[1] 1 −(u2 + v2 −0.3cos(3πu) −0.3cos(3πv)) Rosenbrock(2)[1] 10 −100(y −x2)2 −(1 −x)2 u = 1.6x −0.5, v = 1.6y −0.5 Hartman(3,6)[7] Σi=14αi exp −Σd j=1Aij(xj −Pij)2 Michalewicz(5)[9]−P5 i=1 sin(xi). sin i.x2 i π 20 α1×4, A4×d, P4×d are constants Shekel(4)[7] Σ10 i=1 1 αi+Σj=14(xj−Aji)2 α1×10, A4×10 are constants We deﬁne Π(s, t) to return k + 1 policies, {π(s,t,0), . . . , π(s,t,k)}, where k is the number of experiments running in s. Policy π(s,t,i) is deﬁned so that it waits for i current experiments to ﬁnish, and then uses the ofﬂine IL scheduler to return a schedule. This amounts to adding a small lookahead to the ofﬂine IL scheduler where different amounts of waiting time are considered 1. Note that the deﬁnition of these policies depends on s and t and hence can not be viewed as a ﬁxed set of static policies as used by traditional policy switching. In the initial state s0, π(s0,h,0) corresponds to the ofﬂine IL schedule and hence the above theorem guarantees that we will not perform much worse than the ofﬂine IL, with the expectation of performing much better. Whenever policy switching selects a πi with i > 0 then no new experiments will be started and we wait for the next decision epoch. For i = 0, it will apply the ofﬂine IL scheduler to return a p-safe schedule to start immediately, which may require starting new labs to ensure high probability of completing n experiments. 6 Experiments Implementation of SelectBatch. Given the set of completed experiments O and on-going experiments A, SelectBatch selects k new experiments. We implement SelectBatch based on a recent batch BO algorithm [2], which greedily selects k experiments considering only O. We modify this greedy algorithm to also consider A by forcing the selected batch to include the ongoing experiments plus k additional experiments. SelectBatch makes selections based on a posterior over the unknown function f. We use Gaussian Process with the RBF kernel and the kernel width = 0.01 Pd i=1 li, where li is the input space length in dimension i. Benchmark Functions. We evaluate our scheduling policies using 6 well-known synthetic benchmark functions (shown in Tab. 1 with dimension inside the parenthesis) and two real-world benchmark functions Hydrogen and FuelCell over [0, 1]2 [2]. The Hydrogen data is produced by a study on biosolar hydrogen production [5], where the goal was to maximize hydrogen production of a particular bacteria by optimizing PH and Nitrogen levels. The FuelCell data was collected in our motivating application mentioned in Sect. 1. In both cases, the benchmark function was created by ﬁtting regression models to the available data. Evaluation. We consider a p-safeness guarantee of p = 0.95 and the number of available labs l is 10. For pd(x), we use one sided truncated normal distribution such that x ∈(0, inf) with µ = 1, σ2 = 0.1, and we set the total number of experiments n = 20. We consider three time horizons h of 6, 5, and 4. Given l, n and h, to evaluate policy π using function f (with a set of initial observed experiments), we execute π and get a set X of n or fewer completed experiments. We measure the regret of π as the difference between the optimal value of f (known for all eight functions) and the f value of the predicted best experiment in X. Results. Table 2 shows the results of our proposed ofﬂine and online schedulers. We also include, as a reference point, the result of the un-constrained sequential policy (i.e., selecting one experiment at a time) using SelectBatch, which can be viewed as an effective upper bound on the optimal performance of any constrained scheduler because it ignores the time horizon (h = ∞). The values in the table correspond to the regrets (smaller values are better) achieved by each policy, averaged across 100 independent runs with the same initial experiments (5 for 2-d and 3-d functions and 20 for the rest) for all policies in each run. 1For simplicity our previous discussion of the IL scheduler did not consider states with ongoing experiments, which will occur here. To handle this the scheduler ﬁrst considers using already executing labs taking into account how long they have been running. If more labs are required to ensure p-safeness new ones are added. 7 Table 2: The proposed policies results for different horizons. h=4 h=5 h=6 Functionh = ∞OnFCP OfStaged OfIL OnMEL PS OfStaged OfIL OnMEL PS OfStaged OfIL OnMEL PS Cosines .142 .339 .181 .195 .275 .205 .181 .194 .274 .150 .167 .147 .270 .156 FuelCell .160 .240 .182 .191 .258 .206 .167 .190 .239 .185 .154 .163 .230 .153 Hydro .025 .115 .069 .070 .123 .059 .071 .069 .086 .042 .036 .035 .064 .025 Rosen .008 .013 .010 .009 .013 .008 .009 .008 .011 .008 .007 .009 .010 .009 Hart(3) .037 .095 .070 .069 .096 .067 .055 .064 .081 .045 .045 .050 .070 .038 Michal .465 .545 .509 .508 .525 .502 .500 .510 .521 .494 .477 .460 .502 .480 Shekel .427 .660 .630 .648 .688 .623 .635 .645 .682 .540 .530 .564 .576 .510 Hart(6) .265 .348 .338 .340 .354 .347 .334 .330 .333 .297 .304 .266 .301 .262 CPE 190 55 100 100 66 100 100 100 91 118 133 137 120 138 We ﬁrst note that the two ofﬂine algorithms (OfStages and OfIL) perform similarly across all three horizon settings. This suggests that there is limited beneﬁt in these scenarios to using the more ﬂexible IL schedules, which were primarily introduced for use in the online scheduling context. Comparing with the two online baselines (OnFCP and OnMEL), the ofﬂine algorithms perform signiﬁcantly better. This may seem surprising at ﬁrst because online policies should offer more ﬂexibility than ﬁxed ofﬂine schedules. However, the ofﬂine schedules purposefully wait for experiments to complete before starting up new experiments, which tends to improve the CPE values. To see this, the last row of Table 2 gives the average CPEs of each policy. Both OnFCP and OnMEL yield signiﬁcantly lower CPEs compared to the ofﬂine algorithms, which correlates with their signiﬁcantly larger regrets. Finally, policy switching consistently outperforms other policies (excluding h = ∞) on the medium horizon setting and performs similarly in the other settings. This makes sense since the added ﬂexibility of PS is not as critical for long and short horizons. For short horizons, there is less opportunity for scheduling choices and for longer horizons the scheduling problem is easier and hence the ofﬂine approaches are more competitive. In addition, looking at Table 2, we see that PS achieves a signiﬁcantly higher CPE than ofﬂine approaches in the medium horizon, and is similar to them in the other horizons, again correlating with the regret. Further examination of the schedules produced by PS indicates that although it begins with the same number of labs as OfIL, PS often selects fewer labs in later steps if early experiments are completed sooner than expected, which leads to higher CPE and consequently better performance. Note that the variances of the proposed policies are very small which are shown in the supplementary materials. 7 Summary and Future Work Motivated by real-world applications we introduced a novel setting for Bayesian optimization that incorporates a budget on the total time and number of experiments and allows for concurrent, stochastic-duration experiments. We considered ofﬂine and online approaches for scheduling experiments in this setting, relying on a black box function to intelligently select speciﬁc experiments at their scheduled start times. These approaches aimed to optimize a novel objective function, Cumulative Prior Experiments (CPE), which we empirically demonstrate to strongly correlate with performance on the original optimization problem. Our ofﬂine scheduling approaches signiﬁcantly outperformed some natural baselines and our online approach of policy switching was the best overall performer. For further work we plan to consider alternatives to CPE, which, for example, incorporate factors such as diminishing returns. We also plan to study further extensions to the experimental model for BO and also for active learning. For example, taking into account varying costs and duration distributions across labs and experiments. In general, we believe that there is much opportunity for more tightly integrating scheduling and planning algorithms into BO and active learning to more accurately model real-world conditions. Acknowledgments The authors acknowledge the support of the NSF under grants IIS-0905678. 8 References [1] B. S. Anderson, A. Moore, and D. Cohn. A nonparametric approach to noisy and costly optimization. In ICML, 2000. [2] J. Azimi, A. Fern, and X. Fern. Batch bayesian optimization via simulation matching. In NIPS, 2010. [3] D. Bond and D. Lovley. Electricity production by geobacter sulfurreducens attached to electrodes. Applications of Environmental Microbiology, 69:1548–1555, 2003. [4] E. Brochu, M. Cora, and N. de Freitas. A tutorial on Bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning. Technical Report TR-2009-23, Department of Computer Science, University of British Columbia, 2009. [5] E. H. Burrows, W.-K. Wong, X. Fern, F. W. Chaplen, and R. L. Ely. Optimization of ph and nitrogen for enhanced hydrogen production by synechocystis sp. pcc 6803 via statistical and machine learning methods. Biotechnology Progress, 25:1009–1017, 2009. [6] H. Chang, R. Givan, and E. Chong. Parallel rollout for online solution of partially observable markov decision processes. Discrete Event Dynamic Systems, 14:309–341, 2004. [7] L. Dixon and G. Szeg. The Global Optimization Problem: An Introduction Toward Global Optimization. NorthHolland, Amsterdam, 1978. [8] D. Jones. A taxonomy of global optimization methods based on response surfaces. Journal of Global Optimization, pages 345–383, 2001. [9] Z. Michalewicz. Genetic algorithms + data structures = evolution programs (2nd, extended ed.). Springer-Verlag New York, Inc., New York, NY, USA, 1994. [10] D. Park and J. Zeikus. Improved fuel cell and electrode designs for producing electricity from microbial degradation. Biotechnol.Bioeng., 81(3):348–355, 2003. 9	2011	80
4,431	Data Skeletonization via Reeb Graphs Xiaoyin Ge Issam Safa Mikhail Belkin Yusu Wang Computer Science and Engineering Department The Ohio State University gex,safa,mbelkin,yusu@cse.ohio-state.edu Abstract Recovering hidden structure from complex and noisy non-linear data is one of the most fundamental problems in machine learning and statistical inference. While such data is often high-dimensional, it is of interest to approximate it with a lowdimensional or even one-dimensional space, since many important aspects of data are often intrinsically low-dimensional. Furthermore, there are many scenarios where the underlying structure is graph-like, e.g, river/road networks or various trajectories. In this paper, we develop a framework to extract, as well as to simplify, a one-dimensional ”skeleton” from unorganized data using the Reeb graph. Our algorithm is very simple, does not require complex optimizations and can be easily applied to unorganized high-dimensional data such as point clouds or proximity graphs. It can also represent arbitrary graph structures in the data. We also give theoretical results to justify our method. We provide a number of experiments to demonstrate the effectiveness and generality of our algorithm, including comparisons to existing methods, such as principal curves. We believe that the simplicity and practicality of our algorithm will help to promote skeleton graphs as a data analysis tool for a broad range of applications. 1 Introduction Learning or inferring a hidden structure from discrete samples is a fundamental problem in data analysis, ubiquitous in a broad range of application ﬁelds. With the rapid generation of diverse data all across science and engineering, extracting geometric structure is often a crucial ﬁrst step towards interpreting the data at hand, as well as the underlying process of phenomenon. Recently, there has been a large amount of research in this direction, especially in the machine learning community. In this paper, we consider a simple but important scenario, where the hidden space has a graphlike geometric structure, such as the branching ﬁlamentary structures formed by blood vessels. Our goal is to extract such structures from points sampled on and around them. Graph-like geometric structures arise naturally in many ﬁelds, both in modeling natural phenomena, and in understanding abstract procedures and simulations. However, there has been only limited work on obtaining a general-purpose algorithm to automatically extract skeleton graph structures [2]. In this paper, we present such an algorithm by bringing in a topological concept called the Reeb graph to extract skeleton graphs. Our algorithm is simple, efﬁcient and easy to use. We demonstrate the generality and effectiveness of our algorithm via several applications in both low and high dimensions. Motivation. Geometric graphs are the underlying structures for modeling many natural phenomena from river / road networks, root systems of trees, to blood vessels, and particle trajectories. For example, if we are interested in obtaining the road network of a city, we may send out cars to explore various streets of the city, with each car recording its position using a GPS device. The resulting data is a set of potentially noisy points sampled from the roads in a city. Given these data, the goal is to automatically reconstruct the road network, which is a graph embedded in a two- dimensional space. Indeed, abundant data of this type are available at the open-streets project website [1]. 1 Geometric graphs also arise from many modeling processes, such as molecular simulations. They can sometimes provide a natural platform to study a collection of time-series data, where each timeseries corresponds to a trajectory in the feature space. These trajectories converge and diverge, which can be represented by a graph. This graph in turn can then be used as a starting point for further processing (such as matching) or inference tasks. Generally, there are a number of scenarios where we wish to extract a one-dimensional skeleton from an input space. The goal in this paper is to develop, as well as to demonstrate the use of, a practical and general algorithm to extract a graph structure from input data of any dimensions. New work. Given a set of points P sampling a hidden domain X, we present a simple and practical algorithm to extract a skeleton graph G for X. The input points P do not have to be embedded – we only need their distance matrix or simply a proximity graph as input to our algorithm. Our algorithm is based on using the so-called Reeb graph to model skeleton graphs. Given a continuous function f : X →IR, the Reeb graph tracks the connected components in the level-set f −1(a) of f as we vary the value a. It provides a meaningful abstraction of the scalar ﬁeld f, and has been widely used in graphics, visualization, and computer vision (see [6] for a survey). However, it has not yet been aimed as a tool to analyze high dimensional data from unorganized input data. By bringing the concept of the Reeb graph to machine learning applications, we can leverage the recent algorithms developed to compute and process Reeb graphs [15, 9]. Moreover, combining the Reeb graph with the so-called Rips complex allows us to obtain theoretical guarantees for our algorithm. Our algorithm is simple and efﬁcient. There is only one parameter involved, which intuitively speciﬁes the scale at which we look at the data. Our algorithm always outputs a graph G given data. Furthermore, it also computes a map Φ : P →G, which maps each sample point to G. Hence we can decompose the input data into sets, each corresponding to a single branch in the skeleton graph. Finally, there is a canonical way to measure importance of features in the Reeb graph, which allows us to easily simplify the resulting graph. We summarize our contributions as follows: (1) We bring in Reeb graphs to the learning community for analyzing high dimensional unorganized data sets. We developed an accompanying software to not only extract, but also process skeleton graphs from data. Our algorithm is simple and robust, always extracting a graph from the input. Our algorithm complements principal curve algorithms and can be used in combination with them. (2) We provide certain theoretical guarantees for our algorithm. We also demonstrate both the effectiveness of our software and the usefulness of skeleton graphs via a sets of experiments on diverse datasets. Experimental results show that despite being simple and general, our algorithm compares favorably to existing graph-extracting algorithms in various settings. Related work. At a broad level, the graph-extraction problem is related to manifold learning and non-linear dimensionality reduction which has a rich literature, see e.g [4, 24, 25, 27]. Manifold learning methods typically assume that the hidden domain has a manifold structure. An even more general scenario is that the hidden domain is a stratiﬁed space, which intuitively, can be thought of as a collection of manifolds (strata) glued together. Recently, there have been several approaches to learn stratiﬁed spaces [5, 14]. However, this general problem is hard and requires algorithms both mathematically sophisticated and computationally intensive. In this case, we aim to learn a graph structure, which is simply a one-dimensional stratiﬁed space, allowing for simple approaches. The most relevant previous work related to our graph-extraction problem is a series of results on an elegant concept of principal curves, originally proposed by Hastie and Stuetzle [16, 17]. Intuitively, principal curves are ”self-consistent” curves that pass through the middle of the data. Since its original introduction, there has been much work on analyzing and extending the concept and algorithms as well as on numerous applications. See, e.g, [7, 11, 10, 19, 22, 26, 28, 29] among many others. Below we discuss the results most relevant to the current work. Original principal curves are simple smooth curves with no self-intersections. In [19], K´egl et al. represented principal curves as polygonal lines, and proposed a regularized version of principal curves. They gave a practical algorithm to compute such a polygonal principal curve. This algorithm was later extended in [18] into a principal graph algorithm to compute the skeleton graph of hand-written digits and characters. To the best of our knowledge, this was the ﬁrst algorithm to explicitly allow self-intersections in the output principal curves. However, this principal graph algo2 rithm could only handle 2D images. Very recently in [22], Ozertem and Erdogmus proposed a new deﬁnition for the principal curve associated to the probability density function. Intuitively, imagining the probability density function as a terrain, their principal curves are the mountain ridges. A rigorous deﬁnition can be made in terms of the Hessian of the probability density. Their approach has several nice properties, including connections to the popular mean-shift clustering algorithm. It also allows for certain bifurcations and self-intersections. However, the output of the algorithm is only a collection of points with neither connectivity information, nor the information about which points are junction points (graph nodes) and which points belong to the same arc in the principal graph. Furthermore, the algorithm depends on reliable density estimation from input data, which is a challenging task for high dimensional data. Aanijaneya et al. [2] recently proposed perhaps the ﬁrst general algorithm to approximate a hidden metric graph from an input graph with theoretical guarantees. While the goal of [2] is to approximate a metric graph, their algorithm can also be used to skeletonize data. The algorithm relies on inspecting the local neighborhood of each point to ﬁrst classify whether it should be a “branching point” or an “ edge point”. Although this approach has theoretical guarantees when the sampling is nice and the parameters are chosen correctly, it is often hard to ﬁnd suitable parameters in practice, and such local decisions tend to be less reliable when the input data are not as nice (such as a “fat” junction region). In the section on experimental results we show that our algorithm tends to be more robust in practical applications. Finally we note that the concept of the Reeb graph has been used in a number of applications in graphics, visualization, and computer vision (see [6] for a survey). However, it has been typically used with mesh structures rather than a tool for analyzing unorganized point cloud data, especially in high dimensions, where constructing meshes is prohibitively expensive. An exception is the very recent work[20], where the authors propose to use the Reeb graph for point cloud data and show applications for several data-sets still in 3D. The advantage of our approach is that it is based on the Rips complex, which allows for a general and cleaner Reeb graph reconstruction algorithm with theoretical justiﬁcation (see [9, 15] and Theorem 3.1). 2 Reeb Graphs We now give a very brief description of the Reeb graph; see Section VI.4 of [12] for a more formal discussion of it. Let f : X →IR be a continuous function deﬁned on a domain X. For each scalar value a ∈IR, the level set f −1(a) = {x ∈X \| f(x) = a} may have multiple connected components. The Reeb graph of f, denoted by Rf(X), is obtained by continuously identifying every connected component in a level set to a single point. In other words, Rf(X) is the image of a continuous surjective map Φ : X →Rf(X) where Φ(x) = Φ(y) if and only if x and y come from the same connected component of a level set of f. f X Rf(X) a Φ x y z Φ(z) Φ(x) = Φ(y) Intuitively, as the value a increases, connected components in the level set f −1(a) appear, disappear, split and merge, and the Reeb graph of f tracks such changes. The Reeb graph is an abstract graph. Its nodes indicate changes in the connected components in level sets, and each arc represents the evolution of a connected component before it is merged, killed, or split. See the right ﬁgure for an example, where we show (an embedding of) the Reeb graph of the height function f deﬁned on a topological torus. The Reeb graph Rf(X) provides a simple yet meaningful abstraction of the input domain X w.r.t function f. Computation in discrete setting. Assume the input domain is modeled by a simplicial complex K. Speciﬁcally, a k-dimensional simplex σ is simply the convex combination of k + 1 independent points {v0, . . . , vk}, and any simplex formed by a subset of its vertices is called a face of σ. A simplical complex K is a collection of simplices with the property that if a simplex σ is in K, then any face of it is also in K. A piecewise-linear (PL) function f deﬁned on K is a function with values given at vertices of K and linearly interpolated within each simplex in K. Given a PL-function f on K, its Reeb graph Rf(K) is decided by all the 0, 1 and 2-simplices from K, which are the vertices, edges, and triangles of K. Hence from now on we use only 2-dimensional simplicial complex. 3 p1 p2 p3 p4 p5 p6 p7 f p8 p0 ˜p8 ˜p7 ˜p1 ˜p0 ˜p7 ˜p8 ˜p6 ˜p4 ˜p2 ˜p3 ˜p5 ˜p1 ˜p0 Given a PL function deﬁned on a simplicial complex domain K, its Reeb graph can be computed efﬁciently in O(n log n) expected time by a simple randomized algorithm [15], where n is the size of K. In fact, the algorithm outputs the so-called augmented Reeb graph R, which contains the image of all vertices in K under the surjection map Φ : K →R introduced earlier. See ﬁgure on the right: the Reeb graph (middle) is an abstract graph with four nodes, while the augmented Reeb graph (on the right) shows the image of all vertices (i.e, ˜pis). From the augmented Reeb graph R, we can easily extract junction points (graph nodes), the set of points from the input data that should be mapped to each graph arc, as well as the connectivity between these points along the Reeb graph (e.g, ˜p1˜p4˜p7 form one arc between ˜p1 and ˜p7). 3 Method 3.1 Basic algorithm Step 1: Set up complex K. The input data we consider can be a set of points sampled from a hidden domain or a probabilistic distribution, or it can be the distance matrix, or simply the proximity graph, among a set of points. (So the input points do not have to be embedded.) Our goal is to compute (possibly an embedding of) a skeleton graph from the input data. First, we construct an appropriate space approximating the hidden domain that input points are sampled from. We use a simplicial complex K to model such a space. Speciﬁcally, given input sampled points P and the distance matrix of P, we ﬁrst construct a proximity graph based on either r-neighborhood or k-nearest neighbors(NN) information; that is, a point p ∈P is connected either to all its neighbors within r distance to p, or to its k-NNs. We add all points in P and all edges from this proximity graph to the simplicial complex K we are building. Next, for any three vertices p1, p2, p3 ∈P, if they are pairwise connected in the proximity graph, we add the triangle △p1p2p3 to K. Note that if the proximity graph is already given as the input, then we simply ﬁll in a triangle whenever all its three edges are in the proximity graph to obtain the target simplicial complex K. We remark that there is only one parameter involved in the basic algorithm, which is the parameter r (if we use r-neighborhood) or k (if we use k-NN) to specify the scale with which we look at the input data. Motivation behind this construction. If the proximity graph is built based on r-neighborhood, then the above construction is simply that of the so-called Vietoris-Rips complex, which has been widely used in manifold reconstruction (especially surface reconstruction) community to recover the hidden domain from its point samples. Intuitively, imagine that we grow a ball of radius r around each sample point. The union of these balls roughly captures the hidden domain at scale r. On the other hand, the topological structure of the union of these balls is captured by the so-called ˇCech complex, which mathematically is the nerve of this union of balls. Hence the ˇCech complex captures the topology of the hidden domain when the sampling is reasonable (see e.g., [8, 21]). However, ˇCech complex is hard to compute, and the Vietoris-Rips complex is a practical approximation of the ˇCech complex that is much easier to construct. Furthermore, it has been shown that the Reeb graph of a hidden manifold can be approximated with theoretical guarantees from the Rips complex [9]. b v Step 2: Reeb graph computation. Now we have a simplicial complex K that approximates the hidden domain. In order to extract the skeleton graph using the Reeb graph, we need to deﬁne a function g on K that respects its shape. It is also desirable that this function is intrinsic, given that input points may not be embedded. To this end, we construct the function g as the geodesic distance in K to a certain base point b ∈K. We compute the base point by taking an arbitrary point v ∈K and choosing b as the point furtherest away from v. Intuitively, this base point is an extreme point. If the underlying domain indeed has a branching ﬁlamentary structure, then the geodesic distance to b tends to progress along each ﬁlament, and branch out at junction points. See the right ﬁgure for an example, where the thin curves are level sets of the geodesic distance function to the base point b. 4 Figure 1: Overview of the algorithm. The input points are light (yellow) shades beneath dark curves. (Left): the augmented Reeb graph output by our algorithm. (Center): after iterative smoothing. (Right): ﬁnal output after repairing missing links (e.g top box) and simpliﬁcation (lower box). Since the Reeb graph tracks the evolution of the connected components in the level sets, a branching (splitting in the level set) will happen when the level set passes through point v. In our algorithm, the geodesic distance function g to b in K is approximated by the shortest distance in the proximity graph (i.e, the set of edges in K) to b. We then perform the algorithm from [15] to compute the Reeb graph of K with respect to g, and denote the resulting Reeb graph as R. Recall that this algorithm in fact outputs the augmented Reeb graph R. Hence we not only obtain a graph structure, but also the set of input points (together with their connectivity) that are mapped to every graph arc in this graph structure. Time complexity. The time complexity of the basic algorithm is the summation of time to compute (A) the proximity graph, (B) the complex K from the proximity graph, (C) the geodesic distance and (D) the Reeb graph. (A) is O(n2) for high dimensional data (and can be made near-linear for data in very low dimensions) where n is the number of input points. (B) is O(k3n) if each point takes k neighbors. (C) and (D) takes time O(m log n) = O(k3n log n) where m is the size of K. Hence overall, the time complexity is O(n2 +k3n log n). For high dimensional data sets, this is dominated by the computation of the proximity graph O(n2). Theoretical guarantees. Given a domain X and a function f : X →IR deﬁned on it, the topology (i.e, the number of independent loops) of the Reeb graph Rf(X) may not reﬂect that of the given domain X. However, in our case, we have the following result which offers a partial theoretical guarantee for the basic algorithm. Intuitively, the theorem states that if the hidden space is a graph G, and if our simplicial complex K approximates G both in terms of topology (as captured by homotopy equivalence) and metric (as captured by the ε-approximation), then the Reeb graph captures all loops in G. Below, dY (·, ·) denotes the geodesic distance in domain Y . Theorem 3.1 Suppose K is homotopy equivalent to a graph G, and h : K →G is the corresponding homotopy. Assume that the metric is ε-approximated under h; that is, \|dK(x, y)−dG(h(x), h(y))\| ≤ ε for any x, y ∈K, Let R be the Reeb graph of K w.r.t the geodesic distance function to an arbitrary base point b ∈K. If ε < l/4, where l is the length of the shortest arc in G, we have that there is a one-to-one correspondence between loops in R and loops in G. The proof can be found in the full version [13]. It relies on results and observations from [9]. The above result can be made even stronger: (i) There is not only a one-to-one correspondence between loops in R and in G, the ranges of each pair of corresponding loops are also close. Here, the range of a loop γ w.r.t. a function f is the interval [minx∈γ f(x), maxx∈γ f(x)]. (ii) The condition on ε < l/4 can be relaxed. Furthermore, even when ε does not satisfy this condition, the reconstructed Reeb graph R can still preserve all loops in G whose range is larger than 2ε. 3.2 Embedding and Visualization The Reeb graph is an abstract graph. To visualize the skeleton graph, we need to embed it in a reasonable way that reﬂects the geometry of hidden domain. To this end, if points are not already embedded in 2D or 3D, we project the input points P to IR3 using any standard dimensionality reduction algorithm. We then connect projected points based on their connectivity given in the 5 augmented Reeb graph R. Each arc of the Reeb graph is now embedded as a polygonal curve. To further improve the quality of this curve, we ﬁx its endpoints, and iteratively smooth it by repeatedly assigning a point’s position to be the average of its neighbors’ positions. See Figure 1 for an example. 3.3 Further Post-processing In practice, data can be noisy, and there may be spurious branches or loops in the Reeb graph R constructed no matter how we choose parameter r or k to decide the scale. Following [3], there is a natural way to deﬁne “features” in a Reeb graph and measure their “importance”. Speciﬁcally, given a function f : X →IR, imagine we plot its Reeb graph Rf(X) such that the height of each point z ∈Rf(X) is the function value of all those points in X mapped to z. Now we sweep the Reeb graph bottom-up in increasing order of the function values. As we sweep through a point z, we inspect what happens to the part of Reeb graph that we already swept, denoted by Rz f := {w ∈ Rf(X) \| f(w) ≤f(z)}. When we sweep past a down-fork saddle s, there are two possibilities: s C1 C2 h (i). The two branches merged by s belong to different connected components, say C1 and C2, in Rs f. In such case, we have a branch-feature, where two disjoint lower-branches in Rs f will be merged at s. The importance of this feature is the smaller height of the lower-branches being merged. Intuitively, this is the amount we have to perturb function f in order to remove this branch-feature. See the right ﬁgure, where the height h of C2 is the importance of this branch-feature. s γ (ii). The two branches merged by s are already connected below s in Rs f. In such case, when s connects them again, we create a family of new loops. This is called a loop-feature. Its size is measured as smallest height of any loop formed by s in Rs f, where the height of a loop γ is deﬁned as maxz∈γ f(z) −minz∈γ f(z). See the right ﬁgure, where the dashed loop γ is the thinnest loop created by s. Now if we sweep Rf(X) top-down, we will also obtain branch-features and loop-features captured by up-fork saddles in a symmetric manner. It turns out that these features (and their sizes) correspond to the so-called extended persistence of the Reeb graph Rf(X) with respect to function f [12]. The size of each feature is called its persistence, as it indicates how persistent this feature is as we perturb the function f. These features and their persistence can be computed in O(n log2 n) time, where n is the number of nodes and arcs in the Reeb graph [3]. We can now simplify the Reeb graph by merging features whose persistence value is smaller than a given threshold. This simpliﬁcation step not only removes noise, but can also be used as a way to look at features at larger scales. Finally, there may also be missing data causing missing links in the constructed skeleton graph. Hence in post-processing the user can also choose to ﬁrst ﬁll some missing links before the simpliﬁcation step. This is achieved by connecting pairs of degree-1 nodes (x, y) in the Reeb graph whose distances d(x, y) is smaller than certain distance threshold. Here d(x, y) is the input distance between x and y (if the input points are embedded, or the distance matrix is given), not the distance in the simplicial complex K constructed by our algorithm. Connecting x and y may either connect two disjoint component in the Reeb graph, thus creating new branch-features; or form new loopfeatures. See Figure 1. We do not check the size of the new features created when connecting pairs of vertices. Small newly-created features will be removed in the subsequent simpliﬁcation step. 4 Experimental Results In this section we ﬁrst provide comparisons of our algorithm to three existing methods. We then present three sets of experiments to demonstrate the effectiveness of our software and show potential applications of skeleton graph extraction for data analysis. Experimental Comparisons. We compare our approach with three existing comparable algorithms: (1) the principal graph algorithm (PGA) [18]; (2) the local-density principal curve algorithm (LDPC) [22]; and (3) the metric-graph reconstruction algorithm (MGR) [2]. Note that PGA only works for 2D images. LDPC only outputs point cloud at the center of the input data with no connectivity information. 6 (a) Our output (b) PGA [18] (c) LDPC [22] In the ﬁgure on the right, we show the skeleton graph of the image of a hand-written Chinese character. Our result is shown in (a). PGA [18] is shown in (b), while the output of (the KDE version of) LDPC [22] is shown in (c). We see that the algorithm from [18], speciﬁcally designed for these 2-D applications provides the best output. However, the results of our algorithm, which is completely generic, are comparable. On the other hand, the output of LDPC is a point cloud (rather than a graph). In this example, many points do not belong to the 1D structure1. We do not show the results from MGR [2] as we were not able to produce a satisfactory result for this data using MGR even after tuning the parameters. However, note that the goal of their algorithm is to approximate a graph metric, which is different from extracting a skeleton graph. (a) Our output (b) MGR [2] (c) MGR on nautilus For the second set of comparisons we build a skeleton graph out of an input metric graph. Note that PGA and LDPC cannot handle such graph-type input, and the only comparable algorithm is MGR [2]. We use the image-web data previously used in [2]. Figure (a) on the right is our output and (b) is the output by MGR [2]. The input image web graph is shown in light (gray) color in the background. Finally to provide an additional comparison we apply MGR to image edge detection: (c) above shows the reconstructed edges for the nautilus image used earlier in Figure 1. To be fair, MGR does not provide an embedding, so we should focus on comparing graph structure. Still, MGR collapses the center of nautilus into a single point, while out algorithm is able to recover the structure more accurately2. (a) (b) (c) Figure 2: (a) & (b): Edge detection in images. (c) Sharp feature curves detection. We now proceed with three examples of our algorithms applied to different datasets. Example 1: Image edge detection and surface feature curve reconstruction. In edge detection from images, it is often easy to identify (disconnected) points potentially lying on an edge. We can then use our Reeb-graph algorithm to connect them into curves. See Figure 1 and 2 (a), (b) for some examples. The yellow (light) shades are input potential edge-points computed by a standard edge-detection algorithm based on Roberts edge ﬁlter. Original images are given in the full version [13]. In Figure 2 (c), we are given a set of points sampled from a hidden surface model (gray points), and the goal is to extract (sharp) feature curves from it automatically. We ﬁrst identify points lying around sharp feature lines using a local differential operator (yellow points) and apply our algorithm to connect them into feature lines/graphs (dark curves). 1 Larger σ for kernel density estimation ﬁxes that problem but causes important features to disappear. 2Tuning the parameters of MGR does not seem to help, see the full version [13] for details. 7 Example 2: Speech data. The input speech data contains utterances of single digits by different speakers. Each utterance is sampled every 10msec with a moving frame. Each sample is represented by the ﬁrst 13 coefﬁcients resulting from running the standard Perceptual Linear Prediction (PLP) algorithm on the wave ﬁle. Given this setup, each utterance of a digit is a trajectory in this 13D feature space. −5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2 −0.5 0 0.5 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2 −0.5 0 0.5 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 In the left panel, we show the trajectory of an utterance of digit ‘1’ projected to IR3. The right panel shows the graph reconstructed by our algorithm by treating the input simply as a set of points (i.e, removing the time sequence information). No postprocessing is performed. Note the main portion of the utterance (the large loop) is well-reconstructed. The cluster of points in the right side corresponds to sampling of silence at the beginning and end of the utterance. This indicates that our algorithm can potentially be used to automatically reconstruct trajectories when the time information is lost. −5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2 −1 −0.5 0 0.5 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 −5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2 −1 −0.5 0 0.5 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 Next, we combine three utterances of the digit ‘1’ and construct the graph from the resulting point cloud shown in the left panel. Each color represents the point cloud coming from one utterance of ‘1’. As shown in the right panel, the graph reconstructed by our algorithm automatically aligns these three utterances (curves) in the feature space: well-aligned subcurves are merged into single pieces along the graph skeleton, while divergent portions will appear as branches and loops in the graph (see the loops on the left-side of this picture). We expect that our methods can be used to produce a summary representation for multiple similar trajectories (low and high-dimensional curves), to both align trajectories with no time information and to discover convergent and divergent portions of the trajectories. Example 3: Molecular simulation. The input is a molecular simulation data using the replicaexchange molecular dynamics method [23]. It contains 250K protein conformations, generated by 20 simulation runs, each of which produces a trajectory in the protein conformational space. The ﬁgure on the right shows a 3D-projection of the Reeb graph constructed by our algorithm. Interestingly, ﬁlaments structures can be seen at the beginning of the simulation, which indicates the 20 trajectories at high energy level. As the simulation proceeds, these different simulation runs converge and about 40% of the data points are concentrated in the oval on the right of the ﬁgure, which correspond to low-energy conformations. Ideally, simulations at low energy should provide a good sampling in the protein conformational space around the native structure of this protein. However, it turns out that there are several large loops in the Reeb graph close to the native structure (the conformation with lowest energy). Such loop features could be of interest for further investigation. Combining with principal curve algorithms. Finally, our algorithm can be used in combination with principal curve algorithms. In particular, one way is to use our algorithm to ﬁrst decompose the input data into different arcs of a graph structure, and then use a principal curve algorithm to compute an embedding of this arc in the center of points contributing to it. Alternatively, we can ﬁrst use the LDPC algorithm [22] to move points to the center of the data, and then perform our algorithm to connect them into a graph structure. Some preliminary results on such combination applied to the hand-written Chinese character can be found in the full version [13]. Acknowledgments. The authors thank D. Chen and U. Ozertem for kindly providing their software and for help with using the software. This work was in part supported by the NSF under CCF-0747082, CCF-1048983, CCF-1116258, IIS-1117707, IIS-0643916. 8 References [1] Open street map. http://www.openstreetmap.org/. [2] M. Aanjaneya, F. Chazal, D. Chen, M. Glisse, L. Guibas, and D. Morozov. Metric graph reconstruction from noisy data. In Proc. 27th Sympos. Comput. Geom., 2011. [3] P. K. Agarwal, H. Edelsbrunner, J. Harer, and Y. Wang. Extreme elevation on a 2-manifold. Discrete and Computational Geometry (DCG), 36(4):553–572, 2006. [4] M. Belkin and P. Niyogi. Laplacian Eigenmaps for dimensionality reduction and data representation. Neural Comp, 15(6):1373–1396, 2003. [5] P. Bendich, B. Wang, and S. Mukherjee. Local homology transfer and stratiﬁcation learning. 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4,432	MAP Inference for Bayesian Inverse Reinforcement Learning Jaedeug Choi and Kee-Eung Kim bDepartment 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 The difﬁculty in inverse reinforcement learning (IRL) arises in choosing the best reward function since there are typically an inﬁnite number of reward functions that yield the given behaviour data as optimal. Using a Bayesian framework, we address this challenge by using the maximum a posteriori (MAP) estimation for the reward function, and show that most of the previous IRL algorithms can be modeled into our framework. We also present a gradient method for the MAP estimation based on the (sub)differentiability of the posterior distribution. We show the effectiveness of our approach by comparing the performance of the proposed method to those of the previous algorithms. 1 Introduction The objective of inverse reinforcement learning (IRL) is to determine the decision making agent’s underlying reward function from its behaviour data and the model of environment [1]. The signiﬁcance of IRL has emerged from problems in diverse research areas. In animal and human behaviour studies [2], the agent’s behaviour could be understood by the reward function since the reward function reﬂects the agent’s objectives and preferences. In robotics [3], IRL provides a framework for making robots learn to imitate the demonstrator’s behaviour using the inferred reward function. In other areas related to reinforcement learning, such as neuroscience [4] and economics [5], IRL addresses the non-trivial problem of ﬁnding an appropriate reward function when building a computational model for decision making. In IRL, we generally assume that the agent is an expert in the problem domain and hence it behaves optimally in the environment. Using the Markov decision process (MDP) formalism, the IRL problem is deﬁned as ﬁnding the reward function that the expert is optimizing given the behaviour data of state-action histories and the environment model of state transition probabilities. In the last decade, a number of studies have addressed IRL in a direct (reward learning) and indirect (policy learning by inferring the reward function, i.e., apprenticeship learning) fashions. Ng and Russell [6] proposed a sufﬁcient and necessary condition on the reward functions that guarantees the optimality of the expert’s policy and formulated a linear programming (LP) problem to ﬁnd the reward function from the behaviour data. Extending their work, Abbeel and Ng [7] presented an algorithm for ﬁnding the expert’s policy from its behaviour data with a performance guarantee on the learned policy. Ratliff et al. [8] applied the structured max-margin optimization to IRL and proposed a method for ﬁnding the reward function that maximizes the margin between the expert’s policy and all other policies. Neu and Szepesv´ari [9] provided an algorithm for ﬁnding the policy that minimizes the deviation from the behaviour. Their algorithm uniﬁes the direct method that minimizes a loss function of the deviation and the indirect method that ﬁnds an optimal policy from the learned reward function using IRL. Syed and Schapire [10] proposed a method to ﬁnd a policy that improves the expert’s policy using a game-theoretic framework. Ziebart et al. [11] adopted the principle of the 1 maximum entropy for learning the policy whose feature expectations are constrained to match those of the expert’s behaviour. In addition, Neu and Szepesv´ari [12] provided a (non-Bayesian) uniﬁed view for comparing the similarities and differences among previous IRL algorithms. IRL is an inherently ill-posed problem since there may be an inﬁnite number of reward functions that yield the expert’s policy as optimal. Previous approaches summarized above employ various preferences on the reward function to address the non-uniqueness. For example, Ng and Russell [6] search for the reward function that maximizes the difference in the values of the expert’s policy and the second best policy. More recently, Ramachandran and Amir [13] presented a Bayesian approach formulating the reward preference as the prior and the behaviour compatibility as the likelihood, and proposed a Markov chain Monte Carlo (MCMC) algorithm to ﬁnd the posterior mean of the reward function. In this paper, we propose a Bayesian framework subsuming most of the non-Bayesian IRL algorithms in the literature. This is achieved by searching for the maximum-a-posteriori (MAP) reward function, in contrast to computing the posterior mean. We show that the posterior mean can be problematic for the reward inference since the loss function is integrated over the entire reward space, even including those inconsistent with the behaviour data. Hence, the inferred reward function can induce a policy much different from the expert’s policy. The MAP estimate, however, is more robust in the sense that the objective function (the posterior probability in our case) is evaluated on a single reward function. In order to ﬁnd the MAP reward function, we present a gradient method using the differentiability result of the posterior, and show the effectiveness of our approach through experiments. 2 Preliminaries 2.1 MDPs A Markov decision process (MDP) is deﬁned as a tuple ⟨S, A, T, R, γ, α⟩: S is the ﬁnite set of states; A is the ﬁnite set of actions; T is the state transition function where T(s, a, s′) denotes the probability P(s′\|s, a) of changing to state s′ from state s by taking action a; R is the reward function where R(s, a) denotes the immediate reward of executing action a in state s, whose absolute value is bounded by Rmax; γ ∈[0, 1) is the discount factor; α is the initial state distribution where α(s) denotes the probability of starting in state s. Using matrix notations, the transition function is denoted as an \|S\|\|A\| × \|S\| matrix T , and the reward function is denoted as an \|S\|\|A\|-dimensional vector R. A policy is deﬁned as a mapping π : S →A. The value of policy π is the expected discounted return of executing the policy and deﬁned as V π = E [P∞ t=0 γtR(st, at)\|α, π] where the initial state s0 is determined according to initial state distribution α and action at is chosen by policy π in state st. 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′) such that the value of policy π is calculated by V π = P s α(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′). We can rewrite the equations for the value function and the Q-function in matrix notations as V π = Rπ + γT πV π, Qπ a = Ra + γT aV π (1) where T π is an \|S\| × \|S\| matrix with the (s, s′) element being T(s, π(s), s′), T a is an \|S\| × \|S\| matrix with the (s, s′) element being T(s, a, s′), Rπ is an \|S\|-dimensional vector with the s-th element being R(s, π(s)), Ra is an \|S\|-dimensional vector with the s-th element being R(s, a), and Qπ a is an \|S\|-dimensional vector with the s-th element being Qπ(s, a). An optimal policy π∗maximizes the value function for all the states, and thus should satisfy the Bellman optimality equation: π is an optimal policy if and only if for all s ∈S, π(s) ∈ argmaxa∈A Qπ(s, a). We denote V ∗= V π∗and Q∗= Qπ∗. When the state space is large, the reward function is often linearly parameterized: R(s, a) = Pd i=1 wiφi(s, a) with the basis functions φi : S × A →R and the weight vector w = [w1, w2, · · · , wd]. Each basis function φi has a corresponding basis value V π i of policy π : V π i = E [P∞ t=0 γtφi(st, at)\|α, π]. 2 We also assume that the expert’s behaviour is given as the set X of M trajectories executed by the expert’s policy πE, where the m-th trajectory is an H-step sequence of state-action pairs: {(sm 1 , am 1 ), (sm 2 , am 2 ), · · · , (sm H, am H)}. Given the set of trajectories, the value and the basis value of the expert’s policy πE can be empirically estimated by ˆV E = 1 M PM m=1 PH h=1 γh−1R(sm h , am h ), ˆV E i = 1 M PM m=1 PH h=1 γh−1φi(sm h , am h ). In addition, we can empirically estimate the expert’s policy ˆπE and its state visitation frequency ˆµE from the trajectories: ˆπE(s, a) = PM m=1 PH h=1 1(sm h =s∧am h =a) PM m=1 PH h=1 1(sm h =s) , ˆµE(s) = 1 MH PM m=1 PH h=1 1(sm h =s). In the rest of the paper, we use the notation f(R) or f(x; R) for function f in order to be explicit that f is computed using reward function R. For example, the value function V π(s; R) denotes the value of policy π for state s using reward function R. 2.2 Reward Optimality Condition Ng and Russell [6] presented a necessary and sufﬁcient condition for reward function R of an MDP to guarantee the optimality of policy π: Qπ a(R) ≤V π(R) for all a ∈A. From the condition, we obtain the following corollary (although it is a succinct reformulation of the theorem in [6], the proof is provided in the supplementary material). Corollary 1 Given an MDP\R ⟨S, A, T, γ, α⟩, policy π is optimal if and only if reward function R satisﬁes h I −(IA −γT )(I −γT π)−1Eπi R ≤0, (2) where Eπ is an \|S\| × \|S\|\|A\| matrix with the (s, (s′, a′)) element being 1 if s = s′ and π(s′) = a′, and IA is an \|S\|\|A\| × \|S\| matrix constructed by stacking the \|S\| × \|S\| identity matrix \|A\| times. We refer to Equation (2) as the reward optimality condition w.r.t. policy π. Since the linear inequalities deﬁne the region of the reward functions that yield policy π as optimal, we refer to the region bounded by Equation (2) as the reward optimality region w.r.t. policy π. Note that there exist inﬁnitely many reward functions in the reward optimality region even including constant reward functions (e.g. R = c1 where c ∈[−Rmax, Rmax]). In other words, even when we are presented with the expert’s policy, there are inﬁnitely many reward functions to choose from, including the degenerate ones. To resolve this non-uniqueness in solutions, IRL algorithms in the literature employ various preferences on reward functions. 2.3 Bayesian framework for IRL (BIRL) Ramachandran and Amir [13] proposed a Bayesian framework for IRL by encoding the reward function preference as the prior and the optimality conﬁdence of the behaviour data as the likelihood. We refer to their work as BIRL. Assuming the rewards are i.i.d., the prior in BIRL is computed by P(R) = Q s∈S,a∈A P(R(s, a)). (3) Various distributions can be used as the prior. For example, the uniform prior can be used if we have no knowledge about the reward function other than its range, and a Gaussian or a Laplacian prior can be used if we prefer rewards to be close to some speciﬁc values. The likelihood in BIRL is deﬁned as an independent exponential distribution analogous to the softmax function: P(X\|R) = M Y m=1 H Y h=1 P(am h \|sm h , R) = M Y m=1 H Y h=1 exp(βQ∗(sm h , am h ; R)) P a∈A exp(βQ∗(sm h , a; R)) (4) 3 0 0.2 0.4 0.6 0.8 1 0 0.5 10 0.02 0.04 R(s1) R(s5) P(R(s1), R(s5)\|X) (a) (b) Figure 1: (a) 5-state chain MDP. (b) Posterior for R(s1) and R(s5) of the 5-state chain MDP. where β is a parameter that is equivalent to the inverse of temperature in the Boltzmann distribution. The posterior over the reward function is then formulated by combining the prior and the likelihood, using Bayes theorem: P(R\|X) ∝P(X\|R)P(R). (5) BIRL uses a Markov chain Monte Carlo (MCMC) algorithm to compute the posterior mean of the reward function. 3 MAP Inference in Bayesian IRL In the Bayesian approach to IRL, the reward function can be determined using different estimates, such as the posterior mean, median, or maximum-a-posterior (MAP). The posterior mean is commonly used since it can be shown to be optimal under the mean square error function. However, the problem with the posterior mean in Bayesian IRL is that the error is integrated over the entire space of reward functions, even including inﬁnitely many rewards that induce policies inconsistent with the behaviour data. This can yield a posterior mean reward function with an optimal policy again inconsistent with the data. On the other hand, the MAP does not involve an objective function that is integrated over the reward function space; it is simply a point that maximizes the posterior probability. Hence, it is more robust to inﬁnitely many inconsistent reward functions. We present a simple example that compares the posterior mean and the MAP reward function estimation. Consider an MDP with 5 states arranged in a chain, 2 actions, and the discount factor 0.9. As shown in Figure 1(a), we denote the leftmost state as s1 and the rightmost state as s5. Action a1 moves to the state on the right with probability 0.6 and to the state on the left with probability 0.4. Action a2 always moves to state s1. The true reward of each state is [0.1, 0, 0, 0, 1], hence the optimal policy chooses a1 in every state. Suppose that we already know R(s2), R(s3), and R(s4) which are all 0, and estimate R(s1) and R(s5) from the behaviour data X which contains optimal actions for all the states. We can compute the posterior P(R(s1), R(s5)\|X) using Equations (3), (4), and (5) under the assumption that 0 ≤R ≤1 and priors P(R(s1)) being N(0.1, 1), and P(R(s5)) being N(1, 1). The reward optimality region can be also computed using Equation (2). Figure 1(b) presents the posterior distribution of the reward function. The true reward, the MAP reward, and the posterior mean reward are marked with the black star, the blue circle, and the red cross, respectively. The black solid line is the boundary of the reward optimality region. Although the prior mean is set to the true reward, the posterior mean is outside the reward optimality region. An optimal policy for the posterior mean reward function chooses action a2 rather than action a1 in state s1, while an optimal policy for the MAP reward function is identical to the true one. The situation gets worse when using the uniform prior. An optimal policy for the posterior mean reward function chooses action a2 in states s1 and s2, while an optimal policy for the MAP reward function is again identical to the true one. In the rest of this section, we additionally show that most of the IRL algorithms in the literature can be cast as searching for the MAP reward function in Bayesian IRL. The main insight comes from the fact that these algorithms try to optimize an objective function consisting of a regularization term for the preference on the reward function and an assessment term for the compatibility of the reward function with the behaviour data. The objective function is naturally formulated as the posterior in a Bayesian framework by encoding the regularization into the prior and the data compatibility into the likelihood. In order to subsume different approaches used in the literature, we generalize the 4 Table 1: IRL algorithms and their equivalent f(X; R) and prior for the Bayesian formulation. q ∈ {1, 2} is for representing L1 or L2 slack penalties. Previous algorithm f(X; R) Prior Ng and Russell’s IRL from sampled trajectories [6] fV Uniform MMP without the loss function [8] (fV )q Gaussian MWAL [10] fG Uniform Policy matching [9] fJ Uniform MaxEnt [11] fE Uniform likelihood in Equation (4) to the following: P(X\|R) ∝exp(βf(X; R)) where β is a parameter for scaling the likelihood and f(X; R) is a function which will be deﬁned appropriately to encode the data compatibility assessment used in each IRL algorithm. We then have the following result (the proof is provided in the supplementary material): Theorem 1 IRL algorithms listed in Table 1 are equivalent to computing the MAP estimates with the prior and the likelihood using f(X; R) deﬁned as follows: • fV (X; R) = ˆV E(R) −V ∗(R) • fG(X; R) = mini h V π∗(R) i −ˆV E i i • fJ(X; R) = −P s,a ˆµE(s) (J(s, a; R) −ˆπE(s, a))2 • fE(X; R) = log PMaxEnt(X\|T , R) where π∗(R) is an optimal policy induced by the reward function R, J(s, a; R) is a smooth mapping from reward function R to a greedy policy such as the soft-max function, and PMaxEnt is the distribution on the behaviour data (trajectory or path) satisfying the principle of maximum entropy. The MAP estimation approach provides a rich framework for explaining the previous non-Bayesian IRL algorithms in a uniﬁed manner, as well as encoding various types of a priori knowledge into the prior distribution. Note that this framework can exploit the insights behind apprenticeship learning algorithms even if they do not explicitly learn a reward function (e.g., MWAL [10]). 4 A Gradient Method for Finding the MAP Reward Function We have proposed a unifying framework for Bayesian IRL and suggested that the MAP estimate can be a better solution to the IRL problem. We can then reformulate the IRL problem into the posterior optimization problem, which is ﬁnding RMAP that maximizes the (log unnormalized) posterior: RMAP = argmaxR P(R\|X) = argmaxR [log P(X\|R) + log P(R)] Before presenting a gradient method for the optimization problem, we show that the generalized likelihood is differentiable almost everywhere. The likelihood is deﬁned for measuring the compatibility of the reward function R with the behaviour data X. This is often accomplished using the optimal value function V ∗or the optimal Q-function Q∗w.r.t. R. For example, the empirical value of X is compared with V ∗[6, 8], X is directly compared to the learned policy (e.g. the greedy policy from Q∗) [9], or the probability of following the trajectories in X is computing using Q∗[13]. Thus, we generally assume that P(X\|R) = g(X, V ∗(R)) or g(X, Q∗(R)) where g is differentiable w.r.t. V ∗or Q∗. The remaining question is the differentiability of V ∗and Q∗w.r.t. R, which we address in the following two theorems (The proofs are provided in the supplementary material.): Theorem 2 V ∗(R) and Q∗(R) are convex. Theorem 3 V ∗(R) and Q∗(R) are differentiable almost everywhere. Theorems 2 and 3 relate to the previous work on gradient methods for IRL. Neu and Szepesv´ari [9] showed that Q∗(R) is Lipschitz continuous, and except on a set of measure zero (almost everywhere), it is Fr´echet differentiable by Rademacher’s theorem. We have obtained the same result 5 based on the reward optimality region, and additionally identiﬁed the condition for which V ∗(R) and Q∗(R) are non-differentiable (refer to the proof for details). Ratliff et al. [8] used a subgradient of their objective function because it involves differentiating V ∗(R). Using Theorem 3 for computing the subgradient of their objective function yields an identical result. Assuming a differentiable prior, we can compute the gradient of the posterior using the result in Theorem 3 and the chain rule. If the posterior is convex, we will ﬁnd the MAP reward function. Otherwise, as in [9], we will obtain a locally optimal solution. In the next section, we will experimentally show that the locally optimal solutions are nonetheless better than the posterior mean in practice. This is due to the property that they are generally found within the reward optimality region w.r.t. the policy consistent with the behaviour data. The gradient method uses the update rule Rnew ←R + δt∇RP(R\|X) where δt is an appropriate step-size (or learning rate). Since computing ∇RP(R\|X) involves computing an optimal policy for the current reward function and a matrix inversion, caching these results helps reduce repetitive computation. The idea is to compute the reward optimality region for checking whether we can reuse the cached result. If Rnew is inside the reward optimality region of an already visited reward function R′, they share the same optimal policy and hence the same ∇RV π(R) or ∇RQπ(R). Given policy π, the reward optimality region is deﬁned by Hπ = I −(IA −γT )(I −γT π)−1Eπ, and we can reuse the cached result if Hπ · Rnew ≤0. The gradient method using this idea is presented in Algorithm 1. Algorithm 1 Gradient method for MAP inference in Bayesian IRL Input: MDP\R, behaviour data X, step-size sequence {δt}, number of iterations N 1: Initialize R 2: π ←solveMDP(R) 3: H π ←computeRewardOptRgn(π) 4: Π ←{⟨π, H π⟩} 5: for t = 1 to N do 6: Rnew ←R + δt∇RP(R\|X) 7: if isNotInRewardOptRgn(Rnew, H π) then 8: ⟨π, H π⟩←ﬁndRewardOptRgn(Rnew, Π) 9: if isEmpty(⟨π, H π⟩) then 10: π ←solveMDP(Rnew) 11: H π ←computeRewardOptRgn(π) 12: Π ←Π ∪{⟨π, H π⟩} 13: end if 14: end if 15: R ←Rnew 16: end for 5 Experimental Results The ﬁrst set of experiments was conducted in N × N gridworlds [7]. The agent can move west, east, north, or south, but with probability 0.3, it fails and moves in a random direction. The grids are partitioned into M × M non-overlapping regions, so there are ( N M )2 regions. The basis function is deﬁned by a 0-1 indicator function for each region. The linearly parameterized reward function is determined by the weight vector w sampled i.i.d. from a zero mean Gaussian prior with variance 0.1 and \|wi\| ≤1 for all i. The discount factor is set to 0.99. We compared the performance of our gradient method to those of other IRL algorithms in the literature: Maximum Margin Planning (MMP) [8], Maximum Entropy (MaxEnt) [11], Policy Matching with natural gradient (NatPM) and the plain gradient (PlainPM) [9], and Bayesian Inverse Reinforcement Learning (BIRL) [13]. We executed our gradient method for ﬁnding MAP using three different choices of the likelihood: B denotes the BIRL likelihood, and E and J denote the likelihood with fE and fJ, respectively. For the Bayesian IRL algorithms (BIRL and MAP), two types of the prior are prepared: U denotes the uniform prior and G denotes the true Gaussian prior. We evaluated the performance of the algorithms using the difference between V ∗(the value of the expert’s policy) and V L(the value of the optimal policy induced by the learned weight wL measured on the true weight w∗) and the difference between w∗and wL using L2 norm. 6 Table 2: Results in the gridworld problems. ∥w∗−wL ∥2 V ∗−V L \|S\| 24 × 24 32 × 32 24 × 24 32 × 32 dim(w) 36 144 576 64 256 1024 36 144 576 64 256 1024 NatPM 3.04 6.84 16.83 3.50 8.88 21.25 2.49 8.97 8.74 1.08 12.84 10.97 PlainPM 3.77 6.63 16.60 5.21 9.05 17.36 0.15 0.67 0.51 0.41 1.28 1.91 MaxEnt 6.05 11.98 22.11 7.91 15.48 25.52 0.33 0.60 0.60 0.95 2.22 2.91 MMP 0.85 1.26 2.38 0.83 1.61 3.17 10.74 16.32 13.72 13.58 10.59 8.87 BIRL-U 3.27 5.67 n.a. 3.78 7.89 n.a. 1.38 0.80 n.a. 0.35 2.24 n.a. BIRL-G 0.86 1.36 n.a. 0.98 1.71 n.a. 2.21 0.54 n.a. 0.50 0.90 n.a. MAP-B-U 4.45 8.46 13.87 5.68 10.50 18.21 0.13 0.57 1.06 1.63 1.34 2.17 MAP-B-G 0.83 1.30 2.40 0.94 1.62 3.17 0.16 0.45 0.40 0.41 0.77 0.87 MAP-E-G 0.83 1.22 2.33 0.76 1.53 3.13 0.19 0.44 0.42 0.43 1.29 1.88 MAP-J-G 0.48 1.10 2.30 0.65 1.51 3.11 0.17 0.42 0.37 0.38 0.90 1.21 0 20 40 60 80 1 2 3 CPU time (sec) ∥w∗−wL ∥2 0 20 40 60 80 0 5 10 15 CPU time (sec) V ∗−V L 0 200 400 600 800 1 2 3 4 5 CPU time (sec) ∥w∗−wL ∥2 0 200 400 600 800 0 10 20 CPU time (sec) V ∗−V L BIRL MAP−B (a) (b) Figure 2: CPU timing results of BIRL and MAP-B in 24 × 24 gridworld problem. (a) dim(w) = 36. (b) dim(w) = 144. We used training data with 10 trajectories of 50 time steps, collected from the simulated runs of the expert’s policy. Table 2 shows the average performance over 10 training data. Most of the algorithms found the weight that induces an optimal policy whose performance is as good as that of the expert’s policy (i.e., small V ∗−V L) except for MMP and NatPM. The poor performance of MMP was due to the small size in the training data, as already noted in [14]. The poor performance of NatPM may be due to the ineffectiveness of pseudo-metric in high dimensional reward spaces, since PlainPM was able produce good performance. Regarding the learned weights, the algorithms using the true prior (MMP, BIRL, and the variants of MAP) found the weight close to the true one (i.e., small \|\|w∗−wL\|\|2). Comparing BIRL and MAP-B is especially meaningful since they share the same prior and likelihood. The only difference was in computing the mean versus MAP from the posterior. MAP-B was consistently better than BIRL in terms of both \|\|w∗−wL\|\|2 and V ∗−V L. Finally, we note that the correct prior yields small \|\|w∗−wL\|\|2 and V ∗−V L when we compare PlainPM, MaxEnt, BIRL-U, and MAP-B-U (uniform prior) to MAP-J-G, MAP-E-G, BIRL-G, and MAP-B-G (Gaussian prior), respectively. Figure 2 compares the CPU timing results of the MCMC algorithm in BIRL and the gradient method in MAP-B for the 24×24 gridworld with 36 and 144 basis functions. BIRL takes much longer CPU time to converge than MAP-B since the former takes much larger number of iterations to converge, and in addition, each iteration requires solving an MDP with a sampled reward function. The CPU time gap gets larger as we increase the dimension of the reward function. Caching the optimal policies and gradients sped up the gradient method by factors of 1.5 to 4.2 until convergence, although not explicitly shown in the ﬁgure. The second set of experiments was performed on a simpliﬁed car race problem, modiﬁed from [14]. The racetrack is shown in Figure 3. The shaded and white cells indicate the off-track and on-track locations, respectively. The state consists of the location and velocity of the car. The velocities in the vertical and horizontal directions are represented as 0, 1, or 2, and the net velocity is computed as the squared sum of directional velocities. The net velocity is regarded as high if greater than 2, zero if 0, and low otherwise. The car can increase, decrease, or maintain one of the directional velocities. The control of the car succeeds with p=0.9 if the net velocity is low, but p=0.6 if high. If the control fails, the velocity is maintained, and if the car attempts to move outside the racetrack, it remains in the previous location with velocity 0. The basis functions are 0-1 indicator functions for the goal locations, off-track locations, and 3 net velocity values (zero, low, high) while the car is on track. Hence, there are 3150 states, 5 actions, and 5 basis functions. The discount factor is set to 0.99. 7 Table 3: True and learned weights in the car race problem. Goal Off-track Velocity while on track Zero Low High Fast expert 1.00 0.00 0.00 0.00 0.10 BIRL 0.96±0.02 -0.20±0.03 -0.04±0.01 -0.12±0.02 0.32±0.02 MAP-B 1.00±0.00 -0.19±0.02 -0.03±0.01 -0.13±0.01 0.29±0.01 Table 4: Statistics of the policies simulated in the car race problem. Avg. steps Avg. steps in locations Avg. steps in velocity to goal Off-track On-track Zero Low High Fast expert 20.41 1.56 17.85 2.01 3.40 12.44 BIRL 32.98±6.42 2.13±0.60 29.85±6.03 3.33±0.52 4.34±0.79 22.18±4.84 MAP-B 24.77±1.92 1.68±0.26 22.09±1.71 2.70±0.16 3.38±0.18 16.01±1.48 We designed two experts. The slow expert prefers low velocity and avoids the off-track locations, w = [1, −0.1, 0, 0.1, 0]. The fast expert prefers high velocity, w = [1, 0, 0, 0, 0.1]. We compared the posterior mean and the MAP using the prior P(w1)=N(1, 1) and P(w2)=P(w3)=P(w4)=P(w5)= N(0, 1) assuming that we do not know the experts’ preference on the locations nor the velocity, but we know the experts’ ultimate goal is to reach one of the goal locations. We used BIRL for the posterior mean and MAP-B for the MAP estimation, hence using the identical prior and likelihood. We used 10 training data, each consisting of 5 trajectories. We omit S S G G G G Figure 3: Racetrack. the results regarding the slow expert since both BIRL and MAPB successfully found the weight similar with the true one, which induced the slow expert’s policy as optimal. However for the fast expert, MAP-B was signiﬁcantly better than BIRL.1 Table 3 shows the true and learned weights, and Table 4 shows some statistics characterizing the expert’s and learned policies. The policy from BIRL tends to remain in high speed on the track for signiﬁcantly more steps than the one from MAP-B since BIRL converged to a larger ratio of w5 to w1. 6 Conclusion We have argued that, when using a Bayesian framework for learning reward functions in IRL, the MAP estimate is preferable over the posterior mean. Experimental results conﬁrmed the effectiveness of our approach. We have also shown that the MAP estimation approach subsumes nonBayesian IRL algorithms in the literature, and allows us to incorporate various types of a priori knowledge about the reward functions and the measurement of the compatibility with behaviour data. We proved that the generalized posterior is differentiable almost everywhere, and proposed a gradient method to ﬁnd a locally optimal solution to the MAP estimation. We provided the theoretical result equivalent to the previous work on gradient methods for non-Bayesian IRL, but used a different proof based on the reward optimality region. Our work could be extended in a number of ways. For example, the IRL algorithm for partially observable environments in [15] mostly relies on Ng and Russell [6]’s heuristics for MDPs, but our work opens up new opportunities to leverage the insight behind other IRL algorithms for MDPs. Acknowledgments This work was supported by National Research Foundation of Korea (Grant# 2009-0069702) and the Defense Acquisition Program Administration and the Agency for Defense Development of Korea (Contract# UD080042AD) 1All the results in Table 4 except for the average number of steps in the off-track locations are statistically signiﬁcant at the 95% conﬁdence level. 8 References [1] S. Russell. Learning agents for uncertain environments (extended abstract). In Proceedings of COLT, 1998. [2] P. R. Montague and G. S. Berns. Neural economics and the biological substrates of valuation. Neuron, 36(2), 2002. [3] B. D. Argall, S. Chernova, M. Veloso, and B. Browning. A survey of robot learning from demonstration. Robotics and Autonomous Systems, 57(5), 2009. [4] Y. Niv. Reinforcement learning in the brain. Journal of Mathematical Psychology, 53(3), 2009. [5] E. Hopkins. Adaptive learning models of consumer behavior. Journal of Economic Behavior and Organization, 64(3–4), 2007. [6] A. Y. Ng and S. Russell. Algorithms for inverse reinforcement learning. In Proceedings of ICML, 2000. [7] P. Abbeel and A. Y. Ng. Apprenticeship learning via inverse reinforcement learning. In Proceedings of ICML, 2004. [8] N. D. Ratliff, J. A. Bagnell, and M. A. Zinkevich. Maximum margin planning. In Proceedings of ICML, 2006. [9] G. Neu and C. Szepesv´ari. Apprenticeship learning using inverse reinforcement learning and gradient methods. In Proceedings of UAI, 2007. [10] U. Syed and R. E. Schapire. A game-theoretic approach to apprenticeship learning. In Proceedings of NIPS, 2008. [11] B. D. Ziebart, A. Maas, J. A. Bagnell, and A. K. Dey. Maximum entropy inverse reinforcement learning. In Proceedings of AAAI, 2008. [12] G. Neu and C. Szepesv´ari. Training parsers by inverse reinforcement learning. Machine Learning, 77(2), 2009. [13] D. Ramachandran and E. Amir. Bayesian inverse reinforcement learning. In Proceedings of IJCAI, 2007. [14] A. Boularias and B. Chaib-Draa. Bootstrapping apprenticeship learning. In Proceedings of NIPS, 2010. [15] J. Choi and K. Kim. Inverse reinforcement learning in partially observable environments. In Proceedings of IJCAI, 2009. 9	2011	82
4,433	Learning Sparse Representations of High Dimensional Data on Large Scale Dictionaries Zhen James Xiang Hao Xu Peter J. Ramadge Department of Electrical Engineering, Princeton University Princeton, NJ 08544, USA {zxiang,haoxu,ramadge}@princeton.edu Abstract Learning sparse representations on data adaptive dictionaries is a state-of-the-art method for modeling data. But when the dictionary is large and the data dimension is high, it is a computationally challenging problem. We explore three aspects of the problem. First, we derive new, greatly improved screening tests that quickly identify codewords that are guaranteed to have zero weights. Second, we study the properties of random projections in the context of learning sparse representations. Finally, we develop a hierarchical framework that uses incremental random projections and screening to learn, in small stages, a hierarchically structured dictionary for sparse representations. Empirical results show that our framework can learn informative hierarchical sparse representations more efﬁciently. 1 Introduction Consider approximating a p-dimensional data point x by a linear combination x ≈Bw of m (possibly linearly dependent) codewords in a dictionary B = [b1, b2, . . . , bm]. Doing so by imposing the additional constraint that w is a sparse vector, i.e., x is approximated as a weighted sum of only a few codewords in the dictionary, has recently attracted much attention [1]. As a further reﬁnement, when there are many data points xj, the dictionary B can be optimized to make the representations wj as sparse as possible. This leads to the following problem. Given n data points in Rp organized as matrix X = [x1, x2, . . . , xn] ∈Rp×n, we want to learn a dictionary B = [b1, b2, . . . , bm] ∈Rp×m and sparse representation weights W = [w1, w2, . . . , wn] ∈Rm×n so that each data point xj is well approximated by Bwj with wj a sparse vector: min B,W 1 2∥X −BW∥2 F + λ∥W∥1 s.t. ∥bi∥2 2 ≤1, ∀i = 1, 2, . . . , m. (1) Here ∥·∥F and ∥·∥1 denote the Frobenius norm and element-wise l1-norm of a matrix, respectively. There are two advantages to this representation method. First, the dictionary B is adapted to the data. In the spirit of many modern approaches (e.g. PCA, SMT [2], tree-induced bases [3,4]), rather than ﬁxing B a priori (e.g. Fourier, wavelet, DCT), problem (1) assumes minimal prior knowledge and uses sparsity as a cue to learn a dictionary adapted to the data. Second, the new representation w is obtained by a nonlinear mapping of x. Algorithms such as Laplacian eigenmaps [5] and LLE [6], also use nonlinear mappings x 7→w. By comparison, l1-regularization enjoys a simple formulation with a single tuning parameter (λ). In many other approaches (including [2–4]), although the codewords in B are cleverly chosen, the new representation w is simply a linear mapping of x, e.g. w = B†x. In this case, training a linear model on w cannot learn nonlinear structure in the data. As a ﬁnal point, we note that the human visual cortex uses similar mechanisms to encode visual scenes [7] and sparse representation has exhibited superior performance on difﬁcult computer vision problems such as face [8] and object [9] recognition. 1 The challenge, however, is that solving the non-convex optimization problem(1) is computationally expensive. Most state-of-the-art algorithms solve (1) by iteratively optimizing W and B. For a ﬁxed B, optimizing W requires solving n, p-dimensional, lasso problems of size m. Using LARS [10] with a Cholesky-based implementation, each lasso problem has a computation cost of O(mpκ + mκ2), where κ is the number of nonzero coefﬁcients [11]. For a ﬁxed W, optimizing B is a least squares problem of pm variables and m constraints. In an efﬁcient algorithm [12], the dual formulation has only m variables but still requires inverting m × m matrices (O(m3) complexity). To address this challenge, we examine decomposing a large dictionary learning problem into a set of smaller problems. First (§2), we explore dictionary screening [13,14], to select a subset of codewords to use in each Lasso optimization. We derive two new screening tests that are signiﬁcantly better than existing tests when the data points and codewords are highly correlated, a typical scenario in sparse representation applications [15]. We also provide simple geometric intuition for guiding the derivation of screening tests. Second (§3), we examine projecting data onto a lower dimensional space so that we can control information ﬂow in our hierarchical framework and solve sparse representations with smaller p. We identify an important property of the data that’s implicitly assumed in sparse representation problems (scale indifference) and study how random projection preserves this property. These results are inspired by [16] and related work in compressed sensing. Finally (§4), we develop a framework for learning a hierarchical dictionary (similar in spirit to [17] and DBN [18]). To do so we exploit our results on screening and random projection and impose a zero-tree like structured sparsity constraint on the representation. This constraint is similar to the formulation in [19]. The key difference is that we learn the sparse representation stage-wise in layers and use the exact zero-tree sparsity constraint to utilize the information in previous layers to simplify the computation, whereas [19] uses a convex relaxation to approximate the structured sparsity constraint and learns the sparse representation (of all layers) by solving a single large optimization problem. Our idea of using incremental random projections is inspired by the work in [20, 21]. Finally, unlike [12] (that addresses the same computational challenge), we focus on a high level reorganization of the computations rather than improving basic optimization algorithms. Our framework can be combined with all existing optimization algorithms, e.g. [12], to attain faster results. 2 Reducing the Dictionary By Screening In this section we assume that all data points and codewords are normalized: ∥xj∥2 = ∥bi∥2 = 1, 1 ≤j ≤n, 1 ≤i ≤m (we discuss the implications of this assumption in §3). When B is ﬁxed, ﬁnding the optimal W in (1) requires solving n subproblems. The jth subproblem ﬁnds wj for xj. For notational simplicity, in this section we drop the index j and denote x = xj, w = wj = [w1, w2, . . . , wm]T . Each subproblem is then of the form: min w1,w2,...,wm 1 2∥x − m X i=1 wibi∥2 2 + λ m X i=1 \|wi\|. (2) To address the challenge of solving (2) for large m, we ﬁrst explore simple screening tests that identify and discard codewords bi guaranteed to have optimal solution ˜wi = 0. El Ghaoui’s SAFE rule [13] is an example of a simple screening test. We introduce some simple geometric intuition for screening and use this to derive new tests that are signiﬁcantly better than existing tests for the type of problems of interest here. To this end, it will help to consider the dual problem of (2): max θ 1 2∥x∥2 2 −λ2 2 ∥θ −x λ∥2 2 s.t. \|θT bi\| ≤1 ∀i = 1, 2, . . . , m. (3) As is well known (see the supplemental material), the optimal solution of the primal problem ˜w = [ ˜w1, ˜w2, . . . , ˜wm]T and the optimal solution of the dual problem ˜θ are related through: x = m X i=1 ˜wibi + λ˜θ, ˜θ T bi ∈ {sign ˜wi} if ˜wi ̸= 0, [−1, 1] if ˜wi = 0. (4) The dual formulation gives useful geometric intuition. Since ∥x∥2 = ∥bi∥2 = 1, x and all bi lie on the unit sphere Sp−1 (Fig.1(a)). For y on Sp−1, P(y) = {z : zT y = 1} is the tangent hyperplane 2 Feasible Region 0 x Sp−1 b* P(b) b1 P(b1) b2 P(b2) x/λmax→ x/λ θ (a) 0 Sp−1 q (b) Sp−1 0 x b x/λ x/λmax (c) 0.6 0.8 1 0 0.2 0.4 0.6 0.8 h / hmax Discarding Threshold hmax = 0.8 ST2, our new test. ST1/SAFE 0.6 0.8 1 0 0.2 0.4 0.6 0.8 h / hmax Discarding Threshold hmax = 0.9 ST2, our new test. ST1/SAFE (d) Figure 1: (a) Geometry of the dual problem. (b) Illustration of a sphere test. (c) The solid red, dotted blue and solid magenta circles leading to sphere tests ST1/SAFE, ST2, ST3, respectively. (d) The thresholds in ST2 and ST1/SAFE when λmax = 0.8 (top) and λmax = 0.9 (bottom). A higher threshold yields a better test. of Sp−1 at y and H(y) = {z : zT y ≤1} is the corresponding closed half space containing the origin. The constraints in (3) indicate that feasible θ must be in H(bi) and H(−bi) for all i. To ﬁnd ˜θ that maximizes the objective in (3), we must ﬁnd a feasible θ closest to x/λ. By (4), if ˜θ is not on P(bi) or P(−bi), then ˜wi = 0 and we can safely discard bi from problem (2). Let λmax = maxi \|xT bi\| and b∗∈{±bi}m i=1 be selected so that λmax = xT b∗. Note that θ = x/λmax is a feasible solution for (3). λmax is also the largest λ for which (2) has a nonzero solution. If λ > λmax, then x/λ itself is feasible, making it the optimal solution. Since it is not on any hyperplane P(bi) or P(−bi), ˜wi = 0, i = 1, . . . , m. Hence we assume that λ ≤λmax. These observations can be used for screening as follows. If we know that ˜θ is within a region R, then we can discard those bi for which the tangent hyperplanes P(bi) and P(−bi) don’t intersect R, since by (4) the corresponding ˜wi will be 0. Moreover, if the region R is contained in a closed ball (e.g. the shaded blue area in Fig.1(b)) centered at q with radius r, i.e., {θ: ∥θ −q∥2 ≤r}, then one can discard all bi for which \|qT bi\| is smaller than a threshold determined by the common tangent hyperplanes of the spheres ∥θ −q∥2 = r and Sp−1. This “sphere test” is made precise in the following lemma (All lemmata are proved in the supplemental material). Lemma 1. If the solution ˜θ of (3) satisﬁes ∥˜θ −q∥2 ≤r, then \|qT bi\| < (1 −r) ⇒˜wi = 0. El Ghaoui’s SAFE rule [13] is a sphere test of the simplest form. To see this, note that x/λmax is a feasible point of (3), so the optimal θ cannot be further away from x/λ than x/λmax. Therefore we have the constraint : ∥˜θ −x/λ∥2 ≤1/λ−1/λmax (solid red ball in Fig.1(c)). Plugging in q = x/λ and r = 1/λ −1/λmax into Lemma 1 yields El Ghaoui’s SAFE rule: Sphere Test # 1 (ST1/SAFE): If \|xT bi\| < λ −1 + λ/λmax, then ˜wi = 0. Note that the SAFE rule is weakest when λmax is large, i.e., when the codewords are very similar to the data points, a frequent situation in applications [15]. To see that there is room for improvement, consider the constraint: θT b∗≤1. This puts ˜θ in the intersection of the previous closed ball (solid red) and H(b∗). This is indicated by the shaded green region in Fig. 1(c). Since this intersection is small when λmax is large, a better test results by selecting R to be the shaded green region. However, to simplify the test, we relax R to a closed ball and use the sphere test of Lemma 1. Two relaxations, the solid magenta ball and the dotted blue ball in Fig. 1(c), are detailed in the following lemma. Lemma 2. If θ satisﬁes (a) ∥θ −x/λ∥2 ≤1/λ −1/λmax and (b) θT b∗≤1, then θ satisﬁes ∥θ −(x/λ −(λmax/λ −1)b∗∥2 ≤ p 1/λ2max −1(λmax/λ −1), and (5) ∥θ −x/λmax∥2 ≤ 2 p 1/λ2max −1(λmax/λ −1). (6) By Lemma 2, since ˜θ satisﬁes (a) and (b), it satisﬁes (5) and (6). We start with (6) because of its similarity to the closed ball constraint used to derive ST1/SAFE (solid red ball). Plugging q = x/λmax and r = 2 p 1/λ2max −1(λmax/λ −1) into Lemma 1 yields our ﬁrst new test: 3 Sphere Test # 2 (ST2): If \|xT bi\| < λmax(1 −2 p 1/λ2max −1(λmax/λ −1)), then ˜wi = 0. Since ST2 and ST1/SAFE both test \|xT bi\| against thresholds, we can compare the tests by plotting their thresholds. We do so for λmax = 0.8, 0.9 in Fig.1(d). The thresholds must be positive and large to be useful. ST2 is most useful when λmax is large. Indeed, we have the following lemma: Lemma 3. When λmax > √ 3/2, if ST1/SAFE discards bi, then ST2 also discards bi. Finally, to use the closed ball constraint (5), we plug in q = x/λ −(λmax/λ −1)b∗and r = p 1/λ2max −1(λmax/λ −1) into Lemma 1 to obtain a second new test: Sphere Test # 3 (ST3): If \|xT bi −(λmax −λ)bT ∗bi\| < λ(1 − p 1/λ2max −1(λmax/λ −1)), then ˜wi = 0. ST3 is slightly more complex. It requires ﬁnding b∗and computing a weighted sum of inner products. But ST3 is always better than ST2 since its sphere lies strictly inside that of ST2: Lemma 4. Given any x, b∗and λ, if ST2 discards bi, then ST3 also discards bi. To summarize, ST3 completely outperforms ST2, and when λmax is larger than √ 3/2 ≈0.866, ST2 completely outperforms ST1/SAFE. Empirical comparisons are given in §5. By making two passes through the dictionary, the above tests can be efﬁciently implemented on large-scale dictionaries that can’t ﬁt in memory. The ﬁrst pass holds x, u, bi ∈Rp in memory at once and computes u(i) = xT bi. By simple bookkeeping, after pass one we have b∗and λmax. The second pass holds u, b∗, bi in memory at once, computes bT ∗bi and executes the test. 3 Random Projections of the Data In §4 we develop a framework for learning a hierarchical dictionary and this involves the use of random data projections to control information ﬂow to the levels of the hierarchy. The motivation for using random projections will become clear, and is speciﬁcally discussed, in §4. Here we lay some groundwork by studying basic properties of random projections in learning sparse representations. We ﬁrst revisit the normalization assumption ∥xj∥2 = ∥bi∥2 = 1, 1 ≤j ≤n, 1 ≤i ≤m in §2. The assumption that all codewords are normalized: ∥bi∥2 = 1, is necessary for (1) to be meaningful, otherwise we can increase the scale of bi and inversely scale the ith row of W to lower the loss. The assumption that all data points are normalized: ∥xj∥2 = 1, warrants a more careful examination. To see this, assume that the data {xj}n j=1 are samples from an underlying low dimensional smooth manifold X and that one desires a correspondence between codewords and local regions on X. Then we require the following scale indifference (SI) property to hold: Deﬁnition 1. X satisﬁes the SI property if ∀x1, x2 ∈X, with x1 ̸= x2, and ∀γ ̸= 0, x1 ̸= γx2. Intuitively, SI means that X doesn’t contain points differing only in scale and it implies that points x1, x2 from distinct regions on X will use different codewords in their representation. SI is usually implicitly assumed [9,15] but it will be important for what follows to make the condition explicit. SI is true in many typical applications of sparse representation. For example, for image signals when we are interested in the image content regardless of image luminance. When SI holds we can indeed normalize the data points to Sp−1 = {x : ∥x∥2 = 1}. Since a random projection of the original data doesn’t preserve the normalization ∥xj∥2 = 1, it’s important for the random projection to preserve the SI property so that it is reasonable to renormalize the projected data. We will show that this is indeed the case under certain assumptions. Suppose we use a random projection matrix T ∈Rd×p, with orthonormal rows, to project the data to Rd (d < p) and use TX as the new data matrix. Such T can be generated by running the GramSchmidt procedure on d, p-dimensional random row vectors with i.i.d. Gaussian entries. It’s known that for certain sets X, with high probability random projection preserves pairwise distances: (1 −ϵ) p d/p ≤∥Tx1 −Tx2∥2 ∥x1 −x2∥2 ≤(1 + ϵ) p d/p. (7) For example, when X contains only κ-sparse vectors, we only need d ≥O(κ ln(p/κ)) and when X is a K-dimensional Riemannian submanifold, we only need d ≥O(K ln p) [16]. We will show that when the pairwise distances are preserved as in (7), the SI property will also be preserved: 4 Theorem 1. Deﬁne S(X) = {z : z = γx, x ∈X, \|γ\| ≤1}. If X satisﬁes SI and ∀(x1, x2) ∈ S(X) × S(X) (7) is satisﬁed, then T(X) = {z : z = Tx, x ∈X} also satisﬁes SI. Proof. If T(X) doesn’t satisfy SI, then by Deﬁnition 1, ∃(x1, x2) ∈X × X, γ /∈{0, 1} s.t.: Tx1 = γTx2. Without loss of generality we can assume that \|γ\| ≤1 (otherwise we can exchange the positions of x1 and x2). Since x1 and γx2 are both in S(X), using (7) gives that ∥x1 −γx2∥2 ≤ ∥Tx1 −γTx2∥2/((1 −ϵ) p d/p) = 0. So x1 = γx2. This contradicts the SI property of X. For example, if X contains only κ-sparse vectors, so does S(X). If X is a Riemannian submanifold, so is S(X). Therefore applying random projections to these X will preserve SI with high probability. For the case of κ-sparse vectors, under some strong conditions, we can prove that random projection always preserves SI. (Proofs of the theorems below are in the supplemental material.) Theorem 2. If X satisﬁes SI and has a κ-sparse representation using dictionary B, then the projected data T(X) satisﬁes SI if (2κ −1)M(TB) < 1, where M(·) is matrix mutual coherence. Combining (7) with Theorem 1 or 2 provides an important insight: the projected data TX contains rough information about the original data X and we can continue to use the formulation (1) on TX to extract such information. Actually, if we do this for a Riemannian submanifold X, then we have: Theorem 3. Let the data points lie on a K-dimensional compact Riemannian submanifold X ⊂Rp with volume V , conditional number 1/τ, and geodesic covering regularity R (see [16]). Assume that in the optimal solution of (1) for the projected data (replacing X with TX), data points Tx1 and Tx2 have nonzero weights on the same set of κ codewords. Let wj be the new representation of xj and µi = ∥Txj −Bwj∥2 be the length of the residual (j = 1, 2). With probability 1 −ρ: ∥x1 −x2∥2 2 ≤(p/d)(1 + ϵ1)(1 + ϵ2)(∥w1 −w2∥2 2 + 2µ2 1 + 2µ2 2) ∥x1 −x2∥2 2 ≥(p/d)(1 −ϵ1)(1 −ϵ2)(∥w1 −w2∥2 2, (8) with ϵ1 = O(( K ln(NV Rτ −1) ln(1/ρ) d )0.5−η) (for any small η > 0) and ϵ2 = (κ −1)M(B). Therefore the distances between the sparse representation weights reﬂect the original data point distances. We believe a similar result should also hold when X contains only κ-sparse vectors. 4 Learning a Hierarchical Dictionary Our hierarchical framework decomposes a large dictionary learning problem into a sequence of smaller hierarchically structured dictionary learning problems. The result is a tree of dictionaries. High levels of the tree give course representations, deeper levels give more detailed representations, and the codewords at the leaves form the ﬁnal dictionary. The tree is grown top-down in l levels by reﬁning the dictionary at the previous level to give the dictionary at the next level. Random data projections are used to control the information ﬂow to different layers. We also enforce a zero-tree constraint on the sparse representation weights so that the zero weights in the previous level will force the corresponding weights in the next level to be zero. At each stage we combine this zero-tree constraint with our new screening tests to reduce the size of Lasso problems that must be solved. In detail, we use l random projections Tk ∈Rdk×p (1≤k ≤l) to extract information incrementally from the data in l stages. Each Tk has orthonormal rows and the rows of distinct Tk are orthogonal. At level k we learn a dictionary Bk ∈Rdk×mk and weights Wk ∈Rmk×n by solving a small sparse representation problem similar to (1): min Bk,Wk 1 2∥TkX −BkWk∥2 F + λk∥Wk∥1 s.t. ∥b(k) i ∥2 2 ≤1, ∀i = 1, 2, . . . , mk. (9) Here b(k) i is the ith column of matrix Bk and mk is assumed to be a multiple of mk−1, so the number of codewords mk increases with k. We solve (9) for level k = 1, 2, . . . , l sequentially. An additional constraint is required to enforce a tree structure. Denote the ith element of the jth column of Wk by w(k) j (i) and organize the weights at level k > 1 in mk−1 groups, one per level 5 k −1 codeword. The ith group has mk/mk−1 weights: {w(k) j (rmk−1 + i), 0 ≤r < mk/mk−1}, and has weight w(k−1) j (i) as its parent weight. To enforce a tree structure we require that a child weight is zero if its parent weight is zero. So for every level k ≥2, data point j (1≤j ≤n), group i (1≤i≤mk−1), and weight w(k) j (rmk−1 + i) (0≤r<mk/mk−1), we enforce: w(k−1) j (i) = 0 ⇒ w(k) j (rmk−1 + i) = 0. (10) This imposed tree structure is analogous to a “zero-tree” in EZW wavelet compression [22]. In addition, (10) means that the weights of the previous layer select a small subset of codewords to enter the Lasso optimization. When solving for wk j , (10) reduces the number of codewords from mk to (mk/mk−1)∥w(k−1) j ∥0, a considerable reduction since w(k−1) j is sparse. Thus the screening rules in §2 and the imposed screening rule (10) work together to reduce the effective dictionary size. The tree structure in the weights introduces a similar hierarchical tree structure in the dictionaries {Bk}l k=1: the codewords {b(k) rmk−1+i, 0≤r<mk/mk−1} are the children of codeword b(k−1) i . This tree structure provides a heuristic way of updating Bk. When k > 1, the mk codewords in layer k are naturally divided into mk−1 groups, so we can solve Bk by optimizing each group sequentially. This is similar to block coordinate descent. For i = 1, 2, . . . , mk−1, let B′ = [b(k) rmk−1+i]mk/mk−1−1 r=0 denote the codewords in group i. Let W′ be the submatrix of W containing only the (rmk−1 +i)th rows of W, r = 0, 1, . . . , mk/mk−1 −1. W′ is the weight matrix for B′. Denote the remaining codewords and weights by B′′ and W′′. For all mk−1 groups in random order, we ﬁx B′′ and update B′ by solving (1) for data matrix TkX −B′′W′′. This reduces the complexity from O(mq k) to O(mq k/mq−1 k−1) where O(mq) is the complexity of updating a dictionary with size m. Since q≥3, this offers big computational savings but might yield a suboptimal solution of (9). After ﬁnalizing Wk and Bk, we can solve an unconstrained QP to ﬁnd Ck = arg minC∥X −CWk∥2 F . Ck is useful for visualization purposes; it represents the points on the original data manifold corresponding to Bk. In principle, our framework can use any orthogonal projection matrix Tk. We choose random projections because they’re simple and, more importantly, because they provide a mechanism to control the amount of information extracted at each layer. If all Tk are randomly generated independently of X, then on average, the amount of information in TkX is proportional to dk. This allows us to control the ﬂow of information to each layer so that we avoid using all the information in one layer. 5 Experiments We tested our framework on: (a) the COIL rotational image data set [23], (b) the MNIST digit classiﬁcation data set [24], and (c) the extended Yale B face recognition data set [25] [26]. The basic sparse representation problem (1) is solved using the toolbox provided in [12] to iteratively optimize B and W until an iteration results in a loss function reduction of less than 0.01%. COIL Rotational Image Data Set: This is intended as a small scale illustration of our framework. We use the 72, 128x128 color images of object No. 80 rotating around a circle in 15 degreeincrements (18 images shown in Fig.2(a)). We ran the traditional sparse representation algorithm to compare the three screening tests in §2. The dictionary size is m = 16 and we vary λ. As shown in Fig.2(c), ST3 discards a larger fraction of codewords than ST2 and ST2 discards a larger fraction than ST1/SAFE. We ran the same algorithms on 200 random data projections and the results are almost identical. The average λmax for these two situations is 0.98. Next we test our hierarchical framework using two layers. We set (d2, m2) = (200, 16) so that the second layer solves a problem of the same scale as in the previous paragraph. We demonstrate how the result of the ﬁrst layer, with (d1, m1, λ1) = (100, 4, 0.5), helps the second layer discard more codewords when the tree constraint (10) is imposed. Fig.2(b) illustrates this constraint: the 16 second layer codewords are organized in 4 groups of 4 (only 2 groups shown). The weight on any codeword in a group has to be zero if the parent codeword in the ﬁrst layer has weight zero. This imposed constraint discards many more codewords in the screening stage than any of the three tests in §2. (Fig.2(d)). Finally, the illustrated codewords and weights in Fig.2(b) are the actual values in 6 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 h Average percentage of discarded codewords in the prescreening. Use constraint (13) and our new bound Use our new bound Use constraint (13) and El Ghaoui et al. 2010 Use El Ghaoui et al. 2010 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 h Average percentage of discarded codewords in the prescreening. Use our new bound on the origianl data Use our new bound on the projected data Use El Ghaoui et al. 2010 on the original data Use El Ghaoui et al. 2010 on the projected data !"#$%&'()#&& +,-./&'()*#&& 0(1& 021& 0,1& 0/1& 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 λ Average % of codewords discarded Learning non−hierarchical sparse representation ST3, original data ST3, projected data ST2, original data ST2, projected data ST1/SAFE, original data ST1/SAFE, projected data 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 λ Average % of codewords discarded Learning the second layer sparse representation (10) + ST3 ST3 only (10) + ST2 ST2 only (10) + ST1/SAFE ST1/SAFE only Figure 2: (a): Example images of the data set. (b): Illustration of a two layer hierarchical sparse representation. (c): Comparison of the three screening tests for sparse representation. (d): Screening performance in the second layer of our hierarchical framework using combinations of screening criteria. The imposed constraint (10) helps to discard signiﬁcantly more codewords when λ is small. 2 3 5 10 20 30 91 92 93 94 95 96 97 Average encoding time for a testing image (ms) Classification accuracy (%) on testing set Traditional sparse representation: m=64, with 6 different λ settings m=128, with 6 λ (same as above) m=192, with 6 λ m=256, with 6 λ m=512, with 6 λ Our hierarchical framework: m1=32, m2=512, with 6 λ m1=64, m2=2048, with 6 λ m1=16, m2=256, m3=4096, with 6 λ Baseline: the same linear classifier using 250 principal components using original pixel values 32(0.1%) 64(0.2%) 128(0.4%) 256(0.8%) 50 60 70 80 90 100 # of random projections (percentage of image size) to use Recognition rate (%) on testing set Traditional sparse representation Our hierarchical framework Our framework with PCA projections Linear classifier Wright et al., 2008, SRC 32(0.1%) 64(0.2%) 128(0.4%) 256(0.8%) 0 20 40 60 80 # of random projections (percentage of image size) to use Average encoding time (ms) Traditional sparse representation Our hierarchical framework Our framework with PCA projections Linear classifier Figure 3: Left: MNIST: The tradeoff between classiﬁcation accuracy and average encoding time for various sparse representation methods. Our hierarchical framework yields better performance in less time. The average encoding time doesn’t apply to baseline methods. The performance of traditional sparse representation is consistent with [9]. Right: Face Recognition: The recognition rate (top) and average encoding time (bottom) for various methods. Traditional sparse representation has the best accuracy and is very close to a similar method SRC in [8] (SRC’s recognition rate is cited from [8] but data on encoding time is not available). Our hierarchical framework achieves a good tradeoff between the accuracy and speed. Using PCA projections in our framework yields worse performance since these projections do not spread information across the layers. C2 and W2 when λ2 = 0.4 (the marked point in Fig.2(d)). The sparse representation gives a multiresolution representation of the rotational pattern: the ﬁrst layer encodes rough orientation and the second layer reﬁnes it. The next two experiments evaluate the performance of sparse representation by (1) the accuracy of a classiﬁcation task using the columns in W (or in [WT 1 , WT 2 , . . . , WT l ]T for our framework) as features, and (2) the average encoding time required to obtain these weights for a testing data point. This time is highly correlated with the total time needed for iterative dictionary learning. We used linear SVM (liblinear [27]) with parameters tuned by 10-fold cross-validations on the training set. 7 MNIST Digit Classiﬁcation: This data set contains 70,000 28x28 hand written digit images (60,000 training, 10,000 testing). We ran the traditional sparse representation algorithm for dictionary size m ∈{64, 128, 192, 256} and λ ∈Λ = {0.06, 0.08, 0.11, 0.16, 0.23, 0.32}. In Fig.3 left panel, each curve contains settings with the same m but with different λ. Points to the right correspond to smaller λ values (less sparse solutions and more difﬁcult computation). There is a tradeoff between speed (x-axis) and classiﬁcation performance (y-axis). To see where our framework stands, we tested the following settings: (a) 2 layers with (d1, d2) = (200, 500), (m1, m2) = (32, 512), λ1 = 0.23 and λ2 ∈Λ, (b) (m1, m2) = (64, 2048) and everything else in (a) unchanged, (c) 3 layers with (d1, d2, d3) = (100, 200, 400), (m1, m2, m3) = (16, 256, 4096), (λ1, λ2) = (0.16, 0.11) and λ3 ∈ Λ. The plot shows that compared to the traditional sparse representation, our hierarchical framework achieves roughly a 1% accuracy improvement given the same encoding time and a roughly 2X speedup given the same accuracy. Using 3 layers also offers competitive performance but doesn’t outperform the 2 layer setting. Face Recognition: For each of 38 subjects we used 64 cropped frontal face views under differing lighting conditions, randomly divided into 32 training and 32 testing images. This set-up mirrors that in [8]. In this experiment we start with the random projected data (p ∈{32, 64, 128, 256} random projections of the original 192x128 data) and use this data as follows: (a) learn a traditional non-hierarchical sparse representation, (b) our framework, i.e., sample the data in two stages using orthogonal random projections and learn a 2 layer hierarchical sparse representation, (c) use PCA projections to replace random projections in (b), (d) directly apply a linear classiﬁer without ﬁrst learning a sparse representation. For (a) we used m = 1024, λ = 0.030 for p = 32, 64 and λ = 0.029 for p = 128, 256 (tuned to yield the same average sparsity for different p). For (b) we used (m1, m2) = (32, 1024), (d1, d2) = ( 3 8p, 5 8p), λ1 = 0.02 and λ2 the same as λ in (a). For (c) we used the same setting in (b) except random projection matrices T1, T2 in our framework are now set to the PCA projection matrices (calculate SVD X = USVT with singular values in descending order, then use the ﬁrst d1 columns of U as the rows in T1 and the next d2 columns of U as the rows in T2). The results in Fig.3 right panel suggest that our framework strikes a good balance between speed and accuracy. The PCA variant of our framework has worse performance because the ﬁrst 3 8p projections contain too much information, leaving the second layer too little information (which also drags down the speed for lack of sparsity and structure). This reinforces our argument at the end of §4 about the advantage of random projections. The fact that a linear SVM performs well given enough random projections suggests this data set does not have a strong nonlinear structure. Finally, at any iteration, the average λmax for all data points ranges from 0.76 to 0.91 in all settings in the MNIST experiment and ranges from 0.82 to nearly 1 in the face recognition experiment (except for the second layer in the PCA variant, in which average λmax can be as low as 0.54). As expected, λmax is large, a situation that favors our new screening tests (ST2, ST3). 6 Conclusion Our theoretical results and algorithmic framework effectively make headway on the computational challenge of learning sparse representations on large size dictionaries for high dimensional data The new screening tests greatly reduce the size of the lasso problems to be solved and the tests are proven, both theoretically and empirically, to be much more effective than the existing ST1/SAFE test. We have shown that under certain conditions, random projection preserves the scale indifference (SI) property with high probability, thus providing an opportunity to learn informative sparse representations with data fewer dimensions. Finally, the new hierarchical dictionary learning framework employs random data projections to control the ﬂow of information to the layers, screening to eliminate unnecessary codewords, and a tree constraint to select a small number of candidate codewords based on the weights leant in the previous stage. By doing so, it can deal with large m and p simultaneously. The new framework exhibited impressive performance on the tested data sets, achieving equivalent classiﬁcation accuracy with less computation time. Acknowledgements This research was partially supported by the NSF grant CCF-1116208. Zhen James Xiang thanks Princeton University for support through the Charlotte Elizabeth Procter honoriﬁc fellowship. 8 References [1] M. Elad. 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4,434	Eﬃcient Online Learning via Randomized Rounding Nicol`o Cesa-Bianchi DSI, Universit`a degli Studi di Milano Italy nicolo.cesa-bianchi@unimi.it Ohad Shamir Microsoft Research New England USA ohadsh@microsoft.com Abstract Most online algorithms used in machine learning today are based on variants of mirror descent or follow-the-leader. In this paper, we present an online algorithm based on a completely diﬀerent approach, which combines “random playout” and randomized rounding of loss subgradients. As an application of our approach, we provide the ﬁrst computationally eﬃcient online algorithm for collaborative ﬁltering with trace-norm constrained matrices. As a second application, we solve an open question linking batch learning and transductive online learning. 1 Introduction Online learning algorithms, which have received much attention in recent years, enjoy an attractive combination of computational eﬃciency, lack of distributional assumptions, and strong theoretical guarantees. However, it is probably fair to say that at their core, most of these algorithms are based on the same small set of fundamental techniques, in particular mirror descent and regularized follow-the-leader (see for instance [14]). In this work we revisit, and signiﬁcantly extend, an algorithm which uses a completely diﬀerent approach. This algorithm, known as the Minimax Forecaster, was introduced in [9, 11] for the setting of prediction with static experts. It computes minimax predictions in the case of known horizon, binary outcomes, and absolute loss. Although the original version is computationally expensive, it can easily be made eﬃcient through randomization. We extend the analysis of [9] to the case of non-binary outcomes and arbitrary convex and Lipschitz loss functions. The new algorithm is based on a combination of “random playout” and randomized rounding, which assigns random binary labels to future unseen instances, in a way depending on the loss subgradients. Our resulting Randomized Rounding (R2) Forecaster has a parameter trading oﬀregret performance and computational complexity, and runs in polynomial time (for T predictions, it requires computing O(T 2) empirical risk minimizers in general, as opposed to O(T) for generic follow-the-leader algorithms). The regret of the R2 Forecaster is determined by the Rademacher complexity of the comparison class. The connection between online learnability and Rademacher complexity has also been explored in [2, 1]. However, these works focus on the information-theoretically achievable regret, as opposed to computationally eﬃcient algorithms. The idea of “random playout”, in the context of online learning, has also been used in [16, 3], but we apply this idea in a diﬀerent way. We show that the R2 Forecaster can be used to design the ﬁrst eﬃcient online learning algorithm for collaborative ﬁltering with trace-norm constrained matrices. While this is a well-known setting, a straightforward application of standard online learning approaches, such as mirror descent, appear to give only trivial performance guarantees. Moreover, our 1 regret bound matches the best currently known sample complexity bound in the batch distribution-free setting [21]. As a diﬀerent application, we consider the relationship between batch learning and transductive online learning. This relationship was analyzed in [16], in the context of binary prediction with respect to classes of bounded VC dimension. Their main result was that eﬃcient learning in a statistical setting implies eﬃcient learning in the transductive online setting, but at an inferior rate of T 3/4 (where T is the number of rounds). The main open question posed by that paper is whether a better rate can be obtained. Using the R2 Forecaster, we improve on those results, and provide an eﬃcient algorithm with the optimal √ T rate, for a wide class of losses. This shows that eﬃcient batch learning not only implies eﬃcient transductive online learning (the main thesis of [16]), but also that the same rates can be obtained, and for possibly non-binary prediction problems as well. We emphasize that the R2 Forecaster requires computing many empirical risk minimizers (ERM’s) at each round, which might be prohibitive in practice. Thus, while it does run in polynomial time whenever an ERM can be eﬃciently computed, we make no claim that it is a “fully practical” algorithm. Nevertheless, it seems to be a useful tool in showing that eﬃcient online learnability is possible in various settings, often working in cases where more standard techniques appear to fail. Moreover, we hope the techniques we employ might prove useful in deriving practical online algorithms in other contexts. 2 The Minimax Forecaster We start by introducing the sequential game of prediction with expert advice —see [10]. The game is played between a forecaster and an adversary, and is speciﬁed by an outcome space Y, a prediction space P, a nonnegative loss function ℓ: P × Y →R, which measures the discrepancy between the forecaster’s prediction and the outcome, and an expert class F. Here we focus on classes F of static experts, whose prediction at each round t does not depend on the outcome in previous rounds. Therefore, we think of each f ∈F simply as a sequence f = (f1, f2, . . . ) where each ft ∈P. At each step t = 1, 2, . . . of the game, the forecaster outputs a prediction pt ∈P and simultaneously the adversary reveals an outcome yt ∈Y. The forecaster’s goal is to predict the outcome sequence almost as well as the best expert in the class F, irrespective of the outcome sequence y = (y1, y2, . . . ). The performance of a forecasting strategy A is measured by the worst-case regret VT (A, F) = sup y∈YT T X t=1 ℓ(pt, yt) −inf f∈F T X t=1 ℓ(ft, yt) ! (1) viewed as a function of the horizon T. To simplify notation, let L(f, y) = PT t=1 ℓ(ft, yt). Consider now the special case where the horizon T is ﬁxed and known in advance, the outcome space is Y = {−1, +1}, the prediction space is P = [−1, +1], and the loss is the absolute loss ℓ(p, y) = \|p −y\|. We will denote the regret in this special case as Vabs T (A, F). The Minimax Forecaster —which is based on work presented in [9] and [11], see also [10] for an exposition— is derived by an explicit analysis of the minimax regret infA Vabs T (A, F), where the inﬁmum is over all forecasters A producing at round t a prediction pt as a function of p1, y1, . . . pt−1, yt−1. For general online learning problems, the analysis of this quantity is intractable. However, for the speciﬁc setting we focus on (absolute loss and binary outcomes), one can get both an explicit expression for the minimax regret, as well as an explicit algorithm, provided inff∈F PT t=1 ℓ(ft, yt) can be eﬃciently computed for any sequence y1, . . . , yT . This procedure is akin to performing empirical risk minimization (ERM) in statistical learning. A full development of the analysis is out of scope, but is outlined in Appendix A of the supplementary material. In a nutshell, the idea is to begin by calculating the optimal prediction in the last round T, and then work backwards, calculating the optimal prediction at round T −1, T −2 etc. Remarkably, the value of infA Vabs T (A, F) is exactly the Rademacher complexity RT (F) of the class F, which is known to play a crucial role in understanding the sample complexity in statistical learning [5]. In this paper, we 2 deﬁne it as1: RT (F) = E " sup f∈F T X t=1 σtft # (2) where σ1, . . . , σT are i.i.d. Rademacher random variables, taking values −1, +1 with equal probability. When RT (F) = o(T), we get a minimax regret infA Vabs T (A, F) = o(T) which implies a vanishing per-round regret. In terms of an explicit algorithm, the optimal prediction pt at round t is given by a complicated-looking recursive expression, involving exponentially many terms. Indeed, for general online learning problems, this is the most one seems able to hope for. However, an apparently little-known fact is that when one deals with a class F of ﬁxed binary sequences as discussed above, then one can write the optimal prediction pt in a much simpler way. Letting Y1, . . . , YT be i.i.d. Rademacher random variables, the optimal prediction at round t can be written as pt = E inf f∈F L (f, y1 · · · yt−1 (−1) Yt+1 · · · YT ) −inf f∈F L (f, y1 · · · yt−1 1 Yt+1 · · · YT ) . (3) In words, the prediction is simply the expected diﬀerence between the minimal cumulative loss over F, when the adversary plays −1 at round t and random values afterwards, and the minimal cumulative loss over F, when the adversary plays +1 at round t, and the same random values afterwards. We refer the reader to Appendix A of the supplementary material for how this is derived. We denote this optimal strategy (for absolute loss and binary outcomes) as the Minimax Forecaster (mf): Algorithm 1 Minimax Forecaster (mf) for t = 1 to T do Predict pt as deﬁned in Eq. (3) Receive outcome yt and suﬀer loss \|pt −yt\| end for The relevant guarantee for mf is summarized in the following theorem. Theorem 1. For any class F ⊆[−1, +1]T of static experts, the regret of the Minimax Forecaster (Algorithm 1) satisﬁes Vabs T (mf, F) = RT (F). 2.1 Making the Minimax Forecaster Eﬃcient The Minimax Forecaster described above is not computationally eﬃcient, as the computation of pt requires averaging over exponentially many ERM’s. However, by a martingale argument, it is not hard to show that it is in fact suﬃcient to compute only two ERM’s per round. Algorithm 2 Minimax Forecaster with eﬃcient implementation (mf) for t = 1 to T do For i = t + 1, . . . , T, let Yi be a Rademacher random variable Let pt := inff∈F L (f, y1 . . . yt−1 (−1) Yt+1 . . . YT ) −inff∈F L (f, y1 . . . yt−1 1 Yt+1 . . . YT ) Predict pt, receive outcome yt and suﬀer loss \|pt −yt\| end for Theorem 2. For any class F ⊆[−1, +1]T of static experts, the regret of the randomized forecasting strategy mf (Algorithm 2) satisﬁes Vabs T (mf, F) ≤RT (F) + p 2T ln(1/δ) 1In the statistical learning literature, it is more common to scale this quantity by 1/T, but the form we use here is more convenient for stating cumulative regret bounds. 3 with probability at least 1 −δ. Moreover, if the predictions p = (p1, . . . , pT ) are computed reusing the random values Y1, . . . , YT computed at the ﬁrst iteration of the algorithm, rather than drawing fresh values at each iteration, then it holds that E L(p, y) −inf f∈F L(f, y) ≤RT (F) for all y ∈{−1, +1}T . Proof sketch. To prove the second statement, note that E[pt]−yt = E \|pt−yt\| for any ﬁxed yt ∈{−1, +1} and pt bounded in [−1, +1], and use Thm. 1. To prove the ﬁrst statement, note that \|pt −yt\| − Ept[pt] −yt for t = 1, . . . , T is a martingale diﬀerence sequence with respect to p1, . . . , pT , and apply Azuma’s inequality. The second statement in the theorem bounds the regret only in expectation and is thus weaker than the ﬁrst one. On the other hand, it might have algorithmic beneﬁts. Indeed, if we reuse the same values for Y1, . . . , YT , then the computation of the inﬁma over f in mf are with respect to an outcome sequence which changes only at one point in each round. Depending on the speciﬁc learning problem, it might be easier to re-compute the inﬁmum after changing a single point in the outcome sequence, as opposed to computing the inﬁmum over a diﬀerent outcome sequence in each round. 3 The R2 Forecaster The Minimax Forecaster presented above is very speciﬁc to the absolute loss ℓ(f, y) = \|f −y\| and for binary outcomes Y = {−1, +1}, which limits its applicability. We note that extending the forecaster to other losses or diﬀerent outcome spaces is not trivial: indeed, the recursive unwinding of the minimax regret term, leading to an explicit expression and an explicit algorithm, does not work as-is for other cases. Nevertheless, we will now show how one can deal with general (convex, Lipschitz) loss functions and outcomes belonging to any real interval [−b, b]. The algorithm we propose essentially uses the Minimax Forecaster as a subroutine, by feeding it with a carefully chosen sequence of binary values zt, and using predictions ft which are scaled to lie in the interval [−1, +1]. The values of zt are based on a randomized rounding of values in [−1, +1], which depend in turn on the loss subgradient. Thus, we denote the algorithm as the Randomized Rounding (R2) Forecaster. To describe the algorithm, we introduce some notation. For any scalar f ∈[−b, b], deﬁne ef = f/b to be the scaled versions of f into the range [−1, +1]. For vectors f, deﬁne ef = (1/b)f. Also, we let ∂ptℓ(pt, yt) denote any subgradient of the loss function ℓwith respect to the prediction pt. The pseudocode of the R2 Forecaster is presented as Algorithm 3 below, and its regret guarantee is summarized in Thm. 3. The proof is presented in Appendix B of the supplementary material. Theorem 3. Suppose ℓis convex and ρ-Lipschitz in its ﬁrst argument. For any F ⊆[−b, b]T the regret of the R2 Forecaster (Algorithm 3) satisﬁes VT (R2, F) ≤ρ RT (F) + ρ b r1 η + 2 s 2T ln 2T δ (4) with probability at least 1 −δ. The prediction pt which the algorithm computes is an empirical approximation to b EYt+1,...,YT inf f∈F L ef, z1 . . . zt−1 0 Yt+1 . . . YT −inf f∈F L ef, z1 · · · zt−1 1 Yt+1 · · · YT by repeatedly drawing independent values to Yt+1, . . . , YT and averaging. The accuracy of the approximation is reﬂected in the precision parameter η. A larger value of η improves the regret bound, but also increases the runtime of the algorithm. Thus, η provides a trade-oﬀ between the computational complexity of the algorithm and its regret guarantee. We note 4 Algorithm 3 The R2 Forecaster Input: Upper bound b on \|ft\|, \|yt\| for all t = 1, . . . , T and f ∈F; upper bound ρ on supp,y∈[−b,b] ∂pℓ(p, y) ; precision parameter η ≥1 T . for t = 1 to T do pt := 0 for j = 1 to η T do For i = t, . . . , T, let Yi be a Rademacher random variable Draw ∆:= inf f∈F L ef, z1 . . . zt−1 (−1) Yt+1 . . . YT −inf f∈F L ef, z1 . . . zt−1 1 Yt+1 . . . YT Let pt := pt + b η T ∆ end for Predict pt Receive outcome yt and suﬀer loss ℓ(pt, yt) Let rt := 1 2 1 −1 ρ∂ptℓ(pt, yt) ∈[0, 1] Let zt := 1 with probability rt, and zt := −1 with probability 1 −rt end for that even when η is taken to be a constant fraction, the resulting algorithm still runs in polynomial time O(T 2c), where c is the time to compute a single ERM. In subsequent results pertaining to this Forecaster, we will assume that η is taken to be a constant fraction. We end this section with a remark that plays an important role in what follows. Remark 1. The predictions of our forecasting strategies do not depend on the ordering of the predictions of the experts in F. In other words, all the results proven so far also hold in a setting where the elements of F are functions f : {1, . . . , T} →P, and the adversary has control on the permutation π1, . . . , πT of {1, . . . , T} that is used to deﬁne the prediction f(πt) of expert f at time t.2 Also, Thm. 1 implies that the value of Vabs T (F) remains unchanged irrespective of the permutation chosen by the adversary. 4 Application 1: Transductive Online Learning The ﬁrst application we consider is a rather straightforward one, in the context of transductive online learning [6]. In this model, we have an arbitrary sequence of labeled examples (x1, y1), . . . , (xT , yT ), where only the set {x1, . . . , xT } of unlabeled instances is known to the learner in advance. At each round t, the learner must provide a prediction pt for the label of yt. The true label yt is then revealed, and the learner incurs a loss ℓ(pt, yt). The learner’s goal is to minimize the transductive online regret PT t=1 ℓ(pt, yt) −inff∈F ℓ(f(xt), yt) with respect to a ﬁxed class of predictors F of the form {x 7→f(x)}. The work [16] considers the binary classiﬁcation case with zero-one loss. Their main result is that if a class F of binary functions has bounded VC dimension d, and there exists an eﬃcient algorithm to perform empirical risk minimization, then one can construct an eﬃcient randomized algorithm for transductive online learning, whose regret is at most O(T 3/4p d ln(T)) in expectation. The signiﬁcance of this result is that eﬃcient batch learning (via empirical risk minimization) implies eﬃcient learning in the transductive online setting. This is an important result, as online learning can be computationally harder than batch learning —see, e.g., [8] for an example in the context of Boolean learning. A major open question posed by [16] was whether one can achieve the optimal rate O( √ dT), matching the rate of a batch learning algorithm in the statistical setting. Using the R2 Forecaster, we can easily achieve the above result, as well as similar results in a strictly more general setting. This shows that eﬃcient batch learning not only implies eﬃcient transductive online learning (the main thesis of [16]), but also that the same rates can be obtained, and for possibly non-binary prediction problems as well. 2Formally, at each step t: (1) the adversary chooses and reveals the next element πt of the permutation; (2) the forecaster chooses pt ∈P and simultaneously the adversary chooses yt ∈Y. 5 Theorem 4. Suppose we have a computationally eﬃcient algorithm for empirical risk minimization (with respect to the zero-one loss) over a class F of {0, 1}-valued functions with VC dimension d. Then, in the transductive online model, the eﬃcient randomized forecaster mf* achieves an expected regret of O( √ dT) with respect to the zero-one loss. Moreover, for an arbitrary class F of [−b, b]-valued functions with Rademacher complexity RT (F), and any convex ρ-Lipschitz loss function, if there exists a computationally eﬃcient algorithm for empirical risk minimization, then the R2 Forecaster is computationally eﬃcient and achieves, in the transductive online model, a regret of ρRT (F)+O(ρb p T ln(T/δ)) with probability at least 1 −δ. Proof. Since the set {x1, . . . , xT } of unlabeled examples is known, we reduce the online transductive model to prediction with expert advice in the setting of Remark 1. This is done by mapping each function f ∈F to a function f : {1, . . . , T} →P by t 7→f(xt), which is equivalent to an expert in the setting of Remarks 1. When F maps to {0, 1}, and we care about the zero-one loss, we can use the forecaster mf* to compute randomized predictions and apply Thm. 2 to bound the expected transductive online regret with RT (F). For a class with VC dimension d, RT (F) ≤O( √ dT) for some constant c > 0, using Dudley’s chaining method [12], and this concludes the proof of the ﬁrst part of the theorem. The second part is an immediate corollary of Thm. 3. We close this section by contrasting our results for online transductive learning with those of [7] about standard online learning. If F contains {0, 1}-valued functions, then the optimal regret bound for online learning is order of √ d′T, where d′ is the Littlestone dimension of F. Since the Littlestone dimension of a class is never smaller than its VC dimension, we conclude that online learning is a harder setting than online transductive learning. 5 Application 2: Online Collaborative Filtering We now turn to discuss the application of our results in the context of collaborative ﬁltering with trace-norm constrained matrices, presenting what is (to the best of our knowledge) the ﬁrst computationally eﬃcient online algorithms for this problem. In collaborative ﬁltering, the learning problem is to predict entries of an unknown m × n matrix based on a subset of its observed entries. A common approach is norm regularization, where we seek a low-norm matrix which matches the observed entries as best as possible. The norm is often taken to be the trace-norm [22, 19, 4], although other norms have also been considered, such as the max-norm [18] and the weighted trace-norm [20, 13]. Previous theoretical treatments of this problem assumed a stochastic setting, where the observed entries are picked according to some underlying distribution (e.g., [23, 21]). However, even when the guarantees are distribution-free, assuming a ﬁxed distribution fails to capture important aspects of collaborative ﬁltering in practice, such as non-stationarity [17]. Thus, an online adversarial setting, where no distributional assumptions whatsoever are required, seems to be particularly well-suited to this problem domain. In an online setting, at each round t the adversary reveals an index pair (it, jt) and secretely chooses a value yt for the corresponding matrix entry. After that, the learner selects a prediction pt for that entry. Then yt is revealed and the learner suﬀers a loss ℓ(pt, yt). Hence, the goal of a learner is to minimize the regret with respect to a ﬁxed class W of prediction matrices, PT t=1 ℓ(pt, yt) −infW ∈W PT t=1 ℓ Wit,jt, yt . Following reality, we will assume that the adversary picks a diﬀerent entry in each round. When the learner’s performance is measured by the regret after all T = mn entries have been predicted, the online collaborative ﬁltering setting reduces to prediction with expert advice as discussed in Remark 1. As mentioned previously, W is often taken to be a convex class of matrices with bounded trace-norm. Many convex learning problems, such as linear and kernel-based predictors, as well as matrix-based predictors, can be learned eﬃciently both in a stochastic and an online setting, using mirror descent or regularized follow-the-leader methods. However, 6 for reasonable choices of W, a straightforward application of these techniques can lead to algorithms with trivial bounds. In particular, in the case of W consisting of m × n matrices with trace-norm at most r, standard online regret bounds would scale like O r √ T . Since for this norm one typically has r = O √mn , we get a per-round regret guarantee of O( p mn/T). This is a trivial bound, since it becomes “meaningful” (smaller than a constant) only after all T = mn entries have been predicted. On the other hand, based on general techniques developed in [15] and greatly extended in [1], it can be shown that online learnability is information-theoretically possible for such W. However, these techniques do not provide a computationally eﬃcient algorithm. Thus, to the best of our knowledge, there is currently no eﬃcient (polynomial time) online algorithm, which attain non-trivial regret. In this section, we show how to obtain such an algorithm using the R2 Forecaster. Consider ﬁrst the transductive online setting, where the set of indices to be predicted is known in advance, and the adversary may only choose the order and values of the entries. It is readily seen that the R2 Forecaster can be applied in this setting, using any convex class W of ﬁxed matrices with bounded entries to compete against, and any convex Lipschitz loss function. To do so, we let {ik, jk}T k=1 be the set of entries, and run the R2 Forecaster with respect to F = {t 7→Wit,jt : W ∈W}, which corresponds to a class of experts as discussed in Remark 1. What is perhaps more surprising is that the R2 Forecaster can also be applied in a nontransductive setting, where the indices to be predicted are not known in advance. Moreover, the Forecaster doesn’t even need to know the horizon T in advance. The key idea to achieve this is to utilize the non-asymptotic nature of the learning problem —namely, that the game is played over a ﬁnite m × n matrix, so the time horizon is necessarily bounded. The algorithm we propose is very simple: we apply the R2 Forecaster as if we are in a setting with time horizon T = mn, which is played over all entries of the m × n matrix. By Remark 1, the R2 Forecaster does not need to know the order in which these m × n entries are going to be revealed. Whenever W is convex and ℓis a convex function, we can ﬁnd an ERM in polynomial time by solving a convex problem. Hence, we can implement the R2 Forecaster eﬃciently. To show that this is indeed a viable strategy, we need the following lemma, whose proof is presented in Appendix C of the supplementary material. Lemma 1. Consider a (possibly randomized) forecaster A for a class F whose regret after T steps satisﬁes VT (A, F) ≤G with probability at least 1−δ > 1 2. Furthermore, suppose the loss function is such that inf p′∈P sup y∈Y inf p∈P ℓ(p, y) −ℓ(p′, y) ≥0. Then max t=1,...,T Vt(A, F) ≤G with probability at least 1 −δ. Note that a simple suﬃcient condition for the assumption on the loss function to hold, is that P = Y and ℓ(p, y) ≥ℓ(y, y) for all p, y ∈P. Using this lemma, the following theorem exempliﬁes how we can obtain a regret guarantee for our algorithm, in the case of W consisting of the convex set of matrices with bounded trace-norm and bounded entries. For the sake of clarity, we will consider n × n matrices. Theorem 5. Let ℓbe a loss function which satisﬁes the conditions of Lemma 1. Also, let W consist of n × n matrices with trace-norm at most r = O(n) and entries at most b = O(1), suppose we apply the R2 Forecaster over time horizon n2 and all entries of the matrix. Then with probability at least 1 −δ, after T rounds, the algorithm achieves an average per-round regret of at most O n3/2 + n p ln(n/δ) T ! uniformly over T = 1, . . . , n2. Proof. In our setting, where the adversary chooses a diﬀerent entry at each round, [21, Theorem 6] implies that for the class W′ of all matrices with trace-norm at most r = O(n), 7 it holds that RT (W′)/T ≤O(n3/2/T). Therefore, Rn2(W′) ≤O(n3/2). Since W ⊆W′, we get by deﬁnition of the Rademacher complexity that Rn2(W) = O(n3/2) as well. By Thm. 3, the regret after n2 rounds is O(n3/2 + n p ln(n/δ)) with probability at least 1 −δ. Applying Lemma 1, we get that the cumulative regret at the end of any round T = 1, . . . , n2 is at most O(n3/2 + n p ln(n/δ)), as required. This bound becomes non-trivial after n3/2 entries are revealed, which is still a vanishing proportion of all n2 entries. While the regret might seem unusual compared to standard regret bounds (which usually have rates of 1/ √ T for general losses), it is a natural outcome of the non-asymptotic nature of our setting, where T can never be larger than n2. In fact, this is the same rate one would obtain in a batch setting, where the entries are drawn from an arbitrary distribution. Moreover, an assumption such as boundedness of the entries is required for currently-known guarantees even in a batch setting —see [21] for details. Acknowledgments The ﬁrst author acknowledges partial support by the PASCAL2 NoE under EC grant FP7216886. References [1] K. Sridharan A. Rakhlin and A. Tewari. Online learning: Random averages, combinatorial parameters, and learnability. In NIPS, 2010. [2] J. Abernethy, P. Bartlett, A. Rakhlin, and A. Tewari. Optimal strategies and minimax lower bounds for online convex games. In COLT, 2009. [3] J. Abernethy and M. Warmuth. Repeated games against budgeted adversaries. In NIPS, 2010. [4] F. Bach. Consistency of trace-norm minimization. Journal of Machine Learning Research, 9:1019–1048, 2008. [5] P. Bartlett and S. Mendelson. Rademacher and Gaussian complexities: Risk bounds and structural results. In COLT, 2001. [6] S. Ben-David, E. Kushilevitz, and Y. Mansour. Online learning versus oﬄine learning. Machine Learning, 29(1):45–63, 1997. [7] S. Ben-David, D. P´al, and S. Shalev-Shwartz. Agnostic online learning. In COLT, 2009. [8] A. Blum. 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4,435	Spatial distance dependent Chinese restaurant processes for image segmentation Soumya Ghosh1, Andrei B. Ungureanu2, Erik B. Sudderth1, and David M. Blei3 1Department of Computer Science, Brown University, {sghosh,sudderth}@cs.brown.edu 2Morgan Stanley, andrei.b.ungureanu@gmail.com 3Department of Computer Science, Princeton University, blei@cs.princeton.edu Abstract The distance dependent Chinese restaurant process (ddCRP) was recently introduced to accommodate random partitions of non-exchangeable data [1]. The ddCRP clusters data in a biased way: each data point is more likely to be clustered with other data that are near it in an external sense. This paper examines the ddCRP in a spatial setting with the goal of natural image segmentation. We explore the biases of the spatial ddCRP model and propose a novel hierarchical extension better suited for producing “human-like” segmentations. We then study the sensitivity of the models to various distance and appearance hyperparameters, and provide the ﬁrst rigorous comparison of nonparametric Bayesian models in the image segmentation domain. On unsupervised image segmentation, we demonstrate that similar performance to existing nonparametric Bayesian models is possible with substantially simpler models and algorithms. 1 Introduction The Chinese restaurant process (CRP) is a distribution on partitions of integers [2]. When used in a mixture model, it provides an alternative representation of a Bayesian nonparametric Dirichlet process mixture—the data are clustered and the number of clusters is determined via the posterior distribution. CRP mixtures assume that the data are exchangeable, i.e., their order does not affect the distribution of cluster structure. This can provide computational advantages and simplify approximate inference, but is often an unrealistic assumption. The distance dependent Chinese restaurant process (ddCRP) was recently introduced to model random partitions of non-exchangeable data [1]. The ddCRP clusters data in a biased way: each data point is more likely to be clustered with other data that are near it in an external sense. For example, when clustering time series data, points that closer in time are more likely to be grouped together. Previous work [1] developed the ddCRP mixture in general, and derived posterior inference algorithms based on Gibbs sampling [3]. While they studied the ddCRP in time-series and sequential settings, ddCRP models can be used with any type of distance and external covariates. Recently, other researchers [4] have also used the ddCRP in non-temporal settings. In this paper, we study the ddCRP in a spatial setting. We use a spatial distance function between pixels in natural images and cluster them to provide an unsupervised segmentation. The spatial distance encourages the discovery of connected segments. We also develop a region-based hierarchical generalization, the rddCRP. Analogous to the hierarchical Dirichlet process (HDP) [5], the rddCRP clusters groups of data, where cluster components are shared across groups. Unlike the HDP, however, the rddCRP allows within-group clusterings to depend on external distance measurements. To demonstrate the power of this approach, we develop posterior inference algorithms for segmenting images with ddCRP and rddCRP mixtures. Image segmentation is an extensively studied area, 1 C Removing C leaves clustering unchanged Adding C leaves the clustering unchanged C Removing C splits the cluster Adding C merges the cluster Figure 1: Left: An illustration of the relationship between the customer assignment representation and the table assignment representation. Each square is a data point (a pixel or superpixel) and each arrow is a customer assignment. Here, the distance window is of length 1. The corresponding table assignments, i.e., the clustering of these data, is shown by the color of the data points. Right: Intuitions behind the two cases considered by the Gibbs sampler. Consider the link from node C. When removed, it may leave the clustering unchanged or split a cluster. When added, it may leave the clustering unchanged or merge two clusters. which we will not attempt to survey here. Inﬂuential existing methods include approaches based on kernel density estimation [6], Markov random ﬁelds [3, 7], and the normalized cut spectral clustering algorithm [8, 9]. A recurring difﬁculty encountered by traditional methods is the need to determine an appropriate segment resolution for each image; even among images of similar scene types, the number of observed objects can vary widely. This has usually been dealt via heuristics with poorly understood biases, or by simplifying the problem (e.g., partially specifying each image’s segmentation via manual user input [7]). Recently, several promising segmentation algorithms have been proposed based on nonparametric Bayesian methods [10, 11, 12]. In particular, an approach which couples Pitman-Yor mixture models [13] via thresholded Gaussian processes [14] has lead to very promising initial results [10], and provides a baseline for our later experiments. Expanding on the experiments in [10], we analyze 800 images of different natural scene types, and show that the comparatively simpler ddCRP-based algorithms perform similarly to this work. Moreover, unlike previous nonparametric Bayesian approaches, the structure of the ddCRP allows spatial connectivity of the inferred segments to (optionally) be enforced. In some applications, this is a known property of all reasonable segmentations. Our results demonstrate the practical utility of spatial ddCRP and hierarchical rddCRP models. We also provide the ﬁrst rigorous comparison of nonparametric Bayesian image segmentation models. 2 Image Segmentation with Distance Dependent CRPs Our goal is to develop a probabilistic method to segment images of complex scenes. Image segmentation is the problem of partitioning an image into self-similar groups of adjacent pixels. Segmentation is an important step towards other tasks in image understanding, such as object recognition, detection, or tracking. We model images as observed collections of “superpixels” [15], which are small blocks of spatially adjacent pixels. Our goal is to associate the features xi in the ith superpixel with some cluster zi; these clusters form the segments of that image. Image segmentation is thus a special kind of clustering problem where the desired solution has two properties. First, we hope to ﬁnd contiguous regions of the image assigned to the same cluster. Due to physical processes such as occlusion, it may be appropriate to ﬁnd segments that contain two or three contiguous image regions, but we do not want a cluster that is scattered across individual image pixels. Traditional clustering algorithms, such as k-means or probabilistic mixture models, do not account for external information such as pixel location and are not biased towards contiguous regions. Image locations have been heuristically incorporated into Gaussian mixture models by concatenating positions with appearance features in a vector [16], but the resulting bias towards elliptical regions often produces segmentation artifacts. Second, we would like a solution that deter2 mines the number of clusters from the image. Image segmentation algorithms are typically applied to collections of images of widely varying scenes, which are likely to require different numbers of segments. Except in certain restricted domains such as medical image analysis, it is not practical to use an algorithm that requires knowing the number of segments in advance. In the following sections, we develop a Bayesian algorithm for image segmentation based on the distance dependent Chinese restaurant process (ddCRP) mixture model [1]. Our algorithm ﬁnds spatially contiguous segments in the image and determines an image-speciﬁc number of segments from the observed data. 2.1 Chinese restaurant process mixtures The ddCRP mixture is an extension of the traditional Chinese restaurant process (CRP) mixture. CRP mixtures provide a clustering method that determines the number of clusters from the data— they are an alternative formulation of the Dirichlet process mixture model. The assumed generative process is described by imagining a restaurant with an inﬁnite number of tables, each of which is endowed with a parameter for some family of data generating distributions (in our experiments, Dirichlet). Customers enter the restaurant in sequence and sit at a randomly chosen table. They sit at the previously occupied tables with probability proportional to how many customers are already sitting at each; they sit at an unoccupied table with probability proportional to a scaling parameter. After the customers have entered the restaurant, the “seating plan” provides a clustering. Finally, each customer draws an observation from a distribution determined by the parameter at the assigned table. Conditioned on observed data, the CRP mixture provides a posterior distribution over table assignments and the parameters attached to those tables. It is a distribution over clusterings, where the number of clusters is determined by the data. Though described sequentially, the CRP mixture is an exchangeable model: the posterior distribution over partitions does not depend on the ordering of the observed data. Theoretically, exchangeability is necessary to make the connection between CRP mixtures and Dirichlet process mixtures. Practically, exchangeability provides efﬁcient Gibbs sampling algorithms for posterior inference. However, exchangeability is not an appropriate assumption in image segmentation problems—the locations of the image pixels are critical to providing contiguous segmentations. 2.2 Distance dependent CRPs The distance dependent Chinese Restaurant Process (ddCRP) is a generalization of the Chinese restaurant process that allows for a non-exchangeable distribution on partitions [1]. Rather than represent a partition by customers assigned to tables, the ddCRP models customers linking to other customers. The seating plan is a byproduct of these links—two customers are sitting at the same table if one can reach the other by traversing the customer assignments. As in the CRP, tables are endowed with data generating parameters. Once the partition is determined, the observed data for each customer are generated by the per-table parameters. As illustrated in Figure 1, the generative process is described in terms of customer assignments ci (as opposed to partition assignments or tables, zi). The distribution of customer assignments is p (ci = j \| D, f, α) ∝ f(dij) j ̸= i, α j = i. (1) Here dij is a distance between data points i and j and f(d) is called the decay function. The decay function mediates how the distance between two data points affects their probability of connecting to each other, i.e., their probability of belonging to the same cluster. Details of the ddCRP are found in [1]. We note that the traditional CRP is an instance of a ddCRP. However, in general, the ddCRP does not correspond to a model based on a random measure, like the Dirichlet process. The ddCRP is appropriate for image segmentation because it can naturally account for the spatial structure of the superpixels through its distance function. We use a spatial distance between pixels to enforce a bias towards contiguous clusters. Though the ddCRP has been previously described in general, only time-based distances are studied in [1]. 3 Figure 2: Comparison of distance-dependent segmentation priors. From left to right, we show segmentations produced by the ddCRP with a = 1, the ddCRP with a = 2, the ddCRP with a = 5, and the rddCRP with a = 1. Restaurants represent images, tables represent segment assignment, and customers represent superpixels. The distance between superpixels is modeled as the number of hops required to reach one superpixel from another, with hops being allowed only amongst spatially neighboring superpixels. A “window” decay function of width a, f(d) = 1[d ≤a], determines link probabilities. If a = 1, superpixels can only directly connect to adjacent superpixels. Note this does not explicitly restrict the size of segments, because any pair of pixels for which one is reachable from the other (i.e., in the same connected component of the customer assignment graph) are in the same image segment. For this special case segments are guaranteed with probability one to form spatially connected subsets of the image, a property not enforced by other Bayesian nonparametric models [10, 11, 12]. The full generative process for the observed features x1:N within a N-superpixel image is as follows: 1. For each table, sample parameters φk ∼G0. 2. For each customer, sample a customer assignment ci ∼ddCRP(α, f, D). This indirectly determines the cluster assignments z1:N, and thus the segmentation. 3. For each superpixel, independently sample observed data xi ∼P(· \| φzi). The customer assignments are sampled using the spatial distance between pixels. The partition structure, derived from the customer assignments, is used to sample the observed image features. Given an image, the posterior distribution of the customer assignments induces a posterior over the cluster structure; this provides the segmentation. See Figure 1 for an illustration of the customer assignments and their derived table assignments in a segmentation setting. As in [10], the data generating distribution for the observed features studied in Section 4 is multinomial, with separate distributions for color and texture. We place conjugate Dirichlet priors on these cluster parameters. 2.3 Region-based hierarchical distance dependent CRPs The ddCRP model, when applied to an image with window size a = 1, produces a collection of contiguous patches (tables) homogeneous in color and texture features (Figure 2). While such segmentations are useful for various applications [16], they do not reﬂect the statistics of manual human segmentations, which contain larger regions [17]. We could bias our model to produce such regions by either increasing the window size a, or by introducing a hierarchy wherein the produced patches are grouped into a small number of regions. This region level model has each patch (table) associated with a region k from a set of potentially inﬁnite regions. Each region in turn is associated with an appearance model φk. The corresponding generative process is described as follows: 1. For each customer, sample customer assignments ci ∼ddCRP (α, f, D). This determines the table assignments t1:N. 2. For each table t, sample region assignments kt ∼CRP (γ). 3. For each region, sample parameters φk ∼G0. 4. For each superpixel, independently sample observed data xi ∼P (· \| φzi), where zi = kti. Note that this region level rddCRP model is a direct extension of the Chinese restaurant franchise (CRF) representation of the HDP [5], with the image partition being drawn from a ddCRP instead 4 of a CRP. In contrast to prior applications of the HDP, our region parameters are not shared amongst images, although it would be simple to generalize to this case. Figure 3 plots samples from the rddCRP and ddCRP priors with increasing a. The rddCRP produces larger partitions than the ddCRP with a = 1, while avoiding the noisy boundaries produced by a ddCRP with large a (see Figure 2). 3 Inference with Gibbs Sampling A segmentation of an observed image is found by posterior inference. The problem is to compute the conditional distribution of the latent variables—the customer assignments c1:N—conditioned on the observed image features x1:N, the scaling parameter α, the distances between pixels D, the window size a, and the base distribution hyperparameter λ: p(c1:N \| x1:N, α, d, a, λ) = QN i=1 p(ci \| D, a, α) p(x1:N \| z(c1:N), λ) P c1:N QN i=1 p(ci \| D, a, α) p(x1:N \| z(c1:N), λ) (2) where z(c1:N) is the cluster representation that is derived from the customer representation c1:N. Notice again that the prior term uses the customer representation to take into account distances between data points; the likelihood term uses the cluster representation. The posterior in Equation (2) is not tractable to directly evaluate, due to the combinatorial sum in the denominator. We instead use Gibbs sampling [3], a simple form of Monte Carlo Markov chain (MCMC) inference [18]. We deﬁne the Markov chain by iteratively sampling each latent variable ci conditioned on the others and the observations, p(ci \| c−i, x1:N, D, α, λ) ∝p(ci \| D, α)p(x1:N \| z(c1:N), λ). (3) The prior term is given in Equation (1). We can decompose the likelihood term as follows: p(x1:N \| z(c1:N), λ) = K(c1:N) Y k=1 p(xz(c1:N )=k \| z(c1:N), λ). (4) We have introduced notation to more easily move from the customer representation—the primary latent variables of our model—and the cluster representation. Let K(c1:N) denote the number of unique clusters in the customer assignments, z(c1:N) the cluster assignments derived from the customer assignments, and xz(c1:N)=k the collection of observations assigned to cluster k. We assume that the cluster parameters φk have been analytically marginalized. This is possible when the base distribution G0 is conjugate to the data generating distribution, e.g. Dirichlet to multinomial. Sampling from Equation (3) happens in two stages. First, we remove the customer link ci from the current conﬁguration. Then, we consider the prior probability of each possible value of ci and how it changes the likelihood term, by moving from p(x1:N \| z(c−i), λ) to p(x1:N \| z(c1:N), λ). In the ﬁrst stage, removing ci either leaves the cluster structure intact, i.e., z(cold 1:N) = z(c−i), or splits the cluster assigned to data point i into two clusters. In the second stage, randomly reassigning ci either leaves the cluster structure intact, i.e., z(c−i) = z(c1:N), or joins the cluster assigned to data point i to another. See Figure 1 for an illustration of these cases. Via these moves, the sampler explores the space of possible segmentations. Let ℓand m be the indices of the tables that are joined to index k. We ﬁrst remove ci, possibly splitting a cluster. Then we sample from p(ci \| c−i, x1:N, D, α, λ) ∝ p(ci \| D, α)Γ(x, z, λ) if ci joins ℓand m; p(ci \| D, α) otherwise, (5) where Γ(x, z, λ) = p(xz(c1:N)=k \| λ) p(xz(c1:N)=ℓ\| λ)p(xz(c1:N )=m \| λ). (6) This deﬁnes a Markov chain whose stationary distribution is the posterior of the spatial ddCRP deﬁned in Section 2. Though our presentation is slightly different, this algorithm is equivalent to the one developed for ddCRP mixtures in [1]. 5 In the rddCRP, the algorithm for sampling the customer indicators is nearly the same, but with two differences. First, when ci is removed, it may spawn a new cluster. In that case, the region identity of the new table must be sampled from the region level CRP. Second, the likelihood term in Equation (4) depends only on the superpixels in the image assigned to the segment in question. In the rddCRP, it also depends on other superpixels assigned to segments that are assigned to the same region. Finally, the rddCRP also requires resampling of region assignments as follows: p(kt = ℓ\| k−t, x1:N, t(c1:N), γ, λ) ∝ m−t ℓp(xt \| x−t, λ) if ℓis used; γp(xt \| λ) if ℓis new. (7) Here, xt is the set of customers sitting at table t, x−t is the set of all customers associated with region ℓexcluding xt, and m−t ℓ is the number of tables associated with region ℓexcluding xt. 4 Empirical Results We compare the performance of the ddCRP to manual segmentations of images drawn from eight natural scene categories [19]. Non-expert users segmented each image into polygonal shapes, and labeled them as distinct objects. The collection, which is available from LabelMe [17], contains a total of 2,688 images.1 We randomly select 100 images from each category. This image collection has been previously used to analyze an image segmentation method based on spatially dependent Pitman-Yor (PY) processes [10], and we compare both methods using an identical feature set. Each image is ﬁrst divided into approximately 1000 superpixels [15, 20]2 using the normalized cut algorithm [9].3 We describe the texture of each superpixel via a local texton histogram [21], using band-pass ﬁlter responses quantized to 128 bins. A 120-bin HSV color histogram is also computed. Each superpixel i is summarized via these histograms xi. Our goal is to make a controlled comparison to alternative nonparametric Bayesian methods on a challenging task. Performance is assessed via agreement with held out human segmentations, via the Rand index [22]. We also present segmentation results for qualitative evaluation in Figures 3 and 4 . 4.1 Sensitivity to Hyperparameters Our models are governed by the CRP concentration parameters γ and α, the appearance base measure hyperparameter λ = (λ0, ...λ0), and the window size a. Empirically, γ has little impact on the segmentation results, due to the high-dimensional and informative image features; all our experiments set γ = 1. α and λ0 induce opposing biases: a small α encourages larger segments, while a large λ0 encourages larger segments. We found α = 10−8 and λ0 = 20 to work well. The most inﬂuential prior parameter is the window size a, the effect of which is visualized in Figure 3. For the ddCRP model, setting a = 1 (ddCRP1) produces a set of small but contiguous segments. Increasing to a = 2 (ddCRP2) results in fewer segments, but the produced segments are typically spatially fragmented. This phenomenon is further exacerbated with larger values of a. The rddCRP model groups segments produced by a ddCRP. Because it is hard to recover meaningful partitions if these initial segments are poor, the rddCRP performs best when a = 1. 4.2 Image Segmentation Performance We now quantitatively measure the performance of our models. The ddCRP and the rddCRP samplers were run for 100 and 500 iterations, respectively. Both samplers displayed rapid mixing and often stabilized withing the ﬁrst 50 iterations. Note that similar rapid mixing has been observed in other applications of the ddCRP [1]. We also compare to two previous models [10]: a PY mixture model with no spatial dependence (pybof20), and a PY mixture with spatial coupling induced via thresholded Gaussian processes (pydist20). To control the comparison as much as possible, the PY models are tested with identical features and base measure β, and other hyperparameters as in [10]. We also compare to the nonspatial PY with λ0 = 1, the best bag-of-feature model in our experiments (pybof). We employ 1http://labelme.csail.mit.edu/browseLabelMe/ 2http://www.cs.sfu.ca/˜mori/ 3http://www.eecs.berkeley.edu/Research/Projects/CS/vision/ 6 Models Images Figure 3: Segmentations produced by various Bayesian nonparametric methods. From left to right, the columns display natural images, segmentations for the ddCRP with a = 1, the ddCRP with a = 2, the rddCRP with a = 1, and thresholded Gaussian processes (pydist20) [10]. The top row displays partitions sampled from the corresponding priors, which have 130, 54, 5, and 6 clusters, respectively. Figure 4: Top left: Average segmentation performance on the database of natural scenes, as measured by the Rand index (larger is better), and those pairs of methods for which a Wilcoxon’s signed rank test indicates comparable performance with 95% conﬁdence. In the binary image, dark pixels indicate pairs that are statistically indistinguishable. Note that the rddCRP, spatial PY, and mean shift methods are statistically indistinguishable, and signiﬁcantly better than all others. Bottom left: Scatter plots comparing the pydist20 and rddCRP methods in the Mountain and Street scene categories. Right: Example segmentations produced by the rddCRP. non-hierarchical versions of the PY models, so that each image is analyzed independently, and perform inference via the previously developed mean ﬁeld variational method. Finally, from the vision literature we also compare to the normalized cuts (Ncuts) [8] and mean shift (MS) [6] segmentation algorithms.4 4We used the EDISON implementation of mean shift. The parameters of mean shift and normalized cuts were tuned by performing a grid search over a training set containing 25 images from each of the 8 categories. For normalized cuts the optimal number of segments was determined to be 5. For mean shift we held the spatial 7 Quantitative performance is summarized in Figure 4. The rddCRP outscores both versions of the ddCRP model, in terms of Rand index. Nevertheless, the patchy ddCRP1 segmentations are interesting for applications where segmentation is an intermediate step rather than the ﬁnal goal. The bag of features model with λ0 = 20 performs poorly; with optimized λ0 = 1 it is better, but still inferior to the best spatial models. In general, the spatial PY and rddCRP perform similarly. The scatter plots in Fig. 4, which show Rand indexes for each image from the mountain and street categories, provide insights into when one model outperforms the other. For the street images rddCRP is better, while for images containing mountains spatial PY is superior. In general, street scenes contain more objects, many of which are small, and thus disfavored by the smooth Gaussian processes underlying the PY model. To most fairly compare priors, we have tested a version of the spatial PY model employing a covariance function that depends only on spatial distance. Further performance improvements were demonstrated in [10] via a conditionally speciﬁed covariance, which depends on detected image boundaries. Similar conditional speciﬁcation of the ddCRP distance function is a promising direction for future research. Finally, we note that the ddCRP (and rddCRP) models proposed here are far simpler than the spatial PY model, both in terms of model speciﬁcation and inference. The ddCRP models only require pairwise superpixel distances to be speciﬁed, as opposed to the positive deﬁnite covariance function required by the spatial PY model. Furthermore, the PY model’s usage of thresholded Gaussian processes leads to a complex likelihood function, for which inference is a signiﬁcant challenge. In contrast, ddCRP inference is carried out through a straightforward sampling algorithm,5 and thus may provide a simpler foundation for building rich models of visual scenes. 5 Discussion We have studied the properties of spatial distance dependent Chinese restaurant processes, and applied them to the problem of image segmentation. We showed that the spatial ddCRP model is particularly well suited for segmenting an image into a collection of contiguous patches. Unlike previous Bayesian nonparametric models, it can produce segmentations with guaranteed spatial connectivity. To go from patches to coarser, human-like segmentations, we developed a hierarchical region-based ddCRP. This hierarchical model achieves performance similar to state-of-the-art nonparametric Bayesian segmentation algorithms, using a simpler model and a substantially simpler inference algorithm. References [1] D. M. Blei and P. I. Frazier. Distant dependent chinese restaurant processes. Journal of Machine Learning Research, 12:2461–2488, August 2011. [2] J. Pitman. Combinatorial Stochastic Processes. Lecture Notes for St. Flour Summer School. Springer-Verlag, New York, NY, 2002. [3] S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on pattern analysis and machine intelligence, 6(6):721–741, November 1984. [4] Richard Socher, Andrew Maas, and Christopher D. Manning. Spectral chinese restaurant processes: Nonparametric clustering based on similarities. In Fourteenth International Conference on Artiﬁcial Intelligence and Statistics (AISTATS), 2011. [5] Y. W. Teh, M. I. Jordan, M. J. Beal, and D. M. Blei. Hierarchical Dirichlet processes. Journal of American Statistical Association, 25(2):1566 – 1581, 2006. [6] D. Comaniciu and P. Meer. Mean shift: A robust approach toward feature space analysis. IEEE Transactions on pattern analysis and machine intelligence, pages 603–619, 2002. bandwidth constant at 7, and found optimal values of feature bandwidth and minimum region size to be 25 and 4000 pixels, respectively. 5In our Matlab implementations, the core ddCRP code was less than half as long as the corresponding PY code. For the ddCRP, the computation time was 1 minute per iteration, and convergence typically happened after only a few iterations. The PY code, which is based on variational approximations, took 12 minutes per image. 8 [7] C. Rother, V. Kolmogorov, and A. Blake. Grabcut: Interactive foreground extraction using iterated graph cuts. In ACM Transactions on Graphics (TOG), volume 23, pages 309–314, 2004. [8] J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Trans. PAMI, 22(8):888– 905, 2000. [9] C. Fowlkes, D. Martin, and J. Malik. Learning afﬁnity functions for image segmentation: Combining patch-based and gradient-based approaches. CVPR, 2:54–61, 2003. [10] E. B. Sudderth and M. I. Jordan. Shared segmentation of natural scenes using dependent pitman-yor processes. NIPS 22, 2008. [11] P. Orbanz and J. M. Buhmann. Smooth image segmentation by nonparametric Bayesian inference. In ECCV, volume 1, pages 444–457, 2006. [12] Lan Du, Lu Ren, David Dunson, and Lawrence Carin. A bayesian model for simultaneous image clustering, annotation and object segmentation. In NIPS 22, pages 486–494. 2009. [13] J. Pitman and M. Yor. The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Annals of Probability, 25(2):855–900, 1997. [14] J. A. Duan, M. Guindani, and A. E. Gelfand. Generalized spatial Dirichlet process models. Biometrika, 94(4):809–825, 2007. [15] X. Ren and J. Malik. Learning a classiﬁcation model for segmentation. ICCV, 2003. [16] C. Carson, S. Belongie, H. Greenspan, and J. Malik. Blobworld: Image segmentation using expectation-maximization and its application to image querying. PAMI, 24(8):1026–1038, August 2002. [17] B. C. Russell, A. Torralba, K. P. Murphy, and W. T. Freeman. Labelme: A database web-based tool for image annotation. IJCV, 77:157–173, 2008. [18] C. Robert and G. Casella. Monte Carlo Statistical Methods. Springer Texts in Statistics. Springer-Verlag, New York, NY, 2004. [19] A. Oliva and A. Torralba. Modeling the shape of the scene: A holistic representation of the spatial envelope. IJCV, 42(3):145 – 175, 2001. [20] G. Mori. Guiding model search using segmentation. ICCV, 2005. [21] D. R. Martin, C.C. Fowlkes, and J. Malik. Learning to detect natural image boundaries using local brightness, color, and texture cues. IEEE Trans. PAMI, 26(5):530–549, 2004. [22] W.M. Rand. Objective criteria for the evaluation of clustering methods. Journal of the American Statistical Association, pages 846–850, 1971. 9	2011	85
4,436	History distribution matching method for predicting effectiveness of HIV combination therapies Jasmina Bogojeska Max-Planck Institute for Computer Science Campus E1 4 66123 Saarbr¨ucken, Germany jasmina@mpi-inf.mpg.de Abstract This paper presents an approach that predicts the effectiveness of HIV combination therapies by simultaneously addressing several problems affecting the available HIV clinical data sets: the different treatment backgrounds of the samples, the uneven representation of the levels of therapy experience, the missing treatment history information, the uneven therapy representation and the unbalanced therapy outcome representation. The computational validation on clinical data shows that, compared to the most commonly used approach that does not account for the issues mentioned above, our model has signiﬁcantly higher predictive power. This is especially true for samples stemming from patients with longer treatment history and samples associated with rare therapies. Furthermore, our approach is at least as powerful for the remaining samples. 1 Introduction According to [18], more than 33 million people worldwide are infected with the human immunodeﬁciency virus (HIV), for which there exists no cure. HIV patients are treated by administration of combinations of antiretroviral drugs, which succeed in suppressing the virus much longer than the monotherapies based on a single drug. Eventually, the drug combinations also become ineffective and need to be replaced. On such occasion, the very large number of potential therapy combinations makes the manual search for an effective therapy increasingly impractical. The search is particulary challenging for patients in the mid to late stages of antiretroviral therapy because of the accumulated drug resistance from all previous therapies. The availability of large clinical data sets enables the development of statistical methods that offer an automated procedure for predicting the outcome of potential antiretroviral therapies. An estimate of the therapy outcome can assist physicians in choosing a successful regimen for an HIV patient. However, the HIV clinical data sets suffer from several problems. First of all, the clinical data comprise therapy samples that originate from patients with different treatment backgrounds. Also the various levels of therapy experience ranging from therapy-na¨ıve to heavily pretreated are represented with different sample abundances. Second, the samples on different combination therapies have widely differing frequencies. In particular, many therapies are only represented with very few data points. Third, the clinical data do not necessarily have the complete information on all administered HIV therapies for all patients and the information on whether all administered therapies is available or not is also missing for many of the patients. Finally, the imbalance between the effective and the ineffective therapies is increasing over time: due to the knowledge acquired from HIV research and clinical practice the quality of treating HIV patients has largely increased in the recent years rendering the amount of effective therapies in recently collected data samples much larger than the amount of ineffective ones. These four problems create bias in the data sets which might negatively affect the usefulness of the derived statistical models. 1 In this paper we present an approach that addresses all these problems simultaneously. To tackle the issues of the uneven therapy representation and the different treatment backgrounds of the samples, we use information on both the current therapy and the patient’s treatment history. Additionally, our method uses a distribution matching approach to account for the problems of missing information in the treatment history and the growing gap between the abundances of effective and ineffective HIV therapies over time. The performance of our history distribution matching approach is assessed by comparing it with two common reference methods in the so called time-oriented validation scenario, where all models are trained on data from the more distant past, while their performance is assessed on data from the more recent past. In this way we account for the evolving trends in composing drug combination therapies for treating HIV patients. Related work. Various statistical learning methods, including artiﬁcial neural networks, decision trees, random forests, support vector machines (SVMs) and logistic regression [19, 11, 14, 10, 16, 1, 15], have been used to predict the effectiveness of HIV combination therapies from clinical data. None of these methods considers the problems affecting the available clinical data sets: different treatment backgrounds of the samples, uneven representations of therapies and therapy outcomes, and incomplete treatment history information. Some approaches [2, 4] deal with the uneven therapy representation by training a separate model for each combination therapy on all available samples with properly derived sample weights. The weights reﬂect the similarities between the target therapy and all training therapies. However, the therapy-speciﬁc approaches do not address the bias originating from the different treatment backgrounds of the samples, or the missing treatment history information. 2 Problem setting Let z denote a therapy sample that comprises the viral genotype g represented as a binary vector indicating the occurrence of a set of resistance-relevant mutations, the therapy combination z encoded as a binary vector that indicates the individual drugs comprising the current therapy, the binary vector h representing the drugs administered in all known previous therapies, and the label y indicating the success (1) or failure (−1) of the therapy z. Let D = {(g1, z1, h1, y1), . . . , (gm, zm, hm, ym)} denote the training set and let s refer to the therapy sample of interest. Let start(s) refer to the point of time when the therapy s was started and patient(s) refer to the patient identiﬁer corresponding to the therapy sample s. Then: r(s) = {z \| (start(z) ≤start(s)) and (patient(z) = patient(s))} denotes the complete treatment data associated with the therapy sample s and will be referred to as therapy sequence. It contains all known therapies administered to patient(s) not later than start(s) ordered by their corresponding starting times. We point out that each therapy sequence also contains the current therapy, i.e., the most recent therapy in the therapy sequence r(s) is s. Our goal is to train a model f(g, s, h) that addresses the different types of bias associated with the available clinical data sets when predicting the outcome of the therapy s. In the rest of the paper we denote the set of input features (g, s, h) by x. 3 History distribution matching method The main idea behind the history distribution matching method we present in this paper is that the predictions for a given patient should originate from a model trained using samples from patients with treatment backgrounds similar as the one of the target patient. The details of this method are summarized in Algorithm 1. In what follows, we explain each step of this algorithm. 3.1 Clustering based on similarities of therapy sequences Clustering partitions a set of objects into clusters, such that the objects within each cluster are more similar to one another than to the objects assigned to a different cluster [7]. In the ﬁrst step of Algorithm 1, all available training samples are clustered based on the pairwise dissimilarity of their corresponding therapy sequences. In the following, we ﬁrst describe a similarity measure for therapy sequences and then present the details of the clustering. 2 Algorithm 1: History distribution matching method 1. Cluster the training samples by using the pairwise dissimilarities of their corresponding therapy sequences. 2. For each (target) cluster: • Compute sample weights that match the distribution of all available training samples to the distribution of samples in the target cluster. • Train a sample-weighted logistic regression model using the sample weights computed in the previous distribution matching step. Similarity of therapy sequences. In order to quantify the pairwise similarity of therapy sequences we use a slightly modiﬁed version of the alignment similarity measure introduced in [5]. It adapts sequence alignment techniques [13] to the problem of aligning therapy sequences by considering the speciﬁc therapies given to a patient, their respective resistance-relevant mutations, the order in which they were applied and the length of the therapy history. The alphabet used for the therapy sequence alignment comprises all distinct drug combinations making up the clinical data set. The pairwise similarities between the different drug combinations are quantiﬁed with the resistance mutations kernel [5], which uses the table of resistance-associated mutations of each drug afforded by the International AIDS society [8]. First, binary vectors indicating resistance-relevant mutations for the set of drugs occurring in a combination are calculated for each therapy. Then, the similarity score of two therapies of interest is computed as normalized inner product between their corresponding resistance mutation vectors. In this way, the therapy similarity also accounts for the similarity of the genetic ﬁngerprint of the potential latent virus populations of the compared therapies. Each therapy sequence ends with the current (most recent) therapy – the one that determines the label of the sample and the sequence alignment is adapted such that the most recent therapies are always matched. Therefore, it also accounts for the problem of uneven representation of the different therapies in the clinical data. It has one parameter that speciﬁes the linear gap cost penalty. For the history distribution matching method, we modiﬁed the alignment similarity kernel described in the paragraph above such that it also takes the importance of the different resistance-relevant mutations into account. This is achieved by updating the resistance mutations kernel, where instead of using binary vectors that indicate the occurrence of a set of resistance-relevant mutations, we use vectors that indicate their importance. If two or more drugs from a certain drug group, that comprise a target therapy share a resistance mutation, then we consider its maximum importance score. Importance scores for the resistance-relevant mutations are derived from in-vivo experiments and can be obtained from the Stanford University HIV Drug Resistance Database [12]. Furthermore, we want to keep the cluster similarity measure parameter-free, such that in the process of model selection the clustering Step 1 in Algorithm 1 is decoupled from the Step 2 and is computed only once. This is achieved by computing the alignments with zero gap costs and ensures time-efﬁcient model selection procedure. However, in this case only the similarities of the matched therapies comprising the two compared therapy sequences contribute to the similarity score and thus the differing lengths of the therapy sequences are not accounted for. Having a clustering similarity measure that addresses the differing therapy lengths is important for tackling the uneven sample representation with respect to the level of therapy experience. In order to achieve this we normalize each pairwise similarity score with the length of the longer therapy sequence. This yields pairwise similarity values in the interval [0, 1] which can easily be converted to dissimilarity values in the same range by subtracting them from 1. Clustering. Once we have a measure of dissimilarity of therapy sequences, we cluster our data using the most popular version of K-medoids clustering [7], referred to as partitioning around medoids (PAM) [9]. The main reason why we choose this approach instead of the simpler K-means clustering [7] is that it can use any precomputed dissimilarity matrix. We select the number of clusters with the silhouette validation technique [17], which uses the so-called silhouette value to assess the quality of the clustering and select the optimal number of clusters. 3 3.2 Cluster distribution matching The clustering step of our method groups the training data into different bins based on their therapy sequences. However, the complete treatment history is not necessarily available for all patients in our clinical data set. Therefore, by restricting the prediction model for a target sample only to the data from its corresponding cluster, the model might ignore relevant information from the other clusters. The approach we use to deal with this issue is inspired by the multi-task learning with distribution matching method introduced in [2]. In our current problem setting, the goal is to train a prediction model fc : x →y for each cluster c of similar treatment sequences, where x denotes the input features and y denotes the label. The straightforward approach to achieve this is to train a prediction model by using only the samples in cluster c. However, since the available treatment history for some samples might be incomplete, totally excluding the samples from all other clusters (̸= c) ignores relevant information about the model fc. Furthermore, the cluster-speciﬁc tasks are related and the samples from the other clusters – especially those close to the cluster boundaries of cluster c – also carry valuable information for the model fc. Therefore, we use a multi-task learning approach where a separate model is trained for each cluster by not only using the training samples from the target cluster, but also the available training samples from the remaining clusters with appropriate sample-speciﬁc weights. These weights are computed by matching the distribution of all samples to the distribution of the samples of the target cluster and they thereby reﬂect the relevance of each sample for the target cluster. In this way, the model for the target cluster uses information from the input features to extract relevant knowledge from the other clusters. More formally, let D = {(x1, y1, c1), . . . , (xm, ym, cm)} denote the training data, where ci denotes the cluster associated with the training sample (xi, yi) in the history-based clustering. The training data are governed by the joint training distribution P c p(c)p(x, y\|c). The most accurate model for a given target cluster t minimizes the loss with respect to the conditional probability p(x, y\|t) referred to as the target distribution. In [2] it is shown that: E(x,y)∼p(x,y\|t)[ℓ(ft(x))] = E(x,y)∼P c p(c)p(x,y\|c)[rt(x, y)ℓ(ft(x))], (1) where: rt(x, y) = p(x, y\|t) P c p(c)p(x, y\|c). (2) In other words, by using sample-speciﬁc weights rt(x, y) that match the training distribution P c p(c)p(x, y\|c) to the target distribution p(x, y\|t) we can minimize the expected loss with respect to the target distribution by minimizing the expected loss with respect to the training distribution. The weighted training data are governed by the correct target distribution p(x, y\|t) and the sample weights reﬂect the relevance of each training sample for the target model. The weights are derived based on information from the input features. If a sample was assigned to the wrong cluster due to the incompleteness of the treatment history, by matching the training to the target distribution it can still receive high sample weight for the model of its correct cluster. In order to avoid the estimation of the high-dimensional densities p(x, y\|t) and p(x, y\|c) in Equation 2, we follow the example of [3, 2] and compute the sample weights rt(x, y) using a discriminative model for a conditional distribution with a single variable: rt(x, y) = p(t\|x, y) p(t) , (3) where p(t\|x, y) quantiﬁes the probability that a sample (x, y) randomly drawn from the training set D belongs to the target cluster t. p(t) is the prior probability which can easily be estimated from the training data. As in [2], p(t\|x, y) is modeled for all clusters jointly using a kernelized version of multi-class logistic regression with a feature mapping that separates the effective from the ineffective therapies: Φ(x, y) = δ(y, +1)x δ(y, −1)x , (4) where δ is the Kronecker delta (δ(a, b) = 1, if a = b, and δ(a, b) = 0, if a ̸= b). In this way, we can train the cluster-discriminative models for the effective and the ineffective therapies independently, 4 and thus, by proper time-oriented model selection address the increasing imbalance in their representation over time. Formally, the multi-class model is trained by maximizing the log-likelihood over the training data using a Gaussian prior on the model parameters: arg max v X (xi,yi,ci)∈Dc log(p(ci\|xi, yi, v)) + vTΣ−1v, where v are the model parameters (a concatenation of the cluster speciﬁc parameters vc), and Σ is the covariance matrix of the Gaussian prior. 3.3 Sample-weighted logistic regression method As described in the previous subsection, we use a multi-task distribution matching procedure to obtain sample-speciﬁc weights for each cluster, which reﬂect the relevance of each sample for the corresponding cluster. Then, a separate logistic regression model that uses all available training data with the proper sample weights is trained for each cluster. More formally, let t denote the target cluster and let rt(x, y) denote the weight of the sample (x, y) for the cluster t. Then, the prediction model for the cluster t that minimizes the loss over the weighted training samples is given by: arg min wt 1 \|D\| X (xi,yi)∈D rt(xi, y)γ · ℓ(ft(xi), yi) + σwT t wt, (5) where wt are the model parameters, σ is the regularization parameter, γ is a smoothing parameter for the sample-speciﬁc weights and ℓ(f(x, wt), y) = ln(1 + exp(−ywT t x)) is the loss of linear logistic regression. All in all, our method ﬁrst clusters the training data based on their corresponding therapy sequences and then learns a separate model for each cluster by using relevant data from the remaining clusters. By doing so it tackles the problems of the different treatment backgrounds of the samples and the uneven sample representation in the clinical data sets with respect to the level of therapy experience. Since the alignment kernel considers the most recent therapy and the drugs comprising this therapy are encoded as a part of the input feature space, our method also deals with the differing therapy abundances in the clinical data sets. Once we have the models for each cluster, we use them to predict the label of a given test sample x as follows: First of all, we use the therapy sequence of the target sample to calculate its dissimilarity to the therapy sequences of each of the cluster centers. Then, we assign the sample x to the cluster c with the closest cluster center. Finally, we use the logistic regression model trained for cluster c to predict the label y for the target sample x. 4 Experiments and results 4.1 Data The clinical data for our model are extracted from the EuResist [16] database that contains information on 93014 antiretroviral therapies administered to 18325 HIV (subtype B) patients from several countries in the period from 1988 to 2008. The information employed by our model is extracted from these data: the viral sequence g assigned to each therapy sample is obtained shortly before the respective therapy was started (up to 90 days before); the individual drugs of the currently administered therapy z; all available (known) therapies administered to each patient h, r(z); and the response to a given therapy quantiﬁed with a label y (success or failure) based on the virus load values (copies of viral RNA per ml blood plasma) measured during its course (for more details see [4] and the Supplementary material). Finally, our training set comprises 6537 labeled therapy samples from 690 distinct therapy combinations. 4.2 Validation setting Time-oriented validation scenario. The trends of treating HIV patients change over time as a result of the gathered practical experience with the drugs and the introduction of new antiretroviral drugs. In order to account for this phenomenon we use the time-oriented validation scenario [4] which makes a time-oriented split when selecting the training and the test set. First, we order all 5 available training samples by their corresponding therapy starting dates. We then make a timeoriented split by selecting the most recent 20% of the samples as the test set and the rest as the training set. For the model selection we split the training set further in a similar manner. We take the most recent 25% of the training set for selecting the best model parameters (see Supplementary material) and refer to this set as tuning set. In this way, our models are trained on the data from the more distant past, while their performance is measured on the data from the more recent past. This scenario is more realistic than other scenarios since it captures how a given model would perform on the recent trends of combining the drugs. The details of the data sets resulting from this scenario are given in Table 1, where one can also observe the large gap between the abundances of the effective and ineffective therapies, especially for the most recent data. Table 1: Details on the data sets generated in the time-oriented validation scenario. Data set training tuning test Sample count 3596 1634 1307 Success rate 69% 79% 83% The search for an effective HIV therapy is particulary challenging for patients in the mid to late stages of antiretroviral therapy when the number of therapy options is reduced and effective therapies are increasingly hard to ﬁnd because of the accumulated drug resistance mutations from all previous therapies. The therapy samples gathered in the HIV clinical data sets are associated with patients whose treatment histories differ in length: while some patients receive their ﬁrst antiretroviral treatment, others are heavily pretreated. These different sample groups, from treatment na¨ıve to heavily pretreated, are represented unevenly in the HIV clinical data with fewer samples associated to therapy-experienced patients (see Figure 1 (a) in the Supplementary material). In order to assess the ability of a given target model to address this problem, we group the therapy samples in the test set into different bins based on the number of therapies administered prior to the therapy of interest – the current therapy (see Table 1 in the Supplementary material). Then, we assess the quality of a given target model by reporting its performance for each of the bins. In this way we can assess the predictive power of the models in dependence on the level of therapy experience. Another important property of an HIV model is its ability to address the uneven representation of the different therapies (see Figure 1 (b) in the Supplementary material). In order to achieve this we group the therapies in the test set based on the number of samples they have in the training set, and then we measure the model performance on each of the groups. The details on the sample counts in each of the bins are given in Table 2 of the Supplementary material. In this manner we can evaluate the performance of the models for the rare therapies. Due to the lack of data and practical experience for the rare HIV combination therapies, predicting their efﬁciency is more challenging compared to estimating the efﬁciency of the frequent therapies. Reference methods. In our computational experiments we compare the results of our history distribution matching approach, denoted as transfer history clustering validation scenario, to those of three reference approaches, namely the one-for-all validation scenario, the history-clustering validation scenario, and the therapy-speciﬁc validation scenario. The one-for-all method mimics the most common approaches in the ﬁeld [16, 1, 19] that train a single model (here logistic regression) on all available therapy samples in the data set. The information on the individual drugs comprising the target (most recent) therapy and the drugs administered in all its available preceding therapies are encoded in a binary vector and supplied as input features. The history-clustering method implements a modiﬁed version of Algorithm 1 that skips the distribution matching step. In other words, a separate model is trained for each cluster by using only the data from the respective cluster. We introduce this approach to assess the importance of the distribution matching step. The therapy-speciﬁc scenario implements the drugs kernel therapy similarity model described in [4]. It represents the approaches that train a separate model for each combination therapy by using not only the samples from the target therapy but also the available samples from similar therapies with appropriate sample-importance weights. Performance measures. The performance of all considered methods is assessed by reporting their corresponding accuracies (ACC) and AUCs (Area Under the ROC Curve). The accuracy reﬂects the ability of the methods to make correct predictions, i.e., to discriminate between successful and failing HIV combination therapies. With the AUC we are able to assess the quality of the ranking based 6 on the probability of therapy success. For this reason, we carry out the model selection based on both accuracy and AUC and then use accuracy or AUC, respectively, to assess the model performance. In order to compare the performance of two methods on a separate test set, the signiﬁcance of the difference of two accuracies as well as their standard deviations are calculated based on a paired t-test. The standard deviations of the AUC values and the signiﬁcance of the difference of two AUCs used for the pairwise method comparison are estimated as described in [6]. 4.3 Experimental results According to the results from the silhouette validation technique [17] displayed in Figure 2 in the Supplementary material, the ﬁrst clustering step of Algorithm 1 divides our training data into two clusters – one comprises the samples with longer therapy sequences (with average treatment history length of 5.507 therapies), and the other one those with shorter therapy sequences (with average treatment history length of 0.308 therapies). Thus, the transfer history distribution matching method trains two models, one for each cluster. The clustering results are depicted in Figure 3 in the Supplementary material. In what follows, we ﬁrst present the results of the time-oriented validation scenario stratiﬁed for the length of treatment history, followed by the results stratiﬁed for the abundance of the different therapies. In both cases we report both the accuracies and the AUCs for all considered methods. The computational results for the transfer history method and the three reference methods stratiﬁed for the length of the therapy history are summarized in Figure 1, where (a) depicts the accuracies, and (b) depicts the AUCs. For samples with a small number (≤5) of previously administered therapies, i.e., with short treatment histories, all considered models have comparable accuracies. For test samples from patients with longer (> 5) treatment histories, the transfer history clustering approach achieves signiﬁcantly better accuracy (p-values ≤0.004) compared to those of the reference methods. According to the paired difference test described in [6], the transfer history approach has signiﬁcantly better AUC performance for test samples with longer (> 5) treatment histories compared to the one-for-all (p-value = 0.043) and the history-clustering (p-value = 0.044) reference methods. It also has better AUC performance compared to the one of the therapy-speciﬁc model, yet this improvement is not signiﬁcant (p-value = 0.253). Furthermore, the transfer history approach achieves better AUCs for test samples with less than ﬁve previously administered therapies compared to all reference methods. However, the improvement is only signiﬁcant for the one-for-all method (p-value = 0.007). The corresponding p-values for the history-clustering method and the therapy-speciﬁc method are 0.080 and 0.178, respectively. 0−5 >5 transfer history clustering history clustering therapy specific one−for−all Number of preceding treatments ACC 0.5 0.6 0.7 0.8 0.9 (a) 0−5 >5 transfer history clustering history clustering therapy specific one−for−all Number of preceding treatments AUC 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 (b) Figure 1: Accuracy (a) and AUC (b) results of the different models obtained on the test set in the time-oriented validation scenario. Error bars indicate the standard deviations of each model. The test samples are grouped based on their corresponding number of known previous therapies. The experimental results, stratiﬁed for the abundance of the therapies summarizing the accuracies and AUCs for all considered methods, are depicted in Figure 2 (a) and (b), respectively. As can 7 be observed from Figure 2 (a), all considered methods have comparable accuracies for the test therapies with more than seven samples. The transfer history method achieves signiﬁcantly better accuracy (p-values ≤0.0001) compared to all reference methods for the test therapies with few (0 −7) available training samples. Considering the AUC results in Figure 2 (b), the transfer history approach outperforms all the reference models for the rare test therapies (with 0 −7 training samples) with estimated p-values of 0.05 for the one-for-all, 0.042 for the therapy-speciﬁc and 0.1 for the history-clustering model. The one-for-all and the therapy-speciﬁc models have slightly better AUC performance compared to the transfer history and the history-clustering approaches for test therapies with 8 −30 available training samples. However, according to the paired difference test described in [6], the improvements are not signiﬁcant with p-values larger than 0.141 for all pairwise comparisons. Moreover, considering the test therapies with more than 30 training samples the transfer history approach signiﬁcantly outperforms the one-for-all approach with estimated p-value of 0.037. It also has slightly better AUC performance than the history-clustering model and the therapy-speciﬁc model, however these improvements are not signiﬁcant with estimated p-values of 0.064 and 0.136, respectively. 0−7 8−30 >30 transfer history clustering history clustering therapy specific one−for−all Number of available training samples ACC 0.5 0.6 0.7 0.8 0.9 1.0 (a) 0−7 8−30 >30 transfer history clustering history clustering therapy specific one−for−all Number of available training samples AUC 0.5 0.6 0.7 0.8 (b) Figure 2: Accuracy (a) and AUC (b) results of the different models obtained on the test set in the time-oriented validation scenario. Error bars indicate the standard deviations of each model. The test samples are grouped based on the number of available training examples for their corresponding therapy combinations. 5 Conclusion This paper presents an approach that simultaneously considers several problems affecting the available HIV clinical data sets: the different treatment backgrounds of the samples, the uneven representation of the different levels of therapy experience, the missing treatment history information, the uneven therapy representation and the unbalanced therapy outcome representation especially pronounced in recently collected samples. The transfer history clustering model has its prime advantage for samples stemming from patients with long treatment histories and for samples associated with rare therapies. In particular, for these two groups of test samples it achieves signiﬁcantly better accuracy than all considered reference approaches. Moreover, the AUC performance of our method for these test samples is also better than all reference methods and signiﬁcantly better compared to the one-for-all method. For the remaining test samples both the accuracy and the AUC performance of the transfer history method are at least as good as the corresponding performances of all considered reference methods. Acknowledgments We gratefully acknowledge the EuResist EEIG for providing the clinical data. We thank Thomas Lengauer for the helpful comments and for supporting this work. We also thank Levi Valgaerts for the constructive suggestions. This work was funded by the Cluster of Excellence (Multimodal Computing and Interaction). 8 References [1] A. Altmann, M. D¨aumer, N. Beerenwinkel, E. Peres, Y. Sch¨ulter, A. B¨uch, S. Rhee, A. S¨onnerborg, WJ. Fessel, M. Shafer, WR. Zazzi, R. Kaiser, and T. Lengauer. 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4,437	Non-Asymptotic Analysis of Stochastic Approximation Algorithms for Machine Learning Francis Bach INRIA - Sierra Project-team Ecole Normale Sup´erieure, Paris, France francis.bach@ens.fr Eric Moulines LTCI Telecom ParisTech, Paris, France eric.moulines@enst.fr Abstract We consider the minimization of a convex objective function deﬁned on a Hilbert space, which is only available through unbiased estimates of its gradients. This problem includes standard machine learning algorithms such as kernel logistic regression and least-squares regression, and is commonly referred to as a stochastic approximation problem in the operations research community. We provide a non-asymptotic analysis of the convergence of two well-known algorithms, stochastic gradient descent (a.k.a. Robbins-Monro algorithm) as well as a simple modiﬁcation where iterates are averaged (a.k.a. Polyak-Ruppert averaging). Our analysis suggests that a learning rate proportional to the inverse of the number of iterations, while leading to the optimal convergence rate in the strongly convex case, is not robust to the lack of strong convexity or the setting of the proportionality constant. This situation is remedied when using slower decays together with averaging, robustly leading to the optimal rate of convergence. We illustrate our theoretical results with simulations on synthetic and standard datasets. 1 Introduction The minimization of an objective function which is only available through unbiased estimates of the function values or its gradients is a key methodological problem in many disciplines. Its analysis has been attacked mainly in three communities: stochastic approximation [1, 2, 3, 4, 5, 6], optimization [7, 8], and machine learning [9, 10, 11, 12, 13, 14, 15]. The main algorithms which have emerged are stochastic gradient descent (a.k.a. Robbins-Monro algorithm), as well as a simple modiﬁcation where iterates are averaged (a.k.a. Polyak-Ruppert averaging). Traditional results from stochastic approximation rely on strong convexity and asymptotic analysis, but have made clear that a learning rate proportional to the inverse of the number of iterations, while leading to the optimal convergence rate in the strongly convex case, is not robust to the wrong setting of the proportionality constant. On the other hand, using slower decays together with averaging robustly leads to optimal convergence behavior (both in terms of rates and constants) [4, 5]. The analysis from the convex optimization and machine learning literatures however has focused on differences between strongly convex and non-strongly convex objectives, with learning rates and roles of averaging being different in these two cases [11, 12, 13, 14, 15]. A key desirable behavior of an optimization method is to be adaptive to the hardness of the problem, and thus one would like a single algorithm to work in all situations, favorable ones such as strongly convex functions and unfavorable ones such as non-strongly convex functions. In this paper, we unify the two types of analysis and show that (1) a learning rate proportional to the inverse of the number of iterations is not suitable because it is not robust to the setting of the proportionality constant and the lack of strong convexity, (2) the use of averaging with slower decays allows (close to) optimal rates in all situations. More precisely, we make the following contributions: −We provide a direct non-asymptotic analysis of stochastic gradient descent in a machine learning context (observations of real random functions deﬁned on a Hilbert space) that includes 1 kernel least-squares regression and logistic regression (see Section 2), with strong convexity assumptions (Section 3) and without (Section 4). −We provide a non-asymptotic analysis of Polyak-Ruppert averaging [4, 5], with and without strong convexity (Sections 3.3 and 4.2). In particular, we show that slower decays of the learning rate, together with averaging, are crucial to robustly obtain fast convergence rates. −We illustrate our theoretical results through experiments on synthetic and non-synthetic examples in Section 5. Notation. We consider a Hilbert space H with a scalar product ⟨·, ·⟩. We denote by ∥· ∥the associated norm and use the same notation for the operator norm on bounded linear operators from H to H, deﬁned as ∥A∥= sup∥x∥⩽1 ∥Ax∥(if H is a Euclidean space, then ∥A∥is the largest singular value of A). We also use the notation “w.p.1” to mean “with probability one”. We denote by E the expectation or conditional expectation with respect to the underlying probability space. 2 Problem set-up We consider a sequence of convex differentiable random functions (fn)n⩾1 from H to R. We consider the following recursion, starting from θ0 ∈H: ∀n ⩾1, θn = θn−1 −γnf ′ n(θn−1), (1) where (γn)n⩾1 is a deterministic sequence of positive scalars, which we refer to as the learning rate sequence. The function fn is assumed to be differentiable (see, e.g., [16] for deﬁnitions and properties of differentiability for functions deﬁned on Hilbert spaces), and its gradient is an unbiased estimate of the gradient of a certain function f we wish to minimize: (H1) Let (Fn)n⩾0 be an increasing family of σ-ﬁelds. θ0 is F0-measurable, and for each θ ∈H, the random variable f ′ n(θ) is square-integrable, Fn-measurable and ∀θ ∈H, ∀n ⩾1, E(f ′ n(θ)\|Fn−1) = f ′(θ), w.p.1. (2) For an introduction to martingales, σ-ﬁelds, and conditional expectations, see, e.g., [17]. Note that depending whether F0 is a trivial σ-ﬁeld or not, θ0 may be random or not. Moreover, we could restrict Eq. (2) to be satisﬁed only for θn−1 and θ∗(which is a global minimizer of f). Given only the noisy gradients f ′ n(θn−1), the goal of stochastic approximation is to minimize the function f with respect to θ. Our assumptions include two usual situations, but also include many others (e.g., potentially, active learning): −Stochastic approximation: in the so-called Robbins-Monro setting, for all θ ∈H and n ⩾1, fn(θ) may be expressed as fn(θ) = f(θ)+⟨εn, θ⟩, where (εn)n⩾1 is a square-integrable martingale difference (i.e., such that E(εn\|Fn−1) = 0), which corresponds to a noisy observation f ′(θn−1) + εn of the gradient f ′(θn−1). −Learning from i.i.d. observations: for all θ ∈H and n ⩾1, fn(θ) = ℓ(θ, zn) where zn is an i.i.d. sequence of observations in a measurable space Z and ℓ: H × Z is a loss function. Then f(θ) is the generalization error of the predictor deﬁned by θ. Classical examples are leastsquares or logistic regression (linear or non-linear through kernel methods [18, 19]), where fn(θ) = 1 2(⟨xn, θ⟩−yn)2, or fn(θ) = log[1 + exp(−yn ⟨xn, θ⟩)], for xn ∈H, and yn ∈R (or {−1, 1} for logistic regression). Throughout this paper, unless otherwise stated, we assume that each function fn is convex and smooth, following the traditional deﬁnition of smoothness from the optimization literature, i.e., Lipschitz-continuity of the gradients (see, e.g., [20]). However, we make two slightly different assumptions: (H2) where the function θ 7→E(f ′ n(θ)\|Fn−1) is Lipschitz-continuous in quadratic mean and a strengthening of this assumption, (H2’) in which θ 7→f ′ n(θ) is almost surely Lipschitzcontinuous. (H2) For each n ⩾1, the function fn is almost surely convex, differentiable, and: ∀n ⩾1, ∀θ1, θ2 ∈H, E(∥f ′ n(θ1) −f ′ n(θ2)∥2\|Fn−1) ⩽L2∥θ1 −θ2∥2 , w.p.1. (3) (H2’) For each n ⩾1, the function fn is almost surely convex, differentiable with Lipschitzcontinuous gradient f ′ n, with constant L, that is: ∀n ⩾1, ∀θ1, θ2 ∈H, ∥f ′ n(θ1) −f ′ n(θ2)∥⩽L∥θ1 −θ2∥, w.p.1. (4) 2 If fn is twice differentiable, this corresponds to having the operator norm of the Hessian operator of fn bounded by L. For least-squares or logistic regression, if we assume that (E∥xn∥4)1/4 ⩽ R for all n ∈N, then we may take L = R2 (or even L = R2/4 for logistic regression) for assumption (H2), while for assumption (H2’), we need to have an almost sure bound ∥xn∥⩽R. 3 Strongly convex objectives In this section, following [21], we make the additional assumption of strong convexity of f, but not of all functions fn (see [20] for deﬁnitions and properties of such functions): (H3) The function f is strongly convex with respect to the norm ∥·∥, with convexity constant µ > 0. That is, for all θ1, θ2 ∈H, f(θ1) ⩾f(θ2) + ⟨f ′(θ2), θ1 −θ2⟩+ µ 2 ∥θ1 −θ2∥2. Note that (H3) simply needs to be satisﬁed for θ2 = θ∗being the unique global minimizer of f (such that f ′(θ∗) = 0). In the context of machine learning (least-squares or logistic regression), assumption (H3) is satisﬁed as soon as µ 2 ∥θ∥2 is used as an additional regularizer. For all strongly convex losses (e.g., least-squares), it is also satisﬁed as soon as the expectation E(xn ⊗xn) is invertible. Note that this implies that the problem is ﬁnite-dimensional, otherwise, the expectation is a compact covariance operator, and hence non-invertible (see, e.g., [22] for an introduction to covariance operators). For non-strongly convex losses such as the logistic loss, f can never be strongly convex unless we restrict the domain of θ (which we do in Section 3.2). Alternatively to restricting the domain, replacing the logistic loss u 7→log(1 + e−u) by u 7→log(1 + e−u) + εu2/2, for some small ε > 0, makes it strongly convex in low-dimensional settings. By strong convexity of f, if we assume (H3), then f attains its global minimum at a unique vector θ∗∈H such that f ′(θ∗) = 0. Moreover, we make the following assumption (in the context of stochastic approximation, it corresponds to E(∥εn∥2\|Fn−1) ⩽σ2): (H4) There exists σ2 ∈R+ such that for all n ⩾1, E(∥f ′ n(θ∗)∥2\|Fn−1) ⩽σ2, w.p.1. 3.1 Stochastic gradient descent Before stating our ﬁrst theorem (see proof in [23]), we introduce the following family of functions ϕβ : R+ \ {0} →R given by: ϕβ(t) = tβ−1 β if β ̸= 0, log t if β = 0. The function β 7→ϕβ(t) is continuous for all t > 0. Moreover, for β > 0, ϕβ(t) < tβ β , while for β < 0, we have ϕβ(t) < 1 −β (both with asymptotic equality when t is large). Theorem 1 (Stochastic gradient descent, strong convexity) Assume (H1,H2,H3,H4). Denote δn = E∥θn −θ∗∥2, where θn ∈H is the n-th iterate of the recursion in Eq. (1), with γn = Cn−α. We have, for α ∈[0, 1]: δn ⩽        2 exp 4L2C2ϕ1−2α(n) exp −µC 4 n1−α δ0 + σ2 L2 + 4Cσ2 µnα , if 0 ⩽α < 1, exp(2L2C2) nµC δ0 + σ2 L2 + 2σ2C2 ϕµC/2−1(n) nµC/2 , if α = 1. (5) Sketch of proof. Under our assumptions, it can be shown that (δn) satisﬁes the following recursion: δn ⩽(1 −2µγn + 2L2γ2 n)δn−1 + 2σ2γ2 n. (6) Note that it also appears in [3, Eq. (2)] under different assumptions. Using this deterministic recursion, we then derive bounds using classical techniques from stochastic approximation [2], but in a non-asymptotic way, by deriving explicit upper-bounds. Related work. To the best of our knowledge, this non-asymptotic bound, which depends explicitly upon the parameters of the problem, is novel (see [1, Theorem 1, Electronic companion paper] for a simpler bound with no such explicit dependence). It shows in particular that there is convergence in quadratic mean for any α ∈(0, 1]. Previous results from the stochastic approximation literature have focused mainly on almost sure convergence of the sequence of iterates. Almost-sure convergence requires that α > 1/2, with counter-examples for α < 1/2 (see, e.g., [2] and references therein). 3 Bound on function values. The bounds above imply a corresponding a bound on the functions values. Indeed, under assumption (H2), it may be shown that E[f(θn) −f(θ∗)] ⩽L 2 δn (see proof in [23]). Tightness for quadratic functions. Since the deterministic recursion in Eq. (6) is an equality for quadratic functions fn, the result in Eq. (5) is optimal (up to constants). Moreover, our results are consistent with the asymptotic results from [6]. Forgetting initial conditions. Bounds depend on the initial condition δ0 = E ∥θ0 −θ∗∥2 and the variance σ2 of the noise term. The initial condition is forgotten sub-exponentially fast for α ∈(0, 1), but not for α = 1. For α < 1, the asymptotic term in the bound is 4Cσ2 µnα . Behavior for α = 1. For α = 1, we have ϕµC/2−1(n) nµC/2 ⩽ 1 µC/2−1 1 n if Cµ > 2, ϕµC/2−1(n) nµC/2 = log n n if Cµ = 2 and ϕµC/2−1(n) nµC/2 ⩽ 1 1−µC/2 1 nµC/2 if Cµ > 2. Therefore, for α = 1, the choice of C is critical, as already noticed by [8]: too small C leads to convergence at arbitrarily small rate of the form n−µC/2, while too large C leads to explosion due to the initial condition. This behavior is conﬁrmed in simulations in Section 5. Setting C too large. There is a potentially catastrophic term when C is chosen too large, i.e., exp 4L2C2ϕ1−2α(n) , which leads to an increasing bound when n is small. Note that for α < 1, this catastrophic term is in front of a sub-exponentially decaying factor, so its effect is mitigated once the term in n1−α takes over ϕ1−2α(n), and the transient term stops increasing. Moreover, the asymptotic term is not involved in it (which is also observed in simulations in Section 5). Minimax rate. Note ﬁnally, that the asymptotic convergence rate in O(n−1) matches optimal asymptotic minimax rate for stochastic approximation [24, 25]. Note that there is no explicit dependence on dimension; this dependence is implicit in the deﬁnition of the constants µ and L. 3.2 Bounded gradients In some cases such as logistic regression, we also have a uniform upper-bound on the gradients, i.e., we assume (note that in Theorem 2, this assumption replaces both (H2) and (H4)). (H5) For each n ⩾1, almost surely, the function fn if convex, differentiable and has gradients uniformly bounded by B on the ball of center 0 and radius D, i.e., for all θ ∈H and all n > 0, ∥θ∥⩽D ⇒∥f ′ n(θ)∥⩽B. Note that no function may be strongly convex and Lipschitz-continuous (i.e., with uniformly bounded gradients) over the entire Hilbert space H. Moreover, if (H2’) is satisﬁed, then we may take D = ∥θ∗∥and B = LD. The next theorem shows that with a slight modiﬁcation of the recursion in Eq. (1), we get simpler bounds than the ones obtained in Theorem 1, obtaining a result which already appeared in a simpliﬁed form [8] (see proof in [23]): Theorem 2 (Stochastic gradient descent, strong convexity, bounded gradients) Assume (H1,H3,H5). Denote δn = E ∥θn −θ∗∥2 , where θn ∈H is the n-th iterate of the following recursion: ∀n ⩾1, θn = ΠD[θn−1 −γnf ′ n(θn−1)], (7) where ΠD is the orthogonal projection operator on the ball {θ : ∥θ∥⩽D}. Assume ∥θ∗∥⩽D. If γn = Cn−α, we have, for α ∈[0, 1]: δn ⩽    δ0 + B2C2ϕ1−2α(n) exp −µC 2 n1−α + 2B2C2 µnα , if α ∈[0, 1) ; δ0n−µC + 2B2C2n−µCϕµC−1(n), if α = 1 . (8) The proof follows the same lines than for Theorem 1, but with the deterministic recursion δn ⩽ (1−2µγn)δn−1 +B2γ2 n. Note that we obtain the same asymptotic terms than for Theorem 1 (but B replaces σ). Moreover, the bound is simpler (no explosive multiplicative factors), but it requires to know D in advance, while Theorem 1 does not. Note that because we have only assumed Lipschitzcontinuity, we obtain a bound on function values of order O(n−α/2), which is sub-optimal. For bounds directly on function values, see [26]. 4 3.3 Polyak-Ruppert averaging We now consider ¯θn = 1 n Pn−1 k=0 θk and, following [4, 5], we make extra assumptions regarding the smoothness of each fn and the fourth-order moment of the driving noise: (H6) For each n ⩾1, the function fn is almost surely twice differentiable with Lipschitz-continuous Hessian operator f ′′ n, with Lipschitz constant M. That is, for all θ1, θ2 ∈H and for all n ⩾1, ∥f ′′ n(θ1) −f ′′ n(θ2)∥⩽M∥θ1 −θ2∥, where ∥· ∥is the operator norm. Note that (H6) needs only to be satisﬁed for θ2 = θ∗. For least-square regression, we have M = 0, while for logistic regression, we have M = R3/4. (H7) There exists τ ∈R+, such that for each n ⩾1, E(∥f ′ n(θ∗)∥4\|Fn−1) ⩽τ 4 almost surely. Moreover, there exists a nonnegative self-adjoint operator Σ such that for all n, E(f ′ n(θ∗) ⊗ f ′ n(θ∗)\|Fn−1) ≼Σ almost-surely. The operator Σ (which always exists as soon as τ is ﬁnite) is here to characterize precisely the variance term, which will be independent of the learning rate sequence (γn), as we now show: Theorem 3 (Averaging, strong convexity) Assume (H1, H2’, H3, H4, H6, H7). Then, for ¯θn = 1 n Pn−1 k=0 θk and α ∈(0, 1), we have: E∥¯θn −θ∗∥21/2 ⩽ tr f ′′(θ∗)−1Σf ′′(θ∗)−11/2 √n + 6σ µC1/2 1 n1−α/2 + MCτ 2 2µ3/2 (1+(µC)1/2)ϕ1−α(n) n + 4LC1/2 µ ϕ1−α(n)1/2 n + 8A nµ1/2 1 C + L δ0 + σ2 L2 1/2 + 5MC1/2τ 2nµ A exp 24L4C4 δ0 + µE ∥θ0 −θ∗∥4 20Cτ2 + 2τ 2C3µ + 8τ 2C21/2 , (9) where A is a constant that depends only on µ, C, L and α. Sketch of proof. Following [4], we start from Eq. (1), write it as f ′ n(θn−1) = 1 γn (θn−1 −θn), and notice that (a) f ′ n(θn−1) ≈f ′ n(θ∗) + f ′′(θ∗)(θn−1 −θ∗), (b) f ′ n(θ∗) has zero mean and behaves like an i.i.d. sequence, and (c) 1 n Pn k=1 1 γk (θk−1 −θk) turns out to be negligible owing to a summation by parts and to the bound obtained in Theorem 1. This implies that ¯θn −θ∗behaves like −1 n Pn k=1 f ′′(θ∗)−1f ′ k(θ∗). Note that we obtain a bound on the root mean square error. Forgetting initial conditions. There is no sub-exponential forgetting of initial conditions, but rather a decay at rate O(n−2) (last two lines in Eq. (9)). This is a known problem which may slow down the convergence, a common practice being to start averaging after a certain number of iterations [2]. Moreover, the constant A may be large when LC is large, thus the catastrophic terms are more problematic than for stochastic gradient descent, because they do not appear in front of sub-exponentially decaying terms (see [23]). This suggests to take CL small. Asymptotically leading term. When M >0 and α>1/2, the asymptotic term for δn is independent of (γn) and of order O(n−1). Thus, averaging allows to get from the slow rate O(n−α) to the optimal rate O(n−1). The next two leading terms (in the ﬁrst line) have order O(nα−2) and O(n−2α), suggesting the setting α=2/3 to make them equal. When M =0 (quadratic functions), the leading term has rate O(n−1) for all α∈(0, 1) (with then a contribution of the ﬁrst term in the second line). Case α = 1. We get a simpler bound by directly averaging the bound in Theorem 1, which leads to an unchanged rate of n−1, i.e., averaging is not key for α = 1, and does not solve the robustness problem related to the choice of C or the lack of strong convexity. Leading term independent of (γn). The term in O(n−1) does not depend on γn. Moreover, as noticed in the stochastic approximation literature [4], in the context of learning from i.i.d. observations, this is exactly the Cramer-Rao bound (see, e.g., [27]), and thus the leading term is asymptotically optimal. Note that no explicit Hessian inversion has been performed to achieve this bound. Relationship with prior work on online learning. There is no clear way of adding a bounded gradient assumption in the general case α ∈(0, 1), because the proof relies on the recursion without projections, but for α = 1, the rate of O(n−1) (up to a logarithmic term) can be achieved in the more general framework of online learning, where averaging is key to deriving bounds for stochastic approximation from regret bounds. Moreover, bounds are obtained in high probability rather than simply in quadratic mean (see, e.g., [11, 12, 13, 14, 15]). 5 4 Non-strongly convex objectives In this section, we do not assume that the function f is strongly convex, but we replace (H3) by: (H8) The function f attains its global minimum at a certain θ∗∈H (which may not be unique). In the machine learning scenario, this essentially implies that the best predictor is in the function class we consider.1 In the following theorem, since θ∗is not unique, we only derive a bound on function values. Not assuming strong convexity is essential in practice to make sure that algorithms are robust and adaptive to the hardness of the learning or optimization problem (much like gradient descent is). 4.1 Stochastic gradient descent The following theorem is shown in a similar way to Theorem 1; we ﬁrst derive a deterministic recursion, which we analyze with novel tools compared to the non-stochastic case (see details in [23]), obtaining new convergence rates for non-averaged stochastic gradient descent : Theorem 4 (Stochastic gradient descent, no strong convexity) Assume (H1,H2’,H4,H8). Then, if γn = Cn−α, for α ∈[1/2, 1], we have: E [f(θn) −f(θ∗)] ⩽1 C δ0 + σ2 L2 exp 4L2C2ϕ1−2α(n) 1 + 4L3/2C3/2 min{ϕ1−α(n), ϕα/2(n)}. (10) When α = 1/2, the bound goes to zero only when LC < 1/4, at rates which can be arbitrarily slow. For α ∈(1/2, 2/3), we get convergence at rate O(n−α/2), while for α ∈(2/3, 1), we get a convergence rate of O(nα−1). For α = 1, the upper bound is of order O((log n)−1), which may be very slow (but still convergent). The rate of convergence changes at α = 2/3, where we get our best rate O(n−1/3), which does not match the minimax rate of O(n−1/2) for stochastic approximation in the non-strongly convex case [25]. These rates for stochastic gradient descent without strong convexity assumptions are new and we conjecture that they are asymptotically minimax optimal (for stochastic gradient descent, not for stochastic approximation). Nevertheless, the proof of this result falls out of the scope of this paper. If we further assume that we have all gradients bounded by B (that is, we assume D = ∞in (H5)), then, we have the following theorem, which allows α ∈(1/3, 1/2) with rate O(n−3α/2+1/2): Theorem 5 (Stochastic gradient descent, no strong convexity, bounded gradients) Assume (H1, H2’, H5, H8). Then, if γn = Cn−α, for α ∈[1/3, 1], we have: E [f(θn) −f(θ∗)] ⩽    δ0 + B2C2ϕ1−2α(n) 1+4L1/2C1/2 C min{ϕ1−α(n),ϕα/2(n)}, if α ∈[1/2, 1], 2 C (δ0 + B2C2)1/2 (1+4L1/2BC3/2) (1−2α)1/2ϕ3α/2−1/2(n), if α ∈[1/3, 1/2]. (11) 4.2 Polyak-Ruppert averaging Averaging in the context of non-strongly convex functions has been studied before, in particular in the optimization and machine learning literature, and the following theorems are similar in spirit to earlier work [7, 8, 13, 14, 15]: Theorem 6 (averaging, no strong convexity) Assume (H1,H2’,H4,H8). Then, if γn = Cn−α, for α ∈[1/2, 1], we have E f(¯θn) −f(θ∗) ⩽1 C δ0 + σ2 L2 exp 2L2C2ϕ1−2α(n) n1−α h 1+(2LC)1+ 1 α i + σ2C 2n ϕ1−α(n). (12) If α = 1/2, then we only have convergence under LC < 1/4 (as in Theorem 4), with potentially slow rate, while for α > 1/2, we have a rate of O(n−α), with otherwise similar behavior than for the strongly convex case with no bounded gradients. Here, averaging has allowed the rate to go from O(max{nα−1, n−α/2}) to O(n−α). 1For least-squares regression with kernels, where fn(θ) = 1 2(yn −⟨θ, Φ(xn)⟩)2, with Φ(xn) being the feature map associated with a reproducing kernel Hilbert space H with universal kernel [28], then we need that x 7→E(Y \|X = x) is a function within the RKHS. Taking care of situations where this is not true is clearly of importance but out of the scope of this paper. 6 0 1 2 3 4 5 −4 −3 −2 −1 0 1 log(n) log[f(θn)−f∗] power 2 sgd − 1/3 ave − 1/3 sgd − 1/2 ave − 1/2 sgd − 2/3 ave − 2/3 sgd − 1 ave − 1 0 1 2 3 4 5 −6 −4 −2 0 2 log(n) log[f(θn)−f∗] power 4 sgd − 1/3 ave − 1/3 sgd − 1/2 ave − 1/2 sgd − 2/3 ave − 2/3 sgd − 1 ave − 1 Figure 1: Robustness to lack of strong convexity for different learning rates and stochastic gradient (sgd) and Polyak-Ruppert averaging (ave). From left to right: f(θ) = \|θ\|2 and f(θ) = \|θ\|4, (between −1 and 1, afﬁne outside of [−1, 1], continuously differentiable). See text for details. 0 2 4 −5 0 5 log(n) log[f(θn)−f∗] α = 1/2 sgd − C=1/5 ave − C=1/5 sgd − C=1 ave − C=1 sgd − C=5 ave − C=5 0 2 4 −5 0 5 log(n) log[f(θn)−f∗] α = 1 sgd − C=1/5 ave − C=1/5 sgd − C=1 ave − C=1 sgd − C=5 ave − C=5 Figure 2: Robustness to wrong constants for γn = Cn−α. Left: α = 1/2, right: α = 1. See text for details. Best seen in color. Theorem 7 (averaging, no strong convexity, bounded gradients) Assume (H1,H5,H8). If γn = Cn−α, for α ∈[0, 1], we have E f(¯θn) −f(θ∗) ⩽nα−1 2C (δ0 + C2B2ϕ1−2α(n)) + B2 2n ϕ1−α(n). (13) With the bounded gradient assumption (and in fact without smoothness), we obtain the minimax asymptotic rate O(n−1/2) up to logarithmic terms [25] for α = 1/2, and, for α < 1/2, the rate O(n−α) while for α > 1/2, we get O(nα−1). Here, averaging has also allowed to increase the range of α which ensures convergence, to α ∈(0, 1). 5 Experiments Robustness to lack of strong convexity. Deﬁne f : R →R as \|θ\|q for \|θ\| ⩽1 and extended into a continuously differentiable function, afﬁne outside of [−1, 1]. For all q > 1, we have a convex function with Lipschitz-continuousgradient with constant L = q(q−1). It is strongly convex around the origin for q ∈(1, 2], but its second derivative vanishes for q > 2. In Figure 1, we plot in log-log scale the average of f(θn) −f(θ∗) over 100 replications of the stochastic approximation problem (with i.i.d. Gaussian noise of standard deviation 4 added to the gradient). For q = 2 (left plot), where we locally have a strongly convex case, all learning rates lead to good estimation with decay proportional to α (as shown in Theorem 1), while for the averaging case, all reach the exact same convergence rate (as shown in Theorem 3). However, for q = 4 where strong convexity does not hold (right plot), without averaging, α = 1 is still fastest but becomes the slowest after averaging; on the contrary, illustrating Section 4, slower decays (such as α = 1/2) leads to faster convergence when averaging is used. Note also the reduction in variability for the averaged iterations. Robustness to wrong constants. We consider the function on the real line f, deﬁned as f(θ) = 1 2\|θ\|2 and consider standard i.i.d. Gaussian noise on the gradients. In Figure 2, we plot the average performance over 100 replications, for various values of C and α. Note that for α = 1/2 (left plot), the 3 curves for stochastic gradient descent end up being aligned and equally spaced, corroborating a rate proportional to C (see Theorem 1). Moreover, when averaging for α = 1/2, the error ends up 7 0 1 2 3 4 5 −2.5 −2 −1.5 −1 −0.5 log(n) log[f(θn)−f∗] Selecting rate after n/10 iterations 1/3 − sgd 1/3 − ave 1/2 − sgd 1/2 − ave 2/3 − sgd 2/3 − ave 1 − sgd 1 − ave 0 1 2 3 4 −1.5 −1 −0.5 0 log(n) log[f(θn)−f∗] Selecting rate after n/10 iterations 1/3 − sgd 1/3 − ave 1/2 − sgd 1/2 − ave 2/3 − sgd 2/3 − ave 1 − sgd 1 − ave Figure 3: Comparison on non strongly convex logistic regression problems. Left: synthetic example, right: “alpha” dataset. See text for details. Best seen in color. being independent of C and α (see Theorem 3). Finally, when C is too large, there is an explosion (up to 105), hinting at the potential instability of having C too large. For α = 1 (right plot), if C is too small, convergence is very slow (and not at the rate n−1), as already observed (see, e.g., [8, 6]). Medium-scale experiments with linear logistic regression. We consider two situations where H = Rp: (a) the “alpha” dataset from the Pascal large scale learning challenge (http:// largescale.ml.tu-berlin.de/), for which p = 500 and n = 50000, and (b) a synthetic example where p = 100, n = 100000; we generate the input data i.i.d. from a multivariate Gaussian distribution with mean zero and a covariance matrix sampled from a Wishart distribution with p degrees of freedom (thus with potentially bad condition number), and the output is obtained through a classiﬁcation by a random hyperplane. For different values of α, we choose C in an adaptive way where we consider the lowest test error after n/10 iterations, and report results in Figure 3. In experiments reported in [23], we also consider C equal to 1/L suggested by our analysis to avoid large constants, for which the convergence speed is very slow, suggesting that our global bounds involving the Lipschitz constants may be locally far too pessimistic and that designing a truly adaptive sequence (γn) instead of a ﬁxed one is a fruitful avenue for future research. 6 Conclusion In this paper, we have provided a non-asymptotic analysis of stochastic gradient, as well as its averaged version, for various learning rate sequences of the form γn = Cn−α (see summary of results in Table 1). Following earlier work from the optimization, machine learning and stochastic approximation literatures, our analysis highlights that α = 1 is not robust to the choice of C and to the actual difﬁculty of the problem (strongly convex or not). However, when using averaging with α ∈(1/2, 1), we get, both in strongly convex and non-strongly convex situation, close to optimal rates of convergence. Moreover, we highlight the fact that problems with bounded gradients have better behaviors, i.e., logistic regression is easier to optimize than least-squares regression. Our work can be extended in several ways: ﬁrst, we have focused on results in quadratic mean and we expect that some of our results can be extended to results in high probability (in the line of [13, 3]). Second, we have focused on differentiable objectives, but the extension to objective functions with a differentiable stochastic part and a non-differentiable deterministic (in the line of [14]) would allow an extension to sparse methods. Acknowledgements. Francis Bach was partially supported by the European Research Council (SIERRA Project). We thank Mark Schmidt and Nicolas Le Roux for helpful discussions. SGD SGD SGD SGD Aver. Aver. Aver. α µ, L µ, B L L, B µ, L L B (0 , 1/3) α α × × 2α × α (1/3 , 1/2) α α × (3α −1)/2 2α × α (1/2 , 2/3) α α α/2 α/2 1 1 −α 1 −α (2/3 , 1) α α 1 −α 1 −α 1 1 −α 1 −α Table 1: Summary of results: For stochastic gradient descent (SGD) or Polyak-Ruppert averaging (Aver.), we provide their rates of convergence of the form n−β corresponding to learning rate sequences γn = Cn−α, where β is shown as a function of α. For each method, we list the main assumptions (µ: strong convexity, L: bounded Hessian, B: bounded gradients). 8 References [1] M. N. Broadie, D. M. Cicek, and A. Zeevi. General bounds and ﬁnite-time improvement for stochastic approximation algorithms. Technical report, Columbia University, 2009. [2] H. J. Kushner and G. G. Yin. Stochastic approximation and recursive algorithms and applications. Springer-Verlag, second edition, 2003. [3] O. Yu. Kul′chitski˘ı and A. `E. Mozgovo˘ı. An estimate for the rate of convergence of recurrent robust identiﬁcation algorithms. 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4,438	A Non-Parametric Approach to Dynamic Programming Oliver B. Kroemer1,2 Jan Peters1,2 1Intelligent Autonomous Systems, Technische Universität Darmstadt 2Robot Learning Lab, Max Planck Institute for Intelligent Systems {kroemer,peters}@ias.tu-darmstadt.de Abstract In this paper, we consider the problem of policy evaluation for continuousstate systems. We present a non-parametric approach to policy evaluation, which uses kernel density estimation to represent the system. The true form of the value function for this model can be determined, and can be computed using Galerkin’s method. Furthermore, we also present a uniﬁed view of several well-known policy evaluation methods. In particular, we show that the same Galerkin method can be used to derive Least-Squares Temporal Diﬀerence learning, Kernelized Temporal Diﬀerence learning, and a discrete-state Dynamic Programming solution, as well as our proposed method. In a numerical evaluation of these algorithms, the proposed approach performed better than the other methods. 1 Introduction Value functions are an essential concept for determining optimal policies in both optimal control [1] and reinforcement learning [2, 3]. Given the value function of a policy, an improved policy is straightforward to compute. The improved policy can subsequently be evaluated to obtain a new value function. This loop of computing value functions and determining better policies is known as policy iteration. However, the main bottleneck in policy iteration is the computation of the value function for a given policy. Using the Bellman equation, only two classes of systems have been solved exactly: tabular discrete state and action problems [4] as well as linear-quadratic regulation problems [5]. The exact computation of the value function remains an open problem for most systems with continuous state spaces [6]. This paper focuses on steps toward solving this problem. As an alternative to exact solutions, approximate policy evaluation methods have been developed in reinforcement learning. These approaches include Monte Carlo methods, temporal diﬀerence learning, and residual gradient methods. However, Monte Carlo methods are well-known to have an excessively high variance [7, 2], and tend to overﬁt the value function to the sampled data [2]. When using function approximations, temporal diﬀerence learning can result in a biased solution[8]. Residual gradient approaches are biased unless multiple samples are taken from the same states [9], which is often not possible for real continuous systems. In this paper, we propose a non-parametric method for continuous-state policy evaluation. The proposed method uses a kernel density estimate to represent the system in a ﬂexible manner. Model-based approaches are known to be more data eﬃcient than direct methods, and lead to better policies [10, 11]. We subsequently show that the true value function for this model has a Nadaraya-Watson kernel regression form [12, 13]. Using Galerkin’s projection method, we compute a closed-form solution for this regression problem. The 1 resulting method is called Non-Parametric Dynamic Programming (NPDP), and is a stable as well as consistent approach to policy evaluation. The second contribution of this paper is to provide a uniﬁed view of several sample-based algorithms for policy evaluation, including the NPDP algorithm. In Section 3, we show how Least-Squares Temporal Diﬀerence learning (LSTD) in [14], Kernelized Temporal Diﬀerence learning (KTD) in [15], and Discrete-State Dynamic Programming (DSDP) in [4, 16] can all be derived using the same Galerkin projection method used to derive NPDP. In Section 4, we compare these methods using empirical evaluations. In reinforcement learning, the uncontrolled system is usually represented by a Markov Decision Process (MDP). An MDP is deﬁned by the following components: a set of states S; a set of actions A; a transition distribution p(s′\|a, s), where s′ ∈S is the next state given action a ∈A in state s ∈S; a reward function r, such that r(s, a) is the immediate reward obtained for performing action a in state s; and a discount factor γ ∈[0, 1) on future rewards. Actions a are selected according to the stochastic policy π(a\|s). The goal is to maximize the discounted rewards that are obtained; i.e., max P∞ t=0γtr(st, at). The term system will refer jointly to the agent’s policy and the MDP. The value of a state V (s), for a speciﬁc policy π, is deﬁned as the expected discounted sum of rewards that an agent will receive after visiting state s and executing policy π; i.e., V (s) = E P∞ t=0γtr(st, at) s0 = s, π . (1) By using the Markov property, Eq. (1) can be rewritten as the Bellman equation V (s) = ´ A ´ Sπ (a\|s) p (s′\|s, a) [r (s, a) + γV π (s′)] ds′da. (2) The advantage of using the Bellman equation is that it describes the relationship between the value function at one state s and its immediate follow-up states s′ ∼p(s′\|s, a). In contrast, the direct computation of Eq. (1) relies on the rewards obtained from entire trajectories. 2 Non-Parametric Model-based Dynamic Programming We begin describing the NPDP approach by introducing the kernel density estimation framework used to represent the system. The true value function for this model has a kernel regression form, which can be computed by using Galerkin’s projection method. We subsequently discuss some of the properties of this algorithm, including its consistency. 2.1 Non-Parametric System Modeling The dynamics of a system are compactly represented by the joint distribution p(s, a, s′). Using Bayes rule and marginalization, one can compute the transition probabilities p(s′\|s, a) and the current policy π(a\|s) from this joint distribution; e.g. p(s′\|s, a) = p(s, a, s′)/ ´ p(s, a, s′)ds′. Rather than assuming that certain prior information is given, we will focus on the problem where only sampled information of the system is available. Hence, the system’s joint distribution is modeled from a set of n samples obtained from the real system. The ith sample includes the current state si ∈S, the selected action ai ∈A, and the follow-up state s′ i ∈S, as well as the immediate reward ri ∈R. The state space S and the action space A are assumed to be continuous. We propose using kernel density estimation to represent the joint distribution [17, 18] in a non-parametric manner. Unlike parametric models, non-parametric approaches use the collected data as features, which leads to accurate representations of arbitrary functions [19]. The system’s joint distribution is therefore modeled as p(s, a, s′) = n−1Pn i=1ψi (s′) ϕi (a) φi (s), where ψi (s′) = ψ (s′, s′ i), ϕi (a) = ϕ (a, ai), and φi (s) = φ (s, si) are symmetric kernel functions. In practice, the kernel functions ψ and φ will often be the same. To ensure a valid probability density, each kernel must integrate to one; i.e., ´ φi (s) ds = 1, ∀i, and similarly for ψ and ϕ. As an additional constraint, the kernel must always be positive; i.e., ψi (s′) ϕi (a) φi (s) ≥0, ∀s ∈S. This representation implies a factorization into separate ψi (s′), ϕi (a), and φi (s) kernels. As a result, an individual sample cannot express correlations between s′, a, and s. However, the representation does allow multiple samples to express correlations between these components in p(s, a, s′). 2 The reward function r(s, a) must also be represented. Given the kernel density estimate representation, the expected reward for a state-action pair is denoted as [12] r(s, a) = E[r\|s, a] = Pn k=1rkϕk (a) φk (s) Pn i=1ϕi (a) φi (s) . Having speciﬁed the model of the system dynamics and rewards, the next step is to derive the corresponding value function. 2.2 Resulting Solution In this section, we propose an approach to computing the value function for the continuous model speciﬁed in Section 2.1. Every policy has a unique value function, which fulﬁlls the Bellman equation, Eq. (2), for all states [2, 20]. Hence, the goal is to solve the Bellman equation for the entire state space, and not just at the sampled states. This goal can be achieved by using the Galerkin projection method to compute the value function for the model [21]. The Galerkin method involves ﬁrst projecting the integral equation into the space spanned by a set of basis functions. The integral equation is then solved in this projected space. To begin, the Bellman equation, Eq. (2), is rearranged as V (s) = ´ A ´ Sπ (a\|s) r (s, a) p (s′\|s, a) ds′da + ´ S ´ Ap (s′\|s, a) γV (s′) π (a\|s) dads′, p (s) V (s) = ˆ A p (a, s) r (s, a) da + γ ˆ S p (s′, s) V (s′) ds′. (3) Before applying the Galerkin method, we derive the exact form of the value function. Expanding the reward function and joint distributions, as deﬁned in Section 2.1, gives p (s) V (s) = n−1 ˆ A Pn k=1ϕk (a) φk (s) Pn i=1riϕi (a) φi (s) Pn j=1ϕj (a) φj (s) da + γ ˆ S p (s′, s) V (s′) ds′, p (s) V (s) = ˆ A n−1Pn i=1riϕi (a) φi (s) da + γ ˆ S n−1Pn i=1ψi (s′) φi (s) V (s′) ds′, p (s) V (s) = n−1Pn i=1riφi (s) + n−1Pn i=1γ ˆ S ψi (s′) φi (s) V (s′) ds′, Therefore, p(s)V (s) = n−1Pn i=1θiφi (s), where θ are value weights. Given that p(s) = n−1Pn j=1φj (s), the true value function of the kernel density estimate system has a Nadaraya-Watson kernel regression [12, 13] form V (s) = Pn i=1θiφi (s) Pn j=1φj (s) . (4) Having computed the true form of the value function, the Galerkin projection method can be used to compute the value weights θ. The projection is performed by taking the expectation of the integral equation with respect to each of the n basis function φi. The resulting n simultaneous equations can be written as the vector equation ˆ S φ (s) p(s)V (s)ds = ˆ S φ (s) n−1φ (s)T rds+γ ˆ S ˆ S φ (s) n−1 φ (s)T ψ (s′) V (s′)ds′ds, where the ith elements of the vectors are given by [r]i = ri, [φ (s)]i = φi (s), and [ψ (s′)]i = ψi (s′). Expanding the value functions gives ˆ S φ (s) φ (s)T θds = ˆ S φ (s) φ (s)T rds + γ ˆ S ˆ S φ (s) φ (s)T ψ (s′) φ (s′)T θ Pn i=1φi (s′)ds′ds, Cθ = Cr + γCλθ, where C = ´ S φ (s) φ (s)T ds, and λ = ´ S(Pn i=1φi (s′))−1ψ (s′) φ (s′)T ds′ is a stochastic matrix; i.e., a transition matrix. The matrix C can become singular if two basis functions 3 Algorithm 1 Non-Parametric Dynamic Programming Input: Computation: n system samples: Reward vector: state si, next state s′ i, and reward ri [r]i = ri Kernel functions: Transition matrix: φi (sj) = φ (si, sj), and ψi s′ j = ψ s′ i, s′ j [λ]i,j = ´ S φj(s′)ψi(s′) Pn k=1φk(s′)ds′ Discount factor: Value weights: 0 ≤γ < 1 θ = (I −γλ)−1r Output: Value function: V (s) = Pn i=1θiφi(s) Pn j=1φj(s) are coincident. In such cases, there exists an inﬁnite set of solutions for θ. However, all of the solutions result in identical values. The NPDP algorithm uses the solution given by θ = (I −γλ)−1r, (5) which always exists for any stochastic matrix λ. Thus, the derivation has shown that the exact value function for the model in Section 2.1 has a Nadaraya-Watson kernel regression form, as shown in Eq. (4), with weights θ given by Eq. (5). The non-parametric dynamic programming algorithm is summarized in Alg. 1. The NPDP algorithm ultimately requires only the state information s and s′, and not the actions a. In Section 3, we will show how this form of derivation can also be used to derive the LSTD, KTD, and DSDP algorithms. 2.3 Properties of the NPDP Algorithm In this section, we discuss some of the key properties of the proposed NPDP algorithm, including precision, accuracy, and computational complexity. Precision refers to how close the predicted value function is to the true value function of the model, while accuracy refers to how close the model is to the true system. One of the key contributions of this paper is providing the true form of the value function for policy evaluation with the non-parametric model described in Section 2.1. The parameters of this value function can be computed precisely by solving Eq. (5). Even if λ is evaluated numerically, a high level of precision can still be obtained. As a non-parametric method, the accuracy of the NPDP algorithm depends on the number of samples obtained from the system. It is important that the model, and thus the value function, converges to that of the true system as the number of samples increases; i.e., that the model is statistically consistent. In fact, kernel density estimation can be proven to have almost sure convergence to the true distribution for a wide range of kernels [22]. Given that λ is a stochastic matrix and 0 ≤γ < 1, it is well-known that the inversion of (I −γλ) is well-deﬁned [16]. The inversion can therefore also be expanded according to the Neumann series; i.e., θ = P∞ i=0[γλ]ir. Similar to other kernel-based policy evaluation methods [23, 24], NPDP has a computational complexity of O(n3) when performed naively. However, by taking advantage of sparse matrix computations, this complexity can be reduced to O(nz), where z is the number of non-zero elements in (I −γλ). 3 Relation to Existing Methods The second contribution of this paper is to provide a uniﬁed view of Least Squares Temporal Diﬀerence learning (LSTD), Kernelized Temporal Diﬀerence learning (KTD), Discrete-State Dynamic Programming (DSDP), and the proposed Non-Parametric Dynamic Programming (NPDP). In this section, we utilize the Galerkin methodology from Section 2.2 to re-derive the LSTD, KTD, and DSDP algorithms, and discuss how these methods compare to NPDP. A numerical comparison is given in Section 4. 4 3.1 Least Squares Temporal Diﬀerence Learning The LSTD algorithm allows the value function V (s) to be represented by a set of m arbitrary basis functions ˆφi(s), see [14]. Hence, V (s) = Pm i=1ˆθi ˆφi (s) = ˆφ (s)T ˆθ, where ˆθ is a vector of coeﬃcients learned during policy evaluation, and [ˆφ (s)]i = ˆφi (s). In order to re-derive the LSTD policy evaluation, the joint distribution is represented as a set of delta functions p (s, a, s′) = n−1 Pn i=1 δi(s, a, s′), where δi(s, a, s′) is a Dirac delta function centered on (si, ai, s′ i). Using Galerkin’s method, the integral equation is projected into the space of the basis functions ˆφ (s). Thus, Eq. (3) becomes ˆ S ˆφ (s) p (s) ˆφ (s)T ˆθds = ˆ A ˆ S ˆφ (s) p (s, a) r (s, a) dsda+γ ˆ S ˆφ (s) p (s, s′) ˆφ (s′)T ˆθds′ds, n X i=1 ˆφ (si) ˆφ (si)T ˆθ = n X j=1 r (sj, aj) ˆφ (sj) + γ n X k=1 ˆφ (sk) ˆφ (s′ k)T ˆθ, n X i=1 ˆφ (si) ˆφ (si)T −γ ˆφ (s′ i)T ˆθ = n X j=1 r (sj, aj) ˆφ (sj) , and thus Aˆθ = b, where A = Pn i=1 ˆφ (si) (ˆφ (si)T −γ ˆφ (s′ i)T ) and b = Pn j=1 r (sj, aj) ˆφ (sj). The ﬁnal weights are therefore given by ˆθ = A−1b. This equation is also solved by LSTD, including the incremental updates of A and b as new samples are acquired [14]. Therefore, LSTD can be seen as computing the transitions between the basis functions using a Monte Carlo approach. However, Monte Carlo methods rely on large numbers of samples to obtain accurate results. A key disadvantage of the LSTD method is the need to select a speciﬁc set of basis functions. The computed value function will always be a projection of the true value function into the space of these basis functions [8]. If the true value function does not lie within the space of these basis functions, the resulting approximation may be arbitrarily inaccurate, regardless of the number of acquired samples. However, using predeﬁned basis functions only requires inverting an m × m matrix, which results in a lower computational complexity than NPDP. The LSTD may also need to be regularized, as the inversion of A becomes ill-posed if the basis functions are too densely spaced. Regularization has a similar eﬀect to changing the transition probabilities of the system [25]. 3.2 Kernelized Temporal Diﬀerence Learning Methods The proposed approach is of course not the ﬁrst to use kernels for policy evaluation. Methods such as kernelized least-squares temporal diﬀerence learning [24] and Gaussian process temporal diﬀerence learning [23] have also employed kernels in policy evaluation. Taylor and Parr demonstrated that these methods diﬀer mainly in their use of regularization [15]. The uniﬁed view of these methods is referred to as Kernelized Temporal Diﬀerence learning. The KTD approach assumes that the reward and value functions can be represented by kernelized linear least-squares regression; i.e., r(s) = k(s)T K−1r and V (s) = k(s)T ˆθ, where [k(s)]i = k(s, si), [K]ij = k(si, sj), [r]i = ri, and ˆθ is a weight vector. In order to derive KTD using Galerkin’s method, it is necessary to again represent the joint distribution as p (s, a, s′) = n−1 Pn i=1 δi(s, a, s′). The Galerkin method projects the integral equation into the space of the Kronecker delta functions [ˇδ(s)]i = ˇδi(s, ai, s′ i), where ˇδi(s, a, s′) = 1 if s′ = s′ i, a = ai, and s = si; otherwise ˇδi(s, a, s′) = 0. Thus, Eq. (3) becomes ˆ S ˇδ (s) p (s) k(s)T ˆθds = ˆ S ˇδ (s) p (s) r (s) ds + γ ˆ S ˇδ (s) p (s, s′) k(s′)T ˆθds′ds, 5 By substituting p(s, a, s′) and applying the sifting property of delta functions, this equation becomes n X i=1 ˇδ(si)k(si)T ˆθ = n X j=1 ˇδ(sj)k(sj)T K−1r + γ n X k=1 ˇδ(sk)k(s′ k)T ˆθ, and thus K ˆθ = r+γK′ ˆθ, where [K′]ij = k(s′ i, sj). The value function weights are therefore ˆθ = (K −γK′)−1r, which is identical to the solution found by the KTD approach [15]. In this manner, the KTD approach computes a weighting ˆθ such that the diﬀerence in the value at si and the discounted value at s′ i equals the observed empirical reward ri. Thus, only the ﬁnite set of sampled states are regarded for policy evaluation. Therefore, some KTD methods, e.g. Gaussian process temporal diﬀerence learning [23], require that the samples are obtained from a single trajectory to ensure that s′ i = si+1. A key diﬀerence between KTD and NPDP is the representation of the value function V (s). The form of the value function is a direct result of the representation used to embody the state transitions. In the original paper [15], the KTD algorithm represents the transitions by using linear kernelized regression ˆk(s′) = k(s)T K−1K′, where [ˆk(s′)]i = E[k(s′, si)]. The value function V (s) = k(s)T ˆθ is the correct form for this transition model. However, the transition model does not explicitly represent a conditional distribution and can lead to inaccurate predictions. For example, consider two samples that start at s1 = 0 and s2 = 0.75 respectively, and both transition to s′ = 0.75. For clarity, we use a box-cart kernel with a width of one k(si, sj) = 1 iﬀ∥si −sj∥≤0.5 and 0 otherwise. Hence, K = I and each row of K’ corresponds to (0, 1). In the region 0.25 ≤s ≤0.5, where the two kernels overlap, the transition model would then predict ˆk(s) = k(s)T K−1K′ = [ 0 2 ]. This prediction is however impossible as it requires that E[k(s′, s2)] > maxs k(s, s2). In comparison, NPDP would predict the distribution ψ(s′) ≡ψ1(s′) ≡ψ2(s′) for all states in the range −0.5 ≤s ≤1.25. Similar as for LSTD, the matrix (K −γK′) may become singular and thus not be invertible. As a result, KTD usually needs to be regularized [15]. Given that KTD requires inverting an n × n matrix, this approach has a computational complexity similar to NPDP. 3.3 Discrete-State Dynamic Programming The standard tabular DSDP approach can also be derived using the Galerkin method. Given a system with q discrete states, the value function has the form V (s) = ˇδ(s)T v, where ˇδ(s) is a vector of q Kronecker delta functions centered on the discrete states. The corresponding reward function is r(s) = ˇδ(s)T ¯r. The joint distribution is given by p(s′, s) = q−1δ(s)T P δ(s′), where P is a stochastic matrix Pq j=1[P ]ij = 1, ∀i and hence p(s) = q−1 Pq i=1 δi(s). Galerkin’s method projects the integral equation into the space of the states ˇδ(s). Thus, Eq. (3) becomes ˆ S ˇδ (s) p (s) ˇδ(s)T vds = ˆ S ˇδ (s) p (s) ˇδ(s)T ¯rds + γ ˆ S ˇδ (s) p (s, s′) ˇδ(s′)T vds′ds, Iv = I¯r + γ ˆ S ˇδ (s) δ(s)T P δ(s′)ˇδ(s′)T vds′ds, v = ¯r + γP v, v = (I −γP )−1¯r, (6) which is the same computation used by DSDP [16]. The DSDP and NPDP methods actually use similar models to represent the system. While NPDP uses a kernel density estimation, the DSDP algorithm uses a histogram representation. Hence, DSDP can be regarded as a special case of NPDP for discrete state systems. The DSDP algorithm has also been the basis for continuous-state policy evaluation algorithms [26, 27]. These algorithms ﬁrst use the sampled states as the discrete states of an MDP and compute the corresponding values. The computed values are then generalized, under a smoothness assumption, to the rest of the state-space using local averaging. Unlike these methods, NPDP explicitly performs policy evaluation for a continuous set of states. 6 4 Numerical Evaluation In this section, we compare the diﬀerent policy evaluation methods discussed in the previous section, with the proposed NPDP method, on an illustrative benchmark system. 4.1 Benchmark Problem and Setup In order to compare the LSTD, KTD, DSDP, and NPDP approaches, we evaluated the methods on a discrete-time continuous-state system. A standard linear-Gaussian system was used for the benchmark problem, with transitions given by s′ = 0.95s + ω where ω is Gaussian noise N(µ = 0, σ = 0.025). The initial states are restricted to the range 0.95 to 1. The reward functions consist of three Gaussians, as shown by the black line in Fig. 1. The KTD method was implemented using a Gaussian kernel function and regularization. The LSTD algorithm was implemented using 15 uniformly-spaced normalized Gaussian basis functions, and did not require regularization. The DSDP method was implemented by discretizing the state-space into 10 equally wide regions. The NPDP method was also implemented using Gaussian kernels. The hyper-parameters of all four methods, including the number of basis functions for LSTD and DSDP, were carefully tuned to achieve the best performance. As a performance base-line, the values of the system in the range 0 < s < 1 were computed using a Monte Carlo estimate based on 50000 trajectories. The policy evaluations performed by the tested methods were always based on only 500 samples; i.e. 100 times less samples than the baseline. The experiment was run 500 times using independent sets of 500 samples. The samples were not drawn from the same trajectory. 4.2 Results The performance of the diﬀerent methods were compared using three performance measures. Two of the performance measures are based on the weighted Mean Squared Error (MSE) [2] E(V ) = ´ 1 0 W(s) (V (s) −V ⋆(s))2 ds where V ⋆is the true value function and W(s) ≥0, for all states, is a weighting distribution ´ 1 0 W(s)ds = 1. The ﬁrst performance measure Eunif corresponds to the MSE where W(s) = 1 for all states in the range zero to one. The second performance measure Esamp corresponds to the MSE where W(s) = n−1Σn i=1δi(s) respectively. Thus, Esamp is an indicator of the accuracy in the space of the samples, while Eunif is an indicator of how well the computed value function generalizes to the entire state space. The third performance measure Emax is given by the maximum error in the value function. This performance measure is the basis of a bound on the overall value function approximation [20]. The results of the experiment are shown in Table 1. The performance measures were averaged over the 500 independent trials of the experiment. For all three performance measures, the NPDP algorithm achieved the highest levels of performance, while the DSDP approach consistently led to the worst performance. Eunif Esamp Emax NPDP 0.5811 ± 0.0333 0.7185 ± 0.0321 1.4971 ± 0.0309 LSTD 0.6898 ± 0.0443 0.8932 ± 0.0412 1.5591 ± 0.0382 KTD 0.7585 ± 0.0460 0.8681 ± 0.0270 2.5329 ± 0.0391 DSDP 1.6979 ± 0.0332 2.1548 ± 0.1082 2.9985 ± 0.0449 Table 1: Each row corresponds to one of the four tested algorithms for policy evaluation. The columns indicate the performance of the approaches during the experiment. The performance indexes include the mean squared error evaluated uniformly over the zero to one range, the mean squared error evaluated at the 500 sampled points, and the maximum error. The results are averaged over 500 trials. The standard errors of the means are also given. 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 State Value True Value Reward LSTD KTD 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 State Value True Value Reward DSDP NPDP Figure 1: Value functions obtained by the evaluated methods. The black lines show the reward function. The blue lines show the value function computed from the trajectories of 50,000 uniformly sampled points. The LSTD, KTD, DSDP, and NPDP methods evaluated the policy using only 500 points. The presentation was divided into two plots for improved clarity 4.3 Discussion The LSTD algorithm achieved a relatively low Eunif value, which indicates that the tuned basis functions could accurately represent the true value function. However, the performance of LSTD is sensitive to the choice of basis functions and the number of samples per basis function. Using 20 basis functions instead of 15 reduces the performance of LSTD to Eunif = 2.8705 and Esamp = 1.0256 as a result of overﬁtting. The KTD method achieved the second best performance for Esamp, as a result of using a non-parametric representation. However, the value tended to drop in sparsely-sampled regions, which lead to relatively high Eunif and Emax values. The discretization of states for DSDP is generally a disadvantage when modeling continuous systems, and resulted in poor overall performance for this evaluation. The NPDP approach out-performed the other methods in all three performance measures. The performance of NPDP could be further improved by using adaptive kernel density estimation [28] to locally adapt the kernels’ bandwidths according to the sampling density. However, all methods were restricted to using a single global bandwidth for the purpose of this comparison. 5 Conclusion This paper presents two key contributions to continuous-state policy evaluation. The ﬁrst contribution is the Non-Parametric Dynamic Programming algorithm for policy evaluation. The proposed method uses a kernel density estimate to generate a consistent representation of the system. It was shown that the true form of the value function for this model is given by a Nadaraya-Watson kernel regression. The NPDP algorithm provides a solution for calculating the value function. As a kernel-based approach, NPDP simultaneously addresses the problems of function approximation and policy evaluation. The second contribution of this paper is providing a uniﬁed view of Least-Squares Temporal Diﬀerence learning, Kernelized Temporal Diﬀerence learning, and discrete-state Dynamic Programming, as well as NPDP. All four approaches can be derived from the Bellman equation using the Galerkin projection method. These four approaches were also evaluated and compared on an empirical problem with a continuous state space and non-linear reward function, wherein the NPDP algorithm out-performed the other methods. Acknowledgements The project receives funding from the European Community’s Seventh Framework Programme under grant agreement n° ICT- 248273 GeRT and n° 270327 Complacs. 8 References [1] Dimitri P. Bertsekas. Dynamic Programming and Optimal Control, Vol. II. Athena Scientiﬁc, 2007. [2] R. S. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. 1998. [3] H. Maei, C. Szepesvari, S. Bhatnagar, D. Precup, D. Silver, and R. Sutton. Convergent temporal-diﬀerence learning with arbitrary smooth function approximation. In NIPS, pages 1204–1212, 2009. [4] Richard Bellman. Bottleneck problems and dynamic programming. Proceedings of the National Academy of Sciences of the United States of America, 39(9):947–951, 1953. [5] R.E. Kalman. Contributions to the theory of optimal control, 1960. 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4,439	Extracting Speaker-Speciﬁc Information with a Regularized Siamese Deep Network Ke Chen and Ahmad Salman School of Computer Science, The University of Manchester Manchester M13 9PL, United Kingdom {chen,salmana}@cs.manchester.ac.uk Abstract Speech conveys different yet mixed information ranging from linguistic to speaker-speciﬁc components, and each of them should be exclusively used in a speciﬁc task. However, it is extremely difﬁcult to extract a speciﬁc information component given the fact that nearly all existing acoustic representations carry all types of speech information. Thus, the use of the same representation in both speech and speaker recognition hinders a system from producing better performance due to interference of irrelevant information. In this paper, we present a deep neural architecture to extract speaker-speciﬁc information from MFCCs. As a result, a multi-objective loss function is proposed for learning speaker-speciﬁc characteristics and regularization via normalizing interference of non-speaker related information and avoiding information loss. With LDC benchmark corpora and a Chinese speech corpus, we demonstrate that a resultant speaker-speciﬁc representation is insensitive to text/languages spoken and environmental mismatches and hence outperforms MFCCs and other state-of-the-art techniques in speaker recognition. We discuss relevant issues and relate our approach to previous work. 1 Introduction It is well known that speech conveys various yet mixed information where there are linguistic information, a major component, and non-verbal information such as speaker-speciﬁc and emotional components [1]. For human communication, all the information components in speech turn out to be very useful and exclusively used for different tasks. For example, one often recognizes a speaker regardless of what is spoken for speaker recognition, while it is effortless for him/her to understand what is exactly spoken by different speakers for speech recognition. In general, however, there is no effective way to automatically extract an information component of interest from speech signals so that the same representation has to be used in different speech information tasks. The interference of different yet entangled speech information components in most existing acoustic representations hinders a speech or speaker recognition system from achieving better performance [1]. For speaker-speciﬁc information extraction, two main efforts have been made so far; one is the use of data component analysis [2], e.g., PCA or ICA, and the other is the use of adaptive ﬁltering techniques [3]. However, the aforementioned techniques either fail to associate extracted data components with speaker-speciﬁc information as such information is non-predominant over speech or obtain features overﬁtting to a speciﬁc corpus since it is unlikely that speaker-speciﬁc information is statically resided in ﬁxed frequency bands. Hence, the problem is still unsolved in general [4]. Recent studies suggested that learning deep architectures (DAs) provides a new way for tackling complex AI problems [5]. In particular, representations learned by DAs greatly facilitate various recognition tasks and constantly lead to the improved performance in machine perception [6]-[9]. On the other hand, the Siamese architecture originally proposed in [10] uses supervised yet contrastive 1 1t x 2t x ˆ1t x ˆ2t x CS CS ! ! ! , ; 1 2 D CS X CS X " Figure 1: Regularized Siamese deep network (RSDN) architecture. learning to explore intrinsic similarity/disimilarity underlying an unknown data space. Incorporated by DAs, the Siamese architecture has been successfully applied to face recognition [11] and dimensionality reduction [12]. Inspired by the aforementioned work, we present a regularized Siamese deep network (RSDN) to extract speaker-speciﬁc information from a spectral representation, Mel Frequency Cepstral Coefﬁcients (MFCCs), commonly used in both speech and speaker recognition. A multi-objective loss function is proposed for learning speaker-speciﬁc characteristics, normalizing interference of non-speaker related information and avoiding information loss. Our RSDN learning adopts the famous two-phase deep learning strategy [5],[13]; i.e., greedy layer-wise unsupervised learning for initializing its component deep neural networks followed by global supervised learning based on the proposed loss function. With LDC benchmark corpora [14] and a Chinese corpus [15], we demonstrate that a generic speaker-speciﬁc representation learned by our RSDN is insensitive to text and languages spoken and, moreover, applicable to speech corpora unseen during learning. Experimental results in speaker recognition suggest that a representation learned by the RSDN outperforms MFCCs and that by the CDBN [9] that learns a generic speech representation without speaker-speciﬁc information extraction. To our best knowledge, the work presented in this paper is the ﬁrst attempt on speaker-speciﬁc information extraction with deep learning. In the reminder of this paper, Sect. 2 describes our RSDN architecture and proposes a loss function. Sect. 3 presents a two-phase learning algorithm to train the RSDN. Sect. 4 reports our experimental methodology and results. The last section discusses relevant issues and relates our approach to previous work in deep learning. 2 Model Description In this section, we ﬁrst describe our RSDN architecture and then propose a multi-objective loss function used to train the RSDN for learning speaker-speciﬁc characteristics. 2.1 Architecture As illustrated in Figure 1, our RSDN architecture consists of two subnets, and each subnet is a fully connected multi-layered perceptron of 2K+1 layers, i.e., an input layer, 2K-1 hidden layers and a visible layer at the top. If we stipulate that layer 0 is input layer, there are the same number of neurons in layers k and 2K-k for k = 0, 1, · · · , K. In particular, the Kth hidden layer is used as code layer, and neurons in this layer are further divided into two subsets. As depicted in Figure 1, those neurons in the box named CS and colored in red constitute one subset for encoding speakerspeciﬁc information and all remaining neurons in the code layer form the other subset expected to 2 accommodate non-speaker related information. The input to each subnet is an MFCC representation of a frame after a short-term analysis that a speech segment is divided into a number of frames and the MFCC representation is achieved for each frame. As depicted in Figure 1, xit is the MFCC feature vector of frame t in Xi, input to subnet i (i=1,2), where Xi = {xit}TB t=1 collectively denotes MFCC feature vectors for a speech segment of TB frames. During learning, two identical subsets are coupled at their coding layers via neurons in CS with an incompatibility measure deﬁned on two speech segments of equal length, X1 and X2, input to two subnets, which will be presented in 2.2. After learning, we achieve two identical subnets and hence can use either of them to produce a new representation for a speech frame. For input x to a subnet, only the bottom K layers of the subnet are used and the output of neurons in CS at the code layer or layer K, denoted by CS(x), is its new representation, as illustrated by the dash box in Figure 1. 2.2 Loss Function Let CS(xit) be the output of all neurons in CS of subnet i (i=1,2) for input xit ∈Xi and CS(Xi) = {CS(xit)}TB t=1, which pools output of neurons in CS for TB frames in Xi, as illustrated in Figure 1. As statistics of speech signals is more likely to capture speaker-speciﬁc information [5], we deﬁne the incompatibility measure based on the 1st- and 2nd-order statistics of a new representation to be learned as D[CS(X1), CS(X2); Θ] = \|\|µ(1) −µ(2)\|\|2 2 + \|\|Σ(1) −Σ(2)\|\|2 F , (1) where µ(i) = 1 TB TB X t=1 CS(xit), Σ(i) = 1 TB −1 TB X t=1 [CS(xit) −µ(i)][CS(xit) −µ(i)]T , i = 1, 2. In Eq. (1), \|\| · \|\|2 and \|\| · \|\|F are the L2 norm and the Frobenius norm, respectively. Θ is a collective notation of all connection weights and biases in the RSDN. Intuitively, two speech segments belonging to different speakers lead to different statistics and hence their incompatibility score measured by (1) should be large after learning. Otherwise their score is expected to be small. For a corpus of multiple speakers, we can construct a training set so that an example be in the form: (X1, X2; I) where I is the label deﬁned as I = 1 if two speech segments, X1 and X2, are spoken by the same speaker or I = 0 otherwise. Using such training examples, we apply the energy-based model principle [16] to deﬁne a loss function as L(X1, X2; Θ) = α[LR(X1; Θ) + LR(X2; Θ)] + (1 −α)LD(X1, X2; Θ), (2) where LR(Xi; Θ) = 1 TB TB X t=1 \|\|xit −ˆxit\|\|2 2 (i=1, 2), LD(X1, X2; Θ) = ID + (1 −I)(e−Dm λm + e−DS λS ). Here Dm = \|\|µ(1) −µ(2)\|\|2 2 and DS = \|\|Σ(1) −Σ(2)\|\|2 F . λm and λS are the tolerance bounds of incompatibility scores in terms of Dm and DS, which can be estimated from a training set. In LD(X1, X2; Θ), we drop explicit parameters of D[CS(X1), CS(X2); Θ] to simplify presentation. Eq. (2) deﬁnes a multi-objective loss function where α (0 < α < 1) is a parameter used to tradeoff between two objectives LR(Xi; Θ) and LD(X1, X2; Θ). The motivation for two objectives are as follows. By nature, both speaker-speciﬁc and non-speaker related information components are entangled over speech [1],[5]. When we tend to extract speaker-speciﬁc information, the interference of non-speaker related information is inevitable and appears in various forms. LD(X1, X2; Θ) measures errors responsible for wrong speaker-speciﬁc statistics on a representation learned by a Siamese DA in different situations. However, using LD(X1, X2; Θ) only to train a Siamese DA cannot cope with enormous variations of non-speaker related information, in particular, linguistic information (a predominant information component in speech), which often leads to overﬁtting to a training corpus according to our observations. As a result, we use LR(Xi; Θ) to measure reconstruction errors to monitor information loss during speaker-speciﬁc information extraction. By minimizing reconstruction errors in two subnets, the code layer leads to a speaker-speciﬁc representation with the output of neurons in CS while the remaining neurons are used to regularize various interference by capturing some invariant properties underlying them for good generalization. In summary, we anticipate that minimizing the multi-objective loss function deﬁned in Eq. (2) will enable our RSDN to extract speaker-speciﬁc information by encoding it through a generic speakerspeciﬁc representation. 3 3 Learning Algorithm In this section, we apply the two-phase deep learning strategy [5],[13] to derive our learning algorithm, i.e., pre-training for initializing subnets and discriminative learning for learning a speakerspeciﬁc representation. We ﬁrst present the notation system used in our algorithm. Let hkj(xit) denote the output of the jth neuron in layer k for k=0,1,· · · ,K,· · · ,2K. hk(xit) = ¡ hkj(xit) ¢\|hk\| j=1 is a collective notation of the output of all neurons in layer k of subnet i (i=1,2) where \|hk\| is the number of neurons in layer k. By this notation, k=0 refers to the input layer with h0(xit) = xit, and k=2K refers to the top layer producing the reconstruction ˆxit. In the coding layer, i.e., layer K, CS(xit) = ¡ hKj(xit) ¢\|CS\| j=1 is a simpliﬁed notation for output of neurons in CS. Let W(i) k and b(i) k denote the connection weight matrix between layers k-1 and k and the bias vector of layer k in subnet i (i=1,2), respectively, for k=1,· · · ,2K. Then output of layer k is hk(xit) = σ[uk(xit)] for k=1,· · · ,2K-1, where uk(xit) = W (i) k hk−1(xit) + b(i) k and σ(z) = ¡ (1 + e−zj)−1¢\|z\| j=1. Note that we use the linear transfer function in the top layer, i.e., layer 2K, to reconstruct the original input. 3.1 Pre-training For pre-training, we employ the denoising autoencoder [17] as a building block to initialize biases and connection weight matrices of a subnet. A denoising autoencoder is a three-layered perceptron where the input, ˜x, is a distorted version of the target output, x. For a training example, (˜x, x), the output of the autoencoder is a restored version, ˆx. Since MFCCs fed to the ﬁrst hidden layer and its intermediate representation input to all other hidden layers are of continuous value, we always distort input, x, by adding Gaussian noise to form a distorted version, ˜x. The restoration learning is done by minimizing the MSE loss between x and ˆx with respect to the weight matrix and biases. We apply the stochastic back-propagation (SBP) algorithm to train denoising autoencoders, and the greedy layer-wise learning procedure [5],[13] leads to initial weight matrices for the ﬁrst K hidden layers, as depicted in a dash box in Figure 1, i.e., W1, · · · , WK of a subnet. Then, we set WK+k = W T K−k+1 for k=1,· · · ,K to initialize WK+1, · · · , W2K of the subnet. Finally, the second subnet is created by simply duplicating the pre-trained one. 3.2 Discriminative Learning For discriminative learning, we minimizing the loss function in Eq. (2) based on pre-trained subnets for speaker-speciﬁc information extraction. Given our loss function is deﬁned on statistics of TB frames in a speech segment, we cannot update parameters until we have TB output of neurons in CS at the code layer. Fortunately, the SBP algorithm perfectly meets our requirement; In the SBP algorithm, we always set the batch size to the number of frames in a speech segment. To simplify the presentation, we shall drop explicit parameters in our derivation if doing so causes no ambiguities. In terms of the reconstruction loss, LR(Xi; Θ), we have the following gradients. For layer k = 2K, ∂LR ∂u2K(xit) = 2(ˆxit −xit), i=1, 2. (3) For all hidden layers, k=2K-1,· · · ,1, applying the chain rule and (3) leads to ∂LR ∂uk(xit) = µ ∂LR ∂hkj(xit)hkj(xit)[1−hkj(xit)] ¶\|hk\| j=1 , ∂LR ∂hk(xit) = £ W (i) k+1 ¤T ∂LR ∂uk+1(xit). (4) As the contrastive loss, LD(X1, X2; Θ), deﬁned on neurons in CS at code layers of two subnets, its gradients are determined only by parameters related to K hidden layers in two subnets, as depicted by dash boxes in Figure 1. For layer k=K and subnet i=1, 2, after a derivation (see the appendix for details), we obtain ∂LD ∂uK(xit) = ³¡ [I −λ−1 m (1 −I)e−Dm λm ]ψj(xit) ¢\|CS\| j=1, ¡ 0 ¢\|hK\| j=\|CS\|+1 ´ + ³¡ [I −λ−1 S (1 −I)e−DS λS ]ξj(xit) ¢\|CS\| j=1, ¡ 0 ¢\|hK\| j=\|CS\|+1 ´ . (5) 4 Here, ψj(xit)=p(i) j ¡ CS(xit) ¢ j £ 1− ¡ CS(xit) ¢ j ¤ and ξj(xit)=qj(xit) ¡ CS(xit) ¢ j £ 1− ¡ CS(xit) ¢ j ¤ , where p(i) = 2 TB sign(1.5−i)(µ(1)−µ(2)), q(xit)= 4 TB−1sign(1.5−i)(Σ(1)−Σ(2))[CS(xit)−µ(i)] and ¡ CS(xit) ¢ j is output of the jth neuron in CS for input xit. For layers k=K-1, · · · ,1, we have ∂LD ∂uk(xit) = µ ∂LD ∂hkj(xit)hkj(xit)[1−hkj(xit)] ¶\|hk\| j=1 , ∂LD ∂hk(xit) = £ W (i) k+1 ¤T ∂LR ∂uk+1(xit). (6) Given a training example, ¡ {x1t}TB t=1, {x2t}TB t=1; I ¢ , we use gradients achieved from Eqs. (3)-(6) to update all the parameters in the RSDN. For layers k=K+1, · · · , 2K, their parameters are updated by W (i) k ←W (i) k −ϵα TB TB X t=1 2 X r=1 ∂LR ∂uk(xrt)[hk−1(xrt)]T , b(i) k ←b(i) k −ϵα TB TB X t=1 2 X r=1 ∂LR ∂uk(xrt). (7) For layers k=1, · · · , K, their weight matrices and biases are updated with W (i) k ←W (i) k −ϵ TB TB X t=1 2 X r=1 ³ α ∂LR ∂uk(xrt) +(1 −α) ∂LD ∂uk(xrt) ´ [hk−1(xrt)]T , (8a) b(i) k ←b(i) k −ϵ TB TB X t=1 2 X r=1 ³ α ∂LR ∂uk(xrt) +(1 −α) ∂LD ∂uk(xrt) ´ . (8b) In Eqs. (7) and (8), ϵ is a learning rate. Here we emphasize that using sum of gradients caused by two subnets in update rules guarantees that two subsets are always kept identical during learning. 4 Experiment In this section, we describe our experimental methodology and report experiments results in visualization of vowel distributions, speaker comparison and speaker segmentation. We employ two LDC benchmark corpora [14], KING and TIMIT, and a Chinese speech corpus [15], CHN, in our experiments. KING, including wide-band and narrow-band sets, consists of 51 speakers whose utterances were recorded in 10 sessions. By convention, its narrow-band set is called NKING while KING itself is often referred to its wide-band set. There are 630 speakers in TIMIT and 59 speakers in CHN of three sessions, respectively. All corpora were collected especially for evaluating a speaker recognition system. The same feature extraction procedure is applied to all three corpora; i.e., after a short-term analysis suggested in [18], including silence removal with an energy-based method, pre-emphasis with the ﬁlter H(z) = 1−0.95z−1 as well as Hamming windowing with the size of 20 ms and 10 ms shift, we extract 19-order MFCCs [1] for each frame. For the RSDN learning, we use utterances of all 49 speakers recorded in sessions 1 and 2 in KING. Furthermore, we distort all the utterances by the additive white noise channel with SNR of 10dB and the Rayleigh fading channel with 5 Hz Doppler shift [19] to simulate channel effects. Thus our training set consists of clean utterances and their corrupted versions. We randomly divide all utterances into speech segments of a length TB (1 sec≤TB ≤2 sec) and then exhaustively combine them to form training examples as described in Sect. 2.2. With a validation set of all the utterances recorded in session 3 in KING, we select a structure of K=4 (100, 100, 100 and 200 neurons in layers 1-4 and \|CS\|=100 in the code layer or layer 4) from candidate models of 2<K<5 and 501000 neurons in a hidden layer. Parameters used in our learning are as follows: Gaussian noise of N(0, 0.1σ) used in denoising autoencoder, α=0.2, λm=100 and λS=2.5 in the loss function deﬁned in Eq. (2), and learning rates ϵ=0.01 and 0.001 for pre-training and discriminative learning. After learning, the RSDN is used to yield a 100-dimensional representation, CS, from 19-order MFCCs. For any speaker recognition tasks, speaker modeling (SM) is inevitable. In our experiments, we use the 1st- and 2nd-order statistics of a speech segment based on a representation, SM = {µ, Σ}, for SM. Furthermore, we employ a speaker distance metric: d(SM1, SM2) = tr[(Σ−1 1 + Σ−1 2 )(µ1 − µ2)(µ1 −µ2)T ], where SMi = {µi, Σi} (i = 1, 2) are two speaker models (SMs). This distance metric is derived from the divergence metric for two normal distributions [20] by dropping the term concerning only covariance matrices based on our observation that covariance matrices often vary considerably for short segments and the original divergence metric often leads to poor performance for various representations including MFCCs and ours. In contrast, the one deﬁned above is stable irrespective of utterance lengths and results in good performance for different representations. 5 (a) /aa/, /iy/, /aw/, /ay/ /ae/, /aw/, /iy/, /ix/ (b) /iy/, /ih/, /eh/, /ix/ /ae/, /aa/, /aw/, /ay/ (c) Figure 2: Visualization of all 20 vowels. (a) CS representation. (b) CS representation. (c) MFCCs. 4.1 Visualization Vowels have been recognized to be a main carrier of speaker-speciﬁc information [1],[4],[18],[20]. TIMIT [14] provides phonetic transcription of all 10 utterances containing all 20 vowels in English for every speaker. As all the vowels may appear in 10 different utterances, up to 200 vowel segments in length of 0.1-0.5 sec are available for a speaker, which enables us to investigate vowel distributions in a representation space for different speakers. Here, we merely visualize mean feature vectors of up to 200 segments for a speaker in terms of a speciﬁc representation with the t-SNE method [21], which is likely to reﬂect intrinsic manifolds, by projecting them onto a two-dimensional plane. In the code layer of our RSDN, output of neurons 1-100 forms a speaker-speciﬁc representation, CS, and that of remaining 100 neurons becomes a non-speaker related representation, dubbed CS. For a noticeable effect, we randomly choose only ﬁve speakers (four females and one male) and visualize their vowel distributions in Figure 2 in terms of CS, CS and MFCC representations, respectively, where a maker/color corresponds to a speaker. It is evident from Figure 2(a) that, by using the CS representation, most vowels spoken by a speaker are tightly grouped together while vowels spoken by different speakers are well separated. For the CS representation, close inspection on Figure 2(b) reveals that the same vowels spoken by different speakers are, to a great extent, co-located. Moreover, most of phonetically correlated vowels, as circled and labeled, are closely located in dense regions independent of speakers and genders. For comparison, we also visualize the same by using their original MFCCs in Figure 2(c) and observe that most of phonetically correlated vowels are also co-located, as circled and labeled, whilst others scatter across the plane and their positions are determined mainly by vowels but affected by speakers. In particular, most of vowels spoken by the male, marked by □and colored by green, are grouped tightly but isolated from those by all females. Thus, visualization in Figure 2 demonstrates how our RSDN learning works and could lend an evidence to justiﬁcation on why MFCCs can be used in both speech and speaker recognition [1]. 4.2 Speaker Comparison Speaker comparison (SC) is an essential process involved in any speaker recognition tasks by comparing two speaker models to collect evidence for decision-making, which provides a direct way to evaluate representations/speaker modeling without addressing decision-making issues [22]. In our SC experiments, we employ NKING [14], a narrow-band corpus, of many variabilities. During data collection, there was a “great divide” between sessions 1-5 and 6-10; both recording device and environments changed, which alters spectral features of 26 speakers and leads to 10dB SNR reduction on average. As suggested in [18], we conduct two experiments: within-divide where SMs built on utterances in session 1 are compared to SMs on those in sessions 2-5 and cross-divide where SMs built on utterances in session 1 are compared with those in sessions 6-10. As short utterances poses a greater challenge for speaker recognition [4],[18],[20], utterances are partitioned into short segments of a certain length and SMs built on segments of the same length are always used for SC. For a thorough evaluation, we apply the SM technique in question to our representation, MFCCs, and a representation (i.e., the better one of those yielded by two layers) learned by the CDBN [9] on all 10 sessions in NKING, and name them SM-RSDN, SM-MFCC and SM-CDBN hereinafter. In addition, we also compare them to GMMs trained on MFCCs (GMM-MFCC), a state-of-the-art SM technique that provides the baseline performance [4],[20], where for each speaker a GMM-based SM consisting of 32 Gaussian components is trained on his/her utterances of 60 sec in sessions 1-2 with the EM algorithm [18]. For the CDBN learning [9] and the GMM training [18], we strictly follow their suggested parameter settings in our experiments (see [9],[18] for details). 6 0.1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.4 0.6 0.8 1 False Alarm Probability Miss Probability SM−MFCC GMM−MFCC SM−CDBN SM−RSDN 0.1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.4 0.6 0.8 1 False Alarm Probability Miss Probability SM−MFCC GMM−MFCC SM−CDBN SM−RSDN 0.1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.4 0.6 0.8 1 False Alarm Probability Miss Probability SM−MFCC GMM−MFCC SM−CDBN SM−RSDN 0.1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.4 0.6 0.8 1 False Alarm Probability Miss Probability SM−MFCC GMM−MFCC SM−CDBN SM−RSDN (a) 0.1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.4 0.6 0.8 1 False Alarm Probability Miss Probability SM−MFCC GMM−MFCC SM−CDBN SM−RSDN (b) 0.1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.4 0.6 0.8 1 False Alarm Probability Miss Probability SM−MFCC GMM−MFCC SM−CDBN SM−RSDN (c) Figure 3: Performance of speaker comparison (DET) in the within-divide (upper row) and the crossdivide (lower row) experiments for different segment lengths. (a) 1 sec. (b) 3 sec. (d) 5 sec. Table 1: Performance (mean±std)% of speaker segmentation on TIMIT and CHN audio streams. Index TIMIT Audio Stream CHN Audio Stream BIC-MFCC Dist-MFCC Dist-RSDN BIC-MFCC Dist-MFCC Dist-RSDN FAR 26±09 22±11 18±11 46±04 27±11 24±11 MDR 26±14 22±12 18±10 46±10 27±17 24±17 F1 67±12 74±11 79±09 44±08 68±17 72±17 We use Detection Error Trade-off (DET) curves as the performance index in SC. From Figure 3, it is evident that SM-RSDN outperforms SM-MFCC, SM-CDBN and GMM-MFCC, a baseline system trained on much longer utterances, as it always yields a smaller operating region, i.e., all possible errors, in all the settings. In contrast, SM-MFCC performs better in within-divide settings while SM-CDBN is always inferior to the baseline system. Relevant issues will be discussed later on. 4.3 Speaker Segmentation Speaker segmentation (SS) is a task of detecting speaker change points in an audio stream to split it into acoustically homogeneous segments so that every segment contains only one speaker [23]. Following the same protocol used in previous work [23], we utilize utterances in TIMIT and CHN corpora to simulate audio conversations. As a result, we randomly select 250 speakers from TIMIT to create 25 audio streams where the duration of speakers ranges from 1.6 to 7.0 sec and 50 speakers from CHN to create 15 audio streams where the duration of speakers is from 3.0 to 8.3 sec. In the absence of prior knowledge, the distance-based and the BIC techniques are two main approaches to SS [23]. In our simulations, we apply the distance-based method [23] to our representation and MFCCs, dubbed Dist-RSDN and Dist-MFCC, where the same parameters, including sliding window of 1.5 sec and tolerance level of 0.5 sec, are used. In addition, we also apply the BIC method [23] to MFCCs (BIC-MFCC). Note that the BIC method is inapplicable to our representation since it uses only covariance information but the high dimensionality of our representation and the use of a small sliding window in the BIC result in unstable performance, as pointed out early in this section. For evaluation, we use three common indexes [23], i.e., False Alarm Rate (FAR), Miss Detection Rate (MDR) and F1 measure deﬁned based on both precision and recall rates. Moreover, we only report results as FAR equals MDR to avoid addressing decision-making issues [23]. Table 1 tabulates SS performance where, as boldfaced, results by our representation are superior to those by MFCCs regardless of SS methods and corpora for creating audio streams used in our simulations. In summary, visualization of vowels and results in SC and SS suggest that our RSDN successfully extracts speaker-speciﬁc information; its resultant representation can be generalized to unseen corpora during learning and is insensitive to text and languages spoken and environmental changes. 7 5 Discussion As pointed out earlier, speech carries different yet mixed information and speaker-speciﬁc information is minor in comparison to predominant linguistic information. Our empirical studies suggest that our success in extracting speaker-speciﬁc information is attributed to both unsupervised pretraining and supervised discriminative learning with a contrastive loss. In particular, the use of data regularization in discriminative learning and distorted data in two learning phases plays a critical role in capturing intrinsic speaker-speciﬁc characteristics and variations caused by miscellaneous mismatches. Our results not reported here, due to limited space, indicate that without the pretraining in Sect. 3.1, a randomly initialized RSDN leads to unstable performance often considerably worse than that of using the pre-training in general. Without discriminative learning, a DA working on unsupervised learning only, e.g., the CDBN [9], tends to yield a new representation that redistributes different information but does not highlight minor speaker-speciﬁc information given the fact that the CDBN trained on all 10 sessions in NKING leads to a representation that fails to yield satisfactory SC performance on the same corpus but works well for various audio classiﬁcation tasks [9]. If we do not use the regularization term, LR(Xi; Θ), in the loss function in Eq. (2), our RSDN is boiled down to a standard Siamese architecture [10]. Our results not reported here show that such an architecture learns a representation often overﬁtting to the training corpus due to interference of predominant non-speaker related information, which is not a problem in predominant information extraction. The previous work in face recognition [11] could lend an evidence to support our argument where a Siamese DA without regularization successfully captures predominant identity characteristics from facial images as, we believe, facial expression and other non-identity information are minor in this situation. While the use of distorted data in pre-training is in the same spirit of self-taught learning [24], our results including those not reported here reveal that the use of distorted data in pre-training but not in discriminative learning yields results worse than the baseline performance in the cross-divide SC experiment. Hence, sufﬁcient training data reﬂecting mismatches are also required in discriminative learning for speaker-speciﬁc information extraction. Our RSDN architecture resembles the one proposed in [12] for dimensionality reduction of handwritten digits via learning a nonlinear embedding. However, ours distinguishes from theirs in the use of different building blocks in our DAs, loss functions and motivations. The DA in [12] uses the RBM [13] as a building block to construct a deep belief subnet in their Siamese DA and the NCA [25] as their contrastive loss function to minimize the intra-class variability. However, the NCA does not meet our requirements as there are so many examples in one class. Instead we propose a contrastive loss to minimize both intra- and inter-class variabilities simultaneously. On the other hand, intrinsic topological structures of a handwritten digit convey predominant information given the fact that without using the NCA loss a deep belief autoencoder already yields a good representation [7],[12],[13],[26]. Thus, the use of the NCA in [12] simply reinforces the topological invariance by minimizing other variabilities with a small amount of labeled data [12]. In our work, however, speaker-speciﬁc information is non-predominant in speech and hence a large amount of labeled data reﬂecting miscellaneous variabilities are required during discriminative learning despite the pre-training. Finally, our code layer yields an overcomplete representation to facilitate nonpredominant information extraction. In contrast, a parsimonious representation seems more suitable for extracting predominant information since dimensionality reduction is likely to discover “principal” components that often associate with predominant information, as are evident in [11],[12]. To conclude, we propose a deep neural architecture for speaker-speciﬁc information extraction and demonstrate that its resultant speaker-speciﬁc representation outperforms the state-of-the-art techniques. It should also be stated that our work presented here is limited to speech corpora available at present. In our ongoing work, we are employing richer training data towards learning a universal speaker-speciﬁc representation. In a broader sense, our work presented in this paper suggests that speech information component analysis (ICA) becomes critical in various speech information processing tasks; the use of proper speech ICA techniques would result in task-speciﬁc speech representations to improve their performance radically. Our work demonstrates that speech ICA is feasible via learning. Moreover, deep learning could be a promising methodology for speech ICA. Acknowledgments Authors would like to thank H. Lee for providing their CDBN code [9] and L. 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(2009) Exploring strategies for training deep neural networks. Journal of Machine Learning Research 10(1): pp. 1-40. [8] Boureau, Y., Bach, F., LeCun, Y. & Ponce, J. (2010) Learning mid-level features for recognition. Proc. CVPR, IEEE Press. [9] Lee, H., Largman, Y., Pham, P. & Ng, A. (2009) Unsupervised feature learning for audio classiﬁcation using convolutional deep belief networks. In Advances in Neural Information Processing Systems 22, Cambridge, MA: MIT Press. [10] Bromley, J., Guyon, I., LeCun, Y., Sackinger, E. & Shah, R. (1994) Signature veriﬁcation using a Siamese time delay neural network. In Advances in Neural Information Processing Systems 5, Morgan Kaufmann. [11] Chopra, S., Hadsell, R. & LeCun, Y. (2005) Learning a similarity metric discriminatively, with application to face veriﬁcation. In Proc. CVPR, IEEE Press. [12] Salakhutdinov, R. & Hinton, G. (2007) Learning a non-linear embedding by preserving class neighborhood structure. In Proc. 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[21] van der Maaten, L. & Hinton, G. (2008) Visualizing data using t-SNE. Journal of Machine Learning Research 9: 2579-2605. [22] Campbell, W. & Karam, Z. (2009) Speaker comparison with inner product discriminant functions. In Advances in Neural Information Processing Systems 22, Cambridge, MA: MIT Press. [23] Kotti, M., Moschou, V. & Kotropoulos, C. (2008) Speaker segmentation and clustering. Signal Processing 88(8): 1091-1124. [24] Raina, R., Battle, A., Lee, H., Packer, B. & Ng, A. (2007) Self-taught learning: transfer learning from unlabeled data. Proc. ICML, ACM press. [25] Goldberger, J., Roweis, S., Hinton, G. & Salakhutdinov, R., (2005) Neighbourhood component analysis. In Advances in Neural Information Processing Systems 17, Cambridge, MA: MIT Press. [26] Hinton, G. & Salakhutdinov, R. (2006) Reducing the dimensionality of data with neural networks. Science 313: 504-507. 9	2011	89
4,440	Variational Learning for Recurrent Spiking Networks Danilo Jimenez Rezende Brain Mind Institute ´Ecole Polytechnique F´ed´erale de Lausanne 1015 Lausanne EPFL, Switzerland danilo.rezende@epfl.ch Daan Wierstra School of Computer and Communication Sciences, Brain Mind Institute ´Ecole Polytechnique F´ed´erale de Lausanne 1015 Lausanne EPFL, Switzerland daan.wierstra@epfl.ch Wulfram Gerstner School of Computer and Communication Sciences, Brain Mind Institute ´Ecole Polytechnique F´ed´erale de Lausanne 1015 Lausanne EPFL, Switzerland wulfram.gerstner@epfl.ch Abstract We derive a plausible learning rule for feedforward, feedback and lateral connections in a recurrent network of spiking neurons. Operating in the context of a generative model for distributions of spike sequences, the learning mechanism is derived from variational inference principles. The synaptic plasticity rules found are interesting in that they are strongly reminiscent of experimental Spike Time Dependent Plasticity, and in that they differ for excitatory and inhibitory neurons. A simulation conﬁrms the method’s applicability to learning both stationary and temporal spike patterns. 1 Introduction This study considers whether recurrent networks of spiking neurons can be seen as a generative model not only of stationary patterns but also of temporal sequences. More precisely, we derive a model that learns to adapt its spontaneously spike sequences to conform as closely as possible to the empirical distribution of actual spike sequences caused by inputs impinging upon the sensory layer of the network. A generative model is a model of the joint distribution of percepts and hidden causes in the world. Since the world has complex temporal relationships, we need a model that is able to both recognize and predict temporal patterns. Behavioural studies (e.g., [1]) support the assumption that the brain is performing approximate Bayesian inference. More recently, evidence for this hypothesis was found in electro-physiological work as well [2]. Various abstract Bayesian models have been proposed to account for this phenomenon [3, 4, 5, 6, 7]. However, it remains an open question whether optimization in abstract Bayesian models can be translated into plausible learning rules for synapses in networks of spiking neurons. In this paper, we show that the derivation of spike-based plasticity rules from statistical learning principles yields learning dynamics for a generative spiking network model which are akin to those 1   Figure 1: A network of spiking neurons, divided into observed and latent pools of neurons. of Spike-Time Dependent Plasticity (STDP) [8]. Our learning rule is derived from a variational optimization process. Typically, optimization in recurrent Bayesian networks involves both forward and backward propagation steps. We propose a plasticity rule that approximates backward steps by the introduction of delayed updates in the synaptic weights and dynamics. The theory is supported by simulations in which we demonstrate that the learning mechanism is able to capture the hidden causes behind the observed spiking patterns. We use the Spike Response Model (SRM) [9, 10], in which spikes are generated stochastically depending on the neuronal membrane potential. The SRM is an example of a generalized linear model (GLM). It is closely related to the integrate-and-ﬁre model, and has been successfully used to explain neuronal spike trains [11, 12]. In this model, the membrane potential of a neuron i at time t, expressed as ui(t) is given by ⌧˙ui(t) = −ui(t) + bi + X j Wi,jXj(t), (1) where bi is a bias which represents a constant external input to the neuron, and Xj(t) is the spike train of the jth neuron deﬁned by Xj(t) = P tf j 2{t1 j,...,tN j } δ(t −tf j ), where {t1 j, . . . , tN j } is the set of spike timings. The diagonal elements of the synaptic matrix are kept ﬁxed to a negative value Wi,i = −⌘0 with ⌘0 = 1.0, which implements a reset of the membrane potential after each spike and is a simple way to take into account neuronal refractoriness [9, 13]. The time constant is taken to be ⌧= 10ms as in [13]. The spike generation process is stochastic with time-dependent ﬁring intensity ⇢i(t) which depends on the membrane potential ui(t): ⇢i(t) = ⇢0 exp (ui(t)) . (2) An exponential dependence of the ﬁring intensity upon the membrane potential agrees with experimental results [12]. The set of equations (2) and (1) captures the simpliﬁed dynamics of a spiking neuron with stochastic spike timing. In the following sections, we will introduce the theoretical framework and the approximations used in this paper. The basic learning mechanism is introduced and derived, followed by a simulation illustrating that our proposed learning rule is able to learn spatio-temporal features in the input spike trains and reproduce them in its spontaneous activity. 2 Principled Framework We consider a network consisting of two distinct sets of neurons, observed neurons ( also called visible neurons or V) and latent neurons ( also called hidden or H), as illustrated in Figure 1. The activities of the observed neurons represent the quantity of interest to be modelled, while the latent neurons fulﬁll a mediating role representing the hidden causes of the observed spike train. Learning in the context of this neuronal network consists of changing the synaptic strengths between neurons. We postulate that the underlying principle behind learning relies on learning distributions of spike trains evoked by either sensory inputs or more complicated sequences of cognitive events. In statistics, learning distributions involves minimizing a measure of distance between the model (that is, our neuronal network) and a target distribution (e.g. observations). A principled measure of distance between two distributions p and pempirical is the Kullback-Leibler divergence [14] deﬁned as KL(pempirical\|\|p) = Z DXpempirical(X) log pempirical(X) p(X) . (3) 2 where individual X represent entire spike trains. DX is a measure of integration over spike trains. Our learning mechanism tries to minimize the KL divergence between the distribution deﬁned by our network p(X) and the observed spike timings distribution pempirical that is evoked by an unknown external process. Note that minimizing the KL divergence entails maximizing the likelihood that the observed spike trains XV could have been generated by the model. In order to derive the learning dynamics of our model in the next section, we need to evaluate the gradient of the likelihood (3) with respect to the free parameters of our model, i.e. the synaptic efﬁcacies Wi,j and biases bi. The joint likelihood of a particular spike train of both the observed XV and the latent neurons XH under our neuronal model can be written as [13] log p(XV, XH) = X i2V[H Z T 0 d⌧[log ⇢i(⌧)Xi(⌧) −⇢i(⌧)] (4) Since we have a neuronal network including latent units (that is, neurons not receiving external inputs), the actual observation likelihood is an effective quantity obtained by integrating over all possible latent spike trains XH, p(XV) = Z DXHp(XV, XH). (5) The gradient of (5) is given by an expectation conditioned on the observed neurons’ history: r log p(XV) = r log Z DXHp(X) = hr log p(X)ip(XH\|XV) where hf(X)ip = R DXf(x)p(x). This is difﬁcult to evaluate since it conditions an entire latent spike train on an entire observed spike train. In other words, the posterior distribution of spiketimings of the latent neurons depends on both past and future of the observed neurons’ spike train. 2.1 Weak Coupling Approximation In order to render the model more tractable, we introduce an approximation on the dynamics based on the weak coupling approximation [15], which amounts to replacing (1) by ⌧˙ui(t) = −ui(t) + bi + X j Wi,j⇢j(t) + zj(t), (6) where zi(t) is a Gaussian process with mean zero and inverse variance 1 λi(t) given by λ−1 i (t) = σ0 + 1 ⌧2 X j W 2 i,j⇢j(t), (7) where σ0 is intrinsic noise which we have added to regularize the simulations (we assume σ0 = 0.1). Note that λi(t) is a function of both the network state and synaptic efﬁcacies. Our network model deﬁnes a joint distribution between observed input spike trains and membrane potentials given by log p(XV, u) = X i2V Z dt [Xi(t)ui(t) −⇢0 exp(ui(t))] − X i2V[H Z dtλi(t) 2 ( ˙ui(t) −fi(t))2, (8) where terms not depending on the model parameters and latent states have been dropped out as they do not contribute to the gradients we are interested in and fi(t) is the drift of the Gaussian process of the membrane potentials and can be read from equation (6). It is given by fi(t) = 1 ⌧ 0 @−ui(t) + bi + X j Wi,j⇢j(t) 1 A (9) 1The variance of ˙u due to the external input can be obtained by noting that ui(t+dt) = ui(t) exp(−dt/⌧)+ R t+dt t ds exp((s −t −dt)/⌧)(bi + P j Wi,jXj(t))/⌧. Thus, in the weak coupling regime V ar(u(t + dt)\|u(t)) = X j W 2 i,j Z t+dt t ds exp(2(s −t −dt)/⌧)(⇢j(t))/⌧2 = dt ⌧2 X j W 2 i,j⇢j(t) 3 The weak coupling approximation amounts to replacing spikes of the latent neurons by intensities plus Gaussian noise. Note that in this approximated model, the latent variables are non longer the latent spike trains, but the membrane potentials. However, we emphasize that in the end the intensities can be substituted by spikes as we will see below. 2.2 Variational Approximation of the Posterior Membrane Potential p(u\|XV) The variational approach in statistics is a method to approximate some complex distribution p by a family of simpler distributions q . Variational methods have been applied to spiking neural networks in many different contexts, such as in connectivity or external source inference [20, 21]. In the following, we try to interpret the neural activity and plasticity together as an approximate form of variational learning. We approximate the posterior p(u\|XV) by the Gaussian process log q(u) = X i Z dtλi(t) 2 ( ˙ui(t) −hi(t))2 + c (10) where the hi(t) are variational parameters representing the drift of the ith membrane potential at time t in the posterior process and c is a normalization constant. Note that the parameters λi(t) of the posterior process are taken to be the same as the network dynamics noise in (6). This is necessary in order to have a ﬁnite KL-divergence between the prior and the posterior processes [22]. Finding a good approximation for the variational parameters hi(t) amounts to minimizing the quantity KL(q(u) k p(XV, u)), which is given by KL(q k p) = Z dt * − X i2V [Xi(t)ui(t) −⇢0 exp(ui(t))] + X i2V[H λi(t) 2 ( ˙ui(t) −fi(t))2 − X i2V[H λi(t) 2 ( ˙ui(t) −hi(t))2 + q(u) (11) Although (11) can be written analytically in terms of the instantaneous mean and covariance of the posterior process, we adopt a simpler mean-ﬁeld approximation, i.e. hF(ui(t))i ⇡F(hui(t)i). We write the mean hui(t)i = ¯ui(t) as ¯ui(t) = ¯ui(0) + Z t 0 dshi(s) (12) where the hi plays the role of the ’drift’ or the derivative of ¯ui. Note that δ¯ui(t) δhj(t0) = ⇥(t −t0)δi,j, where ⇥(x) is the Heaviside step function. As a result, the KL-divergence becomes KL(q k p) ⇡ Z dt X i ⇢ −[Xi(t)¯ui(t) −⇢0 exp(¯ui(t))] δi2V + λi(t) 2 (hi(t) −fi(t))2 , (13) The drifts hi(t) of the variational approximation can be updated using gradient descent δ δhk(t0) KL = − Z dt [Xk(t) −⇢0 exp(¯uk(t))] ⇥(t −t0)δk2V + λk(t0)(hk(t0) −fi(t0)) − Z dt X i λi(t)(hi(t) −fi(t)) δfi(t) δhk(t0) + 1 2 X i Z dt δλi(t) δhk(t0)(hi(t) −fi(t))2, (14) where δfi(t) δhk(t0) = 1 ⌧(−δi,k + Wi,k⇢k(t)) ⇥(t −t0) (15) δλi(t) δhk(t0) = −1 ⌧2 λ2 i (t)W 2 i,k⇢k(t)⇥(t −t0) (16) 4 Figure 2: Posterior ﬁring intensity for two simple networks: (a) A network with 4 neurons, simulated with mean ﬁeld approximation. (b) From top to bottom: the observed spike train, the ﬁring intensity for the three latent neurons and the posterior inverse variance. The green neuron has a direct connection to the observed neuron, and as such has a much stronger modulation of its ﬁring rate than the other two latent neurons. (c) A network with two pools of 20 neurons, the observed and the latent pools. (d) Simulation results. From top to bottom: observed spike trains, spike trains in the latent pool and mean ﬁring intensities of the latent neurons over different realizations of the network. The rate of the latent pool increases just before the spikes of the observed neurons. Note that the spiking implementation of the model has the same rates as the mathematical rate model. There are few key points to note regarding (14). First, in the absence of observations, the best approximating hi(t) is simply given by fi(t), that is the posterior and the prior processes become equal. Second, the ﬁrst, third and fourth terms in (14) are backward terms, that is, they correspond to corrections in the “belief” about the past states generated by new inputs. This implies that in order to estimate the drift hi(t) of the posterior membrane potential of neuron i at time t, we need to know the observations X(t0) at time t0 > t. Third, the fourth term in equation (14) is a contribution to the gradient that comes from the fact that the inverse variance λi(t) deﬁned in equation (7) is also a function of the network state. This is an important feature of the model, since it implies that the amount of noise in the dynamics is also being adapted to better explain the observed spike trains. 2.3 Towards Spike-time Dependent Plasticity We learn the parameters of our network, that is, the synaptic weights and the neural ‘biases’ by gradient descent with learning rate ⌘: ∆bi = −⌘δ δbi KL = −⌘ Z dtλi(t) ⌧ (hi(t) −fi(t)) (17) ∆Wk,l = −⌘ δ δWk,l KL = −⌘ Z dtλk(t) ⌧ (hk(t) −fk(t))⇢l(t) +⌘1 2 X i Z dtδλi(t) δWk,l (hi(t) −fi(t))2, (18) where δλi(t) δWk,l = −2 1 ⌧2 λ2 i (t)Wi,l⇢l(t)δk,i. Note that once the posterior drift hi(t) is known, the computation of ∆b and ∆W can be done purely locally. 5 A long ‘backward window’ would, of course, be biologically implausible. However, on-line approximations to the backward terms provide a reasonable approximation by taking small backwards ﬁlters of up to 50ms. Mechanistically, applications of ∆W can operate with a small delay, which is required to calculate the backwards correction term. In biology such delays indeed exist, as the weights are switched to a new value only some time after the stimulation that induces the change [23, 24] More precisely, using a small backward window amounts to approximating the gradient of the posterior drift hi(t) by cutting off the time integrals using a ﬁnite time horizon, i.e., in equation (14) we replace integral R dt by R t0+∆T t0 dt where ∆T is the size of the “backward window” used to approximate the gradient. The expression (14) can now be written as a delayed update equation δhk(t −∆T) / − Z t t−∆T ds [Xk(s) −⇢0 exp(¯uk(s))] δk2V + λk(t −∆T)(hk(t −∆T) −fk(t −∆T)) − Z t t−∆T ds X i λi(s)(hi(s) −fi(s)) δfi(s) δhk(t −∆T) + 1 2 X i Z t t−∆T ds δλi(s) δhk(t −∆T)(hi(s) −fi(s))2, (19) The resulting update for the variable hk is used in the learning equation 18. The simulation shown in Figure 2 provides a conceptual illustration of how the posterior ﬁring intensity ⇢l(t) propagates information backward from observed into latent neurons, a process that is essential for learning temporal patterns. Note that ⇢l is the ﬁring rate of the presynaptic neuron l and as such it is information that is not directly available at the site of the synapse which has only access to spike arrivals (but not the underlying ﬁring rate). However, spike arrivals do provide a reasonable estimate of the rate. Indeed Figure 2c and d show that a simulation of a network of pools of spiking neurons where updates are only based on spike times (rather than rates) gives qualitatively the same information as the rate formula derived above. In equations (20,15) we could therefore replace the pre-synaptic ﬁring intensity ⇢j(t) by temporally ﬁltered spike trains which constitute a good approximation to ⇢j(t). 2.4 STDP Window From our learning equation for the synaptic weight (18), we can extract an STDP-like learning window by rewriting the plasticity rules as ∆Wi,j = R dt∆Wi,j(t), where ∆Wi,j(t) = λi(t) ⌧ (hi(t) −fi(t))⇢j(t) + 1 2 X k δλk(t) δWi,j (hk(t) −fk(t))2 (20) ∆Wi,j(t) is the expected change in ∆Wi,j at time t under the posterior. As before, we replace the ﬁring intensity ⇢j in a given trial by the spikes. Assuming a spike of the observed neuron at t = 0, we have evaluated h(t) and f(t) and plot the weight change λk(t0)(hk(t0) −fk(t0)) that would occur if the latent neuron ﬁres at t0 cf. equation (18). We show the resulting Spike-time Dependent Plasticity for a simple network of two neurons in Figure 3. Note that the shape of ∆Wi,j(t) is remarkably reminiscent of the experimentally found measurements for STDP [8]. In particular, the shape of the STDP curve depends on the type of neuron and is different for connections from excitatory to excitatory than from excitatory to inhibitory or inhibitory to inhibitory neurons (Figure 3). 3 Simulations In order to demonstrate the method’s ability to capture both stationary and temporal patterns, we performed simulations on two tasks. The ﬁrst one involves the formation of a temporal chain, while the second one involves a stationary pattern generator. Both simulations were done using a discretetime (Euler method) version of the equations (14, 17, 18 and 19) with dt = 1ms. The backward window size was taken to be ∆T = 50ms, and a learning rate of 0.02 was used. 6                                                     Figure 3: Spike-time Dependent Plasticity in a simple network composed of two neurons. Weight change ∆Wi,j(t) (vertical axis) as a function of spike timing of the neuron at the top (the latent neuron), given that the bottom (observed) neuron produces a spike at t = 0 (horizontal axis). Shown are all permutations of excitatory (e) and inhibitory (i) neuron types, with the left and right learning windows next to each network corresponding to the downward and upward synapses, respectively. The ﬁrst task consisted of learning a periodic chain, in which three pools of observed neurons were successively activated as shown in Figure 4a. A time lag was introduced between the third and the ﬁrst pattern so as to force the network to form temporal hidden cause representations that are capable of capturing time dependencies without obvious observable instantaneous clues – during a blank moment, the only way a network can tell which pattern comes next is by actively using the latent neurons. After learning, the spontaneously patterns in the observable neurons developed a clear resemblance to the patterns provided during training, although a slightly larger amount of noise was present, as shown in Figure 4b. If the noise level of the model network is reduced, a noise-free “cleared-up concept” of the observed patterns is generated (Figure 4d) which clearly demonstrates that the recurrent network has indeed learned the task. The way learning has conﬁgured the network in the sequence task can be understood if we study the connectivity pattern of the latent neurons. The latent neuron are active during the whole sequence (Figure 4c). We have reordered the labels of the neurons so that the structure of the connectivity matrix becomes as visble. There are subsets of latent neurons that are particularly active during each of the three ’subpatterns’ in the sequence task, and other latent neurons that become active while the observable units are quiescent (Figure 4i). The lateral connectivity between the latent neurons has an asymmetry in the forward direction of the chain. The second task aimed at learning to randomly generate one of three statinonary patterns every 10ms. Successfull learning of this task requires both the learning of the stationary patterns and the stochastic transitions between them. Figure 4d–g shows the results on this task. 4 Discussion Some models have recently been proposed where STDP-like learning rules derive from ‘ﬁrst principles’ (e.g., [25, 26, 13]). However, these models have either difﬁculty dealing with recurrent latent dynamics, or they do not account for non-factorial latent representations. In this work, we have proposed a plausible derivation for synaptic plasticity in a network consisting of spiking neurons, which can both capture time dependencies in observed spike trains and process combinatorial features. Using a generative model comprising both latent and observed neurons, the mechanism utilizes implicit (that is, short-term delayed) backward iterations that arise naturally from variational inference. A plasticity mechanism emerges that closely resembles that of the familiar STDP mechanism found in experimental studies. In our simulations we show that the plasticity rules are capable of learning both a temporal and a stationary pattern generator. Future work will attempt to further elucidate the possible biological plausibility of the approach, and its connection to Spike-Time Dependent Plasticity. Acknowledgments Support was provided by the SNF grant (CRSIK0 122697), the ERC grant (268689) and the SystemsX IPhD grant. 7 Figure 4: Simulation results. Sequence task a–d, i: a 20ms-periodic sequence with a network of 30 observed neurons and 15 latent neurons having 50% of inhibitory neurons (chosen randomly). The connections between the observed neurons have been set to zero in order to illustrate the use of latent-to-latent recurrent connections. (a) A sample of the periodic input pattern. Note the long waiting time after each sequence 1 −2 −3 (1 −2 − 3−wait−1−2−3−. . . ). (b) Simulations from the network with the ﬁrst 20ms clamped to the data. (c) Latent neurons sample. (d) Sample simulation of the network with the same parameters but with less noise, in order to better show the underlying dynamics. This is achieved by the transformation ⇢i(t) ! ⇢i(t)β with β = 2. Random jump task e–h: learning to produce one of three patterns (4ms long) every 10ms. (e) A sample input pattern (f) One realization from the network with ﬁrst the 20ms clamped to the data. (g) Sample latent pattern. (h) Sample simulation of the network with the same parameters but with less noise. Note that decreasing the level of noise is actually an impairment in performance for this task. 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4,441	Variance Penalizing AdaBoost Pannagadatta K. Shivaswamy Department of Computer Science Cornell University, Ithaca NY pannaga@cs.cornell.edu Tony Jebara Department of Compter Science Columbia University, New York NY jebara@cs.columbia.edu Abstract This paper proposes a novel boosting algorithm called VadaBoost which is motivated by recent empirical Bernstein bounds. VadaBoost iteratively minimizes a cost function that balances the sample mean and the sample variance of the exponential loss. Each step of the proposed algorithm minimizes the cost efﬁciently by providing weighted data to a weak learner rather than requiring a brute force evaluation of all possible weak learners. Thus, the proposed algorithm solves a key limitation of previous empirical Bernstein boosting methods which required brute force enumeration of all possible weak learners. Experimental results conﬁrm that the new algorithm achieves the performance improvements of EBBoost yet goes beyond decision stumps to handle any weak learner. Signiﬁcant performance gains are obtained over AdaBoost for arbitrary weak learners including decision trees (CART). 1 Introduction Many machine learning algorithms implement empirical risk minimization or a regularized variant of it. For example, the popular AdaBoost [4] algorithm minimizes exponential loss on the training examples. Similarly, the support vector machine [11] minimizes hinge loss on the training examples. The convexity of these losses is helpful for computational as well as generalization reasons [2]. The goal of most learning problems, however, is not to obtain a function that performs well on training data, but rather to estimate a function (using training data) that performs well on future unseen test data. Therefore, empirical risk minimization on the training set is often performed while regularizing the complexity of the function classes being explored. The rationale behind this regularization approach is that it ensures that the empirical risk converges (uniformly) to the true unknown risk. Various concentration inequalities formalize the rate of convergence in terms of the function class complexity and the number of samples. A key tool in obtaining such concentration inequalities is Hoeffding’s inequality which relates the empirical mean of a bounded random variable to its true mean. Bernstein’s and Bennett’s inequalities relate the true mean of a random variable to the empirical mean but also incorporate the true variance of the random variable. If the true variance of a random variable is small, these bounds can be signiﬁcantly tighter than Hoeffding’s bound. Recently, there have been empirical counterparts of Bernstein’s inequality [1, 5]; these bounds incorporate the empirical variance of a random variable rather than its true variance. The advantage of these bounds is that the quantities they involve are empirical. Previously, these bounds have been applied in sampling procedures [6] and in multiarmed bandit problems [1]. An alternative to empirical risk minimization, called sample variance penalization [5], has been proposed and is motivated by empirical Bernstein bounds. A new boosting algorithm is proposed in this paper which implements sample variance penalization. The algorithm minimizes the empirical risk on the training set as well as the empirical variance. The two quantities (the risk and the variance) are traded-off through a scalar parameter. Moreover, the 1 algorithm proposed in this article does not require exhaustive enumeration of the weak learners (unlike an earlier algorithm by [10]). Assume that a training set (Xi, yi)n i=1 is provided where Xi ∈X and yi ∈{±1} are drawn independently and identically distributed (iid) from a ﬁxed but unknown distribution D. The goal is to learn a classiﬁer or a function f : X →{±1} that performs well on test examples drawn from the same distribution D. In the rest of this article, G : X →{±1} denotes the so-called weak learner. The notation Gs denotes the weak learner in a particular iteration s. Further, the two indices sets Is and Js, respectively, denote examples that the weak learner Gs correctly classiﬁed and misclassiﬁed, i.e., Is := {i\|Gs(Xi) = yi} and Js := {j\|Gs(Xj) ̸= yj}. Algorithm 1 AdaBoost Require: (Xi, yi)n i=1, and weak learners H Initialize the weights: wi ←1/n for i = 1, . . . , n; Initialize f to predict zero on all inputs. for s ←1 to S do Estimate a weak learner Gs(·) from training examples weighted by (wi)n i=1. αs = 1 2 log P i:Gs(Xi)=yi wi / P j:Gs(Xj)̸=yj wj if αs ≤0 then break end if f(·) ←f(·) + αsGs(·) wi ←wi exp(−yiGs(Xi)αs)/Zs where Zs is such that Pn i=1 wi = 1. end for Algorithm 2 VadaBoost Require: (Xi, yi)n i=1, scalar parameter 1 ≥λ ≥0, and weak learners H Initialize the weights: wi ←1/n for i = 1, . . . , n; Initialize f to predict zero on all inputs. for s ←1 to S do ui ←λnw2 i + (1 −λ)wi Estimate a weak learner Gs(·) from training examples weighted by (ui)n i=1. αs = 1 4 log P i:Gs(Xi)=yi ui / P j:Gs(Xj)̸=yj uj if αs ≤0 then break end if f(·) ←f(·) + αsGs(·) wi ←wi exp(−yiGs(Xi)αs)/Zs where Zs is such that Pn i=1 wi = 1. end for 2 Algorithms In this section, we brieﬂy discuss AdaBoost [4] and then propose a new algorithm called the VadaBoost. The derivation of VadaBoost will be provided in detail in the next section. AdaBoost (Algorithm 1) assigns a weight wi to each training example. In each step of the AdaBoost, a weak learner Gs(·) is obtained on the weighted examples and a weight αs is assigned to it. Thus, AdaBoost iteratively builds PS s=1 αsGs(·). If a training example is correctly classiﬁed, its weight is exponentially decreased; if it is misclassiﬁed, its weight is exponentially increased. The process is repeated until a stopping criterion is met. AdaBoost essentially performs empirical risk minimization: minf∈F 1 n Pn i=1 e−yif(Xi) by greedily constructing the function f(·) via PS s=1 αsGs(·). Recently an alternative to empirical risk minimization has been proposed. This new criterion, known as the sample variance penalization [5] trades-off the empirical risk with the empirical variance: arg min f∈F 1 n n X i=1 l(f(Xi), yi) + τ s ˆV[l(f(X), y)] n , (1) where τ ≥0 explores the trade-off between the two quantities. The motivation for sample variance penalization comes from the following theorem [5]: 2 Theorem 1 Let (Xi, yi)n i=1 be drawn iid from a distribution D. Let F be a class of functions f : X →R. Then, for a loss l : R × Y →[0, 1], for any δ > 0, w.p. at least 1 −δ, ∀f ∈F E[l(f(X), y)] ≤1 n n X i=1 l(f(Xi), yi) + 15 ln(M(n)/δ) (n −1) + s 18 ˆV[l(f(X), y)] ln(M(n)/δ) n , (2) where M(n) is a complexity measure. From the above uniform convergence result, it can be argued that future loss can be minimized by minimizing the right hand side of the bound on training examples. Since the variance ˆV[l(f(X), y)] has a multiplicative factor involving M(n), δ and n, for a given problem, it is difﬁcult to specify the relative importance between empirical risk and empirical variance a priori. Hence, sample variance penalization (1) necessarily involves a trade-off parameter τ. Empirical risk minimization or sample variance penalization on the 0 −1 loss is a hard problem; this problem is often circumvented by minimizing a convex upper bound on the 0 −1 loss. In this paper, we consider the exponential loss l(f(X), y) := e−yf(X). With the above loss, it was shown by [10] that sample variance penalization is equivalent to minimizing the following cost, n X i=1 e−yif(Xi) !2 + λ  n n X i=1 e−2yif(Xi) − n X i=1 e−yif(Xi) !2 . (3) Theorem 1 requires that the loss function be bounded. Even though the exponential loss is unbounded, boosting is typically performed only for a ﬁnite number of iterations in most practical applications. Moreover, since weak learners typically perform only slightly better than random guessing, each αs in AdaBoost (or in VadaBoost) is typically small thus limiting the range of the function learned. Furthermore, experiments will conﬁrm that sample variance penalization results in a signiﬁcant empirical performance improvement over empirical risk minimization. Our proposed algorithm is called VadaBoost1 and is described in Algorithm 2. VadaBoost iteratively performs sample variance penalization (i.e., it minimizes the cost (3) iteratively). Clearly, VadaBoost shares the simplicity and ease of implementation found in AdaBoost. 3 Derivation of VadaBoost In the sth iteration, our objective is to choose a weak learner Gs and a weight αs such that Ps t=1 αtGt(·) reduces the cost (3). Denote by wi the quantity e−yi Ps−1 t=1 αtGt(xi)/Zs. Given a candidate weak learner Gs(·), the cost (3) for the function Ps−1 t=1 αtGt(·) + αGs(·) can be expressed as a function of α: V (α; w, λ, I, J) :=  X i∈I wie−α + X j∈J wjeα   2 +λ   n X i∈I w2 i e−2α + n X j∈J w2 je2α−  X i∈I wie−α + X j∈J wjeα   2  . (4) up to a multiplicative factor. In the quantity above, I and J are the two index sets (of correctly classiﬁed and incorrectly classiﬁed examples) over Gs. Let the vector w whose ith component is wi denote the current set of weights on the training examples. Here, we have dropped the subscripts/superscripts s for brevity. Lemma 2 The update of αs in Algorithm 2 minimizes the cost U(α; w, λ, I, J) := X i∈I λnw2 i + (1 −λ)wi ! e−2α +  X j∈J λnw2 j + (1 −λ)wj  e2α. (5) 1The V in VadaBoost emphasizes the fact that Algorithm 2 penalizes the empirical variance. 3 Proof By obtaining the second derivative of the above expression (with respect to α), it is easy to see that it is convex in α. Thus, setting the derivative with respect to α to zero gives the optimal choice of α as shown in Algorithm 2. Theorem 3 Assume that 0 ≤λ ≤1 and Pn i=1 wi = 1 (i.e. normalized weights). Then, V (α; w, λ, I, J) ≤U(α; w, λ, I, J) and V (0; w, λ, I, J) = U(0; w, λ, I, J). That is, U is an upper bound on V and the bound is exact at α = 0. Proof Denoting 1 −λ by ¯λ, we have: V (α; w, λ, I, J)=  X i∈I wie−α+ X j∈J wjeα   2 + λ   n X i∈I w2 i e−2α+ n X j∈J w2 je2α −  X i∈I wie−α+ X j∈J wjeα   2   = ¯λ  X i∈I wie−α + X j∈J wjeα   2 + λ  n X i∈I w2 i e−2α + n X j∈J w2 je2α   = λ  n X i∈I w2 i e−2α+ n X j∈J w2 je2α  + ¯λ    X i∈I wi !2 e−2α +  X j∈J wj   2 e2α + 2 X i∈I wi ! X j∈J wj      = λ  n X i∈I w2 i e−2α + n X j∈J w2 je2α  + ¯λ   X i∈I wi !  1 − X j∈J wj  e−2α +  X j∈J wj   1 − X i∈I wi ! e2α  + 2¯λ X i∈I wi !  X j∈J wj   = X i∈I λnw2 i + ¯λwi ! e−2α +  X j∈J λnw2 j + ¯λwj  e2α + ¯λ X i∈I wi !  X j∈J wj  −e2α −e−2α + 2 ≤ X i∈I λnw2 i + ¯λwi ! e−2α +  X j∈J λnw2 j + ¯λwj  e2α = U(α; w, λ, I, J). On line two, terms were simply regrouped. On line three, the square term from line two was expanded. On the next line, we used the fact that P i∈I wi + P j∈J = Pn i=1 wi = 1. On the ﬁfth line, we once again regrouped terms; the last term in this expression (which is e2α + e−2α −2) can be written as (eα−e−α)2. When α = 0 this term vanishes. Hence the bound is exact at α = 0. Corollary 4 VadaBoost monotonically decreases the cost (3). The above corollary follows from: V (αs; w, λ, I, J) ≤U(αs; w, λ, I, J) < U(0; w, λ, I, J) = V (0; w, λ, I, J). In the above, the ﬁrst inequality follows from Theorem (3). The second strict inequality holds because αs is a minimizer of U from Lemma (2); it is not hard to show that U(αs; w, λ, I, J) is strictly less than U(0; w, λ, I, J) from the termination criterion of VadaBoost. The third equality again follows from Theorem (3). Finally, we notice that V (0; w, λ, I, J) merely corresponds to the cost (3) at Ps−1 t=1 αtGt(·). Thus, we have shown that taking a step αs decreases the cost (3). 4 0 0.3 0.6 0.9 1 2 α Cost Actual Cost:V Upper Bound:U 0 0.3 0.6 0.9 1 2 3 α Cost Actual Cost:V Upper Bound:U Figure 1: Typical Upper bound U(α; w, λ, I, J) and the actual cost function V (α; w, λ, I, J) values under varying α. The bound is exact at α = 0. The bound gets closer to the actual function value as λ grows. The left plot shows the bound for λ = 0 and the right plot shows it for λ = 0.9 We point out that we use a different upper bound in each iteration since V and U are parameterized by the current weights in the VadaBoost algorithm. Also note that our upper bound holds only for 0 ≤λ ≤1. Although the choice 0 ≤λ ≤1 seems restrictive, intuitively, it is natural to have a higher penalization on the empirical mean rather than the empirical variance during minimization. Also, a closer look at the empirical Bernstein inequality in [5] shows that the empirical variance term is multiplied by p 1/n while the empirical mean is multiplied by one. Thus, for large values of n, the weight on the sample variance is small. Furthermore, our experiments suggest that restricting λ to this range does not signiﬁcantly change the results. 4 How good is the upper bound? First, we observe that our upper bound is exact when λ = 1. Also, our upper bound is loosest for the case λ = 0. We visualize the upper bound and the true cost for two settings of λ in Figure 1. Since the cost (4) is minimized via an upper bound (5), a natural question is: how good is this approximation? We evaluate the tightness of this upper bound by considering its impact on learning efﬁciency. As is clear from ﬁgure (1), when λ = 1, the upper bound is exact and incurs no inefﬁciency. In the other extreme when λ = 0, the cost of VadaBoost coincides with AdaBoost and the bound is effectively at its loosest. Even in this extreme case, VadaBoost derived through an upper bound only requires at most twice the number of iterations as AdaBoost to achieve a particular cost. The following theorem shows that our algorithm remains efﬁcient even in this worst-case scenario. Theorem 5 Let OA denote the squared cost obtained by AdaBoost after S iterations. For weak learners in any iteration achieving a ﬁxed error rate ϵ < 0.5, VadaBoost with the setting λ = 0 attains a cost at least as low as OA in no more than 2S iterations. Proof Denote the weight on the example i in sth iteration by ws i . The weighted error rate of the sth classiﬁer is ϵs = P j∈Js ws j. We have, for both algorithms, wS+1 i = wS i exp(−yiαSGS(Xi)) Zs = exp(−yi PS s=1 αsGs(Xi)) n QS s=1 Zs . (6) The value of the normalization factor in the case of AdaBoost is Za s = X j∈js ws jeαs + X i∈Is ws i e−αs = 2 p ϵs(1 −ϵs). (7) Similarly, the value of the normalization factor for VadaBoost is given by Zv s = X j∈Js ws jeαs + X i∈Is ws i e−αs = ((ϵs)(1 −ϵs)) 1 4 (√ϵs + √ 1 −ϵs). (8) 5 The squared cost function of AdaBoost after S steps is given by OA = n X i=1 exp(−yi S X s=1 αsyiGs(X)) !2 = n S Y s=1 Za s n X i=1 ws+1 i !2 = n2 S Y s=1 Za s !2 = n2 S Y s=1 4ϵs(1 −ϵs). We used (6), (7) and the fact that Pn i=1 wS+1 i = 1 to derive the above expression. Similarly, for λ = 0 the cost of VadaBoost satisﬁes2 OV = n X i=1 exp(−yi S X s=1 αsyiGs(X)) !2 = n S Y s=1 Za s n X i=1 ws+1 i !2 = n2 S Y s=1 Zv s !2 = n2 S Y s=1 (2ϵs(1 −ϵs) + p ϵs(1 −ϵs)). Now, suppose that ϵs = ϵ for all s. Then, the squared cost achieved by AdaBoost is given by n2(4ϵ(1 −ϵ))S. To achieve the same cost value, VadaBoost, with weak learners with the same error rate needs at most S log(4ϵ(1−ϵ)) log(2ϵ(1−ϵ)+√ ϵ(1−ϵ)) times. Within the range of interest for ϵ, the term multiplying S above is at most 2. Although the above worse-case bound achieves a factor of two, for ϵ > 0.4, VadaBoost requires only about 33% more iterations than AdaBoost. To summarize, even in the worst possible scenario where λ = 0 (when the variational bound is at its loosest), the VadaBoost algorithm takes no more than double (a small constant factor) the number of iterations of AdaBoost to achieve the same cost. Algorithm 3 EBBoost: Require: (Xi, yi)n i=1, scalar parameter λ ≥0, and weak learners H Initialize the weights: wi ←1/n for i = 1, . . . , n; Initialize f to predict zero on all inputs. for s ←1 to S do Get a weak learner Gs(·) that minimizes (3) with the following choice of αs: αs = 1 4 log (1−λ)(P i∈Is wi)2+λn P i∈Is w2 i (1−λ)(P i∈Js wi)2+λn P i∈Js w2 i if αs < 0 then break end if f(·) ←f(·) + αsGs(·) wi ←wi exp(−yiGs(Xi)αs)/Zs where Zs is such that Pn i=1 wi = 1. end for 5 A limitation of the EBBoost algorithm A sample variance penalization algorithm known as EBBoost was previously explored [10]. While this algorithm was simple to implement and showed signiﬁcant improvements over AdaBoost, it suffers from a severe limitation: it requires enumeration and evaluation of every possible weak learner per iteration. Recall the steps implementing EBBoost in Algorithm 3. An implementation of EBBoost requires exhaustive enumeration of weak learners in search of the one that minimizes cost (3). It is preferable, instead, to ﬁnd the best weak learner by providing weights on the training examples and efﬁciently computing the rule whose performance on that weighted set of examples is guaranteed to be better than random guessing. However, with the EBBoost algorithm, the weight on all the misclassiﬁed examples is P i∈Js w2 i + P i∈Js wi 2 and the weight on correctly classiﬁed examples is P i∈Is w2 i + P i∈Is wi 2; these aggregate weights on misclassiﬁed examples and correctly classiﬁed examples do not translate into weights on the individual examples. Thus, it becomes necessary to exhaustively enumerate weak learners in Algorithm 3. While enumeration of weak learners is possible in the case of decision stumps, it poses serious difﬁculties in the case of weak learners such as decision trees, ridge regression, etc. Thus, VadaBoost is the more versatile boosting algorithm for sample variance penalization. 2The cost which VadaBoost minimizes at λ = 0 is the squared cost of AdaBoost, we do not square it again. 6 Table 1: Mean and standard errors with decision stump as the weak learner. Dataset AdaBoost EBBoost VadaBoost RLP-Boost RQP-Boost a5a 16.15 ± 0.1 16.05 ± 0.1 16.22 ± 0.1 16.21 ± 0.1 16.04 ± 0.1 abalone 21.64 ± 0.2 21.52 ± 0.2 21.63 ± 0.2 22.29 ± 0.2 21.79 ± 0.2 image 3.37 ± 0.1 3.14 ± 0.1 3.14 ± 0.1 3.18 ± 0.1 3.09 ± 0.1 mushrooms 0.02 ± 0.0 0.02 ± 0.0 0.01 ± 0.0 0.01 ± 0.0 0.00 ± 0.0 musk 3.84 ± 0.1 3.51 ± 0.1 3.59 ± 0.1 3.60 ± 0.1 3.41 ± 0.1 mnist09 0.89 ± 0.0 0.85 ± 0.0 0.84 ± 0.0 0.98 ± 0.0 0.88 ± 0.0 mnist14 0.64 ± 0.0 0.58 ± 0.0 0.60 ± 0.0 0.68 ± 0.0 0.63 ± 0.0 mnist27 2.11 ± 0.1 1.86 ± 0.1 2.01 ± 0.1 2.06 ± 0.1 1.95 ± 0.1 mnist38 4.45 ± 0.1 4.12 ± 0.1 4.32 ± 0.1 4.51 ± 0.1 4.25 ± 0.1 mnist56 2.79 ± 0.1 2.56 ± 0.1 2.62 ± 0.1 2.77 ± 0.1 2.72 ± 0.1 ringnorm 13.16 ± 0.6 11.74 ± 0.6 12.46 ± 0.6 13.02 ± 0.6 12.86 ± 0.6 spambase 5.90 ± 0.1 5.64 ± 0.1 5.78 ± 0.1 5.81 ± 0.1 5.75 ± 0.1 splice 8.83 ± 0.2 8.33 ± 0.1 8.48 ± 0.1 8.55 ± 0.2 8.47 ± 0.1 twonorm 3.16 ± 0.1 2.98 ± 0.1 3.09 ± 0.1 3.29 ± 0.1 3.07 ± 0.1 w4a 2.60 ± 0.1 2.38 ± 0.1 2.50 ± 0.1 2.44 ± 0.1 2.36 ± 0.1 waveform 10.99 ± 0.1 10.96 ± 0.1 10.75 ± 0.1 10.95 ± 0.1 10.60 ± 0.1 wine 23.62 ± 0.2 23.52 ± 0.2 23.41 ± 0.1 24.16 ± 0.1 23.61 ± 0.1 wisc 5.32 ± 0.3 4.38 ± 0.2 5.00 ± 0.2 4.96 ± 0.3 4.72 ± 0.3 Table 2: Mean and standard errors with CART as the weak learner. Dataset AdaBoost VadaBoost RLP-Boost RQP-Boost a5a 17.59 ± 0.2 17.16 ± 0.1 18.24 ± 0.1 17.99 ± 0.1 abalone 21.87 ± 0.2 21.30 ± 0.2 22.16 ± 0.2 21.84 ± 0.2 image 1.93 ± 0.1 1.98 ± 0.1 1.99 ± 0.1 1.95 ± 0.1 mushrooms 0.01 ± 0.0 0.01 ± 0.0 0.02 ± 0.0 0.01 ± 0.0 musk 2.36 ± 0.1 2.07 ± 0.1 2.40 ± 0.1 2.29 ± 0.1 mnist09 0.73 ± 0.0 0.72 ± 0.0 0.76 ± 0.0 0.71 ± 0.0 mnist14 0.52 ± 0.0 0.50 ± 0.0 0.55 ± 0.0 0.52 ± 0.0 mnist27 1.31 ± 0.0 1.24 ± 0.0 1.32 ± 0.0 1.29 ± 0.0 mnist38 1.89 ± 0.1 1.72 ± 0.1 1.88 ± 0.1 1.87 ± 0.1 mnist56 1.23 ± 0.1 1.17 ± 0.0 1.20 ± 0.0 1.19 ± 0.1 ringnorm 7.94 ± 0.4 7.78 ± 0.4 8.60 ± 0.5 7.84 ± 0.4 spambase 6.14 ± 0.1 5.76 ± 0.1 6.25 ± 0.1 6.03 ± 0.1 splice 4.02 ± 0.1 3.67 ± 0.1 4.03 ± 0.1 3.97 ± 0.1 twonorm 3.40 ± 0.1 3.27 ± 0.1 3.50 ± 0.1 3.38 ± 0.1 w4a 2.90 ± 0.1 2.90 ± 0.1 2.90 ± 0.1 2.90 ± 0.1 waveform 11.09 ± 0.1 10.59 ± 0.1 11.11 ± 0.1 10.82 ± 0.1 wine 21.94 ± 0.2 21.18 ± 0.2 22.44 ± 0.2 22.18 ± 0.2 wisc 4.61 ± 0.2 4.18 ± 0.2 4.63 ± 0.2 4.37 ± 0.2 6 Experiments In this section, we evaluate the empirical performance of the VadaBoost algorithm with respect to several other algorithms. The primary purpose of our experiments is to compare sample variance penalization versus empirical risk minimization and to show that we can efﬁciently perform sample variance penalization for weak learners beyond decision stumps. We compared VadaBoost against EBBoost, AdaBoost, regularized LP and QP boost algorithms [7]. All the algorithms except AdaBoost have one extra parameter to tune. Experiments were performed on benchmark datasets that have been previously used in [10]. These datasets include a variety of tasks including all digits from the MNIST dataset. Each dataset was divided into three parts: 50% for training, 25% for validation and 25% for test. The total number of examples was restricted to 5000 in the case of MNIST and musk datasets due to computational restrictions of solving LP/QP. The ﬁrst set of experiments use decision stumps as the weak learners. The second set of experiments used Classiﬁcation and Regression Trees or CART [3] as weak learners. A standard MATLAB implementation of CART was used without modiﬁcation. For all the datasets, in both experiments, 7 AdaBoost, VadaBoost and EBBoost (in the case of stumps) were run until there was no drop in the error rate on the validation for the next 100 consecutive iterations. The values of the parameters for VadaBoost and EBBoost were chosen to minimize the validation error upon termination. RLP-Boost and RQP-Boost were given the predictions obtained by AdaBoost. Their regularization parameter was also chosen to minimize the error rate on the validation set. Once the parameter values were ﬁxed via the validation set, we noted the test set error corresponding to that parameter value. The entire experiment was repeated 50 times by randomly selecting train, test and validation sets. The numbers reported here are average from these runs. The results for the decision stump and CART experiments are reported in Tables 1 and 2. For each dataset, the algorithm with the best percentage test error is represented by a dark shaded cell. All lightly shaded entries in a row denote results that are not signiﬁcantly different from the minimum error (according to a paired t-test at a 1% signiﬁcance level). With decision stumps, both EBBoost and VadaBoost have comparable performance and signiﬁcantly outperform AdaBoost. With CART as the weak learner, VadaBoost is once again signiﬁcantly better than AdaBoost. 100 200 300 400 500 600 700 10 0 10 1 10 2 10 3 10 4 10 5 Iteration cost + 1 AdaBoost EBBoost λ=0.5 VadaBoost λ=0 VadaBoost λ=0.5 Figure 2: 1+ cost vs the number of iterations. We gave a guarantee on the number of iterations required in the worst case for Vadaboost (which approximately matches the AdaBoost cost (squared) in Theorem 5). An assumption in that theorem was that the error rate of each weak learner was ﬁxed. However, in practice, the error rates of the weak learners are not constant over the iterations. To see this behavior in practice, we have shown the results with the MNIST 3 versus 8 classiﬁcation experiment. In ﬁgure 2 we show the cost (plus 1) for each algorithm (the AdaBoost cost has been squared) versus the number of iterations using a logarithmic scale on the Y-axis. Since at λ = 0, EBBoost reduces to AdaBoost, we omit its plot at that setting. From the ﬁgure, it can be seen that the number of iterations required by VadaBoost is roughly twice the number of iterations required by AdaBoost. At λ = 0.5, there is only a minor difference in the number of iterations required by EBBoost and VadaBoost. 7 Conclusions This paper identiﬁed a key weakness in the EBBoost algorithm and proposed a novel algorithm that efﬁciently overcomes the limitation to enumerable weak learners. VadaBoost reduces a well motivated cost by iteratively minimizing an upper bound which, unlike EBBoost, allows the boosting method to handle any weak learner by estimating weights on the data. The update rule of VadaBoost has a simplicity that is reminiscent of AdaBoost. Furthermore, despite the use of an upper bound, the novel boosting method remains efﬁcient. Even when the bound is at its loosest, the number of iterations required by VadaBoost is a small constant factor more than the number of iterations required by AdaBoost. Experimental results showed that VadaBoost outperforms AdaBoost in terms of classiﬁcation accuracy and efﬁciently applying to any family of weak learners. The effectiveness of boosting has been explained via margin theory [9] though it has taken a number of years to settle certain open questions [8]. Considering the simplicity and effectiveness of VadaBoost, one natural future research direction is to study the margin distributions it obtains. Another future research direction is to design efﬁcient sample variance penalization algorithms for other problems such as multi-class classiﬁcation, ranking, and so on. Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. 1117631, by a Google Research Award, and by the Department of Homeland Security under Grant No. N66001-09-C-0080. 8 References [1] J-Y. Audibert, R. Munos, and C. Szepesv´ari. Tuning bandit algorithms in stochastic environments. In ALT, 2007. [2] P. L. Bartlett, M. I. Jordan, and J. D. McAuliffe. Convexity, classiﬁcation, and risk bounds. Journal of the American Statistical Association, 101(473):138–156, 2006. [3] L. Breiman, J.H. Friedman, R.A. Olshen, and C.J. Stone. Classiﬁcation and Regression Trees. Chapman and Hall, New York, 1984. [4] Y. Freund and R. 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] A. Maurer and M. Pontil. Empirical Bernstein bounds and sample variance penalization. In COLT, 2009. [6] V. Mnih, C. Szepesv´ari, and J-Y. Audibert. Empirical Bernstein stopping. In COLT, 2008. [7] G. Raetsch, T. Onoda, and K.-R. Muller. Soft margins for AdaBoost. Machine Learning, 43:287–320, 2001. [8] L. Reyzin and R. Schapire. How boosting the margin can also boost classiﬁer complexity. In ICML, 2006. [9] R. E. Schapire, Y. Freund, P. L. Bartlett, and W. S. Lee. Boosting the margin: a new explanation for the effectiveness of voting methods. Annals of Statistics, 26(5):1651–1686, 1998. [10] P. K. Shivaswamy and T. Jebara. Empirical Bernstein boosting. In AISTATS, 2010. [11] V. Vapnik. The Nature of Statistical Learning Theory. Springer, New York, NY, 1995. 9	2011	90
4,442	Autonomous Learning of Action Models for Planning Neville Mehta Prasad Tadepalli Alan Fern School of Electrical Engineering and Computer Science Oregon State University, Corvallis, OR 97331, USA. {mehtane,tadepall,afern}@eecs.oregonstate.edu Abstract This paper introduces two new frameworks for learning action models for planning. In the mistake-bounded planning framework, the learner has access to a planner for the given model representation, a simulator, and a planning problem generator, and aims to learn a model with at most a polynomial number of faulty plans. In the planned exploration framework, the learner does not have access to a problem generator and must instead design its own problems, plan for them, and converge with at most a polynomial number of planning attempts. The paper reduces learning in these frameworks to concept learning with one-sided error and provides algorithms for successful learning in both frameworks. A speciﬁc family of hypothesis spaces is shown to be efﬁciently learnable in both the frameworks. 1 Introduction Planning research typically assumes that the planning system is provided complete and correct models of the actions. However, truly autonomous agents must learn these models. Moreover, model learning, planning, and plan execution must be interleaved, because agents need to plan long before perfect models are learned. This paper formulates and analyzes the learning of deterministic action models used in planning for goal achievement. It has been shown that deterministic STRIPS actions with a constant number of preconditions can be learned from raw experience with at most a polynomial number of plan prediction mistakes [8]. In spite of this positive result, compact action models in fully observable, deterministic action models are not always efﬁciently learnable. For example, action models represented as arbitrary Boolean functions are not efﬁciently learnable under standard cryptographic assumptions such as the hardness of factoring [2]. Learning action models for planning is different from learning an arbitrary function from states and actions to next states, because one can ignore modeling the effects of some actions in certain contexts. For example, most people who drive do not ever learn a complete model of the dynamics of their vehicles; while they might accurately know the stopping distance or turning radius, they could be oblivious to many aspects that an expert auto mechanic is comfortable with. To capture this intuition, we introduce the concept of an adequate model, that is, a model that is sound and sufﬁciently complete for planning for a given class of goals. When navigating a city, any spanning tree of the transportation network connecting the places of interest would be an adequate model. We deﬁne two distinct frameworks for learning adequate models for planning and then characterize sufﬁcient conditions for success in these frameworks. In the mistake-bounded planning (MBP) framework, the goal is to continually solve user-generated planning problems while learning action models and guarantee at most a polynomial number of faulty plans or mistakes. We assume that in addition to the problem generator, the learner has access to a sound and complete planner and a simulator (or the real world). We also introduce a more demanding planned exploration (PLEX) framework, where the learner needs to generate its own problems to reﬁne its action model. This requirement translates to an experiment-design problem, where the learner needs to design problems in a goal language to reﬁne the action models. 1 The MBP and PLEX frameworks can be reduced to over-general query learning, concept learning with strictly one-sided error, where the learner is only allowed to make false positive mistakes [7]. This is ideally suited for the autonomous learning setting in which there is no oracle who can provide positive examples of plans or demonstrations, but negative examples are observed when the agent’s plans fail to achieve their goals. We introduce mistake-bounded and exact learning versions of this learning framework and show that they are strictly more powerful than the recently introduced KWIK framework [4]. We view an action model as a set of state-action-state transitions and ensure that the learner always maintains a hypothesis which includes all transitions in some adequate model. Thus, a sound plan is always in the learner’s search space, while it may not always be generated. As the learner gains more experience in generating plans, executing them on the simulator, and receiving observations, the hypothesis is incrementally reﬁned until an adequate model is discovered. To ground our analysis, we show that a general family of hypothesis spaces is learnable in polynomial time in the two frameworks given appropriate goal languages. This family includes a generalization of propositional STRIPS operators with conditional effects. 2 Over-General Query Learning We ﬁrst introduce a variant of a concept-learning framework that serves as formal underpinning of our model-learning frameworks. This variant is motivated by the principle of “optimism under uncertainty”, which is at the root of several related algorithms in reinforcement learning [1, 3]. A concept is a set of instances. An hypothesis space H is a set of strings or hypotheses, each of which represents a concept. The size of the concept is the length of the smallest hypothesis that represents it. Without loss of generality, H can be structured as a (directed acyclic) generalization graph, where the nodes correspond to sets of equivalent hypotheses representing a concept and there is a directed edge from node n1 to node n2 if and only if the concept at n1 is strictly more general than (a strict superset of) that at n2. Deﬁnition 2.1. The height of H is a function of n and is the length of the longest path from a root node to any node representing concepts of size n in the generalization graph of H. Deﬁnition 2.2. A hypothesis h is consistent with a set of negative examples Z if h∩Z = ∅. Given a set of negative examples Z consistent with a target hypothesis h, the version space of action models is the subset of all hypotheses in H that are consistent with Z and is denoted as M(Z). Deﬁnition 2.3. H is well-structured if, for any negative example set Z which has a consistent target hypothesis in H, the version space M(Z) contains a most general hypothesis mgh(Z). Further, H is efﬁciently well-structured if there exists an algorithm that can compute mgh(Z ∪{z}) from mgh(Z) and a new example z in time polynomial in the size of mgh(Z) and z. Lemma 2.1. Any ﬁnite hypothesis space H is well-structured if and only if it is closed under union. Proof. (If) Let Z be a set of negative examples and let H0 = S h∈M(Z) h represent the unique union of all concepts represented by hypotheses in M(Z). Because H is closed under union and ﬁnite, H0 must be in H. If ∃z ∈H0 ∩Z, then z ∈h∩Z for some h ∈M(Z). This is a contradiction, because all h ∈M(Z) are consistent with Z. Consequently, H0 is consistent with Z, and is in M(Z). It is more general than (is a superset of) every other hypothesis in M(Z) because it is their union. (Only if) Let h1, h2 be any two hypotheses in H and Z be the set of all instances not included in either h1 and h2. Both h1 and h2 are consistent with examples in Z. As H is well-structured, mgh(Z) must also be in the version space M(Z), and consequently in H. However, mgh(Z) = h1 ∪h2 because it cannot include any element without h1 ∪h2 and must include all elements within. Hence, h1 ∪h2 is in H, which makes it closed under union. In the over-general query (OGQ) framework, the teacher selects a target concept c ∈H. The learner outputs a query in the form of a hypothesis h ∈H, where h must be at least as general as c. The teacher responds with yes if h ≡c and the episode ends; otherwise, the teacher gives a counterexample x ∈h −c. The learner then outputs a new query, and the cycle repeats. Deﬁnition 2.4. A hypothesis space is OGQ-learnable if there exists a learning algorithm for the OGQ framework that identiﬁes the target c with the number of queries and total running time that is polynomial in the size of c and the size of the largest counterexample. Theorem 1. H is learnable in the OGQ framework if and only if H is efﬁciently well-structured and its height is a polynomial function. 2 Proof. (If) If H is efﬁciently well-structured, then the OGQ learner can always output the mgh, guaranteed to be more general than the target concept, in polynomial time. Because the maximum number of hypothesis reﬁnements is bounded by the polynomial height of H, it is learnable in the OGQ framework. (Only if) If H is not well-structured, then ∃h1, h2 ∈H, h1 ∪h2 /∈H. The teacher can delay picking its target concept, but always provide counterexamples from outside both h1 and h2. At some point, these counterexamples will force the learner to choose between h1 or h2, because their union is not in the hypothesis space. Once the learner makes its choice, the teacher can choose the other hypothesis as its target concept c, resulting in the learner’s hypothesis not being more general than c. If H is not efﬁciently well-structured, then there exists Z and z such that computing mgh(Z ∪{z}) from mgh(Z) and a new example z cannot be done in polynomial time. If the teacher picks mgh(Z ∪{z}) as the target concept and only provides counterexamples from Z ∪{z}, then the learner cannot have polynomial running time. Finally, the teacher can always provide counterexamples that forces the learner to take the longest path in H’s generalization graph. Thus, if H does not have polynomial height, then the number of queries will not be polynomial. 2.1 A Comparison of Learning Frameworks In order to compare the OGQ framework to other learning frameworks, we ﬁrst deﬁne the overgeneral mistake-bounded (OGMB) learning framework, in which the teacher selects a target concept c from H and presents an arbitrary instance x from the instance space to the learner for a prediction. An inclusion mistake is made when the learner predicts x ∈c although x /∈c; an exclusion mistake is made when the learner predicts x /∈c although x ∈c. The teacher presents the true label to the learner if a mistake is made, and then presents the next instance to the learner, and so on. Deﬁnition 2.5. A hypothesis space is OGMB-learnable if there exists a learning algorithm for the OGMB framework that never makes any exclusion mistakes and its number of inclusion mistakes and the running time on each instance are both bounded by polynomial functions of the size of the target concept and the size of the largest instance seen by the learner. In the following analysis, we let the name of a framework denote the set of hypothesis spaces learnable in that framework. Theorem 2. OGQ ⊊OGMB. Proof. We can construct an OGMB learner from the OGQ learner as follows. When the OGQ learner makes a query h, we use h to make predictions for the OGMB learner. As h is guaranteed to be over-general, it never makes an exclusion mistake. Any instance x on which it makes an inclusion mistake must be in h −c and this is returned to the OGQ learner. The cycle repeats with the OGQ learner providing a new query. Because the OGQ learner makes only a polynomial number of queries and takes polynomial time for query generation, the simulated OGMB learner makes only a polynomial number of mistakes and runs in at most polynomial time per instance. The converse does not hold in general because the queries of the OGQ learner are restricted to be “proper”, that is, they must belong to the given hypothesis space. While the OGMB learner can maintain the version space of all consistent hypotheses of a polynomially-sized hypothesis space, the OGQ learner can only query with a single hypothesis and there may not be any hypothesis that is guaranteed to be more general than the target concept. If the learner is allowed to ask queries outside H, such as queries of the form h1 ∪· · · ∪hn for all hi in the version space, then over-general learning is possible. In general, if the learner is allowed to ask about any polynomially-sized, polynomial-time computable hypothesis, then it is as powerful as OGMB, because it can encode the computation of the OGMB learner inside a polynomial-sized circuit and query with that as the hypothesis. We call this the OGQ+ framework and claim the following theorem (the proof is straightforward). Theorem 3. OGQ+ = OGMB. The Knows-What-It-Knows (KWIK) learning framework [4] is similar to the OGMB framework with one key difference: it does not allow the learner to make any prediction when it does not know the correct answer. In other words, the learner either makes a correct prediction or simply abstains from making a prediction and gets the true label from the teacher. The number of abstentions is 3 bounded by a polynomial in the target size and the largest instance size. The set of hypothesis spaces learnable in the mistake-bound (MB) framework is a strict subset of that learnable in the probably-approximately-correct (PAC) framework [5], leading to the following result. Theorem 4. KWIK ⊊OGMB ⊊MB ⊊PAC. Proof. OGMB ⊊MB: Every hypothesis space that is OGMB-learnable is MB-learnable because the OGMB learner is additionally constrained to not make an exclusion mistake. However, every MBlearnable hypothesis space is not OGMB-learnable. Consider the hypothesis space of conjunctions of n Boolean literals (positive or negative). A single exclusion mistake is sufﬁcient for an MB learner to learn this hypothesis space. In contrast, after making an inclusion mistake, the OGMB learner can only exclude that example from the candidate set. As there is exactly one positive example, this could force the OGMB learner to make an exponential number of mistakes (similar to guessing an unknown password). KWIK ⊊OGMB: If a concept class is KWIK-learnable, it is also OGMB-learnable — when the KWIK learner does not know the true label, the OGMB learner simply predicts that the instance is positive and gets corrected if it is wrong. However, every OGMB-learnable hypothesis space is not KWIK-learnable. Consider the hypothesis space of disjunctions of n Boolean literals. The OGMB learner begins with a disjunction over all possible literals (both positive and negative) and hence predicts all instances as positive. A single inclusion mistake is sufﬁcient for the OGMB learner to learn this hypothesis space. On the other hand, the teacher can supply the KWIK learner with an exponential number of positive examples, because the KWIK learner cannot ever know that the target does not include all possible instances; this implies that the number of abstentions is not polynomially bounded. This theorem demonstrates that KWIK is too conservative a framework for model learning — any prediction that might be a mistake is disallowed. This makes it impossible to learn even simple concept classes such as pure disjunctions. 3 Planning Components A factored planning domain P is a tuple (V, D, A, T), where V = {v1, . . . , vn} is the set of variables, D is the domain of the variables in V , and A is the set of actions. S = Dn represents the state space and T ⊂S ×A×S is the transition relation, where (s, a, s′) ∈T signiﬁes that taking action a in state s results in state s′. As we only consider learning deterministic action models, the transition relation is in fact a function, although the learner’s hypothesis space may include nondeterministic models. The domain parameters, n, \|D\|, and \|A\|, characterize the size of P and are implicit in all claims of complexity in the rest of this paper. Deﬁnition 3.1. An action model is a relation M ⊆S × A × S. A planning problem is a pair (s0, g), where s0 ∈S and the goal condition g is an expression chosen from a goal language G and represents a set of states in which it evaluates to true. A state s satisﬁes a goal g if and only if g is true in s. Given a planning problem (s0, g), a plan is a sequence of states and actions s0, a1, . . . , ap, sp, where the state sp satisﬁes the goal g. The plan is sound with respect to (M, g) if (si−1, ai, si) ∈M for 1 ≤i ≤p. Deﬁnition 3.2. A planner for the hypothesis space and goal language pair (H, G) is an algorithm that takes M ∈H and (s0, g ∈G) as inputs and outputs a plan or signals failure. It is sound with respect to (H, G) if, given any M and (s0, g), it produces a sound plan with respect to (M, g) or signals failure. It is complete with respect to (H, G) if, given any M and (s0, g), it produces a sound plan whenever one exists with respect to (M, g). We generalize the deﬁnition of soundness from its standard usage in the literature in order to apply to nondeterministic action models, where the nondeterminism is “angelic” — the planner can control the outcome of actions when multiple outcomes are possible according to its model [6]. One way to implement such a planner is to do forward search through all possible action and outcome sequences and return an action sequence if it leads to a goal under some outcome choices. Our analysis is agnostic to plan quality or plan length and applies equally well to suboptimal planners. This is motivated by the fact that optimal planning is hard for most domains, but suboptimal planning such as hierarchical planning can be quite efﬁcient and practical. 4 Deﬁnition 3.3. A planning mistake occurs if either the planner signals failure when a sound plan exists with respect to the transition function T or when the plan output by the planner is not sound with respect to T. Deﬁnition 3.4. Let P be a planning domain and G be a goal language. An action model M is adequate for G in P if M ⊆T and the existence of a sound plan with respect to (T, g ∈G) implies the existence of a sound plan with respect to (M, g). H is adequate for G if ∃M ∈H such that M is adequate for G. An adequate model may be partial or incomplete in that it may not include every possible transition in the transition function T. However, the model is sufﬁcient to produce a sound plan with respect to (T, g) for every goal g in the desired language. Thus, the more limited the goal language, the more incomplete the adequate model can be. In the example of a city map, if the goal language excludes certain locations, then the spanning tree could exclude them as well, although not necessarily so. Deﬁnition 3.5. A simulator of the domain is always situated in the current state s. It takes an action a as input, transitions to the state s′ resulting from executing a in s, and returns the current state s′. Given a goal language G, a problem generator generates an arbitrary problem (s0, g ∈G) and sets the state of the simulator to s0. 4 Mistake-Bounded Planning Framework This section constructs the MBP framework that allows learning and planning to be interleaved for user-generated problems. It actualizes the teacher of the OGQ framework by combining a problem generator, a planner, and a simulator, and interfaces with the OGQ learner to learn action models as hypotheses over the space of possible state transitions for each action. It turns out that the one-sided mistake property is needed for autonomous learning because the learner can only learn by generating plans and observing the results; if the learner ever makes an exclusion error, there is no guarantee of ﬁnding a sound plan even when one exists and the learner cannot recover from such mistakes. Deﬁnition 4.1. Let G be a goal language such that H is adequate for it. H is learnable in the MBP framework if there exists an algorithm A that interacts with a problem generator over G, a sound and complete planner with respect to (H, G), and a simulator of the planning domain P, and outputs a plan or signals failure for each planning problem while guaranteeing at most a polynomial number of planning mistakes. Further, A must respond in time polynomial in the domain parameters and the length of the longest plan generated by the planner, assuming that a call to the planner, simulator, or problem generator takes O(1) time. The goal language is picked such that the hypothesis space is adequate for it. We cannot bound the time for the convergence of A, because there is no limit on when the mistakes are made. Theorem 5. H is learnable in the MBP framework if H is OGQ-learnable. Algorithm 1 MBP LEARNING SCHEMA Input: Goal language G 1: M ←OGQ-LEARNER() // Initial query 2: loop 3: (s, g) ←PROBLEMGENERATOR(G) 4: plan ←PLANNER(M, (s, g)) 5: if plan ̸= false then 6: for (ˆs, a, ˆs′) in plan do 7: s′ ←SIMULATOR(a) 8: if s′ ̸= ˆs′ then 9: M ←OGQ-LEARNER((s, a, ˆs′)) 10: print mistake 11: break 12: s ←s′ 13: if no mistake then 14: print plan Proof. Algorithm 1 is a general schema for action model learning in the MBP framework. The model M begins with the initial query from OGQ-LEARNER. PROBLEMGENERATOR provides a planning problem and initializes the current state of SIMULATOR. Given M and the planning problem, PLANNER always outputs a plan if one exists because H is adequate for G (it contains a “target” adequate model) and M is at least as general as every adequate model. If PLANNER signals failure, then there is no plan for it. Otherwise, the plan is executed through SIMULATOR until an observed transition conﬂicts with the predicted transition. If such a transition is found, it is supplied to OGQ-LEARNER and M is updated with the next query; otherwise, the plan is output. If H is OGQ-learnable, then OGQ-LEARNER will only be called a polynomial number of times, every call taking polynomial time. As the number of planning mistakes is 5 polynomial and every response of Algorithm 1 is polynomial in the runtime of OGQ-LEARNER and the length of the longest plan, H is learnable in the MBP framework. The above result generalizes the work on learning STRIPS operator models from raw experience (without a teacher) in [8] to arbitrary hypotheses spaces by identifying sufﬁciency conditions. (A family of hypothesis spaces considered later in this paper subsumes propositional STRIPS by capturing conditional effects.) It also clariﬁes the notion of an adequate model, which can be much simpler than the true transition model, and the inﬂuence of the goal language on the complexity of learning action models. 5 Planned Exploration Framework The MBP framework is appropriate when mistakes are permissible on user-given problems as long as their total number is limited and not for cases where no mistakes are permitted after the training period. In the planned exploration (PLEX) framework, the agent seeks to learn an action model for the domain without an external problem generator by generating planning problems for itself. The key issue here is to generate a reasonably small number of planning problems such that solving them would identify a deterministic action model. Learning a model in the PLEX framework involves knowing where it is deﬁcient and then planning to reach states that are informative, which entails formulating planning problems in a goal language. This framework provides a polynomial sample convergence guarantee which is stronger than a polynomial mistake bound of the MBP framework. Without a problem generator that can change the simulator’s state, it is impossible for the simulator to transition freely between strongly connected components (SCCs) of the transition graph. Hence, we make the assumption that the transition graph is a disconnected union of SCCs and require only that the agent learn the model for a single SCC that contains the initial state of the simulator. Deﬁnition 5.1. Let P be a planning domain whose transition graph is a union of SCCs. (H, G) is learnable in the PLEX framework if there exists an algorithm A that interacts with a sound and complete planner with respect to (H, G) and the simulator for P and outputs a model M ∈H that is adequate for G within the SCC that contains the initial state s0 of the simulator after a polynomial number of planning attempts. Further, A must run in polynomial time in the domain parameters and the length of the longest plan output by the planner, assuming that every call to the planner and the simulator takes O(1) time. A key step in planned exploration is designing appropriate planning problems. We call these experiments because the goal of solving these problems is to disambiguate nondeterministic action models. In particular, the agent tries to reach an informative state where the current model is nondeterministic. Deﬁnition 5.2. Given a model M, the set of informative states is I(M) = {s : (s, a, s′), (s, a, s′′) ∈ M ∧s′ ̸= s′′}, where a is said to be informative in s. Deﬁnition 5.3. A set of goals G is a cover of a set of states R if S g∈G{s : s satisﬁes g} = R. Given the goal language G and a model M, the problem of experiment design is to ﬁnd a set of goals G ⊆G such that the sets of states that satisfy the goals in G collectively cover all informative states I(M). If it is possible to plan to achieve one of these goals, then either the plan passes through a state where the model is nondeterministic or it executes successfully and the agent reaches the ﬁnal goal state; in either case, an informative action can be executed and the observed transition is used to reﬁne the model. If none of the goals in G can be successfully planned for, then no informative states for that action are reachable. We formalize these intuitions below. Deﬁnition 5.4. The width of (H, G) is deﬁned as max M∈H min G⊆G:G is a cover of I(M) \|G\|, where minG \|G\| = ∞if there is no G ⊆G to cover a nonempty I(M). Deﬁnition 5.5. (H, G) permits efﬁcient experiment design if, for any M ∈H, 1⃝there exists an algorithm (EXPERIMENTDESIGN) that takes M and G as input and outputs a polynomial-sized cover of I(M) in polynomial time and 2⃝there exists an algorithm (INFOACTIONSTATES) that takes M and a state s as input and outputs an informative action and two (distinct) predicted next states according to M in polynomial time. If (H, G) permits efﬁcient experiment design, then it has polynomial width because no algorithm can always guarantee to output a polynomial-sized cover otherwise. 6 Theorem 6. (H, G) is learnable in the PLEX framework if it permits efﬁcient experiment design, and H is adequate for G and is OGQ-learnable. Algorithm 2 PLEX LEARNING SCHEMA Input: Initial state s, goal language G Output: Model M 1: M ←OGQ-LEARNER() // Initial query 2: loop 3: G ←EXPERIMENTDESIGN(M, G) 4: if G = ∅then 5: return M 6: for g ∈G do 7: plan ←PLANNER(M, (s, g)) 8: if plan ̸= false then 9: break 10: if plan = false then 11: return M 12: for (ˆs, a, ˆs′) in plan do 13: s′ ←SIMULATOR(a) 14: s ←s′ 15: if s′ ̸= ˆs′ then 16: M ←OGQ-LEARNER((s, a, ˆs′)) 17: break 18: if M has not been updated then 19: (a, ˆS′) ←INFOACTIONSTATES(M, s) 20: s′ ←SIMULATOR(a) 21: M ←OGQ-LEARNER((s, a, ˆs′ ∈ˆS′ −{s′})) 22: s ←s′ 23: return M Proof. Algorithm 2 is a general schema for action model learning in the PLEX framework. The model M begins with the initial query from OGQ-LEARNER. Given M and G, EXPERIMENTDESIGN computes a polynomial-sized cover G. If G is empty, then the model cannot be reﬁned further; otherwise, given M and a goal g ∈G, PLANNER may signal failure if either no state satisﬁes g or states satisfying g are not reachable from the current state of the simulator. If PLANNER signals failure on all of the goals, then none of the informative states are reachable and M cannot be reﬁned further. If PLANNER does output a plan, then the plan is executed through SIMULATOR until an observed transition conﬂicts with the predicted transition. If such a transition is found, it is supplied to OGQ-LEARNER and M is updated with the next query. If the plan executes successfully, then INFOACTIONSTATES provides an informative action with the corresponding set of two resultant states according to M; OGQ-LEARNER is supplied with the transition of the goal state, the informative action, and the incorrectly predicted next state, and M is updated with the new query. A new cover is computed every time M is updated, and the process continues until all experiments are exhausted. If (H, G) permits efﬁcient experiment design, then every cover can be computed in polynomial time and INFOACTIONSTATES is efﬁcient. If H is OGQ-learnable, then OGQ-LEARNER will only be called a polynomial number of times and it can output a new query in polynomial time. As the number of failures per successful plan is bounded by a polynomial in the width w of (H, G), the total number of calls to PLANNER is polynomial. Further, as the innermost loop of Algorithm 2 is bounded by the longest length l of a plan, its running time is a polynomial in the domain parameters and l. Thus, (H, G) is learnable in the PLEX framework. 6 A Hypothesis Family for Action Modeling This section proves the learnability of a hypothesis-space family for action modeling in the MBP and PLEX frameworks. Let U = {u1, u2, . . .} be a polynomial-sized set of polynomially computable basis hypotheses (polynomial in the relevant parameters), where ui represents a deterministic set of transition tuples. Let Power(U) = {S u∈H u : H ⊆U} and Pairs(U) = {u1 ∪u2 : u1, u2 ∈U}. Lemma 6.1. Power(U) is OGQ-learnable. Proof. Power(U) is efﬁciently well-structured, because it is closed under union by deﬁnition and the new mgh can be computed by removing any basis hypotheses that are not consistent with the counterexample; this takes polynomial time as U is of polynomial size. At the root of the generalization graph of Power(U) is the hypothesis S u∈U u and at the leaf is the empty hypothesis. Because U is of polynomial size and the longest path from the root to the leaf involves removing a single component at a time, the height of Power(U) is polynomial. Lemma 6.2. Power(U) is learnable in the MBP framework. Proof. This follows from Lemma 6.1 and Theorem 5. 7 Lemma 6.3. For any goal language G, (Power(U), G) permits efﬁcient experiment design if (Pairs(U), G) permits efﬁcient experiment design. Proof. Any informative state for a hypothesis in Power(U) is an informative state for some hypothesis in Pairs(U), and vice versa. Hence, a cover for (Pairs(U), G) would be a cover for (Power(U), G). Consequently, if (Pairs(U), G) permits efﬁcient experiment design, then the efﬁcient algorithms EXPERIMENTDESIGN and INFOACTIONSTATES are directly applicable to (Power(U), G). Lemma 6.4. For any goal language G, (Power(U), G) is learnable in the PLEX framework if (Pairs(U), G) permits efﬁcient experiment design and Power(U) is adequate for G. Proof. This follows from Lemmas 6.1 and 6.3, and Theorem 6. We now deﬁne a hypothesis space that is a concrete member of the family. Let an action production r be deﬁned as “act : pre →post”, where act(r) is an action and the precondition pre(r) and postcondition post(r) are conjunctions of “variable = value” literals. Deﬁnition 6.1. A production r is triggered by a transition (s, a, s′) if s satisﬁes the precondition pre(r) and a = act(r). A production r is consistent with (s, a, s′) if either 1⃝r is not triggered by (s, a, s′) or 2⃝s′ satisﬁes the post(r) and all variables not mentioned in post(r) have the same values in both s and s′. A production represents the set of all consistent transitions that trigger it. All the variables in pre(r) must take their speciﬁed values in a state to trigger r; when r is triggered, post(r) deﬁnes the values in the next state. An example of an action production is “Do : v1 = 0, v2 = 1 →v1 = 2, v3 = 1”. It is triggered only when the Do action is executed in a state in which v1 = 0 and v2 = 1, and deﬁnes the value of v1 to be 2 and v3 to be 1 in the next state, with all other variables staying unchanged. Let k-SAP be the hypothesis space of models represented by a set of action productions (SAP) with no more than k variables per production. If U is the set of productions, then \|U\| = O \|A\| Pk i=1 n i (\|D\| + 1)2i = O(\|A\|nk\|D\|2k), because a production can have one of \|A\| actions, up to k relevant variables ﬁguring on either side of the production, and each variable set to a value in its domain. As U is of polynomial size, k-SAP is an instance of the family of basis action models. Moreover, if Conj is the goal language consisting of all goals that can be expressed as conjunctions of “variable = value” literals, then (Pairs(k-SAP), Conj) permits efﬁcient experiment design. Lemma 6.5. (k-SAP, Conj) is learnable in the PLEX framework if k-SAP is adequate for Conj. 7 Conclusion The main contributions of the paper are the development of the MBP and PLEX frameworks for learning action models and the characterization of sufﬁcient conditions for efﬁcient learning in these frameworks. It also provides results on learning a family of hypothesis spaces that is, in some ways, more general than standard action modeling languages. For example, unlike propositional STRIPS operators, k-SAP captures the conditional effects of actions. While STRIPS-like languages served us well in planning research by creating a common useful platform, they are not designed from the point of view of learnability or planning efﬁciency. Many domains such as robotics and real-time strategy games are not amenable to such clean and simple action speciﬁcation languages. This suggests an approach in which the learner considers increasingly complex models as dictated by its planning needs. For example, the model learner might start with small values of k in k-SAP and then incrementally increase k until a value is found that is adequate for the goals encountered. In general, this motivates a more comprehensive framework in which planning and learning are tightly integrated, which is the premise of this chapter. Another direction is to investigate better exploration methods that go beyond using optimistic models to include Bayesian and utility-guided optimal exploration. 8 Acknowledgments We thank the reviewers for their helpful feedback. This research is supported by the Army Research Ofﬁce under grant number W911NF-09-1-0153. 8 References [1] R. Brafman and M. Tennenholtz. R-MAX — A General Polynomial Time Algorithm for NearOptimal Reinforcement Learning. Journal of Machine Learning Research, 3:213–231, 2002. [2] M. Kearns and L. Valiant. Cryptographic Limitations on Learning Boolean Formulae and Finite Automata. In Annual ACM Symposium on Theory of Computing, 1989. [3] L. Li. A Unifying Framework for Computational Reinforcement Learning Theory. PhD thesis, Rutgers University, 2009. [4] L. Li, M. Littman, and T. Walsh. Knows What It Knows: A Framework for Self-Aware Learning. In ICML, 2008. [5] N. Littlestone. Mistake Bounds and Logarithmic Linear-Threshold Learning Algorithms. PhD thesis, U.C. Santa Cruz, 1989. [6] B. Marthi, S. Russell, and J. Wolfe. Angelic Semantics for High-Level Actions. In ICAPS, 2007. [7] B. K. Natarajan. On Learning Boolean Functions. In Annual ACM Symposium on Theory of Computing, 1987. [8] T. Walsh and M. Littman. Efﬁcient Learning of Action Schemas and Web-Service Descriptions. In AAAI, 2008. 9	2011	91
4,443	Hierarchically Supervised Latent Dirichlet Allocation Adler Perotte Nicholas Bartlett No´emie Elhadad Frank Wood Columbia University, New York, NY 10027, USA {ajp9009@dbmi,bartlett@stat,noemie@dbmi,fwood@stat}.columbia.edu Abstract We introduce hierarchically supervised latent Dirichlet allocation (HSLDA), a model for hierarchically and multiply labeled bag-of-word data. Examples of such data include web pages and their placement in directories, product descriptions and associated categories from product hierarchies, and free-text clinical records and their assigned diagnosis codes. Out-of-sample label prediction is the primary goal of this work, but improved lower-dimensional representations of the bagof-word data are also of interest. We demonstrate HSLDA on large-scale data from clinical document labeling and retail product categorization tasks. We show that leveraging the structure from hierarchical labels improves out-of-sample label prediction substantially when compared to models that do not. 1 Introduction There exist many sources of unstructured data that have been partially or completely categorized by human editors. In this paper we focus on unstructured text data that has been, at least in part, manually categorized. Examples include but are not limited to webpages and curated hierarchical directories of the same [1], product descriptions and catalogs, and patient records and diagnosis codes assigned to them for bookkeeping and insurance purposes. In this work we show how to combine these two sources of information using a single model that allows one to categorize new text documents automatically, suggest labels that might be inaccurate, compute improved similarities between documents for information retrieval purposes, and more. The models and techniques that we develop in this paper are applicable to other data as well, namely, any unstructured representations of data that have been hierarchically classiﬁed (e.g., image catalogs with bag-of-feature representations). There are several challenges entailed in incorporating a hierarchy of labels into the model. Among them, given a large set of potential labels (often thousands), each instance has only a small number of labels associated to it. Furthermore, there are no naturally occurring negative labeling in the data, and the absence of a label cannot always be interpreted as a negative labeling. Our work operates within the framework of topic modeling. Our approach learns topic models of the underlying data and labeling strategies in a joint model, while leveraging the hierarchical structure of the labels. For the sake of simplicity, we focus on “is-a” hierarchies, but the model can be applied to other structured label spaces. We extend supervised latent Dirichlet allocation (sLDA) [6] to take advantage of hierarchical supervision. We propose an efﬁcient way to incorporate hierarchical information into the model. We hypothesize that the context of labels within the hierarchy provides valuable information about labeling. We demonstrate our model on large, real-world datasets in the clinical and web retail domains. We observe that hierarchical information is valuable when incorporated into the learning and improves our primary goal of multi-label classiﬁcation. Our results show that a joint, hierarchical model outperforms a classiﬁcation with unstructured labels as well as a disjoint model, where the topic model and the hierarchical classiﬁcation are inferred independently of each other. 1 Figure 1: HSLDA graphical model The remainder of this paper is as follows. Section 2 introduces hierarchically supervised LDA (HSLDA), while Section 3 details a sampling approach to inference in HSLDA. Section 4 reviews related work, and Section 5 shows results from applying HSLDA to health care and web retail data. 2 Model HSLDA is a model for hierarchically, multiply-labeled, bag-of-word data. We will refer to individual groups of bag-of-word data as documents. Let wn,d ∈Σ be the nth observation in the dth document. Let wd = {w1,d, . . . , w1,Nd} be the set of Nd observations in document d. Let there be D such documents and let the size of the vocabulary be V = \|Σ\|. Let the set of labels be L = l1, l2, . . . , l\|L\| . Each label l ∈L, except the root, has a parent pa(l) ∈L also in the set of labels. We will for exposition purposes assume that this label set has hard “is-a” parent-child constraints (explained later), although this assumption can be relaxed at the cost of more computationally complex inference. Such a label hierarchy forms a multiply rooted tree. Without loss of generality we will consider a tree with a single root r ∈L. Each document has a variable yl,d ∈{−1, 1} for every label which indicates whether the label is applied to document d or not. In most cases yi,d will be unobserved, in some cases we will be able to ﬁx its value because of constraints on the label hierarchy, and in the relatively minor remainder its value will be observed. In the applications we consider, only positive labels are observed. The constraints imposed by an is-a label hierarchy are that if the lth label is applied to document d, i.e., yl,d = 1, then all labels in the label hierarchy up to the root are also applied to document d, i.e., ypa(l),d = 1, ypa(pa(l)),d = 1, . . . , yr,d = 1. Conversely, if a label l′ is marked as not applying to a document then no descendant of that label may be applied to the same. We assume that at least one label is applied to every document. This is illustrated in Figure 1 where the root label is always applied but only some of the descendant labelings are observed as having been applied (diagonal hashing indicates that potentially some of the plated variables are observed). In HSLDA, documents are modeled using the LDA mixed-membership mixture model with global topic estimation. Label responses are generated using a conditional hierarchy of probit regressors. The HSLDA graphical model is given in Figure 1. In the model, K is the number of LDA “topics” (distributions over the elements of Σ), φk is a distribution over “words,” θd is a document-speciﬁc distribution over topics, β is a global distribution over topics, DirK(·) is a K-dimensional Dirichlet distribution, NK(·) is the K-dimensional Normal distribution, IK is the K dimensional identity matrix, 1d is the d-dimensional vector of all ones, and I(·) is an indicator function that takes the value 1 if its argument is true and 0 otherwise. The following procedure describes how to generate from the HSLDA generative model. 2 1. For each topic k = 1, . . . , K • Draw a distribution over words φk ∼DirV (γ1V ) 2. For each label l ∈L • Draw a label application coefﬁcient ηl \| µ, σ ∼NK(µ1K, σIK) 3. Draw the global topic proportions β \| α′ ∼DirK (α′1K) 4. For each document d = 1, . . . , D • Draw topic proportions θd \| β, α ∼DirK (αβ) • For n = 1, . . . , Nd – Draw topic assignment zn,d \| θd ∼Multinomial(θd) – Draw word wn,d \| zn,d, φ1:K ∼Multinomial(φzn,d) • Set yr,d = 1 • For each label l in a breadth ﬁrst traversal of L starting at the children of root r – Draw al,d \| ¯zd, ηl, ypa(l),d ∼ N(¯zT d ηl, 1), ypa(l),d = 1 N(¯zT d ηl, 1)I(al,d < 0), ypa(l),d = −1 – Apply label l to document d according to al,d yl,d \| al,d = 1 if al,d > 0 −1 otherwise Here ¯zT d = [¯z1, . . . , ¯zk, . . . , ¯zK] is the empirical topic distribution for document d, in which each entry is the percentage of the words in that document that come from topic k, ¯zk = N −1 d PNd n=1 I(zn,d = k). The second half of step 4 is a substantial part of our contribution to the general class of supervised LDA models. Here, each document is labeled generatively using a hierarchy of conditionally dependent probit regressors [14]. For every label l ∈L, both the empirical topic distribution for document d and whether or not its parent label was applied (i.e. I(ypa(l),d = 1)) are used to determine whether or not label l is to be applied to document d as well. Note that label yl,d can only be applied to document d if its parent label pa(l) is also applied (these expressions are speciﬁc to is-a constraints but can be modiﬁed to accommodate different constraints). The regression coefﬁcients ηl are independent a priori, however, the hierarchical coupling in this model induces a posteriori dependence. The net effect of this is that label predictors deeper in the label hierarchy are able to focus on ﬁnding speciﬁc, conditional labeling features. We believe this to be a signiﬁcant source of the empirical label prediction improvement we observe experimentally. We test this hypothesis in Section 5. Note that the choice of variables al,d and how they are distributed were driven at least in part by posterior inference efﬁciency considerations. In particular, choosing probit-style auxiliary variable distributions for the al,d’s yields conditional posterior distributions for both the auxiliary variables (3) and the regression coefﬁcients (2) which are analytic. This simpliﬁes posterior inference substantially. In the common case where no negative labels are observed (like the example applications we consider in Section 5), the model must be explicitly biased towards generating data that has negative labels in order to keep it from learning to assign all labels to all documents. This is a common problem in modeling unbalanced data. To see how this model can be biased in this way we draw the reader’s attention to the µ parameter and, to a lesser extent, the σ parameter above. Because ¯zd is always positive, setting µ to a negative value results in a bias towards negative labelings, i.e. for large negative values of µ, all labels become a priori more likely to be negative (yl,d = −1). We explore the ability of µ to bias out-of-sample label prediction performance in Section 5. 3 Inference In this section we provide the conditional distributions required to draw samples from the HSLDA posterior distribution using Gibbs sampling and Markov chain Monte Carlo. Note that, like in collapsed Gibbs samplers for LDA [16], we have analytically marginalized out the parameters φ1:K 3 and θ1:D in the following expressions. Let a be the set of all auxiliary variables, w the set of all words, η the set of all regression coefﬁcients, and z\zn,d the set z with element zn,d removed. The conditional posterior distribution of the latent topic indicators is p (zn,d = k \| z\zn,d, a, w, η, α, β, γ) ∝ ck,−(n,d) (·),d + αβk ck,−(n,d) wn,d,(·) +γ ck,−(n,d) (·),(·) +V γ Q l∈Ld exp −(¯zT d ηl−al,d) 2 2 (1) where ck,−(n,d) v,d is the number of words of type v in document d assigned to topic k omitting the nth word of document d. The subscript (·)’s indicate to sum over the range of the replaced variable, i.e. ck,−(n,d) wn,d,(·) = P d ck,−(n,d) wn,d,d . Here Ld is the set of labels which are observed for document d. The conditional posterior distribution of the regression coefﬁcients is given by p(ηl \| z, a, σ) = N(ˆµl, ˆΣ) (2) where ˆµl = ˆΣ 1µ σ + ¯ZT al ˆΣ−1 = Iσ−1 + ¯ZT ¯Z. Here ¯Z is a D × K matrix such that row d of ¯Z is ¯zd, and al = [al,1, al,2, . . . , al,D]T . The simplicity of this conditional distribution follows from the choice of probit regression [4]; the speciﬁc form of the update is a standard result from Bayesian normal linear regression [14]. It also is a standard probit regression result that the conditional posterior distribution of al,d is a truncated normal distribution [4]. p (al,d \| z, Y, η) ∝ exp −1 2 al,d −ηT l ¯zd I (al,dyl,d > 0) I(al,d < 0), ypa(l),d = −1 exp −1 2 al,d −ηT l ¯zd I (al,dyl,d > 0) , ypa(l),d = 1 (3) Note that care must be taken to initialize the Gibbs sampler in a valid state. HSLDA employs a hierarchical Dirichlet prior over topic assignments (i.e., β is estimated from data rather than ﬁxed a priori). This has been shown to improve the quality and stability of inferred topics [26]. Sampling β is done using the “direct assignment” method of Teh et al. [25] β \| z, α′, α ∼Dir m(·),1 + α′, m(·),2 + α′, . . . , m(·),K + α′. (4) Here md,k are auxiliary variables that are required to sample the posterior distribution of β. Their conditional posterior distribution is sampled according to p md,k = m \| z, m−(d,k), β = Γ (αβk) Γ αβk + ck (·),d s ck (·),d, m (αβk)m (5) where s (n, m) represents stirling numbers of the ﬁrst kind. The hyperparameters α, α′, and γ are sampled using Metropolis-Hastings. 4 Related Work In this work we extend supervised latent Dirichlet allocation (sLDA) [6] to take advantage of hierarchical supervision. sLDA is latent Dirichlet allocation (LDA) [7] augmented with per document “supervision,” often taking the form of a single numerical or categorical label. It has been demonstrated that the signal provided by such supervision can result in better, task-speciﬁc document models and can also lead to good label prediction for out-of-sample data [6]. It also has been demonstrated that sLDA has been shown to outperform both LASSO (L1 regularized least squares regression) and LDA followed by least squares regression [6]. sLDA can be applied to data of the type we consider in this paper; however, doing so requires ignoring the hierarchical dependencies amongst the labels. In Section 5 we constrast HSLDA with sLDA applied in this way. Other models that incorporate LDA and supervision include LabeledLDA [23] and DiscLDA [18]. Various applications of these models to computer vision and document networks have been explored [27, 9] . None of these models, however, leverage dependency structure in the label space. 4 In other work, researchers have classiﬁed documents into a hierarchy (a closely related task) with naive Bayes classiﬁers and support vector machines. Most of this work has been demonstrated on relatively small datasets, small label spaces, and has focused on single label classiﬁcation without a model of documents such as LDA [21, 11, 17, 8]. 5 Experiments We applied HSLDA to data from two domains: predicting medical diagnosis codes from hospital discharge summaries and predicting product categories from Amazon.com product descriptions. 5.1 Data and Pre-Processing 5.1.1 Discharge Summaries and ICD-9 Codes Discharge summaries are authored by clinicians to summarize patient hospitalization courses. The summaries typically contain a record of patient complaints, ﬁndings and diagnoses, along with treatment and hospital course. For each hospitalization, trained medical coders review the information in the discharge summary and assign a series of diagnoses codes. Coding follows the ICD-9-CM controlled terminology, an international diagnostic classiﬁcation for epidemiological, health management, and clinical purposes.1 The ICD-9 codes are organized in a rooted-tree structure, with each edge representing an is-a relationship between parent and child, such that the parent diagnosis subsumes the child diagnosis. For example, the code for “Pneumonia due to adenovirus” is a child of the code for “Viral pneumonia,” where the former is a type of the latter. It is worth noting that the coding can be noisy. Human coders sometimes disagree [3], tend to be more speciﬁc than sensitive in their assignments [5], and sometimes make mistakes [13]. The task of automatic ICD-9 coding has been investigated in the clinical domain. Methods range from manual rules to online learning [10, 15, 12]. Other work had leveraged larger datasets and experimented with K-nearest neighbor, Naive Bayes, support vector machines, Bayesian Ridge Regression, as well as simple keyword mappings, all with promising results [19, 24, 22, 20]. Our dataset was gathered from the NewYork-Presbyterian Hospital clinical data warehouse. It consists of 6,000 discharge summaries and their associated ICD-9 codes (7,298 distinct codes overall), representing all the discharges from the hospital in 2009. All included discharge summaries had associated ICD-9 Codes. Summaries have 8.39 associated ICD-9 codes on average (std dev=5.01) and contain an average of 536.57 terms after preprocessing (std dev=300.29). We split our dataset into 5,000 discharge summaries for training and 1,000 for testing. The text of the discharge summaries was tokenized with NLTK.2 A ﬁxed vocabulary was formed by taking the top 10,000 tokens with highest document frequency (exclusive of names, places and other identifying numbers). The study was approved by the Institutional Review Board and follows HIPAA (Health Insurance Portability and Accountability Act) privacy guidelines. 5.1.2 Product Descriptions and Categorizations Amazon.com, an online retail store, organizes its catalog of products in a mulitply-rooted hierarchy and provides textual product descriptions for most products. Products can be discovered by users through free-text search and product category exploration. Top-level product categories are displayed on the front page of the website and lower level categories can be discovered by choosing one of the top-level categories. Products can exist in multiple locations in the hierarchy. In this experiment, we obtained Amazon.com product categorization data from the Stanford Network Analysis Platform (SNAP) dataset [2]. Product descriptions were obtained separately from the Amazon.com website directly. We limited our dataset to the collection of DVDs in the product catalog. Our dataset contains 15,130 product descriptions for training and 1,000 for testing. The product descriptions are shorter than the discharge summaries (91.89 terms on average, std dev=53.08). 1http://www.cdc.gov/nchs/icd/icd9cm.htm 2http://www.nltk.org 5 Overall, there are 2,691 unique categories. Products are assigned on average 9.01 categories (std dev=4.91). The vocabulary consists of the most frequent 30,000 words omitting stopwords. 5.2 Comparison Models We evaluated HSLDA along with two closely related models against the two datasets. The comparison models included sLDA with independent regressors (hierarchical constraints on labels ignored) and HSLDA ﬁt by ﬁrst performing LDA then ﬁtting tree-conditional regressions. These models were chosen to highlight several aspects of HSLDA including performance in the absence of hierarchical constraints, the effect of the combined inference, and regression performance attributable solely to the hierarchical constraints. sLDA with independent regressors is the most salient comparison model for our work. The distinguishing factor between HSLDA and sLDA is the additional structure imposed on the label space, a distinction that we hypothesized would result in a difference in predictive performance. There are two components to HSLDA, LDA and a hierarchically constrained response. The second comparison model is HSLDA ﬁt by performing LDA ﬁrst followed by performing inference over the hierarchically constrained label space. In this comparison model, the separate inference processes do not allow the responses to inﬂuence the low dimensional structure inferred by LDA. Combined inference has been shown to improve performance in sLDA [6]. This comparison model examines not the structuring of the label space, but the beneﬁt of combined inference over both the documents and the label space. For all three models, particular attention was given to the settings of the prior parameters for the regression coefﬁcients. These parameters implement an important form of regularization in HSLDA. In the setting where there are no negative labels, a Gaussian prior over the regression parameters with a negative mean implements a prior belief that missing labels are likely to be negative. Thus, we evaluated model performance for all three models with a range of values for µ, the mean prior parameter for regression coefﬁcients (µ ∈{−3, −2.8, −2.6, . . . , 1}). The number of topics for all models was set to 50, the prior distributions of p (α), p (α′), and p (γ) were gamma distributed with a shape parameter of 1 and a scale parameters of 1000. 0.0 0.2 0.4 0.6 0.8 1.0 1-Speciﬁcity 0.0 0.2 0.4 0.6 0.8 1.0 Sensitivity HSLDA sLDA LDA + conditional regression (a) Clinical data performance. 0.0 0.2 0.4 0.6 0.8 1.0 1-Speciﬁcity 0.0 0.2 0.4 0.6 0.8 1.0 Sensitivity HSLDA sLDA LDA + conditional regression (b) Retail product performance. Figure 2: ROC curves for out-of-sample label prediction varying µ, the prior mean of the regression parameters. In both ﬁgures, solid is HSLDA, dashed are independent regressors + sLDA (hierarchical constraints on labels ignored), and dotted is HSLDA ﬁt by running LDA ﬁrst then running tree-conditional regressions. 5.3 Evaluation and Results We evaluated our model, HSLDA, against the comparison models with a focus on predictive performance on held-out data. Prediction performance was measured with standard metrics – sensitivity (true positive rate) and 1-speciﬁcity (false positive rate). 6 0.0 0.2 0.4 0.6 0.8 1.0 1-Speciﬁcity 0.0 0.2 0.4 0.6 0.8 1.0 Sensitivity HSLDA sLDA LDA + conditional regression Figure 3: ROC curve for out-of-sample ICD-9 code prediction varying auxiliary variable threshold. µ = −1.0 for all three models in this ﬁgure. The gold standard for comparison was derived from the testing set in each dataset. To make the comparison as fair as possible among models, ancestors of observed nodes in the label hierarchy were ignored, observed nodes were considered positive and descendents of observed nodes were considered to be negative. Note that this is different from our treatment of the observations during inference. Since the sLDA model does not enforce the hierarchical constraints, we establish a more equal footing by considering only the observed labels as being positive, despite the fact that, following the hierarchical constraints, ancestors must also be positive. Such a gold standard will likely inﬂate the number of false positives because the labels applied to any particular document are usually not as complete as they could be. ICD-9 codes, for instance, lack sensitivity and their use as a gold standard could lead to correctly positive predictions being labeled as false positives [5]. However, given that the label space is often large (as in our examples) it is a moderate assumption that erroneous false positives should not skew results signiﬁcantly. Predictive performance in HSLDA is evaluated by p yl, ˆd \| w1:N ˆ d, ˆd, w1:Nd,1:D, yl∈L,1:D for each test document, ˆd. For efﬁciency, the expectation of this probability distribution was estimated in the following way. Expectations of ¯z ˆd and ηl were estimated with samples from the posterior. Using these expectations, we performed Gibbs sampling over the hierarchy to acquire predictive samples for the documents in the test set. The true positive rate was calculated as the average expected labeling for gold standard positive labels. The false positive rate was calculated as the average expected labeling for gold standard negative labels. As sensitivity and speciﬁcity can always be traded off, we examined sensitivity for a range of values for two different parameters – the prior means for the regression coefﬁcients and the threshold for the auxiliary variables. The goal in this analysis was to evaluate the performance of these models subject to more or less stringent requirements for predicting positive labels. These two parameters have important related functions in the model. The prior mean in combination with the auxiliary variable threshold together encode the strength of the prior belief that unobserved labels are likely to be negative. Effectively, the prior mean applies negative pressure to the predictions and the auxiliary variable threshold determines the cutoff. For each model type, separate models were ﬁt for each value of the prior mean of the regression coefﬁcients. This is a proper Bayesian sensitivity analysis. In contrast, to evaluate predictive performance as a function of the auxiliary variable threshold, a single model was ﬁt for each model type and prediction was evaluated based on predictive samples drawn subject to different auxiliary variable thresholds. These methods are signiﬁcantly different since the prior mean is varied prior to inference, and the auxiliary variable threshold is varied following inference. Figure 2(a) demonstrates the performance of the model on the clinical data as an ROC curve varying µ. For instance, a hyperparameter setting of µ = −1.6 yields the following performance: the full HSLDA model had a true positive rate of 0.57 and a false positive rate of 0.13, the sLDA model had 7 a true positive rate of 0.42 and a false positive rate of 0.07, and the HSLDA model where LDA and the regressions were ﬁt separately had a true positive rate of 0.39 and a false positive rate of 0.08. These points are highlighted in Figure 2(a). These results indicate that the full HSLDA model predicts more of the the correct labels at a cost of an increase in the number of false positives relative to the comparison models. Figure 2(b) demonstrates the performance of the model on the retail product data as an ROC curve also varying µ. For instance, a hyperparameter setting of µ = −2.2 yields the following performance: the full HSLDA model had a true positive rate of 0.85 and a false positive rate of 0.30, the sLDA model had a true positive rate of 0.78 and a false positive rate of 0.14, and the HSLDA model where LDA and the regressions were ﬁt separately had a true positive rate of 0.77 and a false positive rate of 0.16. These results follow a similar pattern to the clinical data. These points are highlighted in Figure 2(b). Figure 3 shows the predictive performance of HSLDA relative to the two comparison models on the clinical dataset as a function of the auxiliary variable threshold. For low values of the auxiliary variable threshold, the models predict labels in a more sensitive and less speciﬁc manner, creating the points in the upper right corner of the ROC curve. As the auxiliary variable threshold is increased, the models predict in a less sensitive and more speciﬁc manner, creating the points in the lower left hand corner of the ROC curve. HSLDA with full joint inference outperforms sLDA with independent regressors as well as HSLDA with separately trained regression. 6 Discussion The SLDA model family, of which HSLDA is a member, can be understood in two different ways. One way is to see it as a family of topic models that improve on the topic modeling performance of LDA via the inclusion of observed supervision. An alternative, complementary way is to see it as a set of models that can predict labels for bag-of-word data. A large diversity of problems can be expressed as label prediction problems for bag-of-word data. A surprisingly large amount of that kind of data possess structured labels, either hierarchically constrained or otherwise. That HSLDA directly addresses this kind of data is a large part of the motivation for this work. That it outperforms more straightforward approaches should be of interest to practitioners. Variational Bayes has been the predominant estimation approach applied to sLDA models. Hierarchical probit regression makes for tractable Markov chain Monte Carlo SLDA inference, a beneﬁt that should extend to other sLDA models should probit regression be used for response variable prediction there too. The results in Figures 2(a) and 2(b) suggest that in most cases it is better to do full joint estimation of HSLDA. An alternative interpretation of the same results is that, if one is more sensitive to the performance gains that result from exploiting the structure of the labels, then one can, in an engineering sense, get nearly as much gain in label prediction performance by ﬁrst ﬁtting LDA and then ﬁtting a hierarchical probit regression. There are applied settings in which this could be advantageous. 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4,444	Agnostic Selective Classiﬁcation Ran El-Yaniv and Yair Wiener Computer Science Department Technion – Israel Institute of Technology {rani,wyair}@{cs,tx}.technion.ac.il Abstract For a learning problem whose associated excess loss class is (β, B)-Bernstein, we show that it is theoretically possible to track the same classiﬁcation performance of the best (unknown) hypothesis in our class, provided that we are free to abstain from prediction in some region of our choice. The (probabilistic) volume of this rejected region of the domain is shown to be diminishing at rate O(Bθ( √ 1/m)β), where θ is Hanneke’s disagreement coefﬁcient. The strategy achieving this performance has computational barriers because it requires empirical error minimization in an agnostic setting. Nevertheless, we heuristically approximate this strategy and develop a novel selective classiﬁcation algorithm using constrained SVMs. We show empirically that the resulting algorithm consistently outperforms the traditional rejection mechanism based on distance from decision boundary. 1 Introduction Is it possible to achieve the same test performance as the best classiﬁer in hindsight? The answer to this question is “probably not.” However, when changing the rules of the standard game it is possible. Indeed, consider a game where our classiﬁer is allowed to abstain from prediction, without penalty, in some region of our choice. For this case, and assuming a noise free “realizable” setting, it was shown in [1] that there is a “perfect classiﬁer.” This means that after observing only a ﬁnite labeled training sample, the learning algorithm outputs a classiﬁer that, with certainty, will never err on any test point. To achieve this, this classiﬁer must refuse to classify in some region of the domain. Perhaps surprisingly it was shown that the volume of this rejection region is bounded, and in fact, this volume diminishes with increasing training set sizes (under certain conditions). An open question, posed in [1], is what would be an analogous notion of perfection in an agnostic, noisy setting. Is it possible to achieve any kind of perfection in a real world scenario? The setting under consideration, where classiﬁers can abstain from prediction, is called classiﬁcation with a reject option [2, 3], or selective classiﬁcation [1]. Focusing on this model, in this paper we present a blend of theoretical and practical results. We ﬁrst show that the concept of “perfect classiﬁcation” that was introduced for the realizable case in [1], can be extended to the agnostic setting. While pure perfection is impossible to accomplish in a noisy environment, a more realistic objective is to perform as well as the best hypothesis in the class within a region of our choice. We call this type of learning “weakly optimal” selective classiﬁcation and show that a novel strategy accomplishes this type of learning with diminishing rejection rate under certain Bernstein type conditions (a stronger notion of optimality is mentioned later as well). This strategy relies on empirical risk minimization, which is computationally difﬁcult. In the practical part of the paper we present a heuristic approximation algorithm, which relies on constrained SVMs, and mimics the optimal behavior. We conclude with numerical examples that examine the empirical performance of the new algorithm and compare its performance with that of the widely used selective classiﬁcation method for rejection, based on distance from decision boundary. 1 2 Selective classiﬁcation and other deﬁnitions Consider a standard agnostic binary classiﬁcation setting where X is some feature space, and H is our hypothesis class of binary classiﬁers, h : X →{±1}. Given a ﬁnite training sample of m labeled examples, Sm = {(xi, yi)}m i=1, assumed to be sampled i.i.d. from some unknown underlying distribution P(X, Y ) over X × {±1}, our goal is to select the best possible classiﬁer from H. For any h ∈H, its true error, R(h), and its empirical error, ˆR(h), are, R(h) ≜ Pr (X,Y )∼P {h(X) ̸= Y } , ˆR(h) ≜1 m m ∑ i=1 I (h(xi) ̸= yi) . Let ˆh ≜arg infh∈H ˆR(h) be the empirical risk minimizer (ERM), and h∗≜arg infh∈H R(h), the true risk minimizer. In selective classiﬁcation [1], given Sm we need to select a binary selective classiﬁer deﬁned to be a pair (h, g), with h ∈H being a standard binary classiﬁer, and g : X →{0, 1} is a selection function deﬁning the sub-region of activity of h in X. For any x ∈X, (h, g)(x) ≜ { reject, if g(x) = 0; h(x), if g(x) = 1. (1) Selective classiﬁcation performance is characterized in terms of two quantities: coverage and risk. The coverage of (h, g) is Φ(h, g) ≜E [g(X)] . For a bounded loss function ℓ: Y × Y →[0, 1], the risk of (h, g) is deﬁned as the average loss on the accepted samples, R(h, g) ≜E [ℓ(h(X), Y ) · g(X)] Φ(h, g) . As pointed out in [1], the trade-off between risk and coverage is the main characteristic of a selective classiﬁer. This trade-off is termed there the “risk-coverage curve” (RC curve)1 Let G ⊆H. The disagreement set [4, 1] w.r.t. G is deﬁned as DIS(G) ≜{x ∈X : ∃h1, h2 ∈G s.t. h1(x) ̸= h2(x)} . For any hypothesis class H, target hypothesis h ∈H, distribution P, sample Sm, and real r > 0, deﬁne V(h, r) = {h′ ∈H : R(h′) ≤R(h) + r} and ˆV(h, r) = { h′ ∈H : ˆR(h′) ≤ˆR(h) + r } . (2) Finally, for any h ∈H we deﬁne a ball in H of radius r around h [5]. Speciﬁcally, with respect to class H, marginal distribution P over X, h ∈H, and real r > 0, deﬁne B(h, r) ≜ { h′ ∈H : Pr X∼P {h′(X) ̸= h(X)} ≤r } . 3 Perfect and weakly optimal selective classiﬁers The concept of perfect classiﬁcation was introduced in [1] within a realizable selective classiﬁcation setting. Perfect classiﬁcation is an extreme case of selective classiﬁcation where a selective classiﬁer (h, g) achieves R(h, g) = 0 with certainty; that is, the classiﬁer never errs on its region of activity. Obviously, the classiﬁer must compromise sufﬁciently large part of the domain X in order to achieve this outstanding performance. Surprisingly, it was shown in [1] that not-trivial perfect classiﬁcation exists in the sense that under certain conditions (e.g., ﬁnite hypothesis class) the rejected region diminishes at rate Ω(1/m), where m is the size of the training set. In agnostic environments, as we consider here, such perfect classiﬁcation appears to be out of reach. In general, in the worst case no hypothesis can achieve zero error over any nonempty subset of the 1Some authors refer to an equivalent variant of this curve as “Accuracy-Rejection Curve” or ARC. 2 domain. We consider here the following weaker, but still extremely desirable behavior, which we call “weakly optimal selective classiﬁcation.” Let h∗∈H be the true risk minimizer of our problem. Let (h, g) be a selective classiﬁer selected after observing the training set Sm. We say that (h, g) is a weakly optimal selective classiﬁer if, for any 0 < δ < 1, with probability of at least 1 −δ over random choices of Sm, R(h, g) ≤R(h∗, g). That is, with high probability our classiﬁer is at least as good as the true risk minimizer over its region of activity. We call this classiﬁer ‘weakly optimal’ because a stronger requirement would be that the classiﬁer should achieve the best possible error among all hypotheses in H restricted to the region of activity deﬁned by g. 4 A learning strategy We now present a strategy that will be shown later to achieve non-trivial weakly optimal selective classiﬁcation under certain conditions. We call it a “strategy” rather than an “algorithm” because it does not include implementation details. Let’s begin with some motivation. Using standard concentration inequalities one can show that the training error of the true risk minimizer, h∗, cannot be “too far” from the training error of the empirical risk minimizer, ˆh. Therefore, we can guarantee, with high probability, that the class of all hypothesis with “sufﬁciently low” empirical error includes the true risk minimizer h∗. Selecting only subset of the domain, for which all hypothesis in that class agree, is then sufﬁcient to guarantee weak optimality. Strategy 1 formulates this idea. In the next section we analyze this strategy and show that it achieves this optimality with non trivial (bounded) coverage. Strategy 1 Learning strategy for weakly optimal selective classiﬁers Input: Sm, m, δ, d Output: a selective classiﬁer (h, g) such that R(h, g) = R(h∗, g) w.p. 1 −δ 1: Set ˆh = ERM(H, Sm), i.e., ˆh is any empirical risk minimizer from H 2: Set G = ˆV ( ˆh, 4 √ 2 d(ln 2me d )+ln 8 δ m ) (see Eq. (2)) 3: Construct g such that g(x) = 1 ⇐⇒x ∈{X \ DIS (G)} 4: h = ˆh 5 Analysis We begin with a few deﬁnitions. Consider an instance of a binary learning problem with hypothesis class H, an underlying distribution P over X × Y, and a loss function ℓ(Y, Y). Let h∗= arg infh∈H {Eℓ(h(X), Y )} be the true risk minimizer. The associated excess loss class [6] is deﬁned as F ≜{ℓ(h(x), y) −ℓ(h∗(x), y) : h ∈H} . Class F is said to be a (β, B)-Bernstein class with respect to P (where 0 < β ≤1 and B ≥1), if every f ∈F satisﬁes Ef 2 ≤B(Ef)β. Bernstein classes arise in many natural situations; see discussions in [7, 8]. For example, if the probability P(X, Y ) satisﬁes Tsybakov’s noise conditions then the excess loss function is a Bernstein [8, 9] class. In the following sequence of lemmas and theorems we assume a binary hypothesis class H with VC-dimension d, an underlying distribution P over X ×{±1}, and ℓis the 0/1 loss function. Also, F denotes the associated excess loss class. Our results can be extended to losses other than 0/1 by similar techniques to those used in [10]. Lemma 5.1. If F is a (β, B)-Bernstein class with respect to P, then for any r > 0 V(h∗, r) ⊆B ( h∗, Brβ) . Proof. If h ∈V(h∗, r) then, by deﬁnition E {I(h(X) ̸= Y )} ≤E {I(h∗(X) ̸= Y )} + r. 3 Using the linearity of expectation we have, E {I(h(X) ̸= Y ) −I(h∗(X) ̸= Y )} ≤r. (3) Since F is a (β, B)-Bernstein class, E {I(h(X) ̸= h∗(X))} = E {\|I(h(X) ̸= Y ) −I(h∗(X) ̸= Y )\|} = E { (ℓ(h(X), Y ) −ℓ(h∗(X), Y ))2} = Ef 2 ≤B(Ef)β = B (E {I(h(X) ̸= Y ) −I(h∗(X) ̸= Y )})β . By (3), for any r > 0, E {I(h(X) ̸= h∗(X))} ≤Brβ. Therefore, by deﬁnition, h ∈B ( h∗, Brβ) . Throughout this section we denote σ(m, δ, d) ≜2 √ 2d ( ln 2me d ) + ln 2 δ m . Theorem 5.2 ([11]). For any 0 < δ < 1, with probability of at least 1 −δ over the choice of Sm from P m, any hypothesis h ∈H satisﬁes R(h) ≤ˆR(h) + σ(m, δ, d). Similarly ˆR(h) ≤R(h) + σ(m, δ, d) under the same conditions. Lemma 5.3. For any r > 0, and 0 < δ < 1, with probability of at least 1 −δ, ˆV(ˆh, r) ⊆V (h∗, 2σ(m, δ/2, d) + r) . Proof. If h ∈ˆV(ˆh, r), then, by deﬁnition, ˆR(h) ≤ˆR(ˆh)+r. Since ˆh minimizes the empirical error, we have, ˆR(ˆh) ≤ˆR(h∗). Using Theorem 5.2 twice, and applying the union bound, we know that w.p. of at least 1 −δ, R(h) ≤ˆR(h) + σ(m, δ/2, d) ∧ ˆR(h∗) ≤R(h∗) + σ(m, δ/2, d). Therefore, R(h) ≤R(h∗) + 2σ(m, δ/2, d) + r, and h ∈V (h∗, 2σ(m, δ/2, d) + r). For any G ⊆H, and distribution P we deﬁne, ∆G ≜Pr {DIS(G)}. Hanneke introduced a complexity measure for active learning problems termed the disagreement coefﬁcient [5]. The disagreement coefﬁcient of h with respect to H under distribution P is, θh ≜sup r>ϵ ∆B(h, r) r , (4) where ϵ = 0. The disagreement coefﬁcient of the hypothesis class H with respect to P is deﬁned as θ ≜lim sup k→∞ θh(k), where { h(k)} is any sequence of h(k) ∈H with R(h(k)) monotonically decreasing. Theorem 5.4. Assume that H has disagreement coefﬁcient θ and that F is a (β, B)-Bernstein class w.r.t. P. Then, for any r > 0 and 0 < δ < 1, with probability of at least 1 −δ, ∆ˆV(ˆh, r) ≤Bθ (2σ(m, δ/2, d) + r)β . Proof. Applying Lemmas 5.3 and 5.1 we get that with probability of at least 1 −δ, ˆV(ˆh, r) ⊆B ( h∗, B (2σ(m, δ/2, d) + r)β) . Therefore ∆ˆV(ˆh, r) ≤∆B ( h∗, B (2σ(m, δ/2, d) + r)β) . By the deﬁnition of the disagreement coefﬁcient, for any r′ > 0, ∆B(h∗, r′) ≤θr′. 4 Theorem 5.5. Assume that H has disagreement coefﬁcient θ and that F is a (β, B)-Bernstein class w.r.t. P. Let (h, g) be the selective classiﬁer chosen by Algorithm 1. Then, with probability of at least 1 −δ, Φ(h, g) ≥1 −Bθ (4σ(m, δ/4, d))β ∧ R(h, g) = R(h∗, g). Proof. Applying Theorem 5.2 we get that with probability of at least 1 −δ/4, ˆR(h∗) ≤R(h∗) + σ(m, δ/4, d). Since h∗minimizes the true error, wet get that R(h∗) ≤R(ˆh). Applying again Theorem 5.2 we know that with probability of at least 1 −δ/4, R(ˆh) ≤ˆR(ˆh) + σ(m, δ/4, d). Applying the union bound we have that with probability of at least 1 −δ/2, ˆR(h∗) ≤ˆR(ˆh) + 2σ(m, δ/4, d). Hence, with probability of at least 1 −δ/2, h∗∈ˆV ( ˆh, 2σ(m, δ/4, d) ) = G. We note that the selection function g(x) equals one only for x ∈X \DIS (G) . Therefore, for any x ∈X, for which g(x) = 1, all the hypotheses in G agree, and in particular h∗and ˆh agree. Thus, R(ˆh, g) = E{I(ˆh(X) ̸= Y ) · g(X)} E{g(X)} = E {I(h∗(X) ̸= Y ) · g(X)} E{g(X)} = R(h∗, g). Applying Theorem 5.4 and the union bound we therefore know that with probability of at least 1−δ, Φ(ˆh, g) = E{g(X)} = 1 −∆G ≥1 −Bθ (4σ(m, δ/4, d))β . Hanneke introduced, in his original work [5], an alternative deﬁnition of the disagreement coefﬁcient θ, for which the supermum in (4) is taken with respect to any ﬁxed ϵ > 0. Using this alternative definition it is possible to show that fast coverage rates are achievable, not only for ﬁnite disagreement coefﬁcients (Theorem 5.5), but also if the disagreement coefﬁcient grows slowly with respect to 1/ϵ (as shown by Wang [12], under sufﬁcient smoothness conditions). This extension will be discussed in the full version of this paper. 6 A disbelief principle and the risk-coverage trade-off Theorem 5.5 tells us that the strategy presented in Section 4 not only outputs a weakly optimal selective classiﬁer, but this classiﬁer also has guaranteed coverage (under some conditions). As emphasized in [1], in practical applications it is desirable to allow for some control on the trade-off between risk and coverage; in other words, we would like to be able to develop the entire riskcoverage curve for the classiﬁer at hand and select ourselves the cutoff point along this curve in accordance with other practical considerations we may have. How can this be achieved? The following lemma facilitates a construction of a risk-coverage trade-off curve. The result is an alternative characterization of the selection function g, of the weakly optimal selective classiﬁer chosen by Strategy 1. This result allows for calculating the value of g(x), for any individual test point x ∈X, without actually constructing g for the entire domain X. Lemma 6.1. Let (h, g) be a selective classiﬁer chosen by Strategy 1 after observing the training sample Sm. Let ˆh be the empirical risk minimizer over Sm. Let x be any point in X and ehx ≜argmin h∈H { ˆR(h) \| h(x) = −sign ( ˆh(x) )} , an empirical risk minimizer forced to label x the opposite from ˆh(x). Then g(x) = 0 ⇐⇒ ˆR(ehx) −ˆR(ˆh) ≤2σ(m, δ/4, d). Proof. According to the deﬁnition of ˆV (see Eq. (2)), ˆR(ehx) −ˆR(ˆh) ≤2σ(m, δ/4, d) ⇐⇒ eh ∈ˆV ( ˆh, 2σ(m, δ/4, d) ) Thus, ˆh,ehx ∈ˆV. However, by construction, ˆh(x) = −eh(x), so x ∈DIS(ˆV) and g(x) = 0. 5 Lemma 6.1 tells us that in order to decide if point x should be rejected we need to measure the empirical error ˆR(ehx) of a special empirical risk minimizer, ehx, which is constrained to label x the opposite from ˆh(x). If this error is sufﬁciently close to ˆR(ˆh) our classiﬁer cannot be too sure about the label of x and we must reject it. This result strongly motivates the following deﬁnition of a “disbelief index” for each individual point. Deﬁnition 6.2 (disbelief index). For any x ∈X, deﬁne its disbelief index w.r.t. Sm and H, D(x) ≜D(x, Sm) ≜ˆR(ehx) −ˆR(ˆh). Observe that D(x) is large whenever our model is sensitive to label of x in the sense that when we are forced to bend our best model to ﬁt the opposite label of x, our model substantially deteriorates, giving rise to a large disbelief index. This large D(x) can be interpreted as our disbelief in the possibility that x can be labeled so differently. In this case we should deﬁnitely predict the label of x using our unforced model. Conversely, if D(x) is small, our model is indifferent to the label of x and in this sense, is not committed to its label. In this case we should abstain from prediction at x. This “disbelief principle” facilitates an exploration of the risk-coverage trade-off curve for our classiﬁer. Given a pool of test points we can rank these test points according to their disbelief index, and points with low index should be rejected ﬁrst. Thus, this ranking provides the means for constructing a risk-coverage trade-off curve. A similar technique of using an ERM oracle that can enforce an arbitrary number of example-based constraints was used in [13, 14] in the context of active learning. As in our disbelief index, the difference between the empirical risk (or importance weighted empirical risk [14]) of two ERM oracles (with different constraints) is used to estimate prediction conﬁdence. 7 Implementation At this point in the paper we switch from theory to practice, aiming at implementing rejection methods inspired by the disbelief principle and see how well they work on real world (well, ..., UCI) problems. Attempting to implement a learning algorithm driven by the disbelief index we face a major bottleneck because the calculation of the index requires the identiﬁcation of ERM hypotheses. To handle this computationally difﬁcult problem, we “approximate” the ERM as follows. Focusing on SVMs we use a high C value (105 in our experiments) to penalize more on training errors than on small margin. In this way the solution to the optimization problem tend to get closer to the ERM. Another problem we face is that the disbelief index is a noisy statistic that highly depends on the sample Sm. To overcome this noise we use robust statistics. First we generate 11 different samples (S1 m, S2 m, . . . S11 m ) using bootstrap sampling. For each sample we calculate the disbelief index for all test points and for each point take the median of these measurements as the ﬁnal index. We note that for any ﬁnite training sample the disbelief index is a discrete variable. It is often the case that several test points share the same disbelief index. In those cases we can use any conﬁdence measure as a tie breaker. In our experiments we use distance from decision boundary to break ties. In order to estimate ˆR(ehx) we have to restrict the SVM optimizer to only consider hypotheses that classify the point x in a speciﬁc way. To accomplish this we use a weighted SVM for unbalanced data. We add the point x as another training point with weight 10 times larger than the weight of all training points combined. Thus, the penalty for misclassiﬁcation of x is very large and the optimizer ﬁnds a solution that doesn’t violate the constraint. 8 Empirical results Focusing on SVMs with a linear kernel we compared the RC (Risk-Coverage) curves achieved by the proposed method with those achieved by SVM with rejection based on distance from decision boundary. This latter approach is very common in practical applications of selective classiﬁcation. For implementation we used LIBSVM [15]. Before presenting these results we wish to emphasize that the proposed method leads to rejection regions fundamentally different than those obtained by the traditional distance-based technique. In 6 Figure 1 we depict those regions for a training sample of 150 points sampled from a mixture of two identical normal distributions (centered at different locations). The height map reﬂects the “conﬁdence regions” of each technique according to its own conﬁdence measure. (a) (b) Figure 1: conﬁdence height map using (a) disbelief index; (b) distance from decision boundary. We tested our algorithm on standard medical diagnosis problems from the UCI repository, including all datasets used in [16]. We transformed nominal features to numerical ones in a standard way using binary indicator attributes. We also normalized each attribute independently so that its dynamic range is [0, 1]. No other preprocessing was employed. In each iteration we choose uniformly at random non overlapping training set (100 samples) and test set (200 samples) for each dataset.SVM was trained on the entire training set and test samples were sorted according to conﬁdence (either using distance from decision boundary or disbelief index). Figure 2 depicts the RC curves of our technique (red solid line) and rejection based on distance from decision boundary (green dashed line) for linear kernel on all 6 datasets. All results are averaged over 500 iterations (error bars show standard error). 0 0.2 0.4 0.6 0.8 1 0 0.01 0.02 0.03 c Hypo 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 c test error Pima 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 c test error Hepatitis 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 c Haberman 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 c test error BUPA 0 0.2 0.4 0.6 0.8 1 0 0.01 0.02 0.03 c test error Breast Figure 2: RC curves for SVM with linear kernel. Our method in solid red, and rejection based on distance from decision boundary in dashed green. Horizntal axis (c) represents coverage. With the exception of the Hepatitis dataset, in which both methods were statistically indistinguishable, in all other datasets the proposed method exhibits signiﬁcant advantage over the traditional approach. We would like to highlight the performance of the proposed method on the Pima dataset. While the traditional approach cannot achieve error less than 8% for any rejection rate, in our approach the test error decreases monotonically to zero with rejection rate. Furthermore, a clear advantage for our method over a large range of rejection rates is evident in the Haberman dataset.2. 2The Haberman dataset contains survival data of patients who had undergone surgery for breast cancer. With estimated 207,090 new cases of breast cancer in the united states during 2010 [17] an improvement of 1% affects the lives of more than 2000 women. 7 For the sake of fairness, we note that the running time of our algorithm (as presented here) is substantially longer than the traditional technique. The performance of our algorithm can be substantially improved when many unlabeled samples are available. Details will be provided in the full paper. 9 Related work The literature on theoretical studies of selective classiﬁcation is rather sparse. El-Yaniv and Wiener [1] studied the performance of a simple selective learning strategy for the realizable case. Given an hypothesis class H, and a sample Sm, their method abstain from prediction if all hypotheses in the version space do not agree on the target sample. They were able to show that their selective classiﬁer achieves perfect classiﬁcation with meaningful coverage under some conditions. Our work can be viewed as an extension of the above algorithm to the agnostic case. Freund et al. [18] studied another simple ensemble method for binary classiﬁcation. Given an hypothesis class H, the method outputs a weighted average of all the hypotheses in H, where the weight of each hypothesis exponentially depends on its individual training error. Their algorithm abstains from prediction whenever the weighted average of all individual predictions is close to zero. They were able to bound the probability of misclassiﬁcation by 2R(h∗) + ϵ(m) and, under some conditions, they proved a bound of 5R(h∗) + ϵ(F, m) on the rejection rate. Our algorithm can be viewed as an extreme variation of the Freund et al. method. We include in our “ensemble” only hypotheses with sufﬁciently low empirical error and we abstain if the weighted average of all predictions is not deﬁnitive ( ̸= ±1). Our risk and coverage bounds are asymptotically tighter. Excess risk bounds were developed by Herbei and Wegkamp [19] for a model where each rejection incurs a cost 0 ≤d ≤1/2. Their bound applies to any empirical risk minimizer over a hypothesis class of ternary hypotheses (whose output is in {±1, reject}). See also various extensions [20, 21]. A rejection mechanism for SVMs based on distance from decision boundary is perhaps the most widely known and used rejection technique. It is routinely used in medical applications [22, 23, 24]. Few papers proposed alternative techniques for rejection in the case of SVMs. Those include taking the reject area into account during optimization [25], training two SVM classiﬁers with asymmetric cost [26], and using a hinge loss [20]. Grandvalet et al. [16] proposed an efﬁcient implementation of SVM with a reject option using a double hinge loss. They empirically compared their results with two other selective classiﬁers: the one proposed by Bartlett and Wegkamp [20] and the traditional rejection based on distance from decision boundary. In their experiments there was no statistically signiﬁcant advantage to either method compared to the traditional approach for high rejection rates. 10 Conclusion We presented and analyzed a learning strategy for selective classiﬁcation that achieves weak optimality. We showed that the coverage rate directly depends on the disagreement coefﬁcient, thus linking between active learning and selective classiﬁcation. Recently it has been shown that, for the noise-free case, active learning can be reduced to selective classiﬁcation [27]. We conjecture that such a reduction also holds in noisy settings. Exact implementation of our strategy, or exact computation of the disbelief index may be too difﬁcult to achieve or even obtain with approximation guarantees. We presented one algorithm that heuristically approximate the required behavior and there is certainly room for other, perhaps better methods and variants. Our empirical examination of the proposed algorithm indicate that it can provide signiﬁcant and consistent advantage over the traditional rejection technique with SVMs. This advantage can be of great value especially in medical diagnosis applications and other mission critical classiﬁcation tasks. The algorithm itself can be implemented using off-the-shelf packages. Acknowledgments This work was supported in part by the IST Programme of the European Community, under the PASCAL2 Network of Excellence, IST-2007-216886. This publication only reﬂects the authors’ views. 8 References [1] R. El-Yaniv and Y. Wiener. On the foundations of noise-free selective classiﬁcation. JMLR, 11:1605– 1641, 2010. [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] S. Hanneke. A bound on the label complexity of agnostic active learning. In ICML, pages 353–360, 2007. [5] S. Hanneke. Theoretical Foundations of Active Learning. PhD thesis, Carnegie Mellon University, 2009. [6] P.L. Bartlett, S. Mendelson, and P. Philips. Local complexities for empirical risk minimization. In COLT: Proceedings of the Workshop on Computational Learning Theory, Morgan Kaufmann Publishers, 2004. [7] V. Koltchinskii. 2004 IMS medallion lecture: Local rademacher complexities and oracle inequalities in risk minimization. Annals of Statistics, 34:2593–2656, 2006. [8] P.L. Bartlett and S. Mendelson. Discussion of ”2004 IMS medallion lecture: Local rademacher complexities and oracle inequalities in risk minimization” by V. koltchinskii. Annals of Statistics, 34:2657–2663, 2006. [9] A.B. Tsybakov. Optimal aggregation of classiﬁers in statistical learning. Annals of Mathematical Statistics, 32:135–166, 2004. [10] 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. [11] O. Bousquet, S. Boucheron, and G. Lugosi. Introduction to statistical learning theory. In Advanced Lectures on Machine Learning, volume 3176 of Lecture Notes in Computer Science, pages 169–207. Springer, 2003. [12] L. Wang. Smoothness, disagreement coefﬁcient, and the label complexity of agnostic active learning. JMLR, pages 2269–2292, 2011. [13] S. Dasgupta, D. Hsu, and C. Monteleoni. A general agnostic active learning algorithm. In NIPS, 2007. [14] A. Beygelzimer, D. Hsu, J. Langford, and T. Zhang. Agnostic active learning without constraints. Advances in Neural Information Processing Systems 23, 2010. [15] 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”. [16] Y. Grandvalet, A. Rakotomamonjy, J. Keshet, and S. Canu. Support vector machines with a reject option. In NIPS, pages 537–544. MIT Press, 2008. [17] American Cancer Society. Cancer facts and ﬁgures. 2010. [18] Y. Freund, Y. Mansour, and R.E. Schapire. Generalization bounds for averaged classiﬁers. Annals of Statistics, 32(4):1698–1722, 2004. [19] R. Herbei and M.H. Wegkamp. Classiﬁcation with reject option. The Canadian Journal of Statistics, 34(4):709–721, 2006. [20] P.L. Bartlett and M.H. Wegkamp. Classiﬁcation with a reject option using a hinge loss. Journal of Machine Learning Research, 9:1823–1840, 2008. [21] M.H. Wegkap. Lasso type classiﬁers with a reject option. Electronic Journal of Statistics, 1:155–168, 2007. [22] S. Mukherjee, P. Tamayo, D. Slonim, A. Verri, T. Golub, J. P. Mesirov, and T. Poggio. Support vector machine classiﬁcation of microarray data. Technical report, AI Memo 1677, Massachusetts Institute of Technology, 1998. [23] I. Guyon, J. Weston, S. Barnhill, and V. Vapnik. Gene selection for cancer classiﬁcation using support vector machines. Machine Learning, pages 389–422, 2002. [24] S. Mukherjee. Chapter 9. classifying microarray data using support vector machines. In of scientists from the University of Pennsylvania School of Medicine and the School of Engineering and Applied Science. Kluwer Academic Publishers, 2003. [25] G. Fumera and F. Roli. Support vector machines with embedded reject option. In Pattern Recognition with Support Vector Machines: First International Workshop, pages 811–919, 2002. [26] R. Sousa, B. Mora, and J.S. Cardoso. An ordinal data method for the classiﬁcation with reject option. In ICMLA, pages 746–750. IEEE Computer Society, 2009. [27] R. El-Yaniv and Y. Wiener. Active learning via perfect selective classiﬁcation. Accepted to JMLR, 2011. 9	2011	93
4,445	Additive Gaussian Processes David Duvenaud Department of Engineering Cambridge University dkd23@cam.ac.uk Hannes Nickisch MPI for Intelligent Systems T¨ubingen, Germany hn@tue.mpg.de Carl Edward Rasmussen Department of Engineering Cambridge University cer54@cam.ac.uk Abstract We introduce a Gaussian process model of functions which are additive. An additive function is one which decomposes into a sum of low-dimensional functions, each depending on only a subset of the input variables. Additive GPs generalize both Generalized Additive Models, and the standard GP models which use squared-exponential kernels. Hyperparameter learning in this model can be seen as Bayesian Hierarchical Kernel Learning (HKL). We introduce an expressive but tractable parameterization of the kernel function, which allows efﬁcient evaluation of all input interaction terms, whose number is exponential in the input dimension. The additional structure discoverable by this model results in increased interpretability, as well as state-of-the-art predictive power in regression tasks. 1 Introduction Most statistical regression models in use today are of the form: g(y) = f(x1)+f(x2)+· · ·+f(xD). Popular examples include logistic regression, linear regression, and Generalized Linear Models [1]. This family of functions, known as Generalized Additive Models (GAM) [2], are typically easy to ﬁt and interpret. Some extensions of this family, such as smoothing-splines ANOVA [3], add terms depending on more than one variable. However, such models generally become intractable and difﬁcult to ﬁt as the number of terms increases. At the other end of the spectrum are kernel-based models, which typically allow the response to depend on all input variables simultaneously. These have the form: y = f(x1, x2, . . . , xD). A popular example would be a Gaussian process model using a squared-exponential (or Gaussian) kernel. We denote this model as SE-GP. This model is much more ﬂexible than the GAM, but its ﬂexibility makes it difﬁcult to generalize to new combinations of input variables. In this paper, we introduce a Gaussian process model that generalizes both GAMs and the SE-GP. This is achieved through a kernel which allow additive interactions of all orders, ranging from ﬁrst order interactions (as in a GAM) all the way to Dth-order interactions (as in a SE-GP). Although this kernel amounts to a sum over an exponential number of terms, we show how to compute this kernel efﬁciently, and introduce a parameterization which limits the number of hyperparameters to O(D). A Gaussian process with this kernel function (an additive GP) constitutes a powerful model that allows one to automatically determine which orders of interaction are important. We show that this model can signiﬁcantly improve modeling efﬁcacy, and has major advantages for model interpretability. This model is also extremely simple to implement, and we provide example code. We note that a similar breakthrough has recently been made, called Hierarchical Kernel Learning (HKL) [4]. HKL explores a similar class of models, and sidesteps the possibly exponential number of interaction terms by cleverly selecting only a tractable subset. However, this method suffers considerably from the fact that cross-validation must be used to set hyperparameters. In addition, the machinery necessary to train these models is immense. Finally, on real datasets, HKL is outperformed by the standard SE-GP [4]. 1 −4 −2 0 2 4 −4 −2 0 2 4 0 0.2 0.4 0.6 0.8 1 + −4 −2 0 2 4 −4 −2 0 2 4 0 0.2 0.4 0.6 0.8 1 = −4 −2 0 2 4 −4 −2 0 2 4 0 0.2 0.4 0.6 0.8 1 −4 −2 0 2 4 −4 −2 0 2 4 0 0.2 0.4 0.6 0.8 1 k1(x1) k2(x2) k1(x1) + k2(x2) k1(x1)k2(x2) 1D kernel 1D kernel 1st order kernel 2nd order kernel ↓ ↓ ↓ ↓ −4 −2 0 2 4 −4 −2 0 2 4 −1.5 −1 −0.5 0 0.5 1 1.5 + −4 −2 0 2 4 −4 −2 0 2 4 0 0.5 1 1.5 2 = −4 −2 0 2 4 −4 −2 0 2 4 −1 0 1 2 3 4 −4 −2 0 2 4 −4 −2 0 2 4 −3 −2 −1 0 1 2 f1(x1) f2(x2) f1(x1) + f2(x2) f(x1, x2) draw from draw from draw from draw from 1D GP prior 1D GP prior 1st order GP prior 2nd order GP prior Figure 1: A ﬁrst-order additive kernel, and a product kernel. Left: a draw from a ﬁrst-order additive kernel corresponds to a sum of draws from one-dimensional kernels. Right: functions drawn from a product kernel prior have weaker long-range dependencies, and less long-range structure. 2 Gaussian Process Models Gaussian processes are a ﬂexible and tractable prior over functions, useful for solving regression and classiﬁcation tasks [5]. The kind of structure which can be captured by a GP model is mainly determined by its kernel: the covariance function. One of the main difﬁculties in specifying a Gaussian process model is in choosing a kernel which can represent the structure present in the data. For small to medium-sized datasets, the kernel has a large impact on modeling efﬁcacy. Figure 1 compares, for two-dimensional functions, a ﬁrst-order additive kernel with a second-order kernel. We can see that a GP with a ﬁrst-order additive kernel is an example of a GAM: Each function drawn from this model is a sum of orthogonal one-dimensional functions. Compared to functions drawn from the higher-order GP, draws from the ﬁrst-order GP have more long-range structure. We can expect many natural functions to depend only on sums of low-order interactions. For example, the price of a house or car will presumably be well approximated by a sum of prices of individual features, such as a sun-roof. Other parts of the price may depend jointly on a small set of features, such as the size and building materials of a house. Capturing these regularities will mean that a model can conﬁdently extrapolate to unseen combinations of features. 3 Additive Kernels We now give a precise deﬁnition of additive kernels. We ﬁrst assign each dimension i ∈{1 . . . D} a one-dimensional base kernel ki(xi, x′ i). We then deﬁne the ﬁrst order, second order and nth order additive kernel as: kadd1(x, x′) = σ2 1 D X i=1 ki(xi, x′ i) (1) kadd2(x, x′) = σ2 2 D X i=1 D X j=i+1 ki(xi, x′ i)kj(xj, x′ j) (2) kaddn(x, x′) = σ2 n X 1≤i1<i2<...<in≤D N Y d=1 kid(xid, x′ id) (3) 2 where D is the dimension of our input space, and σ2 n is the variance assigned to all nth order interactions. The nth covariance function is a sum of D n terms. In particular, the Dth order additive covariance function has D D = 1 term, a product of each dimension’s covariance function: kaddD(x, x′) = σ2 D D Y d=1 kd(xd, x′ d) (4) In the case where each base kernel is a one-dimensional squared-exponential kernel, the Dth-order term corresponds to the multivariate squared-exponential kernel: kaddD(x, x′) = σ2 D D Y d=1 kd(xd, x′ d) = σ2 D D Y d=1 exp −(xd −x′ d)2 2l2 d = σ2 D exp − D X d=1 (xd −x′ d)2 2l2 d (5) also commonly known as the Gaussian kernel. The full additive kernel is a sum of the additive kernels of all orders. 3.1 Parameterization The only design choice necessary in specifying an additive kernel is the selection of a onedimensional base kernel for each input dimension. Any parameters (such as length-scales) of the base kernels can be learned as usual by maximizing the marginal likelihood of the training data. In addition to the hyperparameters of each dimension-wise kernel, additive kernels are equipped with a set of D hyperparameters σ2 1 . . . σ2 D controlling how much variance we assign to each order of interaction. These “order variance” hyperparameters have a useful interpretation: The dth order variance hyperparameter controls how much of the target function’s variance comes from interactions of the dth order. Table 1 shows examples of normalized order variance hyperparameters learned on real datasets. Table 1: Relative variance contribution of each order in the additive model, on different datasets. Here, the maximum order of interaction is set to 10, or smaller if the input dimension less than 10. Values are normalized to sum to 100. Order of interaction 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th pima 0.1 0.1 0.1 0.3 1.5 96.4 1.4 0.0 liver 0.0 0.2 99.7 0.1 0.0 0.0 heart 77.6 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.1 22.0 concrete 70.6 13.3 13.8 2.3 0.0 0.0 0.0 0.0 pumadyn-8nh 0.0 0.1 0.1 0.1 0.1 0.1 0.1 99.5 servo 58.7 27.4 0.0 13.9 housing 0.1 0.6 80.6 1.4 1.8 0.8 0.7 0.8 0.6 12.7 On different datasets, the dominant order of interaction estimated by the additive model varies widely. An additive GP with all of its variance coming from the 1st order is equivalent to a GAM; an additive GP with all its variance coming from the Dth order is equivalent to a SE-GP. Because the hyperparameters can specify which degrees of interaction are important, the additive GP is an extremely general model. If the function we are modeling is, in fact, decomposable into a sum of low-dimensional functions, our model can discover this fact (see Figure 5) and exploit it. If this is not the case, the hyperparameters can specify a suitably ﬂexible model. 3.2 Interpretability As noted by Plate [6], one of the chief advantages of additive models such as GAM is their interpretability. Plate also notes that by allowing high-order interactions as well as low-order interactions, one can trade off interpretability with predictive accuracy. In the case where the hyperparameters indicate that most of the variance in a function can be explained by low-order interactions, it is useful and easy to plot the corresponding low-order functions, as in Figure 2. 3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Water Strength −1 0 1 2 3 4 5 6 7 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Age Strength 0 2 4 6 −2 −1 0 1 2 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 Age Water Strength Figure 2: Low-order functions on the concrete dataset. Left, Centre: By considering only ﬁrst-order terms of the additive kernel, we recover a form of Generalized Additive Model, and can plot the corresponding 1-dimensional functions. Green points indicate the original data, blue points are data after the mean contribution from the other dimensions’ ﬁrst-order terms has been subtracted. The black line is the posterior mean of a GP with only one term in its kernel. Right: The posterior mean of a GP with only one second-order term in its kernel. 3.3 Efﬁcient Evaluation of Additive Kernels An additive kernel over D inputs with interactions up to order n has O(2n) terms. Na¨ıvely summing over these terms quickly becomes intractable. In this section, we show how one can evaluate the sum over all terms in O(D2). The nth order additive kernel corresponds to the nth elementary symmetric polynomial [7] [8], which we denote en. For example: if x has 4 input dimensions (D = 4), and if we let zi = ki(xi, x′ i), then kadd1(x, x′) = e1(z1, z2, z3, z4) = z1 + z2 + z3 + z4 kadd2(x, x′) = e2(z1, z2, z3, z4) = z1z2 + z1z3 + z1z4 + z2z3 + z2z4 + z3z4 kadd3(x, x′) = e3(z1, z2, z3, z4) = z1z2z3 + z1z2z4 + z1z3z4 + z2z3z4 kadd4(x, x′) = e4(z1, z2, z3, z4) = z1z2z3z4 The Newton-Girard formulae give an efﬁcient recursive form for computing these polynomials. If we deﬁne sk to be the kth power sum: sk(z1, z2, . . . , zD) = PD i=1 zk i , then kaddn(x, x′) = en(z1, . . . , zD) = 1 n n X k=1 (−1)(k−1)en−k(z1, . . . , zD)sk(z1, . . . , zD) (6) Where e0 ≜1. The Newton-Girard formulae have time complexity O(D2), while computing a sum over an exponential number of terms. Conveniently, we can use the same trick to efﬁciently compute all of the necessary derivatives of the additive kernel with respect to the base kernels. We merely need to remove the kernel of interest from each term of the polynomials: ∂kaddn ∂zj = en−1(z1, . . . , zj−1, zj+1, . . . zD) (7) This trick allows us to optimize the base kernel hyperparameters with respect to the marginal likelihood. 3.4 Computation The computational cost of evaluating the Gram matrix of a product kernel (such as the SE kernel) is O(N 2D), while the cost of evaluating the Gram matrix of the additive kernel is O(N 2DR), where R is the maximum degree of interaction allowed (up to D). In higher dimensions, this can be a signiﬁcant cost, even relative to the ﬁxed O(N 3) cost of inverting the Gram matrix. However, as our experiments show, typically only the ﬁrst few orders of interaction are important for modeling a given function; hence if one is computationally limited, one can simply limit the maximum degree of interaction without losing much accuracy. 4 1 2 3 4 12 13 14 23 24 34 123 124 134 234 ∅ 1234 1 2 3 4 12 13 14 23 24 34 123 124 134 234 ∅ 1234 1 2 3 4 12 13 14 23 24 34 123 124 134 234 ∅ 1234 1 2 3 4 12 13 14 23 24 34 123 124 134 234 ∅ 1234 HKL kernel GP-GAM kernel Squared-exp GP Additive GP kernel kernel Figure 3: A comparison of different models. Nodes represent different interaction terms, ranging from ﬁrst-order to fourth-order interactions. Far left: HKL can select a hull of interaction terms, but must use a pre-determined weighting over those terms. Far right: the additive GP model can weight each order of interaction seperately. Neither the HKL nor the additive model dominate one another in terms of ﬂexibility, however the GP-GAM and the SE-GP are special cases of additive GPs. Additive Gaussian processes are particularly appealing in practice because their use requires only the speciﬁcation of the base kernel. All other aspects of GP inference remain the same. All of the experiments in this paper were performed using the standard GPML toolbox1; code to perform all experiments is available at the author’s website.2 4 Related Work Plate [6] constructs a form of additive GP, but using only the ﬁrst-order and Dth order terms. This model is motivated by the desire to trade off the interpretability of ﬁrst-order models, with the ﬂexibility of full-order models. Our experiments show that often, the intermediate degrees of interaction contribute most of the variance. A related functional ANOVA GP model [9] decomposes the mean function into a weighted sum of GPs. However, the effect of a particular degree of interaction cannot be quantiﬁed by that approach. Also, computationally, the Gibbs sampling approach used in [9] is disadvantageous. Christoudias et al. [10] previously showed how mixtures of kernels can be learnt by gradient descent in the Gaussian process framework. They call this Bayesian localized multiple kernel learning. However, their approach learns a mixture over a small, ﬁxed set of kernels, while our method learns a mixture over all possible products of those kernels. 4.1 Hierarchical Kernel Learning Bach [4] uses a regularized optimization framework to learn a weighted sum over an exponential number of kernels which can be computed in polynomial time. The subsets of kernels considered by this method are restricted to be a hull of kernels.3 Given each dimension’s kernel, and a pre-deﬁned weighting over all terms, HKL performs model selection by searching over hulls of interaction terms. In [4], Bach also ﬁxes the relative weighting between orders of interaction with a single term α, computing the sum over all orders by: ka(x, x′) = v2 D D Y d=1 (1 + αkd(xd, x′ d)) (8) which has computational complexity O(D). However, this formulation forces the weight of all nth order terms to be weighted by αn. Figure 3 contrasts the HKL hull-selection method with the Additive GP hyperparameter-learning method. Neither method dominates the other in ﬂexibility. The main difﬁculty with the approach 1Available at http://www.gaussianprocess.org/gpml/code/ 2http://mlg.eng.cam.ac.uk/duvenaud/ 3In the setting we are considering in this paper, a hull can be deﬁned as a subset of all terms such that if term Q j∈J kj(x, x′) is included in the subset, then so are all terms Q j∈J/i kj(x, x′), for all i ∈J. For details, see [4]. 5 of [4] is that hyperparameters are hard to set other than by cross-validation. In contrast, our method optimizes the hyperparameters of each dimension’s base kernel, as well as the relative weighting of each order of interaction. 4.2 ANOVA Procedures Vapnik [11] introduces the support vector ANOVA decomposition, which has the same form as our additive kernel. However, they recommend approximating the sum over all D orders with only one term “of appropriate order”, presumably because of the difﬁculty of setting the hyperparameters of an SVM. Stitson et al. [12] performed experiments which favourably compared the support vector ANOVA decomposition to polynomial and spline kernels. They too allowed only one order to be active, and set hyperparameters by cross-validation. A closely related procedure from the statistics literature is smoothing-splines ANOVA (SS-ANOVA) [3]. An SS-ANOVA model is estimated as a weighted sum of splines along each dimension, plus a sum of splines over all pairs of dimensions, all triplets, etc, with each individual interaction term having a separate weighting parameter. Because the number of terms to consider grows exponentially in the order, in practice, only terms of ﬁrst and second order are usually considered. Learning in SS-ANOVA is usually done via penalized-maximum likelihood with a ﬁxed sparsity hyperparameter. In contrast to these procedures, our method can easily include all D orders of interaction, each weighted by a separate hyperparameter. As well, we can learn kernel hyperparameters individually per input dimension, allowing automatic relevance determination to operate. 4.3 Non-local Interactions By far the most popular kernels for regression and classiﬁcation tasks on continuous data are the squared exponential (Gaussian) kernel, and the Mat´ern kernels. These kernels depend only on the scaled Euclidean distance between two points, both having the form: k(x, x′) = f(PD d=1 (xd −x′ d)2 /l2 d). Bengio et al. [13] argue that models based on squared-exponential kernels are particularily susceptible to the curse of dimensionality. They emphasize that the locality of the kernels means that these models cannot capture non-local structure. They argue that many functions that we care about have such structure. Methods based solely on local kernels will require training examples at all combinations of relevant inputs. 1st order interactions 2nd order interactions 3rd order interactions All interactions k1 + k2 + k3 k1k2 + k2k3 + k1k3 k1k2k3 (Squared-exp kernel) (Additive kernel) Figure 4: Isocontours of additive kernels in 3 dimensions. The third-order kernel only considers nearby points relevant, while the lower-order kernels allow the output to depend on distant points, as long as they share one or more input value. Additive kernels have a much more complex structure, and allow extrapolation based on distant parts of the input space, without spreading the mass of the kernel over the whole space. For example, additive kernels of the second order allow strong non-local interactions between any points which are similar in any two input dimensions. Figure 4 provides a geometric comparison between squaredexponential kernels and additive kernels in 3 dimensions. 6 5 Experiments 5.1 Synthetic Data Because additive kernels can discover non-local structure in data, they are exceptionally well-suited to problems where local interpolation fails. Figure 5 shows a dataset which demonstrates this feature −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 x1 f1(x1) −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 x2 f2(x2) True Function Squared-exp GP Additive GP Additive GP & data locations posterior mean posterior mean 1st-order functions Figure 5: Long-range inference in functions with additive structure. of additive GPs, consisting of data drawn from a sum of two axis-aligned sine functions. The training set is restricted to a small, L-shaped area; the test set contains a peak far from the training set locations. The additive GP recovered both of the original sine functions (shown in green), and inferred correctly that most of the variance in the function comes from ﬁrst-order interactions. The ability of additive GPs to discover long-range structure suggests that this model may be well-suited to deal with covariate-shift problems. 5.2 Experimental Setup On a diverse collection of datasets, we compared ﬁve different models. In the results tables below, GP Additive refers to a GP using the additive kernel with squared-exp base kernels. For speed, we limited the maximum order of interaction in the additive kernels to 10. GP-GAM denotes an additive GP model with only ﬁrst-order interactions. GP Squared-Exp is a GP model with a squaredexponential ARD kernel. HKL4 was run using the all-subsets kernel, which corresponds to the same set of kernels as considered by the additive GP with a squared-exp base kernel. For all GP models, we ﬁt hyperparameters by the standard method of maximizing training-set marginal likelihood, using L-BFGS [14] for 500 iterations, allowing ﬁve random restarts. In addition to learning kernel hyperparameters, we ﬁt a constant mean function to the data. In the classiﬁcation experiments, GP inference was done using Expectation Propagation [15]. 5.3 Results Tables 2, 3, 4 and 5 show mean performance across 10 train-test splits. Because HKL does not specify a noise model, it could not be included in the likelihood comparisons. Table 2: Regression Mean Squared Error Method bach concrete pumadyn-8nh servo housing Linear Regression 1.031 0.404 0.641 0.523 0.289 GP GAM 1.302 0.142 0.602 0.281 0.179 HKL 0.199 0.147 0.346 0.199 0.151 GP Squared-exp 0.045 0.159 0.317 0.124 0.092 GP Additive 0.045 0.097 0.317 0.110 0.102 The model with best performance on each dataset is in bold, along with all other models that were not signiﬁcantly different under a paired t-test. The additive model never performs signiﬁcantly worse than any other model, and sometimes performs signiﬁcantly better than all other models. The 4Code for HKL available at http://www.di.ens.fr/˜fbach/hkl/ 7 Table 3: Regression Negative Log Likelihood Method bach concrete pumadyn-8nh servo housing Linear Regression 2.430 1.403 1.881 1.678 1.052 GP GAM 1.746 0.433 1.167 0.800 0.563 GP Squared-exp −0.131 0.412 0.843 0.425 0.208 GP Additive −0.131 0.181 0.843 0.309 0.161 Table 4: Classiﬁcation Percent Error Method breast pima sonar ionosphere liver heart Logistic Regression 7.611 24.392 26.786 16.810 45.060 16.082 GP GAM 5.189 22.419 15.786 8.524 29.842 16.839 HKL 5.377 24.261 21.000 9.119 27.270 18.975 GP Squared-exp 4.734 23.722 16.357 6.833 31.237 20.642 GP Additive 5.566 23.076 15.714 7.976 30.060 18.496 Table 5: Classiﬁcation Negative Log Likelihood Method breast pima sonar ionosphere liver heart Logistic Regression 0.247 0.560 4.609 0.878 0.864 0.575 GP GAM 0.163 0.461 0.377 0.312 0.569 0.393 GP Squared-exp 0.146 0.478 0.425 0.236 0.601 0.480 GP Additive 0.150 0.466 0.409 0.295 0.588 0.415 difference between all methods is larger in the case of regression experiments. The performance of HKL is consistent with the results in [4], performing competitively but slightly worse than SE-GP. Because the additive GP is a superset of both the GP-GAM model and the SE-GP model, instances where the additive GP model performs signiﬁcantly worse are presumably due to over-ﬁtting, or due to the hyperparameter optimization becoming stuck in a local maximum. Additive GP performance can be expected to beneﬁt signiﬁcantly from integrating out the kernel hyperparameters. 6 Conclusion We present additive Gaussian processes: a simple family of models which generalizes two widelyused classes of models. Additive GPs also introduce a tractable new type of structure into the GP framework. Our experiments indicate that such additive structure is present in real datasets, allowing our model to perform better than standard GP models. In the case where no such structure exists, our model can recover arbitrarily ﬂexible models, as well. In addition to improving modeling efﬁcacy, the additive GP also improves model interpretability: the order variance hyperparameters indicate which sorts of structure are present in our model. Compared to HKL, which is the only other tractable procedure able to capture the same types of structure, our method beneﬁts from being able to learn individual kernel hyperparameters, as well as the weightings of different orders of interaction. Our experiments show that additive GPs are a state-of-the-art regression model. Acknowledgments The authors would like to thank John J. Chew and Guillaume Obozonksi for their helpful comments. 8 References [1] J.A. Nelder and R.W.M. Wedderburn. Generalized linear models. Journal of the Royal Statistical Society. Series A (General), 135(3):370–384, 1972. [2] T.J. Hastie and R.J. Tibshirani. Generalized additive models. Chapman & Hall/CRC, 1990. [3] G. Wahba. Spline models for observational data. Society for Industrial Mathematics, 1990. [4] Francis Bach. High-dimensional non-linear variable selection through hierarchical kernel learning. CoRR, abs/0909.0844, 2009. [5] C.E. Rasmussen and CKI Williams. Gaussian Processes for Machine Learning. The MIT Press, Cambridge, MA, USA, 2006. [6] T.A. Plate. Accuracy versus interpretability in ﬂexible modeling: Implementing a tradeoff using Gaussian process models. Behaviormetrika, 26:29–50, 1999. [7] I.G. Macdonald. Symmetric functions and Hall polynomials. Oxford University Press, USA, 1998. [8] R.P. Stanley. Enumerative combinatorics. Cambridge University Press, 2001. [9] C.G. Kaufman and S.R. Sain. Bayesian functional anova modeling using Gaussian process prior distributions. Bayesian Analysis, 5(1):123–150, 2010. [10] M. Christoudias, R. Urtasun, and T. Darrell. Bayesian localized multiple kernel learning. 2009. [11] V.N. Vapnik. Statistical learning theory, volume 2. Wiley New York, 1998. [12] M. Stitson, A. Gammerman, V. Vapnik, V. Vovk, C. Watkins, and J. Weston. Support vector regression with ANOVA decomposition kernels. Advances in kernel methods: Support vector learning, pages 285–292, 1999. [13] Y. Bengio, O. Delalleau, and N. Le Roux. The curse of highly variable functions for local kernel machines. Advances in neural information processing systems, 18, 2006. [14] J. Nocedal. Updating quasi-newton matrices with limited storage. Mathematics of computation, 35(151):773–782, 1980. [15] T.P. Minka. Expectation propagation for approximate Bayesian inference. In Uncertainty in Artiﬁcial Intelligence, volume 17, pages 362–369, 2001. 9	2011	94
4,446	A Collaborative Mechanism for Crowdsourcing Prediction Problems Jacob Abernethy Division of Computer Science University of California at Berkeley jake@cs.berkeley.edu Rafael M. Frongillo Division of Computer Science University of California at Berkeley raf@cs.berkeley.edu Abstract Machine Learning competitions such as the Netﬂix Prize have proven reasonably successful as a method of “crowdsourcing” prediction tasks. But these competitions have a number of weaknesses, particularly in the incentive structure they create for the participants. We propose a new approach, called a Crowdsourced Learning Mechanism, in which participants collaboratively “learn” a hypothesis for a given prediction task. The approach draws heavily from the concept of a prediction market, where traders bet on the likelihood of a future event. In our framework, the mechanism continues to publish the current hypothesis, and participants can modify this hypothesis by wagering on an update. The critical incentive property is that a participant will proﬁt an amount that scales according to how much her update improves performance on a released test set. 1 Introduction The last several years has revealed a new trend in Machine Learning: prediction and learning problems rolled into prize-driven competitions. One of the ﬁrst, and certainly the most well-known, was the Netﬂix prize released in the Fall of 2006. Netﬂix, aiming to improve the algorithm used to predict users’ preferences on its database of ﬁlms, released a dataset of 100M ratings to the public and asked competing teams to submit a list of predictions on a test set withheld from the public. Netﬂix offered $1,000,000 to the ﬁrst team achieving prediction accuracy exceeding a given threshold, a goal that was eventually met. This competitive model for solving a prediction task has been used for a range of similar competitions since, and there is even a new company (kaggle.com) that creates and hosts such competitions. Such prediction competitions have proven quite valuable for a couple of important reasons: (a) they leverage the abilities and knowledge of the public at large, commonly known as “crowdsourcing”, and (b) they provide an incentivized mechanism for an individual or team to apply their own knowledge and techniques which could be particularly beneﬁcial to the problem at hand. This type of prediction competition provides a nice tool for companies and institutions that need help with a given prediction task yet can not afford to hire an expert. The potential leverage can be quite high: the Netﬂix prize winners apparently spent more than $1,000,000 in effort on their algorithm alone. Despite the extent of its popularity, is the Netﬂix competition model the ideal way to “crowdsource” a learning problem? We note several weaknesses: It is anti-collaborative. Competitors are strongly incentivized to keep their techniques private. This is in stark contrast to many other projects that rely on crowdsourcing – Wikipedia being a prime example, where participants must build off the work of others. Indeed, in the case of the Netﬂix prize, not only do leading participants lack incentives to share, but the work of non-winning competitors is effectively wasted. 1 The incentives are skewed and misaligned. The winner-take-all prize structure means that second place is as good as having not competed at all. This ultimately leads to an equilibrium where only a few teams are actually competing, and where potential new teams never form since catching up seems so unlikely. In addition, the ﬁxed achievement benchmark, set by Netﬂix as a 10% improvement in prediction RMSE over a baseline, leads to misaligned incentives. Effectively, the prize structure implies that an improvement of %9.9 percent is worth nothing to Netﬂix, whereas a 20% improvement is still only worth $1,000,000 to Netﬂix. This is clearly not optimal. The nature of the competition precludes the use of proprietary methods. By requiring that the winner reveal the winning algorithm, potential competitors utilizing non-open software or proprietary techniques will be unwilling to compete. By participating in the competition, a user must effectively give away his intellectual property. In this paper we describe a new and very general mechanism to crowdsource prediction/learning problems. Our mechanism requires participants to place bets, yet the space they are betting over is the set of hypotheses for the learning task at hand. At any given time the mechanism publishes the current hypothesis w and participants can wager on a modiﬁcation of w to w′, upon which the modiﬁed w′ is posted. Eventually the wagering period ﬁnishes, a set of test data is revealed, and each participant receives a payout according to their bets. The critical property is that every trader’s proﬁt scales according to how well their modiﬁcation improved the solution on the test data. The framework we propose has many qualities similar to that of an information or prediction market, and many of the ideas derive from recent research on the design of automated market makers [7, 8, 3, 4, 1]. Many information markets already exist; at sites like Intrade.com and Betfair.com, individuals can bet on everything ranging from election outcomes to geopolitical events. There has been a burst of interest in such markets in recent years, not least of which is due to their potential for combining large amounts of information from a range of sources. In the words of Hanson et al [9]: “Rational expectations theory predicts that, in equilibrium, asset prices will reﬂect all of the information held by market participants. This theorized information aggregation property of prices has lead economists to become increasingly interested in using securities markets to predict future events.” In practice, prediction markets have proven impressively accurate as a forecasting tool [11, 2, 12]. The central contribution of the present paper is to take the framework of a prediction market as a tool for information aggregation and to apply this tool for the purpose of “aggregating” a hypothesis (classiﬁer, predictor, etc.) for a given learning problem. The crowd of ML researchers, practitioners, and domain experts represents a highly diverse range of expertise and algorithmic tools. In contrast to the Netﬂix prize, which pitted teams of participants against each other, the mechanism we propose allows for everyone to contribute whatever knowledge they may have available towards the ﬁnal solution. In a sense, this approach decentralizes the process of solving the task, as individual experts can potentially apply their expertise to a subset of the problem on which they have an advantage. Whereas a market price can be thought of as representing a consensus estimate of the value of an asset, our goal is to construct a consensus hypothesis reﬂecting all the knowledge and capabilities about a particular learning problem1. Layout: We begin in Section 2.1 by introducing the simple notion of a generalized scoring rule L(·, ·) representing the “loss function” of the learning task at hand. In Section 2.2 we describe our proposed Crowdsourced Learning Mechanism (CLM) in detail, and discuss how to structure a CLM for a particular scoring function L, in order that the traders are given incentives to minimize L. In Section 3 we give an example based on the design of Huffman codes. In Section 4 we discuss previous work on the design of prediction markets using an automated prediction market maker (APMM). In Section 5 we ﬁnish by considering two learning settings (e.g. linear regression) and we construct a CLM for each. The proofs have been omitted throughout, but these are available in the full version of the present paper. Notation: Given a smooth strictly convex function R : Rd →R, and points x, y ∈dom(R), we deﬁne the Bregman divergence DR(x, y) as the quantity R(x) −R(y) −∇R(y) · (x −y). For any convex function R, we let R∗denote the convex conjugate of R, that is R∗(y) := supx∈dom(R) y · x −R(x). We shall use ∆(S) to refer to the set of integrable probability distributions over the set 1It is worth noting that Barbu and Lay utilized concepts from prediction markets to design algorithms for classiﬁer aggregation [10], although their approach was unrelated to crowdsourcing. 2 S, and ∆n to refer to the set of probability vectors p ∈Rn. The function H : ∆n →R shall denote the entropy function, that is H(p) := −Pn i=1 p(i) log p(i). We use the notation KL(p; q) to describe the relative entropy or Kullback-Leibler divergence between distributions p, q ∈∆n, that is KL(p; q) := Pn i=1 p(i) log p(i) q(i). We will also use ei ∈Rn to denote the ith standard basis vector, having a 1 in the ith coordinate and 0’s elsewhere. 2 Scoring Rules and Crowdsourced Learning Mechanisms 2.1 Generalized Scoring Rules For the remainder of this section, we shall let H denote some set of hypotheses, which we will assume is a convex subset of Rn. We let O be some arbitrary set of outcomes. We use the symbol X to refer to either an element of O, or a random variable taking values in O. We recall the notion of a scoring rule, a concept that arises frequently in economics and statistics [6]. Deﬁnition 1. Let P ⊆∆(O) be some convex set of distributions on an outcome space O. A scoring rule is a function S : P × O →R where, for all P ∈P, P ∈argmaxQ∈P EX∼P S(Q, X). In other words, if you are paid S(P, X) upon stating belief P ∈P and outcome X occurring, then you maximize your expected utility by stating your true belief. We offer a much weaker notion: Deﬁnition 2. Given a convex hypothesis space H ⊂Rn and an outcome space O, let L : H ×O → R be a continuous function. Given any P ∈∆(O), let WL(P) := argminw∈H EX∼P [L(w; X)]. Then we say that L is a Generalized Scoring Rule (GSR) if WL(P) is a nonempty convex set for every P ∈∆(O). The generalized scoring rule shall represent the “loss function” for the learning problem at hand, and in Section 2.2 we will see how L is utilized in the mechanism. The hypothesis w shall represent the advice we receive from the crowd, X shall represent the test data to be revealed at the close of the mechanism, and L(w; X) shall represent the loss of the advised w on the data X. Notice that we do not deﬁne L to be convex in its ﬁrst argument as this does not hold for many important cases. Instead, we require the weaker condition that EX[L(w; X)] is minimized on a convex set for any distribution on X. Our scoring rule differs from traditional scoring rules in an important way. Instead of starting with the desire know about the true value of X, and then designing a scoring rule which incentivizes participants to elicit their belief P ∈P, our objective is precisely to minimize our scoring rule. In other words, traditional scoring rules were a means to an end (eliciting P) but our generalized scoring rule is the end itself. One can recover the traditional scoring rule deﬁnition by setting H = P and imposing the constraint that P ∈WL(P). A useful class of GSRs L are those based on a Bregman divergence. Deﬁnition 3. We say that a GSR L : H × O →R is divergence-based if there exists an alternative hypothesis space H′ ⊂Rm, for some m, where we can write L(w; X) ≡DR(ρ(X), ψ(w)) + f(X) (1) for arbitrary maps ρ : O →H′, f : O →R, and ψ : H →H′, and any closed strictly convex R : H′ →R whose convex conjugate R∗is ﬁnite on all of Rm. This property allows us to think of L(w; X) as a kind of distance between ρ(X) and ψ(w). Clearly then, the minimum value of L for a given X will be attained when ψ(w) = ρ(X), given that DR(x, x) = 0 for any Bregman divergence. In fact, as the following proposition shows, we can even think of the expected value E[L(w; X)], as a distance between E[ρ(X)] and ψ(w). Proposition 1. Given a divergence-based GSR L(w; X) = DR(ρ(X), ψ(w)) + f(X) and a belief distribution P on O, we have WL(P) = ψ−1EX∼P [ρ(X)] . We now can see that the divergence-based property greatly simpliﬁes the task of minimizing L; instead of worrying about E[L(·; X)] one can simply base the hypothesis directly on the expectation E[ρ(X)]. As we will see in section 4, this also leads to efﬁcient prediction markets and crowdsourcing mechanisms. 3 2.2 The Crowdsourced Learning Mechanism We will now deﬁne our actual mechanism rigorously. Deﬁnition 4. A Crowdsourced Learning Mechanism (CLM) is the procedure in Algorithm 1 as deﬁned by the tuple (H, O, Cost, Payout). The function Cost : H × H →R sets the cost charged to a participant that makes a modiﬁcation to the posted hypothesis. The function Payout : H×H× O →R determines the amount paid to each participant when the outcome is revealed to be X. Algorithm 1 Crowdsourced Learning Mechanism for (H, O, Cost, Payout) 1: Mechanism sets initial hypothesis to some w0 ∈H 2: for rounds t = 0, 1, 2, . . . do 3: Mechanism posts current hypothesis wt ∈H 4: Some participant places a bid on the update wt 7→w′ 5: Mechanism charges participant Cost(wt, w′) 6: Mechanisms updates hypothesis wt+1 ←w′ 7: end for 8: Market closes after T rounds and the outcome (test data) X ∈O is revealed 9: for each t do 10: Participant responsible for the update wt 7→wt+1 receives Payout(wt, wt+1; X) 11: end for The above procedure describes the process by which participants can provide advice to the mechanism to select a good w, and the proﬁt they earn by doing so. Of course, this proﬁt will precisely determine the incentives of our mechanism, and hence a key question is: how can we design Cost and Payout so that participants are incentivized to provide good hypotheses? The answer is that we shall structure the incentives around a GSR L(w; X) chosen by the mechanism designer. Deﬁnition 5. For a CLM A = (H, O, Cost, Payout), denote the ex-post proﬁt for the bid (w 7→ w′) when the outcome is X ∈O by Profit(w, w′; X) := Payout(w, w′; X) −Cost(w, w′). We say that A implements a GSR L : H′ × O →R if there exists a surjective map ϕ : H →H′ such that for all w1, w2 ∈H and X ∈O, Profit(w1, w2; X) = L(ϕ(w1); X) −L(ϕ(w2); X). (2) If additionally H′ = H and ϕ = idH, we call A an L-CLM and say that A is L-incentivized. When a CLM implements a given L, the incentives are structured in order that the participants will work to minimize L(w; X). Of course, the input X is unknown to the participants, yet we can assume that the mechanism has provided a public “training set” to use in a learning algorithm. The participants are thus asked not only to propose a “good” hypothesis wt but to wager on whether the update wt−1 7→wt improves generalization error. It is worth making clear that knowledge of the true distribution on X provides a straightforward optimal strategy. Proposition 2. Given a GSR L : H ×O →R and an L-CLM (Cost, Payout), any participant who knows the true distribution P ∈P over X will maximize expected proﬁt by modifying the hypothesis to any w ∈WL(P). Cost of operating a CLM. It is clear that the agent operating the mechanism must pay the participants at the close of the competition, and is thus at risk of losing money (in fact, it is possible he may gain). How much money is lost depends on the bets (wt 7→wt+1) made by the participants, and of course the ﬁnal outcome X. The agent has a clear interest in knowing precisely the potential cost – fortunately this cost is easy to compute. The loss to the agent is clearly the total ex-post proﬁt earned by the participants, and by construction this sum telescopes: PT t=0 Profit(wt, wt+1; X) = L(w0; X) −L(wT ; X). This is a simple yet appealing property of the CLM: the agent pays only as much in reward to the participants as it beneﬁts from the improvement of wT over the initial w0. It is worth noting that this value could be negative when wT is actually “worse” than w0; in this case, as we shall see in section 3, the CLM can act as an insurance policy with respect to the mistakes of the participants. A more typical scenario, of course, is where the participants provide an improved hypothesis, in which case the CLM will run at a cost. We can compute the WorstCaseLoss(L-CLM) := maxw∈H,X∈O (L(w0; X) −L(w; X)). Given a budget 4 of size $B, the mechanism can always rescale L in order that WorstCaseLoss(L-CLM) = B. This requires, of course, that the WorstCaseLoss is ﬁnite. Computational efﬁciency of operating a CLM. We shall say that a CLM has the efﬁcient computation (EC) property if both Cost and Payout are efﬁciently computable functions. We shall say a CLM has the tractable trading (TT) property if, given a current hypothesis w, a belief P ∈∆(O) and a budget B, one can efﬁciently compute an element of the set argmin w′∈H n EX∼P Profit(w, w′, X) : Cost(w, w′) ≤B o . The EC property ensures that the mechanism operator can run the CLM efﬁciently. The TT property says that participants can compute the optimal hypothesis to bet on given a belief on the outcome and a budget. This is absolutely essential for the CLM to successfully aggregate the knowledge and expertise of the crowd – without it, despite their motivation to lower L(; ), the participants would not be able to compute the optimal bet. Suitable collateral requirements. We say that a CLM has the escrow (ES) property if the Cost and Payout functions are structured in order that, given any wager (w 7→w′), we have that Payout(w, w′; X) ≥0 for all X ∈O. It is clear that, when designing an L-CLM for a particular L, the Payout function is fully speciﬁed once Cost is ﬁxed, since we have the relation Payout(w, w′; X) = L(w; X) −L(w′; X) + Cost(w, w′) for every w, w′ ∈H and X ∈O. A curious reader might ask, why not simply set Cost(w, w′) ≡0 and Payout ≡Profit? The problem with this approach is that potentially Payout(w, w′; X) < 0 which implies that the participant who wagered on (w 7→w′) can be indebted to the mechanism and could default on this obligation. Thus the Cost function should be set in order to require every participant to deposit at least enough collateral in escrow to cover any possible losses. Subsidizing with a voucher pool. One practical weakness of a wagering-based mechanism is that individuals may be hesitant to participate when it requires depositing actual money into the system. This can be allayed to a reasonable degree by including a voucher pool where each of the ﬁrst m participants may receive a voucher in the amount of $C. These candidates need not pay to participate, yet have the opportunity to win. Of course, these vouchers must be paid for by the agent running the mechanism, and hence a value of mC is added to the total operational cost. 3 A Warm-up: Compressing an Unfamiliar Data Stream Let us now introduce a particular setting motivated by a well-known problem in information theory. Imagine a ﬁrm is looking to do compression on an unfamiliar channel, and from this channel the ﬁrm will receive a stream of m characters from an n-sized alphabet which we shall index by [n]. The goal is to select a binary encoding of this alpha in such a way that minimizes the total bits required to store the data, as a cost of $1 is required for each bit. A ﬁrst-order approach to encode such a stream is to assign a probability distribution q ∈∆n to the alphabet, and to select an encoding of character i with a binary word of length log(1/q(i)) (we ignore round-off for simplicity). This can be achieved using Huffman Codes for example, and we refer the reader to Cover and Thomas ([5], Chapter 5) for more details. Thus, given a distribution q, the ﬁrm pays L(q; i) = −log q(i) for each character i. It is easy to see that if the characters are sampled from some “true” distribution p, then the expected cost L(q; p) := Ei∼p [L(q; i)] = KL(p; q) + H(p), which is minimized at q = p. Not knowing the true distribution p, the ﬁrm is thus interested in ﬁnding a q with a low expected cost L(q; p). An attractive option available to the ﬁrm is to crowdsource the task of lowering this cost L(·; ·) by setting up an L-CLM. It is reasonably likely that outside individuals have private information about the behavior of the channel and, in particular, may be able to provide a better estimate q of the true distribution of the characters in the channel. As just discussed, the better the estimate the cheaper the compression. We set H = ∆n and O = [n], where a hypothesis q represents the proposed distribution over the n characters, and X is some character sampled uniformly from the stream after it has been observed. 5 We deﬁne Cost and Payout as Cost(q, q′) := max i∈[n] log(q(i)/q′(i)), Payout(q, q′; i) := log(q(i)/q′(i)) + Cost(q, q′), which is clearly an L-CLM for the loss deﬁned above. It is worth noting that L is a divergence-based GSR if we take R(q) = −H(q), ρ(i) = ei, f ≡0, ψ ≡id∆n, using the convention 0 log 0 = 0 (in fact, L is the LMSR). Finally, the ﬁrm will initially set q0 to be its best guess of p, which we will assume to be uniform (but need not be). We have devised this payout scheme according to the selection of a single character i, and it is worth noting that because this character is sampled uniformly at random from the stream (with private randomness), the participants cannot know which character will be released. This forces the participants to wager on the empirical distribution ˆp of the characters from the stream. A reasonable alternative, and one which lowers the payment variance, is to payout according to the L(q; ˆp), which is also equal to the average of L(q; i) when i is chosen uniformly from the stream. The obvious question to ask is: how does this CLM beneﬁt the ﬁrm that wants to design the encoding? More precisely, if the ﬁrm uses the ﬁnal estimate qT from the mechanism, instead of the initial guess q0, what is the trade-off between the money paid to participants and the money gained by using the crowdsourced hypothesis? At ﬁrst glance, it appears that this trade-off can be arbitrarily bad: the worst case cost of encoding the stream using the ﬁnal estimate qT is supi,qT −log(qT (i)) = ∞. Amazingly, however, by virtue of the aligned incentives, the ﬁrm has a very strong control of its total cost (the CLM cost plus the encoding cost). Suppose the ﬁrm scales L by a parameter α, to separate the scale of the CLM from the scale of the encoding cost (which we assumed to be $1 per bit). Then given any initial estimate q0 and ﬁnal estimate qT , the expected total cost over p is Total expected cost = Encoding cost of using qT given p z }\| { H(p) + KL(p; qT ) + Mechanism’s cost of getting advice qT z }\| { α(KL(p; q0) −KL(p; qT )) = H(p) + (1 −α)KL(p; qT ) + αKL(p; q0) Let us spend a moment to analyze the above expression. Imagine that the ﬁrm set α = 1. Then the total cost of the ﬁrm would be H(p) + KL(p; q0), which is bounded by log n for q0 uniform. Notice that this expression does not depend on qT – in fact, this cost precisely corresponds to the scenario where the ﬁrm had not set up a CLM and instead used the initial estimate q0 to encode. In other words, for α = 1, the ﬁrm is entirely neutral to the quality of the estimate qT ; even if the CLM provided an estimate qT which performed worse than q0, the cost increase due to the bad choice of q is recouped from payments of the ill-informed participants. The ﬁrm may not want to be neutral to the estimate of the crowd, however, and under the reasonable assumption that the ﬁnal estimate qT will improve upon q0, the ﬁrm should set 0 < α < 1 (of course, positivity is needed for nonzero payouts). In this case, the ﬁrm will strictly gain by using the CLM when KL(p; qT ) < KL(p; q0), but still has some insurance policy if the estimate qT is poor. 4 Prediction Markets as a Special Case Let us brieﬂy review the literature for the type of prediction markets relevant to the present work. In such a prediction market, we imagine a future event to reveal one of n uncertain outcomes. Hanson [7, 8] proposed a framework in which traders make “reports” to the market about their internal belief in the form of a distribution p ∈∆n. Each trader would receive a reward (or loss) based on a function of their proposed belief and the belief of the previous trader, and the function suggested by Hanson was the Logarithmic Market Scoring Rule (LMSR). It was shown later that the LMSR-based market is equivalent to what is known as a cost function based automated market makers, proposed by Chen and Pennock [3]. More recently a much broader equivalence was established by Chen and Wortman Vaughan [4] between markets based on cost functions and those based on scoring rules. The market framework proposed by Chen and Pennock allows traders to buy and sell Arrow-Debreu securities (equivalently: shares, contracts), where an Arrow-Debreu security corresponding to outcome i pays out $1 if and only if i is realized. All shares are bought and sold through an automated market maker, which is the entity managing the market and setting prices. At any time period, traders can purchase bundles of contracts r ∈Rn, where r(i) represents the number of shares purchased on 6 outcome i. The price of a bundle r is set as C(s + r) −C(s), where C is some differentiable convex cost function and s ∈Rn is the “quantity vector” representing the total number of outstanding shares. The LMSR cost function is C(s) := 1 η log (Pn i=1 exp(ηs(i))). This cost function framework was extended by Abernethy et al. [1] to deal with prohibitively large outcome spaces. When the set of potential outcomes O is of exponential size or even inﬁnite, the market designer can offer a restricted number of contracts, say n (≪\|O\|), rather than offer an Arrow-Debreu contract for each member of O. To determine the payout structure, the market designer chooses a function ρ : O →Rn, where contract i returns a payout of ρi(X) and, thus, a contract bundle r pays ρ(X) · r. As with the framework of Chen and Pennock, the contract prices are set according to a cost function C, so that a bundle r has a price of C(s + r) −C(s). The design of the function C is addressed at length in Abernethy et al., to which we refer the reader. For the remainder of this section we shall discuss the prediction market template of Abernethy et al. as it provides the most general model; we shall refer to such a market as an Automated Prediction Market Maker. We now precisely state the ingredients of this framework. Deﬁnition 6. An Automated Prediction Market Maker (APMM) is deﬁned by a tuple (S, O, ρ, C) where S is the share space of the market, which we will assume to be the linear space Rn; O is the set of outcomes; C : S →R is a smooth and convex cost function with ∇C(S) = relint(∇C(S)) (here, we use ∇C(S) := {∇C(s) \| s ∈S} to denote the derivative space of C); and ρ : O →∇C(S) is a payoff function2. Fortunately, we need not provide a full description of the procedure of the APMM mechanism: The APMM is precisely a special case of a CLM! Indeed, the APMM framework can be described as a CLM (H, O, Cost, Payout) where H = S(= Rn) Cost(s, s′) = C(s′) −C(s) Payout(s, s′; X) = ρ(X) · (s′ −s). (3) Hence we can think of APMM prediction markets in terms of our learning mechanism. Markets of this form are an important special class of CLMs – in particular, we can guarantee that they are efﬁcient to work with, as we show in the following proposition. Proposition 3. An APMM (S, O, ρ, C) with efﬁciently computable C satisﬁes EC and TT. We now ask, what is the learning problem that the participants of an APMM are trying to solve? More precisely, when we think of an APMM as a CLM, does it implement a particular L? Theorem 1. Given APMM A := (S, O, ρ, C), then A implements L : ∇C(S) × O →R deﬁned by L(w; X) = DC∗(ρ(X), w), (4) where C∗is the conjugate dual of the function C. There is another more subtle beneﬁt to APMMs – and, in fact, to most prediction market mechanisms in practice – which is that participants make bets via purchasing of shares or share bundles. When a trader makes a bet, she purchases a contract bundle r, is charged C(s + r) −C(s) (when the current quantity vector is s), and shall receive payout ρ(X) · r if and when X is realized. But at any point before X is observed and trading is open, the trader can sell off this bundle, to the APMM or another trader, and hence neutralize her risk. In this sense bets made in an APMM are stateless, whereas for an arbitrary CLM this may not be the case: the wager deﬁned by (wt 7→wt+1) can not necessarily be sold back to the mechanism, as the posted hypothesis may no longer remain at wt+1. Given a learning problem deﬁned by the GSR L : H × O →R, it is natural to ask whether we can design a CLM which implements this L and has this “share-based property” of APMMs. More precisely, under what conditions is it possible to implement L with an APMM? Theorem 2. For any divergence-based GSR L(w; X) = DR(ρ(X), ψ(w)) + f(X), with ψ : H → H′ one-to-one, H′ = relint(H′), and ρ(O) ⊆ψ(H), there exists an APMM which implements L. We point out, as a corollary, that if an APMM implements some arbitrary L, then we must be able to write L as a divergence function. This fully speciﬁes the class of problems solvable using APMMs. 2The conditions that ρ(O) ⊆∇C(S) and ∇C(S) = relint(∇C(S)) are technical but important, and we do not address these details in the present extended abstract although they will be considered in the full version. More relevant discussion can also be found in Abernethy et al. [1]. 7 Corollary 1. If APMM (S, O, ρ, C) implements a GSR L : H×O →R, then L is divergence-based. Theorem 1 establishes a strong connection between prediction markets and a natural class of GSRs. One interpretation of this result is that any GSR based on a Bregman divergence has a “dual” characterization as a share-based market, where participants buy and sell shares rather than directly altering the share prices (the hypothesis). This has many advantages for prediction markets, not least of which is that shares are often easier to think about than the underlying hypothesis space. Our notion of a CLM offers another interpretation. In light of Proposition 3, any machine learning problem whose hypotheses can be evaluated in terms of a divergence leads to a tractable crowdsourcing mechanism, as was the case in Section 3. Moreover, this theorem does not preclude efﬁcient yet non-divergence-based loss functions as we see in the next section. 5 Example CLMs for Typical Machine Learning Tasks Regression. We now construct a CLM for a typical regression problem. We let H be the ℓ2-norm ball of radius 1 in Rd, and we shall let an outcome be a batch of a data, that is X := {(x1, y1), . . . , (xn, yn)} where for each i we have xi ∈Rd, yi ∈[−1, 1], and we assume ∥xi∥2 ≤1. We construct a GSR according to the mean squared error, L(w; {(xi, yi)}n i=1) = α 2n Pn i=1(w · xi −yi)2 for some parameter α > 0. It is worth noting that L is not divergence-based. In order to satisfy the escrow property (ES), we can set Cost(w, w′) := 2α∥w −w′∥2 because the function L(w; X) is 2α-lipschitz with respect to w for any X. To ensure that the CLM is L-incentivized, we must set Payout(w, w′; X) := Cost(w, w′) + L(w; X) −L(w′; X). If we set the initial hypothesis w0 = 0, it is easy to check that WorstCaseLoss = α/2. It remains to check whether this CLM is tractable. It’s clear that we can efﬁciently compute Cost and Payout, hence the EC property holds. Given how Cost is deﬁned, it is clear that the set {w′ : Cost(w, w′) ≤B} is just an ℓ2-norm ball. Also, since L is convex in w for each X, so is the function EX∼P Profit(w, w′, X) for every P. A budget-constrained proﬁt-maximizing participant must simply solve a convex optimization problem, and hence the TT property holds. Betting Directly on the Labels. Let us return our attention to the Netﬂix Prize model as discussed in the Introduction. For this style of competition a host releases a dataset for a given prediction task. The host then requests participants to provide predictions on a speciﬁed set of instances on which it has correct labels. For every submission the agent computes an error measure, say the MSE, and reports this to the participants. Of course, the correct labels are withheld throughout. Our CLM framework is general enough to apply to this problem framework as well. Deﬁne H = O = Km where K ⊆R bounded is the set of valid labels, and m is the number of requested test set predictions. For some w ∈H and y ∈O, w(k) speciﬁes the kth predicted label, and y(k) speciﬁes the true label. A natural scoring function is the total squared loss, L(w; y) := Pm k=1(w(k)−y(k))2. Of course, this approach is quite different from the Netﬂix Prize model, in two key respects: (a) the participants have to wager on their predictions and (b) by participating in the mechanism they are required to reveal their modiﬁcation to all of the other players. Hence while we have structured a competitive process the participants are de facto forced to collaborate on the solution. A reasonable critique of this collaborative mechanism approach to a Netﬂix-style competition is that it does not provide the instant feedback of the “leaderboard” where individuals observe performance improvements in real time. However, we can adjust our mechanism to be online with a very simple modiﬁcation of the CLM protocol, which we sketch here. Rather than make payouts in a large batch at the end, the competition designer could perform a mini-payout at the end of each of a sequence of time intervals. At each interval, the designer could select a (potentially random) subset S of user/movie pairs in the remaining test set, freeze updates on the predictions w(k) for all k ∈S, and perform payouts to the participants on only these labels. What makes this possible, of course, is that the generalized scoring rule we chose decomposes as a sum over the individual labels. Acknowledgments. We gratefully acknowledge the support of the NSF under award DMS0830410, a Google University Research Award, and the National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a. 8 References [1] J. Abernethy, Y. Chen, and J. Wortman Vaughan. An optimization-based framework for automated market-making. 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4,447	Sparse Bayesian Multi-Task Learning C´edric Archambeau, Shengbo Guo, Onno Zoeter Xerox Research Centre Europe {Cedric.Archambeau, Shengbo.Guo, Onno.Zoeter}@xrce.xerox.com Abstract We propose a new sparse Bayesian model for multi-task regression and classiﬁcation. The model is able to capture correlations between tasks, or more speciﬁcally a low-rank approximation of the covariance matrix, while being sparse in the features. We introduce a general family of group sparsity inducing priors based on matrix-variate Gaussian scale mixtures. We show the amount of sparsity can be learnt from the data by combining an approximate inference approach with type II maximum likelihood estimation of the hyperparameters. Empirical evaluations on data sets from biology and vision demonstrate the applicability of the model, where on both regression and classiﬁcation tasks it achieves competitive predictive performance compared to previously proposed methods. 1 Introduction Learning multiple related tasks is increasingly important in modern applications, ranging from the prediction of tests scores in social sciences and the classiﬁcation of protein functions in systems biology to the categorisation of scenes in computer vision and more recently to web search and ranking. In many real life problems multiple related target variables need to be predicted from a single set of input features. A problem that attracted considerable interest in recent years is to label an image with (text) keywords based on the features extracted from that image [26]. In general, this multi-label classiﬁcation problem is challenging as the number of classes is equal to the vocabulary size and thus typically very large. While capturing correlations between the labels seems appealing it is in practice difﬁcult as it rapidly leads to numerical problems when estimating the correlations. A naive solution is to learn a model for each task separately and to make predictions using the independent models. Of course, this approach is unsatisfactory as it does not take advantage of all the information contained in the data. If the model is able to capture the task relatedness, it is expected to have generalisation capabilities that are drastically increased. This motivated the introduction of the multi-task learning paradigm that exploits the correlations amongst multiple tasks by learning them simultaneously rather than individually [12]. More recently, the abundant literature on multi-task learning demonstrated that performance indeed improves when the tasks are related [6, 31, 2, 14, 13]. The multi-task learning problem encompasses two main settings. In the ﬁrst one, for every input, every task produces an output. If we restrict ourselves to multiple regression for the time being, the most basic multi-task model would consider P correlated tasks1, the vector of covariates and targets being respectively denoted by xn ∈RD and yn ∈RP : yn = Wxn + µ + ϵn, ϵn ∼N(0, Σ), (1) where W ∈RP ×D is the matrix of weights and µ ∈RP the task offsets and ϵn ∈RP the vector residual errors with covariance Σ ∈RP ×P . In this setting, the output of all tasks is observed for 1While it is straightfoward to show that the maximum likelihood estimate of W would be the same as when considering uncorrelated noise, imposing any prior on W would lead to a different solution. 1 every input. In the second setting, the goal is to learn from a set of observed tasks and to generalise to a new task. This approach views the multi-task learning problem as a transfer learning problem, where it is assumed that the various tasks belong in some sense to the same environment and share common properties [23, 5]. In general only a single task output is observed for every input. A recent trend in multi-task learning is to consider sparse solutions to facilitate the interpretation. Many formulate the sparse multi-task learning problem in a (relaxed) convex optimization framework [5, 22, 35, 23]. If the regularization constant is chosen using cross-validation, regularizationbased approaches often overestimate the support [32] as they select more features than the set that generated the data. Alternatively, one can adopt a Bayesian approach to sparsity in the context of multi-task learning [29, 21]. The main advantage of the Bayesian formalism is that it enables us to learn the degree of sparsity supported by the data and does not require the user to specify the type of penalisation in advance. In this paper, we adopt the ﬁrst setting for multi-task learning, but we will consider a hierarchical Bayesian model where the entries of W are correlated so that the residual errors are uncorrelated. This is similar in spirit as the approach taken by [18], where tasks are related through a shared kernel matrix. We will consider a matrix-variate prior to simultaneously model task correlations and group sparsity in W. A matrix-variate Gaussian prior was used in [35] in a maximum likelihood setting to capture task correlations and feature correlations. While we are also interested in task correlations, we will consider matrix-variate Gaussian scale mixture priors centred at zero to drive entire blocks of W to zero. The Bayesian group LASSO proposed in [30] is a special case. Group sparsity [34] is especially useful in presence of categorical features, which are in general represented as groups of “dummy” variables. Finally, we will allow the covariance to be of low-rank so that we can deal with problems involving a very large number of tasks. 2 Matrix-variate Gaussian prior Before starting our discussion of the model, we introduce the matrix variate Gaussian as it plays a key role in our work. For a matrix W ∈RP ×D, the matrix-variate Gaussian density [16] with mean matrix M ∈RP ×D, row covariance Ω∈RD×D and column covariance Σ ∈RP ×P is given by N(M, Ω, Σ) ∝e−1 2 vec(W−M)⊤(Ω⊗Σ)−1vec(W−M) ∝e−1 2 tr{Ω−1(W−M)⊤Σ−1(W−M)}. (2) If we let Σ = E(W −M)(W −M)⊤, then Ω= E(W −M)⊤(W −M)/c where c ensures the density integrates to one. While this introduces a scale ambiguity between Σ and Ω(easily removed by means of a prior), the use of a matrix-variate formulation is appealing as it makes explicit the structure vec(W), which is a vector formed by the concatenation of the columns of W. This structure is reﬂected in its covariance matrix which is not of full rank, but is obtained by computing the Kronecker product of the row and the column covariance matrices. It is interesting to compare a matrix-variate prior for W in (1) with the classical multi-level approach to multiple regression from statistics (see e.g. [20]). In a standard multi-level model, the rows of W are drawn iid from a multivariate Gaussian with mean m and covariance S, and m is further drawn from zero mean Gaussian with covariance R. Integrating out m leads then to a Gaussian distributed vec(W) with mean zero and with a covariance matrix that has the block diagonal elements equal to S + R and all off-diagonal elements equal to R. Hence, the standard multi-level model assumes a very different covariance structure than the one based on (2) and incidentally cannot learn correlated and anti-correlated tasks simultaneously. 3 A general family of group sparsity inducing priors We seek a solution for which the expectation of W is sparse, i.e., blocks of W are driven to zero. A straightforward way to induce sparsity, and which would be equivalent to ℓ1-regularisation on blocks of W, is to consider a Laplace prior (or double exponential). Although applicable in a penalised likelihood framework, the Laplace prior would be computationally hard in a Bayesian setting as it is not conjugate to the Gaussian likelihood. Hence, naively using this prior would prevent us from computing the posterior in closed form, even in a variational setting. In order to circumvent this problem, we take a hierarchical Bayesian approach. 2 N Q τ V σ2 Zi Wi yn tn ω, χ, φ γi Ωi υ, λ Figure 1: Graphical model for sparse Bayesian multiple regression (when excluding the dashed arrow) and sparse Bayesian multiple classiﬁcation (when considering all arrows). We assume that the marginal prior, or effective prior, on each block Wi ∈RP ×Di has the form of a matrix-variate Gaussian scale mixture, a generalisation of the multivariate Gaussian scale mixture [3]: p(Wi) = Z ∞ 0 N(0, γ−1 i Ωi, Σ) p(γi) dγi, Q X i=1 Di = D, (3) where Ωi ∈RDi×Di, Σ ∈RP ×P and γi > 0 is the latent precision (i.e., inverse scale) associated to block Wi. A sparsity inducing prior for Wi can then be constructed by choosing a suitable hyperprior for γi. We impose a generalised inverse Gaussian prior (see Supplemental Appendix A for a formal deﬁnition with special cases) on the latent precision variables: γi ∼N −1(ω, χ, φ) = χ−ω √χφ ω 2Kω(√χφ) γω−1 i e−1 2 (χγ−1 i +φγi), (4) where Kω(·) is the modiﬁed Bessel function of the second kind, ω is the index, √χφ deﬁnes the concentration of the distribution and p χ/φ deﬁnes its scale. The effective prior is then a symmetric matrix-variate generalised hyperbolic distribution: p(Wi) ∝ Kω+ P Di 2 q χ(φ + tr{Ω−1 i W⊤ i Σ−1Wi}) r φ+tr{Ω−1 i W⊤ i Σ−1Wi} χ !ω+ P Di 2 . (5) The marginal (5) has fat tails compared to the matrix-variate Gaussian. In particular, the family contains the matrix-variate Student-t, the matrix-variate Laplace and the matrix-variate VarianceGamma as special cases. Several of the multivariate equivalents have recently been used as priors to induce sparsity in the Bayesian paradigm, both in the context of supervised [19, 11] and unsupervised linear Gaussian models [4]. 4 Sparse Bayesian multiple regression We view {Wi}Q i=1, {Ωi}Q i=1 and {γ1, . . . , γD1, . . . , γ1, . . . , γDQ} as latent variables that need to be marginalised over. This is motivated by the fact that overﬁtting is avoided by integrating out all parameters whose cardinality scales with the model complexity, i.e., the number of dimensions and/or the number of tasks. We further introduce a latent projectoin matrix V ∈RP ×K and a set of latent matrices {Zi}Q i=1 to make a low-rank approximation of the column covariance Σ as explained below. Note also that Ωi captures the correlations between the rows of group i. 3 The complete probabilistic model is given by yn\|W, xn ∼N(Wxn, σ2IP ), V ∼N(0, τIP , IK), (6) Wi\|V, Zi, Ωi, γi ∼N(VZi, γ−1 i Ωi, τIP ), Ωi ∼W−1(υ, λIDi), Zi\|Ωi, γi ∼N(0, γ−1 i Ωi, IK), γi ∼N −1(ω, χ, φ), where σ2 is the residual noise variance and τ is residual variance associated to W. The graphical model is shown in Fig. 1. We reparametrise the inverse Wishart distribution and deﬁne it as follows: Ω∼W−1(υ, Λ) = \|Λ\| D+υ−1 2 \|Ω−1\| 2D+υ 2 2 (D+υ−1)D 2 ΓD( D+υ−1 2 ) e−1 2 tr{ΛΩ−1}, υ > 0, where Γp(z) = π p(p−1) 4 Qp j=1 Γ(z + 1−j 2 ). Using the compact notations W = (W1, . . . , WQ), Z = (Z1, . . . , ZQ), Ω= diag{Ω1, . . . , ΩQ} and Γ = diag{γ1, . . . , γD1, . . . , γ1, . . . , γDQ}, we can compute the following marginal: p(W\|V, Ω) ∝ ZZ N(VZ, Γ−1Ω, τIP )N(0, Γ−1Ω, IK)p(Γ)dZdΓ = Z N(0, Γ−1Ω, VV⊤+ τIP )p(Γ)dΓ. Thus, the probabilistic model induces sparsity in the blocks of W, while taking correlations between the task parameters into account through the random matrix Σ ≈VV⊤+ τIP . This is especially useful when there is a very large number of tasks. The latent variables Z = {W, V, Z, Ω, Γ} are infered by variational EM [27], while the hyperparameters ϑ = {σ2, τ, υ, λ, ω, χ, φ} are estimated by type II ML [8, 25]). Using variational inference is motivated by the fact that deterministic approximate inference schemes converge faster than traditional sampling methods such as Markov chain Monte Carlo (MCMC), and their convergence can easily be monitored. The choice of learning the hyperparameters by type II ML is preferred to the option of placing vague priors over them, although this would also be a valid option. In order to ﬁnd a tractable solution, we assume that the variational posterior q(Z) = q(W, V, Z, Ω, Γ) factorises as q(W)q(V)q(, Z)q(Ω)q(Γ) given the data D = {(yn, xn)}N n=1 [7]. The variational EM combined to the type II ML estimation of the hyperparameters cycles through the following two steps until convergence: 1. Update of the approximate posterior of the latent variables and parameters for ﬁxed hyperparameters. The update for W is given by q(W) ∝e⟨ln p(D,Z\|ϑ)⟩q(Z/W), (7) where Z/W is the set Z with W removed and ⟨·⟩q denotes the expectation with respect to q. The posteriors of the other latent matrices have the same form. 2. Update of the hyperparameters for ﬁxed variational posteriors: ϑ ←argmax ϑ ⟨ln p(D, Z, \|ϑ)⟩q(Z) . (8) Variational EM converges to a local maximum of the log-marginal likelihood. The convergence can be checked by monitoring the variational lower bound, which monotonically increases during the optimisation. Next, we give the explicit expression of the variational EM steps and the updates for the hyperparameters, whereas we show that of the variational bound in the Supplemental Appendix D. 4.1 Variational E step (mean ﬁeld) Asssuming a factorised posterior enables us to compute it in closed form as the priors are each conjugate to the Gaussian likelihood. The approximate posterior is given by q(Z) = N(MW , ΩW , SW )N(MV , ΩV , SV )N(MZ, ΩZ, SZ) (9) × Y i W−1(υi, Λi)N −1(ωi, χi, φi). The expression of posterior parameters are given in Supplemental Appendix C. The computational bottleneck resides in the inversion of ΩW which is O(D3) per iteration. When D > N, we can use the Woodbury identity for a matrix inversion of complexity O(N 3) per iteration. 4 4.2 Hyperparameter updates To learn the degree of sparsity from data we optimise the hyperparameters. There are no closed form updates for {ω, χ, φ}. Hence, we need to ﬁnd the root of the following expressions, e.g., by line search: ω : Q ln s φ χ −Qd ln Kω(√χφ) dω X i ⟨ln γi⟩= 0, (10) χ : Qω χ −Q 2 s φ χRω( p χφ) + 1 2 X i ⟨γ−1 i ⟩= 0, (11) φ : Q rχ φRω( p χφ) − X i ⟨γi⟩= 0, (12) where (??) was invoked. Unfortunately, the derivative in the ﬁrst equation needs to be estimated numerically. When considering special cases of the mixing density such as the Gamma or the inverse Gamma simpliﬁed updates are obtained and no numerical differentiation is required. Due to space constraints, we omit the type II ML updates for the other hyperparameters. 4.3 Predictions Predictions are performed by Bayesian averaging. The predictive distribution is approximated as follows: p(y∗\|x∗) ≈ R p(y∗\|W, x∗)q(W)dW = N(MW x∗, (σ2 + x⊤ ∗ΩW x∗)IP ). 5 Sparse Bayesian multiple classiﬁcation We restrict ourselves to multiple binary classiﬁers and consider a probit model in which the likelihood is derived from the Gaussian cumulative density. A probit model is equivalent to a Gaussian noise and a step function likelihood [1]. Let tn ∈RP be the class label vectors, with tnp ∈{−1, +1} for all n. The likelihood is replaced by tn\|yn ∼ Y p I(tnpynp), yn\|W, xn ∼N(Wxn, σ2IP ), (13) where I(z) = 1 for z ⩾0 and 0 otherwise. The rest of the model is as before; we will set σ = 1. The latent variables to infer are now Y and Z. Again, we assume a factorised posterior. We further assume the variational posterior q(Y) is a product of truncated Gaussians (see Supplemental Appendix B): q(Y) ∝ Y n Y p I(tnpynp)N(νnp, 1) = Y tnp=+1 N+(νnp, 1) Y tnp=−1 N−(νnp, 1), (14) where νnp is the pth entry of νn = MW xn. The other variational and hyperparameter updates are unchanged, except that Y is replaced by matrix ν±. The elements of ν± are deﬁned in (??). 5.1 Bayesian classiﬁcation In Bayesian classiﬁcation the goal is to predict the label with highest posterior probability. Based on the variational approximation we propose the following classiﬁcation rule: ˆt∗= arg max t∗ P(t∗\|T) ≈arg max t∗ Y p Z Nt∗p(ν∗p, 1)dy∗p = arg max t∗ Y p Φ (t∗pν∗p) , (15) where ν∗= MW x∗. Hence, to decide whether the label t∗p is −1 or +1 it is sufﬁcient to use the sign of ν∗p as the decision rule. However, the probability P(t∗p\|T) tells us also how conﬁdent we are in the prediction we make. 5 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 Training set size Average Squared Test Error SPBMRC Ordinary Least Squares Predict with ground truth W Estimated task covariance True task covariance Sparsity pattern 5 10 15 20 25 30 35 40 45 50 0 0.2 0.4 0.6 0.8 1 E γ−1 Feature index SBMR estimated weight matrix OLS estimated weight matrix True weight matrix Figure 2: Results for the ground truth data set. Top left: Prediction accuracy on a test set as a function of training set size. Top right: estimated and true Σ (top), true underlying sparsity pattern (middle) and inverse of the posterior mean of {γi}i showing that the sparsity is correctly captured (bottom). Bottom diagrams: Hinton diagram of true W (bottom), ordinary least squares learnt W (middle) and the sparse Bayesian multi-task learnt W (top). The ordinary least squares learnt W contains many non-zero elements. 6 A model study with ground truth data To understand the properties of the model we study a regression problem with known parameters. Figure 2 shows the results for 5 tasks and 50 features. Matrix W is drawn using V = [ √ .9 √ .9 √ .9 − √ .9 − √ .9]⊤and τ = 0.1, i.e. the covariance for vec(W) has 1’s on the diagonal and ±.9 on the off-diagonal elements. The ﬁrst three tasks and the last two tasks are positively correlated. There is a negative correlation between the two groups. The active features are randomly selected among the 50 candidate features. We evaluate the models with 104 test points and repeated the experiment 25 times. Gaussian noise was added to the targets (σ = 0.1). It can be observed that the proposed model performs better and converges faster to the optimal performance when the data set size increases compared ordinary least squares. Note also that both Σ and the sparsity pattern are correctly identiﬁed. 6 Table 1: Performance (with standard deviation) of classiﬁcation tasks on Yeast and Scene data sets in terms of accuracy and AUC. LR: Bayesian logistic regression; Pooling: pooling all data and learning a single model; Xue: the matrix stick-breaking process based multi-task learning model proposed in [33]. K = 10 for the proposed models (i.e., Laplace, Student-t, and ARD). Note that the ﬁrst ﬁve rows for Yeast and Scene data sets are reported in [29]. The reported performances are averaged over ﬁve randomized repetitions. Model Yeast Scene Accuracy AUC Accuracy AUC LR 0.5047 0.5049 0.7362 0.6153 Pool 0.4983 0.5112 0.7862 0.5433 Xue [33] 0.5106 0.5105 0.7765 0.5603 Model-1 [29] 0.5212 0.5244 0.7756 0.6325 Model-2 [29] 0.5424 0.5406 0.7911 0.6416 Chen [15] NA 0.7987±0.0044 NA 0.9160±0.0038 Laplace 0.7987±0.0017 0.8349±0.0020 0.8892±0.0038 0.9188±0.0041 Student 0.7988±0.0017 0.8349±0.0019 0.8897±0.0034 0.9183±0.0041 ARD 0.7987±0.0020 0.8349±0.0020 0.8896±0.0044 0.9187±0.0042 7 Multi-task classiﬁcation experiments In this section, we evaluate the proposed model on two data sets: Yeast [17] and Scene [9], which have been widely used as testbeds to evaluate multi-task learning approaches [28, 29, 15]. To demonstrate the superiority of the proposed models, we conduct systematic empirical evaluations including the comparisons with (1) Bayesian logistic regression (BLR) that learns tasks separately, (2) a pooling model that pools data together and learns a single model collectively, and (3) the state-of-the-art multi-task learning methods proposed in [33, 29, 15]. We follow the experimental setting as introduced in [29] for fair comparisons, and omit the detailed setting due to space limitation. We evaluate all methods for the classiﬁcation task using two metrics: (1) overall accuracy at a threshold of zero and (2) the average area under the curve (AUC). Results on the Yeast and Scence data sets using these two metrics are reported in Table 7. It is interesting to note that even for small values of K (fewer parameters in the column covariance) the proposed model achieves good results. We also study how the performances vary with different K on a tuning set, and observe that there are no signiﬁcant differences on performances using different K (not shown in the paper). The results in Table 7 were produced with K = 10. The proposed models (Laplace, Student-t, ARD) signiﬁcantly outperform the Bayesian logistic regression approach that learns each task separately. This observation agrees with the previous work [6, 31, 2, 5] demonstrating that the multi-task approach is beneﬁcial over the naive approach of learning tasks separately. For the Yeast data set, the proposed models are signiﬁcantly better than “Xue” [33], Model-1 and Model-2 [29], and the best performing model in [15]. For the Scene data set, our models and the model in [15] show comparable results. The advantage of using hierarchical priors is particularly evident in a low data regime. To study the impact of training set size on performance, we report the accuracy and AUC as functions of the training set sizes in Figure 3. For this experiment, we use a single test set of size 1196, which replicates the experimental setup in [29]. Figure 3 shows that the proposed Bayesian methods perform well overall, but that the performances are not signiﬁcantly impacted when the number of data is small. Similar results were obtained for the Yeast data set. 8 Conclusion In this work we proposed a Bayesian multi-task learning model able to capture correlations between tasks and to learn the sparsity pattern of the data features simultaneously. We further proposed a low-rank approximation of the covariance to handle a very large number of tasks. Combining lowrank and sparsity at the same time has been a long open standing issue in machine learning. Here, we are able to achieve this goal by exploiting the special structure of the parameters set. Hence, the 7 400 600 800 1000 0.5 0.6 0.7 0.8 0.9 Scene data set, K=10 Number of training samples Accuracy ARD Student−t Laplace Model−2 Model−1 BLR 400 600 800 1000 0.5 0.6 0.7 0.8 0.9 Scene data set, K=10 Number of training samples AUC Figure 3: Model comparisons in terms of classiﬁcation accuracy and AUC on the Scene data set for K = 10. Error bars represent 3 times the standard deviation. Results for Bayesian logistic regression (BLR), Model-1 and Model-2 are obtained based on the measurements using a ruler from Figure 2 in [29], for which no error bars are given. proposed model combines sparsity and low-rank in a different manner than in [10], where a sum of a sparse and low-rank matrix is considered. By considering a matrix-variate Gaussian scale mixture prior we extended the Bayesian group LASSO to a more general family of group sparsity inducing priors. This suggests the extension of current Bayesian methodology to learn structured sparsity from data in the future. 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4,448	Orthogonal Matching Pursuit with Replacement Prateek Jain Microsoft Research India Bangalore, INDIA prajain@microsoft.com AmbujTewari The University of Texas at Austin Austin, TX ambuj@cs.utexas.edu Inderjit S. Dhillon The University of Texas at Austin Austin, TX inderjit@cs.utexas.edu Abstract In this paper, we consider the problem of compressed sensing where the goal is to recover all sparse vectors using a small number offixed linear measurements. For this problem, we propose a novel partial hard-thresholding operator that leads to a general family of iterative algorithms. While one extreme of the family yields well known hard thresholding algorithms like ITI and HTP[17, 10], the other end of the spectrum leads to a novel algorithm that we call Orthogonal Matching Pursnit with Replacement (OMPR). OMPR, like the classic greedy algorithm OMP, adds exactly one coordinate to the support at each iteration, based on the correlation with the current residnal. However, unlike OMP, OMPR also removes one coordinate from the support. This simple change allows us to prove that OMPR has the best known guarantees for sparse recovery in terms of the Restricted Isometry Property (a condition on the measurement matrix). In contrast, OMP is known to have very weak performance guarantees under RIP. Given its simple structore, we are able to extend OMPR using locality sensitive hashing to get OMPR-Hasb, the first provably sub-linear (in dimensionality) algorithm for sparse recovery. Our proof techniques are novel and flexible enough to also permit the tightest known analysis of popular iterative algorithms such as CoSaMP and Subspace Pursnit. We provide experimental results on large problems providing recovery for vectors of size up to million dimensions. We demonstrste that for large-scale problems our proposed methods are more robust and faster than existing methods. 1 Introduction We nowadays routinely face high-dimensional datasets in diverse application areas such as biology, astronomy, and finance. The associated curse of dimensionality is often alleviated by prior knowledge that the object being estimsted has some structore. One of the most natorsl and well-stodied structural assumption for vectors is sparsity. Accordingly, a huge amount of recent work in machine learning, statistics and signal processing has been devoted to finding better ways to leverage sparse structures. Compressed sensing, a new and active branch of modem signal processing, deals with the problem of designing measurement matrices and recovery algorithms, such that almost all sparse signals can be recovered from a smalI number of measurements. It has important applications in imsging, computer vision and machine learning (see, for example, [9,24, 14]). In this paper, we focus on the compressed sensing setting [3, 7] where we want to design a measurement matrix A E R=xn such that a sparse vector x* E Rn with Ilxllo := I BUpp(X)I ::; k < n can be efficiently recovered from the measurements b = Ax* E R=. Initial work focused on various random ensembles of matrices A such that, if A was chosen randomly from that ensemble, one would be able to recover all or almost all sparse vectors x* from Ax. Candes and Tao[3] isolated a key property called the restricted Isometry property (RIP) and proved that, as long as the measurement matrix A satisfies RIP, the true sparse vector can be obtained by solving an i,-optimization problem, min Ilxll, S.t. Ax = b. The above problem can be easily formulated as a linear program and is hence efficiently solvable. We recall for the reader that a matrix A is said to satisfY RIP of order k if there is some Ok E 10,1) such that, for all x with Ilxllo ::; k, we have I Several random matrix ensembles are known to satisfY 00> < {} with high probability provided one chooses m ~ 0 (~ log ~) measurements. It was shown in [2] that i,-minimization recovers all k-sparse vectors provided A satisfies t.k < 0.414 although the conditioohas been recently intproved to 02k < 0.473 [11]. Note that, in compressed sensing, the goal is to recover all, or most, k-sparse signals using the same measurement matrix A. Hence, weaker cooditioos such as restricted coovexity [20] studied in the statistical literature (where the aint is to recover a single sparse vector from noisy linear measurements) typically do not suffice. In fact, if RIP is not satisfied then multiple sparse vectors x can lead to the sante observatioo b, hence making recovery of the true sparse vector intpossible. Based on its RIP guarantees, i,-minimizatioo can guarantee recovery using just O(k log(n/ k») measurements, but it has been observed in practice that i,-minimization is too expensive in large scale applications [8], for example, when the dimensionality is in the millions. This has sparked a huge interest in other iterative methods for sparse recovery. An early classic iterative method is Orthogooal Matching Pursuit (OMP) [21, 6] that greedily chooses elements to add to the support. It is a natural, easy-to-intplement and fast method but unfortuoately lacks stroug theoretical guarantees. Indeed, it is known that, if run for k iterations, OMP cannot uoiformly recover all k-sparse vectors assumiug RIP cooditioo of the form 02k :'0 IJ [22, 18]. However, Zhang [26] showed that OMP, if run for 30k iterations, recovers the optimal solution when 03'k :'0 1/3; a significantly more restrictive cooditioo than the ones required by other methods like i,-minimization. Several other iterative approaches have been proposed that include Iterative Soft Thresholding (1ST) [17], Iterative Hard Thresholding (!BT) [I], Compressive Santpling Matching Pursuit (CoSaMP) [19], Subspace Pursuit (SP) [4], Iterative Thresholding with Inversion (IT!) [16], Hard Thresholding Pursuit (HTP) [10] and many others. In the family ofiterative hard thresholding algorithms, we can identifY two major subfamilies [17]: one- and two-stage algorithms. As their nantes suggest, the distiuctioo is based on the number of stages in each iteration of the algorithm. One-stage algorithms such as IHT, m and HTP, decide on the choice of the next support set and then usually solve a least squares problem on the updated support. The one-stage methods always set the support set to have size k, where k is the target sparsity level. On the other hand, two-stage algorithms, notable examples being CoSaMP and SP, first enlarge the support set, solve a least squares 00 it, and then reduce the support set back again to the desired size. A secood least squares problem is then solved 00 the reduced support. These algorithms typically enlarge and reduce the support set by k or 2k elements. An exceptioo is the two-stage algorithm FoBa [25] that adds and removes single elements from the support. However, it differs from our proposed methods as its analysis requires very restrictive RIP cooditioos (08k < 0.1 as quoted in [14]) and the connection to locality sensitive hashing (see below) is not made. Another algorithm with replacentent steps was studied by Shalev-Shwartz et al. [23]. However, the algorithm and the settiug under which it is analyzed are different from ours. In this paper, we present and provide a unified analysis for a family of one-stage iterative hard thresholding algorithms. The family is parameterized by a positive integer I :'0 k. At the extrente value I ~ k, we recover the algorithm ITIIHTP. At the other extrente k ~ 1, we get a novel algorithm that we call Orthogonal Matching Pursuit with Replacement (OMPR). OMPR can be thought of as a sintple modification of the classic greedy algorithm OMP: instead of sintply adding an element to the existiug support, it replaces an existiug support element with a new one. Surprisingly, this change allows us to prove sparse recovery under the condition 02k < 0.499. This is the best 02k based RIP condition under which any method, including i, -minimization, is (currently) known to provably perform sparse recovery. OMPR also lends itself to a faster intplententatioo using locality sensitive hashing (LSH). This allows us to provide recovery guarantees using an algorithm whose run-time is provably sub-linear in n, the number of dimensions. An added advantage of OMPR, unlike many iterative methods, is that no careful tuning of the step-size parameter is required even under noisy settiugs or even when RIP does not hold. The default step-size of 1 is always guaranteed to converge to at least a local optimum. Finally, we show that our proof techniques used in the analysis of the OMPR family are useful in tightening the analysis of two-stage algorithms, such as CoSaMP and SP, as well. As a result, we are able to prove better recovery guarantees for these algorithms: 04k < 0.35 for CoSaMP, and 03k < 0.35 for SP. We hope that this unified analysis sheds more light on the interrelationships between the various kinds of iterative hard thresholding algorithms. In summary, the contributions of this paper are as follows . • We present a family of iterative hard thresholding algorithms that on one end of the spectrum includes existing methods such as ITIIHTP while on the other end gives OMPR. OMPR is an intproventent over the classical OMP method as it enjoys better theoretical guarantees and is also better in practice as shown in our experiments . • Unlike other intprovements over OMP, such as CoSaMP or SP, OMPR changes ouly ooe elentent of the support at a tinte. This allows us to use Locality Sensitive Hashing (LSH) to speed it up resultiug in the first provably sub-linear (in the ambient dimensionality n) time sparse recovery algorithm. 2 Algorithm 1 OMPR 1: Input: matrix A, vector b, sparsity level k 2: Parameter: s1ep size 1/ > 0 3: Initialize Xl S.t I supp(xl)1 = k, h = supp(XI) 4: for t = 1 to T do 5: zHI <- x' + 1/AT(b - Ax') 6 . I HII : Jt+l +- argmaxj~It Zj 7: J'+1 <- I, U {iHI} 8: yt+l +- H (zt+l) k Jt +l 9: It+1 <- supp(y'+1) 10 xHI A \b xHI 0 : [Hl +It+l , it+l +11: end for Algorithm 2 OMPR (I) 1: Input: matrix A, vector b, sparsity level k 2: Parameter: step size 1/ > 0, replacement budget 1 3: Initialize Xl S.t I supp(xl)1 = k, h = supp(xl ) 4: fort = ltoTdo 5: zHI <- x' + 1/AT(b - Ax') 6: tOPHI <- indices of top 1 elements of Iz};'"11 7: J'+1 <- I, U tOPHI 8: yt+l +- Hk (z~~:J 9: IHI <- supp(yHI) 10: XHI <- A \b x'.+1 <- 0 I t+1 It+l' 1t+l 11: end for • We provide a general proof for all the algorithms in our partial hard thresholding based family. In particular, we can guarantee recovery using OMPR, under both noiseless and noisy settings, provided 02' < 0.499. This is the least restrictive 02. cooditioo under which any efficient sparse recovery method is known to work. Furthermore, our proof technique can be used to provide a general theorem that provides the least restrictive known guarantees for all the two-stage algorithms such as CoSaMP and SP (see Appendix D). All proofs omitted from the main body of the paper can be found in the appendix. 2 Orthogonal Matching PUl"lIuit with Replacement Orthogonal matching pursuit (OMP), is a classic iterative algorithm for sparse recovery. At every stage, it selecta a coordinate to include in the current support set by maximizing the inner product between columns of the measurement matrix A and the current residnal b - Ax'. Doce the new coordinate has been added, it solves a least squares problem to fully miuimize the error on the current support set As a result, the residnal becomes orthogonal to the columos of A that correspond to the current support set. Thus, the least squares s1ep is also referred to as orthogonalization by some authors [5]. Let us briefly explain some of our notation. We use the MATI..AB notation: A\b:= argmin IIAx bl1 2 • z The hard thresholding operator H.O sorts its argument vector in decreasing order (in absolute value) and retains ooly the top k entries. It is defined formally in the next sectioo. Also, we use subscripts to denote sub-vectors and submatrices, e.g. if I <;; Inl is a set of cardinality k and x ERn, XI E R' denotes the sub-vector of X indexed by I. Similarly, AI for a matrix A E Rmx n denotes a sub-matrix of size m x k with columns indexed by I. The complement of set I is denoted by I and x I denotes the subvector not indexed by I. The support (indices of non-zero entries) of a vector x is denoted by supp(x). Our new algorithm called Orthogooal Matching Pursuit with Replacement (OMPR), shown as Algorithm 1, differs from OMP in two respects. First, the selection of the coordinate to include is based not just on the magnitude of entries in AT (b - Ax') but instead on a weighted combination x' + 1/AT (b - Ax') with the s1ep-size 1/ cootrolling the relative importance of the two addends. Second, the selected coordinate replaces one of the existing elements in the support, namely the one corresponding to the minimum magnitude entry in the weighted combination mentioned above. Doce the support IHI of the next iterate has been determined, the actna1 iterate XHI is obtained by solving the least squares problem: XHI = argmin IIAx - bli2 . x: supp(z)=It+l Note that if the matrix A satisfies RIP of order k or larger, the above problem will be well conditioned and can be solved quickly and reliably using an iterative least squares solver. We will show that OMPR, uulike OMP, recovers any k-sparse vector under the RIP based cooditioo 02. :<:; 0.499. This appears to be the least restrictive recovery condition (i.e., best known coodition) under which any method, be it basis pursuit (ll-minimizatioo) or some iterative algorithm, is guaranteed to recover all k-sparse vectors. In the literature on sparse recovery, RIP based cooditioos of a different order other than 2k are often provided. It is seldom possible to directly compare two conditions, say, one based on 62• and the other based on 63 •• Foucart [10] has 3 given a heuristic to compare such RIP conditions based on the number of samples it takes in the Gaussian ensemble to satisfy a given RIP condition. This heuristic says that an RIP condition of the form lic' < 9 is less restrictive if the ratio c/92 is smaller. For the OMPR condition Ii,. < 0.499, this ratio is 2/0.4992 "" 8 which makes it heuristically the least restrictive RIP condition for sparse recovery. The following summarize our main results on OMPR. Theorem 1 (Noiseless Case). Suppose the vector x E IRn is k-sparse and the matrix A satisfies 1i2• < 0.499 and Ii, < 0.002. Then OMPR converges to an E approximate solution (i.e. 1/211Ax bl12 ~ E) from measurements b ~ Ax* in O(klog(k/E)) iterations. Theorem 2 (Noisy Case). Suppose the vector x* E IRn is k-sparse and the matrix A satisfies 1i2• < 0.499 and Ii, < 0.002. Then OMPR converges to a (C,E) approximate solution (i.e. 1/211Ax - bll' ~ ~llell' + E) from measurements b ~ Ax* + e in O(k log((k + IleI1 2)/E)) iterations. Here C > 1 is a constant dependent only on 1i2 •. The above theorems are special cases of our convergence results for a family of algorithms that contains OMPR as a special case. We now tum our attention to this family. We note that the condition 1i2 < 0.002 is very mild and will typically hold for standard random matrix ensembles as soon as the number of rows sampled is larger than a fixed universal constant 3 A New Family of Iterative Algorithms In this section we show that OMPR is one particular member of a family of algorithms parameterized by a single integer 1 E {I, ... , k}. The I-th member of this family, OMPR (I), showo in Algorithm 2, replaces at most 1 elements of the curreot support with new elements. OMPR corresponds to the choice 1 ~ 1. Hence, OMPR and OMPR (1) refer to the same algorithm. Our first result in this section conoects the OMPR family to hard thresholding. Given a set I of cardinality k, define the partial hard thresholding operator Hk (z; I, I):~ argmin Ily - zll . (I) hlo:S;k I supp(y)\II5:l As is clear from the definition, the above operator tries to find a vector V close to a given vector z under two constraints: (i) the vector V should have bounded support (1lvllo ~ k), and (ii) its support should not include more than 1 new elements outside a given support I. The name partial hard thresholding operator is justified because of the following reasoning. When 1 ~ k, the constraint I supp(Y)\I1 ~ 1 is trivially implied by IIYllo ~ k and hence the operator becomes independent of!. In fact, itbecomes identical to the standard hard thresholding operator H. (z; I, k) ~ H. (z) :~ argmin Ily - zll . 11.1109 (2) Even though the definition of Hk (z) seems to involve searching through GJ subsets, it can in fact be computed efficiently by simply sorting the vector z by decreasing absolute value and retaming the top k entries. The following result shows that even the partial hard thresholding operator is easy to compute. In fact, lines 6-8 in Algorithm 2 precisely compute H. (zt+1; It, I). Proposition 3. Let III ~ k and z be given. Then Y ~ H. (z;I, I) can be computed using the sequence of operations top ~ indices of top 1 elements oflzll, J ~ I U top, V ~ Hk (ZJ) . The proof of this proposition is straightforward and elementary. However, using it, we can now see that the OMPR (I) algorithm has a simple conceptoa1 s1ructore. In each iteration (with current iterate x' having support It ~ supp(xt», we do the following: 1. (Gradient Descent) Fonn zHI ~ xt - '1AT(Axt - b). Note that AT(Axt - b) is the gradient of the objective function ~IIAx - bll' at x'. 2. (partial Hard Thresholding) Form VH1 by partially hard thresholding zHI using the operator H. (.; It, I). 3. (Least Squares) Form the next iterate XHI by solving a least squares problem on the support IHI ofyHI. A nice property enjoyed by the entire OMPR family is guaranteed sparse recovery under RIP based conditions. Note from below that the condition under which OMPR (I) recovers sparse vectors becomes more restrictive as I increases. This could be an artifact of our analysis, as in experiments, we do not see any degradation in recovery ability as I is increased. 4 Theorem 4 (Noiseless Case). Suppose the vector x' E IRn is k-sporse. Then OMPR (I) converges to an < approximation solution (i.e. 1/211Ax - bl12 :5 <)from measurements b = Ax* in O( ~ log(k/<» iterations provided we choose a step size 1'/ that satisfies 1'/(1 + 02.) < 1 and 1'/(1 - 02.) > 1/2. Theorem S (Noisy Case). Suppose the vector x' E IRn is k-sparse. Then OMPR (I) converges to a (C, <) approximate solution (i.e., 1/211Ax - bl12 :5 t IIell2 + <) from measurements b = Ax' + e in O( t log«k + IleI12)1<) iterations provided we choose a step size 1'/ that satisfies 1'/(1 + 02,) < 1 and 1'/(1 - 02.) > 1/2. Here C > 1 is a constant dependent only on 02., 02 •. Proof Here we provide a rough sketch of the proof of Theorem 4; the complete proof is giveo in Appeodix A. Our proof uses the following crucial observatioo regarding the structure of the vector zH1 = x' - 1'/AT (Ax' - b) . Due to the least squares step of the previous iteration, the curreot residual Ax' - b is orthogoual to columns of AI,. This meaos that ZH1 - x' z~+1 = -nA'!' (Ax' - b) . It It' It " It (3) As the algorithm proceeds, elemeots come in and move out of the curreot set I,. Let us give names to the set offound and lost elements as we move from I, to 1'+1: (found): F, = IH1 \I" Heoce, using (3) and updates for YH1: Y~;' = Z~;' = -1'/A~,A(x' - x'), and Z~;' = xL. Now let J(x) = 1/211Ax - b112, theo using upper RIP and the fact that I supp(yH1 - x')1 = IF, U L,I :5 21, we can sbow that (details are in the Appeodix A): J(yH1) - J(x'):5 C ~02' - D IIyWII2 + 1 ~02'llxUI2. (4) Furthermore, since yH1 is choseo based on the k largest eotries in z~;:" we have: IIY~;'112 = Ilz~;'112 ~ Ilz~;'112 = IlxL 112 . Plugging this into (4), we get: J(yH1) - J(x'):5 (1 +O2'-~) M;'112. (5) Since J(xH1 ) :5 J(yH1) :5 J(x'), the above expression shows that if 1'/ < 1':." then our method moootonically decreases the objective function and converges to a local optimum even if RIP is not satisfied (note that upper RIP bound is indepeodeot oflower RIP bound, and can always be satisfied by nurma1izing the matrix appropriately). However, to prove convergeoce to the global optimum, we need to show that at least ooe new elemeot is added at each step, i.e., IF,I ~ 1. Furthermore, we need to show sufficieot decrease, i.e, IIY~;'112 ~ elJ(x'). We show both these conditions for global coovergeoce in Lemma 6, whose proof is giveo in Appeodix A. Lemma 6. Let 02k < 1 2~ and 1/2 < 1'/ < 1. Then assuming J(x') > 0, at least one new element is found i.e. F, '" 0. Furthermore, IIY~;'11 > teJ(x'), where e = min(41'/(1 - 1'/),,2(21'/- 1-~"» > 0 is a constant. Assunling Lemma 6, (5) shows that at each iteration OMPR (I) reduces the objective functioo value by at least a constant fractioo. Furthermore, if XO is choseo to have eotries bounded by 1, theo J(XO) :5 (1 + 02k)k. Heoce, afier O(k/llog(k/<» iteratioos, the optimal solution x* would be obtained within < error. D Speeial Cases: We have already observed that the OMPR algorithm of the previous sectioo is simply OMPR (1). Also note that Theorem I immediately follows from Theorem 4. The algorithm at the other extreme of 1 = k has appeared at least three times in the receot literature: as Iterative (hard) Thresholding with Inversioo (IT!) in [16], as SVP-Newton (in its matrix avatar) in [15], and as Hard Thresholding Pursuit (HTP) in [10]). Let us call it IHT-Newton as the least squares step can be viewed as a Newton step for the quadrstic objective. The above geoera1 result for the OMPR family immediately implies that it recovers sparse vectors as soon as the measuremeot matrix A satisfies 02, < 1/3. CoroUary 7. Suppose the vector x' E an is k-sparse and the matrix A satisfies 02k < 1/3. Then IlIT-Newton recovers x* from measurements b = Ax' in O(1og(k» iterations. 5 4 Tighter Analysis of Two Stage Hard Thresholding Algorithms Recently, Maleki and Donoho [17] proposed a novel family of algorithms, namely two-stage hard thresholding algorithms. Doring each iteration, these algorithms add a fixed nwnber (say l) of elements to the current iterate's support set. A least squares problem is solved over the larger support set and then I elements with smallest magnitude are dropped to form next iterate's support set. Next iterate is then obtained by agaiu solviug the least squares over next iterate's support set. See Appendix D for a more detailed description of the algorithm. Usiug proof techniques developed for our proof of Theorem 4, we can obtain a simple proof for the entire spectrum of algorithms iu the two-stage hard thresholding family. Theorem 8. Suppose the vector x* E {-I, 0, l}n is k-sparse. Then the 7Wo-stage Hard Thresholding algorithm with replacement size I recovers x* from measurements b = Ax* in O(k) iterations provided: 6.H1 :::; .35. Note that CoSaMP [19] and Subspace Pursuit(SP) [4] are popular special cases of the two-stage family. Usiug our general analysis, we are able to provide significantly less restrictive RIP conditions for recovery. CoroUary 9. CoSaMP[l9] recovers k-sparse x* E {-1,0, l}n from measurements b = Ax* provided 64k :::; 0.35. CoroUary 10. Subspace Pursuit[4] recovers k-sparse x* E {-I, 0, I}n from measurements b = Ax* provided 63k :::; 0.35. Note that CoSaMP's analysis given by [19] requires 64k :::; 0.1 while Subspace Pursuit's analysis given by [4] requires 63k :::; 0.205. See Appendix Diu the supplementary material for proofs of the ahove theorem and coroUaries. 5 Fast Implementation Using Hashing In this section, we discuss a fast implementation of the OMPR method usiug locality-sensitive hashiug. The mall iutuition behind our approach is that the OMPR method selects at most one element at each step (given by argmax, IAT(Ax' - b) I); hence, selection of the top most element is equivalent to finding the column Ai that is most "similar" (iu magnitude) to r, = Ax' - b, i.e., this may be viewed as the similarity search task for queries of the form r, and -r, from a database of N vectors IAI"'" ANI. To this end, we use locality sensitive hashiug (LSH) [12], a well known data-structore for approximate nearestneighbor retrieval. Note that while LSH is designed for nearest neighbor search (iu terms of Euclidean distances) and iu general might not have any guarantees for the similar neighbor search task, we are still able to apply it to our task because we can lower-hound the similarity of the most similar neighbor. We first briefly describe the LSH scheme that we use. LSH generates hash bits for a vector usiug randoruized hash functions that have the property that the probability of collision between two vectors is proportional to the similarity between them. For our problem, we use the following hash function: h,.(a) = sign(uT a), where u ~ N(O, J) is a random hyper-plane generated from the standard multivariate Gaussian distribution. It can be shown that [13] () () I -I ( af a2 ) Pr[hu al = hu a. ] = 1-;;: cos Iladlla211' Now, an .-bit hash key is created by randoruly sampling hash functions h,., i.e., g( a) [hu,(a),hu,(a), ... ,hu.(a)], where each Ui is sampled randoruly from the standard multivariate Gaussian distribution. Next, q hash tables are constructed doring the pre-processiug stage usiug iudependently constructed hash key functions gl, 92, ... , gq' Doring the query stage, a query is iudexed iuto each hash table usiug hash-key functions 91, 92, ... ,9q and then the nearest neighbors are retrieved by doing an exhaustive search over the indexed elements. Below we state the following theorem from [12] that guarantees sub-liuear time nearest neighbor retrieval for LSH. Theorem 11. Let. = O(logn) and q = O(log 1/6)nr1<, then with probability 1 - 6, LSH recovers (I + f)-nearest neighbors, i.e., Ila' - rl12 :::; (1 + f)lla' - rll·, where a' is the nearest neighbor to r and a' is a point retrieved by LSH. However, we cannot directly use the above theorem to guarantee convergence of our hashing based OMPR algorithm as our algorithm requires finding the most similar poiut iu terms of magnitude of the iuner product. Below, we provide appropriate settings of the LSH parameters to guarantee sub-liuear time convergence of our method under a slightly weaker condition on the RIP constant. A detailed proof of the theorem below can be found iu Appendix B. Theorem 12. Let 62• < 1/4 -")' and 'f/ = I -")" where")' > 0 is a small constant, then with probability I - 6, OMPR with hashing converges to the optimal solution in O(kmnl /(1+0(I/k)) log k/6) computational steps. The above theorem shows that the time complexity is sub-liuear iu n. However, currently our guarantees are not particularly strung as for large k the exponent of n will be close to 1. We believe that the exponent can be improved by more careful analysis and our empirical results iudicate that LSH does speed up the OMPR method significantly. 6 (a)OMPR (b)OMP (c) nIT-Newton Figure 1: Phase Transition Diagrams for different methods. Red represents high probability of success while blue represents low probability of success. Clearly, OMPR recovers correct solution for a much larger region of the plot than OMP and is comparable to nIT-Newton. (Best viewed in color) 6 Experimental Results In this section we present empirical results to demonstrate accurate and fast recovery by our OMPR method. In the first set of experiments, we present a phase transition diagram for OMPR and compare it to the phase transition diagrams of OMP and nIT-Newton with step size 1. For the second set of experiments, we demonstrate robostoess of OMPR compared to many existiog methods when measurements are noisy or smaller in number than what is required for exact recovery. For the third set of experiments, we demonstrate efficiency of our LSH based implementation by comparing recovery error and time required for our method with OMP and nIT-Newtoo (with step-size 1 and 1/2). We do not present results for the i,ibasis pursuit methods, as it has a1readybeen shown in several recent papers [10, 17] that the i, relaxation based methods are relatively inefficient for very large scale recovery problems. In all the experiments we generate the measurement matrix by sampling each entry independently from the standard normal distribotion N (0, 1) and then normalize each column to have uuit norm. The underlying k-sparse vectors are generated by randomly selecting a support set of size k and then each entry in the support set is sampled uuiformiy from { +1, -I}. We use our own optimized implementation of OMP and nIT-Newtoo. All the methods are implemented in MATLAB and our hashing routioe uses mex files. 6.1 Phase Transition Diagrams We first compare different methods using phase transition diagrams which are commouly used in compressed sensing literatore to compare different methods [17]. We first fix the number of measurements to be m = 400 and generate different problem sizes by varying p = kim and 6 = min. For each problem size (m, n, k), we generate random m x n Gaussian measurement matrices and k-sparse random vectors. We then estimate the probability of success of each of the method by applying the method to 100 randomly generated instances. A method is considered successful for a particular instance if it recovers the underlying k-sparse vector with at most 1 % relative error. In Figure 1, we show the phase transition diagram of our OMPR method as well as that ofOMP and nIT-Newtoo (with step size 1). The plots shows probability of successful recovery as a function of p = min and 6 = kim. Figure 1 (a) shows color coding of different success probabilities; red represents high probability of success while blue represents low probability of success. Note that for Gaussian measurement matrices, the RIP constant 62• is less than a fixed constant if and ouly ifm = Ck log(nlk), where C is a uuiversal constant This implies that * = Clog p and hence a method that recovers for high 62• will have a large fraction in the phase transition diagram wbere successful recovery probability is high. We observe this phenomenon for both OMPR and nIT-Newton method which is consistent with their respective theoretical goarantees (see Theorem 4). On the other hand, as expected, the phase transition diagram of OMP has a negligible fraction of the plot that shows high recovery probability. 6_2 Performance for Noisy or Under-sampled Observations Next, we empirically compare performance of OMPR to various existing compressed sensing methods. As shown in the phase transition diagrams in Figure 1, OMPR provides comparable recovery to the nIT-Newton method for noiseless cases. Here, we show that OMPR is fairly robust under the noisy settiog as well as in the case of undersampled observations, where the number of observations is much smaller than what is required for exact recovery. For this experiment, we generate random Gaussian measurement matrix of size m = 200, n = 3000. We then generate random binary vector x of sparsity k aod add Gaussian noise to it Figure 2 (a) shows recovery error (1iAx - bll) incurred by various methods for increasing k and noise level of 10%. Clearly, our method outperforms the existing methods, perhaps a consequence of goaranteed convergence to a local minimum for fixed step size 1/ = 1. Similarly, Figure 2 (b) shows recovery error incurred by various methods for fixed k = 50 and varying noise level. Here again, our method outperforms existiog methods and is more robust to noise. Fina11y, in Figure 2 ( c) we show difference in 7 10 Enurvsk(Noi-=10%) 20 30 Sp.lI'IiIy{k) (a) " 50 " _ OMPR Error w NaIM k=SO +OMPR(1rI2 :U . IHT-N ~ _ CoSAMP :ii' + SP ~ 3 ~'.'~~/ "~:----';o,', -'0." -'0." -'0'.-----,10.' Hoi_LewI (b) NOise! 1< 0 ,0 5u 0.00 0.00(0.0) -0.21(0.6) 0.25(0.3) 0.05 0.00(0.0) 0.13(0.3) 0.37(0.3) 0.0 O.OO(u.O) 0.2"(0.3) O. 3 0.4) 0.20 0.03(0.0) 0.62(0.2) 0.58(0.5) U.3U U.1"(U.1) U.92(0.3) O.92(O.b) 0.40 0.31(0.1) 1.19(0.3) 0.84(0.5) 0.50 0.37(0.1) 1.48(0.3) 1.24(0.6) (c) Figure 2: Error in recovery <lIAx - bll) of n = 3000 dimensiooal vectors from m = 200 measurements. (a): Error incurred by various methods as the sparsity level k increases. Note that OMPR incurs the least error as it provably converges to at least a local minimum forfixed step size 1/ = 1. (b): Error incurred by various methods as the noise level increases. Here again OMPR performs significaotly better than the existing methods. (c): Differeoce in error incurred by IHT-Newton aod OMPR . Numbers in bracket dooote confideoce interval at 95% significaoce level. 0." . ,. 0.03 .". o. :.,015 lIo '" ,., .., O. (a) EmrVII n (mIn=O 001 Idm" 1) I:OMPR Huh + 00'" +IHT-NMlanC1 • ~ .... 00 • ncIrOOO) "'" (b) " ., .. TlII'MI wn (rMFO.oD1, k/m=,1 n (x1fOOOO) 000 (c) Figure 3: (a): Error (11Ax - bll) incurred by various methods as k increases. The measuremoots b = Ax are computing by gooerating x with support size milO. (b),(c): Error incurred aod time required by various methods to recover vectors of support size 0.1 mas n increases. IlIT-Newton(1/2) refers to the IHT-Newton method with step size 1/ = 1/2. error incurred along with confideoce interval (at 95% signficaoce level) by IHT-Newton aod OMPR for varying levels of noises aod k. Our method is better thao !HT-Newton (at 95% signficaoce level) in terms of recovery error in arouod 30 cells of the table, aod is not worse in aoy of the cells but one. 6.3 Performance of LSD based implementation Next, we empirically study recovery properties of OMPR-Hasb in the following real-time setop: gooerate a raodom measuremoot matrix from the Gaussiao ensemble aod construct bash tables ollline using hash functioos specified in Section 5. During the reconstruction stage, measurements arrive one at a time and the goal is to recover the underlying sigoal accurately in real-time.For our experimoots, we gooerate measuremoots using raodom sparse vectors aod thoo report recovery error IIAx - bll aod computatiooal time required by each method averaged over 20 runs. In our first set of experimoots, we eropirically study the performaoce of different methods as k increases. Here, we fix m = 500, n = 500, 000 aod gooerate measuremoots using n-dimoosional raodom vectors of support set size milO. We thoo run differeot methods to estimate vectors x of support size k that minimize IIAx - bll. For our OMPR-Hash method, we use 8 = 20 bits bash-keys aod gooerate q = ..;n bash-tables. Figure 3 (a) shows the error incurred by OMPR, OMPR-Hash, aod IHT-Newton for differeot k (recall that k is ao input to both OMPR aod IlIT-Newton). Note that although OMPR-Hash performs ao approximation at each step, it is still able to achieve error similar to OMPR aod !HT-Newton. Also, note that since the number of measure moots are not ooough for exact recovery by the IHT-Newton method, it typically diverges after a few steps. As a result, we use IHT-Newton with step size 1/ = 1/2 which is always goaraoteed to monotonically converge to at least a local minimum (see Theorem 4). In cootrast, in OMPR aod OMPR-Hasb cao always set step size 1/ aggressively to be 1. Next, we evaluate OMPR-Hash as dimoosiooality of the data n increases. For OMPR-Hasb, we use 8 = log2(n) bash-keys aod q = ..;n hash-tables. Figures 3(b) aod (c) compare error incurred aod time required by OMPR-Hash with OMPR aod IHT-Newton. Here again we use step size 1/ = 1/2 for !HT-Newton as it does not converge for 1/ = 1. Note that OMPR-Hash is ao order of magnitude faster thao OMPR while incurring slightly higher error. OMPR-Hash is also nearly 2 times faster thao IHT-Newton. Acknowledgement ISD acknowledges support from the Moncrief Graod Challooge Award. 8 References [I] T. Blumensath and M. E. Davies. Iterative hard thresholding for compressed sensing. Applied and Computational Harmonic Analysis, 27(3):265-274, 2009. [2] E. J. Candes. The restricted isometry property and its implications for compressed sensing. Comptes Rendus Mathematique, 346(9-10):589-592, 2008. [3] E. J. Candes and T. Tao. Decoding by lioear programming. 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4,449	High-Dimensional Graphical Model Selection: Tractable Graph Families and Necessary Conditions Anima Anandkumar Dept. of EECS, Univ. of California Irvine, CA, 92697 a.anandkumar@uci.edu Vincent Y.F. Tan Dept. of ECE, Univ. of Wisconsin Madison, WI, 53706. vtan@wisc.edu Alan S. Willsky Dept. of EECS Massachusetts Inst. of Technology, Cambridge, MA, 02139. willsky@mit.edu Abstract We consider the problem of Ising and Gaussian graphical model selection given n i.i.d. samples from the model. We propose an efﬁcient threshold-based algorithm for structure estimation based on conditional mutual information thresholding. This simple local algorithm requires only loworder statistics of the data and decides whether two nodes are neighbors in the unknown graph. We identify graph families for which the proposed algorithm has low sample and computational complexities. Under some transparent assumptions, we establish that the proposed algorithm is structurally consistent (or sparsistent) when the number of samples scales as n = Ω(J−4 min log p), where p is the number of nodes and Jmin is the minimum edge potential. We also develop novel non-asymptotic techniques for obtaining necessary conditions for graphical model selection. Keywords: Graphical model selection, high-dimensional learning, local-separation property, necessary conditions, typical sets, Fano’s inequality. 1 Introduction The formalism of probabilistic graphical models can be employed to represent dependencies among a large set of random variables in the form of a graph [1]. An important challenge in the study of graphical models is to learn the unknown graph using samples drawn from the graphical model. The general structure estimation problem is NP-hard [2]. In the high-dimensional regime, structure estimation is even more difﬁcult since the number of available observations is typically much smaller than the number of dimensions (or variables). One of the goals is to characterize tractable model classes for which consistent graphical model selection can be guaranteed with low computational and sample complexities. The seminal work by Chow and Liu [3] proposed an efﬁcient algorithm for maximum-likelihood structure estimation in tree-structured graphical models by reducing the problem to a maximum weight spanning tree problem. A more recent approach for efﬁcient structure estimation is based on convex-relaxation [4–6]. The success of such methods typically requires certain “incoherence” conditions to hold. However, these conditions are NP-hard to verify for general graphical models. We adopt an alternative paradigm in this paper and instead analyze a simple local algorithm which requires only low-order statistics of the data and makes decisions on whether two nodes are neighbors in the unknown graph. We characterize the class of Ising and Gaussian graphical models for which we can guarantee efﬁcient and consistent structure estimation using this simple algorithm. The class of graphs is based on a local-separation property and includes many well-known random graph families, including locally-tree like graphs such as large girth graphs, the Erd˝os-R´enyi random graphs [7] and power-law graphs [8], as well as graphs with short cycles such as bounded-degree graphs, and small-world graphs [9]. These graphs are especially relevant in modeling social networks [10,11]. 1 1.1 Summary of Results We propose an algorithm for structure estimation, termed as conditional mutual information thresholding (CMIT), which computes the minimum empirical conditional mutual information of a given node pair over conditioning sets of bounded cardinality η. If the minimum exceeds a given threshold (depending on the number of samples n and the number of nodes p), the node pair is declared as an edge. This test has a low computational complexity of O(pη+2) and requires only low-order statistics (up to order η + 2) when η is small. The parameter η is an upper bound on the size of local vertex-separators in the graph, and is small for many common graph families, as discussed earlier. We establish that under a set of mild and transparent assumptions, structure learning is consistent in high-dimensions for CMIT when the number of samples scales as n = Ω(J−4 min log p), for a p-node graph, where Jmin is the minimum (absolute) edge-potential in the model. We also develop novel techniques to obtain necessary conditions for consistent structure estimation of Erd˝os-R´enyi random graphs. We obtain non-asymptotic bounds on the number of samples n in terms of the expected degree and the number of nodes of the model. The techniques employed are information-theoretic in nature and combine the use of Fano’s inequality and the so-called asymptotic equipartition property. Our results have many ramiﬁcations: we explicitly characterize the tradeoff between various graph parameters such as the maximum degree, girth and the strength of edge potentials for efﬁcient and consistent structure estimation. We draw connections between structure learning and the statistical physical properties of the model: learning is fundamentally related to the absence of long-range dependencies in the model, i.e., the regime of correlation decay. The notion of correlation decay on Ising models has been extensively characterized [12], but its connections to structure learning have only been explored in a few recent works (e.g., [13]). This work establishes that consistent structure learning is feasible under a slightly weaker condition than the usual notion of correlation decay for a rich class of graphs. Moreover, we show that the Gaussian analog of correlation decay is the so-called walk-summability condition [14]. This is a somewhat unexpected and surprising connection since walk-summability is a condition to characterize the performance of inference algorithms such as loopy belief propagation (LBP). Our work demonstrates that both successful inference and learning hinge on similar properties of the Gaussian graphical model. 2 Preliminaries 2.1 Graphical Models A p-dimensional graphical model is a family of p-dimensional multivariate distributions Markov on some undirected graph G=(V, E) [1]. Each node in the graph i ∈V is associated to a random variable Xi taking values in a set X. We consider both discrete (in particular Ising) models where X is a ﬁnite set and Gaussian models where X = R. The set of edges E captures the set of conditional-independence relationships among the random variables. More speciﬁcally, the vector of random variables X := (X1, . . . , Xp) with joint distribution P satisﬁes the global Markov property with respect to a graph G, if for all disjoint sets A, B ⊂V , we have P(xA, xB\|xS) = P(xA\|xS)P(xB\|xS). (1) where set S is a separator1between A and B. The Hammersley-Clifford theorem states that under the positivity condition, given by P(x) > 0 for all x ∈X p [1], the model P satisﬁes the global Markov property according to a graph G if and only if it factorizes according to the cliques of G. We consider the class of Ising models, i.e., binary pairwise models which factorize according to the edges of the graph. More precisely, the probability mass function (pmf) of an Ising model is P(x) ∝exp 1 2xT JGx + hT x , x ∈{−1, 1}p. (2) For Gaussian graphical models, the probability density function (pdf) is of the form, f(x) ∝exp −1 2xT JGx + hT x , x ∈Rp. (3) 1A set S ⊂V is a separator of sets A and B if the removal of nodes in S separates A and B into distinct components. 2 In both the cases, the matrix JG is called the potential or information matrix and h, the potential vector. For both Ising and Gaussian models, the sparsity pattern of the matrix JG corresponds to that of the graph G, i.e., JG(i, j) = 0 if and only if (i, j) /∈G. We assume that the potentials are uniformly bounded above and below as: Jmin ≤\|JG(i, j)\| ≤Jmax, ∀(i, j) ∈G. (4) Our results on structure learning depend on Jmin and Jmax, which is fairly natural – intuitively, models with edge potentials which are “too small” or “too large” are harder to learn than those with comparable potentials, i.e., homogenous models. Notice that the conventional parameterizations for the Ising models in (2) and the Gaussian models in (3) are slightly different. Without loss of generality, for Ising model, we assume that J(i, i) = 0 for all i ∈V . On the other hand, in the Gaussian setting, we assume that the diagonal elements of the inverse covariance (or information) matrix JG are normalized to unity (J(i, i) = 1, i ∈V ), and that JG can be decomposed as JG = I −RG, where RG is the matrix of partial correlation coefﬁcients [14]. We consider the problem of structure learning, which involves the estimation of the edge set of the graph G given n i.i.d. samples X1, . . . , Xn drawn either from the Ising model in (2) or the Gaussian model in (3). We consider the high-dimensional regime, where both p and n grow simultaneously; typically, the growth of p is much faster than that of n. 2.2 Tractable Graph Families We consider the class of graphical models Markov on a graph Gp belonging to some ensemble G(p) of graphs with p nodes. We emphasize that in our formulation the graph ensemble G(p) can either be deterministic or random – in the latter, we also specify a probability measure over the set of graphs in G(p). In the random setting, we use the term almost every (a.e.) graph G ∼G(p) satisﬁes a certain property Q (for example, connectedness) if limp→∞P[Gp satisﬁes Q] = 1. In other words, the property Q holds asymptotically almost surely2 (a.a.s.) with respect to the random graph ensemble G(p). Intuitively, this means that graphs that have a vanishing probability of occurrence as p →∞are ignored. Our conditions and theoretical guarantees will be based on this notion for random graph ensembles. We now characterize the ensemble of graphs amenable for consistent structure estimation. For γ ∈N, let Bγ(i; G) denote the set of vertices within distance γ from node i with respect to graph G. Let Hγ,i := G(Bγ(i; G)) denote the subgraph of G spanned by Bγ(i; G), but in addition, we retain the nodes not in Bγ(i; G) (and remove the corresponding edges). Deﬁnition 1 (γ-Local Separator) Given a graph G, a γ-local separator Sγ(i, j) between i and j, for (i, j) /∈G, is a minimal vertex separator3 with respect to the subgraph Hγ,i. The parameter γ is referred to as the path threshold for local separation. In other words, the γ-local separator Sγ(i, j) separates nodes i and j with respect to paths in G of length at most γ. We now characterize the ensemble of graphs based on the size of local separators. Deﬁnition 2 ((η, γ)-Local Separation Property) An ensemble of graphs G(p; η, γ) satisﬁes (η, γ)-local separation property if for a.e. Gp ∈G(p; η, γ), max (i,j)/∈Gp \|Sγ(i, j)\| ≤η. (5) In Section 3, we propose an efﬁcient algorithm for graphical model selection when the underlying graph belongs to a graph ensemble G(p; η, γ) with sparse local node separators (i.e., with small η). Below we provide examples of three graph families which satisfy (5) for small η. 2Note that the term a.a.s. does not apply to deterministic graph ensembles G(p) where no randomness is assumed, and in this setting, we assume that the property Q holds for every graph in the ensemble. 3A minimal separator is a separator of smallest cardinality. 3 (Example 1) Bounded Degree: Any (deterministic or random) ensemble of degree-bounded graphs GDeg(p, ∆) satisﬁes the (η, γ)-local separation property with η = ∆and every γ ∈N. Thus, our algorithm consistently recovers graphs with small (bounded) degrees (∆= O(1)). This case was considered previously in several works, e.g. [15,16]. (Example 2) Bounded Local Paths: The (η, γ)-local separation property also holds when there are at most η paths of length at most γ in G between any two nodes (henceforth, termed as the (η, γ)-local paths property). In other words, there are at most η −1 overlapping4 cycles of length smaller than 2γ. Thus, a graph with girth g (length of the shortest cycle) satisﬁes the (η, γ)-local separation property with η = 1 and γ = g. For example, the bipartite Ramanujan graph [17, p. 107] and the random Cayley graphs [18] have large girths. The girth condition can be weakened to allow for a small number of short cycles, while not allowing for overlapping cycles. Such graphs are termed as locally tree-like. For instance, the ensemble of Erd˝os-R´enyi graphs GER(p, c/p), where an edge between any node pair appears with a probability c/p, independent of other node pairs, is locally tree-like. It can be shown that GER(p, c/p) satisﬁes (η, γ)-local separation property with η = 2 and γ ≤ log p 4 log c a.a.s. Similar observations apply for the more general scale-free or power-law graphs [8, 19]. Along similar lines, the ensemble of ∆-random regular graphs, denoted by GReg(p, ∆), which is the uniform ensemble of regular graphs with degree ∆has no overlapping cycles of length at most Θ(log∆−1 p) a.a.s. [20, Lemma 1]. (Example 3) Small-World Graphs: The class of hybrid graphs or augmented graphs [8, Ch. 12] consist of graphs which are the union of two graphs: a “local” graph having short cycles and a “global” graph having small average distances between nodes. Since the hybrid graph is the union of these local and global graphs, it simultaneously has large degrees and short cycles. The simplest model GWatts(p, d, c/p), ﬁrst studied by Watts and Strogatz [9], consists of the union of a d-dimensional grid and an Erd˝os-R´enyi random graph with parameter c. One can check that a.e. graph G ∼GWatts(p, d, c/p) satisﬁes (η, γ)-local separation property in (5), with η = d + 2 and γ ≤ log p 4 log c. Similar observations apply for more general hybrid graphs studied in [8, Ch. 12]. 3 Method and Guarantees 3.1 Assumptions (A1) Scaling Requirements: We consider the asymptotic setting where both the number of variables (nodes) p and the number of samples n go to inﬁnity. We assume that the parameters (n, p, Jmin) scale in the following fashion:5 n = ω(J−4 min log p). (6) We require that the number of nodes p →∞to exploit the local separation properties of the class of graphs under consideration. (A2a) Strict Walk-summability for Gaussian Models: The Gaussian graphical model Markov on almost every Gp ∼G(p) is α-walk summable, i.e., ∥RGp∥≤α < 1, (7) where α is a constant (i.e., is not a function of p), RGp := [\|RGp(i, j)\|] is the entry-wise absolute value of the partial correlation matrix RGp. In addition, ∥·∥denotes the spectral norm, which for symmetric matrices, is given by the maximum absolute eigenvalue. (A2b) Bounded Potentials for Ising Models: The Ising model Markov on a.e. Gp ∼G(p) has its maximum absolute potential below a threshold J∗. More precisely, α := tanh Jmax tanh J∗ < 1. (8) Furthermore, the ratio α in (8) is not a function of p. See [21, 22] for an explicit characterization of J∗for speciﬁc graph ensembles. (A3) Local-Separation Property: We assume that the ensemble of graphs G(p; η, γ) satisﬁes the (η, γ)-local separation property with η, γ ∈N satisfying: η = O(1), Jminα−γ = eω(1), (9) 4Two cycles are said to overlap if they have common vertices. 5The notations ω(·), Ω(·), o(·) and O(·) refer to asymptotics as the number of variables p →∞. 4 where α is given by (7) for Gaussian models and by (8) for Ising models.6 We can weaken the second requirement in (9) as Jminα−γ = ω(1) for deterministic graph families (rather than random graph ensembles). (A4) Edge Potentials: The edge potentials {Ji,j, (i, j) ∈G} of the Ising model are assumed to be generically drawn from [−Jmax, −Jmin] ∪[Jmin, Jmax], i.e., our results hold except for a set of Lebesgue measure zero. We also characterize speciﬁc classes of models where this assumption can be removed and we allow for all choices of edge potentials. See [21,22] for details. The above assumptions are very general and hold for a rich class of models. Assumption (A1) stipulates the scaling requirements of number of samples for consistent structure estimation. Assumption (A2) and (A4) impose constraints on the model parameters. Assumption (A3) requires the local-separation property described in Section 2.2 with the path threshold γ satisfying (9). We provide examples of graphs where the above assumptions are met. Gaussian Models on Girth-bounded Graphs: Consider the ensemble of graphs GDeg,Girth(p; ∆, g) with maximum degree ∆and girth g. We now derive a relationship between ∆and g, for the above assumptions to hold. It can be established that for the walk-summability condition in (A2a) to hold for Gaussian models, we require that Jmax = O(1/∆). When the minimum edge potential achieves this bound (Jmin = Θ(1/∆)), a sufﬁcient condition for (A3) to hold is given by ∆αg = o(1). (10) In (10), we notice a natural tradeoff between the girth and the maximum degree of the graph ensemble for successful estimation under our framework: graphs with large degrees can be learned efﬁciently if their girths are large. Indeed, in the extreme case of trees which have inﬁnite girth, in accordance with (10), there is no constraint on the node degrees for consistent graphical model selection and recall that the Chow-Liu algorithm [3] is an efﬁcient method for model selection on tree-structured graphical models. Note that the condition in (10) allows for the maximum degree bound ∆to grow with the number of nodes as long as the girth g also grows appropriately. For example, if the maximum degree scales as ∆= O(poly(log p)) and the girth scales as g = O(log log p), then (10) is satisﬁed. This implies that graphs with fairly large degrees and short cycles can be recovered successfully consistently using the algorithm in Section 3.2. Gaussian Models on Erd˝os-R´enyi and Small-World Graphs: We can also conclude that a.e. Erd˝os-R´enyi graph G ∼GER(p, c/p) satisﬁes (9) with η = 2 when c = O(poly(log p)) under the best possible scaling for Jmin subject to the walk-summability constraint in (7). Similarly, the small-world ensemble GWatts(p, d, c/p) satisﬁes (9) with η = d + 2, when d = O(1) and c = O(poly(log p)). Ising Models: For Ising models, the best possible scaling of the minimum edge potential Jmin is when Jmin = Θ(J∗), for the threshold J∗deﬁned in (8). For the ensemble of graphs GDeg,Girth(p; ∆, g) with degree ∆and girth g, we can establish that J∗= Θ(1/∆). When the minimum edge potential achieves the threshold, i.e., Jmin = Θ(1/∆), we end up with a similar requirement as in (10) for Gaussian models. Similarly, for both the Erd˝os-R´enyi graph ensemble GER(p, c/p) and small-world ensemble GWatts(p, d, c/p), we can establish that the threshold J∗= Θ(1/c), and thus, the observations made for the Gaussian setting hold for the Ising model as well. 3.2 Conditional Mutual Information Threshold Test Our structure learning procedure is known as the Conditional Mutual Information Threshold Test (CMIT). Let CMIT(xn; ξn,p, η) be the output edge set from CMIT given n i.i.d. samples xn, a threshold ξn,p and a constant η ∈N. The conditional mutual information test proceeds as follows: one computes the empirical conditional mutual information7 for each node pair (i, j) ∈V 2 and ﬁnds the conditioning set which achieves the minimum, over all subsets of cardinality at most η, min S⊂V \{i,j},\|S\|≤η bI(Xi; Xj\|XS), (11) where bI(Xi; Xj\|XS) denotes the empirical conditional mutual information of Xi and Xj given XS. If the above minimum value exceeds the given threshold ξn,p, then the node pair is declared as an edge. Recall that the conditional mutual information I(Xi; Xj\|XS) = 0 iff given XS, the random variables Xi and Xj are conditionally independent. 6We say that two sequences f(p), g(p) satisfy f(p) = eω(g(p)), if f(p) g(p) log p →∞as p →∞. 7The empirical conditional mutual information is obtained by ﬁrst computing the empirical distribution and then computing its conditional mutual information. 5 Thus, (11) seeks to identify non-neighbors, i.e., node pairs which can be separated in the unknown graph G. However, since we constrain the conditioning set \|S\| ≤η in (11), the optimal conditioning set may not form an exact separator. Despite this restriction, we establish that the above test can correctly classify the edges and non-neighbors using a suitable threshold ξn,p subject to the assumptions (A1)–(A4). The threshold ξn,p is chosen as a function of the number of nodes p, the number of samples n, and the minimum edge potential Jmin as follows: ξn,p = O(J2 min), ξn,p = ω(α2γ), ξn,p = Ω log p n , (12) where γ is the path-threshold in (5) for (η, γ)-local separation to hold and α is given by (7) and (8). The computational complexity of the CMIT algorithm is O(pη+2). Thus the algorithm is computationally efﬁcient for small η. Moreover, the algorithm only uses statistics of order η + 2 in contrast to the convex-relaxation approaches [4–6] which typically use higher-order statistics. Theorem 1 (Structural consistency of CMIT) Assume that (A1)-(A4) hold. Given a Gaussian graphical model or an Ising model Markov on a graph Gp ∼G(p; η, γ), CMIT(xn; ξn,p, η) is structurally consistent. In other words, lim n,p→∞P [CMIT ({xn}; ξn,p, η) ̸= Gp] = 0. (13) Consistency guarantee The CMIT algorithm consistently recovers the structure of the graphical models with probability tending to one and the probability measure in (4) is with respect to both the graph and the samples. Sample-complexity The sample complexity of the CMIT scales as Ω(J−4 min log p) and is favorable when the minimum edge potential Jmin is large. This is intuitive since the edges have stronger potentials when Jmin is large. On the other hand, Jmin cannot be arbitrarily large due to the assumption (A2). The minimum sample complexity is attained when Jmin achieves this upper bound. It can be established that for both Gaussian and Ising models Markov on a degree-bounded graph ensemble GDeg(p, ∆) with maximum degree ∆and satisfying assumption (A3), the minimum sample complexity is given by n = Ω(∆4 log p) i.e., when Jmin = Θ(1/∆). We can have improved guarantees for the Erd˝os-R´enyi random graphs GER(p, c/p). In the Gaussian setting, the minimum sample complexity can be improved to n = Ω(∆2 log p), i.e., when Jmin = Θ(1/ √ ∆) where the maximum degree scales as ∆= Θ(log p log c) [7]. On the other hand, for Ising models, the minimum sample complexity can be further improved to n = Ω(c4 log p), i.e., when Jmin = Θ(J∗) = Θ(1/c). Note that c/2 is the expected degree of the GER(p, c/p) ensemble. Speciﬁcally, when the Erd˝os-R´enyi random graphs have a bounded average degree (c = O(1)), we can obtain a minimum sample complexity of n = Ω(log p) for structure estimation of Ising models. Recall that the sample complexity of learning tree models is Ω(log p) [23]. Thus, the complexity of learning sparse Erd˝os-R´enyi random graphs is akin to learning trees in certain parameter regimes. The sample complexity of structure estimation can be improved to n = Ω(J−2 min log p) by employing empirical conditional covariances for Gaussian models and empirical conditional variation distances in place of empirical conditional mutual information. However, to present a uniﬁed framework for Gaussian and Ising models, we present the CMIT here. See [21,22] for details. Comparison with convex-relaxation approaches We now compare our approach for structure learning with convex-relaxation methods. The work by Ravikumar et al. in [5] employs an ℓ1-penalized likelihood estimator and under the so-called incoherence conditions, the sample complexity is n = Ω((∆2 +J−2 min) log p). Our sample complexity (using conditional covariances) n = Ω(J−2 min log p) is the same in terms of Jmin, while there is no explicit dependence on the maximum degree ∆. Similarly, we match the neighborhood-based regression method by Meinshausen and Buhlmann in [24] under more transparent conditions. For structure estimation of Ising models, the work in [6] considers ℓ1-penalized logistic regression which has a sample complexity of n = Ω(∆3 log p) for a degree-bounded ensemble GDeg(p, ∆) satisfying certain “incoherence” conditions. The sample complexity of CMIT, given by n = Ω(∆4 log p), is slightly worse, while the modiﬁed algorithm described previously has a sample complexity of n = Ω(∆2 log p), for general degree-bounded ensembles. Additionally, under the CMIT algorithm, we can guarantee an improved sample complexity of n = Ω(c4 log p) for Erd˝os-R´enyi 6 random graphs GER(p, c/p) and small-world graphs GWatts(p, d, c/p), since the average degree c/2 is typically much smaller than the maximum degree ∆. Moreover, note that, the incoherence conditions stated in [6] are NP-hard to establish for general models since they involve the partition function of the model. In contrast, our conditions are transparent and relate to the statistical-physical properties of the model. Moreover, our algorithm is local and requires only low-order statistics, while the method in [6] requires full-order statistics. Proof Outline We ﬁrst analyze the scenario when exact statistics are available. (i) We establish that for any two non-neighbors (i, j) /∈G, the minimum conditional mutual information in (11) (based on exact statistics) does not exceed the threshold ξn,p. (ii) Similarly, we also establish that the conditional mutual information in (11) exceeds the threshold ξn,p for all neighbors (i, j) ∈G. (iii) We then extend these results to empirical versions using concentration bounds. See [21,22] for details. The main challenge in our proof is step (i). To this end, we analyze the conditional mutual information when the conditioning set is a local separator between i and j and establish that it decays as p →∞. The techniques involved to establish this for Ising and Gaussian models are different: for Ising models, we employ the self-avoiding walk (SAW) tree construction [25]. For Gaussian models, we use the techniques from walk-sum analysis [14]. 4 Necessary Conditions for Model Selection In the previous sections, we proposed and analyzed efﬁcient algorithms for learning the structure of graphical models. We now derive the necessary conditions for consistent structure learning. We focus on the ensemble of Erd˝os-R´enyi graphs GER(p, c/p). For the class of degree-bounded graphs GDeg(p, ∆), necessary conditions on sample complexity have been characterized previously [26] by considering a certain (restricted) set of ensembles. However, a na¨ıve application of such bounds (based on Fano’s inequality [27, Ch. 2]) turns out to be too weak for the class of Erd˝os-R´enyi graphs GER(p, c/p). We provide novel necessary conditions for structure learning of Erd˝os-R´enyi graphs. Our techniques may also be applicable to other classes of random graphs. Recall that a graph G is drawn from the ensemble of Erd˝os-R´enyi graphs GER(p, c/p). Given n i.i.d. samples Xn := (X1, . . . , Xn) ∈(X p)n, the task is to estimate G from Xn. Denote the estimated graph as bG := bG(Xn). It is desired to derive tight necessary conditions on the number of samples n (as a function of average degree c/2 and number of nodes p) so that the probability of error P (p) e := P( bG(Xn) ̸= G) →0 as the number of nodes p tends to inﬁnity. Again, note that the probability measure P is with respect to both the Erd˝os-R´enyi graph and the samples. Discrete Graphical Models Let Hb(q) := −q log2 q −(1 −q) log2(1 −q) be the binary entropy function. For the Ising model, or more generally any discrete model where each random variable Xi ∈X = {1, . . ., \|X\|}, we can demonstrate the following: Theorem 2 (Weak Converse for Discrete Models) For a discrete graphical model Markov on G ∼GER(p, c/p), if P (p) e →0, it is necessary for n to satisfy n ≥ 1 p log2 \|X\| p 2 Hb c p ≥ c log2 p 2 log2 \|X\|. (14) The above bound does not involve any asymptotic notation and shows transparently, how n has to depend on p, c and \|X\| for consistent structure learning. Note that if the cardinality of the random variables \|X\| is large, then the necessary sample complexity is small, which makes intuitive sense from a source-coding perspective. Moreover, the above bound states that more samples are required as the average degree c/2 increases. Our bound involves only the average degree c/2 and not the maximum degree of the graph, which is typically much larger than c [7]. Gaussian Graphical Models We now turn out attention to the Gaussian analogue of Theorem 2 under a similar setup. We assume that the α-walk-summability condition in assumption (A2a) holds. We are then able to demonstrate the following: 7 Theorem 3 (Weak Converse for Gaussian Models) For an α-walk summable Gaussian graphical model Markov on G ∼GER(p, c/p) as p →∞, if P (p) e →0, we require n ≥ 2 p log2 h 2πe 1 1−α + 1 i p 2 Hb c p ≥ c log2 p log2 h 2πe 1 1−α + 1 i. (15) As with Theorem 2, the above bound does not involve any asymptotic notation and similar intuitions hold as before. There is a natural logarithmic dependence on p and a linear dependence on the average degree parameter c. Finally, the dependence on α can be explained as follows: any α-walk-summable model is also β-walk-summable for all β > α. Thus, the class of β-walk-summable models contains the class of α-walk-summable models. This results in a looser bound in (15) for large α. Analysis tools Our analysis tools are information-theoretic in nature. A common tool to derive necessary conditions is to resort to Fano’s inequality [27, Ch. 2], which (lower) bounds the probability of error P (p) e as a function of the conditional entropy H(G\|Xn) and the size of the set of all graphs with p nodes. However, a na¨ıve application of Fano’s inequality results in a trivial lower bound as the set of all graphs, which can be realized by GER(p, c/p) is “too large”. To ameliorate this problem, we focus our attention on the typical graphs for applying Fano’s inequality and not all graphs. The set of typical graphs has a small cardinality but high probability when p is large. The novelty of our proof lies in our use of both typicality as well as Fano’s inequality to derive necessary conditions for structure learning. We can show that (i) the probability of the typical set tends to one as p →∞, (ii) the graphs in the typical set are almost uniformly distributed (the asymptotic equipartition property), (iii) the cardinality of the typical set is small relative to the set of all graphs. These properties are used to prove Theorems 2 and 3. 5 Conclusion In this paper, we adopted a novel and a uniﬁed paradigm for graphical model selection. We presented a simple local algorithm for structure estimation with low computational and sample complexities under a set of mild and transparent conditions. This algorithm succeeds on a wide range of graph ensembles such as the Erd˝os-R´enyi ensemble, smallworld networks etc. We also employed novel information-theoretic techniques for establishing necessary conditions for graphical model selection. Acknowledgement The ﬁrst author is supported by the setup funds at UCI and in part by the AFOSR under Grant FA9550-10-1-0310, the second author is supported by A*STAR, Singapore and the third author is supported in part by AFOSR under Grant FA9550-08-1-1080. References [1] S. Lauritzen, Graphical models: Clarendon Press. Clarendon Press, 1996. [2] D. Karger and N. Srebro, “Learning Markov Networks: Maximum Bounded Tree-width Graphs,” in Proc. of ACM-SIAM symposium on Discrete algorithms, 2001, pp. 392–401. [3] C. Chow and C. Liu, “Approximating Discrete Probability Distributions with Dependence Trees,” IEEE Tran. on Information Theory, vol. 14, no. 3, pp. 462–467, 1968. [4] A. d’Aspremont, O. Banerjee, and L. El Ghaoui, “First-order methods for sparse covariance selection,” SIAM. J. Matrix Anal. & Appl., vol. 30, no. 56, 2008. [5] P. Ravikumar, M. Wainwright, G. Raskutti, and B. Yu, “High-dimensional covariance estimation by minimizing ℓ1-penalized log-determinant divergence,” Arxiv preprint arXiv:0811.3628, 2008. [6] P. Ravikumar, M. Wainwright, and J. Lafferty, “High-dimensional Ising Model Selection Using l1-Regularized Logistic Regression,” Annals of Statistics, 2008. 8 [7] B. Bollob´as, Random Graphs. Academic Press, 1985. [8] F. Chung and L. Lu, Complex graphs and network. Amer. Mathematical Society, 2006. [9] D. Watts and S. Strogatz, “Collective dynamics of small-worldnetworks,” Nature, vol. 393, no. 6684, pp. 440– 442, 1998. [10] M. Newman, D. Watts, and S. Strogatz, “Random graph models of social networks,” Proc. of the National Academy of Sciences of the United States of America, vol. 99, no. Suppl 1, 2002. [11] R. Albert and A. Barab´asi, “Statistical mechanics of complex networks,” Reviews of modern physics, vol. 74, no. 1, p. 47, 2002. [12] H. Georgii, Gibbs Measures and Phase Transitions. Walter de Gruyter, 1988. [13] J. Bento and A. 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Virag, “On the girth of random cayley graphs,” Random Structures & Algorithms, vol. 35, no. 1, pp. 100–117, 2009. [19] S. Dommers, C. Giardin`a, and R. van der Hofstad, “Ising models on power-law random graphs,” Journal of Statistical Physics, pp. 1–23, 2010. [20] B. McKay, N. Wormald, and B. Wysocka, “Short cycles in random regular graphs,” The Electronic Journal of Combinatorics, vol. 11, no. R66, p. 1, 2004. [21] A. Anandkumar, V. Y. F. Tan, and A. S. Willsky, “High-Dimensional Structure Learning of Ising Models: Tractable Graph Families,” Preprint, Available on ArXiv 1107.1736, June 2011. [22] ——, “High-Dimensional Gaussian Graphical Model Selection: Tractable Graph Families,” Preprint, ArXiv 1107.1270, June 2011. [23] V. Tan, A. Anandkumar, and A. Willsky, “Learning Markov Forest Models: Analysis of Error Rates,” J. of Machine Learning Research, vol. 12, pp. 1617–1653, May 2011. [24] N. Meinshausen and P. 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4,450	Optimal learning rates for least squares SVMs using Gaussian kernels M. Eberts, I. Steinwart Institute for Stochastics and Applications University of Stuttgart D-70569 Stuttgart {eberts,ingo.steinwart}@mathematik.uni-stuttgart.de Abstract We prove a new oracle inequality for support vector machines with Gaussian RBF kernels solving the regularized least squares regression problem. To this end, we apply the modulus of smoothness. With the help of the new oracle inequality we then derive learning rates that can also be achieved by a simple data-dependent parameter selection method. Finally, it turns out that our learning rates are asymptotically optimal for regression functions satisfying certain standard smoothness conditions. 1 Introduction On the basis of i.i.d. observations D := ((x1, y1) , . . . , (xn, yn)) of input/output observations drawn from an unknown distribution P on X ⇥Y , where Y ⇢R, the goal of non-parametric least squares regression is to ﬁnd a function fD : X ! R such that, for the least squares loss L : Y ⇥R ! [0, 1) deﬁned by L (y, t) = (y −t)2, the risk RL,P (fD) := Z X⇥Y L (y, fD (x)) dP (x, y) = Z X⇥Y (y −fD (x))2 dP (x, y) is small. This means RL,P (fD) has to be close to the optimal risk R⇤ L,P := inf {RL,P (f) \| f : X ! R measureable} , called the Bayes risk with respect to P and L. It is well known that the function f ⇤ L,P : X ! R deﬁned by f ⇤ L,P (x) = EP (Y \|x), x 2 X, is the only function for which the Bayes risk is attained. Furthermore, some simple transformations show RL,P (f) −R⇤ L,P = Z X ""f −f ⇤ L,P ""2 dPX = ##f −f ⇤ L,P ##2 L2(PX) , (1) where PX is the marginal distribution of P on X. In this paper, we assume that X ⇢Rd is a non-empty, open and bounded set such that its boundary @X has Lebesgue measure 0, Y := [−M, M] for some M > 0 and P is a probability measure on X⇥Y such that PX is the uniform distribution on X. In Section 2 we also discuss that this condition can easily be generalized by assuming that PX on X is absolutely continuous with respect to the Lebesgue measure on X such that the corresponding density of PX is bounded away from 0 and 1. Recall that because of the ﬁrst assumption, it sufﬁces to restrict considerations to decision functions f : X ! [−M, M]. To be more precise, if, we denote the clipped value of some t 2 R by Ût, that is Ût := 8 < : −M if t < −M t if t 2 [−M, M] M if t > M , 1 then it is easy to check that RL,P( Ûf ) RL,P (f) , for all f : X ! R. The non-parametric least squares problem can be solved in many ways. Several of them are e.g. described in [1]. In this paper, we use SVMs to ﬁnd a solution for the non-parametric least squares problem by solving the regularized problem fD,λ = arg min f2H λ kfk2 H + RL,D (f) . (2) Here, λ > 0 is a ﬁxed real number, H is a reproducing kernel Hilbert space (RKHS) over X, and RL,D (f) is the empirical risk of f, that is RL,D (f) = 1 n n X i=1 L (yi, f (xi)) . In this work we restrict our considerations to Gaussian RBF kernels kγ on X, which are deﬁned by kγ (x, x0) = exp −kx −x0k2 2 γ2 ! , x, x0 2 X , for some width γ 2 (0, 1]. Our goal is to deduce asymptotically optimal learning rates for the SVMs (2) using the RKHS Hγ of kγ. To this end, we ﬁrst establish a general oracle inequality. Based on this oracle inequality, we then derive learning rates if the regression function is contained in some Besov space. It will turn out, that these learning rates are asymptotically optimal. Finally, we show that these rates can be achieved by a simple data-dependent parameter selection method based on a hold-out set. The rest of this paper is organized as follows: The next section presents the main theorems and as a consequence of these theorems some corollaries inducing asymptotically optimal learning rates for regression functions contained in Sobolev or Besov spaces. Section 3 states some, for the proof of the main statement necessary, lemmata and a version of [2, Theorem 7.23] applied to our special case as well as the proof of the main theorem. Some further proofs and additional technical results can be found in the appendix. 2 Results In this section we present our main results including the optimal rates for LS-SVMs using Gaussian kernels. To this end, we ﬁrst need to introduce some function spaces, which are later assumed to contain the regression function. Let us begin by recalling from, e.g. [3, p. 44], [4, p. 398], and [5, p. 360], the modulus of smoothness: Deﬁnition 1. Let ⌦⇢Rd with non-empty interior, ⌫be an arbitrary measure on ⌦, and f : ⌦! Rd be a function with f 2 Lp (⌫) for some p 2 (0, 1). For r 2 N, the r-th modulus of smoothness of f is deﬁned by !r,Lp(⌫) (f, t) = sup khk2t k4r h (f, · )kLp(⌫) , t ≥0 , where k · k2 denotes the Euclidean norm and the r-th difference 4r h (f, ·) is deﬁned by 4r h (f, x) = (Pr j=0 ,r j (−1)r−j f (x + jh) if x 2 ⌦r,h 0 if x /2 ⌦r,h for h = (h1, . . . , hd) 2 Rd with hi ≥0 and ⌦r,h := {x 2 ⌦: x + sh 2 ⌦8 s 2 [0, r]}. It is well-known that the modulus of smoothness with respect to Lp (⌫) is a nondecreasing function of t and for the Lebesgue measure on ⌦it satisﬁes !r,Lp(⌦) (f, t)  ✓ 1 + t s ◆r !r,Lp(⌦) (f, s) , (3) 2 for all f 2 Lp (⌦) and all s > 0, see e.g. [6, (2.1)]. Moreover, the modulus of smoothness can be used to deﬁne the scale of Besov spaces. Namely, for 1 p, q 1, ↵> 0, r := b↵c + 1, and an arbitrary measure ⌫, the Besov space B↵ p,q (⌫) is B↵ p,q (⌫) := n f 2 Lp (⌫) : \|f\|B↵ p,q(⌫) < 1 o , where, for 1 q < 1, the seminorm \|· \|B↵ p,q(⌫) is deﬁned by \|f\|B↵ p,q(⌫) := ✓Z 1 0 , t−↵!r,Lp(⌫) (f, t) -q dt t ◆1 q , and, for q = 1, it is deﬁned by \|f\|B↵ p,1(⌫) := sup t>0 , t−↵!r,Lp(⌫) (f, t) . In both cases the norm of B↵ p,q (⌫) can be deﬁned by kfkB↵ p,q(⌫) := kfkLp(⌫) + \|f\|B↵ p,q(⌫), see e.g. [3, pp. 54/55] and [4, p. 398]. Finally, for q = 1, we often write B↵ p,1 (⌫) = Lip⇤(↵, Lp (⌫)) and call Lip⇤(↵, Lp (⌫)) the generalized Lipschitz space of order ↵. In addition, it is well-known, see e.g. [7, p. 25 and p. 44], that the Sobolev spaces W ↵ p (Rd) fall into the scale of Besov spaces, namely W ↵ p (Rd) ⇢B↵ p,q(Rd) (4) for ↵2 N, p 2 (1, 1), and max{p, 2} q 1 and especially W ↵ 2 (Rd) = B↵ 2,2(Rd). For our results we need to extend functions f : ⌦! R to functions ˆf : Rd ! R such that the smoothness properties of f described by some Sobolev or Besov space are preserved by ˆf. Recall that Stein’s Extension Theorem guarantees the existence of such an extension, whenever ⌦is a bounded Lipschitz domain. To be more precise, in this case there exists a linear operator E mapping functions f : ⌦! R to functions Ef : Rd ! R with the properties: (a) E (f)\|⌦= f, that is, E is an extension operator. (b) E continuously maps W m p (⌦) into W m p , Rdfor all p 2 [1, 1] and all integer m ≥0. That is, there exist constants am,p ≥0, such that, for every f 2 W m p (⌦), we have kEfkW m p (Rd) am,p kfkW m p (⌦) . (5) (c) E continuously maps B↵ p,q (⌦) into B↵ p,q , Rdfor all p 2 (1, 1), q 2 (0, 1] and all ↵> 0. That is, there exist constants a↵,p,q ≥0, such that, for every f 2 B↵ p,q (⌦), we have kEfkB↵ p,q(Rd) a↵,p,q kfkB↵ p,q(⌦) . For detailed conditions on ⌦ensuring the existence of E, we refer to [8, p. 181] and [9, p. 83]. Property (c) follows by some interpolation argument since B↵ p,q can be interpreted as interpolation space of the Sobolev spaces W m0 p and W m1 p for q 2 [1, 1], p 2 (1, 1), ✓2 (0, 1) and m0, m1 2 N0 with m0 6= m1 and ↵= m0(1 −✓) + m1✓, see [10, pp. 65/66] for more details. In the following, we always assume that we do have such an extension operator E. Moreover, if µ is the Lebesgue measure on ⌦, such that @⌦has Lebesgue measure 0, the canonical extension of µ to Rd is given by eµ(A) := µ(A \ ⌦) for all measurable A ⇢Rd. However, in a slight abuse of notation, we often write µ instead of eµ, since this simpliﬁes the presentation. Analogously, we proceed for the uniform distribution on ⌦and its canonical extension to Rd and the same convention will be applied to measures PX on ⌦that are absolutely continuous w.r.t. the Lebesgue measure. Finally, in order to state our main results, we denote the closed unit ball of the d-dimensional Euclidean space by B`d 2. Theorem 1. Let X ⇢B`d 2 be a domain such that we have an extension operator E in the above sense. Furthermore, let M > 0, Y := [−M, M], and P be a distribution on X ⇥Y such that PX is the uniform distribution on X. Assume that we have ﬁxed a version f ⇤ L,P of the regression 3 function such that f ⇤ L,P (x) = EP (Y \|x) 2 [−M, M] for all x 2 X. Assume that, for ↵≥1 and r := b↵c + 1, there exists a constant c > 0 such that, for all t 2 (0, 1], we have !r,L2(Rd) , Ef ⇤ L,P, t ct↵. (6) Then, for all " > 0 and p 2 (0, 1) there exists a constant K > 0 such that for all n ≥1, ⌧≥1, and λ > 0, the SVM using the RKHS Hγ satisﬁes λ kfD,λk2 Hγ + RL,P( ÛfD,λ) −R⇤ L,P Kλγ−d + Kc2γ2↵+ K γ−(1−p)(1+")d λpn + K⌧ n with probability Pn not less than 1 −e−⌧. With this oracle inequality we can derive learning rates for the learning method (2). Corollary 1. Under the assumptions of Theorem 1 and for " > 0, p 2 (0, 1), and ⌧≥1 ﬁxed, we have, for all n ≥1, λn kfD,λnk2 Hγn + RL,P( ÛfD,λn) −R⇤ L,P Cn− 2↵ 2↵+2↵p+dp+(1−p)(1+")d with probability Pn not less than 1 −e−⌧and with λn = c1n− 2↵+d 2↵+2↵p+dp+(1−p)(1+")d , γn = c2n− 1 2↵+2↵p+dp+(1−p)(1+")d . Here, c1 > 0 and c2 > 0 are user-speciﬁed constants and C > 0 is a constant independent of n. Note that for every ⇢> 0 we can ﬁnd ", p 2 (0, 1) sufﬁciently close to 0 such that the learning rate in Corollary 1 is at least as fast as n− 2↵ 2↵+d +⇢. To achieve these rates, however, we need to set λn and γn as in Corollary 1, which in turn requires us to know ↵. Since in practice we usually do not know this value, we now show that a standard training/validation approach, see e.g. [2, Chapters 6.5, 7.4, 8.2], achieves the same rates adaptively, i.e. without knowing ↵. To this end, let ⇤:= (⇤n) and Γ := (Γn) be sequences of ﬁnite subsets ⇤n, Γn ⇢(0, 1]. For a data set D := ((x1, y1) , . . . , (xn, yn)), we deﬁne D1 := ((x1, y1) , . . . , (xm, ym)) D2 := ((xm+1, ym+1) , . . . , (xn, yn)) where m := ⌅n 2 ⇧ + 1 and n ≥4. We will use D1 as a training set by computing the SVM decision functions fD1,λ,γ := arg min f2Hγ λ kfk2 Hγ + RL,D1 (f) , (λ, γ) 2 ⇤n ⇥Γn and use D2 to determine (λ, γ) by choosing a (λD2, γD2) 2 ⇤n ⇥Γn such that RL,D2 , fD1,λD2,γD2 = min (λ,γ)2⇤n⇥Γn RL,D2 (fD1,λ,γ) . Theorem 2. Under the assumptions of Theorem 1 we ﬁx sequences ⇤:= (⇤n) and Γ := (Γn) of ﬁnite subsets ⇤n, Γn ⇢(0, 1] such that ⇤n is an ✏n-net of (0, 1] and Γn is an δn-net of (0, 1] with ✏n n−1 and δn n− 1 2+d . Furthermore, assume that the cardinalities \|⇤n\| and \|Γn\| grow polynomially in n. Then, for all ⇢> 0, the TV-SVM producing the decision functions fD1,λD2,γD2 learns with the rate n− 2↵ 2↵+d +⇢ (7) with probability Pn not less than 1 −e−⌧. What is left to do is to relate Assumption (6) with the function spaces introduced earlier, such that we can show that the learning rates deduced earlier are asymptotically optimal under some circumstances. 4 Corollary 2. Let X ⇢B`d 2 be a domain such that we have an extension operator E of the form described in front of Theorem 1. Furthermore, let M > 0, Y := [−M, M], and P be a distribution on X ⇥Y such that PX is the uniform distribution on X. If, for some ↵2 N, we have f ⇤ L,P 2 W ↵ 2 (PX), then, for all ⇢> 0, both the SVM considered in Corollary 1 and the TV-SVM considered in Theorem 2 learn with the rate n− 2↵ 2↵+d +⇢ with probability Pn not less than 1 −e−⌧. Moreover, if ↵> d/2, then this rate is asymptotically optimal in a minmax sense. Similar to Corollary 2 we can show assumption (6) and asymptotically optimal learning rates if the regression function is contained in a Besov space. Corollary 3. Let X ⇢B`d 2 be a domain such that we have an extension operator E of the form described in front of Theorem 1. Furthermore, let M > 0, Y := [−M, M], and P be a distribution on X ⇥Y such that PX is the uniform distribution on X. If, for some ↵≥1, we have f ⇤ L,P 2 B↵ 2,1(PX), then, for all ⇢> 0, both the SVM considered in Corollary 1 and the TV-SVM considered in Theorem 2 learn with the rate n− 2↵ 2↵+d +⇢ with probability Pn not less than 1 −e−⌧. Since for the entropy numbers ei( id : B↵ 2,1(PX) ! L2(PX)) ⇠i−↵ d holds (cf. [7, p. 151]) and since B↵ 2,1(PX) = B↵ 2,1(X) is continuously embedded into the space `1(X) of all bounded functions on X, we obtain by [11, Theorem 2.2] that n− 2↵ 2↵+d is the optimal learning rate in a minimax sense for ↵> d (cf. [12, Theorem 13]). Therefore, for ↵> d, the learning rates obtained in Corollary 3 are asymptotically optimal. So far, we always assumed that PX is the uniform distribution on X. This can be generalized by assuming that PX is absolutely continuous w.r.t. the Lebesgue measure µ such that the corresponding density is bounded away from zero and from inﬁnity. Then we have L2(PX) = L2(µ) with equivalent norms and the results for µ hold for PX as well. Moreover, to derive learning rates, we actually only need that the Lebesgue density of PX is upper bounded. The assumption that the density is bounded away from zero is only needed to derive the lower bounds in Corollaries 2 and 3. Furthermore, we assumed γ 2 (0, 1] in Theorem 1, and hence in Corollary 1 and Theorem 2 as well. Note that γ does not need to be restricted by one. Instead γ only needs to be bounded from above by some constant such that estimates on the entropy numbers for Gaussian kernels as used in the proofs can be applied. For the sake of simplicity we have chosen one as upper bound, another upper bound would only have inﬂuence on the constants. There have already been made several investigations on learning rates for SVMs using the least squares loss, see e.g. [13, 14, 15, 16, 17] and the references therein. In particular, optimal rates have been established in [16], if f ⇤ P 2 H, and the eigenvalue behavior of the integral operator associated to H is known. Moreover, if f ⇤ P 62 H [17] and [12] establish both learning rates of the form n−β/(β+p), where β is a parameter describing the approximation properties of H and p is a parameter describing the eigenvalue decay. Furthermore, in the introduction of [17] it is mentioned that the assumption on the eigenvalues and eigenfunctions also hold for Gaussian kernels with ﬁxed width, but this case as well as the more interesting case of Gaussian kernels with variable widths are not further investigated. In the ﬁrst case, where Gaussian kernels with ﬁxed width are considered, the approximation error behaves very badly as shown in [18] and fast rates cannot be expected as we discuss below. In the second case, where variable widths are considered as in our paper, it is crucial to carefully control the inﬂuence of γ on all arising constants which unfortunately has not been worked out in [17], either. In [17] and [12], however, additional assumptions on the interplay between H and L2(PX) are required, and [17] actually considers a different exponent in the regularization term of (2). On the other hand, [12] shows that the rate n−β/(β+p) is often asymptotically optimal in a minmax sense. In particular, the latter is the case for H = W m 2 (X), f 2 W s 2 (X), and s 2 (d/2, m], that is, when using a Sobolev space as the underlying RKHS H, 5 then all target functions contained in a Sobolev of lower smoothness s > d/2 can be learned with the asymptotically optimal rate n− 2s 2s+d . Here we note that the condition s > d/2 ensures by Sobolev’s embedding theorem that W s 2 (X) consists of bounded functions, and hence Y = [−M, M] does not impose an additional assumption on f ⇤ L,P. If s 2 (0, d/2], then the results of [12] still yield the above mentioned rates, but we no longer know whether they are optimal in a minmax sense, since Y = [−M, M] does impose an additional assumption. In addition, note that for Sobolev spaces this result, modulo an extra log factor, has already been proved by [1]. This result suggests that by using a C1-kernel such as the Gaussian RBF kernel, one could actually learn the entire scale of Sobolev spaces with the above mentioned rates. Unfortunately, however, there are good reasons to believe that this is not the case. Indeed, [18] shows that for many analytic kernels the approximation error can only have polynomial decay if f ⇤ L,P is analytic, too. In particular, for Gaussian kernels with ﬁxed width γ and f ⇤ L,P 62 C1 the approximation error does not decay polynomially fast, see [18, Proposition 1.1.], and if f ⇤ L,P 2 W m 2 (X), then, in general, the approximation error function only has a logarithmic decay. Since it seems rather unlikely that these poor approximation properties can be balanced by superior bounds on the estimation error, the above-mentioned results indicate that Gaussian kernels with ﬁxed width may have a poor performance. This conjecture is backed-up by many empirical experience gained throughout the last decade. Beginning with [19], research has thus focused on the learning performance of SVMs with varying widths. The result that is probably the closest to ours is [20]. Although these authors actually consider binary classiﬁcation using convex loss functions including the least squares loss, formulated it is relatively straightforward to translate their ﬁnding to our least squares regression scenario. The result is the learning rate n− m m+2d+2 , again under the assumption f ⇤ L,P 2 W m 2 (X) for some m > 0. Furthermore, [21] treats the case, where X is isometrically embedded into a t-dimensional, connected and compact C1-submanifold of Rd. In this case, it turns out that the resulting learning rate does not depend on the dimension d, but on the intrinsic dimension t of the data. Namely the authors show the rate n− s 8s+4t modulo a logarithmic factor, where s 2 (0, 1] and f ⇤ L,P 2 Lip (s). Another direction of research that can be applied to Gaussian kernels with varying widths are multi-kernel regularization schemes, see [22, 23, 24] for some results in this direction. For example, [22] establishes learning rates of the form n− 2m−d 4(4m−d) +⇢ whenever f ⇤ L,P 2 W m 2 (X) for some m 2 (d/2, d/2 + 2), where again ⇢> 0 can be chosen to be arbitrarily close to 0. Clearly, all these results provide rates that are far from being optimal, so that it seems fair to say that our results represent a signiﬁcant advance. Furthermore, we can conclude that, in terms of asymptotical minmax rates, multi-kernel approaches applied Gaussian RBFs cannot provide any signiﬁcant improvement over a simple training/validation approach for determining the kernel width and the regularization parameter, since the latter already leads to rates that are optimal modulo an arbitrarily small ⇢in the exponent. 3 Proof of the main result To prove Theorem 1 we deduce an oracle inequality for the least squares loss by specializing [2, Theorem 7.23] (cf. Theorem 3). To be ﬁnally able to show Theorem 1 originating from Theorem 3, we have to estimate the approximation error. Lemma 1. Let X ⇢Rd be a domain such that we have an extension operator E of the form described in front of Theorem 1, PX be the uniform distribution on X and f 2 L1 (X). Furthermore, let ˜f be deﬁned by ˜f (x) := , γp⇡ -−d 2 Ef (x) (8) for all x 2 Rd and, for r 2 N and γ > 0, K : Rd ! R be deﬁned by K (·) := r X j=1 ✓r j ◆ (−1)1−j 1 jd ✓ 2 γp⇡ ◆d 2 K jγ p 2 (·) (9) with Kγ (·) := exp −k·k2 2 γ2 ! . 6 Then, for r 2 N, γ > 0, and q 2 [1, 1), we have Ef 2 Lq(ePX) and ###K ⇤˜f −f ### q Lq(PX) Cr,q !q r,Lq(Rd) (Ef, γ/2) , where Cr,q is a constant only depending on r, q and µ(X). In order to use the conclusion of Lemma 1 in the proof of Theorem 1 it is necessary to know some properties of K ⇤˜f. Therefore, we need the next two lemmata. Lemma 2. Let g 2 L2 , Rd, Hγ be the RKHS of the Gaussian RBF kernel kγ over X ⇢Rd and K (x) := r X j=1 ✓r j ◆ (−1)1−j 1 jd ✓ 2 γp⇡ ◆d 2 exp −2 kxk2 2 j2γ2 ! for x 2 Rd and a ﬁxed r 2 N. Then we have K ⇤g 2 Hγ , kK ⇤gkHγ (2r −1) kgkL2(Rd) . Lemma 3. Let g 2 L1 , Rd, Hγ be the RKHS of the Gaussian RBF kernel kγ over X ⇢Rd and K be as in Lemma 2. Then \|K ⇤g (x)\|  , γp⇡ - d 2 (2r −1) kgkL1(Rd) holds for all x 2 X. Additionally, we assume that X is a domain in Rd such that we have an extension operator E of the form described in front of Theorem 1, Y := [−M, M] and, for all x 2 Rd, ˜f (x) := (γp⇡)−d 2 E , f ⇤ L,P (x) , where f ⇤ L,P denotes a version of the conditional expectation such that f ⇤ L,P (x) = EP (Y \|x) 2 [−M, M] for all x 2 X. Then we have ˜f 2 L1 , Rdand \|K ⇤˜f (x) \| a0,1 (2r −1) M for all x 2 X, which implies L(y, K ⇤˜f (x)) 4ra2M 2 for the least squares loss L and all (x, y) 2 X ⇥Y . Next, we modify [2, Theorem 7.23], so that the proof of Theorem 1 can be build upon it. Theorem 3. Let X ⇢B`d 2, Y := [−M, M] ⇢R be a closed subset with M > 0 and P be a distribution on X ⇥Y . Furthermore, let L : Y ⇥R ! [0, 1) be the least squares loss, kγ be the Gaussian RBF kernel over X with width γ 2 (0, 1] and Hγ be the associated RKHS. Fix an f0 2 Hγ and a constant B0 ≥4M 2 such that kL ◦f0k1 B0. Then, for all ﬁxed ⌧≥1, λ > 0, " > 0 and p 2 (0, 1), the SVM using Hγ and L satisﬁes λ kfD,λk2 Hγ + RL,P ⇣ ÛfD,λ ⌘ −R⇤ L,P 9 ⇣ λ kf0k2 Hγ + RL,P (f0) −R⇤ L,P ⌘ + C",p γ−(1−p)(1+")d λpn + , 3456M 2 + 15B0 (ln(3) + 1)⌧ n with probability Pn not less than 1 −e−⌧, where C",p is a constant only depending on ", p and M. With the previous results we are ﬁnally able to prove the oracle inequality declared by Theorem 1. Proof of Theorem 1. First of all, we want to apply Theorem 3 for f0 := K ⇤˜f with K (x) := r X j=1 ✓r j ◆ (−1)1−j 1 jd ✓ 2 γp⇡ ◆d 2 exp −2 kxk2 2 j2γ2 ! 7 and ˜f (x) := , γp⇡ -−d 2 Ef ⇤ L,P (x) for all x 2 Rd. The choice f ⇤ L,P (x) 2 [−M, M] for all x 2 X implies f ⇤ L,P 2 L2 (X) and the latter together with X ⇢B`d 2 and (5) yields k ˜fkL2(Rd) = , γp⇡ -−d 2 kEf ⇤ L,PkL2(Rd)  , γp⇡ -−d 2 a0,2kf ⇤ L,PkL2(X)  ✓ 2 γp⇡ ◆d 2 a0,2M , (10) i.e. ˜f 2 L2 , Rd. Because of this and Lemma 2 f0 = K ⇤˜f 2 Hγ is satisﬁed and with Lemma 3 we have kL ◦f0k1 = sup (x,y)2X⇥Y \|L (y, f0 (x))\| = sup (x,y)2X⇥Y """L ⇣ y, K ⇤˜f (x) ⌘""" 4ra2M 2 =: B0 . Furthermore, (1) and Lemma 1 yield RL,P (f0) −R⇤ L,P = RL,P ⇣ K ⇤˜f ⌘ −R⇤ L,P = ###K ⇤˜f −f ⇤ L,P ### 2 L2(PX) Cr,2 !2 r,L2(Rd) ⇣ Ef ⇤ L,P, γ 2 ⌘ Cr,2 c2γ2↵, where we used the assumption !r,L2(Rd) ⇣ Ef ⇤ L,P, γ 2 ⌘ cγ↵ for γ 2 (0, 1], ↵≥1, r = b↵c + 1 and a constant c > 0 in the last step. By Lemma 2 and (10) we know kf0kHγ = kK ⇤˜fkHγ (2r −1) k ˜fkL2(Rd) (2r −1) ✓ 2 γp⇡ ◆d 2 a0,2M . Therefore, Theorem 3 and the above choice of f0 yield, for all ﬁxed ⌧≥1, λ > 0, " > 0 and p 2 (0, 1), that the SVM using Hγ and L satisﬁes λ kfD,λk2 Hγ + RL,P ⇣ ÛfD,λ ⌘ −R⇤ L,P 9 λ (2r −1)2 ✓ 2 γp⇡ ◆d a2 0,2M 2 + Cr,2c2γ2↵ ! + C",p γ−(1−p)(1+")d λpn + , 3456 + 15 · 4ra2M 2(ln(3) + 1)⌧ n C1λγ−d + 9 Crc2γ2↵+ C",p γ−(1−p)(1+")d λpn + C2⌧ n with probability Pn not less than 1 −e−⌧and with constants C1 := 9 (2r −1)2 2d⇡−d 2 a2 0,2M 2, C2 := (ln(3) + 1) , 3456 + 15 · 4ra2M 2, a := max {a0,1, 1}, Cr := Cr,2 only depending on r and µ(X) and C",p as in Theorem 3. 8 References [1] L. Gy¨orﬁ, M. Kohler, A. Krzy˙zak, and H. Walk. A Distribution-Free Theory of Nonparametric Regression. Springer-Verlag New York, 2002. [2] I. Steinwart and A. Christmann. Support Vector Machines. Springer-Verlag, New York, 2008. [3] R.A. DeVore and G.G. Lorentz. Constructive Approximation. Springer-Verlag Berlin Heidelberg, 1993. [4] R.A. DeVore and V.A. Popov. Interpolation of Besov Spaces. AMS, Volume 305, 1988. 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4,451	Topology Constraints in Graphical Models Marcelo Fiori Universidad de la Rep´ublica, Uruguay mfiori@fing.edu.uy Pablo Mus´e Universidad de la Rep´ublica, Uruguay pmuse@fing.edu.uy Guillermo Sapiro Duke University Durham, NC 27708 guillermo.sapiro@duke.edu Abstract Graphical models are a very useful tool to describe and understand natural phenomena, from gene expression to climate change and social interactions. The topological structure of these graphs/networks is a fundamental part of the analysis, and in many cases the main goal of the study. However, little work has been done on incorporating prior topological knowledge onto the estimation of the underlying graphical models from sample data. In this work we propose extensions to the basic joint regression model for network estimation, which explicitly incorporate graph-topological constraints into the corresponding optimization approach. The ﬁrst proposed extension includes an eigenvector centrality constraint, thereby promoting this important prior topological property. The second developed extension promotes the formation of certain motifs, triangle-shaped ones in particular, which are known to exist for example in genetic regulatory networks. The presentation of the underlying formulations, which serve as examples of the introduction of topological constraints in network estimation, is complemented with examples in diverse datasets demonstrating the importance of incorporating such critical prior knowledge. 1 Introduction The estimation of the inverse of the covariance matrix (also referred to as precision matrix or concentration matrix) is a very important problem with applications in a number of ﬁelds, from biology to social sciences, and is a fundamental step in the estimation of underlying data networks. The covariance selection problem, as introduced by Dempster (1972), consists in identifying the zero pattern of the precision matrix. Let X = (X1 . . . Xp) be a p-dimensional multivariate normal distributed variable, X ∼N(0, Σ), and C = Σ−1 its concentration matrix. Then two coordinates Xi and Xj are conditionally independent given the other variables if and only if C(i, j) = 0 (Lauritzen, 1996). This property motivates the representation of the conditional dependency structure in terms of a graphical model G = (V, E), where the set of nodes V corresponds to the p coordinates and the edges E represent conditional dependency. Note that the zero pattern of the G adjacency matrix coincides with the zero pattern of the concentration matrix. Therefore, the estimation of this graph G from k random samples of X is equivalent to the covariance selection problem. The estimation of G using ℓ1 (sparsity promoting) optimization techniques has become very popular in recent years. This estimation problem becomes particularly interesting and hard at the same time when the number of samples k is smaller than p. Several real life applications lie in this “small k-large p” setting. One of the most studied examples, and indeed with great impact, is the inference of genetic regulatory networks (GRN) from DNA microarray data, where typically the number p of genes is much larger than the number k of experiments. Like in the vast majority of applications, these networks have some very well known topological properties, such as sparsity (each node is connected with only a few other nodes), scale-free behavior, and the presence of hubs (nodes connected with many other vertices). All these properties are shared with many other real life networks like Internet, citation networks, and social networks (Newman, 2010). 1 Genetic regulatory networks also contain a small set of recurring patterns called motifs. The systematic presence of these motifs has been ﬁrst discovered in Escherichia coli (Shen-Orr et al., 2002), where it was found that the frequency of these patterns is much higher than in random networks, and since then they have been identiﬁed in other organisms, from bacteria to yeast, plants and animals. The topological analysis of networks is fundamental, and often the essence of the study. For example, the proper identiﬁcation of hubs or motifs in GRN is crucial. Thus, the agreement of the reconstructed topology with the original or expected one is critical. Sparsity has been successfully exploited via ℓ1 penalization in order to obtain consistent estimators of the precision matrix, but little work has been done with other graph-topological properties, often resulting in the estimation of networks that lack critical known topological structures, and therefore do not look natural. Incorporating such topological knowledge in network estimation is the main goal of this work. Eigenvector centrality (see Section 3 for the precise deﬁnition) is a well-known measure of the importance and the connectivity of each node, and typical centrality distributions are known (or can be estimated) for several types of networks. Therefore, we ﬁrst propose to incorporate this structural information into the optimization procedure for network estimation in order to control the topology of the resulting network. This centrality constraint is useful when some prior information about the graphical model is known, for example, in dynamic networks, where the topology information of the past can be used; in networks which we know are similar to other previously studied graphs; or in networks that model a physical phenomenon for which a certain structure is expected. As mentioned, it has been observed that genetic regulatory networks are conformed by a few geometric patterns, repeated several times. One of these motifs is the so-called feedforward loop, which is manifested as a triangle in the graph. Although it is thought that these important motifs may help to understand more complex organisms, no effort has been made to include this prior information in the network estimation problem. As a second example of the introduction of topological constraints, we propose a simple modiﬁcation to the ℓ1 penalty, weighting the edges according to their local structure, in order to favor the appearance of these motifs in the estimated network. Both developed extensions here presented are very ﬂexible, and they can be combined with each other or with other extensions reported in literature. To recapitulate, we propose several contributions to the network estimation problem: we show the importance of adding topological constraints; we propose an extension to ℓ1 models in order to impose the eigenvector centrality; we show how to transfer topology from one graph to another; we show that even with the centrality estimated from the same data, the proposed extension outperforms the basic model; we present a weighting modiﬁcation to the ℓ1 penalty favoring the appearance of motifs; as illustrative examples, we show how the proposed framework improves the edge and motif detection in the E. coli network, and how the approach is important as well in ﬁnancial applications. The rest of this paper is organized as follows. In Section 2 we describe the basic precision matrix estimation models used in this work. In Section 3 we introduce the eigenvector centrality and describe how to impose it in graph estimation. We propose the weighting method for motifs estimation in Section 4. Experimental results are presented in Section 5, and we conclude in Section 6. 2 Graphical Model Estimation Let X be a k × p matrix containing k independent observations of X, and let us denote by Xi the i-th column of X. Two main families of approaches use sparsity constraints when inferring the structure of the precision matrix. The ﬁrst one is based on the fact that the (i, j) element of Σ−1 is, up to a constant, the regression coefﬁcient βi j in Xi = P l̸=i βi lXl + εi, where εi is uncorrelated with {Xl\|l ̸= i}. Following this property, the neighborhood selection technique by Meinshausen & B¨uhlmann (2006) consists in solving p independent ℓ1 regularized problems (Tibshirani, 1996), arg min βi:βi i=0 1 k \|\|Xi −Xβi\|\|2 + λ\|\|βi\|\|1 , where βi is the vector of βi js. While this is an asymptotically consistent estimator of the Σ−1 zero pattern, βi j and βj i are not necessarily equal since they are estimated independently. Peng et al. (2009) propose a joint regression model which guarantees symmetry. This regression of the form X ≈XB, with B sparse, symmetric, and with null diagonal, allows to control the topology of the graph deﬁned by the non-zero pattern of B, as it will be later exploited in this work. Friedman 2 et al. (2010) also solve a symmetric version of the model by Meinshausen & B¨uhlmann (2006) and incorporate some structure penalties as the grouped lasso by Yuan & Lin (2006). Methods of the second family are based on a maximum likelihood (ML) estimator with an ℓ1 penalty (Yuan & Lin, 2007; Banerjee et al., 2008; Friedman et al., 2008). Speciﬁcally, if S denotes the empirical covariance matrix, the solution is the matrix Θ which solves the optimization problem max Θ≻0 log det Θ −tr(SΘ) −λ X i,j \|Θij\| . An example of an extension to both models (the regression and ML approaches), and the ﬁrst to explicitly consider additional classical network properties, is the work by Liu & Ihler (2011), which modiﬁes the ℓ1 penalty to derive a non-convex optimization problem that favors scale-free networks. A completely different technique for network estimation is the use of the PC-Algorithm to infer acyclic graphs (Kalisch & B¨uhlmann, 2007). This method starts from a complete graph and recursively deletes edges according to conditional independence decisions. In this work, we use this technique to estimate the graph eigenvector centrality. 3 Eigenvector Centrality Model Extension Node degree (the number of connections of a node) is the simplest algebraic property than can be deﬁned over a graph, but it is very local as it only takes into account the neighborhood of the node. A more global measure of the node importance is the so-called centrality, in any of its different variants. In this work, we consider the eigenvector centrality, deﬁned as the dominant eigenvector (the one corresponding to the largest eigenvalue) of the corresponding network connectivity matrix. The coordinates of this vector (which are all non-negatives) are the corresponding centrality of each node, and provide a measure of the inﬂuence of the node in the network (Google’s PageRank is a variant of this centrality measure). Distributions of the eigenvector centrality values are well known for a number of graphs, including scale-free networks as the Internet and GRN (Newman, 2010). In certain situations, we may have at our disposal an estimate of the centrality vector of the network to infer. This may happen, for instance, because we already had preliminary data, or we know a network expected to be similar, or simply someone provided us with some partial information about the graph structure. In those cases, we would like to make use of this important side information, both to improve the overall network estimation and to guarantee that the inferred graph is consistent with our prior topological knowledge. In what follows we propose an extension of the joint regression model which is capable of controlling this topological property of the estimated graph. To begin with, let us remark that as Σ is positive-semideﬁnite and symmetric, all its eigenvalues are non-negative, and thus so are the eigenvalues of Σ−1. By virtue of the Perron-Frobenius Theorem, for any adjacency matrix A, the eigenvalue with largest absolute value is positive. Therefore for precision and graph connectivity matrices it holds that max\|\|v\|\|=1 \|⟨Av, v⟩\| = max\|\|v\|\|=1⟨Av, v⟩, and moreover, the eigenvector centrality is c = arg max\|\|v\|\|=1⟨Av, v⟩. Suppose that we know an estimate of the centrality c ∈Rp, and want the inferred network to have centrality close to it. We start from the basic joint regression model, min B \|\|X −XB\|\|2 F + λ1\|\|B\|\|ℓ1 , s.t. B symmetric, Bii = 0 ∀i, (1) and add the centrality penalty, min B \|\|X −XB\|\|2 F + λ1\|\|B\|\|ℓ1 −λ2⟨Bc, c⟩, s.t. B symmetric, Bii = 0 ∀i (2) where \|\| · \|\|F is the Frobenius norm and \|\|B\|\|ℓ1 = P i,j \|Bij\|. The minus sign is due to the minimization instead of maximization, and since the term ⟨Bc, c⟩is linear, the problem is still convex. Although B is intended to be a good estimation of the precision matrix (up to constants), formulations (1) or (2) do not guarantee that B will be positive-semideﬁnite, and therefore the leading eigenvalue might not be positive. One way to address this is to add the positive-semideﬁnite constraint in the formulation, which keeps the problem convex. However, in all of our experiments with model (2) the spectral radius resulted positive, so we decided to use this simpler formulation due to the power of the available solvers. Note that we are imposing the dominant eigenvector of the graph connectivity matrix A to a nonbinary matrix B. We have exhaustive empirical evidence that the leading eigenvector of the matrix 3 B obtained by solving (2), and the leading eigenvector corresponding to the resulting connectivity matrix (the binarization of B) are very similar (see Section 5.1). In addition, based on Wolf & Shashua (2005), these type of results can be proved theoretically (Zeitouni, 2012). As shown in Section 5, when the correct centrality is imposed, our proposed model outperforms the joint regression model, both in correct reconstructed edge rates and topology. This is still true when we only have a noisy version of c. Even if we do not have prior information at all, and we estimate the centrality from the data with a pre-run of the PC-Algorithm, we obtain improved results. The model extension here presented is general, and the term ⟨Bc, c⟩can be included in maximum likelihood based approaches like Banerjee et al. (2008); Friedman et al. (2008); Yuan & Lin (2007). 3.1 Implementation Following Peng et al. (2009), the matrix optimization (2) can be cast as a classical vector ℓ1 penalty problem. The symmetry and null diagonal constraints are handled considering only the upper triangular sub-matrix of B (excluding the diagonal), and forming a vector θ with its entries: θ = (B12, B13, . . . , B(p−1)p). Let us consider a pk×1 column vector y formed by concatenating all the columns of X. It is easy to ﬁnd a pk×p(p−1)/2 matrix Xt such that \|\|X−XB\|\|2 F = \|\|y−Xtθ\|\|2 2 (see Peng et al. (2009) for details), and trivially \|\|B\|\|ℓ1 = 2\|\|θ\|\|1. The new term in the cost function is ⟨Bc, c⟩, which is linear in B, thus it exists a matrix Ct = Ct(c) such that ⟨Bc, c⟩= ⟨Ct, θ⟩. The construction of Ct is similar to the construction of Xt. The optimization problem (2) then becomes min θ \|\|y −Xtθ\|\|2 2 + λ1\|\|θ\|\|1 −λ2⟨Ct, θ⟩, which can be efﬁciently solved using any modern ℓ1 optimization method (Wright et al., 2009). 4 Favoring Motifs in Graphical Models One of the biggest challenges in bioinformatics is the estimation and understanding of genetic regulatory networks. It has been observed that the structure of these graphs is far from being random: the transcription networks seem to be conformed by a small set of regulation patterns that appear much more often than in random graphs. It is believed that each one of these patterns, called motifs, are responsible of certain speciﬁc regulatory functions. Three basic types of motifs are deﬁned (ShenOrr et al., 2002), the “feedforward loop” being one of the most signiﬁcant. This motif involves three genes: a regulator X which regulates Y, and a gene Z which is regulated by both X and Y. The representation of these regulations in the network takes the form of a triangle with vertices X, Y, Z. Although these triangles are very frequent in GRN, the common algorithms discussed in Section 2 seem to fail at producing them. As these models do not consider any topological structure, and the total number of reconstructed triangles is usually much lower than in transcription networks, it seems reasonable to help in the formation of these motifs by favoring the presence of triangles. In order to move towards a better motif detection, we propose an iterative procedure based on the joint regression model (1). After a ﬁrst iteration of solving (1), a preliminary symmetric matrix B is obtained. Recall that if A is a graph adjacency matrix, then A2 counts the paths of length 2 between nodes. More speciﬁcally, the entry (i, j) of A2 indicates how many paths of length 2 exist from node i to node j. Back to the graphical model estimation, this means that if the entry (B2)ij ̸= 0 (a length 2 path exists between i and j), then by making Bij ̸= 0 (if it is not already), at least one triangle is added. This suggests that by including weights in the ℓ1 penalization, proportionally decreasing with B2, we are favoring those edges that, when added, form a new triangle. Given the matrix B obtained in the preliminary iteration, we consider the cost matrix M such that Mij = e−µ(B2)ij, µ being a positive parameter. This way, if (B2)ij = 0 the weight does not affect the penalty, and if (B2)ij ̸= 0, it favors motifs detection. We then solve the optimization problem min B \|\|X −XB\|\|2 F + λ1\|\|M · B\|\|ℓ1 , (3) where M · B is the pointwise matrix product. The algorithm iterates between reconstructing the matrix B and updating the weight matrix M (initialized as the identity matrix). Usually after two or three iterations the graph stabilizes. 4 5 Experimental Results In this section we present numerical and graphical results for the proposed models, and compare them with the original joint regression one. As discussed in the introduction, there is evidence that most real life networks present scale-free behavior. Therefore, when considering simulated results for validation, we use the model by Barab´asi & Albert (1999) to generate graphs with this property. Namely, we start from a random graph with 4 nodes and add one node at a time, randomly connected to one of the existing nodes. The probability of connecting the new node to the node i is proportional to the current degree of node i. Given a graph with adjacency matrix A, we simulate the data X as follows (Liu & Ihler, 2011): let D be a diagonal matrix containing the degree of node i in the entry Dii, and consider the matrix L = ηD −A with η > 1 so that L is positive deﬁnite. We then deﬁne the concentration matrix Θ = Λ 1 2 LΛ 1 2 , where Λ is the diagonal matrix of L−1 (used to normalize the diagonal of Σ = Θ−1). Gaussian data X is then simulated with distribution N(0, Σ). For each algorithm, the parameters are set such that the resulting graph has the same number of edges as the original one. As the total number of edges is then ﬁxed, the false positive (FP) rate can be deduced from the true positive (TP) rate. We therefore report the TP rate only, since it is enough to compare the different performances. 5.1 Including Actual Centrality In this ﬁrst experiment we show how our model (2) is able to correctly incorporate the prior centrality information, resulting in a more accurate inferred graph, both in detected edges and in topology. The graph of the example in Figure 1 contains 20 nodes. We generated 10 samples and inferred the graph with the joint regression model and with the proposed model (2) using the correct centrality. Figure 1: Comparison of networks estimated with the simple joint model (1) (middle) and with model (2) (right) using the eigenvector centrality. Original graph on left. The following more comprehensive test shows the improvement with respect to the basic joint model (1) when the correct centrality is included. For a ﬁxed value of p = 80, and for each value of k from 30 to 50, we made 50 runs generating scale-free graphs and simulating data X. From these data we estimated the network with the joint regression model with and without the centrality prior. The TP edge rates in Figure 2(a) are averaged over the 50 runs, and count the correctly detected edges over the (ﬁxed) total number of edges in the network. In addition, Figure 2(b) shows a ROC curve. We generated 300 networks and constructed a ROC curve for each one by varying λ1, and we then averaged all the 300 curves. As expected, the incorporation of the known topological property helps in the correct estimation of the graph. 30 40 50 60 70 0.6 0.7 0.8 0.9 k TP edge rate (a) True positive rates for different sample sizes on networks with 80 nodes. 0 0.005 0.01 0.015 0.02 0.025 0.4 0.5 0.6 0.7 0.8 0.9 1 False Positive Rate True Positive Rate (b) Edge detection ROC curve for networks with p = 80 nodes and k = 50. Figure 2: Performance comparison of models 2 and 1. In blue (dashed), the standard joint model (1), and in black the proposed model with centrality (2). In thin lines, curves corresponding to 95% conﬁdence intervals. 5 Following the previous discussion, Figure 3 shows the inner product ⟨vB, vC⟩for several runs of model (2), where vB is the leading eigenvector of the obtained matrix B, C is the resulting connectivity matrix (the binarized version of B), and vC its leading eigenvector. 0 40 80 120 160 200 0 0.2 0.4 0.6 0.8 1 Run number Inner product Figure 3: Inner product ⟨vC, vB⟩for 200 runs. 20 30 40 50 60 70 0.4 0.5 0.6 0.7 0.8 k TP edge rate Figure 4: True positive edge rates for different sample sizes on a network with 100 nodes. Dashed, the joint model (1), dotted, the PC-Algorithm, and solid the model (2) with centrality estimated from data. 5.2 Imposing Centrality Estimated from Data The previous section shows how the performance of the joint regression model (1) can be improved by incorporating the centrality, when this topology information is available. However, when this vector is unknown, it can be estimated from the data, using an independent algorithm, and then incorporated to the optimization in model (2). We use the PC-Algorithm to estimate the centrality (by computing the dominant eigenvector of the resulting graph), and then we impose it as the vector c in model (2). It turns out that even with a technique not specialized for centrality estimation, this combination outperforms both the joint model (1) and the PC-Algorithm. We compare the three mentioned models on networks with p = 100 nodes for several values of k, ranging from 20 to 70. For each value of k, we randomly generated ten networks and simulated data X. We then reconstructed the graph using the three techniques and averaged the edge rate over the ten runs. The parameter λ2 was obtained via cross validation. Figure 4 shows how the model imposing centrality can improve the other ones without any external information. 5.3 Transferring Centrality In several situations, one may have some information about the topology of the graph to infer, mainly based on other data/graphs known to be similar. For instance, dynamic networks are a good example where one may have some (maybe abundant) old data from the network at a past time T1, some (maybe scarce) new data at time T2, and know that the network topology is similar at the different times. This may be the case of ﬁnancial, climate, or any time-series data. Outside of temporal varying networks, this topological transference may be useful when we have two graphs of the same kind (say biological networks), which are expected to share some properties, and lots of data is available for the ﬁrst network but very few samples for the second network are known. We would like to transfer our inferred centrality-based topological knowledge from the ﬁrst network into the second one, and by that improving the network estimation from limited data. For these examples, we have an unknown graph G1 corresponding to a k1×p data matrix X1, which we assume is enough to reasonably estimate G1, and an unknown graph G2 with a k2×p data matrix X2 (with k2 ≪k1). Using X2 only might not be enough to obtain a proper estimate of G2, and considering the whole data together (concatenation of X1 and X2) might be an artiﬁcial mixture or too strong and lead to basically reconstructing G1. What we really want to do is to transfer some high-level structure of G1 into G2, e.g., just the underlying centrality of G1 is transferred to G2. In what follows, we show the comparison of inferring the network G2 using only the data X2 in the joint model (1); the concatenation of X1 and X2 in the joint model (1); and ﬁnally the centrality estimated from X1, imposed in model (2), along with data X2. We ﬁxed the networks size to p = 100 and the size of data for G1 to k1 = 200. Given a graph G1, we construct G2 by randomly changing a certain number of edges (32 and 36 edges in Figure 5). For k2 from 35 to 60, we generate data X2, and we then infer G2 with the methods described above. We averaged over 10 runs. As it can be observed in Figure 5, the performance of the model including the centrality estimated from X1 is better than the performance of the classical model, both when using just the data X2 and the concatenated data X1\|X2. Therefore, we can discard the old data X1 and keep only the structure (centrality) and still be able to infer a more accurate version of G2. 6 35 40 45 50 55 60 0.55 0.65 0.75 k2 TP edge rate (a) G1/G2 differ in 32 edges. 35 40 45 50 55 60 0.55 0.65 0.75 k2 TP edge rate (b) G1/G2 differ in 36 edges. Figure 5: True positive edge rate when estimating the network G2 vs amount of data. In blue, the basic joint model using only X2, in red using the concatenation of X1 and X2, and in black the model (2) using only X2 with centrality estimated from X1 as prior. 5.4 Experiments on Real Data 5.4.1 International Stock Market Data The stock market is a very complicated system, with lots of time-dependent underlying relationships. In this example we show how the centrality constraint can help to understand these relationships with limited data on times of crisis and times of stability. We use the daily closing values (πk) of some relevant stock market indices from U.S., Canada, Australia, Japan, Hong Kong, U.K., Germany, France, Italy, Switzerland, Netherlands, Austria, Spain, Belgium, Finland, Portugal, Ireland, and Greece. We consider 2 time periods containing a crisis, 5/2007-5/2009 and 5/2009-5/2012, each of which was divided into a “pre-crisis” period, and two more sets (training and testing) covering the actual crisis period. We also consider the relatively stable period 6/1997-6/1999, where the division into these three subsets was made arbitrarily. Using as data the return between two consecutive trading days, deﬁned as 100 log( πk πk−1 ), we ﬁrst learned the centrality from the “pre-crisis” period, and we then learned three models with the training sets: a classical least-squares regression (LS), the joint regression model (1), and the centrality model (2) with the estimated eigenvector. For each learned model B we computed the “prediction” accuracy \|\|Xtest −XtestB\|\|2 F in order to evaluate whether the inclusion of the topology improves the estimation. The results are presented in Table 1, illustrating how the topology helps to infer a better model, both in stable and highly changing periods. Additionally, Figure 6 shows a graph learned with the model (2) using the 2009-2012 training data. The discovered relationships make sense, and we can easily identify geographic or socio-economic connections. US CA AU JP HK UK GE FR IT SW NE AT SP BE FN PO IR GR Figure 6: Countries network learned with the centrality model. 97-99 07-09 09-12 LS 2.7 3.5 14.4 Model (1) 2.5 0.9 4.0 Model (2) 1.9 0.6 2.4 Table 1: Mean square error (×10−3) for the different models. 5.4.2 Motif Detection in Escherichia Coli Along this section and the following one, we use as base graph the actual genetic regulation network of the E. coli. This graph contains ≈400 nodes, but for practical issues we selected the sub-graph of all nodes with degree > 1. This sub-graph GE contains 186 nodes and 40 feedforward loop motifs. For the number of samples k varying from 30 to 120, we simulated data X from GE and reconstructed the graph using the joint model (1) and the iterative method (3). We then compared the resulting networks to the original one, both in true positive edge rate (recall that this analysis is sufﬁcient since the total number of edges is made constant), and number of motifs correctly detected. The numerical results are shown in Figure 7, where it can be seen that model (3) correctly detect more motifs, with better TP vs FP motif rate, and without detriment of the true positive edge rate. 5.4.3 Centrality + Motif Detection The simplicity of the proposed models allows to combine them with other existing network estimation extensions. We now show the performance of the two models here presented combined (centrality and motifs constraints), tested on the Escherichia coli network. 7 40 60 80 100 120 0.25 0.35 0.45 0.55 k TP edge rate 40 60 80 100 120 0.1 0.2 0.3 k TP motif rate 40 60 80 100 120 0.06 0.14 0.22 k Pos. pred. value Figure 7: Comparison of model (1) (dashed) with proposed model (3) (solid) for the E. coli network. Left: TP edge rate. Middle: TP motif rate (motifs correctly detected over the total number of motifs in GE). Right: Positive predictive value (motifs correctly detected over the total number of motifs in the inferred graph). We ﬁrst estimate the centrality from the data, as in Section 5.2. Let us assume that we know which ones are the two most central nodes (genes).1 This information can be used to modify the centrality value for these two nodes, by replacing them by the two highest centrality values typical of scalefree networks (Newman, 2010). For the ﬁxed network GE, we simulated data of different sizes k and reconstructed the graph with the model (1) and with the combination of models (2) and (3). Again, we compared the TP edge rates, the percentage of motifs detected, and the TP/FP motifs rate. Numerical results are shown in Figure 8, where it can be seen that, in addition to the motif detection improvement, now the edge rate is also better. Figure 9 shows the obtained graphs for a speciﬁc run. 70 80 90 100 110 0.4 0.46 0.52 0.58 k TP edge rate 70 80 90 100 110 0.14 0.22 0.3 k TP motif rate 70 80 90 100 110 0.12 0.16 0.2 k Pos. pred. value Figure 8: Comparison of model (1) (dashed) with the combination of models (2) and (3) (solid) for the E. coli network. The combination of the proposed extensions is capable of detecting more motifs while also improving the accuracy of the detected edges. Left: TP edge rate. Middle: TP motif rate. Right: Positive predictive value. Figure 9: Comparison of graphs for the E. coli network with k = 80. Original network, inferred with model (1) and with the combination of (2) and (3). Note how the combined model is able to better capture the underlying network topology, as quantitative shown in Figure 8. Correctly detected motifs are highlighted. 6 Conclusions and Future Work We proposed two extensions to ℓ1 penalized models for precision matrix (network) estimation. The ﬁrst one incorporates topological information to the optimization, allowing to control the graph centrality. We showed how this model is able to capture the imposed structure when the centrality is provided as prior information, and we also showed how it can improve the performance of the basic joint regression model even when there is no such external information. The second extension favors the appearance of triangles, allowing to better detect motifs in genetic regulatory networks. We combined both models for a better estimation of the Escherichia coli GRN. There are several other graph-topological properties that may provide important information, making it interesting to study which kind of structure can be added to the optimization problem. An algorithm for estimating with high precision the centrality directly from the data would be a great complement to the methods here presented. It is also important to ﬁnd a model which exploits all the prior information about GRN, including other motifs not explored in this work. Finally, the exploitation of the methods here developed for ℓ1-graphs, is the the subject of future research. 1In this case, it is well known that crp is the most central node, followed by fnr. 8 Acknowledgements Work partially supported by ANII (Uruguay), ONR, NSF, NGA, DARPA, and AFOSR. References Banerjee, O., El Ghaoui, L., and D’Aspremont, A. 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4,452	Bayesian nonparametric models for bipartite graphs Franc¸ois Caron INRIA IMB - University of Bordeaux Talence, France Francois.Caron@inria.fr Abstract We develop a novel Bayesian nonparametric model for random bipartite graphs. The model is based on the theory of completely random measures and is able to handle a potentially inﬁnite number of nodes. We show that the model has appealing properties and in particular it may exhibit a power-law behavior. We derive a posterior characterization, a generative process for network growth, and a simple Gibbs sampler for posterior simulation. Our model is shown to be well ﬁtted to several real-world social networks. 1 Introduction The last few years have seen a tremendous interest in the study, understanding and statistical modeling of complex networks [14, 6]. A network is a set if items, called vertices, with connections between them, called edges. In this article, we shall focus on bipartite networks, also known as twomode, afﬁliation or collaboration networks [16, 17]. In bipartite networks, items are divided into two different types A and B, and only connections between items of different types are allowed. Examples of this kind can be found in movie actors co-starring the same movie, scientists co-authoring a scientiﬁc paper, internet users posting a message on the same forum, people reading the same book or listening to the same song, members of the boards of company directors sitting on the same board, etc. Following the readers-books example, we will refer to items of type A as readers and items of type B as books. An example of bipartite graph is shown on Figure 1(b). An important summarizing quantity of a bipartite graph is the degree distribution of readers (resp. books) [14]. The degree of a vertex in a network is the number of edges connected to that vertex. Degree distributions of real-world networks are often strongly non-Poissonian and exhibit a power-law behavior [15]. A bipartite graph can be represented by a set of binary variables (zij) where zij = 1 if reader i has read book j, 0 otherwise. In many situations, the number of available books may be very large and potentially unknown. In this case, a Bayesian nonparametric (BNP) approach can be sensible, by assuming that the pool of books is inﬁnite. To formalize this framework, it will then be convenient to represent the bipartite graph by a collection of atomic measures Zi, i = 1, . . . , n with Zi = ∞ X j=1 zijδθj (1) where {θj} is the set of books and typically Zi only has a ﬁnite set of non-zero zij corresponding to books reader i has read. Grifﬁths and Ghahramani [8, 9] have proposed a BNP model for such binary random measures. The so-called Indian Buffet Process (IBP) is a simple generative process for the conditional distribution of Zi given Z1, . . . , Zi−1. Such process can be constructed by considering that the binary measures Zi are i.i.d. from some random measure drawn from a beta process [19, 10]. It has found several applications for inferring hidden causes [20], choices [7] or features [5]. Teh and Gor¨ur [18] proposed a three-parameter extension of the IBP, named stable IBP, that enables to 1 model a power-law behavior for the degree distribution of books. Although more ﬂexible, the stable IBP still induces a Poissonian distribution for the degree of readers. In this paper, we propose a novel Bayesian nonparametric model for bipartite graphs that addresses some of the limitations of the stable IBP, while retaining computational tractability. We assume that each book j is assigned a positive popularity parameter wj > 0. This parameter measures the popularity of the book, larger weights indicating larger probability to be read. Similarly, each reader i is assigned a positive parameter γi which represents its ability to read books. The higher γi, the more books the reader i is willing to read. Given the weights wj and γi, reader i reads book j with probability 1 −exp(−γiwj). We will consider that the weights wj and/or γi are the points of a Poisson process with a given L´evy measure. We show that depending on the choice of the L´evy measure, a power-law behavior can be obtained for the degree distribution of books and/or readers. Moreover, using a set of suitably chosen latent variables, we can derive a generative process for network growth, and an efﬁcient Gibbs sampler for approximate inference. We provide illustrations of the ﬁt of the proposed model on several real-world bipartite social networks. Finally, we discuss some potentially useful extensions of our work, in particular to latent factor models. 2 Statistical Model 2.1 Completely Random Measures We ﬁrst provide a brief overview of completely random measures (CRM) [12, 13] before describing the BNP model for bipartite graphs in Section 2.2. Let Θ be a measurable space. A CRM is a random measure G such that for any collection of disjoint measurable subsets A1, . . . , An of Θ, the random masses of the subsets G(A1), . . . , G(An) are independent. CRM can be decomposed into a sum of three independent parts: a non-random measure, a countable collection of atoms with ﬁxed locations, and a countable collection of atoms with randoms masses at random locations. In this paper, we will be concerned with models deﬁned by CRMs with random masses at random locations, i.e. G = P∞ j=1 wjδθj. The law of G can be characterized in terms of a Poisson process over the point set {(wj, θj), j = 1, . . . , ∞} ⊂R+ × Θ. The mean measure Λ of this Poisson process is known as the L´evy measure. We will assume in the following that the L´evy measure decomposes as a product of two non-atomic densities, i.e. that G is a homogeneous CRM Λ(dw, dθ) = λ(w)h(θ)dwdθ with h : Θ →[0, +∞) and R Θ h(θ)dθ = 1. It implies that the locations of the atoms in G are independent of the masses, and are i.i.d. from h, while the masses are distributed according to a Poisson process over R+ with mean intensity λ. We will further assume that the total mass G(Θ) = P∞ j=1 wj is positive and ﬁnite with probability one, which is guaranteed if the following conditions are satisﬁed Z ∞ 0 λ(w)dw = ∞ and Z ∞ 0 (1 −exp(−w))λ(w)dw < ∞ (2) and note g(x) its probability density function evaluated at x. We will refer to λ as the L´evy intensity in the following, and to h as the base density of G, and write G ∼CRM(λ, h). We will also note ψλ(t) = −log E [exp(−tG(Θ))] = Z ∞ 0 (1 −exp(−tw))λ(w)dw (3) eψλ(t, b) = Z ∞ 0 (1 −exp(−tw))λ(w) exp(−bw)dw (4) κ(n, z) = Z ∞ 0 λ(w)wne−zwdw (5) As a notable particular example of CRM, we can mention the generalized gamma process (GGP) [1], whose L´evy intensity is given by λ(w) = α Γ(1 −σ)w−σ−1e−wτ GGP encompasses the gamma process (σ = 0), the inverse Gaussian process (σ = 0.5) and the stable process (τ = 0) as special cases. Table ?? in supplementary material provides the expressions of λ, ψ and κ for these processes. 2 2.2 A Bayesian nonparametric model for bipartite graphs Let G ∼CRM(λ, h) where λ satisﬁes conditions (2). A draw G takes the form G = ∞ X j=1 wjδθj (6) where {θj} is the set of books and {wj} the set of popularity parameters of books. For i = 1, . . . , n, let consider the latent exponential process Vi = ∞ X j=1 vijδθj (7) deﬁned for j = 1, . . . , ∞by vij\|wj ∼Exp(wjγi) where Exp(a) denotes the exponential distribution of rate a. The higher wj and/or γi, the lower vij. We then deﬁne the binary process Zi conditionally on Vi by Zi = ∞ X j=1 zijδθj with zij = 1 if vij < 1 zij = 0 otherwise (8) By integrating out the latent variables vij we clearly have p(zij = 1\|wj, γi) = 1 −exp(−γiwj). Proposition 1 Zi is marginally characterized by a Poisson process over the point set {(θ∗ j ), j = 1, . . . , ∞} ⊂Θ, of intensity measure ψλ(γi)h(θ∗). Hence, the total mass Zi(Θ) = P∞ j=1 zij, which corresponds to the total number of books read by reader i is ﬁnite with probability one and admits a Poisson(ψλ(γi)) distribution, where ψλ(z) is deﬁned in Equation (3), while the locations θ∗ j are i.i.d. from h. The proof, which makes use of Campbell’s theorem for point processes [13] is given in supplementary material. As an example, for the gamma process we have Zi(Θ) ∼Poisson α log 1 + γi τ . It will be useful in the following to introduce a censored version of the latent process Vi, deﬁned by Ui = ∞ X j=1 uijδθj (9) where uij = min(vij, 1), for i = 1, . . . , n and j = 1, . . . , ∞. Note that Zi can be obtained deterministically from Ui. 2.3 Characterization of the conditional distributions The conditional distribution of G given Z1, . . . , Zn cannot be obtained in closed form1. We will make use of the latent process Ui. In this section, we derive the formula for the conditional laws P(U1, . . . , Un\|G), P(U1, . . . , Un) and P(G\|U1, . . . , Un) . Based on these results, we derive in Section 2.4 a generative process and in Section 2.5 a Gibbs sampler for our model, that both rely on the introduction of these latent variables. Assume that K books {θ1, . . . , θK} have appeared. We write Ki = Zi(Θ) = P∞ j=1 zij the degree of reader i (number of books read by reader i) and mj = Pn i=1 Zi({θj}) = Pn i=1 zij the degree of book j (number of people having read book j). The conditional likelihood of U1, . . . Un given G is given by P(U1, . . . Un\|G) = n Y i=1      K Y j=1 γzij i wzij j exp (−γiwjuij)  exp (−γiG(Θ\{θ1, . . . , θK}))    = n Y i=1 γKi i !   K Y j=1 wmj j exp −wj n X i=1 γi(uij −1) ! exp − n X i=1 γi ! G(Θ) ! (10) 1In the case where γi = γ, it is possible to derive P(Z1, . . . , Zn) and P(Zn+1\|Z1, . . . , Zn) where the random measure G and the latent variables U are marginalized out. This particular case is described in supplementary material. 3 Proposition 2 The marginal distribution P(U1, . . . Un) is given by P(U1, . . . Un) = n Y i=1 γKi i ! exp " −ψλ n X i=1 γi !# K Y j=1 h(θj)κ mj, n X i=1 γiuij ! (11) where ψλ and κ are resp. deﬁned by Eq. (3) and (5). Proof. The proof, detailed in supplementary material, is obtained by an application of the Palm formula for CRMs [3, 11], and is the same as that of Theorem 1 in [2]. Proposition 3 The conditional distribution of G given the latent processes U1, . . . Un can be expressed as G = G∗+ K X j=1 wjδθj (12) where G∗and (wj) are mutually independent with G∗∼CRM(λ∗, h) λ∗(w) = λ(w) exp −w n X i=1 γi ! (13) and the masses are P(wj\|rest) = λ(wj)wmj j exp (−wj Pn i=1 γiUij) κ(mj, Pn i=1 γiuij) (14) Proof. The proof, based on the application of the Palm formula and detailed in supplementary material, is the same as that of Theorem 2 in [2]. In the case of the GGP, G∗is still a GGP of parameters (α∗= α, σ∗= σ, τ ∗= τ +Pn i=1 γi), while the wj’s are conditionally gamma distributed, i.e. wj\|rest ∼Gamma mj −σ, τ + n X i=1 γiuij ! Corollary 4 The predictive distribution of Zn+1 given the latent processes U1, . . . , Un is given by Zn+1 = Z∗ n+1 + K X j=1 zn+1,jδθj where the zn+1,j are independent of Z∗ n+1 with zn+1,j\|U ∼Ber 1 −κ(mj, τ + γn+1 + Pn i=1 γiuij) κ(mj, τ + Pn i=1 γiuij) where Ber is the Bernoulli distribution and Z∗ n+1 is a homogeneous Poisson process over Θ of intensity measure ψλ∗(γn+1) h(θ). For the GGP, we have Z∗ n+1(Θ) ∼    Poisson α σ h τ + Pn+1 i=1 γi σ −(τ + Pn i=1 γi)σi if σ ̸= 0 Poisson α log 1 + γn+1 τ+Pn i=1 γi if σ = 0 and zn+1,j\|U ∼Ber 1 − 1 + γn+1 τ + Pn i=1 γiuij −mj+σ! . Finally, we consider the distribution of un+1,j\|zn+1,j = 1, u1:n,j. This is given by p(un+1,j\|zn+1,j = 1, u1:n,j) ∝κ(mj + 1, un+1,jγn+1 + n X i=1 γiuij)1un+1,j∈[0,1] (15) In supplementary material, we show how to sample from this distribution by the inverse cdf method for the GGP. 4 Reader 1 Books 18 4 14 ... Reader 2 12 0 8 13 4 ... Reader 3 16 10 0 0 14 9 6 ... (a) A1 A2 A3 B3 B2 B1 B4 B5 B6 B7 (b) Figure 1: Illustration of the generative process described in Section 2.4. 2.4 A generative process In this section we describe the generative process for Zi given (U1, . . . , Ui−1), G being integrated out. This reinforcement process, where popular books will be more likely to be picked, is reminiscent of the generative process for the beta-Bernoulli process, popularized under the name of the Indian buffet process [8]. Let xij = −log(uij) ≥0 be latent positive scores. Consider a set of n readers who successively enter into a library with an inﬁnite number of books. Each reader i = 1, . . . n, has some interest in reading quantiﬁed by a positive parameter γi > 0. The ﬁrst reader picks a number K1 ∼Poisson(ψλ(γ1)) books. Then he assigns a positive score x1j = −log(u1j) > 0 to each of these books, where u1j is drawn from distribution (15). Now consider that reader i enters into the library, and knows about the books read by previous readers and their scores. Let K be the total number of books chosen by the previous i −1 readers, and mj the number of times each of the K books has been read. Then for each book j = 1, . . . , K, reader i will choose this book with probability 1 −κ(mj, τ + γi + Pi−1 k=1 γkukj) κ(mj, τ + Pi−1 k=1 γkukj) and then will choose an additional number of K+ i books where K+ i ∼Poisson eψλ γi, i−1 X k=1 γk !! Reader i will then assign a score xij = −log uij > 0 to each book j he has read, where uij is drawn from (15). Otherwise he will set the default score xij = 0. This generative process is illustrated in Figure 1 together with the underlying bipartite graph . In Figure 2 are represented draws from this generative process with a GGP with parameters γi = 2 for all i, τ = 1, and different values for α and σ. 2.5 Gibbs sampling From the results derived in Proposition 3, a Gibbs sampler can be easily derived to approximate the posterior distribution P(G, U\|Z). The sampler successively updates U given (w, G∗(Θ)) then (w, G∗(Θ)) given U. We present here the conditional distributions in the GGP case. For i = 1, . . . , n, j = 1, . . . , K, set uij = 1 if zij = 0, otherwise sample uij\|zij, wj, γi ∼rExp(γiwj, 1) where rExp(λ, a) is the right-truncated exponential distribution of pdf λ exp(−λx)/(1 − exp(−λa))1x∈[0,a] from which we can sample exactly. For j = 1, . . . , K, sample wj\|U, γi ∼Gamma mj −σ, τ + n X i=1 γiuij ! and the total mass G∗(Θ) follows a distribution g∗(w) ∝g(w) exp (−w Pn i=1 γi) where g(w) is the distribution of G(Θ). In the case of the GGP, g∗(w) is an exponentially tilted stable distribution for which exact samplers exist [4]. In the particular case of the gamma process, we have the simple update G∗(Θ) ∼Gamma (α, τ + Pn i=1 γi) . 5 Books Readers 20 40 60 80 5 10 15 20 25 30 (a) α = 1, σ = 0 Books Readers 20 40 60 80 5 10 15 20 25 30 (b) α = 5, σ = 0 Books Readers 20 40 60 80 5 10 15 20 25 30 (c) α = 10, σ = 0 Books Readers 20 40 60 80 5 10 15 20 25 30 (d) α = 2, σ = 0.1 Books Readers 20 40 60 80 5 10 15 20 25 30 (e) α = 2, σ = 0.5 Books Readers 20 40 60 80 5 10 15 20 25 30 (f) α = 2, σ = 0.9 Figure 2: Realisations from the generative process of Section 2.4 with a GGP of parameters γ = 2, τ = 1 and various values of α and σ. 3 Update of γi and other hyperparameters We may also consider the weight parameters γi to be unknown and estimate them from the graph. We can assign a gamma prior γi ∼Gamma(aγ, bγ) with parameters (aγ > 0, bγ > 0) and update it conditionally on other variables with γi\|G, U ∼Gamma  aγ + K X j=1 zij, bγ + K X j=1 wjuij + G∗(Θ)   In this case, the marginal distribution of Zi(Θ), hence the degree distribution of books, follows a continuous mixture of Poisson distributions, which offers more ﬂexibility in the modelling. We may also go a step further and consider that there is an inﬁnite number of readers with weights γi associated to a given CRM Γ ∼CRM(λγ, hγ) and a measurable space of readers eΘ. We then have Γ = P∞ i=1 γiδeθi. This provides a lot of ﬂexibility in the modelling of the distribution of the degree of readers, allowing in particular to obtain a power-law behavior, as shown in Section 5. We focus here on the case where Γ is drawn from a generalized gamma process of parameters (αγ, σγ, τγ) for simplicity. Conditionally on (w, G∗(Θ), U), we have Γ = Γ∗+ Pn i=1 γiδeθi where for i = 1, . . . , n, γi\|G, U ∼Gamma   K X j=1 zij −σγ, τ + K X j=1 wjuij + G∗(Θ)   and Γ∗∼CRM(λ∗ γ, hγ) with λ∗ γ(γ) = λγ(γ) exp −γ PK j=1 wj + G∗(Θ) . In this case, the update for (w, G∗) conditional on (U, γ, Γ(eΘ)) is now for j = 1, . . . , K wj\|U, Γ ∼Gamma mj −σ, τ + n X i=1 γiuij + Γ∗(eΘ) ! and G∗ ∼ CRM(λ∗, h) with λ∗(w) = λ(w) exp −w Pn i=1 γi + Γ∗(eΘ) . Note that there is now symmetry in the treatment of books/readers. For the scale parameter α of the GGP, we can assign a gamma prior α ∼Gamma(aα, bα) and update it with α\|γ ∼ Gamma aα + K, bα + ψλ Pn i=1 γi + Γ∗(eΘ) . Other parameters of the GGP can be updated using a Metropolis-Hastings step. 6 4 Discussion Power-law behavior. We now discuss some of the properties of the model, in the case of the GGP. The total number of books read by n readers is O(nσ). Moreover, for σ > 0, the degree distribution follows a power-law distribution: asymptotically, the proportion of books read by m readers is O(m−1−σ) (details in supplementary material). These results are similar to those of the stable IBP [18]. However, in our case, a similar behavior can be obtained for the degree distribution of readers when assigning a GGP to it, while it will always be Poisson for the stable IBP. Connection to IBP. The stable beta process [18] is a particular case of our construction, obtained by setting weights γi = γ and L´evy measure λ(w) = α Γ(1 + c) Γ(1 −σ)Γ(c + σ)γ(1 −e−γw)−σ−1e−γw(c+σ) (16) The proof is obtained by a change of variable from the L´evy measure of the stable beta process. Extensions to latent factor models. So far, we have assumed that the binary matrix Z was observed. The proposed model can also be used as a prior for latent factor models, similarly to the IBP. As an example of the potential usefulness of our model compared to IBP, consider the extraction of features from time series of different lengths. Longer time series are more likely to exhibit more features than shorter ones, and it is sensible in this case to assume different weights γi. In a more general setting, we may want γi to depend on a set of metadata associated to reader i. Inference for latent factor models is described in supplementary material. 5 Illustrations on real-world social networks We now consider estimating the parameters of our model and evaluating its predictive performance on six bipartite social networks of various sizes. We ﬁrst provide a short description of these networks. The dataset ‘Boards’ contains information about members of the boards of Norwegian companies sitting at the same board in August 20112. ‘Forum’ is a forum network about web users contributing to the same forums3. ‘Books’ concerns data collected from the Book-Crossing community about users providing ratings on books4 where we extracted the bipartite network from the ratings. ‘Citations’ is the co-authorship network based on preprints posted to Condensed Matter section of ArXiv between 1995 and 1999 [15]. ‘Movielens100k’ contains information about users rating particular movies5 from which we extracted the bipartite network. Finally, ‘IMDB’ contains information about actors co-starring a movie6. The sizes of the different networks are given in Table 1. Dataset n K Edges Board 355 5766 1746 Forum 899 552 7089 Books 5064 36275 49997 Citations 16726 22016 58595 Movielens100k 943 1682 100000 IMDB 28088 178074 341313 S-IBP SG IG GGP 9.82 8.3 -145.1 -68.6 (29.8) (30.8) (81.9) (31.9) -6.7e3 -6.7e3 -5.5e3 -5.6e3 83.1 214 4.6e4 4.4e4 -3.7e4 -3.7e4 -3.1e4 -3.4e4 -6.7e4 -6.7e4 -5.5e4 -5.5e4 -1.5e5 -1.5e5 -1.1e5 -1.1e5 Table 1: Size of the different datasets and test log-likelihood of four different models. We evaluate the ﬁt of four different models on these datasets. First, the stable IBP [18] with parameters (αIBP , τIBP , σIBP ) (S-IBP). Second, our model where the parameter γ is the same over different readers, and is assigned a ﬂat prior (SG). Third our model where each γi ∼Gamma(aγ, bγ) where (aγ, bγ) are unknown parameters with ﬂat improper prior (IG). Finally, our model with a GGP model for γi, with parameters (αγ, σγ, τγ) (GGP). We divide each dataset between a training 2Data can be downloaded from http://www.boardsandgender.com/data.php 3Data for the forum and citation datasets can be downloaded from http://toreopsahl.com/datasets/ 4http://www.informatik.uni-freiburg.de/ cziegler/BX/ 5The dataset can be downloaded from http://www.grouplens.org 6The dataset can be downloaded from http://www.cise.uﬂ.edu/research/sparse/matrices/Pajek/IMDB.html 7 10 0 10 2 10 0 10 1 10 2 10 3 Degree Model Data (a) S-IBP 10 0 10 2 10 0 10 1 10 2 10 3 Degree Model Data (b) GS 10 0 10 2 10 0 10 1 10 2 10 3 Degree Model Data (c) IG 10 0 10 2 10 0 10 1 10 2 10 3 10 4 Degree Model Data (d) GGP 10 0 10 0 10 1 10 2 10 3 10 4 10 5 Degree Model Data (e) S-IBP 10 0 10 0 10 1 10 2 10 3 10 4 10 5 Degree Model Data (f) GS 10 0 10 0 10 1 10 2 10 3 10 4 10 5 Degree Model Data (g) IG 10 0 10 0 10 1 10 2 10 3 10 4 10 5 Degree Model Data (h) GGP Figure 3: Degree distributions for movies (a-d) and actors (e-h) for the IMDB movie-actor dataset with four different models. Data are represented by red plus and samples from the model by blue crosses. 10 0 10 2 10 0 10 1 10 2 10 3 Degree Model Data (a) S-IBP 10 0 10 2 10 0 10 1 10 2 10 3 Degree Model Data (b) GS 10 0 10 2 10 0 10 1 10 2 10 3 Degree Model Data (c) IG 10 0 10 2 10 0 10 1 10 2 10 3 Degree Model Data (d) GGP 10 0 10 0 10 1 10 2 10 3 10 4 10 5 Degree Model Data (e) S-IBP 10 0 10 0 10 1 10 2 10 3 10 4 10 5 Degree Model Data (f) GS 10 0 10 0 10 1 10 2 10 3 10 4 10 5 Degree Model Data (g) IG 10 0 10 0 10 1 10 2 10 3 10 4 10 5 Degree Model Data (h) GGP Figure 4: Degree distributions for readers (a-d) and books (e-h) for the BX books dataset with four different models. Data are represented by red plus and samples from the model by blue crosses. set containing 3/4 of the readers and a test set with the remaining. For each model, we approximate the posterior mean of the unknown parameters (respectively (αIBP , τIBP , σIBP ), γ, (aγ, bγ) and (αγ, σγ, τγ) for S-IBP, SG, IG and GGP) given the training network with a Gibbs sampler with 10000 burn-in iterations then 10000 samples; then we evaluate the log-likelihood of the estimated model on the test data. For GGP, we use αtest γ = bαγ/3 to take into account the different sample sizes. For ‘Boards’, we do 10 replications with random permutations given the small sample size and report standard deviation together with mean value. Table 1 shows the results over the different networks for the different models. Typically, S-IBP and SG give very similar results. This is not surprising, as they share the same properties, i.e. Poissonian degree distribution for readers and power-law degree distribution for books. Both methods perform better solely on the Board dataset, where the Poisson assumption on the number of people sitting on the same board makes sense. On all the other datasets, IG and GGP perform better and similarly, with slightly better performances for IG. These two models are better able to capture the power-law distribution of the degrees of readers. These properties are shown on Figures 3 and 4 which resp. give the empirical degree distributions of the test network and a draw from the estimated models, for the IMDB dataset and the Books dataset. It is clearly seen that the four models are able to capture the power-law behavior of the degree distribution of actors (Figure 3(e-h)) or books (Figure 4(e-h)). However, only IG and GGP are able to capture the power-law behavior of the degree distribution of movies (Figure 3(a-d)) or readers (Figure 4(a-d)). 8 References [1] A. Brix. Generalized gamma measures and shot-noise Cox processes. 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4,453	Meta-Gaussian Information Bottleneck M´elanie Rey Department of Mathematics and Computer Science University of Basel melanie.rey@unibas.ch Volker Roth Department of Mathematics and Computer Science University of Basel volker.roth@unibas.ch Abstract We present a reformulation of the information bottleneck (IB) problem in terms of copula, using the equivalence between mutual information and negative copula entropy. Focusing on the Gaussian copula we extend the analytical IB solution available for the multivariate Gaussian case to distributions with a Gaussian dependence structure but arbitrary marginal densities, also called meta-Gaussian distributions. This opens new possibles applications of IB to continuous data and provides a solution more robust to outliers. 1 Introduction The information bottleneck method (IB) [1] considers the concept of relevant information in the data compression problem, and takes a new perspective to signal compression which was classically treated using rate distortion theory. The IB method formalizes the idea of relevance, or meaningful information, by introducing a relevance variable Y . The problem is then to obtain an optimal compression T of the data X which preserves a maximum of information about Y . Although the IB method beautifully formalizes the compression problem under relevance constraints, the practical solution of this problem remains difﬁcult, particularly in high dimensions, since the mutual informations I(X; T), I(Y ; T) must be estimated. The IB optimization problem has no available analytical solution in the general case. It can be solved iteratively using the generalized BlahutArimoto algorithm which, however, requires us to estimate the joint distribution of the potentially high-dimensional variables X and Y . A formal analysis of the difﬁculties of this estimation problem was conducted in [2]. In the continuous case, estimation of multivariate densities becomes arduous and can be a major impediment to the practical application of IB. A notable exception is the case of joint Gaussian (X, Y ) for which an analytical solution for the optimal representation T exists [3]. The optimal T is jointly Gaussian with (X, Y ) [4] and takes the form of a noisy linear projection to eigenvectors of the normalised conditional covariance matrix. The existence of an analytical solution opens new application possibilities and IB becomes practically feasible in higher dimensions [5]. Finding closed form solutions for other continuous distribution families remains an open challenge. The practical usefulness of the Gaussian IB (GIB), on the other hand, suffers from its missing ﬂexibility and the statistical problem of ﬁnding a robust estimate of the joint covariance matrix of (X, Y ) in high-dimensional spaces. Compression and relevance in IB are deﬁned in terms of mutual information (MI) of two random vectors V and W, which is deﬁned as the reduction in the entropy of V by the conditional entropy of V given W. MI bears an interesting relationship to copulas: mutual information equals negative copula entropy [6]. This relation between two seemingly unrelated concepts might appear surpris1 ing, but it directly follows from the deﬁnition of a copula as the object that captures the “pure” dependency structure of random variables [7]: a multivariate distribution consists of univariate random variables related to each other by a dependence mechanism, and copulas provide a framework to separate the dependence structure from the marginal distributions. In this work we reformulate the IB problem for the continuous variables in terms of copulas and enlighten that IB is completely independent of the marginal distributions of X, Y . The IB problem in the continuous case is in fact to ﬁnd the optimal copula (or dependence structure) of T and X, knowing the copula of X and the relevance variable Y . We focus on the case of Gaussian copula and on the consequences of the IB reformulation for the Gaussian IB. We show that the analytical solution available for GIB can naturally be extended to multivariate distributions with Gaussian copula and arbitrary marginal densities, also called meta-Gaussian densities. Moreover, we show that the GIB solution depends only a correlation matrix, and not on the variance. This allows us to use robust rank correlation estimators instead of unstable covariance estimators, and gives a robust version of GIB. 2 Information Bottleneck and Gaussian IB 2.1 General Information Bottleneck. Consider two random variables X and Y with values in the measurable spaces X and Y. Their joint distribution pXY (x, y) will also be denoted p(x, y) for simplicity. We construct a compressed representation T of X that is most informative about Y by solving the following variational problem: min p(t\|x) L \| L ≡I(X; T) −βI(T; Y ), (1) where the Lagrange parameter β > 0 determines the trade-off between compression of X and preservation of information about Y . Since the compressed representation is conditionally independent of Y given X as illustrated in Figure 1, to fully characterize T we only need to specify its joint distribution with X, i.e. p(x, t). No analytical solution is available for the general problem deﬁned by (1) and this joint distribution must be calculated with an iterative procedure. In the case of discrete variables X and Y , p(x, t) is obtained iteratively by self-consistent determination of p(t\|x), p(t) and p(y\|t) in the generalized Blahut-Arimoto algorithm. The resulting discrete T then deﬁnes (soft) clusters of X. In the case of continuous X and Y , the same set of self-consistent equations for p(t\|x), p(t) and p(y\|t) are obtained. These equations also translate into two coupled eigenvector problems for ∂log p(x\|t)/∂t and ∂log p(y\|t)/∂t, but a direct solution of these problems is very difﬁcult in practice. However, when X and Y are jointly multivariate Gaussian distributed, this problem becomes analytically tractable. Figure 1: Graphical representation of the conditional independence structure of IB. 2.2 Gaussian IB. Consider two joint Gaussian random vectors (rv) X and Y with zero mean: (X, Y ) ∼N 0p+q, Σ = Σx ΣT xy Σxy Σy , (2) where p is the dimension of X, q is the dimension of Y and 0p+q is the zero vector of dimension p+q. In [4] it is proved that the optimal compression T is also jointly Gaussian with X and Y . This implies that T can be expressed as a noisy linear transformation of X: T = AX + ξ, (3) 2 where ξ ∼N(0p, Σξ) is independent of X and A ∈Rp×p. The minimization problem (1) is then reduced to solving: min A,Σξ L\|L ≡I(X; T) −βI(T; Y ). (4) For a given trade-off parameter β, the optimal compression is given by T ∼N(0p, Σt) with Σt = AΣxAT + Σξ and the noise variance can be ﬁxed to the identity matrix Σξ = Ip, as shown in [3]. The transformation matrix A is given by: A =      0T ; . . . ; 0T 0 ≤β ≤βc 1 α1vT 1 ; 0T ; . . . , 0T βc 1 ≤β ≤βc 2 α1vT 1 ; α2vT 2 ; 0T ; . . . ; 0T βc 2 ≤β ≤βc 3 ...      (5) where vT 1 , . . . , vT p are left eigenvectors of Σx\|yΣ−1 x sorted by their corresponding increasing eigenvalues λ1, . . . , λp. The critical β values are βc i = (1 −λi)−1, and the αi coefﬁcients are deﬁned by αi = q β(1−λi)−1 λiri with ri = vT i Σxvi. In the above, 0T is a p-dimensional row vector and semicolons separate rows of A. We can see from equation (5) that the optimal projection of X is a combination of weighted eigenvectors of Σx\|yΣ−1 x . The number of selected eigenvectors, and thus the effective dimension of T, depends on the parameter β. 3 Copula and Information Bottleneck 3.1 Copula and Gaussian copula. A multivariate distribution consists of univariate random variables related to each other by a dependence mechanism. Copulas provide a framework to separate the dependence structure from the marginal distributions. Formally, a d-dimensional copula is a multivariate distribution function C : [0, 1]d →[0, 1] with standard uniform margins. Sklar’s theorem [7] states the relationship between copulas and multivariate distributions. Any joint distribution function F can be represented using its marginal univariate distribution functions and a copula: F (z1, . . . , zd) = C (F1 (z1) , . . . , Fd (zd)) . (6) If the margins are continuous, then this copula is unique. Conversely, if C is a copula and F1, . . . , Fd are univariate distribution functions, then F deﬁned as in (6) is a valid multivariate distribution function with margins F1, . . . , Fd. Assuming that C has d-th order partial derivatives we can deﬁne the copula density function: c(u1, . . . , ud) = ∂C(u1,...,ud) ∂u1...∂ud , u1, . . . , ud ∈[0, 1], The density corresponding to (6) can then be rewritten as a product of the marginal densities and the copula density function: f(z1, . . . , zd) = c (F1(z1), . . . , Fd(zd)) Qd j=1 fj(zj). Gaussian copulas constitute an important class of copulas. If F is a Gaussian distribution Nd (µ, Σ) then the corresponding C fulﬁlling equation (6) is a Gaussian copula. Due to basic invariance properties (cf. [8]), the copula of Nd (µ, Σ) is the same as the copula of Nd (0, P), where P is the correlation matrix corresponding to the covariance matrix Σ. Thus a Gaussian copula is uniquely determined by a correlation matrix P and we denote a Gaussian copula by CP . Using equation (6) with CP , we can construct multivariate distributions with arbitrary margins and a Gaussian dependence structure. These distributions are called meta-Gaussian distributions. Gaussian copulas conveniently have a copula density function: cP (u) = \|P\|−1 2 exp −1 2Φ−1(u)T (P −1 −I)Φ−1(u) , (7) where Φ−1(u) is a short notation for the univariate Gaussian quantile function applied to each component Φ−1(u) = (Φ−1(u1), . . . , Φ−1(ud)). 3.2 Copula formulation of IB. At the heart of the copula formulation of IB is the following identity: for a continuous random vector Z = (Z1, . . . , Zd) with density f(z) and copula density cZ(u) the multivariate mutual information 3 or multi-information is the negative differential entropy of the copula density: I(Z) ≡Dkl(f(z) ∥f0(z)) = Z [0,1]d cZ(u) log cZ(u)du = −H(cZ), (8) where u = (u1, . . . , ud) ∈[0, 1]d, Dkl denotes the Kullback-Leibler divergence, and f0(z) = f1(z1)f2(z2) . . . fd(zd). For continuous multivariate X, Y and T, equation (8) implies that: I(X; T) = Dkl(f(x, t) ∥f0(x, t)) −Dkl(f(x)\|\|f0(x)) −Dkl(f(t)\|\|f0(t)), = −H(cXT ) + H(cX) + H(cT ), I(Y ; T) = −H(cY T ) + H(cY ) + H(cT ), where cXT is the copula density of the vector (X1, . . . , Xp, T1, . . . , Tp). The above derivation then leads to the following proposition. Proposition 3.1. Copula formulation of IB The Information Bottleneck minimization problem (1) can be reformulated as: min cXT L \| L = −H(cXT ) + H(cX) + H(cT ) −β{−H(cY T ) + H(cY ) + H(cT )}. (9) The minimization problem deﬁned in (1) is solved under the assumption that the joint distribution of (X, Y ) is known, this now translates in the assumption that the copula copula density cXY (and thus cX) is assumed to be known. The density cT is entirely determined by cXT , and using the conditional independence structure it is clear that cY T is also determined by cXT when cXY is known. Since the joint density of (X, Y, T) decomposes as: f(x, y, t) = f(t, y\|x)f(x) = f(t\|x)f(y\|x)f(x), (10) the corresponding copula density then also decomposes as: cXY T (ux, uy, ut) = RT \|X(ux, ut)RY \|X(ux, uy)cX(ux), (11) where RT \|X(ux, ut) = cXT (ux, ut) cX(ux) , ux ∈[0, 1]p, uy ∈[0, 1]q, ut ∈[0, 1]p, (12) as shown in [9]. We can ﬁnally rewrite the copula density of (Y, T) as: cY T (uy, ut) = Z cXY T (ux, uy, ut)dux = Z cXT (ux, ut)cXY (ux, uy) cX(ux) dux. (13) The IB optimization problem actually reduces to ﬁnding an optimal copula density cXT . This implies that in order to construct the compression variable T, the only relevant aspect is the copula dependence structure between X, T and Y . 4 Meta-Gaussian IB 4.1 Meta-Gaussian IB formulation. The above reformulation of IB is of great practical interest when we focus on the special case of the Gaussian copula. The only known case for which a simple analytical solution to the IB problem exists is when (X, Y ) are joint Gaussians. Equation (9) shows that actually an optimal solution does not depend of the margins but only on the copula density cXY . From this observation the idea naturally follows that an analytical solution should also exist for any joint distribution of (X, Y ) which has a Gaussian copula, and that regardless of its margins. We show below in proposition 4.1 that this is indeed the case. The notation ˜X and ˜Y is used to represent the normal scores: ˜X = (Φ−1 ◦FX1(X1), . . . , Φ−1 ◦FXp(Xp)). (14) Since copulas are invariant to strictly increasing transformations the normal scores have the same copulas as the original variables X and Y . 4 Proposition 4.1. Optimality of meta-Gaussian IB Consider rv X, Y with a Gaussian dependence structure and arbitrary margins: FX,Y (x, y) ∼CP (FX1(x1), . . . , FXp(xp), FY1(y1), . . . , FYq(yq)), (15) where FXi, FYi are the marginal distributions of X, Y and CP is a Gaussian copula parametrized by a correlation matrix P. Then the optimum of the minimization problem (1) is obtained for T ∈T , where T is the set of all rv T such that (X, Y, T) has a Gaussian copula and T has Gaussian margins. Before proving proposition 4.1 we give a short lemma. Lemma 4.1. T ∈T ⇔( ˜X, ˜Y , T) are jointly Gaussian. Proof. 1. If T ∈T then (X, Y, T) has a Gaussian copula which implies that ( ˜X, ˜Y , T) also has a Gaussian copula. Since ˜X, ˜Y , T all have normally distributed margins it follows that ( ˜X, ˜Y , T) has a joint Gaussian distribution. 2. If ( ˜X, ˜Y , T) are jointly Gaussian then ( ˜X, ˜Y , T) has a Gaussian copula which implies that (X, Y, T) has again a Gaussian copula. Since T has normally distributed margins, it follows that T ∈T . Proposition 4.1 can now be proven by contradiction. Proof of proposition 4.1. Assume there exists T ∗/∈T such that: L(X, Y, T ∗) := I(X; T ∗) −βI(Y ; T ∗) < min p(t\|x),T ∈T I(X; T) −βI(T; Y ) (16) Since ( ˜X, ˜Y , T) has the same copula as (X, Y, T), we have that I( ˜X; T) = I(X; T) and I( ˜Y ; T) = I(Y ; T). Using Lemma 4.1 the right hand part of inequality (16) can be rewritten as : min p(t\|x),T ∈T L(X, Y, T) = min p(t\|x),T ∈T L( ˜X, ˜Y , T) = min p(t\|˜x),( ˜ X, ˜Y ,T )∼N L( ˜X, ˜Y , T). (17) Combining equations (16) and (17) we obtain: I( ˜X; T ∗) −βI( ˜Y ; T ∗) < min p(t\|˜x),( ˜ X, ˜Y ,T )∼N I( ˜X; T) −βI(T; ˜Y ). This is in contradiction with the optimality of Gaussian information bottleneck, which states that the optimal T is jointly Gaussian with (X, Y ). Thus the optimum for meta-Gaussian (X, Y ) is attained for T with normal margins such that (X, Y, T) also is meta-Gaussian. Corollary 4.1. The optimal projection T o obtained for ( ˜X, ˜Y ) is also optimal for (X, Y ). Proof. By the above we know that an optimal compression for (X, Y ) can be obtained in the set of variables T such that ( ˜X, ˜Y , T) is jointly Gaussian, since ˜L = L it is clear that T o is also optimal for (X, Y ). As a consequence of Proposition 4.1, for any random vector (X, Y ) having a Gaussian copula dependence structure, an optimal projection T can be obtained by ﬁrst calculating the vector of the normal scores ( ˜X, ˜Y ) and then computing T = A ˜X + ξ. A is here entirely determined by the covariance matrix of the vector ( ˜X, ˜Y ) which also equals its correlation matrix (the normal scores have unit variance by deﬁnition), and thus the correlation matrix P parametrizing the Gaussian copula CP . In practice the problem is reduced to the estimation the Gaussian copula of (X, Y ). In particular, for the traditional Gaussian case where (X, Y ) ∼N(0, Σ), this means that we actually do not need to estimate the full covariance Σ but only the correlations. 5 4.2 Meta-Gaussian mutual information. The multi-information for a meta-Gaussian random vector Z = (Z1, . . . , Zd) with copula CPz is: I(Z) = I( ˜Z) = −1 2 log \|cov( ˜Z)\| = −1 2 log \|Σ˜z\| = −1 2 log \|corr( ˜Z)\| = −1 2 log \|Pz\|, (18) where \|.\| denotes the determinant. A direct derivation of the multi-information for meta-Gaussian random variables is also given in the supplementary material. The mutual information between X and Y is then I(X; Y ) = −1 2 log \|P\|+ 1 2 log \|Px\|+ 1 2 log \|Py\|, where P = Px Pyx Pxy Py . It is obvious that the formula for the meta-Gaussian is similar to the formula for the Gaussian case IGauss(X; Y ) = −1 2 log \|Σ\|+ 1 2 log \|Σx\|+ 1 2 log \|Σy\|, but uses the correlation matrix parametrizing the copula instead of the data covariance matrix. The two formulas are equivalent when X, Y are jointly Gaussian. 4.3 Semi-parametric copula estimation. Semi-parametric copula estimation has been studied in [10], [11] and [12]. The main idea is to combine non-parametric estimation of the margins with a parametric copula model, in our case the Gaussian copulas family. If the margins F1, . . . , Fd of a random vector Z are known, P can be estimated by the matrix ˆP with elements given by: ˆP(k,l) = 1 n Pn i=1 Φ−1(Fk(zik))Φ−1(Fl(zil)) h 1 n Pn i=1 [Φ−1(Fk(zik))]2 1 n Pn i=1 [Φ−1(Fl(zil))]2i1/2 , (19) where zik denotes the i-th observation of dimension k. ˆP is assured to be positive semi-deﬁnite. If the margins are unknown we can instead use the rescaled empirical cumulative distributions: ˆFj(t) = n n + 1 1 n n X i=1 Izij≤t ! . (20) The estimator resulting from using the rescaled empirical distributions (20) in equation (19) is given in the following deﬁnition. Deﬁnition 4.1 (Normal scores rank correlation coefﬁcient). The normal scores rank correlation coefﬁcient is the matrix ˆP n with elements: ˆP n (k,l) = Pn i=1 Φ−1( R(zik) n+1 )Φ−1( R(zil) n+1 ) Pn i=1 Φ−1( i n+1) 2 , (21) where R(zik) denotes the rank of the i-th observation for dimension k. Robustness properties of the estimator (21) have been studied in [13]. Using (21) we compute an estimate of the correlation matrix P parametrizing cXY and obtain the transformation matrix A as detailed in Algorithm 1. Algorithm 1 Construction of the transformation matrix A 1. Compute the normal scores rank correlation estimate ˆP n of the correlation matrix P parametrizing cXY : for k, l = 1, . . . , p + q do Set the (k, l)-th element of ˆP n to Pn i=1 Φ−1( R(zik) n+1 )Φ−1( R(zil) n+1 ) Pn i=1(Φ−1( i n+1 )) 2 as in equation (21) and where the i-th row of z is the concatenation of the i-th rows of x and y: zi∗= (xi∗, yi∗) ∈Rp+q. end for 2. Compute the estimated conditional covariance matrix of the normal scores: ˆΣ˜x\|˜y = ˆP n x − ˆP n xy( ˆP n y )−1 ˆP n yx. 3. Find the eigenvectors and eigenvalues of ˆΣ˜x\|˜y( ˆP n x )−1. 4. Construct the transformation matrix A as in equation (5). 6 5 Results 5.1 Simulations We tested meta-Gaussian IB (MGIB) in two different setting, ﬁrst when the data is Gaussian but contains outliers, second when the data has a Gaussian copula but non-Gaussian margins. We generated a training sample with n = 1000 observations of X and Y with dimensions ﬁxed to dx = 15 and dy = 15. A covariance matrix was drawn from a Wishart distribution centered at a correlation matrix populated with a few high correlation values to ensure some dependency between X and Y . This matrix was then scaled to obtain the correlation matrix parametrizing the copula. In the ﬁrst setting the data was sampled with N(0, 1) margins. A ﬁxed percentage of outliers, 8%, was then introduced to the sample by randomly drawing a row and a column in the data matrix and replacing the current value with a random draw from the set [−6, −3] ∪[3, 6]. In the second setting data points were drawn from meta-Gaussian distributions with three different type of margins: Student with df = 4, exponential with λ = 1, and beta with α1 = 0.5 = α2. For each training sample two projection matrices AG and AC were computed, AG was calculated based on the sample covariance ˆΣn and AC was obtained using the normal scores rank correlation ˆP n. The compression quality of the projection was then tested on a test sample of n = 10′000 observations generated independently from the same distribution (without outliers). Each experiment was repeated 50 times. Figure 2 shows the information curves obtained by varying β from 0.1 to 200. The mutual informations I(X; T) and (Y ; T) can be reliably estimated on the test sample using (18) and (21). The information curves start with a very steep slope, meaning that a small increase in I(X; T) leads to a signiﬁcant increase in I(Y ; T), and then slowly saturate to reach their asymptotic limit in I(Y ; T). The best information curves are situated in the upper left corner of the ﬁgure, since for a ﬁxed compression value I(X; T) we want to achieve the highest relevant information content (I; T). We clearly see in Figure 2 that MGIB consistently outperforms GIB in that it achieves higher compression rates. 0 10 20 30 0 2 4 6 8 10 12 Gaussian with outliers I(X;T) I(Y;T) MGIB GIB 0 10 20 30 40 0 2 4 6 8 10 12 14 Student margins I(X;T) I(Y;T) MGIB GIB 0 10 20 30 40 0 2 4 6 8 10 12 14 Exponential margins I(X;T) I(Y;T) MGIB GIB 0 10 20 30 40 0 2 4 6 8 10 12 14 Beta margins I(X;T) I(Y;T) MGIB GIB Figure 2: Information curves for Gaussian data with outliers, data with Student, Exponential and Beta margins. Each panel shows 50 curves obtained for repetitions of the MGIB (red) and the GIB (black). The curves stop when they come close to saturation. For higher values of β the information I(X; T) would continue to grow while I(Y ; T) would reach its limit leading to horizontal lines, but such high beta values lead to numerical instability. Since GIB suffers from a model mismatch problem when the margins are not Gaussian, the curves saturate for smaller values of I(Y ; T). 7 5.2 Real data We further applied MGIB to the Communities and Crime data set from the UCI repository 1. The data set contains observations of predictive and target variables. After removing missing values we retained n = 2195 observations. In a pre-processing step we selected the dx = 10 dimensions with the strongest absolute rank correlation to one of the relevance variables. Plotting empirical information curves as in the synthetic examples above was impossible, because even for this setting with drastically decreased dimensionality all mutual information estimates we tried (including the nearest-neighbor graph method in [14]) were too unstable to draw empirical information curves. To still give a graphical representation of our results we show in Figure 3 non-parametric density estimates of the one dimensional compression T split in 5 groups according to corresponding values of the ﬁrst relevance variable. We used GIB, MGIB and Principal Component analysis (PCA) to reduce X to a 1-dimensional variable. For PCA this is the ﬁrst principal component, for GIB and MGIB we independently selected the highest value of β leading to a 1-dimensional compression. It is obvious from Figure 3 that the one-dimensional MGIB compression nicely separates the different target classes, whereas the GIB and PCA projections seem to contain much less information about the target variable. We conclude that similar to our synthetic examples above, the MGIB compression contains more information about the relevance variable than GIB at the same compression rate. Meta−Gaussian IB First component of compression T Gaussian IB First component of compression T PCA first PCA projection Y1 in (−3.5,0) Y1 in (0,0.5) Y1 in (0.5,1) Y1 in (1,1.5) Y1 in (1.5,3.5) Figure 3: Parzen density estimates of the univariate projection of X split in 5 groups according to values of the ﬁrst relevance variable. We see more separation between groups for MGIB than for GIB or PCA, which indicates that the projection is more informative about the relevance variable. 6 Conclusion We present a reformulation of the IB problem in terms of copula which gives new insights into data compression with relevance constraints and opens new possible applications of IB for continuous multivariate data. Meta-Gaussian IB naturally extends the analytical solution of Gaussian IB to multivariate distributions with Gaussian copula and arbitrary marginal density. It can be applied to any type of continuous data, provided the assumption of a Gaussian dependence structure is reasonable, in which case the optimal compression can easily be obtained by semi-parametric copula estimation. Simulated experiments showed that MGIB clearly outperforms GIB when the marginal densities are not Gaussian, and even in the Gaussian case with a tiny amount of outliers MGIB has been shown to signiﬁcantly beneﬁt from the robustness properties of rank estimators. In future work, it would be interesting to see if the copula formulation of IB admits analytical solutions for other copula families. Acknowledgments M. Rey is partially supported by the Swiss National Science Foundation, grant CR32I2 127017 / 1. 1http://archive.ics.uci.edu/ml/ 8 References [1] N. Tishby, F.C. Pereira, and W. Bialek. The information bottleneck method. The 37th annual Allerton Conference on Communication, Control, and Computing, (29-30):368–377, 1999. [2] O. Shamir, S. Sabato, and N. Tishby. Learning and generalization with the information bottleneck. Theor. Comput. Sci., 411(29-30):2696–2711, 2010. [3] G. Chechik, A. Globerson, N. Tishby, and Y. Weiss. Information bottleneck for Gaussian variables. 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4,454	Efﬁcient Monte Carlo Counterfactual Regret Minimization in Games with Many Player Actions Richard Gibson, Neil Burch, Marc Lanctot, and Duane Szafron Department of Computing Science, University of Alberta Edmonton, Alberta, T6G 2E8, Canada {rggibson \| nburch \| lanctot \| dszafron}@ualberta.ca Abstract Counterfactual Regret Minimization (CFR) is a popular, iterative algorithm for computing strategies in extensive-form games. The Monte Carlo CFR (MCCFR) variants reduce the per iteration time cost of CFR by traversing a smaller, sampled portion of the tree. The previous most effective instances of MCCFR can still be very slow in games with many player actions since they sample every action for a given player. In this paper, we present a new MCCFR algorithm, Average Strategy Sampling (AS), that samples a subset of the player’s actions according to the player’s average strategy. Our new algorithm is inspired by a new, tighter bound on the number of iterations required by CFR to converge to a given solution quality. In addition, we prove a similar, tighter bound for AS and other popular MCCFR variants. Finally, we validate our work by demonstrating that AS converges faster than previous MCCFR algorithms in both no-limit poker and Bluff. 1 Introduction An extensive-form game is a common formalism used to model sequential decision making problems containing multiple agents, imperfect information, and chance events. A typical solution concept in games is a Nash equilibrium proﬁle. Counterfactual Regret Minimization (CFR) [12] is an iterative algorithm that, in 2-player zero-sum extensive-form games, converges to a Nash equilibrium. Other techniques for computing Nash equilibria of 2-player zero-sum games include linear programming [8] and the Excessive Gap Technique [6]. Theoretical results indicate that for a ﬁxed solution quality, CFR takes a number of iterations at most quadratic in the size of the game [12, Theorem 4]. Thus, as we consider larger games, more iterations are required to obtain a ﬁxed solution quality. Nonetheless, CFR’s versatility and memory efﬁciency make it a popular choice. Monte Carlo CFR (MCCFR) [9] can be used to reduce the traversal time per iteration by considering only a sampled portion of the game tree. For example, Chance Sampling (CS) [12] is an instance of MCCFR that only traverses the portion of the game tree corresponding to a single, sampled sequence of chance’s actions. However, in games where a player has many possible actions, such as no-limit poker, iterations of CS are still very time consuming. This is because CS considers all possible player actions, even if many actions are poor or only factor little into the algorithm’s computation. Our main contribution in this paper is a new MCCFR algorithm that samples player actions and is suitable for games involving many player choices. Firstly, we provide tighter theoretical bounds on the number of iterations required by CFR and previous MCCFR algorithms to reach a ﬁxed solution quality. Secondly, we use these new bounds to propel our new MCCFR sampling algorithm. By using a player’s average strategy to sample actions, convergence time is signiﬁcantly reduced in large games with many player actions. We prove convergence and show that our new algorithm approaches equilibrium faster than previous sampling schemes in both no-limit poker and Bluff. 1 2 Background A ﬁnite extensive game contains a game tree with nodes corresponding to histories of actions h ∈H and edges corresponding to actions a ∈A(h) available to player P(h) ∈N ∪{c} (where N is the set of players and c denotes chance). When P(h) = c, σc(h, a) is the (ﬁxed) probability of chance generating action a at h. Each terminal history z ∈Z has associated utilities ui(z) for each player i. We deﬁne ∆i = maxz,z′∈Z ui(z) −ui(z′) to be the range of utilities for player i. Non-terminal histories are partitioned into information sets I ∈Ii representing the different game states that player i cannot distinguish between. For example, in poker, player i does not see the private cards dealt to the opponents, and thus all histories differing only in the private cards of the opponents are in the same information set for player i. The action sets A(h) must be identical for all h ∈I, and we denote this set by A(I). We deﬁne \|Ai\| = maxI∈Ii \|A(I)\| to be the maximum number of actions available to player i at any information set. We assume perfect recall that guarantees players always remember information that was revealed to them and the order in which it was revealed. A (behavioral) strategy for player i, σi ∈Σi, is a function that maps each information set I ∈Ii to a probability distribution over A(I). A strategy proﬁle is a vector of strategies σ = (σ1, ..., σ\|N\|) ∈ Σ, one for each player. Let ui(σ) denote the expected utility for player i, given that all players play according to σ. We let σ−i refer to the strategies in σ excluding σi. Let πσ(h) be the probability of history h occurring if all players choose actions according to σ. We can decompose πσ(h) = Q i∈N∪{c} πσ i (h), where πσ i (h) is the contribution to this probability from player i when playing according to σi (or from chance when i = c). Let πσ −i(h) be the product of all players’ contributions (including chance) except that of player i. Let πσ(h, h′) be the probability of history h′ occurring after h, given h has occurred. Furthermore, for I ∈Ii, the probability of player i playing to reach I is πσ i (I) = πσ i (h) for any h ∈I, which is well-deﬁned due to perfect recall. A best response to σ−i is a strategy that maximizes player i’s expected payoff against σ−i. The best response value for player i is the value of that strategy, bi(σ−i) = maxσ′ i∈Σi ui(σ′ i, σ−i). A strategy proﬁle σ is an ϵ-Nash equilibrium if no player can unilaterally deviate from σ and gain more than ϵ; i.e., ui(σ) + ϵ ≥bi(σ−i) for all i ∈N. A game is two-player zero-sum if N = {1, 2} and u1(z) = −u2(z) for all z ∈Z. In this case, the exploitability of σ, e(σ) = (b1(σ2)+b2(σ1))/2, measures how much σ loses to a worst case opponent when players alternate positions. A 0-Nash equilibrium (or simply a Nash equilibrium) has zero exploitability. Counterfactual Regret Minimization (CFR) [12] is an iterative algorithm that, for two-player zero sum games, computes an ϵ-Nash equilibrium proﬁle with ϵ →0. CFR has also been shown to work well in games with more than two players [1, 3]. On each iteration t, the base algorithm, “vanilla” CFR, traverses the entire game tree once per player, computing the expected utility for player i at each information set I ∈Ii under the current proﬁle σt, assuming player i plays to reach I. This expectation is the counterfactual value for player i, vi(I, σ) = P z∈ZI ui(z)πσ −i(z[I])πσ(z[I], z), where ZI is the set of terminal histories passing through I and z[I] is that history along z contained in I. For each action a ∈A(I), these values determine the counterfactual regret at iteration t, rt i(I, a) = vi(I, σt (I→a)) −vi(I, σt), where σ(I→a) is the proﬁle σ except that at I, action a is always taken. The regret rt i(I, a) measures how much player i would rather play action a at I than play σt. These regrets are accumulated to obtain the cumulative counterfactual regret, RT i (I, a) = PT t=1 rt i(I, a), and are used to update the current strategy proﬁle via regret matching [5, 12], σT +1(I, a) = RT,+ i (I, a) P b∈A(I) RT,+ i (I, b) , (1) where x+ = max{x, 0} and actions are chosen uniformly at random when the denominator is zero. It is well-known that in a two-player zero-sum game, if both players’ average (external) regret, RT i T = max σ′ i∈Σi 1 T T X t=1 ui(σ′ i, σt −i) −ui(σt i, σt −i) , is at most ϵ/2, then the average proﬁle ¯σT is an ϵ-Nash equilibrium. During computation, CFR stores a cumulative proﬁle sT i (I, a) = PT t=1 πσt i (I)σt i(I, a) and outputs the average proﬁle 2 ¯σT i (I, a) = sT i (I, a)/ P b∈A(I) sT i (I, b). The original CFR analysis shows that player i’s regret is bounded by the sum of the positive parts of the cumulative counterfactual regrets RT,+ i (I, a): Theorem 1 (Zinkevich et al. [12]) RT i ≤ X I∈I max a∈A(I) RT,+ i (I, a). Regret matching minimizes the average of the cumulative counterfactual regrets, and so player i’s average regret is also minimized by Theorem 1. For each player i, let Bi be the partition of Ii such that two information sets I, I′ are in the same part B ∈Bi if and only if player i’s sequence of actions leading to I is the same as the sequence of actions leading to I′. Bi is well-deﬁned due to perfect recall. Next, deﬁne the M-value of the game to player i to be Mi = P B∈Bi p \|B\|. The best known bound on player i’s average regret is: Theorem 2 (Lanctot et al. [9]) When using vanilla CFR, average regret is bounded by RT i T ≤∆iMi p \|Ai\| √ T . We prove a tighter bound in Section 3. For large games, CFR’s full game tree traversals can be very expensive. Alternatively, one can traverse a smaller, sampled portion of the tree on each iteration using Monte Carlo CFR (MCCFR) [9]. Let Q = {Q1, ..., QK} be a set of subsets, or blocks, of the terminal histories Z such that the union of Q spans Z. For example, Chance Sampling (CS) [12] is an instance of MCCFR that partitions Z into blocks such that two histories are in the same block if and only if no two chance actions differ. On each iteration, a block Qj is sampled with probability qj, where PK k=1 qk = 1. In CS, we generate a block by sampling a single action a at each history h ∈H with P(h) = c according to its likelihood of occurring, σc(h, a). In general, the sampled counterfactual value for player i is ˜vi(I, σ) = X z∈ZI∩Qj ui(z)πσ −i(z[I])πσ(z[I], z)/q(z), where q(z) = P k:z∈Qk qk is the probability that z was sampled. For example, in CS, q(z) = πσ c (z). Deﬁne the sampled counterfactual regret for action a at I to be ˜rt i(I, a) = ˜vi(I, σt (I→a))− ˜vi(I, σt). Strategies are then generated by applying regret matching to ˜RT i (I, a) = PT t=1 ˜rt i(I, a). CS has been shown to signiﬁcantly reduce computing time in poker games [11, Appendix A.5.2]. Other instances of MCCFR include External Sampling (ES) and Outcome Sampling (OS) [9]. ES takes CS one step further by considering only a single action for not only chance, but also for the opponents, where opponent actions are sampled according to the current proﬁle σt −i. OS is the most extreme version of MCCFR that samples a single action at every history, walking just a single trajectory through the tree on each traversal (Qj = {z}). ES and OS converge to equilibrium faster than vanilla CFR in a number of different domains [9, Figure 1]. ES and OS yield a probabilistic bound on the average regret, and thus provide a probabilistic guarantee that ¯σT converges to a Nash equilibrium. Since both algorithms generate blocks by sampling actions independently, we can decompose q(z) = Q i∈N∪{c} qi(z) so that qi(z) is the probability contributed to q(z) by sampling player i’s actions. Theorem 3 (Lanctot et al. [9]) 1 Let X be one of ES or OS (assuming OS also samples opponent actions according to σ−i), let p ∈(0, 1], and let δ = minz∈Z qi(z) > 0 over all 1 ≤t ≤T. When using X, with probability 1 −p, average regret is bounded by RT i T ≤ Mi + p 2\|Ii\|\|Bi\| √p ! 1 δ ∆i p \|Ai\| √ T . 1The bound presented by Lanctot et al. appears slightly different, but the last step of their proof mistakenly used Mi ≥ p \|Ii\|\|Bi\|, which is actually incorrect. The bound we present here is correct. 3 3 New CFR Bounds While Zinkevich et al. [12] bound a player’s regret by a sum of cumulative counterfactual regrets (Theorem 1), we can actually equate a player’s regret to a weighted sum of counterfactual regrets. For a strategy σi ∈Σi and an information set I ∈Ii, deﬁne RT i (I, σi) = P a∈A(I) σi(I, a)RT i (I, a). In addition, let σ∗ i ∈ Σi be a player i strategy such that σ∗ i = arg maxσ′ i∈Σi PT t=1 ui(σ′ i, σt −i). Note that in a two-player game, PT t=1 ui(σ∗ i , σt −i) = Tui(σ∗ i , ¯σT −i), and thus σ∗ i is a best response to the opponent’s average strategy after T iterations. Theorem 4 RT i = X I∈Ii πσ∗ i (I)RT i (I, σ∗ i ). All proofs in this paper are provided in full as supplementary material. Theorem 4 leads to a tighter bound on the average regret when using CFR. For a strategy σi ∈Σi, deﬁne the M-value of σi to be Mi(σi) = P B∈Bi πσ i (B) p \|B\|, where πσ i (B) = maxI∈B πσ i (I). Clearly, Mi(σi) ≤Mi for all σi ∈Σi since πσ i (B) ≤1. For vanilla CFR, we can simply replace Mi in Theorem 2 with Mi(σ∗ i ): Theorem 5 When using vanilla CFR, average regret is bounded by RT i T ≤∆iMi(σ∗ i ) p \|Ai\| √ T . For MCCFR, we can show a similar improvement to Theorem 3. Our proof includes a bound for CS that appears to have been omitted in previous work. Details are in the supplementary material. Theorem 6 Let X be one of CS, ES, or OS (assuming OS samples opponent actions according to σ−i), let p ∈(0, 1], and let δ = minz∈Z qi(z) > 0 over all 1 ≤t ≤T. When using X, with probability 1 −p, average regret is bounded by RT i T ≤ Mi(σ∗ i ) + p 2\|Ii\|\|Bi\| √p ! 1 δ ∆i p \|Ai\| √ T . Theorem 4 states that player i’s regret is equal to the weighted sum of player i’s counterfactual regrets at each I ∈Ii, where the weights are equal to player i’s probability of reaching I under σ∗ i . Since our goal is to minimize average regret, this means that we only need to minimize the average cumulative counterfactual regret at each I ∈Ii that σ∗ i plays to reach. Therefore, when using MCCFR, we may want to sample more often those information sets that σ∗ i plays to reach, and less often those information sets that σ∗ i avoids. This inspires our new MCCFR sampling algorithm. 4 Average Strategy Sampling Leveraging the theory developed in the previous section, we now introduce a new MCCFR sampling algorithm that can minimize average regret at a faster rate than CS, ES, and OS. As we just described, we want our algorithm to sample more often the information sets that σ∗ i plays to reach. Unfortunately, we do not have the exact strategy σ∗ i on hand. Recall that in a two-player game, σ∗ i is a best response to the opponent’s average strategy, ¯σT −i. However, for two-player zero-sum games, we do know that the average proﬁle ¯σT converges to a Nash equilibrium. This means that player i’s average strategy, ¯σT i , converges to a best response of ¯σT −i. While the average strategy is not an exact best response, it can be used as a heuristic to guide sampling within MCCFR. Our new sampling algorithm, Average Strategy Sampling (AS), selects actions for player i according to the cumulative proﬁle and three predeﬁned parameters. AS can be seen as a sampling scheme between OS and ES where a subset of player i’s actions are sampled at each information set I, as opposed to sampling one action (OS) or sampling every action (ES). Given the cumulative proﬁle sT i (I, ·) on iteration T, an exploration parameter ϵ ∈(0, 1], a threshold parameter τ ∈[1, ∞), and a bonus parameter β ∈[0, ∞), each of player i’s actions a ∈A(I) are sampled independently with probability ρ(I, a) = max ( ϵ, β + τsT i (I, a) β + P b∈A(I) sT i (I, b) ) , (2) 4 Algorithm 1 Average Strategy Sampling (Two-player version) 1: Require: Parameters ϵ, τ, β 2: Initialize regret and cumulative proﬁle: ∀I, a : r(I, a) ←0, s(I, a) ←0 3: 4: WalkTree(history h, player i, sample prob q): 5: if h ∈Z then return ui(h)/q end if 6: if h ∈P(c) then Sample action a ∼σc(h, ·), return WalkTree(ha, i, q) end if 7: I ←Information set containing h , σ(I, ·) ←RegretMatching(r(I, ·)) 8: if h /∈P(i) then 9: for a ∈A(I) do s(I, a) ←s(I, a) + (σ(I, a)/q) end for 10: Sample action a ∼σ(I, ·), return WalkTree(ha, i, q) 11: end if 12: for a ∈A(I) do 13: ρ ←max n ϵ, β+τs(I,a) β+P b∈A(I) s(I,b) o , ˜v(a) ←0 14: if Random(0, 1) < ρ then ˜v(a) ←WalkTree(ha, i, q · min{1, ρ}) end if 15: end for 16: for a ∈A(I) do r(I, a) ←r(I, a) + ˜v(a) −P a∈A(I) σ(I, a)˜v(a) end for 17: return P a∈A(I) σ(I, a)˜v(a) or with probability 1 if either ρ(I, a) > 1 or β + P b∈A(I) sT i (I, b) = 0. As in ES, at opponent and chance nodes, a single action is sampled on-policy according to the current opponent proﬁle σT −i and the ﬁxed chance probabilities σc respectively. If τ = 1 and β = 0, then ρ(I, a) is equal to the probability that the average strategy ¯σT i = sT i (I, a)/ P b∈A(I) sT i (I, b) plays a at I, except that each action is sampled with probability at least ϵ. For choices greater than 1, τ acts as a threshold so that any action taken with probability at least 1/τ by the average strategy is always sampled by AS. Furthermore, β’s purpose is to increase the rate of exploration during early AS iterations. When β > 0, we effectively add β as a bonus to the cumulative value sT i (I, a) before normalizing. Since player i’s average strategy ¯σT i is not a good approximation of σ∗ i for small T, we include β to avoid making ill-informed choices early-on. As the cumulative proﬁle sT i (I, ·) grows over time, β eventually becomes negligible. In Section 5, we present a set of values for ϵ, τ, and β that work well across all of our test games. Pseudocode for a two-player version of AS is presented in Algorithm 1. In Algorithm 1, the recursive function WalkTree considers four different cases. Firstly, if we have reached a terminal node, we return the utility scaled by 1/q (line 5), where q = qi(z) is the probability of sampling z contributed from player i’s actions. Secondly, when at a chance node, we sample a single action according to σc and recurse down that action (line 6). Thirdly, at an opponent’s choice node (lines 8 to 11), we again sample a single action and recurse, this time according to the opponent’s current strategy obtained via regret matching (equation (1)). At opponent nodes, we also update the cumulative proﬁle (line 9) for reasons that we describe in a previous paper [2, Algorithm 1]. For games with more than two players, a second tree walk is required and we omit these details. The ﬁnal case in Algorithm 1 handles choice nodes for player i (lines 7 to 17). For each action a, we compute the probability ρ of sampling a and stochastically decide whether to sample a or not, where Random(0,1) returns a random real number in [0, 1). If we do sample a, then we recurse to obtain the sampled counterfactual value ˜v(a) = ˜vi(I, σt (I→a)) (line 14). Finally, we update the regrets at I (line 16) and return the sampled counterfactual value at I, P a∈A(I) σ(I, a)˜v(a) = ˜vi(I, σt). Repeatedly running WalkTree(∅, i, 1) ∀i ∈N provides a probabilistic guarantee that all players’ average regret will be minimized. In the supplementary material, we prove that AS exhibits the same regret bound as CS, ES, and OS provided in Theorem 6. Note that δ in Theorem 6 is guaranteed to be positive for AS by the inclusion of ϵ in equation (2). However, for CS and ES, δ = 1 since all of player i’s actions are sampled, whereas δ ≤1 for OS and AS. While this suggests that fewer iterations of CS or ES are required to achieve the same regret bound compared to OS and AS, iterations for OS and AS are faster as they traverse less of the game tree. Just as CS, ES, and OS 5 have been shown to beneﬁt from this trade-off over vanilla CFR, we will show that in practice, AS can likewise beneﬁt over CS and ES and that AS is a better choice than OS. 5 Experiments In this section, we compare the convergence rates of AS to those of CS, ES, and OS. While AS can be applied to any extensive game, the aim of AS is to provide faster convergence rates in games involving many player actions. Thus, we consider two domains, no-limit poker and Bluff, where we can easily scale the number of actions available to the players. No-limit poker. The two-player poker game we consider here, which we call 2-NL Hold’em(k), is inspired by no-limit Texas Hold’em. 2-NL Hold’em(k) is played over two betting rounds. Each player starts with a stack of k chips. To begin play, the player denoted as the dealer posts a small blind of one chip and the other player posts a big blind of two chips. Each player is then dealt two private cards from a standard 52-card deck and the ﬁrst betting round begins. During each betting round, players can either fold (forfeit the game), call (match the previous bet), or raise by any number of chips in their remaining stack (increase the previous bet), as long as the raise is at least as big as the previous bet. After the ﬁrst betting round, three public community cards are revealed (the ﬂop) and a second and ﬁnal betting round begins. If a player has no more chips left after a call or a raise, that player is said to be all-in. At the end of the second betting round, if neither player folded, then the player with the highest ranked ﬁve-card poker hand wins all of the chips played. Note that the number of player actions in 2-NL Hold’em(k) at one information set is at most the starting stack size, k. Increasing k adds more betting options and allows for more actions before being all-in. Bluff. Bluff(D1, D2) [7], also known as Liar’s Dice, Perduo, and Dudo, is a two-player dice-bidding game played with six-sided dice over a number of rounds. Each player i starts with Di dice. In each round, players roll their dice and look at the result without showing their opponent. Then, players alternate by bidding a quantity q of a face value f of all dice in play until one player claims that the other is blufﬁng (i.e., claims that the bid does not hold). To place a new bid, a player must increase q or f of the current bid. A face value of six is considered “wild” and counts as any other face value. The player calling bluff wins the round if the opponent’s last bid is incorrect, and loses otherwise. The losing player removes one of their dice from the game and a new round begins. Once a player has no more dice left, that player loses the game and receives a utility of −1, while the winning player earns +1 utility. The maximum number of player actions at an information set is 6(D1 + D2) + 1 as increasing Di allows both players to bid higher quantities q. Preliminary tests. Before comparing AS to CS, ES, and OS, we ﬁrst run some preliminary experiments to ﬁnd a good set of parameter values for ϵ, τ, and β to use with AS. All of our preliminary experiments are in two-player 2-NL Hold’em(k). In poker, a common approach is to create an abstract game by merging similar card dealings together into a single chance action or “bucket” [4]. To keep the size of our games manageable, we employ a ﬁve-bucket abstraction that reduces the branching factor at each chance node down to ﬁve, where dealings are grouped according to expected hand strength squared as described by Zinkevich et al. [12]. Firstly, we ﬁx τ = 1000 and test different values for ϵ and β in 2-NL Hold’em(30). Recall that τ = 1000 implies actions taken by the average strategy with probability at least 0.001 are always sampled by AS. Figure 1a shows the exploitability in the ﬁve-bucket abstract game, measured in milli-big-blinds per game (mbb/g), of the proﬁle produced by AS after 1012 nodes visited. Recall that lower exploitability implies a closer approximation to equilibrium. Each data point is averaged over ﬁve runs of AS. The ϵ = 0.05 and β = 105 or 106 proﬁles are the least exploitable proﬁles within statistical noise (not shown). Next, we ﬁx ϵ = 0.05 and β = 106 and test different values for τ. Figure 1b shows the abstract game exploitability over the number of nodes visited by AS in 2-NL Hold’em(30), where again each data point is averaged over ﬁve runs. Here, the least exploitable strategies after 1012 nodes visited are obtained with τ = 100 and τ = 1000 (again within statistical noise). Similar results to Figure 1b hold in 2-NL Hold’em(40) and are not shown. Throughout the remainder of our experiments, we use the ﬁxed set of parameters ϵ = 0.05, β = 106, and τ = 1000 for AS. 6 Exploitability (mbb/g) 100 101 102 103 104 105 106 107 108 109 β 0.01 0.05 0.1 0.2 0.3 0.4 0.5 ε 0 0.2 0.4 0.6 0.8 1 (a) τ = 1000 10-1 100 101 102 1010 1011 1012 Abstract game exploitability (mbb/g) Nodes Visited τ=100 τ=101 τ=102 τ=103 τ=104 τ=105 τ=106 (b) ϵ = 0.05, β = 106 Figure 1: (a) Abstract game exploitability of AS proﬁles for τ = 1000 after 1012 nodes visited in 2-NL Hold’em(30). (b) Log-log plot of abstract game exploitability over the number of nodes visited by AS with ϵ = 0.05 and β = 106 in 2-NL Hold’em(30). For both ﬁgures, units are in milli-big-blinds per hand (mbb/g) and data points are averaged over ﬁve runs with different random seeds. Error bars in (b) indicate 95% conﬁdence intervals. Main results. We now compare AS to CS, ES, and OS in both 2-NL Hold’em(k) and Bluff(D1, D2). Similar to Lanctot et al. [9], our OS implementation is ϵ-greedy so that the current player i samples a single action at random with probability ϵ = 0.5, and otherwise samples a single action according to the current strategy σi. Firstly, we consider two-player 2-NL Hold’em(k) with starting stacks of k = 20, 22, 24, ..., 38, and 40 chips, for a total of eleven different 2-NL Hold’em(k) games. Again, we apply the same ﬁve-bucket card abstraction as before to keep the games reasonably sized. For each game, we ran each of CS, ES, OS, and AS ﬁve times, measured the abstract game exploitability at a number of checkpoints, and averaged the results. Figure 2a displays the results for 2-NL Hold’em(36), a game with approximately 68 million information sets and 5 billion histories (nodes). Here, AS achieved an improvement of 54% over ES at the ﬁnal data points. In addition, Figure 2b shows the average exploitability in each of the eleven games after approximately 3.16 × 1012 nodes visited by CS, ES, and AS. OS performed much worse and is not shown. Since one can lose more as the starting stacks are increased (i.e., ∆i becomes larger), we “normalized” exploitability across each game by dividing the units on the y-axis by k. While there is little difference between the algorithms for the smaller 20 and 22 chip games, we see a signiﬁcant beneﬁt to using AS over CS and ES for the larger games that contain many player actions. For the most part, the margins between AS, CS, and ES increase with the game size. Figure 3 displays similar results for Bluff(1, 1) and Bluff(2, 1), which contain over 24 thousand and 3.5 million information sets, and 294 thousand and 66 million histories (nodes) respectively. Again, AS converged faster than CS, ES, and OS in both Bluff games tested. Note that the same choices of parameters (ϵ = 0.05, β = 106, τ = 1000) that worked well in 2-NL Hold’em(30) also worked well in other 2-NL Hold’em(k) games and in Bluff(D1, D2). 6 Conclusion This work has established a number of improvements for computing strategies in extensive-form games with CFR, both theoretically and empirically. We have provided new, tighter bounds on the average regret when using vanilla CFR or one of several different MCCFR sampling algorithms. These bounds were derived by showing that a player’s regret is equal to a weighted sum of the player’s cumulative counterfactual regrets (Theorem 4), where the weights are given by a best response to the opponents’ previous sequence of strategies. We then used this bound as inspiration for our new MCCFR algorithm, AS. By sampling a subset of a player’s actions, AS can provide faster 7 10-1 100 101 102 103 104 1010 1011 1012 Abstract game exploitability (mbb/g) Nodes Visited CS ES OS AS (a) 2-NL Hold’em(36) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 106 107 108 Abstract game exploitability (mbb/g) / k Game size (# information sets) k=20 k=30 k=40 CS ES AS (b) 2-NL Hold’em(k), k ∈{20, 22, ..., 40} Figure 2: (a) Log-log plot of abstract game exploitability over the number of nodes visited by CS, ES, OS, and AS in 2-NL Hold’em(36). The initial uniform random proﬁle is exploitable for 6793 mbb/g, as indicated by the black dashed line. (b) Abstract game exploitability after approximately 3.16 × 1012 nodes visited over the game size for 2-NL Hold’em(k) with even-sized starting stacks k between 20 and 40 chips. For both graphs, units are in milli-big-blinds per hand (mbb/g) and data points are averaged over ﬁve runs with different random seeds. Error bars indicate 95% conﬁdence intervals. For (b), units on the y-axis are normalized by dividing by the starting chip stacks. 10-5 10-4 10-3 10-2 10-1 100 107 108 109 1010 1011 1012 1013 Exploitability Nodes Visited CS ES OS AS (a) Bluff(1, 1) 10-5 10-4 10-3 10-2 10-1 100 107 108 109 1010 1011 1012 1013 Exploitability Nodes Visited CS ES OS AS (b) Bluff(2, 1) Figure 3: Log-log plots of exploitability over number of nodes visited by CS, ES, OS, and AS in Bluff(1, 1) and Bluff(2, 1). The initial uniform random proﬁle is exploitable for 0.780 and 0.784 in Bluff(1, 1) and Bluff(2, 1) respectively, as indicated by the black dashed lines. Data points are averaged over ﬁve runs with different random seeds and error bars indicate 95% conﬁdence intervals. convergence rates in games containing many player actions. AS converged faster than previous MCCFR algorithms in all of our test games. For future work, we would like to apply AS to games with many player actions and with more than two players. All of our theory still applies, except that player i’s average strategy is no longer guaranteed to converge to σ∗ i . Nonetheless, AS may still ﬁnd strong strategies faster than CS and ES when it is too expensive to sample all of a player’s actions. Acknowledgments We thank the members of the Computer Poker Research Group at the University of Alberta for helpful conversations pertaining to this work. This research was supported by NSERC, Alberta Innovates – Technology Futures, and computing resources provided by WestGrid and Compute Canada. 8 References [1] Nick Abou Risk and Duane Szafron. Using counterfactual regret minimization to create competitive multiplayer poker agents. In Ninth International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pages 159–166, 2010. [2] Richard Gibson, Marc Lanctot, Neil Burch, Duane Szafron, and Michael Bowling. Generalized sampling and variance in counterfactual regret minimization. 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Fast algorithms for ﬁnding randomized strategies in game trees. In Annual ACM Symposium on Theory of Computing (STOC’94), pages 750–759, 1994. [9] Marc Lanctot, Kevin Waugh, Martin Zinkevich, and Michael Bowling. Monte Carlo sampling for regret minimization in extensive games. In Advances in Neural Information Processing Systems 22 (NIPS), pages 1078–1086, 2009. [10] Marc Lanctot, Kevin Waugh, Martin Zinkevich, and Michael Bowling. Monte Carlo sampling for regret minimization in extensive games. Technical Report TR09-15, University of Alberta, 2009. [11] Martin Zinkevich, Michael Johanson, Michael Bowling, and Carmelo Piccione. Regret minimization in games with incomplete information. Technical Report TR07-14, University of Alberta, 2007. [12] Martin Zinkevich, Michael Johanson, Michael Bowling, and Carmelo Piccione. Regret minimization in games with incomplete information. In Advances in Neural Information Processing Systems 20 (NIPS), pages 905–912, 2008. 9	2012	101
4,455	Fusion with Diffusion for Robust Visual Tracking Yu Zhou1∗, Xiang Bai1, Wenyu Liu1, Longin Jan Latecki2 1 Dept. of Electronics and Information Engineering, Huazhong Univ. of Science and Technology, P. R. China 2 Dept. of Computer and Information Sciences, Temple Univ., Philadelphia, USA {zhouyu.hust,xiang.bai}@gmail.com,liuwy@hust.edu.cn,latecki@temple.edu Abstract A weighted graph is used as an underlying structure of many algorithms like semisupervised learning and spectral clustering. If the edge weights are determined by a single similarity measure, then it hard if not impossible to capture all relevant aspects of similarity when using a single similarity measure. In particular, in the case of visual object matching it is beneﬁcial to integrate different similarity measures that focus on different visual representations. In this paper, a novel approach to integrate multiple similarity measures is proposed. First pairs of similarity measures are combined with a diffusion process on their tensor product graph (TPG). Hence the diffused similarity of each pair of objects becomes a function of joint diffusion of the two original similarities, which in turn depends on the neighborhood structure of the TPG. We call this process Fusion with Diffusion (FD). However, a higher order graph like the TPG usually means signiﬁcant increase in time complexity. This is not the case in the proposed approach. A key feature of our approach is that the time complexity of the diffusion on the TPG is the same as the diffusion process on each of the original graphs. Moreover, it is not necessary to explicitly construct the TPG in our framework. Finally all diffused pairs of similarity measures are combined as a weighted sum. We demonstrate the advantages of the proposed approach on the task of visual tracking, where different aspects of the appearance similarity between the target object in frame t −1 and target object candidates in frame t are integrated. The obtained method is tested on several challenge video sequences and the experimental results show that it outperforms state-of-the-art tracking methods. 1 Introduction The considered problem has a simple formulation: Given are multiple similarities between the same set of n data points, each similarity can be represented as a weighted graph. The goal is to combine them to a single similarity measure that best reﬂects the underlying data manifold. Since the set of nodes is the same, it is easy to combine the graphs into a single weighted multigraph, where there are multiple edges between the same pair of vertices representing different similarities. Then our task can be stated as ﬁnding a mapping from the multigraph to a weighted simple graph whose edge weights best represent the similarity of the data points. Of course, this formulation is not precise, since generally the data manifold is unknown, and hence it is hard to quantify the ’best’. However, it is possible to evaluate the quality of the combination experimentally in many applications, e.g., the tracking performance considered in this paper. There are many possible solutions to the considered problem. One of the most obvious ones is a weighted linear combination of the similarities. However, this solution does not consider the similarity dependencies of different data points. The proposed approach aims to utilize the neighborhood structure of the multigraph in the mapping to the weighted simple graph. ∗Part of this work was done while the author was visiting Temple University 1 Given two different similarity measures, we ﬁrst construct their Tensor Product Graph (TPG). Then we jointly diffuse both similarities with a diffusion process on TPG. However, while the original graphs representing the two measures have n nodes, their TPG has n2 nodes, which signiﬁcantly increases the time complexity of the diffusion on TPG. To address this problem, we introduce an iterative algorithm that operates on the original graphs and prove that it is equivalent to the diffusion on TPG. We call this process Fusion with Diffusion (FD). FD is a generalization of the approached in [26], where only a single similarity measure is considered. While the diffusion process on TPG in [26] is used to enhances a single similarity measure, our approach aims at combining two different similarity measures so that they enhance and constrain each others. Although algorithmically very different, our motivation is similar to co-training style algorithms in [5, 23, 24] where multiple cues are fused in an iterative learning process. The proposed approach is also related to the semi-supervised learning in [6, 7, 21, 28, 29]. For online tracking task, we only have the label information from the current frame, which can be regarded as the labeled data, and the label information in the next frame is unavailable, which can be regarded as unlabeled data. In this context, FD jointly propagates two similarities of the unlabeled data to the labeled data. The obtained new diffused similarity, can be then interpreted as the label probability over the unlabeled data. Hence from the point of view of visual tracking, but in the spirit of semi-supervised learning, our approach utilizes the unlabeled data from the next frame for improved visual similarity to the labeled data representing the tracked objets. Visual tracking is an important issue in computer vision and has many practical applications. The challenges in designing a tracking system are often caused by shape deformation, occlusion, viewpoints variances, and background clutter. Different strategies have been proposed to obtain robust tracking systems. In [8, 12, 14, 16, 25, 27], matching based strategy is utilized. Discriminate appearance model of the target is extracted from the current frame, then the optimal target is estimated based on the distance/similatity between the appearance model and the candidate in the hypothesis set. Classiﬁcation based strategies are introduced in [1, 2, 3, 4, 10, 11]. Tracking task is transformed into foreground and background binary classiﬁcation problem in this framework. [15, 20] try to combine both of those two frameworks. In this paper, we focus on improving the distance/similarity measure to improve the matching based tracking strategy. Our motivation is similar to [12], where metric learning is proposed to improve the distance measure. However, different from [12], multiple cues are fused to improve the similarity in our approach. Moreover, the information from the forthcoming frame is also used to improve the similarity. This leads to more stable tracking performance than in [12]. Multiple cues fusion seem to be an effective way to improve the tracking performance. In [13], multiple feature fusion is implemented based on sampling the state space. In [20], the tracking task is formulated as the combination of different trackers, three different trackers are combined into a cascade. Different from those methods, we combine different similarities into a single similarity measure, which makes our method a more general for integrating various appearance models. In summary, we propose a novel framework for integration of multiple similarity measures into a single consistent similarity measure, where the similarity of each pair of data points depends on their similarity to other data points. We demonstrate its superior performance on a challenging task of tracking by visual matching. 2 Problem Formulation The problem of matching based visual tracking boils down to the following simple formulation. Given the target in frame It−1 which can be represented as image patch I1 enclosing the target, and the set of candidate target patches in frame It, C = {In\| n = 2, ..., N}, the goal is to determine which patch in C corresponds to the target in frame It−1. Of course, one can make this setting more complicated, e.g., by considering more frames, but we consider this simple formulation in this paper. The candidate set C is determined by the motion model, which is particularly simple in our setting. The size of all the image patches is ﬁxed and the candidate set is composed of patches in frame It inside a search radius r, i.e. \|\|c(In) −c(I1)\|\| < r, where c is the 2-D coordinate of center position of the image patch. 2 Let S be a similarity measure deﬁned on the set of the image patches V = {I1} ∪C, i.e., S is a function from V × V into positive real numbers. Then our tracking goal can be formally stated as ˆI = arg max X∈C S(I1, X) (1) meaning that the patch in C with most similar appearance to patch I1 is selected as the target location in frame t. Since the appearance of the target object changes, e.g., due to motion and lighting changes, single similarity measure is often not sufﬁcient to identify the target in the next frame. Therefore, we consider a set of similarity measures S = {S1, . . . SQ}, each Sα deﬁned on V × V for α = 1, . . . , Q. For example, in our experimental results, each image patch is represented with three histograms based on three different features, HOG[9], LBP[18], Haar-like feature[4], which lead to three different similarity measures. In other words, each pair of patches can be compared with respect to three different appearance features. We can interpret each similarity measure Sα as the afﬁnity matrix of a graph Gα whose vertex set is V , i.e., Sα a N × N matrix with positive entries, where N is the cardinality of V . Then we can combine the graphs Gα into a single multigraph whose edge weights corresponds to different similarity measures Sα. However, in order to solve Eq. (1), we need a single similarity measure S. Hence we face a question how to combine the measures in S into a single similarity measure. We propose a two stage approach to answer this question. First, we combine pairs of similarity measures Sα and Sβ into a single measure P∗ α,β, which is a matrix of size N × N. P∗ α,β is deﬁned in Section 3 and it is obtained with the proposed process called fusion with diffusion. In the second stage we combine all P∗ α,β for α, β = 1, . . . Q into a single similarity measure S deﬁned as a weighted matrix sum S = X α,β ωαωβP∗ α,β (2) where ωα and ωβ are positive weights associated with measures Sα and Sβ deﬁned in Section 5. We also observe that in contrast to many tracking by matching methods, the combined measure S is not only a function of similarities between I1 and the candidate patches in C, but also of similarities of patches in C to each other. 3 Fusion with Diffusion 3.1 Single Graph on Consecutive Frames Given a single graph Gα = (V, Sα), a reversible Markov chain on V can be constructed with the transition probability deﬁned as Pα(i, j) = Sα(i, j)/Di (3) where Di = PN j=1 Sα(i, j) is the degree of each vertex. Then the transition probability Pα(i, j) inherits the positivity-preserving property PN j=1 Pα(i, j) = 1, i = 1, ..., N. The graph Gα is fully connected graph in many applications. To reduce the inﬂuence of noisy points, i.e., cluttered background patches in tracking, a local transition probability is used: (Pk,α)(i, j) = Pα(i, j) j ∈kNN(i) 0 otherwise (4) Hence the number of non-zero elements in each row is not larger than k, which implies Pn j=1(Pk,α)(i, j) < 1. This inequality is important in our framework, since it guarantees the converge of the diffusion process on the tensor product graph presented in the next section. 3.2 Tensor Product Graph of Two Similarities Given are two graphs Gα = (V, Pk,α) and Gβ = (V, Pk,β) deﬁned in Sec. 3.1, we can deﬁne their Tensor Product Graph (TPG) as Gα ⊗Gβ = (V × V, P), (5) 3 where P = Pk,α ⊗Pk,β is the Kronecker product of matrices deﬁned as P(a, b, i, j) = Pk,α(a, b) Pk,β(i, j). Thus, each entry of P relates four image patches. When Pk,α and Pk,β are two N × N matrices, then P is a N 2 × N 2 matrix. However, as we will see in the next subsection, we actually never compute P explicitly. 3.3 Diffusion Process on Tensor Product Graph We utilize a diffusion process on TPG to combine the two similarity measures Pk,α and Pk,β. We begin with some notations. The vec operator creates a column vector from a matrix M by stacking the column vectors of M below one another. More formally vec : RN×N →RN 2 is deﬁned as vec(M)g = (M)ij, where i = ⌊(g −1)/N⌋+ 1 and j = g mod N. The inverse operator vec−1 that maps a vector into a matrix is often called the reshape operator. We deﬁne a diagonal N × N matrix as ∆(i, i) = 1 i = 1 0 otherwise, (6) Only the entry representing the patch I1 is set to one and all other entries are set to zero in ∆. We observe that P is the adjacency matrix of TPG Gα ⊗Gβ. We deﬁne a q-th iteration of the diffusion process on this graph as q X e=0 (P)evec(∆). (7) As shown in [26], this iterative process is guaranteed to converge to a nontrivial solution given by lim q→∞ q X e=0 (P)evec(∆) = (I −P)−1vec(∆), (8) where I is a identity matrix. Following [26], we deﬁne P∗ α,β = P∗= vec−1((I −P)−1vec(∆)) (9) We observe that our solution P∗is a N × N matrix. We call the diffusion process to compute P∗a Fusion with Diffusion (FD) process, since diffusion on TPG Gα ⊗Gβ is used to fuse two similarity measures Sα and Sβ. Since P is a N 2 × N 2 matrix, FD process on TPG as deﬁned in Eq. (7) may be computationally too demanding. To compute P∗effectively, instead of diffusing on TPG directly, we show in Section 3.4 that FD process on TPG is equivalent to an iterative process on N ×N matrices only. Consequently, instead of an O(n6) time complexity, we obtain an O(n3) complexity. Then in Section 4 we further reduce it to an efﬁcient algorithm with time complexity O(n2), which can be used in real time tracking algorithms. 3.4 Iterative Algorithm for Computing P∗ We deﬁne P1 = P(k,α)P T (k,β) and Pq+1 = Pk,α(Pk,α)q(P T k,β)qP T k,β + ∆. (10) We iterate Eq.(10) until convergence, and as we prove in Proposition 1, we obtain P∗= lim q→∞Pq=vec−1((I −P)−1vec(∆)) (11) The iterative process in Eq.(10) is a generalization of the process introduced in [26]. Consequently, the following properties are simple extensions of the properties derived in [26]. However, we state them explicitly, since we combine two different afﬁnity matrices, while [26] considers only a single matrix. In other words, we consider diffusion on TPG of two different graphs, while diffusion on TPG of a single graph with itself is considered in [26]. Proposition 1 vec lim q→∞P(q+1) = lim q→∞ q−1 X e=0 Pevec(∆) = (I −P)−1vec(∆). (12) 4 Proof: Eq.(10) can be rewritten as P(q+1) = Pk,α (Pk,α)q(P T k,β)q P T k,β + ∆ = Pk,α[Pk,α (Pk,α)(q−1)(P T k,β)(q−1) P T k,β + ∆]P T k,β + ∆ = (Pk,α)2 (Pk,α)(q−1)(P T k,β)(q−1) (P T k,β)2 + Pk,α ∆Pk,β + ∆ = · · · = (Pk,α)q Pk,αP T k,β (P T k,β)q + (Pk,α)q−1 ∆(P T k,β)q−1 + · · · + ∆ = (Pk,α)q Pk,αP T k,β (P T k,β)q + q−1 X e=0 (Pk,α)e ∆(P T k,β)e (13) Lemma 1 limq→∞(Pk,α)q Pk,αP T k,β (P T k,β)q = 0 Proof: It sufﬁces to show that (Pk,α)q and (P T k,β)q go to 0, when q →∞. This is true if and only if every eigenvalue of Pk,α and Pk,β is less than one in absolute value. Since Pk,α and Pk,β has nonnegative entries, this holds if its row sums are all less than one. As described in Sec.3.1, we have PN b=1(Pk,α)a,b < 1 and PN j=1(Pk,β)i,j < 1. Lemma 1 shows that the ﬁrst summand in Eq.(13) converges to zero, and consequently we have lim q→∞P(q+1) = lim q→∞ q−1 X e=0 (Pk,α)e ∆(P T k,β)e. (14) Lemma 2 vec (Pk,α)e ∆(P T k,β)e = (P)evec(∆) for e = 1, 2, . . . . Proof: Our proof is by induction. Suppose (P)lvec(∆)=vec (Pk,α)l ∆(P T k,β)l is true for e = l, then for e = l + 1 we have (P)l+1vec(∆) = P Plvec(∆) = vec Pk,α vec−1(Plvec(∆)) P T k,β = vec Pk,α ((Pk,α)l ∆(P T k,β)l) P T k,β = vec (Pk,α)l+1 ∆(P T k,β)l+1 and the proof of Lemma 2 is complete. By Lemma 1 and Lemma 2, we obtain that vec q−1 X e=0 (Pk,α)e ∆(P T k,β)e ! = q−1 X e=0 (P)evec(∆). (15) The following useful identity holds for the Kronecker Product [22]: vec(Pk,β∆P T k,α) = (Pk,α ⊗Pk,β)vec(∆) = (P)vec(∆) (16) Putting together (14), (15), (16), we obtain vec lim q→∞P(q+1) = vec lim q→∞ q−1 X e=0 (Pk,α)e ∆(P T k,β)e ! (17) = lim q→∞ q−1 X e=0 Pevec(∆) = (I −P)−1vec(∆)=vec(P∗). (18) This proves Proposition 1. We now show how FD could improve the original similarity measures. Suppose we have two similarity measures Sα and Sβ. I1 denotes the image patch enclosing the target in frame t−1. According to Sα, there are many patches in frame t that have nearly equal similarity to I1 with patch In being most similar to I1, while according to Sβ, I1 is clearly more similar to Im in frame t. Then the proposed diffusion will enhance the similarity Sβ(I1, Im), since it will propagate faster the Sβ similarity of I1 to Im than to the other patches. In contrast, the Sα similarities will propagate with similar speed. Consequently, the ﬁnal joint similarity P∗will have Im as the most similar to I1. 5 Algorithm 1: Iterative Fusion with Diffusion Process Input: Two matrices Pk,α, Pk,β ∈RN×N Output: Diffusion result P∗∈RN×N 1 Compute P∗= ∆. 2 Compute uα = ﬁrst column of Pk,α, uβ = ﬁrst column of Pk,β 3 Compute P∗←P∗+ uαuT β . 4 for i = 2, 3, . . . do 5 Compute uα ←Pk,αuα 6 Compute uβ ←Pk,βuβ 7 Compute P∗←P∗+ uαuT β 8 end 4 FD Algorithm To effectively compute P∗, we propose an iterative algorithm that takes the advantage of the structure of matrix ∆. Let uα be a N ×1 vector containing the ﬁrst column of Pk,α. We write Pk,α = [uα\|R] and Pk,α∆= [uα\|0]. It follows then that Pk,α ∆P T k,β = uαuT β . Furthermore, if we denote (Pk,α)j ∆(P T k,β)j = uα,juT β,j, with uα,j being N × 1, and uT β,j being 1 × N, it follows that P j+1 k,α ∆(P T k,β)j+1 = Pk,α(P j k,α ∆(P T k,β)j)P T k,β = Pk,αuα,juT β,jP T k,β = (Pk,αuα,j)(Pk,βuβ,j)T = uα,j+1uT β,j+1. Hence, we replaced one of the two N × N matrix products with one matrix product between an N × N matrix and N × 1 vector, and the other with a product of an N × 1 by an 1 × N vector. This reduces the complexity of our algorithm from O(n3) to O(n2). The ﬁnal algorithm is shown in Alg. 1. 5 Weight Estimation The weight ωα of measure Sα is proportional to how well Sα is able to distinguish the target I1 in frame It−1 from the background surrounding the target. Let {Bh\| h = 1, ..., H} be a set of background patches surrounding the target I1 in frame It−1. The weight of Sα is deﬁned as ωα = 1 1 H PH h=1 Sα(I1, Bh) (19) Thus, the smaller the values of Sα, the larger is the weight ωα. The weights of all similarity measures are normalized so that PQ α=1 ωα = 1. The weights are computed for every frame in order to accommodate appearance changes of the tracked object. 6 Experimental Results We validate our tracking algorithm on eight challenging videos from [4] and [17]: Sylvester, Coke Can, Tiger1, Cliff Bar, Coupon Book, Surfer, and Tiger2, PETS01D1. We compare our method with six famous state-of-the-art tracking algorithms including Multiple Instance Learning tracker (MIL) [4], Fragment tracker(Frag) [1], IVT [19], Online Adaboost tracker (OAB) [10], SemiBoost tracker (Semi) [11], Mean-Shift (MS) tracker, and a simple weighted linear sum of multiple cues (Linear). For the comparison methods, we run source code of Semi, Frag, MIL, IVT and OAB supplied by the authors on the testing videos and use the parameters mentioned in their papers directly. For MS, we implement it based on OpenCV. For Linear, we use three kinds of image features to get the afﬁnity and then simply calculate the average afﬁnity and use the diffusion process mentioned in [26]. Note that all the parameters in our algorithm were ﬁxed for all the experiments. In our experiments, HOG[9], LBP[18] and Haar-like[4] features are used to represent the image patches. Hence each pair of patches is compared with three different similarities based on histograms 6 0 50 100 150 200 250 300 0 20 40 60 80 100 120 140 Coke Can Frame # Center Location Error (pixel) MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our 0 50 100 150 200 250 300 350 0 50 100 150 Cliff Bar Frame # Center Location Error (pixel) MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our 0 50 100 150 200 250 300 350 0 20 40 60 80 100 120 140 160 180 Coupon Book Frame # Center Location Error (pixel) MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our 0 200 400 600 800 1000 1200 1400 0 50 100 150 200 250 Sylvester Frame # Center Location Error (pixel) MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 450 Surfer Frame # Center Location Error (pixel) MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our 0 50 100 150 200 250 300 350 400 0 20 40 60 80 100 120 140 Tiger1 Frame # Center Location Error (pixel) MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our 0 50 100 150 200 250 300 350 400 0 20 40 60 80 100 120 140 160 180 Tiger2 Frame # Center Location Error (pixel) MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our 0 50 100 150 200 250 300 350 400 450 0 100 200 300 400 500 600 PETS01D1 Frame # Center Location Error (pixel) MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our Figure 1: Center Location Error (CLE) versus frame number of HOG, LBP, and Haar-like feature. For the experimental parameters, we set r = 15 pixels, H = 300, k = 12 and the iteration number in Alg. 1 is set to 200. To impartially and comprehensively compare our algorithm with other state-of-the-art trackers, we used two kinds of quantitative comparisons Average Center Location Error (ACLE) and Precision Score [4]. The results are shown in Table 1 and Table 2, respectively. Two kinds of curve evaluation methodologies are also used Center Location Error (CLE) curve and Precision Plots curve1. The results are shown in Fig.1 and Fig.2, respectively. Table 1: Average Center Location Error (ACLE measured in pixels). Red color indicates best performance, Blue color indicates second best, Green color indicates the third best Video MS OAB IVT Semi Frag1 Frag2 Frag3 MIL Linear our Coke Can 43.7 25.0 37.3 40.5 69.1 69.0 34.1 31.9 16.8 15.4 Cliff Bar 43.8 34.6 47.1 57.2 34.7 34.0 44.8 14.2 15.0 6.1 Tiger 1 45.5 39.8 50.2 20.9 39.7 26.7 31.1 7.6 23.8 6.9 Tiger2 47.6 13.2 98.5 39.3 38.6 38.8 51.9 20.6 6.5 5.7 Coup. Book 20.0 17.7 32.2 65.1 55.9 56.1 67.0 19.8 13.6 6.5 Sylvester 20.0 35.0 96.1 21.0 23.0 12.2 10.1 11.4 10.5 9.3 Surfer 17.0 13.4 19.0 9.3 140.1 139.8 138.6 7.7 6.5 5.5 PETS01D1 18.1 7.1 241.8 158.9 6.7 7.2 9.5 11.7 245.4 6.0 Table 2: Precision Score (precision at the ﬁxed threshold of 15). Red color indicates best performance, Blue color indicates second best, Green color indicates the third best. Video MS OAB IVT Semi Frag1 Frag2 Frag3 MIL Linear our Coke Can 0.11 0.21 0.15 0.18 0.09 0.09 0.17 0.24 0.36 0.46 Cliff Bar 0.08 0.21 0.19 0.34 0.20 0.23 0.12 0.79 0.52 0.95 Tiger 1 0.05 0.17 0.03 0.52 0.21 0.38 0.38 0.90 0.54 0.91 Tiger 2 0.06 0.65 0.01 0.44 0.09 0.09 0.12 0.66 0.89 0.95 Coupon Book 0.16 0.18 0.21 0.41 0.39 0.39 0.39 0.23 0.53 1.00 Sylvester 0.46 0.30 0.06 0.53 0.72 0.78 0.81 0.76 0.86 0.90 Surfer 0.59 0.61 0.40 0.89 0.19 0.21 0.23 0.93 1.00 1.00 PETS01D1 0.38 1.00 0.01 0.29 0.99 0.97 0.95 0.80 0.02 1.00 Comparison to matching based methods: MS, IVT, Frag and Linear are all matching based tracking algorithms. In MS, famous Bhattacharyya coefﬁcient is used to measure the distance between histogram distributions; for Frag, we test it under three different measurement strategies: the 1More details about the meaning of Precision Plots can be found in [4] 7 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Cliff Bar Threshold Precision MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Coke Can Threshold Precision MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Coupon Book Threshold Precision MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sylvester Threshold Precision MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Surfer Threshold Precision MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PETS01D1 Threshold Precision MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Tiger1 Threshold Precision MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Tiger2 Threshold Precision MS Frag(KS) Frag(EMD) Frag(Chi) IVT Linear Our Figure 2: Precision Plots. The threshold is set to 15 in our experiments. Kolmogorov-Smirnov statistic, EMD, and Chi-Square distance, represented as Frag1, Frag2, Frag3 in Table 1 and Table 2, respectively. For Linear Combination, the average similarity is used and the diffusion process in [26] is used to improve the similarity measure. Our FD approach clearly outperforms the other approaches, as shown in Table1 and Table2. Our tracking results achieve the best performance in all the testing videos, especially for the Precision Plots shown in Table 2. Even though we set the threshold to 15, which is more challenging for all the trackers, we still get three 1.00 scores. In some videos like sylvester and PETS01D1, Frag achieves comparable results with our method, but it works badly in other videos which means that speciﬁc distance measure can only work on some special cases but our fusion framework is robust for all the challenges that appear in the videos. Our method is always batter than Linear Combination, which means that the fusion with diffusion can really improve the tracking performance. The stability of our method can be clearly seen in the plots of location error as the function of frame number in Fig.1. Our tracking results are always stable, which means that we do not lose the target in the whole tracking process. This is also reﬂected in the fact that our Precision is always batter than all the other methods under different thresholds as shown in Fig.2. Comparison to classiﬁcation based methods: MIL and OAB are both classiﬁcation based tracking algorithms. For OAB, on-line Adaboost is used to train the classiﬁer for the foreground and background classiﬁcation. MIL combines multiple instance learning with on-line Adaboost. Haar-like features are used in both methods. Again our method outperforms those two methods as can be seen in Table1 and Table 2. Comparison to semi-supervised learning based methods: SemiBoost combines semi-supervised learning with on-line Adaboost. Our method is also similar to semi-supervised learning for we build the graph model on consecutive frames, which means that both of our method and SemiBoost use the information from the forthcoming frame. Our method is always better than SemiBoost as shown in Table 1 and Table 2. 7 Conclusions In this paper, a novel Fusion with Diffusion process is proposed for robust visual tracking. Pairs of similarity measures are fused into a single similarity measure with a diffusion process on the tensor product of two graphs determined by the two similarity measures. The proposed method has time complexity of O(n2), which makes it suitable for real time tracking. It is evaluated on several challenging videos, and it signiﬁcantly outperforms a large number of state-of-the-art tracking algorithms. Acknowledgments We would like to thank all the authors for releasing their source codes and testing videos, since they made our experimental evaluation possible. This work was supported by NSF Grants IIS-0812118, BCS-0924164, OIA-1027897, and by the National Natural Science Foundation of China (NSFC) Grants 60903096, 61222308 and 61173120. 8 References [1] A. Adam, E. Rivlin, and I. Shimshoni. Robust fragment-based tracking using the integral histogram. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition(CVPR), pages 798–805, 2006. [2] S. Avidan. Support vector tracking. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(8):1064–1072, 2004. [3] S. Avidan. Ensemble tracking. IEEE Transactions on Pattern Analysis and Machine Intelligence, 29(2):261–271, 2007. [4] B. Babenko, M. Yang, and S. Belongie. Robust object tracking with online multiple instance learning. 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4,456	Convolutional-Recursive Deep Learning for 3D Object Classiﬁcation Richard Socher, Brody Huval, Bharath Bhat, Christopher D. Manning, Andrew Y. Ng Computer Science Department, Stanford University, Stanford, CA 94305, USA richard@socher.org, {brodyh,bbhat,manning}@stanford.edu, ang@cs.stanford.edu Abstract Recent advances in 3D sensing technologies make it possible to easily record color and depth images which together can improve object recognition. Most current methods rely on very well-designed features for this new 3D modality. We introduce a model based on a combination of convolutional and recursive neural networks (CNN and RNN) for learning features and classifying RGB-D images. The CNN layer learns low-level translationally invariant features which are then given as inputs to multiple, ﬁxed-tree RNNs in order to compose higher order features. RNNs can be seen as combining convolution and pooling into one efﬁcient, hierarchical operation. Our main result is that even RNNs with random weights compose powerful features. Our model obtains state of the art performance on a standard RGB-D object dataset while being more accurate and faster during training and testing than comparable architectures such as two-layer CNNs. 1 Introduction Object recognition is one of the hardest problems in computer vision and important for making robots useful in home environments. New sensing technology, such as the Kinect, that can record high quality RGB and depth images (RGB-D) has now become affordable and could be combined with standard vision systems in household robots. The depth modality provides useful extra information to the complex problem of general object detection [1] since depth information is invariant to lighting or color variations, provides geometrical cues and allows better separation from the background. Most recent methods for object recognition with RGB-D images use hand-designed features such as SIFT for 2d images [2], Spin Images [3] for 3D point clouds, or speciﬁc color, shape and geometry features [4, 5]. In this paper, we introduce the ﬁrst convolutional-recursive deep learning model for object recognition that can learn from raw RGB-D images. Compared to other recent 3D feature learning methods [6, 7], our approach is fast, does not need additional input channels such as surface normals and obtains state of the art results on the task of detecting household objects. Fig. 1 outlines our approach. Code for training and testing is available at www.socher.org. Our model starts with raw RGB and depth images and ﬁrst separately extracts features from them. Each modality is ﬁrst given to a single convolutional neural net layer (CNN, [8]) which provides useful translational invariance of low level features such as edges and allows parts of an object to be deformable to some extent. The pooled ﬁlter responses are then given to a recursive neural network (RNN, [9]) which can learn compositional features and part interactions. RNNs hierarchically project inputs into a lower dimensional space through multiple layers with tied weights and nonlinearities. We also explore new deep learning architectures for computer vision. Our previous work on RNNs in natural language processing and computer vision [9, 10] (i) used a different tree structure for each input, (ii) employed a single RNN with one set of weights, (iii) restricted tree structures to be strictly 1 Multiple RNN Multiple RNN Convolution Label: Coffee Mug Filter Responses get pooled K K K filters 4 pooling regions RGB CNN Merging of pooled vectors Merging of pooled vectors K Depth CNN Convolution K Softmax Classifier Figure 1: An overview of our model: A single CNN layer extracts low level features from RGB and depth images. Both representations are given as input to a set of RNNs with random weights. Each of the many RNNs (around 100 for each modality) then recursively maps the features into a lower dimensional space. The concatenation of all the resulting vectors forms the ﬁnal feature vector for a softmax classiﬁer. binary, and (iv) trained the RNN with backpropagation through structure [11, 12]. In this paper, we expand the space of possible RNN-based architectures in these four dimensions by using ﬁxed tree structures and multiple RNNs on the same input and allow n-ary trees. We show that because of the CNN layer, ﬁxing the tree structure does not hurt performance and it allows us to speed up recognition. Similar to recent work [13, 14] we show that performance of RNN models can improve with an increasing number of features. The hierarchically composed RNN features of each modality are concatenated and given to a joint softmax classiﬁer. Most importantly, we demonstrate that RNNs with random weights can also produce high quality features. So far random weights have only been shown to work for convolutional neural networks [15, 16]. Because the supervised training reduces to optimizing the weights of the ﬁnal softmax classiﬁer, a large set of RNN architectures can quickly be explored. By combining the above ideas we obtain a state of the art system for classifying 3D objects which is very fast to train and highly parallelizable at test time. We ﬁrst brieﬂy describe the unsupervised learning of ﬁlter weights and their convolution to obtain low level features. Next we give details of how multiple random RNNs can be used to obtain high level features of the entire image. Then, we discuss related work. In our experiments we show quantitative comparisons of different models, analyze model ablations and describe our state-of-theart results on the RGB-D dataset of Lai et al. [2]. 2 Convolutional-Recursive Neural Networks In this section, we describe our new CNN-RNN model. We ﬁrst learn the CNN ﬁlters in an unsupervised way by clustering random patches and then feed these patches into a CNN layer. The resulting low-level, translationally invariant features are given to recursive neural networks. RNNs compose higher order features that can then be used to classify the images. 2.1 Unsupervised Pre-training of CNN Filters We follow the procedure described by Coates et al. [13] to learn ﬁlters which will be used in the convolution. First, random patches are extracted into two sets, one for each modality (RGB and depth). Each set of patches is then normalized and whitened. The pre-processed patches are clustered by simply running k-means. Fig. 2 shows the resulting ﬁlters for both modalities. They capture standard edge and color features. One interesting result when applying this method to the depth channel is that the edges are much sharper. This is due to the large discontinuities between object boundaries and the background. While the depth channel is often quite noisy most of the features are still smooth. 2 Filters: RGB Depth Gray scale Figure 2: Visualization of the k-means ﬁlters used in the CNN layer after unsupervised pre-training: (left) Standard RGB ﬁlters (best viewed in color) capture edges and colors. When the method is applied to depth images (center) the resulting ﬁlters have sharper edges which arise due to the strong discontinuities at object boundaries. The same is true, though to a lesser extent, when compared to ﬁlters trained on gray scale versions of the color images (right). 2.2 A Single CNN Layer To generate features for the RNN layer, a CNN architecture is chosen for its translational invariance properties. The main idea of CNNs is to convolve ﬁlters over the input image in order to extract features. Our single layer CNN is similar to the one proposed by Jarrett et. al [17] and consists of a convolution, followed by rectiﬁcation and local contrast normalization (LCN). LCN was inspired by computational neuroscience and is used to contrast features within a feature map, as well as across feature maps at the same spatial location [17, 18, 14]. We convolve each image of size (height and width) dI with K square ﬁlters of size dP , resulting in K ﬁlter responses, each of dimensionality dI −dP + 1. We then average pool them with square regions of size dℓand a stride size of s, to obtain a pooled response with width and height equal to r = (dI −dℓ)/s + 1. So the output X of the CNN layer applied to one image is a K × r × r dimensional 3D matrix. We apply this same procedure to both color and depth images separately. 2.3 Fixed-Tree Recursive Neural Networks The idea of recursive neural networks [19, 9] is to learn hierarchical feature representations by applying the same neural network recursively in a tree structure. In our case, the leaf nodes of the tree are K-dimensional vectors (the result of the CNN pooling over an image patch repeated for all K ﬁlters) and there are r2 of them. In our previous RNN work [9, 10, 20] the tree structure depended on the input. While this allows for more ﬂexibility, we found that for the task of object classiﬁcation in conjunction with a CNN layer it was not necessary for obtaining high performance. Furthermore, the search over optimal trees slows down the method considerably as one can not easily parallelize the search or make use of parallelization of large matrix products. The latter could beneﬁt immensely from new multicore hardware such as GPUs. In this work, we focus on ﬁxed-trees which we can design to be balanced. Previous work also only combined pairs of vectors. We generalize our RNN architecture to allow each layer to merge blocks of adjacent vectors instead of only pairs. We start with a 3D matrix X ∈RK×r×r for each image (the columns are K-dimensional). We deﬁne a block to be a list of adjacent column vectors which are merged into a parent vector p ∈RK. In the following we use only square blocks for convenience. Blocks are of size K × b × b. For instance, if we merge vectors in a block with b = 3, we get a total size 128 × 3 × 3 and a resulting list of vectors (x1, . . . , x9). In general, we have b2 many vectors in each block. The neural network 3 for computing the parent vector is p = f   W   x1 ... xb2     , (1) where the parameter matrix W ∈RK×b2K, f is a nonlinearity such as tanh. We omit the bias term which turns out to have no effect in the experiments below. Eq. 1 will be applied to all blocks of vectors in X with the same weights W. Generally, there will be (r/b)2 many parent vectors p, forming a new matrix P1. The vectors in P1 will again be merged in blocks just as those in matrix X using Eq. 1 with the same tied weights resulting in matrix P2. This procedure continues until only one parent vector remains. Fig. 3 shows an example of a pooled CNN output of size K × 4 × 4 and a RNN tree structure with blocks of 4 children. p3 p4 W W W W W K filters p2 p1 p x3 x4 x2 x1 Figure 3: Recursive Neural Network applied to blocks: At each node, the same neural network is used to compute the parent vector of a set of child vectors. The original input matrix is the output of a pooled convolution. The model so far has been unsupervised. However, our original task is to classify each block into one of many object categories. Therefore, we use the top vector Ptop as the feature vector to a softmax classiﬁer. In order to minimize the cross entropy error of the softmax, we could backpropagate through the recursive neural network [12] and convolutional layers [8]. In practice, this is very slow and we will discuss alternatives in the next section. 2.4 Multiple Random RNNs Previous work used only a single RNN. We can actually use the 3D matrix X as input to a number of RNNs. Each of N RNNs will output a Kdimensional vector. After we forward propagate through all the RNNs, we concatenate their outputs to a NK-dimensional vector which is then given to the softmax classiﬁer. Instead of taking derivatives of the W matrices of the RNNs which would require backprop through structure [11], we found that even RNNs with random weights produce high quality feature vectors. Similar results have been found for random weights in the closely related CNNs [16]. Before comparing to other approaches, we brieﬂy review related work. 3 Related Work There has been great interest in object recognition and scene understanding using RGB-D data. Silberman and Fergus have published a 3D dataset for full scene understanding [21]. Koppula et al. also recently provided a new dataset for indoor scene segmentation [4]. The most common approach today for standard object recognition is to use well-designed features based on orientation histograms such as SIFT, SURF [22] or textons and give them as input to a classiﬁer such as a random forest. Despite their success, they have several shortcomings such as being only applicable to one modality (grey scale images in the case of SIFT), not adapting easily to new modalities such as RGB-D or to varying image domains. There have been some attempts to modify these features to colored images via color histograms [23] or simply extending SIFT to the depth channel [2]. More advanced methods that generalize these ideas and can combine several important RGB-D image characteristics such as size, 3D shape and depth edges are kernel descriptors [5]. 4 Another related line of work is about spatial pyramids in object classiﬁcation, in particular the pyramid matching kernel [24]. The similarity is mostly in that our model also learns a hierarchical image representation that can be used to classify objects. Another solution to the above mentioned problems is to employ unsupervised feature learning methods [25, 26, 27] (among many others) which have made large improvements in object recognition. While many deep learning methods exist for learning features from rgb images, few deep learning architectures have yet been investigated for 3D images. Very recently, Blum et al. [6] introduced convolutional k-means descriptors (CKM) for RGB-D data. They use SURF interest points and learn features using k-means similar to [28]. Their work is similar to ours in that they also learn features in an unsupervised way. Very recent work by Bo et al. [7] uses unsupervised feature learning based on sparse coding to learn dictionaries from 8 different channels including grayscale intensity, RGB, depth scalars, and surface normals. Features are then used in hierarchical matching pursuit which consists of two layers. Each layer has three modules: batch orthogonal matching pursuit, pyramid max pooling, and contrast normalization. This results in a very large feature vector size of 188,300 dimensions which is used for classiﬁcation. Lastly, recursive autoencoders have been introduced by Pollack [19] and Socher et al. [10] to which we compare quantitatively in our experiment section. Recursive neural networks have been applied to full scene segmentation [9] but they used hand-designed features. Farabet et al. [29] also introduce a model for scene segmentation that is based on multi-scale convolutional neural networks and learns feature representations. 4 Experiments All our experiments are carried out on the recent RGB-D dataset of Lai et al. [2]. There are 51 different classes of household objects and 300 instances of these classes. Each object instance is imaged from 3 different angles resulting in roughly 600 images per instance. The dataset consists of a total of 207,920 RGB-D images. We subsample every 5th frame of the 600 images resulting in a total of 120 images per instance. In this work we focus on the problem of category recognition and we use the same setup as [2] and the 10 random splits they provide. All development is carried out on a separate split and model ablations are run on one of the 10 splits. For each split’s test set we sample one object from each class resulting in 51 test objects, each with about 120 independently classiﬁed images. This leaves about 34,000 images for training our model. Before the images are given to the CNN they are resized to be dI = 148. Unsupervised pre-training for CNN ﬁlters is performed for all experiments by using k-means on 500,000 image patches randomly sampled from each split’s training set. Before unsupervised pretraining, the 9 × 9 × 3 patches for RGB and 9 × 9 patches for depth are individually normalized by subtracting the mean and divided by the standard deviation of its elements. In addition, ZCA whitening is performed to de-correlate pixels and get rid of redundant features in raw images [30]. A valid convolution is performed with ﬁlter bank size K = 128 and ﬁlter width and height of 9. Average pooling is then performed with pooling regions of size dℓ= 10 and stride size s = 5 to produce a 3D matrix of size 128 × 27 × 27 for each image. Each RNN has non-overlapping child sizes of 3 × 3 applied spatially. This leads to the following matrices at each depth of the tree: X ∈R128×27×27 to P1 ∈R128×9×9 to P2 ∈R128×3×3 to ﬁnally P3 ∈R128. We use 128 randomly initialized RNNs in both modalities. The combination of RGB and depth is done by concatenating the ﬁnal features which have 2 × 1282 = 32, 768 dimensions. 4.1 Comparison to Other Methods In this section we compare our model to related models in the literature. Table 1 lists the main accuracy numbers and compares to the published results [2, 5, 6, 7]. Recent work by Bo et al. [5] investigates multiple kernel descriptors on top of various features, including 3D shape, physical size of the object, depth edges, gradients, kernel PCA, local binary patterns,etc. In contrast, all our features are learned in an unsupervised way from the raw color and depth images. Blum et al. [6] 5 Classiﬁer Extra Features for 3D;RGB 3D RGB Both Linear SVM [2] Spin Images, efﬁcient match kernel (EMK), random Fourier sets, width, depth, height; SIFT, EMK, texton histogram, color histogram 53.1±1.7 74.3±3.3 81.9±2.8 Kernel SVM [2] same as above 64.7±2.2 74.5±3.1 83.9±3.5 Random Forest [2] same as above 66.8±2.5 74.7±3.6 79.6±4.0 SVM [5] 3D shape, physical size of the object, depth edges, gradients, kernel PCA, local binary patterns,multiple depth kernels 78.8±2.7 77.7±1.9 86.2±2.1 CKM [6] SURF interest points – – 86.4±2.3 SP+HMP [7] surface normals 81.2±2.3 82.4±3.1 87.5±2.9 CNN-RNN – 78.9±3.8 80.8±4.2 86.8±3.3 Table 1: Comparison of our CNN-RNN to multiple related approaches. We outperform all approaches except that of Bo et al. which uses an extra input modality of surface normals. also learn feature descriptors and apply them sparsely to interest points. We outperform all methods except that of Bo et al. [7] who perform 0.7% better with a ﬁnal feature vector that requires ﬁve times the amount of memory compared to ours. They make additional use of surface normals and gray scale images on top of RGB and depth channels and also learn features from these inputs with unsupervised methods based on sparse coding. Sparse coding is known to not scale well in terms of speed for large input dimensions [31]. 4.2 Model Analysis We analyze our model through several ablations and model variations. We picked one of the splits as our development fold and focus on RGB images and RNNs with random weights only unless otherwise noted. Two layer CNN. Fig. 4 (left) shows a comparison between our CNN-RNN model and a two layer CNN. We compare a previously recommended architecture for CNNs [17] and one which uses ﬁlters trained with k-means. In both settings, the CNN-RNN outperforms the two layer CNN. Because it also requires many fewer matrix multiplication, it is approximately 4× faster in our experiments compared to a second CNN layer. However, the main bottleneck of our method is still the ﬁrst CNN layer. Both models could beneﬁt from fast GPU implementations [32, 33]. Tree structured neural nets with untied weights. Fig. 4 (left) also gives results when the weights of the random RNNs are untied across layers in the tree (TNN). In other words, different random weights are used at each depth of the tree. Since weights are still tied inside each layer this setting can be seen as a convolution where the stride size is equal to the ﬁlter size. We call this a tree neural network (TNN) because it is technically not a recursive neural network. While this results in a large increase in parameters, it actually hurts performance underlining the fact that tying the weights in RNNs is beneﬁcial. Trained RNN. Another comparison shown in Fig. 4 (left) is between many random RNNs and a single trained RNN. We carefully cross validated the RNN training procedure, objectives (adding reconstruction costs at each layer as in [10], classifying each layer or only at the top node), regularization, layer size etc. The best performance was still lacking compared to 128 random RNNs ( 2% difference) and training time is much longer. With a more efﬁcient GPU-based implementation it might be possible to train many RNNs in the future. Number of random RNNs: Fig. 4 (center) shows that increasing the number of random RNNs improves performance, leveling off at around 64 on this dataset. RGB & depth combinations and features: Fig. 4 (right) shows that combining RGB and depth features from RNNs improves performance. The two modalities complement each other and produce features that are independent enough so that the classiﬁer can beneﬁt from their combination. Global autoencoder on pixels and depth. In this experiment we investigate whether CNN-RNNs learn better features than simply using a single layer of features on raw pixels. Many methods such as those of Coates and Ng [28] show remarkable results with a single very wide layer. The global autoencoder achieves only 61.1%, (it is overﬁtting at 93.3% training accuracy). We cross-validated 6 Filters 2nd Layer Acc. See [17] CNN 77.66 See [17] RNN 77.04 k-means tRNN 78.10 k-means TNN 79.67 k-means CNN 78.65 k-means RNN∗ 80.15 1816 32 64 128 40 50 60 70 80 90 Number of RNNs Accuracy (%) RGB DepthRGB+Depth 70 75 80 85 90 Figure 4: Model analysis on the development split (left and center use rgb only). Left: Comparison of two layer CNN with CNN-RNN with different pre-processing ([17] and [13]). TNN is a tree structured neural net with untied weights across layers, tRNN is a single RNN trained with backpropagation (see text for details). The best performance is achieved with our model of random RNNs (marked with ∗). Center: Increasing the number of random RNNs improves performance. Right: Combining both modalities improves performance to 88% on the development split. over the number of hidden units and sparsity parameters). This shows that even random recursive neural nets can clearly capture more of the underlying class structure in their feature representations than a single layer autoencoder. 4.3 Error Analysis Fig. 5 shows our confusion matrix across all 51 classes. Most model confusions are very reasonable showing that recursive deep learning methods on raw pixels and depth can give provide high quality features. The only class that we consistently misclassify are mushrooms which are very similar in appearance to garlic. Fig. 6 shows 4 pairs of often confused classes. Both garlic and mushrooms have very similar appearances and colors. Water bottles and shampoo bottles in particular are problematic because the IR sensors do not properly reﬂect from see through surfaces. 5 Conclusion We introduced a new model based on a combination of convolutional and recursive neural networks. Unlike previous RNN models, we ﬁx the tree structure, allow multiple vectors to be combined, use multiple RNN weights and keep parameters randomly initialized. This architecture allows for parallelization and high speeds, outperforms two layer CNNs and obtains state of the art performance without any external features. We also demonstrate the applicability of convolutional and recursive feature learning to the new domain of depth images. Acknowledgments We thank Stephen Miller and Alex Teichman for tips on 3D images, Adam Coates for chats about image pre-processing and Ilya Sutskever and Andrew Maas for comments on the paper. We thank the anonymous reviewers for insightful comments. Richard is supported by the Microsoft Research PhD fellowship. The authors gratefully acknowledge the support of the Defense Advanced Research Projects Agency (DARPA) Machine Reading Program under Air Force Research Laboratory (AFRL) prime contract no. FA8750-09-C0181, and the DARPA Deep Learning program under contract number FA8650-10-C-7020. Any opinions, ﬁndings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reﬂect the view of DARPA, AFRL, or the US government. References [1] M. Quigley, S. Batra, S. Gould, E. Klingbeil, Q. Le, A. Wellman, and A.Y. Ng. High-accuracy 3D sensing for mobile manipulation: improving object detection and door opening. In ICRA, 2009. 7 apple ball banana bell pepper binder bowl calculator camera cap cellphone cereal box coffee mug comb dry battery flashlight food bag food box food can food cup food jar garlic glue stick greens hand towel instant noodles keyboard kleenex lemon lightbulb lime marker mushroom notebook onion orange peach pear pitcher plate pliers potato rubber eraser scissors shampoo soda can sponge stapler tomato toothbrush toothpaste water bottle apple ball banana bell pepper binder bowl calculator camera cap cellphone cereal box coffee mug comb dry battery flashlight food bag food box food can food cup food jar garlic glue stick greens hand towel instant noodles keyboard kleenex lemon lightbulb lime marker mushroom notebook onion orange peach pear pitcher plate pliers potato rubber eraser scissors shampoo soda can sponge stapler tomato toothbrush toothpaste water bottle Figure 5: Confusion Matrix of our CNN-RNN model. The ground truth labels are on the y-axis and the predicted labels on the x-axis. Many misclassiﬁcations are between (a) garlic and mushroom (b) food-box and kleenex. Figure 6: Examples of confused classes: Shampoo bottle and water bottle, mushrooms labeled as garlic, pitchers classiﬁed as caps due to shape and color similarity, white caps classiﬁed as kleenex boxes at certain angles. [2] K. Lai, L. Bo, X. Ren, and D. Fox. A Large-Scale Hierarchical Multi-View RGB-D Object Dataset. In ICRA, 2011. [3] A. Johnson. Spin-Images: A Representation for 3-D Surface Matching. PhD thesis, Robotics Institute, Carnegie Mellon University, 1997. [4] H.S. Koppula, A. Anand, T. Joachims, and A. Saxena. Semantic labeling of 3d point clouds for indoor scenes. In NIPS, 2011. [5] L. Bo, X. Ren, and D. Fox. Depth kernel descriptors for object recognition. In IROS, 2011. [6] M. Blum, J. T. Springenberg, J. Wlﬁng, and M. Riedmiller. A Learned Feature Descriptor for Object Recognition in RGB-D Data. 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4,457	Transferring Expectations in Model-based Reinforcement Learning Trung Thanh Nguyen, Tomi Silander, Tze-Yun Leong School of Computing National University of Singapore Singapore, 117417 {nttrung, silander, leongty}@comp.nus.edu.sg Abstract We study how to automatically select and adapt multiple abstractions or representations of the world to support model-based reinforcement learning. We address the challenges of transfer learning in heterogeneous environments with varying tasks. We present an efﬁcient, online framework that, through a sequence of tasks, learns a set of relevant representations to be used in future tasks. Without predeﬁned mapping strategies, we introduce a general approach to support transfer learning across different state spaces. We demonstrate the potential impact of our system through improved jumpstart and faster convergence to near optimum policy in two benchmark domains. 1 Introduction In reinforcement learning (RL), an agent autonomously learns how to make optimal sequential decisions by interacting with the world. The agent’s learned knowledge, however, is task and environment speciﬁc. A small change in the task or the environment may render the agent’s accumulated knowledge useless; costly re-learning from scratch is often needed. Transfer learning addresses this shortcoming by accumulating knowledge in forms that can be reused in new situations. Many existing techniques assume the same state space or state representation in different tasks. While recent efforts have addressed inter-task transfer in different action or state spaces, speciﬁc mapping criteria have to be established through policy reuse [7], action correlation [14], state abstraction [22], inter-space relation [16], or other methods. Such mappings are hard to deﬁne when the agent operates in complex environments with large state spaces and multiple goal states, with possibly different state feature distributions and world dynamics. To efﬁciently accomplish varying tasks in heterogeneous environments, the agent has to learn to focus attention on the crucial features of each environment. We propose a system that tries to transfer old knowledge, but at the same time evaluates new options to see if they work better. The agent gathers experience during its lifetime and enters a new environment equipped with expectations on how different aspects of the world affect the outcomes of the agent’s actions. The main idea is to allow an agent to collect a library of world models or representations, called views, that it can consult to focus its attention in a new task. In this paper, we concentrate on approximating the transition model. The reward model library can be learned in an analogous fashion. Effective utilization of the library of world models allows the agent to capture the transition dynamics of the new environment quickly; this should lead to a jumpstart in learning and faster convergence to a near optimal policy. A main challenge is in learning to select a proper view for a new task in a new environment, without any predeﬁned mapping strategies. We will next formalize the problem and describe the method of collecting views into a library. We will then present an efﬁcient implementation of the proposed transfer learning technique. After 1 discussing related work, we will demonstrate the efﬁcacy of our system through a set of experiments in two different benchmark domains. 2 Method In RL, a task environment is typically modeled as a Markov decision process (MDP) deﬁned by a tuple (S, A, T, R), where S is a set of states; A is a set of actions; T : S × A × S →[0, 1] is transition function, such that T(s, a, s′) = P(s′\|s, a) indicates the probability of transiting to a state s′ upon taking an action a in state s; R : S × A →R is a reward function indicating immediate expected reward after an action a is taken in state s. The goal is then to ﬁnd a policy π that speciﬁes an action to perform in each state so that the expected accumulated future reward (possibly giving higher weights to more immediate rewards) for each state is maximized [18]. In model-based RL, the optimal policy is calculated based on the estimates of the transition model T and the reward model R which are obtained by interacting with the environment. A key idea of this work is that the agent can represent the world dynamics from its sensory state space in different ways. Such different views correspond to the agent’s decisions to focus attention on only some features of the state in order to quickly approximate the state transition function. 2.1 Decomposition of transition model To allow knowledge transfer from one state space to another, we assume that each state s in all the state spaces can be characterized by a d-dimensional feature vector f(s) ∈Rd. The states themselves may or may not be factored. We use the idea in situation calculus [11] to decompose the transition model T in accordance with the possible action effects. In the RL context, an action will stochastically create an effect that determines how the current state changes to the next one [2, 10, 14]. For example, an attempt to move left in a grid world may cause the agent to move one step left or one step forward, with small probabilities. The relative changes in states, “moved left” and “moved forward”, are called effects of the action. Formally, let us call MDP with a decomposed transition model CMDP (situation Calculus MDP). CMDP is deﬁned by a tuple (S, A, E, τ, η, f, R) in which the transition model T has been replaced by the the terms E, τ, η, f, where E is an effect set and f is a function from states to their feature vectors. τ : S×A×E →[0, 1] is an action model such that τ(s, a, e) = P(e \| f(s), a) indicates the probability of achieving effect e upon performing action a at state s. Notice that the probability of effect e depends on state s only through the features f(s). While the agent needs to learn the effects of the action, it is usually assumed to understand the meaning of the effects, i.e., how the effects turn each state into a next state. This knowledge is captured in a deterministic function η : S × E →S. Different effects e will change a state s to a different next state s′ = η(s, e). The MDP transition model T can be reconstructed from the CMDP by the equation: T(s, a, s′; τ) = P(s′ \| f(s), a) = τ(s, a, e), (1) where e is the effect of action a that takes s to s′, if such an e exists, otherwise T(s, a, s′; τ) = 0. The beneﬁt of this decomposition is that while there may be a large number of states, there is usually a limited number of deﬁnable effects of actions, and those are assumed to depend only on some features of the states and not on the actual states themselves. We can therefore turn the learning of the transition model into a supervised online classiﬁcation problem that can be solved by any standard online classiﬁcation method. More speciﬁcally, the classiﬁcation task is to predict the effect e of an action a in a state s with features f(s). 2.2 A multi-view transfer framework In our framework, the knowledge gathered and transferred by the agent is collected into a library T of online effect predictors or views. A view consists of a structure component ¯f that picks the features which should be focused on, and a quantitative component Θ that deﬁnes how these features should be combined to approximate the distribution of action effects. Formally, a view is deﬁned as τ = ( ¯f, Θ), such that P(E\|S, a; τ) = P(E\| ¯f(S), a; Θ) = τ(S, a, E), in which ¯f is an orthogonal projection of f(s) to some subspace 2 of Rd. Each view τ is specialized in predicting the effects of one action a(τ) ∈A and it yields a probability distribution for the effects of the action a in any state. This prediction is based on the features of the state and the parameters Θ(τ) of the view that may be adjusted based on the actual effects observed in the task environment. We denote the subset of views that specify the effects for action a by T a ⊂T . The main challenge is to build and maintain a comprehensive set of views that can be used in new environments likely resembling the old ones, but at the same time allow adaptation to new tasks with completely new transition dynamics and feature distributions. At the beginning of every new task, the existing library is copied into a working library which is also augmented with fresh, uninformed views, one for each action, that are ready to be adapted to new tasks. We then select, for each action, a view with a good track record. This view is is used to estimate the optimal policy based on the transition model speciﬁed in Equation 1, and the policy is used to pick the ﬁrst action a. The action effect is then used to score all the views in the working library and to adjust their parameters. In each round the selection of views is repeated based on their scores, and the new optimal policy is calculated based on the new selections. At the end of the task, the actual library is updated by possibly recruiting the views that have “performed well” and retiring those that have not. A more rigorous version of the procedure is described in Algorithm 1. Algorithm 1 TES: Transferring Expectations using a library of views Input: T = {τ1, τ2, ...}: view library; CMDPj: a new jth task; Φ: view goodness evaluator Let T0 be a set of fresh views - one for each action Ttmp ←T ∪T0 /* THE WORKING LIBRARY FOR THE TASK / for all a ∈A do ˆT[a] ←argmaxτ∈T a Φ(τ, j) end for / SELECTING VIEWS / for t = 0, 1, 2, ... do at ←ˆπ(st), where ˆπ is obtained by solving MDP using transition model ˆT Perform action at and observe effect et for all τ ∈T at tmp ∪T at do Score[τ] ←Score[τ] + log τ(st, at, et) end for for all τ ∈T at tmp do Update view τ based on (f(st), at, et) end for ˆT[at] ←argmaxτ∈T at tmp Score[τ] / SELECTING VIEWS / end for for all a ∈A do τ ∗←argmaxτ∈T a tmp Score[τ]; T a ←growLibrary(T a, τ ∗, Score, j) / UPDATING LIBRARY / end for if \|T \| > M then T ←T −{argminτ∈T Φ(τ, j)} end if / PRUNING LIBRARY / 2.2.1 Scoring the views To assess the quality of a view τ, we measure its predictive performance by a cumulative log-score. This is a proper score [12] that can be effectively calculated online. Given a sequence Da = (d1, d2, . . . , dN) of observations di = (si, a, ei) in which action a has resulted in effect ei in state si, the score for an a-specialized τ is S(τ, Da) = N X i=1 log τ(si, a, ei; θ:i(τ)), where τ(si, a, ei; θ:i(τ)) is the probability of event ei given by the event predictor τ based on the features of state si and the parameters θ:i(τ) that may have been adjusted using previous data (d1, d2, . . . , di−1). 2.2.2 Growing the library After completing a task, the highest scoring new views for each action are considered for recruiting into the actual library. The winning “newbies” are automatically accepted. In this case, the data has most probably come from the distribution that is far from the any current models, otherwise one of the current models would have had an advantage to adapt and win. 3 The winners τ ∗that are adjusted versions of old views ¯τ are accepted as new members if they score signiﬁcantly higher than their original versions, based on the logarithm of the prequential likelihood ratio [5] Λ(τ ∗, ¯τ) = S(τ ∗, Da) −S(¯τ, Da). Otherwise, the original versions ¯τ get their parameters updated to the new values. This procedure is just a heuristic and other inclusion and updating criteria may well be considered. The policy is detailed in Algorithm 2. Algorithm 2 Grow sub-library T a Input: T a, τ ∗, Score, j: task index; c: constant; Hτ ∗= {}: empty history record Output: updated library subset T a and winning histories Hτ ∗ case τ ∗∈T a 0 do T a ←T a ∪{τ ∗} / ADD NEWBIE TO LIBRARY / otherwise do Let ¯τ ∈T be the original, not adapted version of τ ∗ case Score[τ ∗] −Score[¯τ] > c do T a ←T a ∪{τ ∗} otherwise do T a ←T a ∪{τ ∗} −{¯τ} Hτ ∗←H¯τ / INHERIT HISTORY */ Hτ ∗←Hτ ∗∪{j} 2.2.3 Pruning the library To keep the library relatively compact, a plausible policy is to remove views that have not performed well for a long time, possibly because there are better predictors or they have become obsolete in the new tasks or environments. To implement such a retiring scheme, each view τ maintains a list Hτ of task indices that indicates the tasks for which the view has been the best scoring predictor for its specialty action a(τ). We can then calculate the recency weighted track record for each view. In practice, we have adopted the procedure by Zhu et al. [27] that introduces the recency weighted score at time T as Φ(τ, T) = X t∈Hτ e−µ(T −t), where µ controls the speed of decay of past success. Other decay functions could naturally also be used. The pruning can then be done by introducing a threshold for recency weighted score or always maintaining the top M views. 3 A view learning algorithm In TES, a view can be implemented by any probabilistic classiﬁcation model that can be quickly learned online. A popular choice for representing the transition model in factored domains is the dynamic Bayesian network (DBN), but learning DBNs is computationally very expensive. Recent studies [24, 25] have shown encouraging results in learning the structure of logistic regression models that can serve as local structures of DBNs. While these models cannot capture all the conditional distributions, their simplicity allows fast online learning in very high dimensional spaces. We introduce an online sparse multinomial logistic regression algorithm to incrementally learn a view. The proposed algorithm is similar to so called group-lasso [26] which has been recently suggested for feature selection among a very large set of features [25].1 Assuming K classes of vectors x ∈Rd, each class k is represented with a d-dimensional prototype vector Wk. Classiﬁcation of an input vector x in logistic regression is based on how “similar” it is to the prototype vectors. Similarity is measured by the inner product ⟨Wk, x⟩= Pd i=1 Wkixi. The log probability of a class y is deﬁned by log P(y = k\|x; Wk) ∝⟨Wk, x⟩. The classiﬁer can then be parametrized by stacking the Wk vectors as rows into a matrix W = (W1, ..., WK)T . An online learning system usually optimizes its probabilistic classiﬁcation performance by minimizing a total loss function through updating its parameters over time. A typical item-wise loss function of a multinomial logistic regression classiﬁer is l(W) = −log P(y\|x; W), where (y, x) denotes data item observed at time t. To achieve a parsimonious model in a feature-rich domain, we express our a priori belief that most features are superﬂuous by introducing a regularization term 1We report here the details of the method that should allow its replication. A more comprehensive description is available as a separate report in the supplementary material. 4 Ψ(W) = λ Pd i √ K\|\|W·i\|\|2, where \|\|W·i\|\|2 denotes the 2-norm of the ith column of W, and λ is a positive constant. This regularization is similar to that of group lasso [26]. It communicates the idea that it is likely that a whole column of W has zero values (especially, for large λ). A column of all zeros suggests that the corresponding feature is irrelevant for classiﬁcation. The objective function can now be written as PT t=1 l(W t, dt) + Ψ(W t), where W t is the coefﬁcient matrix learned using t −1 previously observed data items. Inspired by the efﬁcient dual averaging method [24] for solving lasso and group lasso [25] logistic regression, we extend the results to the multinomial case. Speciﬁcally, the loss minimizing sequence of parameter matrices W t can be achieved by the following online update scheme. Let Gt ki be the derivatives of function lt(W) with respect to Wki. ¯ Gt is a matrix of average partial derivatives ¯Gt ki = 1 t Pt j=1 Gj ki, where Gj ki = −xj i(I(yj = k) −P(k\|xj; W j−1)). Given a K × d average gradient matrix ¯ Gt, and a regularization parameter λ > 0, the ith column of the new parameter matrix W t+1 can be achieved as follows W t+1 ·i = (⃗0 if \|\| ¯Gt ·i\|\|2 ≤λ √ K, √ t α λ √ K \|\| ¯ Gt ·i\|\|2 −1 ¯Gt ·i otherwise, (2) where α > 0 is a constant. The update rule (2) dictates that when the length of the average gradient matrix column is small enough, the corresponding parameter column should be truncated to zero. This introduces feature selection into the model. 4 Related work The survey by Taylor and Stone [20] offers a comprehensive exposition of recent methods to transfer various forms of knowledge in RL. Not much research, however, has focused on transferring transition models. For example, while superﬁcially similar to our framework, the case-based reasoning approaches [4] [13] focus on collecting good decisions instead of building models of world dynamics. Taylor proposes TIMBREL [19] to transfer observations in a source to a target task via manually tailored inter-task mapping. Fernandez et al. [7] transfers a library of policies learned in previous tasks to bias exploration in new tasks. The method assumes a constant inter-task state space, otherwise a state mapping strategy is needed. Hester and Stone [8] describe a method to learn a decision tree for predicting state relative changes which are similar to our action effects. They learn decision trees online by repeatedly applying batch learning. Such a sequence of classiﬁers forms an effect predictor that could be used as a member of our view library. This work, however, does not directly focus on transfer learning. Multiple models have previously been used to guide behavior in non-stationary environments [6] [15]. Unlike our work, these studies usually assume a common concrete state space. In representation selection, Konidaris and Barto [9] focus on selecting the best abstraction to assist the agent’s skill learning, and Van et al. [21] study using multiple representations together to solve a RL problem. None of these studies, however, solve the problem of transferring knowledge in heterogeneous environments. Atkeson and Santamaria introduce a locally weighted transfer learning technique called LWT to adapt previously learned transition models into a new situation [1]. This study is among the very few that actually consider transferring the transition model to a new task [20]. While their work is conducted in continuous state space using a ﬁxed state similarity measure, it can be adapted to a discrete case. Doing so corresponds to adopting a ﬁxed single view. We will compare our work with this approach in our experiments. This approach could also be extended to be compatible with our work by learning a library of state similarity measures and developing a method to choose among those similarities for each task. Wilson et al. [23] also address the problem of transfer in heterogeneous environments. They formalize the problem as learning a generative Dirichlet process for MDPs and suggest an approximate solution using Gibbs sampling. Our method can be seen as a structure learning enhanced alternative implementation of this generative model. Our online-method is computationally more efﬁcient, but the MCMC estimation should eventually yield more accurate estimates. Both models can also 5 be adjusted to deal with non-stationary task sources. The work by Wilson et al. demonstrates the method for reward models, and it is unclear how to extend the approach for transferring transition models. We will also compare our work with this hierarchical Bayes approach in our experiments. 5 Experiments We examine the performance of our expectation transfer algorithm TES that transfers views to speedup the learning process across different environments in two benchmark domains. We show that TES can efﬁciently: a) learn the appropriate views online, b) select views using the proposed scoring metric, c) achieve a good jump start, and d) perform well in the long run. To better compare with some related work, we evaluate the performance of TES for transferring both transition models and reward models in RL. TES can be adapted to transfer reward models as follows: Assuming that the rewards follow a Gaussian distribution, a view of the expected reward model can be learned similarly as shown in section 3. We use an online sparse linear regression model instead of the multinomial logistic regression. Simply replacing matrix W by a vector w, and using squared loss function, the coefﬁcient update function can be found similar to that in Equation 2 [24]. When studying reward models, the transition models are assumed to be known. 5.1 Learning views for effective transfer In the ﬁrst experiment, we compare TES with the locally weighted LWT approach by Atkeson et al. [1] and the non-parametric hierarchical Bayesian approach HB by Wilson et al. [23] in transferring reward models. We adopt the same domain as described in Wilson et al.’s HB paper, but augment each state with 200 random binary features. The objective is to ﬁnd the optimal route to a known goal state in a color maze. Assuming a deterministic transition model, the highest cumulative reward, determined by the colors around each cell/state, can be achieved on the optimal route. Experiment set-up: Five different reward models are generated by normal Gaussian distributions, each depending on different sets of features. The start state is random. We run experiments on 15 tasks repeatedly 20 times, and conduct leave-one-task-out test. The maximum size M of the views library, initially empty, is set to be 20; threshold c for growing the library is set to be log 300. The parameters for view learning are: λ = 0.05 and α = 2.5. Table 1: Transfer of reward models: Cumulative reward in the ﬁrst episodes; Time to solve 15 tasks (in minutes), in which each is run with 200 episodes. Map sizes vary from 20 × 20 to 30 × 30. Methods Tasks Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 HB -108.01 -85.26 -67.46 -90.17 -130.11 -95.42 -46.23 -77.10 -83.91 -51.01 -131.44 -97.05 -90.11 -48.91 -92.31 77.2 LWT -79.41 -114.28 -83.31 -46.70 -245.11 -156.23 -47.05 -49.52 -105.24 -88.19 -174.15 -85.10 -55.45 -101.24 -86.01 28.6 TES -45.01 -78.23 -62.15 -54.46 -119.76 -115.77 -37.15 -58.09 -167.13 -59.11 -102.46 -45.99 -86.12 -67.23 -81.39 31.5 As seen in Table 1, TES on average wins over HB in 11 and LWT in 12 out of 15 tasks. In the 15 × 20 = 300 runs TES wins over HB 239 and over LWT 279 times, both yielding binomial test pvalues less than 0.05. This demonstrates that TES can successfully learn the views and utilize them in novel tasks. Moreover, TES runs much faster than HB, and just slightly slower than LWT. Since HB does not learn the relevant features for model representation, it may overﬁt, and the knowledge learned cannot be easily generalized. It also needs a costly sampling method. Similarly, the strategy for LWT that tries to learn one common model for transfer in various tasks often does not work well. 5.2 Multi-view transfer in complex environments In the second experiment, we evaluate TES in a more challenging domain, transferring transition models. We consider a grid-based robot navigation problem in which each grid-cell has the surface of either sand, soil, water, brick, or ﬁre. In addition, there may be walls between cells. The surfaces and walls determine the stochastic dynamics of the world. However, the agent also observes numerous other features in the environment. The agent has to learn to focus on the relevant features to quickly achieve its goal. The goal is to reach any exit door in the world consuming as little energy as possible. 6 Experiment set-up: The agent can perform four actions (move up, down, left, right) which will lead it to one of the four states around it, or leave it to its current state if it bumps into a wall. The agent will spend 0.01 units of energy to perform an action. It loses 1 unit if falling into a ﬁre, but gains 1 unit when reaching an exit door. A task ends when the agent reaches any exit door or ﬁre. We design ﬁfteen tasks with grid sizes ranging from 20×20 to 30×30. Each task has a different state space and different terminal states. Each state (cell) also has 200 irrelevant random binary features, besides its surface materials and the walls around it. The tasks may have different dynamics as well as different distributions of the surface materials. In our experiments, the environment transition dynamics is generated using three different sets of multinomial logistic regression models so that every combination of cell surfaces and walls around the cell will lead to a different transition dynamics at the cell. The probability of going through a wall is rounded to zero and the freed probability mass is evenly distributed to other effects. The agent’s starting position is randomly picked in each episode. We represent ﬁve effects of the actions: moved up, left, down, right, did not move. The maximum size M of the view library, initially empty, is set to be 20; threshold c = log 300. In a new environment, the TES-agent mainly relies on its transferred knowledge. However, we allow some ϵ-greedy exploration with ϵ = 0.05. The parameters for view learning algorithm are that λ = 0.05, α = 1.5. We conduct leave-one-out cross-validation experiment with ﬁfteen different tasks. In each scenario the agent is ﬁrst allowed to experience fourteen tasks, over 100 episodes in each, and it is then tested on the remaining one task. No recency weighting is used to calculate the goodness of the views in the library. We next discuss experimental results averaged over 20 runs showing 95% conﬁdence intervals (when practical) for some representative tasks. Transferring expectations between homogeneous tasks. To ensure that TES is capable of basic model transfer, we ﬁrst evaluate it on a simple task to ensure that the learning algorithm in section 3 works. We train and test TES on two environments which have same dynamics and 200 irrelevant binary features that challenge agent’s ability to learn a compact model for transfer. Figure 1a shows how much the other methods lose to TES in terms of accumulated reward in the test task. loreRL is an implementation of TES equipped with the view learning algorithm that does not transfer knowledge. fRmax is the factored Rmax [3] in which the network structures of transition models are provided by an oracle [17]; its parameter m is set to be 10 in all the experiments. fEpsG is a heuristic in which the optimistic Rmax exploration of fRmax is replaced by an ϵ-greedy strategy (ϵ = 0.1). The results show that these oracle methods still have to spend time to learn the model parameters, so they gain less accumulated reward than TES. This also suggests that the transferred view of TES is likely not only compact but also accurate. Figure 1a further shows that loreRL and fEpsG are more effective than fRmax in early episodes. View selection vs. random views. Figure 1b shows how different views lead to different policies and accumulated rewards over the ﬁrst 50 episodes in a given task. The Rands curves show the accumulated reward difference to TES when the agent follows some random combinations of views from the library. For clarity we show only 5 such random combinations. For all these, the difference turns negative fast in the beginning indicating less reward in early episodes. We conclude that our view selection criterion outperforms random selection. -18 -16 -14 -12 -10 -8 -6 -4 -2 0 0 10 20 30 40 50 accumulated reward difference episode fEpsG fRmax loreRL (a) -7 -6 -5 -4 -3 -2 -1 0 0 10 20 30 40 50 accumulated reward difference episode Rands LWT loreRL fEpsG (b) -400 -300 -200 -100 0 100 200 300 0 100 200 300 400 500 accumulated reward episode TES fRmax Rmax (c) Figure 1: Performance difference to TES in early trials in a) homogeneous, b) heterogeneous environments. c) Convergence. 7 Table 2: Cumulative reward after ﬁrst episodes. For example, in Task 1 TES can save (0.616 − 0.113)/0.01 = 50.3 actions compared to LWT. Methods Tasks 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 loreRL -0.681 -0.826 -0.814 -1.068 -0.575 -0.810 -0.529 -0.398 -0.653 -0.518 -0.528 -0.244 -0.173 -1.176 -0.692 LWT 0.113 -0.966 -0.300 0.024 -1.205 -0.345 -1.104 -1.98 -0.057 -0.664 -0.230 -1.228 0.034 0.244 -0.564 TES 0.616 -0.369 0.230 -0.044 -0.541 -0.784 -0.265 0.255 0.001 -0.298 -1.184 -0.077 0.209 0.389 -0.407 Multiple views vs. single view, and non-transfer. We compare the multi-view learning TES agent with a non-transfer agent loreRL, and an LWT agent that tries to learn only one good model for transfer. We also compare with the oracle method fEpsG. As seen in Figure 1b, TES outperforms LWT which, due to differences in the tasks, also performs worse than loreRL. When the earlier training tasks are similar to the test task, the LWT agent performs well. However, the TES agent also quickly picks the correct views, thus we never lose much but often gain a lot. We also notice that TES achieves a higher accumulated reward than loreRL and fEpsG that are bound to make uninformed decisions in the beginning. Table 2 shows the average cumulative reward after the ﬁrst episode (the jumpstart effect) for each test task in the leave-one-out cross-validation. We observe that TES usually outperforms both the non-transfer and the LWT approach. In all 15 × 20 = 300 runs, TES wins over LWT 247 times and it wins over loreRL 263 times yielding p-values smaller than 0.05. We also notice that due to its fast capability of capturing the world dynamics, TES running time is just slightly longer than LWT’s and loreRL’s, which do not perform extra work for view switching but need more time and data to learn the dynamics models. Convergence. To study the asymptotic performance of TES, we compare with the oracle method fRmax which is known to converge to a (near) optimal policy. Notice that in this feature-rich domain, fRmax without the pre-deﬁned DBN structure is just similar to Rmax. Therefore, we also compare with Rmax. For Rmax, the number of visits to any state before it is considered “known” is set to 5, and the exploration probability ϵ for known states starts to decrease from value 0.1. Figure 1c shows the accumulated rewards and their statistical dispersion over episodes. Average performance is reﬂected by the angles of the curves. As seen, TES can achieve a (near) optimal policy very fast and sustain its good performance over the long run. It is only gradually caught up by fRmax and Rmax. This suggests that TES can successfully learn a good library of views in heterogeneous environments and efﬁciently utilize those views in novel tasks. 6 Conclusions We have presented a framework for learning and transferring multiple expectations or views about world dynamics in heterogeneous environments. When the environments are different, the combination of learning multiple views and dynamically selecting the most promising ones yields a system that can learn a good policy faster and gain higher accumulated reward as compared to the common strategy of learning just a single good model and using it in all occasions. Utilizing and maintaining multiple models require additional computation and memory. We have shown that by a clever decomposition of the transition function, model selection and model updating can be accomplished efﬁciently using online algorithms. Our experiments demonstrate that performance improvements in multi-dimensional heterogeneous environments can be achieved with a small computational cost. The current work addresses the question of learning good models, but the problem of learning good policies in large state spaces still remains. Our model learning method is independent of the policy learning task, thus it can well be coupled with any scalable approximate policy learning algorithms. Acknowledgments This research is supported by Academic Research Grants: MOE2010-T2-2-071 and T1 251RES1005 from the Ministry of Education in Singapore. 8 References [1] Atkeson, C., Santamaria, J.: A comparison of direct and model-based reinforcement learning. In: ICRA’97. vol. 4, pp. 3557–3564 (1997) [2] Boutilier, C., Dearden, R., Goldszmidt, M.: Stochastic dynamic programming with factored representations. 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4,458	Modelling Reciprocating Relationships with Hawkes Processes Charles Blundell Gatsby Computational Neuroscience Unit University College London London, United Kingdom c.blundell@gatsby.ucl.ac.uk Katherine A. Heller Duke University Durham, NC, USA kheller@stat.duke.edu Jeffrey M. Beck University of Rochester Rochester, NY, USA jbeck@bcs.rochester.edu Abstract We present a Bayesian nonparametric model that discovers implicit social structure from interaction time-series data. Social groups are often formed implicitly, through actions among members of groups. Yet many models of social networks use explicitly declared relationships to infer social structure. We consider a particular class of Hawkes processes, a doubly stochastic point process, that is able to model reciprocity between groups of individuals. We then extend the Inﬁnite Relational Model by using these reciprocating Hawkes processes to parameterise its edges, making events associated with edges co-dependent through time. Our model outperforms general, unstructured Hawkes processes as well as structured Poisson process-based models at predicting verbal and email turn-taking, and military conﬂicts among nations. 1 Introduction As social animals, people constantly organise themselves into social groups. These social groups can revolve around particular activities, such as sports teams, particular roles, such as store managers, or general social alliances, like gang members. Understanding the dynamics of group interactions is a difﬁcult problem that social scientists strive to address. One basic problem in understanding group behaviour is that groups are often not explicitly deﬁned, and the members must be inferred. How might we infer these groups, and from what data? How can we predict future interactions among individuals based on these inferred groups? A common approach is to infer groups, or clusters, of people based upon a declared relationship between pairs of individuals [1, 2, 3, 4]. For example, data from social networks, where two people declare that they are “friends” or in each others’ social “neighbourhood”, can potentially be used. However these declared relationships are not necessarily readily available, truthful, or pertinent to inferring the social group structure of interest. In this paper we instead propose an approach to inferring social groups based directly on a set of real interactions between people. This approach reﬂects an “actions speak louder than words” philosophy. If we are interested in capturing groups that best reﬂect human behaviour we should be determining the groups from instances of that same behaviour. We develop a model which can learn social group structure based on interactions data. 1 In the work that we present, our data will consist of a sequence of many events, each event reﬂecting one person, the sender, performing some sort of an action towards another person, the recipient, at some particular point in time. As examples, the actions we consider are that of one person sending an email to another, one person speaking to another, or one country engaging in military action towards another. The key property that we leverage to infer social groups is reciprocity. Reciprocity is a common social norm, where one person’s actions towards another increases the probability of the same type of action being returned. For example, if Bob emails Alice, it increases the probability that Alice will email Bob in the near future. Reciprocity widely manifests across many cultures, perhaps most commonly as the golden rule and tit for tat retaliation. When multiple people show a similar pattern of reciprocity, our model will place these people in their own group. The Bayesian nonparametric model we use on these time-series data is generative and accounts for the rate of events between clusters of individuals. It is built upon mutually-exciting point processes, known as Hawkes processes [5, 6]. Pairs of mutually-exciting Hawkes processes are able to capture the causal nature of reciprocal interactions. Here the processes excite one another through their actualised events. Since Poisson processes are a special case of Hawkes processes, our model is also able to capture simpler one-way, non-reciprocal, relationships as well. Our model is also related to the Inﬁnite Relational Model (IRM) [1, 2]. The IRM typically assumes that there is a ﬁxed graph, or social network, which is observed. Here we are interested in inferring the implicit social structure based only on the occurrences of interactions between vertices in the graph. We apply our model to reciprocal behaviour in verbal and email conversations and to military conﬂicts among nations. The remainder of the paper is organised as follows: section 2 discusses using Poisson processes together with the IRM. Section 3 describes our use of self-exciting and pairs of Hawkes processes, and section 4 speciﬁes how they are used to develop our reciprocity clustering model. Section 5 presents an inference algorithm for our model, section 6 discusses related work, and section 7 presents experimental results using our model on synthetic, email, speech and intercountry conﬂict data. 2 Poisson processes with the Inﬁnite Relational Model The Inﬁnite Relational Model (IRM) [1, 2] was developed to model relationships among entities as graphs, based upon previously declared relationships. Let V denote the vertices of the graph, corresponding to individuals, and let euv denote the presence or absence of a relationship between vertices u and v, corresponding to an edge in the graph. The generative process of the IRM is: π ∼CRP(α) (1) λpq ∼Beta(γ, γ) ∀p, q ∈range(π) (2) euv ∼Bernoulli(λπ(u)π(v)) ∀u, v ∈V (3) where π is a partition of the vertices V , distributed according to the Chinese restaurant process (CRP) with concentration parameter α, with p and q indexing clusters of π. Hence vertex u belongs to the cluster given by π(u), and consequently, the clusters in π are given by range(π). The probability of an edge between vertex u and vertex v is then the parameter λpq associated with their pair of clusters. Often in interaction data there are many instances of interactions between the same pair of individuals–this cannot be modelled by the IRM. A straightforward way to modify the IRM to account for this is to use a Gamma-Poisson observation model instead of this usual Beta-Bernoulli model. Unfortunately, a vanilla Gamma-Poisson observation model does not allow us to predict events into the future, outside the observed time window. Therefore we consider using a Poisson process instead. Poisson processes are stochastic counting processes. For an introduction see [7]. We shall consider Poisson processes on [0, ∞), such that the number of events in any interval [s, s′) of the real-half line, denoted N[s, s′), is Poisson distributed with rate λ(s′ −s). 2 Mallory' Bob' Alice' 0 50 100 150 200 250 rate λpq(t) Alice, Bob →Alice, Bob Mallory →Alice, Bob Alice, Bob →Mallory Mallory →Mallory 0.0 0.2 0.4 0.6 0.8 1.0 time t Mallory →Mallory Alice, Bob →Mallory Mallory →Alice, Bob Alice, Bob →Alice, Bob Figure 1: A simple example. The graph in the top left shows the clusters and edge weights learned by our model from the data in the bottom right plot. The top right plot shows the rates of interaction events between clusters. The bottom right plot shows the interaction events. In the graph, the width and temperature (how red the colour is) denotes the expected rate of events between pairs of clusters (using equations (9) and (10)). While in plots on the right, line colours indicates the identity of cluster pairs, and box colours indicate the originator of the event: Alice (red), Bob (blue), Mallory (black). Alice and Bob interact with each other such that they positively reciprocate each others’ actions. Mallory, however, has an asymmetric relationship with both Alice and Bob. Only after many events caused by Mallory do Alice or Bob respond, and when they do respond they both, similarly, respond more sparsely. With Gamma priors on the rate parameter, the full Poisson Process IRM model is: π ∼CRP(α) (4) λpq ∼Gamma(δ, β) ∀p, q ∈range(π) (5) Nuv(·) ∼PoissonProcess(λπ(u)π(v)) ∀u, v ∈V (6) where Nuv(·) is the random counting measure of the Poisson process, and δ and β are respectively the shape and inverse scale parameters of the Gamma prior on the rate of the Poisson processes, λpq. Inference proceeds by conditioning on, Nuv[0, T) = nuv where nuv is the total number of events directed from u to v in the given time interval. Since conjugacy can be maintained, due to the superposition property of Poisson processes, inference in this model is possible in much the same way as in the original IRM [2, 1]. There are two notable deﬁciencies of this model: the rate of events on each edge is independent of every other edge, and conditioned on the time interval containing all observed events, the times of these events are uniformly distributed. This is not the typical pattern we observe in interaction data. If I send an email to someone, it is more likely that I will receive an email from them than had I not sent an email, and the probability of receiving a reply decreases as time advances. In the following sections we will introduce and utilise mutually-exciting Hawkes processes, which are able to exactly model these phenomena. 3 Self-Exciting and Pairs of Mutually-Exciting Hawkes Processes Hawkes [5, 6] introduced a family of self- and mutually-exciting Markov point processes, often called Hawkes processes. These processes are intuitively similar to Poisson processes, but unlike Poisson processes, the rates of Hawkes processes depend upon their own historic events and those of other processes in an excitatory fashion. We shall consider an array of K × K Hawkes processes, where K is the number of clusters in a partition drawn from a CRP restricted to the individuals V . As in the IRM, the CRP allows the 3 number of processes to grow in an unconstrained manner as the number of individuals in the graph grows. However, unlike the IRM, these Hawkes processes will be pairwise-dependent: the Hawkes process governing events from cluster p to cluster q, will depend upon the Hawkes process governing events from cluster q to cluster p. Let Npq be the counting measure of the (p, q)th Hawkes process. Each Hawkes process is a point process whose rate at time t is given by: λpq(t) = γpqnpnq + Z t −∞ gpq(t −s)dNqp(s) (7) where γpq is the base rate of the counting measure of the Hawkes, process, Npq. np and nq are the number of individuals in cluster p and q respectively, and gpq is a non-negative function such that R ∞ 0 gpq(s)ds < 1, ensuring that Npq is stationary. Nqp is the counting measure of the reciprocating Hawkes process of Npq. Intuitively, if Npq governs events from cluster p to cluster q, then Nqp governs events from cluster q to cluster p. Equation (7) shows how the rates of events in these two processes are intimately intertwined. Since Nqp is an atomic measure, whose atoms correspond to the times of events, we can express the rate of Npq given in (7), by conditioning on the events of its reciprocating processes Nqp, as: λpq(t) = γpqnpnq + X i:tqp i <t gpq(t −tqp i ) (8) where tqp i denotes the times of the ith event of process Nqp. Thus the rate of the process Npq at time t is some base rate at which events occur, γpq, plus an additional rate of gpq(t −tqp i ) for each event in the reciprocating process Nqp. Figure 1(top) shows an example of how λpq(t) and λqp(t) vary for these pairs of processes. If gpq(·) = 0 then the process is a Poisson process with rate γpqnpnq. When p = q, the process is self-exciting: its current rate depends solely on its own previous events. In our application, selfexciting processes model interactions within a social group, as they model cohesion in reciprocity: individual reciprocation within a group is as if towards oneself. In the case of p ̸= q, each pair of processes Npq and Nqp mutually excite one another. An event in one increases the probability of an event from the other, and so on. Importantly, the type of reciprocation (parameterised by gpq and gqp, respectively) differs between events from group p to group q and events from group q to group p. This difference in reciprocity is what we would like our model to leverage to learn about social groups. Hawkes processes are an example of doubly stochastic point processes. The rate of events is itself a random variable. By integrating out the events of Nqp we can see that this process is stationary, as its rate does not depend upon time, and also gain further insight into the role of the functions gpq and gqp. For self-exciting Hawkes processes, where p = q, the marginal rate is: E[λpp(t)] = n2 p γpp 1 −Gpp (9) whilst for a pair of mutually-exciting Hawkes processes the marginal rate is: E[λpq(t)] = npnq γpq + γqpGpq 1 −GpqGqp (10) where Gpq = R t −∞gpq(t −u)du which tempers the effect of the rate of events from one process on the rate of the other. The closer Gpq is to zero, the more Poisson-like Hawkes processes behave. Whilst as Gpq approaches one, the rate of events in Npq are increasingly caused by those in Nqp. 4 Hawkes Processes with the Inﬁnite Relational Model We combine Hawkes processes with the IRM as follows. We pick the form for the gpq functions as gpq(δ) = βpqe− δ τpq [5, 6, 8, 9]. Examples of using this parameterisation are shown in Figure 1(top). 4 Due to the memorylessness property of the exponential distribution, inference with Hawkes processes with this parameterisation takes time linear in the number of events [10]. Our generative model is as follows: π ∼CRP(α) (11) λpq(t) = γpqnpnq + βpq Z t −∞ e−t−s τpq dNqp(s) ∀p, q ∈range(π) (12) Npq(·) ∼HawkesProcess(λpq(·)) (13) Nuv(·) ∼Thinning(Nπ(u)π(v)(·)) ∀u, v ∈V (14) where, as before, π is a partition of the individuals, drawn from a Chinese restaurant process (CRP) with concentration parameter α. For each pair of clusters p and q, we associate a time-varying rate λpq(t) which dictates the rate of events from individuals in cluster p to individuals in cluster q, and a Hawkes process Npq. As described in the previous section, this rate depends upon the speciﬁc events sent in the opposite direction, from cluster q to cluster p, whose measure is also random and is denoted Nqp(·). Each random measure Nuv(·) governs events between a particular pair of individuals within clusters p and q respectively. Nuv(·) are drawn by thinning the cluster random measure Npq(·) among all of the edges between individuals in clusters p and q. Thinning means distributing the atoms of Npq(·) among each Nuv(·), such that Npq = P u,v Nuv(·). Constructing the edge measures by thinning means it is sufﬁcient to ensure that R ∞ 0 gpq(u)du < 1 for the process to be stationary. This condition, under the chosen parameterisation, implies that τpqβpq < 1. When all βpq = 0, this model is equivalent to the Poisson process IRM in section 2. Henceforth we will use uniform thinning—each event in Npq(·) is assigned uniformly at random among all Nuv(·) where p = π(u) and q = π(v)—but in principle any thinning scheme may be used. For a Hawkes process Npq, the rate at which no events occurs in the interval [s, s′) is: e− R s′ s λpq(t)dt (15) Suppose we observe the times of all the events in [0, T), {tuv i }nuv i=1 for process Nuv (nuv being the total number of events from u to v in [0, T)). Suppose that individual u is in cluster p and that individual v is in cluster q. Furthermore, assume there are no events before time 0. The likelihood of each edge between individuals u and v is thus: p({tuv i }nuv i=1\|θpq, {tqp i }nqp i=1) = e− 1 npnq R T 0 λpq(t)dt nuv Y i=1 λpq(tuv i ) npnq (16) where θpq = (γpq, βpq, τpq), {tqp i }nqp i=1 are the times of the reciprocal events. We place proper uniform priors on log α, γpq, βpq, and τpq, enforcing the constraint that τpqβpq < 1. 5 Inference We perform posterior inference using Markov chain Monte Carlo. Our model is a departure from previous IRM-based models as there is no conjugate prior for the likelihood. Thus we cannot simply integrate out these parameters, and must sample them. To infer the partition of individuals π, the concentration parameter α, and the parameters of each Hawkes process θpq = (γpq, βpq, τpq), we use Algorithm 5 [11] adapted to the IRM and slice sampling [12] to draw samples from the posterior. We initialise the chain from the prior. Slice sampling is used for α and each of γpq, βpq, and τpq. When setting the bounds of the slice sampler for βpq (τpq) we set the upper bound to 1 τpq ( 1 βpq ) respectively, to ensure that βpqτpq ≤1. 6 Related work Several authors have considered modelling occurrence events [13, 14, 15] using piecewise constant rate Markov point processes for known number of event types. Our work directly models interaction 5 events (where an event is structured to have a sender and recipient) and the number of possible events types is not limited. [16] describes a model of occurrence events as a discrete time-series using a latent ﬁrst-order Markov model. Our model differs in that it considers interaction events in continuous time and requires no ﬁrst-order assumption. The model in Section 2 relates the work of [17] to the IRM [1], yielding a version of their model that learns the number of clusters whilst maintaining conjugacy. However our model does not use a Poisson process to model event times, instead using processes which have a time-varying rate. Simma and Jordan [10] describe a cascade of Poisson processes, forming a marked Hawkes process. Hawkes processes are also the basis of this work, however our work does not use side-channel information to group individuals by imposing ﬁxed marks on the process; instead we learn structure among several co-dependent Hawkes processes and use Bayesian inference for the parameters and structure. Paninski et al [18, 19] describe a process similar to a Hawkes process that uses an additional link function to allow for inhibition amongst neurons. The interest is in modelling the activation and co-activation of neurons and as such they do not directly model cluster structure among the neurons, while our model does model this structure. Learning such structure among neurons is a potential interesting future application of this model. Our model may also be seen as a probabilistic interpretation of the interaction rank of [20], which we leverage to discover global clustering structure. An interesting future direction would be to learn a per-person (i.e., ego-centric) clustering structure. 7 Experiments In our experiments we compared our model to the Poisson process IRM (Section 2), a single Hawkes process and a single Poisson process. These latter two models are equivalent to the ﬁrst two models where just one cluster is used. We compared these models quantitatively by comparing their log predictive densities (with respect to the space of ordered sequences of events) on events falling in the ﬁnal 10% of the total time of the data (Table 2). We normalised the times of all events such that the ﬁrst 90% of total time lay in the interval [0, 1]. We ran our inference algorithm for 5000 iterations and discarded the ﬁrst 500 burn-in samples, repeating each experiment 10 times from different initialisations from the prior. Synthetic data We generated synthetic data to highlight differences between our model and the alternatives. The data involves three individuals and is plotted in Figure 1. Table 1 shows details of the ﬁt of the model to the data, and Table 2 shows the predictive results. The Poisson IRM is uncertain how to cluster individuals as it cannot model the temporal dependence between individuals, while the Hawkes IRM can and so performs better at prediction as well. A single Hawkes process does not model the structure among individuals and so performs worse than the Hawkes IRM, although it is able to model dependence among events. Enron email threads We took the ﬁve longest threads from the Enron 2009 data set [21]. We identiﬁed threads by the set of senders and receivers, and their subject line (after removing common subject line preﬁxes such as “Re:”, “Fwd:” and so on, removing punctuation and making all letters lower case). All of these threads involve two different people so there is little scope for learning much group structure in these data: either both people are in the same cluster, or they are in two separate clusters. However as can be seen in Table 2 these data suggest a predictive advantage to using mutually-exciting Hawkes processes, as automatically determined by our model, instead of a single self-exciting Hawkes process and of both of these approaches over their corresponding Poisson processes model. A self-exciting Hawkes process is unable to mark the sender and receiver of events as differing, whilst Poisson process-based models are unable to model the causal structure of events. Santa Barbara Conversation Corpus We took ﬁve conversations from the Santa Barbara Conversation Corpus [22] involving the largest number of people. These results are labelled “SB conv” followed by the conversation identiﬁer in this corpus, in the results in Tables 1 and 2. These con6 Dan$ Gran$ Rose$ Pa,$ Bern$ X$ Cher$ Paul$$ Many$ Dere$ Kare$ IRQ$ KUW$ USA$ RUS$ CHN$ AFG$ TAW$ Figure 2: Graphs of clusters of individuals inferred by our model. Edge width and temperature (how red the colour is) denotes the expected rate of events between pairs of clusters (using equations (9) and (10); edges whose marginal rate is below 1 are not included). On the left is the graph inferred on the “SB conv 26” data set. On the right is the graph inferred on the “Small MID” data set. versations cover a variety of social situations: questions during a university lecture (12), a book discussion group (23), a meeting among city ofﬁcials (26), a family argument/discussion (33), and a conversation at a family birthday party (49). We modelled the turn-taking behaviour of these groups by taking the times of when one speaker switched to the next. In Figure 2(left) we show the cluster graph found by our model for conversation 26, involving city ofﬁcials discussing a grant application. The identities of participants in all of these data are anonymised, preventing an exact interpretation. However, the model captures the discussive to-and-fro of the meeting, where PATT appears to be the chair of the meeting, and DAN, ROSE and GRAN are the main discussors, all of whom discuss with the chair, initially in question and an answer format, and among themselves, with other members of the audience chipping in sporadically. Correlates of war We use version 3.0 of the Militarized Interstate Disputes (MIDs) data set [23] to model correlates of war. This data set spans the years 1993 to 2001, and consists of MID incidents, along with the countries involved in the incidents. Incidents vary from diplomatic threats of military force to the actual deployment of military force against another state. A detailed description of each incident is available in [24]. The results of all models on the correlates of war data are given in Table 2 with details of the ﬁts in Table 1 in the rows entitles “Small MID” and “Full MID”. The full MID data set consists of 82 countries—yielding a large graph. For exposition purposes, we show the graph (in Figure 2(right)) on part of the MID data set, by restricting to events among the USA, Kuwait, Afghanistan, Taiwan, Russia, China, and Iraq. Thicker and redder lines between clusters (computed from equations 9 and 10) reﬂect a higher rate of incidents directed between the countries along the edge. The results of the clustering given by our model are in keeping with that discussed in [24]. There were three main conﬂicts involving the countries we modelled during the time period this data covers. These conﬂicts involve 1) Russia and Afghanistan, 2) Taiwan (sometimes with support from the USA) and China (sometimes with support from Russia), and 3) Iraq, Kuwait, and the USA. 1) Revolved mostly around border disputes coming out of the Soviet war in Afghanistan, and incidents sometimes involved using former Soviet countries as proxies. 2) Reﬂects conﬂict between Taiwan and China over potential Taiwanese independence. Lastly, 3) deals with conﬂicts between Iraq and either Kuwait or the USA coming out of the Persian Gulf war. It is interesting to note that groups involving smaller countries were found to be more likely to initiate incidents with larger countries in a dispute (e.g. Iraq was almost always the instigator of disputes in their conﬂict with Kuwait and the USA). Since the data ends in 2001, relatively few disputes with Afghanistan involve the USA. 7 Hawkes IRM Poisson IRM N T E[K] log probability E[K] log probability Synthetic 3 239 2.00 594.04±0.01 1.36 533.65±0.00 Small MID 7 57 4.30 33.59±0.02 1.02 -63.99±0.03 Full MID 82 412 13.67 -638.25±1.16 3.93 -1412.49±5.38 Enron 0 2 896 2.00 6724.76±0.01 2.00 4516.77±0.00 Enron 1 2 204 2.00 1202.99±0.02 2.00 692.32±0.00 Enron 2 2 122 2.00 616.37±0.02 2.00 336.02±0.00 Enron 3 2 117 2.00 497.53±0.02 2.00 318.38±0.00 Enron 4 2 85 2.00 252.60±0.02 2.00 192.74±0.00 SB conv 23 18 832 11.87 1581.72±0.12 3.01 599.29±0.42 SB conv 26 11 95 4.26 170.34±0.03 2.00 -51.92±0.14 SB conv 12 12 133 4.11 233.41±0.03 2.53 -59.12±0.15 SB conv 49 11 620 8.85 1728.13±0.07 3.40 990.75±0.15 SB conv 33 10 499 8.44 803.22±0.16 2.03 431.59±0.12 Table 1: Details of data sets and ﬁts of the structured models. N denotes the number of individuals in the data set. T denotes the total number of events in the data set. E[K] is the average number of clusters found in the posterior. Log probability is the average log probability of the training data. Hawkes IRM Poisson IRM Hawkes Poisson Synthetic 43.00±0.00 -6.76±0.02 39.88±0.01 -3.88±0.00 Small MID 12.69±0.04 -50.88±0.02 6.37±0.01 -50.86±0.00 Full MID -134.97±2.98 -355.29±5.61 -188.08±0.00 -302.65±0.00 Enron 0 259.20±0.01 39.33±0.00 233.44±0.00 40.11±0.00 Enron 1 436.66±0.01 133.29±0.00 380.27±0.01 105.71±0.00 Enron 2 139.40±0.01 24.14±0.00 118.86±0.00 22.88±0.00 Enron 3 124.22±0.01 21.06±0.00 101.71±0.01 21.03±0.00 Enron 4 127.82±0.02 28.38±0.00 109.62±0.00 22.08±0.00 SB conv 23 132.57±0.27 -198.34±0.23 30.93±0.00 -213.18±0.00 SB conv 26 -5.85±0.02 -16.83±0.09 -6.05±0.00 -14.54±0.00 SB conv 12 96.07±0.03 -97.89±0.10 33.18±0.00 -128.53±0.00 SB conv 49 220.85±0.09 -116.62±0.12 126.94±0.00 -83.62±0.00 SB conv 33 46.19±0.06 -100.83±0.04 21.71±0.00 -83.79±0.00 Table 2: Average log predictive results for each model with standard errors 8 Discussion We have presented a Bayesian nonparametric approach to learning the structure among collections of co-dependent Hawkes processes, which on several interaction data sets consistently outperforms both unstructured and Poisson-based models in terms of predictive likelihoods. The intuition behind why our model works well is that it captures part of the reciprocal nature of interactions among individuals in social situations, which in turn requires modelling some of the causal relationship of events. By learning this structure, our model is able to make better predictions. There are several future directions. For example, individuals might contribute to groups differently to one another. There may be different kinds of events between individuals and other side-channel information. Both of these artefacts may be modelled by replacing the uniform thinning scheme proposed above, with a detailed model of these effects. It would be interesting to consider other parameterisations of gpq(·) that, for example, include periods of delay between reciprocation; the exponential parameterisation lends itself to efﬁcient computation [10] whilst other parameterisations do not necessarily have this property. But different choices of gpq(·) may yield better statistical models. Another interesting avenue is to explore other structure amongst interaction events using Hawkes processes, beyond reciprocity. Acknowledgements The authors are grateful for helpful comments from the anonymous reviewers, and the support of Josh Tenenbaum, the Gatsby Charitable Foundation, PASCAL2 NoE, NIH award P30 DA028803, and an NSF postdoctoral fellowship, 8 References [1] Charles Kemp, Joshua B. Tenenbaum, Thomas L. Grifﬁths, Takeshi Yamada, and Naonori Ueda. Learning systems of concepts with an inﬁnite relational model. AAAI, 2006. [2] Zhao Xu, Volker Tresp, Kai Yu, and Hans-Peter Kriegel. Inﬁnite hidden relational models. Uncertainty in Artiﬁcial Intelligence (UAI), 2006. [3] Edoardo M. Airoldi, David M. Blei, Stephen E. Fienberg, and Eric P. Xing. Mixed membership stochastic blockmodel. Journal of Machine Learning Research, 9:1981–2014, 2008. [4] Konstantina Palla, David A. Knowles, and Zoubin Ghahramani. An inﬁnite latent attribute model for network data. In Proceedings of the 29th International Conference on Machine Learning, ICML 2012. July 2012. [5] Alan G. Hawkes. Point spectra of some self-exciting and mutually-exciting point processes. Journal of the Royal Statistical Society. Series B (Methodological), 58:83–90, 1971. [6] Alan G. Hawkes. Point spectra of some mutually-exciting point processes. Journal of the Royal Statistical Society. Series B (Methodological), 33(3):438–443, 1971. [7] John F. C. Kingman. Poisson Processes. Oxford University Press, 1993. [8] Alan G. Hawkes and David Oakes. A cluster process representation of a self-exciting process. Journal of Applied Probability, 11(3):493–503, 1974. [9] David Oakes. The Markovian self-exciting process. Journal of Applied Probability, 12(1):69– 77, 1975. [10] Aleskandr Simma and Michael I. Jordan. Modeling events with cascades of poisson processes. Uncertainty in Artiﬁcial Intelligence (UAI), 2010. [11] Radford M. Neal. 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In Proceedings of the 16th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, 2010. [18] Liam Paninski. Maximum likelihood estimation of cascade point-process neural encoding models. Network, 2004. [19] Liam Paninski, Jonathan Pillow, and Jeremy Lewi. Statistical models for neural encoding, decoding, and optimal stimulus design. In Computational Neuroscience: Theoretical Insights Into Brain Function. 2007. [20] Maayan Roth, Assaf Ben-David, David Deutscher, Guy Flysher, Ilan Horn, Ari Leichtberg, Naty Leiser, Yossi Matias, and Ron Merom. Suggesting friends using the implicit social graph. In Proceedings of the 16th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, 2010. [21] Enron 2009 Data set. http://www.cs.cmu.edu/ enron/. [22] John W. DuBois, Wallace L. Chafe, Charles Meyer, and Sandra A. Thompson. Santa Barbara corpus of spoken American English. Linguistic Data Consortium, 2000. [23] Faten Ghosn, Glenn Palmer, and Stuart Bremer. The mid3 data set, 19932001: Procedures, coding rules, and description. Conﬂict Management and Peace Science, 21:133–154, 2004. [24] Dispute Narratives. http://www.correlatesofwar.org/cow2%20data/mids/mid v3.0.narratives.pdf. 9	2012	105
4,459	Ancestor Sampling for Particle Gibbs Fredrik Lindsten Div. of Automatic Control Link¨oping University lindsten@isy.liu.se Michael I. Jordan Dept. of EECS and Statistics University of California, Berkeley jordan@cs.berkeley.edu Thomas B. Sch¨on Div. of Automatic Control Link¨oping University schon@isy.liu.se Abstract We present a novel method in the family of particle MCMC methods that we refer to as particle Gibbs with ancestor sampling (PG-AS). Similarly to the existing PG with backward simulation (PG-BS) procedure, we use backward sampling to (considerably) improve the mixing of the PG kernel. Instead of using separate forward and backward sweeps as in PG-BS, however, we achieve the same effect in a single forward sweep. We apply the PG-AS framework to the challenging class of non-Markovian state-space models. We develop a truncation strategy of these models that is applicable in principle to any backward-simulation-based method, but which is particularly well suited to the PG-AS framework. In particular, as we show in a simulation study, PG-AS can yield an order-of-magnitude improved accuracy relative to PG-BS due to its robustness to the truncation error. Several application examples are discussed, including Rao-Blackwellized particle smoothing and inference in degenerate state-space models. 1 Introduction State-space models (SSMs) are widely used to model time series and dynamical systems. The strong assumptions of linearity and Gaussianity that were originally invoked in state-space inference have been weakened by two decades of research on sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC). These Monte Carlo methods have not, however, led to substantial weakening of a further strong assumption, that of Markovianity. It remains a major challenge to develop inference algorithms for non-Markovian SSMs: xt+1 ∼f(xt+1 \| θ, x1:t), yt ∼g(yt \| θ, x1:t), (1) where θ ∈Θ is a static parameter with prior density p(θ), xt is the latent state and yt is the observation at time t, respectively. Models of this form arise in many different application scenarios, either from direct modeling or via a transformation or marginalization of a larger model. We provide several examples in Section 5. To tackle the challenging problem of inference for non-Markovian SSMs, we work within the framework of particle MCMC (PMCMC), a family of inferential methods introduced in [1]. The basic idea in PMCMC is to use SMC to construct a proposal kernel for an MCMC sampler. Assume that we observe a sequence of measurements y1:T . We are interested in ﬁnding the density p(x1:T , θ \| y1:T ), i.e., the joint posterior density of the state sequence and the parameter. In an idealized Gibbs sampler we would target this density by sampling as follows: (i) Draw θ⋆\| x1:T ∼p(θ \| x1:T , y1:T ); (ii) Draw x⋆ 1:T \| θ⋆∼p(x1:T \| θ⋆, y1:T ). The ﬁrst step of this procedure can be carried out exactly if conjugate priors are used. For non-conjugate models, one option is to replace Step (i) with a Metropolis-Hastings step. However, Step (ii)—sampling from the joint smoothing density p(x1:T \| θ, y1:T )—is in most cases very difﬁcult. In PMCMC, this is addressed by instead sampling a particle trajectory x⋆ 1:T based on an SMC approximation of the joint smoothing density. More precisely, we run an SMC sampler targeting p(x1:T \| θ⋆, y1:T ). We then sample one of the particles 1 at the ﬁnal time T, according to their importance weights, and trace the ancestral lineage of this particle to obtain the trajectory x⋆ 1:T . This overall procedure is referred to as particle Gibbs (PG). The ﬂexibility provided by the use of SMC as a proposal mechanism for MCMC seems promising for tackling inference in non-Markovian models. To exploit this ﬂexibility we must address a drawback of PG in the high-dimensional setting, which is that the mixing of the PG kernel can be very poor when there is path degeneracy in the SMC sampler [2, 3]. This problem has been addressed in the generic setting of SSMs by adding a backward simulation step to the PG sampler, yielding a method denoted PG with backward simulation (PG-BS). It has been found that this considerably improves mixing, making the method much more robust to a small number of particles as well as larger data records [2,3]. Unfortunately, however, the application of backward simulation is problematic for non-Markovian models. The reason is that we need to consider full state trajectories during the backward simulation pass, leading to O(T 2) computational complexity (see Section 4 for details). To address this issue, we develop a novel PMCMC method which we refer to as particle Gibbs with ancestor sampling (PG-AS) that achieves the effect of backward sampling without an explicit backward pass. As part of our development, we also develop a truncation method geared to non-Markovian models. This method is a generic method that is also applicable to PG-BS, but, as we show in a simulation study in Section 6, the effect of the truncation error is much less severe for PG-AS than for PG-BS. Indeed, we obtain up to an order of magnitude increase in accuracy in using PG-AS when compared to PG-BS in this study. Since we assume that it is straightforward to sample the parameter θ of the idealized Gibbs sampler, we will not explicitly include sampling of θ in the subsequent sections to simplify our presentation. 2 Sequential Monte Carlo We ﬁrst review the standard auxiliary SMC sampler, see e.g. [4,5]. Let γt(x1:t) for t = 1, . . . , T be a sequence of unnormalized densities on Xt, which we assume can be evaluated pointwise in linear time. Let ¯γt(x1:t) be the corresponding normalized probability densities. For an SSM we would typically have ¯γt(x1:t) = p(x1:t \| y1:t) and γt(x1:t) = p(x1:t, y1:t). Assume that {xm 1:t−1, wm t−1}N m=1 is a weighted particle system targeting ¯γt−1(x1:t−1). This particle system is propagated to time t by sampling independently from a proposal kernel, Mt(at, xt) = wat t−1νat t−1 P l wl t−1νl t−1 Rt(xt \| xat 1:t−1). (2) In this formulation, the resampling step is implicit and corresponds to sampling the ancestor indices at. Note that am t is the index of the ancestor particle of xm t . When we write xm 1:t we refer to the ancestral path of xm t . The factors νm t = νt(xm 1:t), known as adjustment multiplier weights, are used in the auxiliary SMC sampler to increase the probability of sampling ancestors that better can describe the current observation [5]. The particles are then weighted according to wm t = Wt(xm 1:t), where the weight function is given by Wt(x1:t) = γt(x1:t) γt−1(x1:t−1)νt−1(x1:t−1)Rt(xt \| x1:t−1), (3) for t ≥2. The procedure is initiated by sampling from a proposal density xm 1 ∼R1(x1) and assigning importance weights wm 1 = W1(xm 1 ) with W1(x1) = γ1(x1)/R1(x1). In PMCMC it is instructive to view this sampling procedure as a way of generating a single sample from the density ψ(x1:T , a2:T ) ≜ N Y m=1 R1(xm 1 ) T Y t=2 N Y m=1 Mt(am t , xm t ) (4) on the space XNT × {1, . . . , N}N(T −1). Here we have introduced the boldface notation xt = {x1 t, . . . , xN t } and similarly for the ancestor indices. 3 Particle Gibbs with ancestor sampling PMCMC methods is a class of MCMC samplers in which SMC is used to construct proposal kernels [1]. The validity of these methods can be assessed by viewing them as MCMC samplers on an 2 extended state space in which all the random variables generated by the SMC sampler are seen as auxiliary variables. The target density on this extended space is given by φ(x1:T , a2:T , k) ≜¯γT (xk 1:T ) N T ψ(x1:T , a2:T ) R1(xb1 1 ) QT t=2 Mt(abt t , xbt t ) . (5) By construction, this density admits ¯γT (xk 1:T ) as a marginal, and can thus be used as a surrogate for the original target density ¯γT [1]. Here k is a variable indexing one of the particles at the ﬁnal time point and b1:T corresponds to the ancestral path of this particle: xk 1:T = xb1:T 1:T = {xb1 1 , . . . , xbT T }. These indices are given recursively from the ancestor indices by bT = k and bt = abt+1 t+1 . The PG sampler [1] is a Gibbs sampler targeting φ using the following sweep (note that b1:T = {ab2:T 2:T , bT }), 1. Draw x⋆,−b1:T 1:T , a⋆,−b2:T 2:T ∼φ(x−b1:T 1:T , a−b2:T 2:T \| xb1:T 1:T , b1:T ). 2. Draw k⋆∼φ(k \| x⋆,−b1:T 1:T , a⋆,−b2:T 2:T , xb1:T 1:T , ab2:T 2:T ). Here we have introduced the notation x−m t = {x1 t, . . . , xm−1 t , xm+1 t , . . . , xN t }, x−b1:T 1:T = {x−b1 1 , . . . , x−bT T } and similarly for the ancestor indices. In [1], a sequential procedure for sampling from the conditional density appearing in Step 1 is given. This method is known as conditional SMC (CSMC). It takes the form of an SMC sampler in which we condition on the event that a prespeciﬁed path xb1:T 1:T = x′ 1:T , with indices b1:T , is maintained throughout the sampler (see Algorithm 1 for a related procedure). Furthermore, the conditional distribution appearing in Step 2 of the PG sampler is shown to be proportional to wk T , and it can thus straightforwardly be sampled from. Note that we never sample new values for the variables {xb1:T 1:T , b1:T −1} in this sweep. Hence, the PG sampler is an “incomplete” Gibbs sampler, since it does not loop over all the variables of the model. It still holds that the PG sampler is ergodic, which intuitively can be explained by the fact that the collection of variables that is left out is chosen randomly at each iteration. However, it has been observed that the PG sampler can have very poor mixing, especially when N is small and/or T is large [2,3]. The reason for this poor mixing is that the SMC path degeneracy causes the collections of variables that are left out at any two consecutive iterations to be strongly dependent. We now turn to our new procedure, PG-AS, which aims to address this fundamental issue. Our idea is to sample new values for the ancestor indices b1:T −1 as part of the CSMC procedure1. By adding these variables to the Gibbs sweep, we can considerably improve the mixing of the PG kernel. The CSMC method is a sequential procedure to sample from φ(x−b1:T 1:T , a−b2:T 2:T \| xb1:T 1:T , b1:T ) by sampling according to {x⋆,−bt t , a⋆,−bt t } ∼φ(x−bt t , a−bt t \| x⋆,−b1:t−1 1:t−1 , a⋆,−b2:t−1 2:t−1 , xb1:T 1:T , b1:T ), for t = 1, . . . , T. After having sampled these variables at time t, we add a step in which we generate a new value for bt−1(= abt t ), resulting in the following sweep: 1′. (CSMC with ancestor sampling) For t = 1, . . . , T, draw x⋆,−bt t , a⋆,−bt t ∼φ(x−bt t , a−bt t \| x⋆,−b1:t−1 1:t−1 , a⋆ 2:t−1, xb1:T 1:T , bt−1:T ), (a⋆,bt t =) b⋆ t−1 ∼φ(bt−1 \| x⋆,−b1:t−1 1:t−1 , a⋆ 2:t−1, xb1:T 1:T , bt:T ). 2′. Draw (k⋆=) b⋆ T ∼φ(bT \| x⋆,−b1:T 1:T , a⋆ 2:T , xb1:T 1:T ). It can be veriﬁed that this corresponds to a partially collapsed Gibbs sampler [6] and will thus leave φ invariant. To determine the conditional densities from which the ancestor indices are drawn, consider the following factorization, following directly from (3), γt(x1:t) = Wt(x1:t)νt−1(x1:t−1)Rt(xt \| x1:t−1)γt−1(x1:t−1) ⇒γt(xbt 1:t) = wbt t P l wl t−1νl t−1 wbt−1 t−1 wbt−1 t−1 νbt−1 t−1 P l wl t−1νl t−1 Rt(xbt t \| xbt−1 1:t−1)γt−1(xbt−1 1:t−1) = . . . = wbt t t−1 Y s=1 X l wl sνl s ! R1(xb1 1 ) tY s=2 Mt(abs s , xbs s ). (6) 1Ideally, we would like to include the variables xb1:T 1:T as well, but this is in general not possible since it would be similar to sampling from the original target density (which we assume is infeasible). 3 Furthermore, we have φ(bt \| x1:t, a2:t, xbt+1:T t+1:T , bt+1:T ) ∝φ(x1:t, a2:t, xbt+1:T t+1:T , bt:T ) ∝ γT (xk 1:T )ψ(x1:t, a2:t) R1(xb1 1 ) Qt s=2 Ms(abs s , xbs s ) ∝γt(xbt 1:t) γt(xbt 1:t) γT (xk 1:T ) R1(xb1 1 ) Qt s=2 Ms(abs s , xbs s ) . (7) By plugging (6) into the numerator we get, φ(bt \| x1:t, a2:t, xbt+1:T t+1:T , bt+1:T ) ∝wbt t γT (xk 1:T ) γt(xbt 1:t) . (8) Hence, to sample a new ancestor index for the conditioned path at time t+1, we proceed as follows. Given x′ t+1:T (= xbt+1:T t+1:T ) we compute the backward sampling weights, wm t\|T = wm t γT ({xm 1:t, x′ t+1:T }) γt(xm 1:t) , (9) for m = 1, . . . , N. We then set bt = m with probability proportional to wm t\|T . It follows that the proposed CSMC with ancestor sampling (Step 1′), conditioned on {x′ 1:T , b1:T }, can be realized as in Algorithm 1. The difference between this algorithm and the CSMC sampler derived in [1] lies in the ancestor sampling step 2(b) (where instead, they set abt t = bt−1). By introducing the ancestor sampling, we break the strong dependence between the generated particle trajectories and the path on which we condition. We call the resulting method, deﬁned by Steps 1′ and 2′ above, PG with ancestor sampling (PG-AS). Algorithm 1 CSMC with ancestor sampling, conditioned on {x′ 1:T , b1:T } 1. Initialize (t = 1): (a) Draw xm 1 ∼R1(x1) for m ̸= b1 and set xb1 1 = x′ 1. (b) Set wm 1 = W1(xm 1 ) for m = 1, . . . , N. 2. for t = 2, . . . , T: (a) Draw {am t , xm t } ∼Mt(at, xt) for m ̸= bt and set xbt t = x′ t. (b) Draw abt t with P(abt t = m) ∝wm t−1\|T . (c) Set xm 1:t = {xam t 1:t−1, xm t } and wm t = Wt(xm 1:t) for m = 1, . . . , N. The idea of including the variables b1:T −1 in the PG sampler has previously been suggested by Whiteley [7] and further explored in [2, 3]. This previous work, however, accomplishes this with a explicit backward simulation pass, which, as we discuss in the following section, is problematic for our applications to non-Markovian SSMs. In the PG-AS sampler, instead of requiring distinct forward and backward sequences of Gibbs steps as in PG with backward simulation (PG-BS), we obtain a similar effect via a single forward sweep. 4 Truncation for non-Markovian state-space models We return to the problem of inference in non-Markovian SSMs of the form shown in (1). To employ backward sampling, we need to evaluate the ratio γT (x1:T ) γt(x1:t) = p(x1:T , y1:T ) p(x1:t, y1:t) = T Y s=t+1 g(ys \| x1:s)f(xs \| x1:s−1). (10) In general, the computational cost of computing the backward sampling weights will thus be O(T). This implies that the cost of generating a full backward trajectory is O(T 2). It is therefore computationally prohibitive to employ backward simulation type of particle smoothers, as well as the PG samplers discussed above, for general non-Markovian models. 4 0 50 100 150 200 0 0.2 0.4 0.6 0.8 1 padpt. = 5 Probability 0 50 100 150 200 padpt. = 12 Figure 1: Probability under ePp as a function of the truncation level p for two different systems; one 5 dimensional (left) and one 20 dimensional (right). The N = 5 dotted lines correspond to ePp(m) for m ∈{1, . . . , N}, respectively (N.B. two of the lines overlap in the left ﬁgure). The dashed vertical lines show the value of the truncation level padpt., resulting from the adaption scheme with γ = 0.1 and τ = 10−2. See Section 6.2 for details on the experiments. To make progress, we consider non-Markovian models in which there is a decay in the inﬂuence of the past on the present, akin to that in Markovian models but without the strong Markovian assumption. Hence, it is possible to obtain a useful approximation when the product in (10) is truncated to a smaller number of factors, say p. We then replace (9) with the approximation, ewp,m t\|T = wm t γt+p({xm 1:t, x′ t+1:t+p}) γt(xm 1:t) . (11) The following proposition formalizes our assumption. Proposition 1. Let P and ePp be the probability distributions on {1, . . . , N}, deﬁned by the backward sampling weight (9) and the truncated backward sampling weights (11), respectively. Let hs(k) = g(yt+s \| xk 1:t, x′ t+1:t+s)f(x′ t+s \| xk 1:t, x′ t+1:t−s) and assume that maxk,l (hs(k)/hs(l) −1) ≤A exp(−cs), for some constants A and c > 0. Then, DKLD(P∥ePp) ≤ C exp(−cp) for some constant C, where DKLD is the Kullback-Leibler divergence (KLD). Proof. Provided in the supplemental material. From (11), we see that we can compute the backward weights in constant time under the truncation within the PG-AS framework. The resulting approximation can be quite useful; indeed, in our experiments we have seen that even p = 1 can lead to very accurate inferential results. In general, however, it will not be known a priori how to set the truncation level p for any given problem. To address this problem, we propose to use an adaption of the truncation level. Since the approximative weights (11) can be evaluated sequentially, the idea is to start with p = 1 and then increase p until the weights have, in some sense, converged. In particular, in our experimental work, we have used the following simple approach. Let ePp be the discrete probability measure deﬁned by (11). Let εp = DTV( ePp, ePp−1) be the total variation (TV) distance between the distributions for two consecutive truncation levels. We then compute the exponentially decaying moving average of the sequence εp, with forgetting factor γ ∈ [0, 1], and stop when this falls below some threshold τ ∈[0, 1]. This adaption scheme removes the requirement to specify p directly, but instead introduces the design parameters γ and τ. However, these parameters are much easier to reason about – a small value for γ gives a rapid response to changes in εp whereas a large value gives a more conservative stopping rule, improving the accuracy of the approximation at the cost of higher computational complexity. A similar trade off holds for the threshold τ as well. Most importantly, we have found that the same values for γ and τ can be used for a wide range of models, with very different mixing properties. To illustrate the effect of the adaption rule, and how the distribution ePp typically evolves as we increase p, we provide two examples in Figure 1. These examples are taken from the simulation study provided in Section 6.2. Note that the untruncated distribution P is given for the maximal value of p, i.e., furthest to the right in the ﬁgures. By using the adaptive truncation, we can stop the evaluation of the weights at a much earlier stage, and still obtain an accurate approximation of P. 5 5 Application areas In this section we present examples of problem classes involving non-Markovian SSMs for which the proposed PG-AS sampler can be applied. Numerical illustrations are provided in Section 6. 5.1 Rao-Blackwellized particle smoothing One popular approach to increase the efﬁciency of SMC samplers for SSMs is to marginalize over one component of the state, and apply an SMC sampler in the lower-dimensional marginal space. This leads to what is known as the Rao-Blackwellized particle ﬁlter (RBPF) [8–10]. The same approach has also been applied to state smoothing [11,12], but it turns out that Rao-Blackwellization is less straightforward in this case, since the marginal state-process will be non-Markovian. As an example, a mixed linear/nonlinear Gaussian SSM (see, e.g., [10]) with “nonlinear state” xt and “linear state” zt, can be reduced to xt ∼p(xt \| x1:t−1, y1:t−1) and yt ∼p(yt \| x1:t, y1:t−1). These conditional densities are Gaussian and can be evaluated for any ﬁxed marginal state trajectory x1:t−1 by running a conditional Kalman ﬁlter to marginalize the zt-process. In order to apply a backward-simulation-based method (e.g., a particle smoother) for this model, we need to evaluate the backward sampling weights (9). In a straightforward implementation2, we thus need to run N Kalman ﬁlters for T −t time steps, for each t = 1, . . . , T −1. The computational complexity of this calculation can be reduced by employing the truncation proposed in Section 4. 5.2 Particle smoothing for degenerate state-space models Many dynamical systems are most naturally modelled as degenerate in the sense that the transition kernel of the state process does not admit any dominating measure. For instance, consider a nonlinear system with additive noise of the form, ξt = f(ξt−1) + Gωt−1, yt = g(ξt) + et, (12) where G is a tall matrix, and consequently rank(G) < dim(ξt). That is, the process noise covariance matrix is singular. SMC samplers can straightforwardly be applied to this type of models, but it is more problematic to address the smoothing problem using particle methods. The reason is that the backward kernel also will be degenerate and it cannot be approximated in a natural way by the forward ﬁlter particles, as is normally done in backward-simulation-based particle smoothers. A possible remedy for this issue is to recast the degenerate SSM as a non-Markovian model in a lower-dimensional space. Let G = U [Σ 0]T V T with unitary U and V be a singular value decomposition of G and let, xt zt ≜U Tξt = U Tf(UU Tξt−1) + ΣV Tωt−1 0 . (13) For simplicity we assume that z1 is known. If this is not the case, it can be included in the system state or seen as a static parameter of the model. Hence, the sequence z1:t is σ(x1:t−1)-measurable and we can write zt = zt(x1:t−1). With vt ≜ΣV Tωt and by appropriate deﬁnitions of the functions fx and h, the model (12) can thus be rewritten as, xt = fx(x1:t−1) + vt−1 and yt = h(x1:t) + et, which is a non-degenerate, non-Markovian SSM. By exploiting the truncation proposed in Section 4 we can thus apply PG-AS to do inference in this model. 5.3 Additional problem classes There are many more problem classes in which non-Markovian models arise and in which backwardsimulation-based methods can be of interest. For instance, the Dirichlet process mixture model (DPMM, see, e.g., [13]) is a popular nonparametric Bayesian model for mixtures with an unknown number of components. Using a Polya urn representation, the mixture labels are given by a nonMarkovian stochastic process, and the DPMM can thus be seen as a non-Markovian SSM. SMC has 2For the speciﬁc problem of Rao-Blackwellized smoothing in conditionally Gaussian models, a backward simulator which can be implemented in O(T) computational complexity has recently been proposed in [11]. This is based on the idea of propagating information backward in time as the backward samples are generated. 6 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 10 −2 10 −1 PG w. ancestral sampling PG w. backward simulation Figure 2: Rao-Blackwellized state smoothing using PG. Running RMSEs for ﬁve independent runs of PG-AS (•) and PG-BS (◦), respectively. The truncation level is set to p = 1. The solid line corresponds to a run of an untruncated FF-BS. previously been used for inference in DPMMs [14, 15]. An interesting venue for future work is to use the PG-AS sampler for these models. A second example in Bayesian nonparametrics is Gaussian process (GP) regression and classiﬁcation (see, e.g., [16]). The sample path of the GP can be seen as the state-process in a non-Markovian SSM. We can thus employ PMCMC, and in particular PG-AS, to address these inference problems. An application in genetics, for which SMC has been been successfully applied, is reconstruction of phylogenetic trees [17]. A phylogenetic tree is a binary tree with observation at the leaf nodes. SMC is used to construct the tree in a bottom up fashion. A similar approach has also been used for Bayesian agglomerative clustering, in which SMC is used to construct a binary clustering tree based on Kingman’s coalescent [18]. The generative models for the trees used in [17,18] are in fact Markovian, but the observations give rise to a conditional dependence which destroys the Markov property. To employ backward simulation to these models, we are thus faced with problems of a similar nature as those discussed in Section 4. 6 Numerical evaluation This section contains a numerical evaluation of the proposed method. We consider linear Gaussian systems, which is instructive since the exact smoothing density then is available, e.g., by running a modiﬁed Bryson-Frazier (MBF) smoother [19]. For more details on the experiments, and for additional (nonlinear) examples, see [20]. 6.1 RBPS: Linear Gaussian state-space model As a ﬁrst example, we consider Rao-Blackwellized particle smoothing (RBPS) in a single-output 4th-order linear Gaussian SSM. We generate T = 100 samples from the system and run PG-AS and PG-BS, marginalizing three out of the four states using an RBPF, i.e., dim(xt) = 1. Both methods are run for R = 10000 iterations using N = 5 particles. The truncation level is set to p = 1, leading to a coarse approximation. We discard the ﬁrst 1000 iterations and then compute running means of the state trajectory x1:T . From these, we then compute the running root mean squared errors (RMSEs) ϵr relative to the true posterior means (computed with an MBF smoother). Hence, if no approximation would have been made, we would expect ϵr →0, so any static error can be seen as the effect of the truncation. The results for ﬁve independent runs from both PG samplers are shown in Figure 2. First, we note that both methods give accurate results. Still, the error for PG-AS is close to an order of magnitude less than for PG-BS. Furthermore, it appears as if the error for PG-AS would decrease further, given more iterations, suggesting that the bias caused by the truncation is dominated by the Monte Carlo variance, even after R = 10000 iterations. For further comparison, we also run an untruncated forward ﬁlter/backward simulator (FF-BS) particle smoother [21], using N = 5000 forward ﬁlter particles and M = 500 backward trajectories (with a computational complexity of O(NMT 2)). The resulting RMSE value is shown as a solid line in Figure 2. These results suggest that PMCMC samplers, such as the PG-AS, indeed can be serious competitors to more “standard” particle smoothers. Even with p = 1, PG-AS outperforms 7 10 −3 10 −2 10 −1 p = 1 p = 2 p = 3 Adapt. (3.8) d = 2 10 −3 10 −2 10 −1 p = 1 p = 5 p = 10 Adapt. (5.9) d = 5 10 −3 10 −2 10 −1 p = 1 p = 5 p = 10 Adapt. (10.6) d = 20 PG w. ancestral sampling PG w. backward simulation Figure 3: Box plots of the RMSE errors for PG-AS (black) and PG-BS (gray), for 150 random systems of different dimensions d (left, d = 2; middle, d = 5; right, d = 20). Different values for the truncation level p are considered. The rightmost boxes correspond to an adaptive threshold and the values in parentheses are the average over all systems and MCMC iterations (the same for both methods). The dots within the boxes show the median errors. FF-BS in terms of accuracy and, due to the fact that the ancestor sampling allows us to use as few as N = 5 particles at each iteration, at a lower computational cost. 6.2 Random linear Gaussian systems with rank deﬁcient process noise covariances To see how the PG samplers are affected by the choice of truncation level p and by the mixing properties of the system, we evaluate them on random linear Gaussian SSMs of different orders. We generate 150 random systems, using the MATLAB function drss from the Control Systems Toolbox, with model orders 2, 5 and 20 (50 systems for each model order). The number of outputs are taken as 1, 2 and 4 for the different model orders, respectively. The systems are then simulated for T = 200 time steps, driven by Gaussian process noise entering only on the ﬁrst state component. Hence, the rank of the process noise covariance is 1 for all systems. We run the PG-AS and PG-BS samplers for 10000 iterations using N = 5 particles. We consider different ﬁxed truncation levels, as well as an adaptive level with γ = 0.1 and τ = 10−2. Again, we compute running posterior means (discarding 1000 samples) and RMSE values relative the true posterior mean. Box plots are shown in Figure 3. Since the process noise only enters on one of the state components, the mixing tends to deteriorate as we increase the model order. Figure 1 shows how the probability distributions on {1, . . . , N} change as we increase the truncation level, in two representative cases for a 5th and a 20th order system, respectively. By using an adapted level, we can obtain accurate results for systems of different dimensions, without having to change any settings between the runs. 7 Discussion PG-AS is a novel approach to PMCMC that makes use of backward simulation ideas without needing an explicit backward pass. Compared to PG-BS, a conceptually similar method that does require an explicit backward pass, PG-AS has advantages, most notably for inference in the non-Markovian SSMs that have been our focus here. When using the proposed truncation of the backward weights, we have found PG-AS to be more robust to the approximation error than PG-BS. Furthermore, for non-Markovian models, PG-AS is easier to implement than PG-BS, since it requires less bookkeeping. It can also be more memory efﬁcient, since it does not require us to store intermediate quantities that are needed for a separate backward simulation pass, as is done in PG-BS. Finally, we note that PG-AS can be used as an alternative to PG-BS for other inference problems to which PMCMC can be applied, and we believe that it will prove attractive in problems beyond the non-Markovian SSMs that we have discussed here. Acknowledgments This work was supported by: the project Calibrating Nonlinear Dynamical Models (Contract number: 621-2010-5876) funded by the Swedish Research Council and CADICS, a Linneaus Center also funded by the Swedish Research Council. 8 References [1] C. Andrieu, A. Doucet, and R. Holenstein, “Particle Markov chain Monte Carlo methods,” Journal of the Royal Statistical Society: Series B, vol. 72, no. 3, pp. 269–342, 2010. [2] N. Whiteley, C. Andrieu, and A. Doucet, “Efﬁcient Bayesian inference for switching statespace models using discrete particle Markov chain Monte Carlo methods,” Bristol Statistics Research Report 10:04, Tech. Rep., 2010. [3] F. Lindsten and T. B. Sch¨on, “On the use of backward simulation in the particle Gibbs sampler,” in Proceedings of the 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Kyoto, Japan, Mar. 2012. [4] A. Doucet and A. 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4,460	Interpreting prediction markets: a stochastic approach Rafael M. Frongillo Computer Science Divison University of California, Berkeley raf@cs.berkeley.edu Nicol´as Della Penna Research School of Computer Science The Australian National University me@nikete.com Mark D. Reid Research School of Computer Science The Australian National University & NICTA mark.reid@anu.edu.au Abstract We strengthen recent connections between prediction markets and learning by showing that a natural class of market makers can be understood as performing stochastic mirror descent when trader demands are sequentially drawn from a ﬁxed distribution. This provides new insights into how market prices (and price paths) may be interpreted as a summary of the market’s belief distribution by relating them to the optimization problem being solved. In particular, we show that under certain conditions the stationary point of the stochastic process of prices generated by the market is equal to the market’s Walrasian equilibrium of classic market analysis. Together, these results suggest how traditional market making mechanisms might be replaced with general purpose learning algorithms while still retaining guarantees about their behaviour. 1 Introduction and literature review This paper is part of an ongoing line of research, spanning several authors, into formal connections between markets and machine learning. In [5] an equivalence is shown between the theoretically popular prediction market makers based on sequences of proper scoring rules and follow the regularised leader, a form of no-regret online learning. By modelling the traders that demand the assets the market maker is oﬀering we are able to extend the equivalence to stochastic mirror decent. The dynamics of wealth transfer is studied in [3], for a sequence of markets between agents that behave as Kelly bettors (i.e. have log utilities), and an equivalence to stochastic gradient decent is analysed. More broadly, [9, 2] have analysed how a wide range of machine learning models can be implemented in terms of market equilibria. The literature on the interpretation of prediction market prices [7, 11] has had the goal of relating the equilibrium prices to the distribution of the beliefs of traders. More recent work [8] has looked at a stochastic model, and studied the behavior of simple agents sequentially interacting with the market. We continue this latter path of research, motivated by the observation that the equilibrium price may be a poor predictor of the behavior in a volitile prediction market. As such, we seek a more detailed understanding of the market than the equilibrium point – we would like to know what the “stationary distribution” of the price is, as time goes to inﬁnity. 1 As is standard in the literature, we assume a ﬁxed (product) distribution over traders’ beliefs and wealth. Our model features an automated market maker, following the framework of [1] is becoming a standard framework in the ﬁeld. We obtain two results. First, we prove that under certain conditions the stationary point of our stochastic process deﬁned by the market maker and a belief distribution of traders converges to the Walrasian equilibrium of the market as the market liquidity increases. This result, stated in Theorem 1, is general in the sense that only technical convergence conditions are placed on the demand functions of the traders – as such, we believe it is a generalisation of the stochastic result of [8] to cases where agents are are not limited to linear demands, and leave this precise connection to future work. Second, we show in Corollary 1 that when traders are Kelly bettors, the resulting stochastic market process is equivalent to stochastic mirror descent; see e.g. [6]. This result adds to the growing literature which relates prediction markets, and automated market makers in general, to online learning; see e.g. [1], [5], [3] . This connection to mirror descent seems to suggest that the prices in a prediction market at any given time may be meaningless, as the ﬁnal point in stochastic mirror descent often has poor convergence guarantees. However, standard results suggest that a prudent way to form a “consensus estimate” from a prediction market is to average the prices. The average price, assuming our market model is reasonable, is provably close to the stationary price. In Section 5 we give a natural example that exhibits this behavior. Beyond this, however, Theorem 2 gives us insight into the relationship between the market liquidity and the convergence of prices; in particular it suggests that we should increase liquidity at a rate of √ t if we wish the price to settle down at the right rate. 2 Model Our market model will follow the automated market maker framework of [1]. We will equip our market maker with a strictly convex function C : Rn →R which is twice continuously diﬀerentiable. For brevity we will write ϕ := ∇C. The outcome space is Ω, and the contracts are determined by a payoﬀfunction φ : Ω→Rn such that Π := ϕ(Rn) = ConvHull(φ(Ω)). That is, the derivative space Π of C (the “instantaneous prices”) must be the convex hull of the payoﬀs. A trader purchasing shares at the current prices π ∈Rn pays C(ϕ−1(π) + r) −C(ϕ−1(π)) for the bundle of contracts r ∈Rn. Note that our dependence solely on π limits our model slightly, since in general the share space (domain of C) may contain more information than the current prices (cf. [1]). The bundle r is determined by an agent’s demand function d(C, π) which speciﬁes the bundle to buy given the price π and the cost function C. Our market dynamics are the following. The market maker posts the current price πt, and at each time t = 1 . . . T, a trader is chosen with demand function d drawn i.i.d. from some demand distribution D. Intuitively, these demands are parameterized by latent variables such as the agent’s belief p ∈∆Ωand total wealth W. The price is then updated to πt+1 = ϕ(ϕ−1(πt) + d(C, πt)). (1) After update T, the outcome is revealed and payout φ(ω)i is given for each contract i ∈ {1, . . . , n}. 3 Stationarity and equilibrium We ﬁrst would like to relate our stochastic model (1) to the standard notion of market equilibrium from the Economics literature, which we call the Walrasian equilibrium to avoid confusion. Here prices are ﬁxed, and the equilibrium price is one that clears the market, meaning that the sum of the demands r is 0 ∈Rn. In fact, we will show that the stationary point of our process approaches the Walrasian equilibrium point as the liquidity of the market approaches inﬁnity. 2 First, we must add a liquidity parameter to our market. Following the LMSR (the cost function C(s) = b ln P i esi/b), we deﬁne Cb(s) := b C(s/b). (2) This transformation of a convex function is called a perspective function and is known to preserve convexity [4]. Observe that ϕb(s) := ∇Cb(s) = ∇C(s/b) = ϕ(s/b), meaning that the price under Cb at s is the same as the price under C at s/b. As with the LMSR, we call b the liquidity parameter; this terminology is justiﬁed by noting that one deﬁnition of liquidity, 1/λmax∇2Cb(s) = b/λmax∇2C(s/b) (cf. [1]). In the following, we will consider the limit as b →∞. Second, in order to connect to the Walrasian equilibrium, we need a notion of a ﬁxed-price demand function: if a trader has demand d(C, ·) given C, what would the same trader’s demand be under a market where prices are ﬁxed and do not “change” during a trade? For the sake of generality, we restrict our allowable demand functions to the ones for which the limit d(F, π) := lim b→∞d(Cb, π) (3) exists; this demand d(F, ·) will be the corresponding ﬁxed-price demand for d. We now deﬁne the Walrasion equilibrium point π∗, which is simply the price at which the market clears when traders have demands distributed by D. Formally, this is the following condition:1 Z D d(F, π∗) dD(d) = 0 (4) Note that 0 ∈Rn; the demand for each contract should be balanced. The stationary point of our stochastic process, on the other hand, is the price πs b for which the expected price ﬂuctuation is 0. Formally, we have E d∼D[∆(πs b, d(Cb, πs b))] = 0, (5) where ∆(π, d) := ϕ(ϕ−1(π) + d) −π is the price ﬂuctuation. We now consider the limit of our stochastic process as the market liquidity approaches ∞. Theorem 1. Let C be a strictly convex and α-smooth2 cost function, and assume that ∂ ∂bd(Cb, π) = o(1/b) uniformly in π and all d ∈D. If furthermore the limit (3) is uniform in π and d, then limb→∞πs b = π∗. Proof. Note that by the stationarity condition (5) we may deﬁne π∗and πs b to be the roots of the following “excess demand” functions, respectively: Z(π) := Z D d(F, π) dD(d), Zs b(π) := b E d∼D[∆(π, d(Cb, π))], where we scale the latter by b so that Zs b does not limit to the zero function. Let s = ϕ−1(π) be the current share vector. Then we have lim b→∞b∆(π, d(Cb, π)) = lim b→∞b ϕ ϕ−1(π) + d(Cb, π)/b −π = lim a→0 ϕ s + a d(C1/a, π) −π a = lim a→0 ∇ϕ s + a d(C1/a, π) d(C1/a, π) + a ∂ ∂ad(C1/a, π) = lim b→∞∇ϕ s + 1 b d(Cb, π) d(Cb, π) + 1 b ∂ ∂bd(Cb, π)(−b2) = lim b→∞∇2C(s) d(Cb, π) = ∇2C(s) d(F, π), 1Here and throughout we ignore technical issues of uniqueness. One may simply restrict to the class of demands for which uniqueness is satisﬁed. 2C is α-smooth if λmax∇2C ≤α 3 where we apply L’Hopital’s rule for the third equality. Crucially, the above limit is uniform with respect to both d ∈D and π ∈Π; uniformity in d is by assumption, and uniformity in π follows from α-smoothness of C, since C is dominated by a quadratic. Since the limit is uniform with respect to D, we now have lim b→∞Zs b(π) = lim b→∞b E d∼D[∆(π, d(Cb, π))] = E d∼D lim b→∞b∆(π, d(Cb, π)) = ∇2C(s) E d∼D[d(F, π)] = ∇2C(s) Z(π). As ∇2C(s) is positive deﬁnite by assumption on C, we can conclude that limb→∞Zs b and Z share the same zeroes. Since Z has compact domain and is assumed continuous with a unique zero π∗, for each ϵ ∈(0, ϵmax) there must be some δ > 0 s.t. \|Z(π)\| > ϵ for all π s.t. ∥π −π∗∥> δ (otherwise there would be a sequence of πn →π′ s.t. f(π′) = 0 but π′ ̸= π∗). By uniform convergence there must be a B > 0 s.t. for all b > B we have ∥Zs b −Z∥∞< ϵ/2. In particular, for π s.t. ∥π −π∗∥> δ, \|Zs b(π)\| > ϵ/2. Thus, the corresponding zeros πs b must be within δ of π∗. Hence limb→∞πs b = π∗.3 3.1 Utility-based demands Maximum Expected Utility (MEU) demand functions are a particular kind of demand function derived by assuming a trader has some belief p ∈∆n over the outcomes in Ω, some wealth W ≥0, and a monotonically increasing utility function of money u : R →R. If such a trader buys a bundle r of contracts from a market maker with cost function C and price π, her wealth after ω occurs is Υω(C, W, π, r) := W +φ(ω)·r−[C(ϕ−1(π)+r)−C(ϕ−1(π))]. We ensure traders do not go into debt by requiring that traders only make demands such that this ﬁnal wealth is nonnegative: ∀ω Υω(C, π, r) ≥0. The set of debt-free bundles for wealth W and market C at price π is denoted S(C, W, π) := {r ∈Rn : minω Υω(C, W, π, r) ≥0}. A continuous MEU demand function du W,p(C, π) is then just the demand that maximizes a trader’s expected utility subject to the debt-free constraint. That is, du W,p(C, π) := argmax r∈S(C,W,π) E ω∼p [u (Υω(C, W, π, r))] . (6) We also deﬁne a ﬁxed-price MEU demand function du W,p(F, π) similarly, where Υω(F, W, π, r) := W +φ(ω)·r −π ·r and S(F, W, π) := {r ∈Rn : minω Υω(F, W, π, r) ≥0} are the ﬁxed price analogues to the continuously priced versions above. Using the notation bS := {b r \| r ∈S}, the following relationships between the continuous and ﬁxed price versions of Υ, SW , and the expected utility are a consequence of the convexity of C. Their main purpose is to highlight the relationship between wealth and liquidity in MEU demands. In particular, they show that scaling up of liquidity is equivalent to a scaling down of wealth and that the continuously priced constraints and wealth functions monotonically approach the ﬁxed priced versions. Lemma 1. For any strictly convex cost function C, wealth W > 0, price π, demand r, and liquidity parameter b > 0 the following properties hold: 1. Υω(Cb, W, π, r) = b Υω(C, W/b, π, r/b); 2. S(Cb, W, π) = b S(C, W/b, π); 3. S(C, W, π) is convex for all C; 4. S(C, W, π) ⊆S(Cb, W, π) ⊆S(F, W, π) for all b ≥1. 5. For monotone utilities u, Eω∼p [u (Υω(F, W, π, r))] ≥Eω∼p [u (Υω(C, W, π, r))]. Proof. Property (1) follows from a simple computation: Υω(Cb, W, π, r) = W + φ(ω) · r −b C(ϕ−1(π) + r/b) + b C(ϕ−1(π)) = b W/b + φ(ω) · (r/b) −C(ϕ−1(π) + r/b) + C(ϕ−1(π)) , which equals b Υω(C, W/b, π, r/b) by deﬁnition. We now can see property (2) as well: S(Cb, W, π) = {r : min ω b Υω(C, W/b, π, r/b) ≥0} = {b r : min ω Υω(C, W/b, π, r) ≥0}. 3We thank Avraham Ruderman for a helpful discussion regarding this proof. 4 For (3), deﬁne fC,s,ω(r) = C(s+r)−C(s)−φ(ω)·r, which is the ex-post cost of purchasing bundle r. As C is convex, and fC,s,ω is a shifted and translated version of C plus a linear term, fC,s,ω is convex also. The constraint Υω(C, W, π, r) ≥0 then translates to fC,s,ω(r) ≤ W, and thus the set of r which satisfy the constraint is convex as a sublevel set of a convex function. Now S(C, W, π) is convex as an intersection of convex sets, proving (3). For (4) suppose r satisﬁes fC,s,ω(r) ≤W. Note that fC,s,ω(0) = 0 always. Then by convexity we have for f := fC,s,ω we have f(r/b) = f 1 br + b−1 b 0 ≤1 bf(r) + b−1 b 0 ≤W/b, which implies S(C, W, π) ⊆S(Cb, W, π) when considering (3). To complete (4) note that fC,s,ω dominates fF,s : r 7→(ϕ(s)−φ(ω))·r by convexity of C: C(s+r)−C(s) ≥∇C(s)·r. Finally, proof of (5) is obtained by noting that the convexity of C means that C(ϕ−1(π) + r) −C(ϕ−1(π)) ≥∇C(ϕ−1(π)) · r = π · r and exploting the monotonicty of u. Lemma 1 shows us that MEU demands have a lot of structure, and in particular, properties (4) and (5) suggest that they may satisfy the conditions of Theorem 1; we leave this as an open question for future work. Another interesting aspect of Lemma 1 is the relationship between markets with cost function Cb and wealths W and markets with cost function C and wealths W/b – indeed, properties (1) and (2) suggest that the liquidity limit should in some sense be equivalent to a wealth limit, in that increasing liquidity by a factor b should yield similar dynamics to decreasing the wealths by b. This would relate our model to that of [8], where the authors essentially show a wealth-limit version of Theorem 1 for a binary-outcome market where traders have linear utilities (a special case of (6)). We leave this precise connection for future work. 4 Market making as mirror descent We now explore the surprising relationship between our stochastic price update and standard stochastic optimization techniques. In particular, we will relate our model to a stochastic mirror descent of the form xt+1 = argmin x∈R {η x · ∇F(xt; ξ) + DR(x, xt)}, (7) where at each step ξ ∼Ξ are i.i.d. and R is some strictly convex function. We will refer to an algorithm of the form (7) a stochastic mirror descent of f(x) := Eξ∼Ξ[F(x; ξ)]. Theorem 2. If for all d ∈D we have some F(· ; d) : Rn →Rn such that d(R∗, π) = −∇F(π; d), then the stochastic update of our model (1) is exactly a stochastic mirror descent of f(π) = Ed∼D[F(π; d)]. Proof. By standard arguments, the mirror descent update (7) can be rewritten as xt+1 = ∇R∗(∇R(xt) −∇F(xt; ξ)), where R∗is the conjugate dual of R. Take R = C∗, and let ξ = d ∼D. By assumption, we have ∇F(x; d) = −d(R∗, x) = −d(C, x) for all d. As ∇R∗= ∇C = ϕ, we have ϕ−1 = (∇R∗)−1 = ∇R by duality, and thus our update becomes xt+1 = ϕ ϕ−1(xt) + d(C, xt) , which exactly matches the stochastic update of our model (1). As an example, consider Kelly betters, which correspond to ﬁxed-price demands d(C, π) := dlog W,p(F, π) with utility u(x) = log x as deﬁned in (3). A simple calculation shows that our update becomes πt+1 = ϕ ϕ−1(πt) + W π p −π 1 −π , (8) where W and p are drawn (independently) from P and W. Corollary 1. The stochastic update for ﬁxed-price Kelly betters (8) is exactly a stochastic mirror descent of f(π) = W · KL(p, π), where p and W are the means of P and W, respectively. 5 Proof. We take F(x; dlog W,p) = W · (KL(p, x) + H(p)). Then ∇F(x; dlog W,p) = W −p x + p −1 1 −x = −W x p −x 1 −x = −dlog W,p(F, x). Hence, by Theorem 2 our update is a stochastic mirror descent of: f(x) := E[F(x; dlog W,p)] = E[Wp log x + W(1 −p) log(1 −x)] = W · (KL(p, x) + H(p)) , which of course is equivalent to W · KL(p, x) as the entropy term does not depend on x. Note that while this last result is quite compelling, we have mixed ﬁxed-price demands with a continuous-price market model – see Section 3.1. One could interpret this combination as a model in which the market maker can only adjust the prices after a trade, according to a ﬁxed convex cost function C. This of course diﬀers from the standard model, which adjusts the price continuously during a trade. 4.1 Leveraging existing learning results Theorem 2 not only identiﬁes a fascinating connection between machine learning and our stochastic prediction market model, but it also allows us to use powerful existing techniques to make broad conclusions about the behavior of our model. Consider the following result: Proposition 1 ([6]). If ∥∇F(π; p)∥2 ≤G2 for all p, π, and R is σ-strongly convex, then with probability 1 −δ, f(πT ) ≤min π f(π) + D2 ηT + G2η 2σ 1 + 4 r log 1 δ ! . 0 500 1000 1500 2000 0.50 0.55 0.60 0.65 0.70 Trade number Price of contract 1 Price Avg price Avg belief Figure 1: Price movement for Kelly betters with binomial(q = 0.6, n = 6, α = 0.5) beliefs in the LMSR market with liquidity b = 10. In our context, Proposition 1 says that the average of the prices will be a very good estimate of the minimizer of f, which as suggested by happens to be the underlying mean belief p of the traders! Moreover, as the Kelly demands are linear in both p and W, it is easy to see from Theorem 1 that p is also the stationary point and the Walrasian equilibrium point (the latter was also shown by [11]). On the other hand, as we demonstrate next, it is not hard to come up with an example where the instantaneous price πt is quite far from the equilibrium at any given time period. Before moving to our empirical work, we make one ﬁnal point. The above relationship between our stochastic market model and mirror descent sheds light on an important question: how might an automated market maker adjust the liquidity so that the market actually converges to the mean of the traders’ beliefs? The learning parameter η can be thought of as the inverse of the liquidity, and as such, Proposition 1 suggests that increasing the liquidity as √ t may cause the mean price to converge to the mean belief (assuming a ﬁxed underlying belief distribution). 5 Empirical work Example: biased coin Consider a classic Bayesian setting where a coin has unknown bias Pr[heads] = q, and traders have a prior β(α, α) over q (i.e., traders are α-conﬁdent that the coin is fair). Now suppose each trader independently observes n ﬂips from the coin, and updates her belief; upon seeing k heads, a trader would have posterior β(α + k, α + n −k). When presented with a prediction market with contracts for a single toss of the coin, where and contract 0 pays $1 for tails and contract 1 pays $1 for heads, a trader would purchase 6 0 20 40 60 80 100 0.00 0.02 0.04 0.06 0.08 0.10 Square loss of price to mean belief for State 9 Trades Loss b = 1 Instant Averaged 0 20 40 60 80 100 0.00 0.02 0.04 0.06 0.08 0.10 Square loss of price to mean belief for State 9 Trades Loss b = 3 Instant Averaged 0 20 40 60 80 100 0.00 0.02 0.04 0.06 0.08 0.10 Square loss of price to mean belief for State 9 Trades Loss b = 10 Instant Averaged Figure 2: Mean square loss of average and instantaneous prices relative to the mean belief of 0.26 over 20 simulations for State 9 for b = 1 (left), b = 3 (middle), and b = 10 (right). Bars show standard deviation. contracts as if according to the mean of their posterior. Hence, the belief distribution P of the market assigns weight P(p) = n k qk(1 −q)n−k to belief p = (α + k)/(2α + n), yielding a biased mean belief of (α + nq)/(2α + n). We show a typical simulation of this market in Figure 1, where traders behave as Kelly betters in the ﬁxed-price LMSR. Clearly, after almost every trade, the market price is quite far from the equilibrium/stationary point, and hence the classical supply and demand analysis of this market yields a poor description of the actual behavior, and in particular, of the predictive quality of the price at any given time. However, the mean price is consistently close to the mean belief of the traders, which in turn is quite close to the true parameter q. Election Survey Data We now compare the quality of the running average price versus the instantaneous price as a predictor of the mean belief of a market. We do so by simulating a market maker interacting with traders with unit wealth, log utility, and beliefs drawn from a ﬁxed distribution. The belief distributions are derived from the Princeton election survey data[10]. For each of the 50 US states, participants in the survey were asked to estimate the probability that one of two possible candidates were going to win that state.4 We use these 50 sets of estimates as 50 diﬀerent empirical distributions from which to draw trader beliefs. A simulation is conﬁgured by choosing one of the 50 empirical belief distributions S, a market liquidity parameter b to deﬁne the LMSR cost function C(s) = b ln P i esi/b, and an initial market position vector of (0, 0) – that is, no contracts for either outcome. A conﬁgured simulation is run for T trades. At each trade, a belief p is drawn from S uniformly and with replacement. This belief is used to determine the demand of the trader relative to the current market pricing. The trader purchase a bundle of contracts according to its demand and the market moves its position and price accordingly. The complete price path πt for t = 1, . . . , T of the market is recorded as well as a running average price ¯πt := 1 t Pt i=1 πt for t = 1 . . . , T. For each of the 50 empirical belief distributions we conﬁgured 9 markets with b ∈{1, 2, 3, 5, 10, 15, 20, 30, 50} and ran 20 independent simulations of T = 100 trades. We present a portion of the results for the empirical distributions for states 9 and 11. States 9 and 11 have, respectively, sample sizes of 2,717 and 2,709; means 0.26 and 0.9; and variances 0.04 and 0.02. These are chosen as being representative of the rest of the simulation results: State 9 with mean oﬀ-center and a spread of beliefs (high uncertainty) and State 11 with highly concentrated beliefs around a single outcome (low uncertainty). The results are summarised in Figures 2, 3, and 4. The ﬁrst show the square loss of the average and instaneous prices relative to the mean belief for high uncertainty State 9 for b = 1, 3, 10. Clearly, the average price is a much more reliable estimator of the mean belief for low liquidity (b = 1) and is only outperformed by the instaneous price for higher liquidity (b = 10), but then only early in trading. Similar plots for State 11 are shown in Figure 3 where the advantage of using the average price is signiﬁcantly diminished. 4The original dataset contains conjunctions of wins as well as conditional statements but we only use the single variable results of the survey. 7 0 20 40 60 80 100 0.00 0.02 0.04 0.06 0.08 0.10 Square loss of price to mean belief for State 11 Trades Loss b = 1 Instant Averaged 0 20 40 60 80 100 0.00 0.02 0.04 0.06 0.08 0.10 Square loss of price to mean belief for State 11 Trades Loss b = 3 Instant Averaged 0 20 40 60 80 100 0.00 0.02 0.04 0.06 0.08 0.10 Square loss of price to mean belief for State 11 Trades Loss b = 10 Instant Averaged Figure 3: Mean square loss of average and instantaneous prices relative to the mean belief of 0.9 over 20 simulations for State 11 for b = 1 (left), b = 3 (middle), and b = 10 (right). Bars show standard deviation. Figure 4 shows the improvement the average price has over the instantaneous price in square loss relative to the mean belief for all liquidity settings and highlights that average prices work better in low liquidity settings, consistent with the theory. Similar trends were observed for all the other States, depending on whether they had high uncertainty – in which case average price was a much better estimator – or low uncertainty – in which case instanteous price was better. Trades 20 40 60 80 100 b 10 20 30 40 50 Loss Difference -0.02 0.00 0.02 0.04 0.06 Improvement of Average over Instant Prices for State 9 Trades 20 40 60 80 100 b 10 20 30 40 50 Loss Difference -0.08 -0.06 -0.04 -0.02 0.00 0.02 Improvement of Average over Instant Prices for State 11 Figure 4: An overview of the results for States 9 (left) and 11 (right). For each trade and choice of b, the vertical value shows the improvement of the average price over the instantaneous price as measure by square loss relative to the mean. 6 Conclusion and future work As noted in Section 3.1, there are several open questions with regard to maximum expected utility demands and Theorem 1, as well as the relationship between trader wealth and market liquidity. It would also be interesting to have a application of Theorem 2 to a continuousprice model, which yields a natural minimization as in Corollary 1. The equivalence to mirror decent stablished in Theorem 2 may also lead to a better understanding of the optimal manner in which a automated prediction market ought to increase liquidity so as to maximise eﬃciency. Acknowledgments This work was supported by the Australian Research Council (ARC). 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. The ﬁrst author was partially supported by NSF grant CC-0964033 and by a Google University Research Award. 8 References [1] J. Abernethy, Y. Chen, and J.W. Vaughan. An optimization-based framework for automated market-making. In Proceedings of the 11th ACM conference on Electronic Commerce (EC’11), 2011. [2] A. Barbu and N. Lay. An introduction to artiﬁcial prediction markets for classiﬁcation. Arxiv preprint arXiv:1102.1465, 2011. [3] A. Beygelzimer, J. Langford, and D. Pennock. Learning Performance of Prediction Markets with Kelly Bettors. 2012. [4] S. Boyd and L. Vandenberghe. Convex optimization. Cambridge University Press, 2004. [5] Y. Chen and J.W. Vaughan. A new understanding of prediction markets via no-regret learning, pages 189–198. 2010. [6] J. Duchi, S. Shalev-Shwartz, Y. Singer, and A. Tewari. Composite objective mirror descent. COLT, 2010. [7] C.F. Manski. Interpreting the predictions of prediction markets. Technical report, National Bureau of Economic Research, 2004. [8] A. Othman and T. Sandholm. When do markets with simple agents fail? In Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems: volume 1-Volume 1, pages 865–872. International Foundation for Autonomous Agents and Multiagent Systems, 2010. [9] A. Storkey. Machine learning markets. AISTATS, 2012. [10] G. Wang, S.R. Kulkarni, H.V. Poor, and D.N. Osherson. Aggregating large sets of probabilistic forecasts by weighted coherent adjustment. Decision Analysis, 8(2):128, 2011. [11] J. Wolfers and E. Zitzewitz. Interpreting prediction market prices as probabilities. Technical report, National Bureau of Economic Research, 2006. 9	2012	107
4,461	Nonparanormal Belief Propagation (NPNBP) Gal Elidan Department of Statistics Hebrew University galel@huji.ac.il Cobi Cario School of Computer Science and Engineering Hebrew University cobi.cario@mail.huji.ac.il Abstract The empirical success of the belief propagation approximate inference algorithm has inspired numerous theoretical and algorithmic advances. Yet, for continuous non-Gaussian domains performing belief propagation remains a challenging task: recent innovations such as nonparametric or kernel belief propagation, while useful, come with a substantial computational cost and offer little theoretical guarantees, even for tree structured models. In this work we present Nonparanormal BP for performing efﬁcient inference on distributions parameterized by a Gaussian copulas network and any univariate marginals. For tree structured networks, our approach is guaranteed to be exact for this powerful class of non-Gaussian models. Importantly, the method is as efﬁcient as standard Gaussian BP, and its convergence properties do not depend on the complexity of the univariate marginals, even when a nonparametric representation is used. 1 Introduction Probabilistic graphical models [Pearl, 1988] are widely use to model and reason about phenomena in a variety of domains such as medical diagnosis, communication, machine vision and bioinformatics. The usefulness of such models in complex domains, where exact computations are infeasible, relies on our ability to perform efﬁcient and reasonably accurate inference of marginal and conditional probabilities. Perhaps the most popular approximate inference algoritm for graphical models is belief propagation (BP) [Pearl, 1988]. Guaranteed to be exact for trees, it is the surprising performance of the method when applied to general graphs (e.g., [McEliece et al., 1998, Murphy and Weiss, 1999]) that has inspired numerous works ranging from attempts to shed theoretical light on propagation-based algorithms (e.g., [Weiss and Freeman, 2001, Heskes, 2004, Mooij and Kappen, 2005]) to a wide range of algorithmic variants and generalizations (e.g., [Yedidia et al., 2001, Wiegerinck and Heskes, 2003, Globerson and Jaakkola, 2007]). In most works, the variables are either discrete or the distribution is assumed to be Gaussian [Weiss and Freeman, 2001]. However, many continuous real-world phenomenon are far from Gaussian, and can have a complex multi-modal structure. This has inspired several innovative and practically useful methods speciﬁcally aimed at the continuous non-Gaussian case such as expectation propagation [Minka, 2001], particle BP [Ihler and McAllester, 2009], nonparametric BP [Sudderth et al., 2010b], and kernel BP [Song et al., 2011]. Since these works are aimed at general unconstrained distributions, they all come at a substantial computational price. Further, little can be said a-priori about their expected performance even in tree structured models. Naturally, we would like an inference algorithm that is as general as possible while being as computationally convenient as simple Gaussian BP [Weiss and Freeman, 2001]. In this work we present Nonparanormal BP (NPNBP), an inference method that strikes a balance between these competing desiderata. In terms of generality, we focus on the ﬂexible class of Copula Bayesian Networks (CBNs) [Elidan, 2010] that are deﬁned via local Gaussian copula functions and any univariate densities (possible nonparametric). Utilizing the power of the copula framework [Nelsen, 2007], these models can capture complex multi-modal and heavy-tailed phenomena. 1 Figure 1: Samples from the bivariate Gaussian copula with correlation θ = 0.25. (left) with unit variance Gaussian and Gamma marginals; (right) with a mixture of Gaussian and exponential marginals. Algorithmically, our approach enjoys the beneﬁts of Gaussian BP (GaBP). First, it is guaranteed to converge and return exact results on tree structured models, regardless of the form of the univariate densities. Second, it is computationally comparable to performing GaBP on a graph with the same structure. Third, its convergence properties on general graphs are similar to that of GaBP and, quite remarkably, do not depend on the complexity of the univariate marginals. 2 Background In this section we provide a brief background on copulas in general, the Gaussian copula in particular, and the Copula Bayesian Network model of Elidan [2010]. 2.1 The Gaussian Copula A copula function [Sklar, 1959] links marginal distributions to form a multivariate one. Formally: Deﬁnition 2.1: Let U1, . . . , Un be real random variables marginally uniformly distributed on [0, 1]. A copula function C : [0, 1]n →[0, 1] is a joint distribution Cθ(u1, . . . , un) = P(U1 ≤u1, . . . , Un ≤un), where θ are the parameters of the copula function. Now consider an arbitrary set X = {X1, . . . Xn} of real-valued random variables (typically not marginally uniformly distributed). Sklar’s seminal theorem states that for any joint distribution FX (x), there exists a copula function C such that FX (x) = C(FX1(x1), . . . , FXn(xn)). When the univariate marginals are continuous, C is uniquely deﬁned. The constructive converse, which is of central interest from a modeling perspective, is also true. Since Ui ≡Fi is itself a random variable that is always uniformly distributed in [0, 1], any copula function taking any marginal distributions {Ui} as its arguments, deﬁnes a valid joint distribution with marginals {Ui}. Thus, copulas are “distribution generating” functions that allow us to separate the choice of the univariate marginals and that of the dependence structure, encoded in the copula function C. Importantly, this ﬂexibility often results in a construction that is beneﬁcial in practice. Deﬁnition 2.2: The Gaussian copula distribution is deﬁned by: CΣ(u1, . . . , un) = ΦΣ Φ −1(u1), . . . , Φ −1(un)) , (1) where Φ −1 is the inverse standard normal distribution, and ΦΣ is a zero mean normal distribution with correlation matrix Σ. Example 2.3: The standard Gaussian distribution is mathematically convenient but limited due to its unimodal form and tail behavior. However, the Gaussian copula can give rise to complex varied distribution and offers great ﬂexibility. As an example, Figure 1 shows two bivariate distributions that are constructed using the Gaussian copula and two different sets of univariate marginals. Generally, any univariate marginal, both parametric and nonparametric can be used. Let ϕΣ (x) denote the multivariate normal density with mean zero and covariance Σ, and let ϕ(x) denote the univariate standard normal density. Using the derivative chain rule and the derivative 2 inverse function theorem, the Gaussian copula density c(u1, . . . , un) = ∂nCΣ(u1,...,un) ∂U1,...∂Un is c(u1, . . . , un) = ϕΣ Φ −1(u1), . . . , Φ −1(un) Y i ∂Φ −1(ui) ∂Ui = ϕΣ Φ −1(u1), . . . , Φ −1(un) Q i ϕ(Φ −1(ui)) . For a distribution deﬁned by a Gaussian copula FX (x1, . . . , xn) = CΣ(F1(x1), . . . , Fn(xn)), using ∂Ui/∂Xi = fi, we have fX (x1, . . . , xn) = ∂nCΣ(F1(x1), . . . , Fn(xn)) ∂X1, . . . , ∂Xn = ϕΣ (˜x1, . . . , ˜xn) Q i ϕ(˜xi) Y i fi(xi), (2) where ˜xi ≡Φ −1(ui) ≡Φ −1(Fi(xi)). We will use this compact notation in the rest of the paper. 2.2 Copula Bayesian Networks Let G be a directed acyclic graph (DAG) whose nodes correspond to the random variables X = {X1, . . . , Xn}, and let Pai = {Pai1, . . . , Paiki} be the parents of Xi in G. As for standard BNs, G encodes the independence statements I(G) = {(Xi ⊥NonDescendantsi \| Pai)}, where ⊥denotes the independence relationship, and NonDescendantsi are nodes that are not descendants of Xi in G. Deﬁnition 2.4: A Copula Bayesian Network (CBN) is a triplet C = (G, ΘC, Θf) that deﬁnes fX (x). G encodes the independencies assumed to hold in fX (x). ΘC is a set of local copula functions Ci(ui, upai1, . . . , upaiki ) that are associated with the nodes of G that have at least one parent. In addition, Θf is the set of parameters representing the marginal densities fi(xi) (and distributions ui ≡Fi(xi)). The joint density fX (x) then takes the form fX (x) = n Y i=1 ci(ui, upai1, . . . , upaiki ) ∂KCi(1,upai1,...,upaiki ) ∂Upai1...∂Upaiki fi(xi) ≡ n Y i=1 Rci(ui, upai1, . . . , upaiki )fi(xi) (3) When Xi has no parents in G, Rci (·) ≡1. Note that Rci(·)fi(xi) is always a valid conditional density f(xi \| pai), and can be easily computed. In particular, when the copula density c(·) in the numerator has an explicit form, so does Rci(·). Elidan [2010] showed that a CBN deﬁnes a valid joint density. When the model is tree-structured, Q i Rci(ui, upai1, . . . , upaiki ) deﬁnes a valid copula so that the univariate marginals of the constructed density are fi(xi). More generally, the marginals may be skewed. though in practice only slightly so. In this case the CBN model can be viewed as striking a balance between the ﬁxed marginals and the unconstrained maximum likelihood objectives. Practically, the model leads to substantial generalization advantages (see Elidan [2010] for more details). 3 Nonparanormal Belief Propagation As exempliﬁed in Figure 1, the Gaussian copula can give rise to complex multi-modal joint distributions. When local Gaussian copulas are combined in a high-dimensional Gaussian Copula BN (GCBN), expressiveness is even greater. Yet, as we show in this section, tractable inference in this highly non-Gaussian model is possible, regardless of the form of the univariate marginals. 3.1 Inference for a Single Gaussian Copula We start by showing how inference can be carried out in closed form for a single Gaussian copula. While all that is involved is a simple change of variables, the details are instructive. Let fX (x1, . . . , xn) be a density parameterized by a Gaussian copula. We start with the task of computing the multivariate marginal over a subset of variables Y ⊂X. For convenience and without loss 3 of generality, we assume that Y = {X1, . . . , Xk} with k < n. From Eq. (2), we have fX1,...,XK(x1, . . . , xk) = Z Rn−k fX (x1, . . . , xn)dxk+1 . . . dxn = k Y i=1 fi(xi) ϕ(˜xi) Z ϕΣ Φ −1(F1(x1)), . . . , Φ −1(Fn(xn)) n Y i=k+1 fi(xi) ϕ(Φ −1(Fi(xi))) dxk+1 . . . dxn. Changing the integral variables to Ui and using fi = ∂Ui ∂Xi so that fi(xi)dxi = dui, we have fX1,...,XK(x1, . . . , xk) = k Y i=1 fi(xi) ϕ(˜xi) Z [0,1]n−k " ϕΣ Φ−1(u1), . . . , Φ−1(un) Qn i=k+1 ϕ(Φ−1(ui)) # duk+1 . . . dun. Changing variables once again to ˜xi = Φ −1(ui), and using ∂˜Xi/∂Ui = ϕ(˜xi) −1, we have fX1,...,XK(x1, . . . , xk) = k Y i=1 fi(xi) ϕ(˜xi) Z Rn−k ϕΣ (˜x1, . . . , ˜xn) d˜xk+1 . . . d˜xn. The integral on the right hand side is now a standard marginalization of a multivariate Gaussian (over ˜xi’s) and can be carried out in closed form. Computation of densities conditioned on evidence Z = z can also be easily carried out. Letting W = X \ {Z ∪Y} denote non query or evidence variables, and plugging in the above, we have: fY\|Z(y \| z) = R f(x)dw RR f(x)dwdy = Q i∈Y fi(xi) ϕ(˜xi) R ϕΣ (˜x1, . . . , ˜xn) d ˜w RR ϕΣ (˜x1, . . . , ˜xn) d ˜wd˜y. The conditional density is now easy to compute since a ratio of normal distributions is also normal. The ﬁnal answer, of course, does involve fi(xi). This is not only unavoidable but in fact desirable since we would like to retain the complexity of the desired posterior. 3.2 Tractability of Inference in a Gaussian CBNs We are now ready to consider inference in a Gaussian CBN (GCBN). In this case, the joint density of Eq. (3) takes, after cancellation of terms, the following form: fX (x1, . . . , xn) = Y i fi(xi) ϕ(˜xi) Y i ϕΣi(˜xi, ˜xpai1, . . . , ˜xpaiki ) ϕΣ− i (˜xpai1, . . . , ˜xpaiki ) , where Σ− i is used to denote the i’th local covariance matrix excluding the i’th row and column. When Xi has no parents, the ratio reduces to ϕ(˜xi). When the graph is tree structured, this density is also a copula and its marginals are fi(xi). In this case, the same change of variables as above results in f e X (˜x1, . . . , ˜xn) = Y i ϕΣi(˜xi, ˜xpai1, . . . , ˜xpaiki ) ϕΣ− i (˜xpai1, . . . , ˜xpaiki ) . Since a ratio of Gaussians is also a Gaussian, the entire density is Gaussian in ˜xi space, and computation of any marginal f ˜Y(˜y) is easy. The required marginal in xi space is then recovered using fY(y) = f ˜Y(˜y) Y i∈Y fi(xi) ϕ(˜xi) (4) which essentially summarizes the detailed derivation of the previous section. When we consider a non-tree structured CBN model, as noted in Section 2.2, the marginals may not equal fi(xi), and the above simpliﬁcation is not applicable. However, for the Gaussian case, it is always possible to estimate the local copulas in a topological order so that the univariate marginals are equal to fi(xi) (the model in this case is equivalent to the distribution-free continuous Bayesian belief net model [Kurowicka and Cooke, 2005]). It follows that, for any structure, Corollary 3.1: The complexity of inference in a Gaussian CBN model is the same as that of inference in a multivariate Gaussian model of the same structure. 4 Algorithm 1: Nonparanormal Belief Propagation (NPNBP) for general CBNs. Input: {fk(xk)} for all i, Σi for all nodes with parents. Output: belief bS(xS) for each cluster S. CG ←a valid cluster graph over the following potentials for all nodes i in the graph • ϕΣi(˜xi, ˜xpai1, . . . , ˜xpaiki ) • 1/ϕΣ− i (˜xpai1, . . . , ˜xpaiki ) foreach cluster S in CG // use black-box GaBP in ˜xi space bG(˜xS) ←GaBP belief over cluster S. foreach cluster S in CG // change to xi space bS(xS) = bG(˜xS) Q i∈S fi(xi) ϕ(˜xi) While mathematically this conclusion is quite straightforward, the implications are signiﬁcant. A GCBN model is the only general purpose non-Gaussian continuous graphical model for which exact inference is tractable. At the same time, as is demonstrated in our experimental evaluation, the model is able to capture complex distributions well both qualitatively and quantitatively. A ﬁnal note is worthwhile regarding the (possibly conditional) marginal density. As can be expected, the result of Eq. (4) includes fi(xi) terms for all variables that have not been marginalized out. As noted, this is indeed desirable as we would like to preserve the complexity of the density in the marginal computation. The marginal term, however, is now in low dimension so that quantities of interest (e.g., expectation) can be readily computed using naive grid-based evaluation or, if needed, using more sophisticated sampling schemes (see, for example, [Robert and Cassella, 2005]). 3.3 Belief Propagation for Gaussian CBNs Given the above observations, performing inference in a Gaussian CBN (GCBN) appears to be a solved problem. However, inference in large-scale models can be problematic even in the Gaussian case. First, the large joint covariance matrix may be ill conditioned and inverting it may not be possible. Second, matrix inversion can be slow when dealing with domains of sufﬁcient dimension. A possible alternative is to consider the popular belief propagation algorithm [Pearl, 1988]. For a tree structured model represented as a product of singleton Ψi and pairwise Ψij factors, the method relies on the recursive computation of “messages” mi→j(xj) ←α Z [Ψij(xi, xj)Ψi(xi) Q k∈N(i)\j mk→i(xi)]dxi, where α is a normalization factor and N(i) are the indices of the neighbor nodes of Xi. In the case of a GCBN model, performing belief propagation may seem difﬁcult since Ψi(xi) ≡ fi(xi) can have a complex form. However, the change of variables used in the previous section applies here as well. That is, one can perform inference in ˜xi space using standard Gaussian BP (GaBP) [Weiss and Freeman, 2001], and then perform the needed change of variables. In fact, this is true regardless of the structure of the graph so that loopy GaBP can also be used to perform approximate computations for a general GCBN model in ˜xi space. The approach is summarized in Algorithm 1, where we assume access to a black-box GaBP procedure and a cluster graph construction algorithm. In our experiments we simply use a Bethe approximation construction (see [Koller and Friedman, 2009] for details on BP, GaBP and the cluster graph construction). Generally, little can be said about the convergence of loopy BP or its variants, particularly for nonGaussian domains. Appealingly, the form of our NPNBP algorithm implies that its convergence can be phrased in terms of standard Gaussian BP convergence. In particular: • Observation 1: NPNBP converges whenever GaBP converges for the model deﬁned by Q i Rci. • Observation 2: Convergence of NPNBP depends only on the covariance matrices Σi that parameterize the local copula and does not depend on the univariate form. It follows that convergence conditions identiﬁed for GaBP [Rusmevichientong and Roy, 2000, Weiss and Freeman, 2001, Malioutov et al., 2006] carry over to NPNBP for CBN models. 5 Figure 2: Exact vs. Nonparametric BP marginals for the GCBN model learned from the wine quality dataset. Shown are the marginal densities for the ﬁrst four variables. 4 Experimental Evaluation We now consider the merit of using our NPNBP algorithm for performing inference in a a Gaussian CBN (GCBN) model. We learned a tree structured GCBN using a standard Chow-Liu approach [Chow and Liu, 1968], and a model with up to two parents for each variable using standard greedy structure search. In both cases we use the Bayesian Information Criterion (BIC) [Schwarz, 1978] to guide the structure learning algorithm. For the univariate densities, we use a standard Gaussian kernel density estimator (see, for example, [Bowman and Azzalini, 1997]). Using an identical procedure, we learn a linear Gaussian BN baseline where Xi ∼N(αpai, σi) so that each variable Xi is normally distributed around a linear combination of its parents Pai (see [Koller and Friedman, 2009] for details on this standard approach to structure learning). For the GCBN model, we also compare to Nonparametric BP (NBP) [Sudderth et al., 2010a] using D. Bickson’s code [Bickson, 2008] and A. Ihlers KDE Matlab package (http://www.ics.uci.edu/ ihler/code/kde.html), which relies on a mixture of Gaussians for message representation. In this case, since our univariate densities are constructed using Gaussian kernels, there is no approximation in the NBP representation and all approximations are due to message computations. To carry out message products, we tried all 7 sampling-based methods available in the KDE package. In the experiments below we use only the multiresolution sequential Gibbs sampling method since all other approaches resulted in numerical overﬂows even for small domains. 4.1 Qualitative Assessment We start with a small domain where the qualitative nature of the inferred marginals is easily explored, and consider performance and running time in more substantial domains in the next section. We use the wine quality data set from the UCI repository which includes 1599 measurements of 11 physiochemical properties and a quality variable of red ”Vinho Verde” [Cortez et al., 2009]. We ﬁrst examine a tree structured GCBN model where our NPNBP method allows us to perform exact marginal computations. Figure 2 compares the ﬁrst four marginals to the ones computed by the NBP method. As can be clearly seen, although the NBP marginals are not nonsensical, they are far from accurate (results for the other marginals in the domain are similar). Quantitatively, each NBP marginal is 0.5 to 1.5 bits/instance less accurate than the exact ones. Thus, the accuracy of NPNBP in this case is approximately twice that of NBP per variable, amounting to a substantial per sample advantage. We also note that NBP was approximately an order of magnitude slower than NPNBP in this domain. In the larger domains considered in the next section, NBP proved impractical. Figure 3 demonstrates the quality of the bivariate marginals inferred by our NPNBP method relative to the ones of a linear Gaussian BN model where inference can also be carried out efﬁciently. The middle panel shows a Gaussian distribution constructed only over the two variables and is thus an upper bound on the quality that we can expect from a linear Gaussian BN. Clearly, the Gaussian representation is not sufﬁciently ﬂexible to reasonably capture the distribution of the true samples (left panel). In contrast, the bivariate marginals computed by our algorithm (right panel) demonstrate the power of working with a copula-based construction and an effective inference procedure: in both cases the inferred marginals capture the non-Gaussian distributions quite accurately. Results were qualitatively similar for all other variable pairs (except for the few cases that are approximately Gaussian in the original feature space and for which all models are equally beneﬁcial). 6 Density vs. Alcohol Free vs. Total Sulfur (a) true samples (b) optimal Gaussian (c) CBN marginal Figure 3: The bivariate density for two pairs of variables in a tree structured GCBN model learned from the wine quality dataset. (a) empirical samples; (b) maximum likelihood Gaussian density; (c) exact GCBN marginal computed using our NPNBP algorithm. In Figure 4 we repeat the comparison for another pair of variables in a non-tree GCBN (as before, results were qualitatively similar for all pairs of variables). In this setting, the bivariate marginal computed by our algorithm (d) is approximate and we also compare to the exact marginal (c). As in the case of the tree-structured model, the GCBN model captures the true density quite accurately, even for this multi-modal example. NPNBP dampens some of this accuracy and results in marginal densities that have the correct overall structure but with a reduced variance. This is not surprising since it is well known that GaBP leads to reduced variances [Weiss and Freeman, 2001]. Nevertheless, the approximate result of NPNBP is clearly better than the exact Gaussian model, which assigns very low probability to regions of high density (along the main vertical axis of the density). Finally, Figure 5(left) shows the NPNBP vs. the exact expectations. As can be seen, the inferred values are quite accurate and it is plausible that the differences are due to numerical round-offs. Thus, it is possible that, similarly to the case of standard GaBP [Weiss and Freeman, 2001], the inferred expectations are theoretically exact. The proof for the GaBP case, however, does not carry over to the CBN setting and shedding theoretical light on this issue remains a future challenge. 4.2 Quantitative Assessment We now consider several substantially larger domains with 100 to almost 2000 variables. For each domain we learn a tree structured GCBN, and justify the need for the expressive copula-based model by reporting its average generalization advantage in terms of log-loss/instance over a standard linear Gaussian model. We justify the use of NPNBP for performing inference by comparing the running time of NPNBP to exact computations carried out using matrix inversion. For all datasets, we performed 10-fold cross-validation and report average results. We use the following datasets: • Crime (UCI repository). 100 variables relating to crime ranging from household size to fraction of children born outside of a marriage, for 1994 communities across the U.S. • SP500. Daily changes of value of the 500 stocks comprising the Standard and Poor’s index (S&P 500) over a period of one year. • Gene. A compendium of gene expression experiments used in [Marion et al., 2004]. We chose genes that have only 1, 2, and 3 missing values and only use full observations. This resulted in datasets with 765, 1400, and 1945 variables (genes), and 1088, 956, and 876 samples, respectively. For the 100 variable Crime domain, average test advantage of the GCBN model over the linear Gaussian one was 0.39 bits/instance per variable (as in [Elidan, 2010]). For the 765 variable Gene expression domain the advantage was around 0.1 bits/instance/variable (results were similar for the 7 Sugar level vs. Density (a) true samples (b) optimal Gaussian (c) exact CBN marginal (d) inferred marginal Figure 4: The bivariate density for a pair of variables in a non-tree GCBN model learned from the wine quality dataset. (a) empirical samples; (b) maximum likelihood Gaussian density; (c) exact CBN marginal; (d) marginal density computed by our NPNBP algorithm. Figure 5: (left) exact vs. NPNBP expected values. (right) speedup relative to matrix inversion for a tree structured GCBN model. 765,1400,1945 correspond to the three different datasets extracted from the gene expression compendium. other gene expression datasets). In both cases, the differences are dramatic and each instance is many orders of magnitude more likely given a GCBN model. For the SP500 domain, evaluation of the linear Gaussian model resulted in numerical overﬂows (due to the scarcity of the training data), and the advantage of he GCBN cannot be quantiﬁed. These generalization advantages make it obvious that we would like to perform efﬁcient inference in a GCBN model. As discussed, a GCBN model is itself tractable in that inference can be carried out by ﬁrst constructing the inverse covariance matrix over all variables and then inverting it so as to facilitate marginalization. Thus, using our NPNBP algorithm can only be justiﬁed practically. Figure 5(right) shows the speedup of NPNBP relative to inference based on matrix inversion for the different domains. Although NPNBP is somewhat slower for the small domains (in which inference is carried out in less than a second), the speedup of NPNBP reaches an order of magnitude for the larger gene expression domain. Appealingly, the advantage of NPNBP grows with the domain size due to the growth in complexity of matrix inversion. Finally, we note that we used a Matlab implementation where matrix inversion is highly optimized so that the gains reported are quite conservative. 5 Summary We presented Nonparanormal Belief Propagation (NPNBP), a propagation-based algorithm for performing highly efﬁcient inference in a powerful class of graphical models that are based on the Gaussian copula. To our knowledge, ours is the ﬁrst inference method for an expressive continuous non-Gaussian representation that, like ordinary GaBP, is both highly efﬁcient and provably correct for tree structured models. Appealingly, the efﬁciency and convergence properties of our method do not depend on the choice of univariate marginals, even when a nonparametric representation is used. The Gaussian copula is a powerful model widely used to capture complex phenomenon in ﬁelds ranging from mainstream economics (e.g., Embrechts et al. [2003]) to ﬂood analysis [Zhang and Singh, 2007]. Recent probabilistic graphical models that build on the Gaussian copula open the door for new high-dimensional non-Gaussian applications [Kirshner, 2007, Liu et al., 2010, Elidan, 2010, Wilson and Ghahramani, 2010]. Our method offers the inference tools to make this practical. 8 Acknowledgements G. Elidan and C. Cario were supported in part by an ISF center of research grant. G. Elidan was also supported by a Google grant. Many thanks to O. Meshi and A. Globerson for their comments on an earlier draft. References D. Bickson. Gaussian Belief Propagation: Theory and Application. PhD thesis, The Hebrew University of Jerusalem, Jerusalem, Israel, 2008. A. Bowman and A. Azzalini. Applied Smoothing Techniques for Data Analysis. Oxford Press, 1997. C. K. Chow and C. N. Liu. 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4,462	Adaptive Stratiﬁed Sampling for Monte-Carlo integration of Differentiable functions Alexandra Carpentier Statistical Laboratory, CMS Wilberforce Road, Cambridge CB3 0WB UK a.carpentier@statslab.cam.ac.uk R´emi Munos INRIA Lille - Nord Europe 40, avenue Halley 59000 Villeneuve d’ascq, France remi.munos@inria.fr Abstract We consider the problem of adaptive stratiﬁed sampling for Monte Carlo integration of a differentiable function given a ﬁnite number of evaluations to the function. We construct a sampling scheme that samples more often in regions where the function oscillates more, while allocating the samples such that they are well spread on the domain (this notion shares similitude with low discrepancy). We prove that the estimate returned by the algorithm is almost similarly accurate as the estimate that an optimal oracle strategy (that would know the variations of the function everywhere) would return, and provide a ﬁnite-sample analysis. 1 Introduction In this paper we consider the problem of numerical integration of a differentiable function f : [0, 1]d →R given a ﬁnite budget n of evaluations to the function that can be allocated sequentially. A usual technique for reducing the mean squared error (w.r.t. the integral of f) of a Monte-Carlo estimate is the so-called stratiﬁed Monte Carlo sampling, which considers sampling into a set of strata, or regions of the domain, that form a partition, i.e. a stratiﬁcation, of the domain (see [10][Subsection 5.5] or [6]). It is efﬁcient (up to rounding issues) to stratify the domain, since when allocating to each stratum a number of samples proportional to its measure, the mean squared error of the resulting estimate is always smaller or equal to the one of the crude Monte-Carlo estimate (that samples uniformly the domain). Since the considered functions are differentiable, if the domain is stratiﬁed in K hyper-cubic strata of same measure and if one assigns uniformly at random n/K samples per stratum, the mean squared error of the resulting stratiﬁed estimate is in O(n−1K−2/d). We deduce that if the stratiﬁcation is built independently of the samples (before collecting the samples), and if n is known from the beginning (which is assumed here), the minimax-optimal choice for the stratiﬁcation is to build n strata of same measure and minimal diameter, and to assign only one sample per stratum uniformly at random. We refer to this sampling technique as Uniform stratiﬁed Monte-Carlo. The resulting estimate has a mean squared error of order O(n−(1+2/d)). The arguments that advocate for stratifying in strata of same measure and minimal diameter are closely linked to the reasons why quasi Monte-Carlo methods, or low discrepancy sampling schemes are efﬁcient techniques for integrating smooth functions. See [9] for a survey on these techniques. It is minimax-optimal to stratify the domain in n strata and sample one point per stratum, but it would also be interesting to adapt the stratiﬁcation of the space with respect to the function f. For example, if the function has larger variations in a region of the domain, we would like to discretize the domain in smaller strata in this region, so that more samples are assigned to this region. Since f is initially unknown, it is not possible to design a good stratiﬁcation before sampling. However an efﬁcient algorithm should allocate the samples in order to estimate online the variations of the 1 function in each region of the domain while, at the same time, allocating more samples in regions where f has larger local variations. The papers [5, 7, 3] provide algorithms for solving a similar trade-off when the stratiﬁcation is ﬁxed: these algorithms allocate more samples to strata in which the function has larger variations. It is, however, clear that the larger the number of strata, the more difﬁcult it is to allocate the samples almost optimally in the strata. Contributions: We propose a new algorithm, Lipschitz Monte-Carlo Upper Conﬁdence Bound (LMC-UCB), for tackling this problem. It is a two-layered algorithm. It ﬁrst stratiﬁes the domain in K n strata, and then allocates uniformly to each stratum an initial small amount of samples in order to estimate roughly the variations of the function per stratum. Then our algorithm substratiﬁes each of the K strata according to the estimated local variations, so that there are in total approximately n sub-strata, and allocates one point per sub-stratum. In that way, our algorithm discretizes the domain into more reﬁned strata in regions where the function has higher variations. It cumulates the advantages of quasi Monte-Carlo and adaptive strategies. More precisely, our contributions are the following: • We prove an asymptotic lower bound on the mean squared error of the estimate returned by an optimal oracle strategy that has access to the variations of the function f everywhere and would use the best stratiﬁcation of the domain with hyper-cubes (possibly of heterogeneous sizes). This quantity, since this is a lower-bound on any oracle strategies, is smaller than the mean squared error of the estimate provided by Uniform stratiﬁed Monte-Carlo (which is the non-adaptive minimax-optimal strategy on the class of differentiable functions), and also smaller than crude Monte-Carlo. • We introduce the algorithm LMC-UCB, that sub-stratiﬁes the K strata in hyper-cubic substrata, and samples one point per sub-stratum. The number of sub-strata per stratum is linked to the variations of the function in the stratum. We prove that algorithm LMC-UCB is asymptotically as efﬁcient as the optimal oracle strategy. We also provide ﬁnite-time results when f admits a Taylor expansion of order 2 in every point. By tuning the number of strata K wisely, it is possible to build an algorithm that is almost as efﬁcient as the optimal oracle strategy. The paper is organized as follows. Section 2 deﬁnes the notations used throughout the paper. Section 3 states the asymptotic lower bound on the mean squared error of the optimal oracle strategy. In this Section, we also provide an intuition on how the number of samples into each stratum should be linked to the variation of the function in the stratum in order for the mean squared error of the estimate to be small. Section 4 presents the algorithm LMC-UCB and the ﬁrst Lemma on how many sub-strata are built in the initial strata. Section 5 ﬁnally states that the algorithm LMC-UCB is almost as efﬁcient as the optimal oracle strategy. We ﬁnally conclude the paper. Due to the lack of space, we also provide experiments and proofs in the Supplementary Material (see also [2]). 2 Setting We consider a function f : [0, 1]d →R. We want to estimate as accurately as possible its integral according to the Lebesgue measure, i.e. [0,1]d f(x)dx. In order to do that, we consider algorithms that stratify the domain in two layers of strata, one more reﬁned than the other. The strata of the reﬁned layer are referred to as sub-strata, and we sample in the sub-strata. We will compare the performances of the algorithms we construct, with the performances of the optimal oracle algorithm that has access to the variations \|\|∇f(x)\|\|2 of the function f everywhere in the domain, and is allowed to sample the domain where it wishes. The ﬁrst step is to partition the domain [0, 1]d in K measurable strata. In this paper, we assume that K1/d is an integer1. This enables us to partition, in a natural way, the domain in K hyper-cubic strata (Ωk)k≤K of same measure wk = 1 K . Each of these strata is a region of the domain [0, 1]d, and the K strata form a partition of the domain. We write µk = 1 wk Ωk f(x)dx the mean and σ2 k = 1 wk Ωk f(x)−µk 2dx the variance of a sample of the function f when sampling f at a point chosen at random according to the Lebesgue measure conditioned to stratum Ωk. 1This is not restrictive in small dimension, but it may become more constraining for large d. 2 We possess a budget of n samples (which is assumed to be known in advance), which means that we can sample n times the function at any point of [0, 1]d. We denote by A an algorithm that sequentially allocates the budget by sampling at round t in the stratum indexed by kt ∈{1, . . . , K}, and returns after all n samples have been used an estimate ˆµn of the integral of the function f. We consider strategies that sub-partition each stratum Ωk in hyper-cubes of same measure in Ωk, but of heterogeneous measure among the Ωk. In this way, the number of sub-strata in each stratum Ωk can adapt to the variations f within Ωk. The algorithms that we consider return a sub-partition of each stratum Ωk in Sk sub-strata. We call Nk = (Ωk,i)i≤Sk the sub-partition of stratum Ωk. In each of these sub-strata, the algorithm allocates at least one point2. We write Xk,i the ﬁrst point sampled uniformly at random in sub-stratum Ωk,i. We write wk,i the measure of the sub-stratum Ωk,i. Let us write µk,i = 1 wk,i Ωk,i f(x)dx the mean and σ2 k,i = 1 wk,i Ωk,i f(x) −µk,i 2dx the variance of a sample of f in sub-stratum Ωk,i (e.g. of Xk,i = f(Uk,i) where Uk,i ∼UΩk,i). This class of 2−layered sampling strategies is rather large. In fact it contains strategies that are similar to low discrepancy strategies, and also to any stratiﬁed Monte-Carlo strategy. For example, consider that all K strata are hyper-cubes of same measure 1 K and that each stratum Ωk is partitioned into Sk hyper-rectangles Ωk,i of minimal diameter and same measure 1 KSk . If the algorithm allocates one point per sub-stratum, its sampling scheme shares similarities with quasi Monte-Carlo sampling schemes, since the points at which the function is sampled are well spread. Let us now consider an algorithm that ﬁrst chooses the sub-partition (Nk)k and then allocates deterministically 1 sample uniformly at random in each sub-stratum Ωk,i. We consider the stratiﬁed estimate ˆµn = K k=1 Sk i=1 wk,i Sk Xk,i of µ. We have E(ˆµn) = K k=1 Sk i=1 wk,i Sk µk,i = k≤K Sk i=1 Ωk,i f(x)dx = [0,1]d f(x)dx = µ, and also V(ˆµn) = k≤K Sk i=1 (wk,i Sk )2E(Xk,i −µk,i)2 = k≤K Sk i=1 w2 k,i S2 k σ2 k,i. For a given algorithm A that builds for each stratum k a sub-partition Nk = (Ωk,i)i≤Sk, we call pseudo-risk the quantity Ln(A) = k≤K Sk i=1 w2 k,i S2 k σ2 k,i. (1) Some further insight on this quantity is provided in the paper [4]. Consider now the uniform strategy, i.e. a strategy that divides the domain in K = n hyper-cubic strata. This strategy is a fairly natural, minimax-optimal static strategy, on the class of differentiable function deﬁned on [0, 1]d, when no information on f is available. We will prove in the next Section that its asymptotic mean squared error is equal to 1 12 [0,1]d \|\|∇f(x)\|\|2 2dx 1 n1+ 2 d . This quantity is of order n−1−2/d, which is smaller, as expected, than 1/n: this strategy is more efﬁcient than crude Monte-Carlo. We will also prove in the next Section that the minimum asymptotic mean squared error of an optimal oracle strategy (we call it “oracle” because it builds the stratiﬁcation using the information about the variations \|\|∇f(x)\|\|2 of f in every point x), is larger than 1 12 [0,1]d(\|\|∇f(x)\|\|2) d d+1 dx 2 (d+1) d 1 n1+ 2 d This quantity is always smaller than the asymptotic mean squared error of the Uniform stratiﬁed Monte-Carlo strategy, which makes sense since this strategy assumes the knowledge of the variations of f everywhere, and can thus adapt accordingly the number of samples in each region. We deﬁne Σ = 1 12 [0,1]d(\|\|∇f(x)\|\|2) d d+1 dx 2 (d+1) d . (2) 2This implies that k Sk ≤n. 3 Given this minimum asymptotic mean squared error of an optimal oracle strategy, we deﬁne the pseudo-regret of an algorithm A as Rn(A) = Ln(A) −Σ 1 n1+ 2 d . (3) This pseudo-regret is the difference between the pseudo-risk of the estimate provided by algorithm A, and the lower-bound on the optimal oracle mean squared error. In other words, this pseudo-regret is the price an adaptive strategy pays for not knowing in advance the function f, and thus not having access to its variations. An efﬁcient adaptive strategy should aim at minimizing this gap coming from the lack of informations. 3 Discussion on the optimal asymptotic mean squared error 3.1 Asymptotic lower bound on the mean squared error, and comparison with the Uniform stratiﬁed Monte-Carlo A ﬁrst part of the analysis of the exposed problem consists in ﬁnding a good point of comparison for the pseudo-risk. The following Lemma states an asymptotic lower bound on the mean squared error of the optimal oracle sampling strategy. Lemma 1 Assume that f is such that ∇f is continuous and \|\|∇f(x)\|\|2 2dx < ∞. Let (Ωn k)k≤n n be an arbitrary sequence of partitions of [0, 1]d in n strata such that all the strata are hyper-cubes, and such that the maximum diameter of each stratum goes to 0 as n →+∞(but the strata are allowed to have heterogeneous measures).Let ˆµn be the stratiﬁed estimate of the function for the partition (Ωn k)k≤n when there is one point pulled at random per stratum. Then lim inf n→∞n1+2/dV(ˆµn) ≥Σ. The full proof of this Lemma is in the Supplementary Material, Appendix B (see also [2]). We have also the following equality for the asymptotic mean squared error of the uniform strategy. Lemma 2 Assume that f is such that ∇f is continuous and \|\|∇f(x)\|\|2 2dx < ∞. For any n = ld such that l is an integer (and thus such that it is possible to partition the domain in n hyper-cubic strata of same measure), deﬁne (Ωn k)k≤n n as the sequence of partitions in hyper-cubic strata of same measure 1/n. Let ˆµn be the stratiﬁed estimate of the function for the partition (Ωn k)k≤n when there is one point pulled at random per stratum. Then lim inf n→∞n1+2/dV(ˆµn) = 1 12 [0,1]d \|\|∇f(x)\|\|2 2dx . The proof of this Lemma is substantially similar to the proof of Lemma 1 in the Supplementary Material, Appendix B (see also [2]). The only difference is that the measure of each stratum Ωn k is 1/n and that in Step 2, instead of Fatou’s Lemma, the Theorem of dominated convergence is required. The optimal rate for the mean squared error, which is also the rate of the Uniform stratiﬁed MonteCarlo in Lemma 2, is n−1−2/d and is attained with ideas of low discrepancy sampling. The constant can however be improved (with respect to the constant in Lemma 2), by adapting to the speciﬁc shape of each function. In Lemma 1, we exhibit a lower bound for this constant (and without surprises, 1 12 [0,1]d \|\|∇f(x)\|\|2 2dx ≥Σ). Our aim is to build an adaptive sampling scheme, also sharing ideas with low discrepancy sampling, that attains this lower-bound. There is one main restriction in both Lemma: we impose that the sequence of partitions (Ωn k)k≤n n is composed only with strata that have the shape of an hyper-cube. This assumption is in fact reasonable: indeed, if the shape of the strata could be arbitrary, one could take the level sets (or approximate level sets as the number of strata is limited by n) as strata, and this would lead to limn→∞infΩ n1+2/dV(ˆµn,Ω) = 0. But this is not a fair competition, as the function is unknown, and determining these level sets is actually a much harder problem than integrating the function. The fact that the strata are hyper-cubes appears, in fact, in the bound. If we had chosen other shapes, e.g. l2 balls, the constant 1 12 in front of the bounds in both Lemma would change3. It is however not 3The 1 12 comes from computing the variance of an uniform random variable on [0, 1]. 4 possible to make a ﬁnite partition in l2 balls of [0, 1]d, and we chose hyper-cubes since it is quite easy to stratify [0, 1]d in hyper-cubic strata. The proof of Lemma 1 makes the quantity s∗(x) = (\|\|∇f(x)\|\|2) d d+1 [0,1]d(\|\|∇f(u)\|\|2) d d+1 du appear. This quantity is proposed as “asymptotic optimal allocation”, i.e. the asymptotically optimal number of sub-strata one would ideally create in any small sub-stratum centered in x. This is however not very useful for building an algorithm. The next Subsection provides an intuition on this matter. 3.2 An intuition of a good allocation: Piecewise linear functions In this Subsection, we (i) provide an example where the asymptotic optimal mean squared error is also the optimal mean squared error at ﬁnite distance and (ii) provide explicitly what is, in that case, a good allocation. We do that in order to give an intuition for the algorithm that we introduce in the next Section. We consider a partition in K hyper-cubic strata Ωk. Let us assume that the function f is afﬁne on all strata Ωk, i.e. on stratum Ωk, we have f(x) = θk, x + ρk I {x ∈Ωk}. In that case µk = f(ak) where ak is the center of the stratum Ωk. We then have: σ2 k = 1 wk Ωk (f(x) −f(ak))2dx = 1 wk Ωk θk, (x −ak) 2 dx = 1 wk \|\|θk\|\|2 2 12 w1+2/d k = \|\|θk\|\|2 2 12 w2/d k . We consider also a sub-partition of Ωk in Sk hyper-cubes of same size (we assume that S1/d k is an integer), and we assume that in each sub-stratum Ωk,i, we sample one point. We also have σ2 k,i = \|\|θk\|\|2 2 12 wk Sk 2/d for sub-stratum Ωk,i. For a given k and a given Sk, all the σk,i are equals. The pseudo-risk of an algorithm A that divides each stratum Ωk in Sk sub-strata is thus Ln(A) = k≤K i≤Sk w2 k S2 k \|\|θk\|\|2 2 12 wk Sk 2/d = k≤K w2+2/d k S1+2/d k \|\|θk\|\|2 2 12 = k≤K w2 k S1+2/d k σ2 k. If an unadaptive algorithm A∗has access to the variances σ2 k in the strata, it can choose to allocate the budget in order to minimize the pseudo-risk. After solving the simple optimization problem of minimizing Ln(A) with respect to (Sk)k, we deduce that an optimal oracle strategy on this stratiﬁcation would divide each stratum k in S∗ k = (wkσk) d d+1 i≤K(wiσi) d d+1 n sub-strata4. The pseudo-risk for this strategy is then Ln,K(A∗) = k≤K(wkσk) d d+1 2 (d+1) d n1+2/d = Σ 2 (d+1) d K n1+2/d , (4) where we write ΣK = i≤K(wiσi) d d+1 . We will call in the paper optimal proportions the quantities λK,k = (wkσk) d d+1 i≤K(wiσi) d d+1 . (5) In the speciﬁc case of functions that are piecewise linear, we have ΣK = k≤K(wkσk) d d+1 = k≤K(wk \|\|θk\|\|2 2 √ 3 w1/d k ) d d+1 = [0,1]d (\|\|∇f(x)\|\|2) d d+1 12 d 2(d+1) dx. We thus have Ln,K(A∗) = Σ 1 n1+ 2 d . (6) This optimal oracle strategy attains the lower bound in Lemma 1. We will thus construct, in the next Section, an algorithm that learns and adapts to the optimal proportions deﬁned in Equation 5. 4We deliberately forget about rounding issues in this Subsection. The allocation we provide might not be realizable (e.g. if S∗ k is not an integer), but plugging it in the bound provides a lower bound on any realizable performance. 5 4 The Algorithm LMC-UCB 4.1 Algorithm LMC-UCB We present the algorithm Lipschitz Monte Carlo Upper Conﬁdence Bound (LMC −UCB). It takes as parameter a partition (Ωk)k≤K in K ≤n hyper-cubic strata of same measure 1/K (it is possible since we assume that ∃l ∈N/ld = K). It also takes as parameter an uniform upper bound L on \|\|∇f(x)\|\|2 2, and δ, a (small) probability. The aim of algorithm LMC −UCB is to sub-stratify each stratum Ωk in λK,k = (wkσk) d d+1 K i=1(wiσi) d d+1 n hyper-cubic sub-strata of same measure and sample one point per sub-stratum. An intuition on why this target is relevant was provided in Section 3. Algorithm LMC-UCB starts by sub-stratifying each stratum Ωk in ¯S = n K d d+1 1/d d hypercubic strata of same measure. It is possible to do that since by deﬁnition, ¯S1/d is an integer. We write this ﬁrst sub-stratiﬁcation N k = (Ω k,i)i≤¯S. It then pulls one sample per sub-stratum in N k for each Ωk. It then sub-stratiﬁes again each stratum Ωk using the informations collected. It sub-stratiﬁes each stratum Ωk in Sk = max w d d+1 k ˆσk,K ¯S + A( wk ¯S )1/d 1 ¯S d d+1 K i=1 w d d+1 i ˆσi,K ¯S + A( wi¯S )1/d 1 ¯S d d+1 (n −K ¯S) 1/d d , ¯S (7) hyper-cubic strata of same measure (see Figure 1 for a deﬁnition of A). It is possible to do that because by deﬁnition, S1/d k is an integer. We call this sub-stratiﬁcation of stratum Ωk stratiﬁcation Nk = (Ωk,i)i≤Sk. In the last Equation, we compute the empirical standard deviation in stratum Ωk at time K ¯S as ˆσk,K ¯S = 1 ¯S −1 ¯S i=1 Xk,i −1 ¯S ¯S j=1 Xk,j 2 . (8) Algorithm LMC-UCB then samples in each sub-stratum Ωk,i one point. It is possible to do that since, by deﬁnition of Sk, k Sk + K ¯S ≤n The algorithm outputs an estimate ˆµn of the integral of f, computed with the ﬁrst point in each sub-stratum of partition Nk. We present in Figure 1 the pseudo-code of algorithm LMC-UCB. Input: Partition (Ωk)k≤K, L, δ, set A = 2L √ d log(2K/δ) Initialize: ∀k ≤K, sample 1 point in each stratum of partition N k Main algorithm: Compute Sk for each k ≤K Create partition Nk for each k ≤K Sample a point in Ωk,i ∈Nk for i ≤Sk Output: Return the estimate ˆµn computed when taking the ﬁrst point Xk,i in each sub-stratum Ωk,i of Nk, that is to say ˆµn = K k=1 wk Sk i=1 Xk,i Sk Figure 1: Pseudo-code of LMC-UCB. The deﬁnition of N k, ¯S, Nk, Ωk,i and Sk are in the main text. 4.2 High probability lower bound on the number of sub-strata of stratum Ωk We ﬁrst state an assumption on the function f. Assumption 1 The function f is such that ∇f exists and ∀x ∈[0, 1]d, \|\|∇f(x)\|\|2 2 ≤L. The next Lemma states that with high probability, the number Sk of sub-strata of stratum Ωk, in which there is at least one point, adjusts “almost” to the unknown optimal proportions. Lemma 3 Let Assumption 1 be satisﬁed and (Ωk)k≤K be a partition in K hyper-cubic strata of same measure. If n ≥4K, then with probability at least 1−δ, ∀k, the number of sub-strata satisﬁes Sk ≥max λK,k n −7(L + 1)d3/2 log(K/δ)(1 + 1 ΣK )K 1 d+1 n d d+1 , ¯S . The proof of this result is in the Supplementary Material (Appendix C) (see also [2]). 6 4.3 Remarks A sampling scheme that shares ideas with quasi Monte-Carlo methods: Algorithm LMC − UCB almost manages to divide each stratum Ωk in λK,kn hyper-cubic strata of same measure, each one of them containing at least one sample. It is thus possible to build a learning procedure that, at the same time, estimates the empirical proportions λK,k, and allocates the samples proportionally to them. The error terms: There are two reasons why we are not able to divide exactly each stratum Ωk in λK,kn hyper-cubic strata of same measure. The ﬁrst reason is that the true proportions λK,k are unknown, and that it is thus necessary to estimate them. The second reason is that we want to build strata that are hyper-cubes of same measure. The number of strata Sk needs thus to be such that S1/d k is an integer. We thus also loose efﬁciency because of rounding issues. 5 Main results 5.1 Asymptotic convergence of algorithm LMC-UCB By just combining the result of Lemma 1 with the result of Lemma 3, it is possible to show that algorithm LMC-UCB is asymptotically (when K goes to +∞and n ≥K) as efﬁcient as the optimal oracle strategy of Lemma 1. Theorem 1 Assume that ∇f is continuous, and that Assumption 1 is satisﬁed. Let (Ωn k)n,k≤Kn be an arbitrary sequence of partitions such that all the strata are hyper-cubes, such that 4Kn ≤n, such that the diameter of each strata goes to 0, and such that limn→+∞1 n Kn log(Knn2) d+1 2 = 0. The regret of LMC-UCB with parameter δn = 1 n2 on this sequence of partition, where for sequence (Ωn k)n,k≤Kn it disposes of n points, is such that lim n→∞n1+2/dRn(ALMC−UCB) = 0. The proof of this result is in the Supplementary Material (Appendix D) (see also [2]). 5.2 Under a slightly stronger Assumption We introduce the following Assumption, that is to say that f admits a Taylor expansion of order 2. Assumption 2 f admits a Taylor expansion at the second order in any point a ∈[0, 1]d and this expansion is such that ∀x, \|f(x) −f(a) −∇f, (x −a)\| ≤M\|\|x −a\|\|2 2 where M is a constant. This is a slightly stronger assumption than Assumption 1, since it imposes, additional to Assumption 1, that the variations of ∇f(x) are uniformly bounded for any x ∈[0, 1]d. Assumption 2 implies Assumption 1 since \|\|∇f(x)\|\|2−\|\|∇f(0)\|\|2 ≤M\|\|x−0\|\|2, which implies that \|\|∇f(x)\|\|2 ≤ \|\|∇f(0)\|\|2 + M √ d. This implies in particular that we can consider L = \|\|∇f(0)\|\|2 + M √ d. We however do not need M to tune the algorithm LMC-UCB, as long as we have access to L (although M appears in the bound of next Theorem). We can now prove a bound on the pseudo-regret. Theorem 2 Under Assumptions 1 and 2, if n ≥4K, the estimate returned by algorithm LMC − UCB is such that, with probability 1 −δ, we have Rn(ALMC−UCB) ≤ 1 n d+2 d M(L + 1)4 1 + 3Md Σ 4 650d3/2 log(K/δ)K 1 d+1 n− 1 d+1 + 25d 1 K 1 d+1 . A proof of this result is in the Supplementary Material (Appendix E) (see also [2]). Now we can choose optimally the number of strata so that we minimize the regret. Theorem 3 Under Assumptions 1 and 2, the algorithm LMC −UCB launched on Kn = (√n)1/dd hyper-cubic strata is such that, with probability 1 −δ, we have Rn(ALMC−UCB) ≤ 1 n1+ 2 d + 1 2(d+1) 700M(L + 1)4d3/2 1 + 3Md Σ 4 log(n/δ) . 7 5.3 Discussion Convergence of the algorithm LMC-UCB to the optimal oracle strategy: When the number of strata Kn grows to inﬁnity, but such that limn→+∞1 n Kn log(Knn2) d+1 2 = 0, the pseudoregret of algorithm LMC-UCB converges to 0. It means that this strategy is asymptotically as efﬁcient as (the lower bound on) the optimal oracle strategy. When f admits a Taylor expansion at the ﬁrst order in every point, it is also possible to obtain a ﬁnite-time bound on the pseudo-regret. A new sampling scheme: The algorithm LMC −UCB samples the points in a way that takes advantage of both stratiﬁed sampling and quasi Monte-Carlo. Indeed, LMC-UCB is designed to cumulate (i) the advantages of quasi Monte-Carlo by spreading the samples in the domain and (ii) the advantages of stratiﬁed, adaptive sampling by allocating more samples where the function has larger variations. For these reasons, this technique is efﬁcient on differentiable functions. We illustrate this assertion by numerical experiments in the Supplementary Material (Appendix A) (see also [2]). In high dimension: The bound on the pseudo-regret in Theorem 3 is of order n−1−2 d × poly(d)n− 1 2(d+1) . In order for the pseudo-regret to be negligible when compared to the optimal oracle mean squared error of the estimate (which is of order n−1−2 d ) it is necessary that poly(d)n− 1 2(d+1) is negligible compared to 1. In particular, this says that n should scale exponentially with the dimension d. This is unavoidable, since stratiﬁed sampling shrinks the approximation error to the asymptotic oracle only if the diameter of each stratum is small, i.e. if the space is stratiﬁed in every direction (and thus if n is exponential with d). However Uniform stratiﬁed Monte-Carlo, also for the same reasons, shares this problem5. We emphasize however the fact that a (slightly modiﬁed) version of our algorithm is more efﬁcient than crude Monte-Carlo, up to a negligible term that depends only of poly(log(d)). The bound in Lemma 3 depends of poly(d) only because of rounding issues, coming from the fact that we aim at dividing each stratum Ωk in hyper-cubic sub-strata. The whole budget is thus not completely used, and only k Sk + K ¯S samples are collected. By modifying LMC-UCB so that it allocates the remaining budget uniformly at random on the domain, it is possible to prove that the (modiﬁed) algorithm is always at least as efﬁcient as crude Monte-Carlo. Conclusion This work provides an adaptive method for estimating the integral of a differentiable function f. We ﬁrst proposed a benchmark for measuring efﬁciency: we proved that the asymptotic mean squared error of the estimate outputted by the optimal oracle strategy is lower bounded by Σ 1 n1+2/d . We then proposed an algorithm called LMC-UCB, which manages to learn the amplitude of the variations of f, to sample more points where theses variations are larger, and to spread these points in a way that is related to quasi Monte-Carlo sampling schemes. We proved that algorithm LMC-UCB is asymptotically as efﬁcient as the optimal, oracle strategy. Under the assumption that f admits a Taylor expansion in each point, we provide also a ﬁnite time bound for the pseudo-regret of algorithm LMC-UCB. We summarize in Table 1 the rates and ﬁnite-time bounds for crude Monte-Carlo, Uniform stratiﬁed Monte-Carlo and LMC-UCB. An interesting extension of this work would be to Pseudo-Risk: Sampling schemes Rate Asymptotic constant + Finite-time bound Crude MC 1 n [0,1]d f(x) − [0,1]d f(u)du 2dx +0 Uniform stratiﬁed MC 1 n1+ 2 d 1 12 [0,1]d \|\|∇f(x)\|\|2 2dx +O( d n1+ 2 d + 1 2d ) LMC-UCB 1 n1+ 2 d 1 12 [0,1]d(\|\|∇f(x)\|\|2) d d+1 dx 2 (d+1) d +O( d 11 2 n 1+ 2 d + 1 2(d+1) ) Table 1: Rate of convergence plus ﬁnite time bounds for Crude Monte-Carlo, Uniform stratiﬁed Monte Carlo (see Lemma 2) and LMC-UCB (see Theorems 1 and 3). adapt it to α−H¨older functions that admit a Riemann-Liouville derivative of order α. We believe that similar results could be obtained, with an optimal constant and a rate of order n1+2α/d. 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 270327 (project CompLACS), and by Pascal-2. 5When d is very large and n is not exponential in d, then second order terms, depending on the dimension, take over the bound in Lemma 2 (which is an asymptotic bound) and poly(d) appears in these negligible terms. 8 References [1] J.Y. Audibert, R. Munos, and Cs. Szepesv´ari. Exploration-exploitation tradeoff using variance estimates in multi-armed bandits. Theoretical Computer Science, 410(19):1876–1902, 2009. [2] A. Carpentier and R. Munos. Adaptive Stratiﬁed Sampling for Monte-Carlo integration of Differentiable functions. Technical report, arXiv:0575985, 2012. [3] A. Carpentier and R. Munos. Finite-time analysis of stratiﬁed sampling for monte carlo. In In Neural Information Processing Systems (NIPS), 2011a. [4] A. Carpentier and R. Munos. Finite-time analysis of stratiﬁed sampling for monte carlo. Technical report, INRIA-00636924, 2011b. [5] Pierre Etor´e and Benjamin Jourdain. Adaptive optimal allocation in stratiﬁed sampling methods. Methodol. Comput. Appl. Probab., 12(3):335–360, September 2010. [6] P. Glasserman. Monte Carlo methods in ﬁnancial engineering. Springer Verlag, 2004. ISBN 0387004513. [7] V. Grover. Active learning and its application to heteroscedastic problems. Department of Computing Science, Univ. of Alberta, MSc thesis, 2009. [8] A. Maurer and M. Pontil. Empirical bernstein bounds and sample-variance penalization. In Proceedings of the Twenty-Second Annual Conference on Learning Theory, pages 115–124, 2009. [9] H. Niederreiter. Quasi-monte carlo methods and pseudo-random numbers. Bull. Amer. Math. Soc, 84(6):957–1041, 1978. [10] R.Y. Rubinstein and D.P. Kroese. Simulation and the Monte Carlo method. Wiley-interscience, 2008. ISBN 0470177942. 9	2012	109
4,463	Learning Label Trees for Probabilistic Modelling of Implicit Feedback Andriy Mnih amnih@gatsby.ucl.ac.uk Gatsby Computational Neuroscience Unit University College London Yee Whye Teh ywteh@gatsby.ucl.ac.uk Gatsby Computational Neuroscience Unit University College London Abstract User preferences for items can be inferred from either explicit feedback, such as item ratings, or implicit feedback, such as rental histories. Research in collaborative ﬁltering has concentrated on explicit feedback, resulting in the development of accurate and scalable models. However, since explicit feedback is often difﬁcult to collect it is important to develop effective models that take advantage of the more widely available implicit feedback. We introduce a probabilistic approach to collaborative ﬁltering with implicit feedback based on modelling the user’s item selection process. In the interests of scalability, we restrict our attention to treestructured distributions over items and develop a principled and efﬁcient algorithm for learning item trees from data. We also identify a problem with a widely used protocol for evaluating implicit feedback models and propose a way of addressing it using a small quantity of explicit feedback data. 1 Introduction The rapidly growing number of products available online makes it increasingly difﬁcult for users to choose the ones worth their attention. Recommender systems assist users in making these choices by ranking the products based on inferred user preferences. Collaborative ﬁltering [6] has become the approach of choice for building recommender systems due to its ability to infer complex preference patterns from large collections of user preference data. Most collaborative ﬁltering research deals with inferring preferences from explicit feedback, for example ratings given to items. As a result, several effective methods have been developed for this version of the problem. Matrix factorization based models [13, 5, 12] have emerged as the most popular of these due to their simplicity and superior predictive performance. Such models are also highly scalable because their training algorithms take advantage of the sparsity of the rating matrix, resulting in training times that are linear in the number of observed ratings. However, since explicit feedback is often difﬁcult to collect it is essential to develop effective models that take advantage of the more abundant implicit feedback, such as logs of user purchases, rentals, or clicks. The difﬁculty of modelling implicit feedback comes from the fact that it contains only positive examples, since users explicitly express their interest, by selecting items, but not their disinterest. Note that not selecting a particular item is not necessarily an expression of disinterest, as it might also be due to the obscurity of the item, lack of time, or other reasons. Just like their explicit feedback counterparts, the most successful implicit feedback collaborative ﬁltering (IFCF) methods are based on matrix factorization [4, 10, 9]. However, instead of a highly sparse rating matrix, they approximate a dense binary matrix, where each entry indicates whether or not a particular user selected a particular item. We will collectively refer to such methods as Binary Matrix Factorization (BMF). Since such approaches treat unobserved user/item pairs as fake negative examples which can dominate the much less numerous positive examples, the contribution to the 1 objective function from the zero entries is typically downweighted. The matrix being approximated is no longer sparse, so models of this type are typically trained using batch alternating least squares. As a result, the training time is cubic in the number of latent factors, which makes these models less scalable than their explicit feedback counterparts. Recently [11] introduced a new method, called Bayesian Personalized Ranking (BPR), for modelling implicit feedback that is based on more realistic assumptions than BMF. Instead of assuming that users like the selected items and dislike the unselected ones, it assumes that users merely prefer the former to the latter. The model is presented with selected/unselected item pairs and is trained to rank the selected items above the unselected ones. Since the number of such pairs is typically very large, the unselected items are sampled at random. In this paper we develop a new method that explicitly models the user item selection process using a probabilistic model that, unlike the existing approaches, can generate new item lists. Like BPR it assumes that selected items are more interesting than the unselected ones. Unlike BPR, however, it represents the appeal of items to a user using a probability distribution, producing a complete ordering of items by probability value. In order to scale to large numbers of items efﬁciently, we restrict our attention to tree-structured distributions. Since the accuracy of the resulting models depends heavily on the choice of the tree structure, we develop an algorithm for learning trees from data that takes into account the structure of the model the tree will be used with. We then turn our attention to the task of evaluating implicit feedback models and point out a problem with a widely used evaluation protocol, which stems from the assumption that all items not selected by a user are irrelevant. Our proposed solution involves using a small quantity of explicit feedback to reliably identify the irrelevant items. 2 Modelling item selection We propose a new approach to collaborative ﬁltering with implicit feedback based on modelling the item selection process performed by each user. The identities of the items selected by a user are modelled as independent samples from a user-speciﬁc distribution over all available items. The probability of an item under this distribution reﬂects the user’s interest in it. Training our model amounts to performing multinomial density estimation for each user from the observed user/item pairs without explicitly considering the unobserved pairs. To make the modelling task more manageable we make two simplifying assumptions. First, we assume that user preferences do not change with time and model all items chosen by a user as independent samples from a ﬁxed user-speciﬁc distribution. Second, to keep the model as simple as possible we assume that items are sampled with replacement. We believe that sampling with replacement is a reasonable approximation to sampling without replacement in this case because the space of items is large while the number of items selected by a user is relatively small. These simpliﬁcations allow us to model the identities of the items selected by a user as IID samples. We now outline a simple implementation of the proposed idea which, though impractical for large datasets, will serve as a basis for developing a more scalable model. As is typical for matrix factorization methods in collaborative ﬁltering, we represent users and items with real-valued vectors of latent factors. The factor vectors for user u and item i will be denoted by Uu and Vi respectively. Intuitively, Uu captures the preferences of user u, while Vi encodes the properties of item i. Both user and item factor vectors are unobserved and so have to be learned from the observed user/item pairs. The dot product between Uu and Vi quantiﬁes the preference of user u for item i. We deﬁne the probability of user u choosing item i as P(i\|u) = exp(U ⊤ u Vi + ci) P k exp(U ⊤ u Vk + ck), (1) where ci is the bias parameter that captures the overall popularity of item i and index k ranges over all items in the inventory. The model can be trained using stochastic gradient ascent [2] on the log-likelihood by iterating through the user/item pairs in the training set, updating Uu, Vi, and ci based on the gradient of log P(i\|u). The main weakness of the model is that its training time is linear in the inventory size because computing the gradient of the log-probability of a single item requires explicitly considering all available items. Though linear time complexity might not seem 2 prohibitive, it severely limits the applicability of the model since collaborative ﬁltering tasks with tens or even hundreds of thousands of items are now common. 3 Hierarchical item selection model The linear time complexity of the gradient computation is a consequence of normalization over the entire inventory in Eq. 1, which is required because the space of items is unstructured. We can speed up normalization, and thus learning, exponentially by assuming that the space of items has a known tree structure. We start by supposing that we are given a K-ary tree with items at the leaves and exactly one item per leaf. For simplicity, we will assume that each item is located at exactly one leaf. Such a tree is uniquely determined by specifying for each item the path from the root to the leaf containing the item. Any such path can be represented by the sequence of nodes n = n0, n1, ..., nL it visits, where n0 is always the root node. By making the choice of the next node stochastic, we can induce a distribution over the leaf nodes in the tree and thus over items. To allow each user to have a different distribution over items we make the probability of choosing each child a function of the user’s factor vector. The probability will also depend on the child node’s factor vector and bias the same way the probability of choosing an item in Eq. 1 depends on the item’s factor vector and bias. Let C(n) be the set of children of node n. Then for user u, the probability of moving from node nj to node n on a root-to-leaf tree traversal is given by P(n\|nj, u) = exp U ⊤ u Qn + bn P m∈C(nj) exp (U ⊤ u Qm + bm), (2) if n is a child of nj and 0 otherwise. Here Qn and bn are the factor vector and the bias of node n. The probability of selecting item i is then given by the product of the probabilities of the decisions that lead from the root to the leaf containing i: P(i\|u) = QLi j=1 P(ni j\|ni j−1, u). (3) We will call the model deﬁned by Eq. 3 the Collaborative Item Selection (CIS) model. Given a tree over items, the CIS model can be trained using stochastic gradient ascent in log-likelihood, updating parameters after each user/item pair. While the model can use any tree over items, the choice of the tree affects the model’s efﬁciency and ability to generalize. Since computing the probability of a single item takes time linear in the item’s depth in the tree, we want to avoid trees that are too unbalanced. To produce a model that generalizes well we also want to avoid trees with difﬁcult classiﬁcation problems at the internal nodes [1], which correspond to hard-to-predict item paths. One way to produce a tree that results in relatively easy classiﬁcation problems is to assign similar items to the same class, which is the approach of [7] and [14]. However, the similarity metrics used by these methods are not model-based in the sense that they are not derived from the classiﬁers that will be used at the tree nodes. In Section 5 we will develop a scalable model-based algorithm for learning trees with item paths that are easy to predict using Eq. 2. 4 Related work The use of tree-structured label spaces to reduce the normalization cost has originated in statistical language modelling, where it was used to accelerate neural and maximum-entropy language models [3, 8]. The task of learning trees for efﬁcient probabilistic multiclass classiﬁcation has received surprisingly little attention. The two algorithms most closely related to the one proposed in this paper are [1] and [7]. [1] proposed a fully online algorithm for multinomial density estimation that constructs a binary label tree by inserting the previously unseen labels whenever they are encountered. The location for a new label is found proceeding from the root to a leaf making the left child/right child decisions based on their probability under the model and a tree balancing penalty. This is the only tree-learning algorithm we are aware of that takes into account the probabilistic model the tree is used with. Unfortunately, this approach is very optimistic because it decides on the location for a new label in the tree based on a single training case and never revisits that decision. 3 The algorithm in [7] was developed for learning trees over words for use in probabilistic language models. It constructs such trees by performing top-down hierarchical clustering of words, which are represented by real-valued vectors. The word representations are learned through bootstrapping by training a language model based on a random tree. This algorithm, unlike the one we propose in Section 5, does not take into consideration the model the tree is constructed for. Most work on tree-based multiclass classiﬁcation deals with non-probabilistic models and does not apply to the problem we are concerned with in this paper. Of these approaches our algorithm is most similar to the one in [14], which looks for a tree structure that avoids requiring to discriminate between easily confused items as much as possible. The main weakness of that approach is the need for training a ﬂat classiﬁer to produce the confusion matrix needed by the algorithm. As a result, it is unlikely to scale to large datasets containing tens of thousands of classes. 5 Model-based learning of item trees 5.1 Overview In this section we develop a scalable algorithm for learning trees that takes into account the parametric form of the model the tree will be used with. At the highest level our approach can be seen as top-down model-based hierarchical clustering of items. We chose top-down clustering over bottomup clustering because it is the more scalable option. Since ﬁnding the best tree is intractable, we take a greedy approach that constructs the tree one level at a time, learning the lth node of each item path before ﬁxing it and advancing to the (l + 1)st node. Because our approach is model-based, it learns model parameters, i.e. node biases and factor vectors, jointly with the item paths. As a result, at every point during its execution it speciﬁes a complete probabilistic model of the data, which becomes more expressive with each additional tree level. This makes it possible to monitor the progress of the algorithm by evaluating the predictions made after learning each level. For simplicity, our tree-learning algorithm assumes that user factor vectors are known and ﬁxed. Since these vectors are actually unknown, we learn them by ﬁrst training a CIS model based on a random balanced tree. We then extract the user vectors learned by the model and use them to learn a better tree from the data. Finally, we train a CIS model based on the learned tree, updating all the parameters, including the user vectors. This three-stage approach is similar to the one used in [7] to learn trees over words. However, because our tree-learning algorithm is model-based, we already have a complete probabilistic model at its termination, so we only need to ﬁnetune its parameters instead of learning them from scratch. Finetuning is necessary because the parameters learned while building the tree are based on the ﬁxed user factor vectors from the random-tree-based model. Though it is possible to continue alternating between optimizing over the tree structure and over user vectors, we found the resulting gains too small to be worth the computational cost. 5.2 Learning a level of a tree We now describe how to learn a level of the tree. Suppose we have learned the ﬁrst l −1 nodes of each item path and would like to learn the lth node. Let Ui be the set of users who rated item i in the training set. The contribution made by item i to the log-likelihood is then given by Li = log Q u∈Ui P(i\|u) = P u∈Ui log Q j P(ni j\|ni j−1, u) = P u∈Ui P j log P(ni j\|ni j−1, u). (4) The log-likelihood contribution due to a single observation can be expressed as P j log P(ni j\|ni j−1, u) = Pl−1 j=1 log P(ni j\|ni j−1, u) + log P(ni l\|ni l−1, u)+ (5) PLi j=l+1 log P(ni j\|ni j−1, u). The ﬁrst term on the RHS depends only on the parameters and path nodes that have already been learned, so it can be left out of the objective function. The third term is the log-probability of item i under the subtree rooted at node ni l, which depends on the structure and parameters of that subtree, which we have not learned yet. To emphasize the fact that this term is based on a user-dependent distribution over items under node ni l we will denote it by log P(i\|ni l, u). The overall objective function for learning level l is obtained by adding up the contributions of all items, leaving out the terms that do not depend on the quantities to be learned: Ll = P i P u∈Ui log P(ni l\|ni l−1, u) + P i P u∈Ui log P(i\|ni l, u). (6) 4 The most direct approach to learning the paths would be to alternate between updating the lth node in the paths and the corresponding factor vectors and biases. Since jointly optimizing over the lth node in all item paths is infeasible, we have to resort to incremental updates, maximizing Ll over the lth node in one item path at a time. Unfortunately, even this operation is intractable because evaluating each value of ni l requires knowing the optimal contribution from the still-to-be-learned levels of the tree, which is the second term in Eq. 6. In other words, to ﬁnd the optimal ni l we need to compute ni l = arg maxn∈C(ni l−1) P u∈Ui log P(n\|ni l−1, u) + F(n, ni l−1) , (7) where we left out the terms that do not depend on ni l. The optimal contribution F(ni l, ni l−1) from the future levels is deﬁned as F(ni l, ni l−1) = max Θ P k∈I(ni l−1) P u∈Uk log P(k\|nk l , u), (8) where I(ni l−1) is the set of items that are assigned to node ni l−1, and Θ is the set of node factor vectors, biases, and tree structures that parameterize the set of distributions {P(k\|nk l , u)\|k ∈I(ni l−1)}. 5.3 Approximating the future The value of F(ni l, ni l−1) quantiﬁes the difﬁculty of discriminating between items assigned to node ni l−1 using the best tree structure and parameter setting possible given that item i is assigned to the child ni l of that node. Since F(ni l, ni l−1) in Eq. 8 rules out degenerate solutions where all items below a node are assigned to the same child of it, leaving F(ni l, ni l−1) out to make the optimization problem easier is not an option. We address the intractability of Eq. 7 while avoiding the degenerate solutions by approximating the user-dependent distributions P(k\|nk l , u) by simpler distributions that make it much easier to evaluate F(ni l, ni l−1) for each candidate value for ni l. Since computing F(ni l, ni l−1) requires maximizing over the free parameters of P(k\|nk l , u), choosing a parameterization of P(k\|nk l , u) that makes this maximization easy can greatly speed up this computation. We approximate the tree-structured userdependent P(k\|nk l , u) with a ﬂat user-independent distribution P(k\|nk l ). The main advantage of this parameterization is that the optimal P(k\|nk l ) can be computed by counting the number of times each item assigned to node nk l occurs in the training data and normalizing. In other words, when P(k\|nk l ) is used in Eq. 8, the maximum is achieved at P(i\|nk l ) =    Ni P m∈I(nk l ) Nm if i ∈I(nk l ) 0 otherwise (9) where Ni is the number of times item i occurs in the training set. The corresponding value for F(ni l, ni l−1) is given by F(ni l, ni l−1) = P k∈I(ni l−1) Nk log Nk P m∈I(nk l ) Nm . (10) To show that F(ni l, ni l−1) can be computed in constant time we start by observing that the sum over items under node ni l−1 can be written in terms of sums over items under each of its children: F(ni l, ni l−1) = P c∈C(ni l−1) P k∈I(c) Nk log Nk P m∈I(c) Nm = P c∈C(ni l−1) P k∈I(c) Nk log Nk −P c∈C(ni l−1) Zc log Zc. (11) with Zc = P k∈I(c) Nk. Since adding a constant to F(ni l, ni l−1) has no effect on the solution of Eq. 7 and the ﬁrst term in the equation does not depend on ni l, we can drop it to get ˜F(ni l, ni l−1) = −P c∈C(ni l−1) Zc log Zc. (12) To compute ˜F(ni l, ni l−1) efﬁciently, we store Zc’s and the old ˜F(ni l, ni l−1) value, updating them whenever an item is moved to a different node. Such updates can be performed in constant time. 5 We now show that the ﬁrst term in Eq. 7, corresponding to the contribution of the lth path node for item i, can be computed efﬁciently. Plugging in the deﬁnition of P(n\|nj, u) from Eq. 2 we get P u∈Ui log P(n\|ni l−1, u) = P u∈Ui U ⊤ u Qn + bn + C = P u∈Ui Uu ⊤Qn + \|Ui\|bn + C (13) where C is a term that does not depend on n and so does not have to be considered when maximizing over n. Since we assume that the user factor vectors are known and ﬁxed, we precompute Ri = P u∈Ui Uu for each user, which can be seen as creating a surrogate representation for item i. Finally, plugging Eq. 13 into Eq. 7 gives us the following update for item nodes: ni l = arg maxn∈C(ni l−1) R⊤ i Qn + \|Ui\|bn + ˜F(n, ni l−1) . (14) 6 Evaluating models of implicit feedback Establishing sensible evaluation protocols for machine learning problems is important because they effectively deﬁne what “better” performance means and implicitly guide the development of future methods. Given that the problem of implicit feedback collaborative ﬁltering is relatively new, it is not surprising that the typical evaluation protocol was adopted from information retrieval. However, we believe that this protocol is much less appropriate in collaborative ﬁltering than it is in its ﬁeld of origin. Implicit feedback models are typically evaluated using information retrieval metrics, such as Mean Average Precision (MAP), that require knowing which items are relevant and which are irrelevant to each user. It is typical to assume that the items the user selected are relevant and all others are not [10]. However, this approach is problematic because it fails to distinguish between the items the user really has no interest in (i.e. the truly irrelevant ones) and the relevant items the user simply did not rate. And while the irrelevant items do tend to be far more numerous than the unobserved relevant ones, the effect of the latter can still be strong enough to affect model comparison, as we demonstrate in the next section. To address this issue, we propose using some explicit feedback information to identify a small number of truly irrelevant items for each user and using them in place of items of unknown relevance in the evaluation. Thus the models will be evaluated on their ability to rank the truly relevant items above the truly irrelevant ones, which we believe is the ultimate task of collaborative ﬁltering. Though this approach does require access to explicit feedback, only a small quantity of it is necessary, and it is used only for evaluation. For probabilistic models P(i\|u), the most natural performance metrics are log-probability of the held-out data D and the closely-related perplexity (PPL), the standard metric for language models: PPL = exp −1 \|D\| P (u,i)∈D log P(i\|u) . (15) The model that assigns the correct item probability 1 has the perplexity of 1, while the model that assigns all N items the same probability (1/N) has the perplexity of N. Unlike the ranking metrics above, perplexity is computed based on the selected/relevant items alone and does not require assuming that the unselected items are irrelevant.1 7 Experimental results First we investigated the impact of using tree-structured distributions over items by comparing the performance of tree-based CIS models to that of a ﬂat model deﬁned by Eq. 1. We used MovieLens 1M, which is a fairly small dataset, for the comparison in order to be able to train the ﬂat model within reasonable time. The dataset contains 1M ratings on a scale from 1 to 5 given by 6040 users to 3952 movies. To simulate the implicit feedback setting, where the presence of a user/item pair indicates an expression of interest, we kept only the user/item pairs associated with ratings 4 and above (and discarded the rating values) and split the resulting 575K pairs into a 475K-pair training set, and a validation and test sets of 50K pairs each. We trained three models with 5-dimensional 1The implicit assumption here is that the selected items are more relevant than the unselected ones. 6 Table 1: Test set scores in percent on the MovieLens 10M dataset obtained by treating items with low ratings as irrelevant. Higher scores indicate better performance for all metrics except for perplexity. Model PPL MAP P@1 P@5 P@10 R@1 R@5 R@10 CIS (Random) 921 70.68 74.65 58.02 49.91 20.66 60.02 77.31 CIS (LearnedRI) 822 72.50 76.64 59.29 50.64 21.51 61.24 78.22 CIS (LearnedCI) 820 72.61 76.68 59.37 50.69 21.54 61.31 78.27 BPR 865 72.75 75.75 59.15 50.63 21.50 61.43 78.39 BMF – 70.80 75.66 58.03 49.77 20.94 60.04 77.21 Table 2: Test set scores in percent on the MovieLens 10M dataset obtained by treating all unobserved items as irrelevant. Model MAP P@1 P@5 P@10 R@1 R@5 R@10 BPR 12.73 14.27 11.56 9.89 3.06 11.55 18.86 BMF 16.13 22.10 16.25 12.94 4.66 15.64 23.55 factor vectors: a ﬂat model, a CIS model with a random balanced binary tree, and a CIS model with a learned binary tree (as in Section 5). The ﬂat model took 12 hours to train and had the test set perplexity of 920. Training the random tree model took half an hour, resulting in the perplexity of 975. The training process for the learned-tree model, which included training a random-tree model, learning a tree from the resulting user factor vectors, and ﬁnetuning all the parameters, took 1 hour. The resulting model performed very well, achieving the test set perplexity of 912. These results suggest that even when the number of items is relatively small our tree-based approach to item selection modelling can yield an order-of-magnitude reduction in training times relative to the ﬂat model without hurting the predictive accuracy. We then used the larger MovieLens 10M dataset (0-5 rating scale, 69878 users, 10677 movies) to compare the proposed approach to the existing IFCF methods. As on MovieLens 1M, we kept only the user/item pairs with ratings 4 and above, producing a 4M-pair training set, a 500K-pair validation set, and a 500K-pair test set. We compared the models based on their perplexity and ranking performance as measured by the standard information retrieval metrics: Mean Average Precision (MAP), Precision@k, and Recall@k. We used the evaluation approach described in the previous section, which involved having the models rank only the items with known relevance status. We used the rating values to determine relevance, considering items rated below 3 as irrelevant and items rated 4 and above as relevant. We compared our hierarchical item selection model to two state-of-the-art models for implicit feedback: the Bayesian Personalized Ranking model (BPR) and the Binary Matrix Factorization model (BMF). All models used 25-dimensional factor vectors, as we found that higher-dimensional factor vectors resulted in only marginal improvements. We included three CIS models based on different binary trees (K = 2) to highlight the effect of tree construction methods. The methods are as follows: “Random” generates random balanced trees; “LearnedRI” is the method from Section 5 with randomly initialized item-node assignments; “LearnedCI” is the same method with item-node assignments initialized by clustering surrogate item representations Ri from Section 5.3. Training a ﬂat item selection model on this dataset was infeasible, as a single pass through the data took six hours, compared to a mere two minutes for CIS (LearnedCI). Better performance corresponds to lower values of perplexity and higher values of the other metrics. Table 1 shows the test scores for the resulting models. In terms of perplexity, CIS (Learned) is the top performer, with BPR coming in second and CIS (Random) a distant third. Since BMF does not produce a distribution over items, its performance cannot be naturally measured in terms of PPL. On the ranking metrics, CIS (Learned) and BPR emerge as the best-performing methods, achieving very similar scores. BPR has a slight edge over CIS on MAP, while CIS performs better on Precision@1. BMF and CIS (Random) are the weakest performers, with considerably worse scores than BPR and CIS (Learned) on all metrics. Comparing the results of CIS (Learned) and CIS (Random) shows that the of the tree used has a strong effect on the performance of CIS models and that using trees learned with the proposed algorithm makes CIS competitive with the best collaborative ﬁltering models. The similar results achieved by CIS (LearnedRI) and CIS (LearnedCI) suggest that that the performance 7 of the resulting model is not particularly sensitive to the initialization scheme of the tree-learning algorithm. To understand the behaviour of our tree-learning algorithm better we examined the trees produced by it. The learned trees looked sensible, with neighbouring leaves typically containing movies from the same sub-genre and appealing to the same audience. We then determined how discriminative the decisions were at each level of the tree by replacing the user-dependent distributions under all nodes at a particular depth by the best user-independent approximations (frequencies of items under the node). Comparing the perplexity of a model using the tree truncated at level l and at level l + 1 allowed us to determine how much level l + 1 contributed to the model. In the CIS (Random) model, the ﬁrst few and the last few levels had little effect on perplexity and the medium-depth levels accounted for most of perplexity reduction. In contrast, in the CIS (LearnedRI) model, the effect of a level on perplexity decreased with level depth, with the ﬁrst few levels reducing perplexity the most, which is a consequence of the greedy nature of the tree-learning algorithm. To highlight the importance of excluding items of unknown relevance when evaluating implicit feedback models we recomputed the performance metrics treating all items not rated by a user as irrelevant. As the scores in Table 2 show this seemingly minor modiﬁcation of the evaluation protocol makes BMF appear to outperform BPR by a large margin, which, as Table 1 indicates is not actually the case. In retrospect, these changes in relative performance are not particularly surprising since the training algorithm for BMF treats unobserved items as negative examples, which perfectly matches the assumption the evaluation is based on, namely that unobserved items are irrelevant. This is a clear example of a ﬂawed evaluation protocol favouring an unrealistic modelling assumption. 8 Discussion We proposed a model that in addition to being competitive with the best implicit feedback models in terms of predictive accuracy also provides calibrated item selection probabilities for each user, which quantify the user’s interest in the items. These probabilities allow comparing the degree of interest in an item across users, making it possible to maximize the total user satisfaction when item availability is limited. More generally, the probabilities provided by the model can be used in combination with utility functions for making sophisticated decisions. Although we introduced our tree-learning algorithm in the context of collaborative ﬁltering, it is applicable to several other problems. One such problem is statistical language modelling, where the task is to predict the distribution of the next word in a sentence given its context consisting of several preceding words. While there already exists an algorithm for learning the structure of treebased language models [7], it constructs trees by clustering word representations, not taking into account the form of the model that will use these trees. In contrast, our algorithm optimizes the tree structure and model parameters jointly, which can lead to superior model performance. The proposed algorithm can also be used to learn trees over labels for multinomial regression models. When the number of labels is large, using a label space with a sensible tree structure can lead to much faster training and improved generalization. Our algorithm can be applied in this setting by noticing the correspondence between items and labels, and between user factor vectors and input vectors. However, unlike in collaborative ﬁltering where user factor vectors have to be learned, in this case input vectors are observed, which eliminates the need to train a model based on a random tree before applying the tree-learning algorithm. We believe that evaluation protocols for implicit feedback models deserve more attention than they have received. In this paper we observed that one widely used protocol can produce misleading results due to an unrealistic assumption it makes about item relevance. We proposed using a small quantity of explicit feedback data to directly estimate item relevance in order to avoid having to make that assumption. Acknowledgments We thank Biljana Petreska and Lloyd Elliot for their helpful comments and the Gatsby Charitable Foundation for generous funding. The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 270327. 8 References [1] Alina Beygelzimer, John Langford, Yuri Lifshits, Gregory B. Sorkin, and Alexander L. Strehl. Conditional probability tree estimation analysis and algorithms. In Proceedings of the 25th Conference on Uncertainty in Artiﬁcial Intelligence, 2009. [2] L´eon Bottou. Large-scale machine learning with stochastic gradient descent. In Proceedings of the 19th International Conference on Computational Statistics (COMPSTAT’2010), pages 177–187, 2010. [3] J. Goodman. Classes for fast maximum entropy training. In Proceedings of ICASSP ’01, volume 1, pages 561–564, 2001. [4] Yifan Hu, Yehuda Koren, and Chris Volinsky. Collaborative ﬁltering for implicit feedback datasets. In Proceedings of the 2008 Eighth IEEE International Conference on Data Mining, pages 263–272, 2008. [5] Yehuda Koren. Factorization meets the neighborhood: a multifaceted collaborative ﬁltering model. In Proceeding of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 426–434, 2008. [6] Benjamin Marlin. Collaborative ﬁltering: A machine learning perspective. Master’s thesis, University of Toronto, 2004. [7] Andriy Mnih and Geoffrey Hinton. A scalable hierarchical distributed language model. In Advances in Neural Information Processing Systems, volume 21, 2009. [8] Frederic Morin and Yoshua Bengio. Hierarchical probabilistic neural network language model. In AISTATS’05, pages 246–252, 2005. [9] Rong Pan and Martin Scholz. Mind the gaps: weighting the unknown in large-scale one-class collaborative ﬁltering. In KDD, pages 667–676, 2009. [10] Rong Pan, Yunhong Zhou, Bin Cao, Nathan Nan Liu, Rajan M. Lukose, Martin Scholz, and Qiang Yang. One-class collaborative ﬁltering. In ICDM, pages 502–511, 2008. [11] Steffen Rendle, Christoph Freudenthaler, Zeno Gantner, and Schmidt-Thieme Lars. BPR: Bayesian personalized ranking from implicit feedback. In UAI ’09, pages 452–461, 2009. [12] Ruslan Salakhutdinov and Andriy Mnih. Probabilistic matrix factorization. In Advances in Neural Information Processing Systems, volume 20, 2008. [13] Nathan Srebro, Jason D. M. Rennie, and Tommi Jaakkola. Maximum-margin matrix factorization. In Advances in Neural Information Processing Systems, 2004. [14] Jason Weston, Samy Bengio, and David Grangier. Label embedding trees for large multi-class tasks. In Advances in Neural Information Processing Systems (NIPS), 2010. 9	2012	11
4,464	Bayesian Nonparametric Modeling of Suicide Attempts Francisco J. R. Ruiz Department of Signal Processing and Communications University Carlos III in Madrid franrruiz@tsc.uc3m.es Isabel Valera Department of Signal Processing and Communications University Carlos III in Madrid ivalera@tsc.uc3m.es Carlos Blanco Columbia University College of Physicians and Surgeons Cblanco@nyspi.columbia.edu Fernando Perez-Cruz Department of Signal Processing and Communications University Carlos III in Madrid fernando@tsc.uc3m.es Abstract The National Epidemiologic Survey on Alcohol and Related Conditions (NESARC) database contains a large amount of information, regarding the way of life, medical conditions, etc., of a representative sample of the U.S. population. In this paper, we are interested in seeking the hidden causes behind the suicide attempts, for which we propose to model the subjects using a nonparametric latent model based on the Indian Buffet Process (IBP). Due to the nature of the data, we need to adapt the observation model for discrete random variables. We propose a generative model in which the observations are drawn from a multinomial-logit distribution given the IBP matrix. The implementation of an efﬁcient Gibbs sampler is accomplished using the Laplace approximation, which allows integrating out the weighting factors of the multinomial-logit likelihood model. Finally, the experiments over the NESARC database show that our model properly captures some of the hidden causes that model suicide attempts. 1 Introduction Every year, more than 34,000 suicides occur and over 370,000 individuals are treated for selfinﬂicted injuries in emergency rooms in the U.S., where suicide prevention is one of the top public health priorities [1]. The current strategies for suicide prevention have focused mainly on both the detection and treatment of mental disorders [13], and on the treatment of the suicidal behaviors themselves [4]. However, despite prevention efforts including improvements in the treatment of depression, the lifetime prevalence of suicide attempts in the U.S. has remained unchanged over the past decade [8]. This suggests that there is a need to improve understanding of the risk factors for suicide attempts beyond psychiatric disorders, particularly in non-clinical populations. According to the National Strategy for Suicide Prevention, an important ﬁrst step in a public health approach to suicide prevention is to identify those at increased risk for suicide attempts [1]. Suicide attempts are, by far, the best predictor of completed suicide [12] and are also associated with major morbidity themselves [11]. The estimation of suicide attempt risk is a challenging and complex task, with multiple risk factors linked to increased risk. In the absence of reliable tools for identifying those at risk for suicide attempts, be they clinical or laboratory tests, risk detection still relays mainly on clinical variables. The adequacy of the current predictive models and screening methods has been 1 questioned [12], and it has been suggested that the methods currently used for research on suicide risk factors and prediction models need revamping [9]. Databases that model the behavior of human populations present typically many related questions and analyzing each one of them individually, or a small group of them, do not lead to conclusive results. For example, the National Epidemiologic Survey on Alcohol and Related Conditions (NESARC) samples the U.S. population with nearly 3,000 questions regarding, among others, their way of life, their medical conditions, depression and other mental disorders. It contains yes-or-no questions, and some multiple-choice and questions with ordinal answers. In this paper, we propose to model the subjects in this database using a nonparametric latent model that allows us to seek hidden causes and compact in a few features the immense redundant information. Our starting point is the Indian Buffet Process (IBP) [5], because it allows us to infer which latent features inﬂuence the observations and how many features there are. We need to adapt the observation model for discrete random variables, as the discrete nature of the database does not allow us to use the standard Gaussian observation model. There are several options for modeling discrete outputs given the hidden latent features, like a Dirichlet distribution or sampling from the features, but we prefer a generative model in which the observations are drawn from a multinomial-logit distribution because it is similar to the standard Gaussian observation model, where the observation probability distribution depends on the IBP matrix weighted by some factors. Furthermore, the multinomial-logit model, besides its versatility, allows the implementation of an efﬁcient Gibbs sampler where the Laplace approximation [10] is used to integrate out the weighting factors, which can be efﬁciently computed using the Matrix Inversion Lemma. The IBP model combined with discrete observations has already been tackled in several related works. In [17], the authors propose a model that combines properties from both the hierarchical Dirichlet process (HDP) and the IBP, called IBP compound Dirichlet (ICD) process. They apply the ICD to focused topic modeling, where the instances are documents and the observations are words from a ﬁnite vocabulary, and focus on decoupling the prevalence of a topic in a document and its prevalence in all documents. Despite the discrete nature of the observations under this model, these assumptions are not appropriate for categorical observations such as the set of possible responses to the questions in the NESARC database. Titsias [14] introduced the inﬁnite gamma-Poisson process as a prior probability distribution over non-negative integer valued matrices with a potentially inﬁnite number of columns, and he applies it to topic modeling of images. In this model, each (discrete) component in the observation vector of an instance depends only on one of the active latent features of that object, randomly drawn from a multinomial distribution. Therefore, different components of the observation vector might be equally distributed. Our model is more ﬂexible in the sense that it allows different probability distributions for every component in the observation vector, which is accomplished by weighting differently the latent variables. 2 The Indian Buffet Process In latent feature modeling, each object can be represented by a vector of latent features, and the observations are generated from a distribution determined by those latent feature values. Typically, we have access to the set of observations and the main goal of these models is to ﬁnd out the latent variables that represent the data. The most common nonparametric tool for latent feature modeling is the Indian Buffet Process (IBP). The IBP places a prior distribution over binary matrices where the number of columns (features) K is not bounded, i.e., K →∞. However, given a ﬁnite number of data points N, it ensures that the number of non-zero columns K+ is ﬁnite with probability one. Let Z be a random N × K binary matrix distributed following an IBP, i.e., Z ∼IBP(α), where α is the concentration parameter of the process. The nth row of Z, denoted by zn·, represents the vector of latent features of the nth data point, and every entry nk is denoted by znk. Note that each element znk ∈{0, 1} indicates whether the kth feature contributes to the nth data point. Given a binary latent feature matrix Z, we assume that the N × D observation matrix X, where the nth row contains a D-dimensional observation vector xn·, is distributed according to a probability distribution p(X\|Z). Additionally, x·d stands for the dth column of X, and each element of the 2 matrix is denoted by xnd. For instance, in the standard observation model described in [5], p(X\|Z) is a Gaussian probability density function. MCMC (Markov Chain Monte Carlo) methods have been broadly applied to infer the latent structure Z from a given observation matrix X (see, e.g., [5, 17, 15, 14]). In particular, we focus on the use of Gibbs sampling for posterior inference over the latent variables. The algorithm iteratively samples the value of each element znk given the remaining variables, i.e., it samples from p(znk = 1\|X, Z¬nk) ∝p(X\|Z)p(znk = 1\|Z¬nk), (1) where Z¬nk denotes all the entries of Z other than znk. The distribution p(znk = 1\|Z¬nk) can be readily derived from the exchangeable IBP and can be written as p(znk = 1\|Z¬nk) = m−n,k/N, where m−n,k is the number of data points with feature k, not including n, i.e., m−n,k = P i̸=n zik. 3 Observation model Let us consider that the observations are discrete, i.e., each element xnd ∈{1, . . . , Rd}, where this ﬁnite set contains the indexes to all the possible values of xnd. For simplicity and without loss of generality, we consider that Rd = R, but the following results can be readily extended to a different cardinality per input dimension, as well as mixing continuous variables with discrete variables, since given the latent matrix Z the columns of X are assumed to be independent. We introduce matrices Bd of size K × R to model the probability distribution over X, such that Bd links the hidden latent variables with the dth column of the observation matrix X. We assume that the probability of xnd taking value r (r = 1, . . . , R), denoted by πr nd, is given by the multiplelogistic function, i.e., πr nd = p(xnd = r\|zn·, Bd) = exp (zn·bd ·r) R X r′=1 exp (zn·bd ·r′) , (2) where bd ·r denotes the rth column of Bd. Note that the matrices Bd are used to weight differently the contribution of every latent feature for every component d, similarly as in the standard Gaussian observation model in [5]. We assume that the mixing vectors bd ·r are Gaussian distributed with zero mean and covariance matrix Σb = σ2 BI. The choice of the observation model in Eq. 2, which combines the multiple-logistic function with Gaussian parameters, is based on the fact that it induces dependencies among the probabilities πr nd that cannot be captured with other distributions, such as the Dirichlet distribution [2]. Furthermore, this multinomial-logistic normal distribution has been widely used to deﬁne probability distributions over discrete random variables (see, e.g., [16, 2]). We consider that elements xnd are independent given the latent feature matrix Z and the D matrices Bd. Then, the likelihood for any matrix X can be expressed as p(X\|Z, B1, . . . , BD) = N Y n=1 D Y d=1 p(xnd\|zn·, Bd) = N Y n=1 D Y d=1 πxnd nd . (3) 3.1 Laplace approximation for inference In Section 2, the (heuristic) Gibbs sampling algorithm for the posterior inference over the latent variables of the IBP has been reviewed and it is detailed in [5]. To sample from Eq. 1, we need to integrate out Bd in (3), as sequentially sampling from the posterior distribution of Bd is intractable, for which an approximation is required. We rely on the Laplace approximation to integrate out the parameters Bd for simplicity and ease of implementation. We ﬁrst consider the ﬁnite form of the proposed model, where K is bounded. Recall that our model assumes independence among the observations given the hidden latent variables. Then, the posterior p(B1, . . . , BD\|X, Z) factorizes as p(B1, . . . , BD\|X, Z) = D Y d=1 p(Bd\|x·d, Z) = D Y d=1 p(x·d\|Bd, Z)p(Bd) p(x·d\|Z) . (4) 3 Hence, we only need to deal with each term p(Bd\|x·d, Z) individually. Although the prior p(Bd) is Gaussian, due to the non-conjugacy with the likelihood term, the computation of the posterior p(Bd\|x·d, Z) turns out to be intractable. Following a similar procedure as in Gaussian processes for multiclass classiﬁcation [16], we approximate the posterior p(Bd\|x·d, Z) as a Gaussian distribution using Laplace’s method. In order to obtain the parameters of the Gaussian distribution, we deﬁne ψ(Bd) as the un-normalized log-posterior of p(Bd\|x·d, Z), i.e., ψ(Bd) = log p(x·d\|Bd, Z) + log p(Bd) = trace n Md⊤Bdo − N X n=1 log R X r′=1 exp(zn·bd ·r′) ! − 1 2σ2 B trace n Bd⊤Bdo −RK 2 log(2πσ2 B), (5) where (Md)kr counts the number of data points for which xnd = r and znk = 1, namely, (Md)kr = PN n=1 δ(xnd = r)znk, where δ(·) is the Kronecker delta function. As we prove below, the function ψ(Bd) is a strictly concave function of Bd and therefore it has a unique maximum, which is reached at Bd MAP, denoted by the subscript ‘MAP’ because it coincides with the mean value of the Gaussian distribution in the Laplace’s method (MAP stands for maximum a posteriori). We apply Newton’s method to compute this maximum. By deﬁning (ρd)kr = PN n=1 znkπr nd, the gradient of ψ(Bd) can be derived as ∇ψ = Md −ρd −1 σ2 B Bd. (6) To compute the Hessian, it is easier to deﬁne the gradient ∇ψ as a vector, instead of a matrix, and hence we stack the columns of Bd into βd, i.e., for avid Matlab users, βd = Bd(:). The Hessian matrix can now be readily computed taking the derivatives of the gradient, yielding ∇∇ψ = −1 σ2 B IRK + ∇∇log p(x·d\|βd, Z) = −1 σ2 B IRK − N X n=1 diag(πnd) −(πnd)⊤πnd ⊗(z⊤ n·zn·), (7) where πnd = π1 nd, π2 nd, . . . , πR nd , and diag(πnd) is a diagonal matrix with the values of the vector πnd as its diagonal elements. The posterior p(βd\|x·d, Z) can be approximated as p(βd\|x·d, Z) ≈q(βd\|x·d, Z) = N(βd\|βd MAP, (−∇∇ψ)\|βd MAP), (8) where βd MAP contains all the columns of Bd MAP stacked into a vector. Since p(x·d\|βd, Z) is a log-concave function of βd (see [3, p. 87]), −∇∇ψ is a positive deﬁnite matrix, which guarantees that the maximum of ψ(βd) is unique. Once the maximum Bd MAP has been determined, the marginal likelihood p(x·d\|Z) can be readily approximated by log p(x·d\|Z) ≈log q(x·d\|Z) = −1 2σ2 B trace (Bd MAP)⊤Bd MAP −1 2 log IRK + σ2 B N X n=1 diag(bπnd) −(bπnd)⊤bπnd ⊗(z⊤ n·zn·) + log p(x·d\|Bd MAP, Z), (9) where bπnd is the vector πnd evaluated at Bd = Bd MAP. Similarly as in [5], it is straightforward to prove that the limit of Eq. 9 is well-deﬁned if Z has an unbounded number of columns, i.e., as K →∞. The resulting expression for the marginal likelihood p(x·d\|Z) can be readily obtained from Eq. 9 by replacing K by K+, Z by the submatrix containing only the non-zero columns of Z, and Bd MAP by the submatrix containing the K+ corresponding rows. Through the rest of the paper, let us denote with Z the matrix that contains only the K+ non-zero columns of the full IBP matrix. 4 3.2 Speeding up the matrix inversion The inverse of the Hessian matrix, as well as its determinant in (9), can be efﬁciently carried out if we rearrange the inverse of ∇∇ψ as follows (−∇∇ψ)−1 = D − N X n=1 vnv⊤ n !−1 , (10) where vn = (πnd)⊤⊗z⊤ n· and D is a block-diagonal matrix, in which each diagonal submatrix is Dr = 1 σ2 B IK+ + Z⊤diag (πr ·d) Z, (11) with πr ·d = [ πr 1d, . . . , πr Nd ]⊤. Since vnv⊤ n is a rank-one matrix, we can apply the Woodbury identity [18] N times to invert the matrix −∇∇ψ, similar to the RLS (Recursive Least Squares) updates [7]. At each iteration n = 1, . . . , N, we compute (D(n))−1 = D(n−1) −vnv⊤ n −1 = (D(n−1))−1 + (D(n−1))−1vnv⊤ n (D(n−1))−1 1 −v⊤ n (D(n−1))−1vn . (12) For the ﬁrst iteration, we deﬁne D(0) as the block-diagonal matrix D, whose inverse matrix involves computing the R matrix inversions of size K+ × K+ of the matrices in (11), which can be efﬁciently solved applying the Matrix Inversion Lemma. After N iterations of (12), it turns out that (−∇∇ψ)−1 = (D(N))−1. For the determinant in (9), similar recursions can be applied using the Matrix Determinant Lemma [6], which states that \|D + vu⊤\| = (1 + v⊤Du)\|D\|, and \|D(0)\| = QR r=1 \|Dr\|. 4 Experiments 4.1 Inference over synthetic images We generate a simple example inspired by the experiment in [5, p. 1205] to show that the proposed model works as it should. We deﬁne four base black-and-white images that can be present or absent with probability 0.5 independently of each other (Figure 1a), which are combined to create a binary composite image. We also multiply each pixel independently with equiprobable binary noise, hence each white pixel in the composite image can be turned black 50% of the times, while black pixels always remain black. Several examples can be found in Figure 1c. We generate 200 examples to learn the IBP model. The Gibbs sampler has been initialized with K+ = 2, setting each znk = 1 with probability 1/2, and the hyperparameters have been set to α = 0.5 and σ2 B = 1. After 200 iterations, the Gibbs sampler returns four latent features. Each of the four features recovers one of the base images with a different ordering, which is inconsequential. In Figure 1b, we have plotted the posterior probability for each pixel being white, when only one of the components is active. As expected, the black pixels are known to be black (almost zero probability of being white) and the white pixels have about a 50/50 chance of being black or white, due to the multiplicative noise. The Gibbs sampler has used as many as nine hidden features, but after iteration 60, the ﬁrst four features represent the base images and the others just lock on a noise pattern, which eventually fades away. 4.2 National Epidemiologic Survey on Alcohol and Related Conditions (NESARC) The NESARC was designed to determine the magnitude of alcohol use disorders and their associated disabilities. Two waves of interviews have been ﬁelded for this survey (ﬁrst wave in 2001-2002 and second wave in 2004-2005). For the following experimental results, we only use the data from the ﬁrst wave, for which 43,093 people were selected to represent the U.S. population 18 years of age and older. Public use data are currently available for this wave of data collection. Through 2,991 entries, the NESARC collects data on the background of participants, alcohol and other drug consumption and abuse, medicine use, medical treatment, mental disorders, phobias, 5 (a) (b) (c) (d) 0 20 40 60 80 100 120 140 160 180 200 0 2 4 6 8 Iteration Number of Features (K+) (e) 0 20 40 60 80 100 120 140 160 180 200 −4400 −4200 −4000 −3800 −3600 Iteration log p(X\|Z) (f) Figure 1: Experimental results of the inﬁnite binary multinomial-logistic model over the image data set. (a) The four base images used to generate the 200 observations. (b) Probability of each pixel being white, when a single feature is active (ordered to match the images on the left), computed using Bd MAP. (c) Four data points generated as described in the text. The numbers above each ﬁgure indicate which features are present in that image. (d) Probabilities of each pixel being white after 200 iterations of the Gibbs sampler inferred for the four data points on (c). The numbers above each ﬁgure show the inferred value of zn· for these data points. (e) The number of latent features K+ and (f) the approximate log of p(X\|Z) over the 200 iterations of the Gibbs sampler. family history, etc. The survey includes a question about having attempted suicide as well as other related questions such as ‘felt like wanted to die’ and ‘thought a lot about own death’. In the present paper, we use the IBP with discrete observations for a preliminary study in seeking the latent causes which lead to committing suicide. Most of the questions in the survey (over 2,500) are yes-or-no questions, which have four possible outcomes: ‘blank’ (B), ‘unknown’ (U), ‘yes’ (Y) and ‘no’ (N). If a question is left blank the question was not asked1. If a question is said to be unknown either it was not answered or was unknown to the respondent. In our ongoing study, we want to ﬁnd a latent model that describes this database and can be used to infer patterns of behavior and, speciﬁcally, be able to predict suicide. In this paper, we build an unsupervised model with the 20 variables that present the highest mutual information with the suicide attempt question, which are shown in Table 1 together with their code in the questionnaire. We run the Gibbs sampler over 500 randomly chosen subjects out of the 13,670 that have answered afﬁrmatively to having had a period of low mood. In this study, we use another 9,500 as test cases and have left the remaining samples for further validation. We have initialized the sampler with an active feature, i.e., K+ = 1, and have set znk = 1 randomly with probability 0.5, and ﬁxing α = 1 and σ2 B = 1. After 200 iterations, we obtain seven latent features. In Figure 2, we have plotted the posterior probability for each question when a single feature is active. In these plots, white means 0 and black 1, and each row sums up to one. Feature 1 is active for modeling the ‘blank’ and ‘no’ answers and, fundamentally, those who were not asked Questions 8 and 10. Feature 2 models the ‘yes’ and ‘no’ answers and favors afﬁrmative responses to Questions 1, 2, 5, 9, 11, 12, 17 and 18, which indicates depression. Feature 3 models blank answers for most of the questions and negative responses to 1, 2, 5, 8 and 10, which are questions related to suicide. Feature 4 models the afﬁrmative answers to 1, 2, 5, 9 and 11 and also have higher probability for unknowns in Questions 3, 4, 6 and 7. Feature 5 models the ‘yes’ answer to Questions 3, 4, 6, 7, 8, 1In a questionnaire of this size some questions are not asked when a previous question was answered in a predetermined way to reduce the burden of taking the survey. For example, if a person has never had a period of low mood, the attempt suicide question is not asked. 6 10, 17 and 18, being ambivalent in Questions 1 and 2. Feature 6 favors ‘blank’ and ‘no’ answers in most questions. Feature 7 models answering afﬁrmatively to Questions 15, 16, 19 and 20, which are related to alcohol abuse. We show the percentage of respondents that answered positively to the suicide attempt questions in Table 2, independently for the 500 samples that were used to learn the IBP and the 9,500 hold-out samples, together with the total number of respondents. A dash indicates that the feature can be active or inactive. Table 2 is divided in three parts. The ﬁrst part deals with each individual feature and the other two study some cases of interest. Throughout the database, the prevalence of suicide attempt is 7.83%. As expected, Features 2, 4, 5 and 7 favor suicide attempt risk, although Feature 5 only mildly, and Features 1, 3 and 6 decrease the probability of attempting suicide. From the above description of each feature, it is clear that having Features 4 or 7 active should increase the risk of attempting suicide, while having Features 3 and 1 active should cause the opposite effect. Features 3 and 4 present the lowest and the highest risk of suicide, respectively, and they are studied together in the second part of Table 2, in which we can see that having Feature 3 and not having Feature 4 reduces this risk by an order of magnitude, and that combination is present in 70% of the population. The other combinations favor an increased rate of suicide attempts that goes from doubling (‘11’) to quadrupling (‘00’), to a ten-fold increase (‘01’), and the percentages of population with these features are, respectively, 21%, 6% and 3%. In the ﬁnal part of Table 2, we show combinations of features that signiﬁcantly increase the suicide attempt rate for a reduced percentage of the population, as well as combinations of features that signiﬁcantly decrease the suicide attempt rate for a large chunk of the population. These results are interesting as they can be used to discard signiﬁcant portions of the population in suicide attempt studies and focus on the groups that present much higher risk. Hence, our IBP with discrete observations is being able to obtain features that describe the hidden structure of the NESARC database and makes it possible to pin-point the people that have a higher risk of attempting suicide. # Source Code Description 01 S4AQ4A17 Thought about committing suicide 02 S4AQ4A18 Felt like wanted to die 03 S4AQ17A Stayed overnight in hospital because of depression 04 S4AQ17B Went to emergency room for help because of depression 05 S4AQ4A19 Thought a lot about own death 06 S4AQ16 Went to counselor/therapist/doctor/other person for help to improve mood 07 S4AQ18 Doctor prescribed medicine/drug to improve mood/make you feel better 08 S4CQ15A Stayed overnight in hospital because of dysthymia 09 S4AQ4A12 Felt worthless most of the time for 2+ weeks 10 S4CQ15B Went to emergency room for help because of dysthymia 11 S4AQ52 Had arguments/friction with family, friends, people at work, or anyone else 12 S4AQ55 Spent more time than usual alone because didn’t want to be around people 13 S4AQ21C Used medicine/drug on own to improve low mood prior to last 12 months 14 S4AQ21A Ever used medicine/drug on own to improve low mood/make self feel better 15 S4AQ20A Ever drank alcohol to improve low mood/make self feel better 16 S4AQ20C Drank alcohol to improve mood prior to last 12 months 17 S4AQ56 Couldn’t do things usually did/wanted to do 18 S4AQ54 Had trouble doing things supposed to do -like working, doing schoolwork, etc. 19 S4AQ11 Any episode began after drinking heavily/more than usual 20 S4AQ15IR Only/any episode prior to last 12 months began after drinking/drug use Table 1: Enumeration of the 20 selected questions in the experiments, sorted in decreasing order according to their mutual information with the ‘attempted suicide’ question. 5 Conclusions In this paper, we have proposed a new model that combines the IBP with discrete observations using the multinomial-logit distribution. We have used the Laplace approximation to integrate out the weighting factors, which allows us to efﬁciently run the Gibbs sampler. We have applied our model to the NESARC database to ﬁnd out the hidden features that characterize the suicide attempt risk. We 7 Hidden features Suicide attempt probability Number of cases Train Hold-out Train Hold-out 1 6.74% 5.55% 430 8072 1 10.56% 11.16% 322 6083 1 3.72% 4.60% 457 8632 1 25.23% 22.25% 111 2355 1 8.64% 9.69% 301 5782 1 6.90% 7.18% 464 8928 1 14.29% 14.18% 91 1664 0 0 30.77% 28.55% 26 571 0 1 82.35% 61.95% 17 297 1 0 0.83% 0.87% 363 6574 1 1 14.89% 16.52% 94 2058 0 1 1 100.00% 69.41% 4 85 0 0 1 80.00% 66.10% 5 118 1 1 0 1 0 0.00% 0.25% 252 4739 1 0 0 0.33% 0.63% 299 5543 1 1 0 0.32% 0.41% 317 5807 Table 2: Probabilities of attempting suicide for different values of the latent feature vector, together with the number of subjects possessing those values. The symbol ‘-’ denotes either 0 or 1. The ‘train ensemble’ columns contain the results for the 500 data points used to obtain the model, whereas the ‘hold-out ensemble’ columns contain the results for the remaining subjects. Figure 2: Probability of answering ‘blank’ (B), ‘unknown’ (U), ‘yes’ (Y) and ‘no’ (N) to each of the 20 selected questions, sorted as in Table 1, after 200 iterations of the Gibbs sampler. These probabilities have been obtained with the posterior mean weights Bd MAP , when only one of the seven latent features (sorted from left to right to match the order in Table 2) is active. have analyzed how each of the seven inferred features contributes to the suicide attempt probability. We are developing a variational inference algorithm to be able to extend these remarkable results for larger fractions (subjects and questions) of the NESARC database. Acknowledgments Francisco J. R. Ruiz is supported by an FPU fellowship from the Spanish Ministry of Education, Isabel Valera is supported by the Plan Regional-Programas I+D of Comunidad de Madrid (AGESCM S2010/BMD-2422), and Fernando P´erez-Cruz has been partially supported by a Salvador de Madariaga grant. The authors also acknowledge the support of Ministerio de Ciencia e Innovaci´on of Spain (project DEIPRO TEC2009-14504-C02-00 and program Consolider-Ingenio 2010 CSD200800010 COMONSENS). 8 References [1] Summary of national strategy for suicide prevention: Goals and objectives for action, 2007. Available at: http://www.sprc.org/library/nssp.pdf. [2] D. M. Blei and J. D. Lafferty. A correlated topic model of Science. Annals of Applied Statistics, 1(1):17– 35, August 2007. [3] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, March 2004. [4] G. K. Brown, T. Ten Have, G. R. Henriques, S.X. Xie, J.E. Hollander, and A. T. Beck. 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Advances in Neural Information Processing Systems (NIPS), 19, 2007. [15] J. Van Gael, Y. W. Teh, and Z. Ghahramani. The inﬁnite factorial hidden Markov model. In Advances in Neural Information Processing Systems (NIPS), volume 21, 2009. [16] C. K. I. Williams and D. Barber. Bayesian classiﬁcation with Gaussian Processes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 20:1342–1351, 1998. [17] S. Williamson, C. Wang, K. A. Heller, and D. M. Blei. The IBP Compound Dirichlet Process and its application to focused topic modeling. 11:1151–1158, 2010. [18] M. A. Woodbury. The stability of out-input matrices. Mathematical Reviews, 1949. 9	2012	110
4,465	Non-linear Metric Learning Dor Kedem, Stephen Tyree, Kilian Q. Weinberger Dept. of Comp. Sci. & Engi. Washington U. St. Louis, MO 63130 kedem.dor,swtyree,kilian@wustl.edu Fei Sha Dept. of Comp. Sci. U. of Southern California Los Angeles, CA 90089 feisha@usc.edu Gert Lanckriet Dept. of Elec. & Comp. Engineering U. of California La Jolla, CA 92093 gert@ece.ucsd.edu Abstract In this paper, we introduce two novel metric learning algorithms, χ2-LMNN and GB-LMNN, which are explicitly designed to be non-linear and easy-to-use. The two approaches achieve this goal in fundamentally different ways: χ2-LMNN inherits the computational beneﬁts of a linear mapping from linear metric learning, but uses a non-linear χ2-distance to explicitly capture similarities within histogram data sets; GB-LMNN applies gradient-boosting to learn non-linear mappings directly in function space and takes advantage of this approach’s robustness, speed, parallelizability and insensitivity towards the single additional hyperparameter. On various benchmark data sets, we demonstrate these methods not only match the current state-of-the-art in terms of kNN classiﬁcation error, but in the case of χ2-LMNN, obtain best results in 19 out of 20 learning settings. 1 Introduction How to compare examples is a fundamental question in machine learning. If an algorithm could perfectly determine whether two examples were semantically similar or dissimilar, most subsequent machine learning tasks would become trivial (i.e., a nearest neighbor classiﬁer will achieve perfect results). Guided by this motivation, a surge of recent research [10, 13, 15, 24, 31, 32] has focused on Mahalanobis metric learning. The resulting methods greatly improve the performance of metric dependent algorithms, such as k-means clustering and kNN classiﬁcation, and have gained popularity in many research areas and applications within and beyond machine learning. One reason for this success is the out-of-the-box usability and robustness of several popular methods to learn these linear metrics. So far, non-linear approaches [6, 18, 26, 30] to metric learning have not managed to replicate this success. Although more expressive, the optimization problems are often expensive to solve and plagued by sensitivity to many hyper-parameters. Ideally, we would like to develop easy-to-use black-box algorithms that learn new data representations for the use of established metrics. Further, non-linear transformations should be applied depending on the speciﬁcs of a given data set. In this paper, we introduce two novel extensions to the popular Large Margin Nearest Neighbors (LMNN) framework [31] which provide non-linear capabilities and are applicable out-of-the-box. The two algorithms follow different approaches to achieve this goal: (i) Our ﬁrst algorithm, χ2-LMNN is specialized for histogram data. It generalizes the non-linear χ2-distance and learns a metric that strictly preserve the histogram properties of input data on a probability simplex. It successfully combines the simplicity and elegance of the LMNN objective and the domain-speciﬁc expressiveness of the χ2-distance. 1 (ii) Our second algorithm, gradient boosted LMNN (GB-LMNN) employs a non-linear mapping combined with a traditional Euclidean distance function. It is a natural extension of LMNN from linear to non-linear mappings. By training the non-linear transformation directly in function space with gradient-boosted regression trees (GBRT) [11] the resulting algorithm inherits the positive aspects of GBRT—its insensitivity to hyper-parameters, robustness against overﬁtting, speed and natural parallelizability [28]. Both approaches scale naturally to medium-sized data sets, can be optimized using standard techniques and only introduce a single additional hyper-parameter. We demonstrate the efﬁcacy of both algorithms on several real-world data sets and observe two noticeable trends: i) GB-LMNN (with default settings) achieves state-of-the-art k-nearest neighbor classiﬁcation errors with high consistency across all our data sets. For learning tasks where non-linearity is not required, it reduces to LMNN as a special case. On more complex data sets it reliably improves over linear metrics and matches or out-performs previous work on non-linear metric learning. ii) For data sampled from a simplex, χ2-LMNN is strongly superior to alternative approaches that do not explicitly incorporate the histogram aspect of the data—in fact it obtains best results in 19/20 learning settings. 2 Background and Notation Let {(x1, y1), . . . , (xn, yn)} ∈Rd×C be labeled training data with discrete labels C = {1, . . . , c}. Large margin nearest neighbors (LMNN) [30, 31] is an algorithm to learn a Mahalanobis metric speciﬁcally to improve the classiﬁcation error of k-nearest neighbors (kNN) [7] classiﬁcation. As the kNN rule relies heavily on the underlying metric (a test input is classiﬁed by a majority vote amongst its k nearest neighbors), it is a good indicator for the quality of the metric in use. The Mahalanobis metric can be viewed as a straight-forward generalization of the Euclidean metric, DL(xi, xj) = ∥L(xi −xj)∥2, (1) parameterized by a matrix L ∈Rd×d, which in the case of LMNN is learned such that the linear transformation x →Lx better represents similarity in the target domain. In the remainder of this section we brieﬂy review the necessary terminology and basic framework behind LMNN and refer the interested reader to [31] for more details. Local neighborhoods. LMNN identiﬁes two types of neighborhood relations between an input xi and other inputs in the data set: For each xi, as a ﬁrst step, k dedicated target neighbors are identiﬁed prior to learning. These are the inputs that should ideally be the actual nearest neighbors after applying the transformation (we use the notation j ⇝i to indicate that xj is a target neighbor of xi). A common heuristic for choosing target neighbors is picking the k closest inputs (according to the Euclidean distance) to a given xi within the same class. The second type of neighbors are impostors. These are inputs that should not be among the k-nearest neighbors — deﬁned to be all inputs from a different class that are within the local neighborhood of xi. LMNN optimization. The LMNN objective has two terms, one for each neighborhood objective: First, it reduces the distance between an instance and its target neighbors, thus pulling them closer and making the input’s local neighborhood smaller. Second, it moves impostor neighbors (i.e., differently labeled inputs) farther away so that the distances to impostors should exceed the distances to target neighbors by a large margin. Weinberger et. al [31] combine these two objectives into a single unconstrained optimization problem: min L X i,j:j⇝i DL(xi, xj)2 \| {z } pull target neighbor xj closer + µ X k : yi̸=yk 1 + DL(xi, xj)2 −DL(xi, xk)2 + \| {z } push impostor xk away, beyond target neighbor xj by a large margin ℓ (2) The parameter µ deﬁnes a trade-off between the two objectives and [x]+ is deﬁned as the hinge-loss [x]+ =max(0, x). The optimization (2) can be transformed into a semideﬁnite program (SDP) [31] for which a global solution can be found efﬁciently. The large margin in (2) is set to 1 as its exact value only impacts the scale of L and not the kNN classiﬁer. Dimensionality reduction. As an extension to the original LMNN formulation, [26, 30] show that with L ∈Rr×d with r < d, LMNN learns a projection into a lower-dimensional space Rr that still represents domain speciﬁc similarities. While this low-rank constraint breaks the convexity of the optimization problem, signiﬁcant speed-ups [30] can be obtained when the kNN classiﬁer is applied in the r-dimensional space — especially when combined with special-purpose data structures [33]. 2 3 χ2-LMNN: Non-linear Distance Functions on the Probability Simplex Figure 1: A schematic illustration of the χ2-LMNN optimization. The mapping is constrained to preserve all inputs on the simplex S3 (grey surface). The arrows indicate the push (red and yellow) and pull (blue) forces from the χ2LMNN objective. The original LMNN algorithm learns a linear transformation L ∈Rd×d that captures semantic similarity for kNN classiﬁcation on data in some Euclidean vector space Rd. In this section we extend this formulation to settings in which data are sampled from a probability simplex Sd ={x∈Rd\|x≥0, x⊤1=1}, where 1 ∈Rd denotes the vector of all-ones. Each input xi ∈Sd can be interpreted as a histogram over d buckets. Such data are ubiquitous in computer vision where the histograms can be distributions over visual codebooks [27] or colors [25], in text-data as normalized bag-of-words or topic assignments [3], and many other ﬁelds [9, 17, 21]. Histogram distances. The abundance of such data has sparked the development of several specialized distance metrics designed to compare histograms. Examples are Quadratic-Form distance [16], Earth Mover’s Distance [21], Quadratic-Chi distance family [20] and χ2 histogram distance [16]. We focus explicitly on the latter. Transforming the inputs with a linear transformation learned with LMNN will almost certainly result in a loss of their histogram properties — and the ability to use such distances. In this section, we introduce our ﬁrst non-linear extension for LMNN, to address this issue. In particular, we propose two signiﬁcant changes to the original LMNN formulation: i) we learn a constrained mapping that keeps the transformed data on the simplex (illustrated in Figure 1), and ii) we optimize the kNN classiﬁcation performance with respect to the non-linear χ2 histogram distance directly. χ2 histogram distance. We focus on the χ2 histogram distance, whose origin is the χ2 statistical hypothesis test [19], and which has successfully been applied in many domains [8, 27, 29]. The χ2 distance is a bin-to-bin distance measurement, which takes into account the size of the bins and their differences. Formally, the χ2 distance is a well-deﬁned metric χ2 : Sd →[0, 1] deﬁned as [20] χ2(xi, xj) = 1 2 d X f=1 ([xi]f −[xj]f)2 [xi]f + [xj]f , (3) where [xi]f indicates the f th feature value of the vector xi. Generalized χ2 distance. First, analogous to the generalized Euclidean metric in (1), we generalize the χ2 distance with a linear transformation and introduce the pseudo-metric χ2 L(xi, xj), deﬁned as χ2 L(xi, xj) = χ2(Lxi, Lxj). (4) The χ2 distance is only a well-deﬁned metric within the simplex Sd and therefore we constrain L to map any x onto Sd. We deﬁne the set of such simplex-preserving linear transformations as P ={L∈Rd×d : ∀x ∈Sd, Lx ∈Sd}. χ2-LMNN Objective. To optimize the transformation L with respect to the χ2 histogram distance directly, we replace the Mahalanobis distance DL in (2) with χ2 L and obtain the following: min L∈P X i,j: j⇝i χ2 L(xi, xj) + µ X k: yi̸=yk ℓ+ χ2 L(xi, xj) −χ2 L(xi, xk) +. (5) Besides the substituted distance function, there are two important changes in the optimization problem (5) compared to (2). First, as mentioned before, we have an additional constraint L∈P. Second, because (4) is not linear in L⊤L, different values for the margin parameter ℓlead to truly different solutions (which differ not just up to a scaling factor as before). We therefore can no longer arbitrarily set ℓ= 1. Instead, ℓbecomes an additional hyper-parameter of the model. We refer to this algorithm as χ2-LMNN. Optimization. To learn (5), it can be shown L∈P if and only if L is element-wise non-negative, i.e., L≥0, and each column is normalized, i.e., P i Lij =1, ∀j. These constraints are linear with respect 3 to L and we can optimize (5) efﬁciently with a projected sub-gradient method [2]. As an even faster optimization method, we propose a simple change of variables to generate an unconstrained version of (5). Let us deﬁne f : Rd×d →P to be the column-wise soft-max operator [f(A)]ij = eAij P k eAkj . (6) By design, all columns of f(A) are normalized and every matrix entry is non-negative. The function f(·) is continuous and differentiable. By deﬁning L=f(A) we obtain L∈P for any choice of A∈ Rd×d. This allows us to minimize (5) with respect to A using unconstrained sub-gradient descent1. We initialize the optimization with A = 10 I + 0.01 11⊤(where I denotes the identity matrix) to approximate the non-transformed χ2 histogram distance after the change of variable (f(A)≈I). Dimensionality Reduction. Analogous to the original LMNN formulation (described in Section 2), we can restrict from a square matrix to L ∈Rr×d with r < d. In this case χ2-LMNN learns a projection into a lower dimensional simplex L : Sd →Sr. All other parts of the algorithm change analogously. This extension can be very valuable to enable faster nearest neighbor search [33] especially for time-sensitive applications, e.g., object recognition tasks in computer vision [27]. In section 6 we evaluate this version of χ2-LMNN under a range of settings for r. 4 GB-LMNN: Non-linear Transformations with Gradient Boosting Whereas section 3 focuses on the learning scenario where a linear transformation is too general, in this section we explore the opposite case where it is too restrictive. Afﬁne transformations preserve collinearity and ratios of distances along lines — i.e., inputs on a straight line remain on a straight line and their relative distances are preserved. This can be too restrictive for data where similarities change locally (e.g., because similar data lie on non-linear sub-manifolds). Chopra et al. [6] pioneered non-linear metric learning, using convolutional neural networks to learn embeddings for faceveriﬁcation tasks. Inspired by their work, we propose to optimize the LMNN objective (2) directly in function space with gradient boosted CART trees [11]. Combining the learned transformation φ(x) : Rd →Rd with a Euclidean distance function has the capability to capture highly non-linear similarity relations. It can be optimized using standard techniques, naturally scales to large data sets while only introducing a single additional hyper-parameter in comparison with LMNN. Generalized LMNN. To generalize the LMNN objective 2 to a non-linear transformation φ(·), we denote the Euclidean distance after the transformation as Dφ(xi, xj) = ∥φ(xi) −φ(xj)∥2, (7) which satisﬁes all properties of a well-deﬁned pseudo-metric in the original input space. To optimize the LMNN objective directly with respect to Dφ, we follow the same steps as in Section 3 and substitute Dφ for DL in (2). The resulting unconstrained loss function becomes L(φ) = X i,j: j⇝i ∥φ(xi)−φ(xj)∥2 2 + µ X k: yi̸=yk 1 + ∥φ(xi)−φ(xj)∥2 2 −∥φ(xi)−φ(xk)∥2 2 + . (8) In its most general form, with an unspeciﬁed mapping φ, (8) uniﬁes most of the existing variations of LMNN metric learning. The original linear LMNN mapping [31] is a special case where φ(x)=Lx. Kernelized versions [5, 12, 26] are captured by φ(x) = Lψ(x), producing the kernel K(xi, xj) = φ(xi)⊤φ(xj) = ψ(xi)⊤L⊤Lψ(xj). The embedding of Globerson and Roweis [14] corresponds to the most expressive mapping function φ(xi)=zi, where each input xi is transformed independently to a new location zi to satisfy similarity constraints — without out-of-sample extensions. GB-LMNN. The previous examples vary widely in expressiveness, scalability, and generalization, largely as a consequence of the mapping function φ. It is important to ﬁnd the right non-linear form for φ, and we believe an elegant solution lies in gradient boosted regression trees. Our method, termed GB-LMNN, learns a global non-linear mapping. The construction of the mapping, an ensemble of multivariate regression trees selected by gradient boosting [11], minimizes the general LMNN objective (8) directly in function space. Formally, the GB-LMNN transformation 1The set of all possible matrices f(A) is slightly more restricted than P, as it reaches zero entries only in the limit. However, given ﬁnite computational precision, this does not seem to be a problem in practice. 4 Approximated Gradient Itera/on 1 Itera/on 10 Itera/on 20 Itera/on 40 Itera/on 100 True Gradient Figure 2: GB-LMNN illustrated on a toy data set sampled from two concentric circles of different classes (blue and red dots). The ﬁgure depicts the true gradient (top row) with respect to each input and its least squares approximation (bottom row) with a multi-variate regression tree (depth, p=4). is an additive function φ = φ0 + α PT t=1 ht initialized by φ0 and constructed by iteratively adding regression trees ht of limited depth p [4], each weighted by a learning rate α. Individually, the trees are weak learners and are capable of learning only simple functions, but additively they form powerful ensembles with good generalization to out-of-sample data. In iteration t, the tree ht is selected greedily to best minimize the objective upon its addition to the ensemble, φt(·) = φt−1(·) + αht(·), where ht ≈argmin h∈T p L(φt−1 + αh). (9) Here, T p denotes the set of all regression trees of depth p. The (approximately) optimal tree ht is found by a ﬁrst-order Taylor approximation of L. This makes the optimization akin to a steepest descent step in function space, where ht is selected to approximate the negative gradient gt of the objective L(φt−1) with respect to the transformation learned at the previous iteration φt−1. Since we learn an approximation of gt as a function of the training data, sub-gradients are computed with respect to each training input xi, and approximated by the tree ht(·) in the least-squared sense, ht(·) = argmin h∈T p n X i=1 (gt(xi) −ht(xi))2, where: gt(xi)= ∂L(φt−1) ∂φt−1(xi). (10) Intuitively, at each iteration, the tree ht(·) of depth p splits the input space into 2p axis-aligned regions. All inputs that fall into one region are translated by a constant vector — consequently, the inputs in different regions are shifted in different directions. We learn the trees greedily with a modiﬁed version of the public-domain CART implementation pGBRT [28]. Optimization details. Since (8) is non-convex with respect to φ, we initialize with the linear transformation learned by LMNN, φ0 =Lx, making our method a non-linear reﬁnement of LMNN. The only additional hyperparameter to the optimization is the maximum tree depth p to which the algorithm is not particularly sensitive (we set p=6). 2 Figure 2 depicts a simple toy-example with concentric circles of inputs from two different classes. By design, the inputs are sampled such that the nearest neighbor for any given input is from the other class. A linear transformation is incapable of separating the two classes. However GB-LMNN produces a mapping with the desired separation. The ﬁgure illustrates the actual gradient (top row) and its approximation (bottom row). The limited-depth regression trees are unable to capture the gradient for all inputs in a single iteration. But by greedily focusing on inputs with the largest gradients or groups of inputs with the most easily encoded gradients, the gradient boosting process additively constructs the transformation function. At iteration 100, the gradients with respect to most inputs vanish, indicating that a local minimum of L(φ) is almost reached — the inputs from the two classes are separated by a large margin. 2Here, we set the step-size, a common hyper-parameter across all variations of LMNN, to α=0.01. 5 Dimensionality reduction. Like linear LMNN and χ2-LMNN, it is possible to learn a non-linear transformation to a lower dimensional space, φ(x) : Rd →Rr, r ≤d. Initialization is made with the rectangular matrix output of the dimensionality-reduced LMNN transformation, φ0 = Lx with L∈Rr×d. Training proceeds by learning trees with r- rather than d-dimensional outputs. 5 Related Work There have been previous attempts to generalize learning linear distances to nonlinear metrics. The nonlinear mapping φ(x) of eq. (7) can be implemented with kernels [5, 12, 18, 26]. These extensions have the advantages of maintaining computational tractability as convex optimization problems. However, their utility is limited inherently by the sizes of kernel matrices .Weinberger et. al [30] propose M 2-LMNN, a locally linear extension to LMNN. They partition the space into multiple regions, and jointly learn a separate metric for each region—however, these local metrics do not give rise to a global metric and distances between inputs within different regions are not well-deﬁned. Neural network-based approaches offer the ﬂexibility of learning arbitrarily complex nonlinear mappings [6]. However, they often demand high computational expense, not only in parameter ﬁtting but also in model selection and hyper-parameter tuning. Of particular relevance to our GB-LMNN work is the use of boosting ensembles to learn distances between bit-vectors [1, 23]. Note that their goals are to preserve distances computed by locality sensitive hashing to enable fast search and retrieval. Ours are very different: we alter the distances discriminatively to minimize classiﬁcation error. Our work on χ2-LMNN echoes the recent interest in learning earth-mover-distance (EMD) which is also frequently used in measuring similarities between histogram-type data [9]. Despite its name, EMD is not necessarily a metric [20]. Investigating the link between our work and those new advances is a subject for future work. 6 Experimental Results We evaluate our non-linear metric learning algorithms against several competitive methods. The effectiveness of learned metrics is assessed by kNN classiﬁcation error. Our open-source implementations are available for download at http://www.cse.wustl.edu/˜kilian/code/code.html. GB-LMNN We compare the non-linear global metric learned by GB-LMNN to three linear metrics: the Euclidean metric and metrics learned by LMNN [31] and Information-Theoretic Metric Learning (ITML) [10]. Both optimize similar discriminative loss functions. We also compare to the metrics learned by Multi-Metric LMNN (M 2-LMNN) [30]. M 2-LMNN learns \|C\| linear metrics, one for each the input labels. We evaluate these methods and our GB-LMNN on several medium-sized data sets: ISOLET, USPS and Letters from the UCI repository. ISOLET and USPS have predeﬁned test sets, otherwise results are averaged over 5 train/test splits (80%/20%). A hold-out set of 25% of the training set3 is used to assign hyper-parameters and to determine feature pre-processing (i.e., feature-wise normalization). We set k = 3 for kNN classiﬁcation, following [31]. Table 1 reports the means and standard errors of each approach (standard error is omitted for data with pre-deﬁned test sets), with numbers in bold font indicating the best results up to one standard error. On all three datasets, GB-LMNN outperforms methods of learning linear metrics. This shows the beneﬁt of learning nonlinear metrics. On Letters, GB-LMNN outperforms the second-best method M 2-LMNN by signiﬁcant margins. On the other two, GB-LMNN is as good as M 2-LMNN. We also apply GB-LMNN to four datasets with histogram data — setting the stage for an interesting comparison to χ2-LMNN below. The results are displayed on the right side of the table. These datasets are popularly used in computer vision for object recognition [22]. Data instances are 800bin histograms of visual codebook entries. There are ten common categories to the four datasets and we use them for multiway classiﬁcation with kNN. Neither method evaluated so far is speciﬁcally adapted to histogram features. 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Table 1: kNN classiﬁcation error (in %, ± standard error where applicable), for general methods (top section) and histogram methods (bottom section). Best results up to one standard error in bold. Best results among general methods for simplex data in red italics. data types may encode. As shown in the table, GB-LMNN consistently outperforms the linear methods and M 2-LMNN. χ2-LMNN In Table 1, we compare χ2-LMNN to other methods for computing distances on histogram features: χ2-distance without transformation (equivalent to our parameterized distance χ2 L distance with the transformation L being the identity matrix), Quadratic-Chi-Squared (QCS) and Quadratic-Chi-Normalized (QCN) distances, deﬁned in [20]. For QCS and QCN, we use histogram intersection as the ground distance. Unlike our approach, none of these is discriminatively learned from data. χ2-LMNN outperforms all other methods signiﬁcantly. It is also instructive to compare the results to the performance of non-histogram speciﬁc methods. We observe that LMNN performs better than the standard χ2-distance on Amazon and Caltech. This seems to suggest that for those two datasets, linear metrics may be adequate and GB-LMNN’s nonlinear mapping might not be able to provide extra expressiveness and beneﬁts. This is conﬁrmed in Table 1: GB-LMNN improves performance less signiﬁcantly for Amazon and Caltech than for the other two datasets, DSLR and Webcam. For the latter two, on the contrary, LMNN performs worse than χ2-distance. In such cases, GB-LMNN’s nonlinear mapping seems more beneﬁcial. It provides a signiﬁcant performance boost, and matches the performance of χ2-distance (up to one standard-error). Nonetheless, despite learning a nonlinear mapping, GB-LMNN still underperforms χ2-LMNN. In other words, it is possible that no matter how ﬂexible a nonlinear mapping could be, it is still best to use metrics that respect the semantic features of the data. Dimensionality reduction. GB-LMNN and χ2-LMNN are both capable of performing dimensionality reduction. We compare these with three dimensionality reduction methods (PCA, LMNN, and M 2-LMNN) on the histogram datasets and the larger UCI datasets. Each dataset is reduced to an output dimensionality of r = 10, 20, 40, 80 features. As we can see from the results in Table 6, it is fair to say that GB-LMNN performs comparably with LMNN and M 2-LMNN, whereas χ2-LMNN obtains at times phenomenally low kNN error rates on the histograms data sets (e.g., Webcam). This suggests that dimensionality reduction of histogram data can be highly effective, if the data properties are carefully incorporated in the process. We do not apply dimensionality reduction to Letters as it already lies in a low-dimensional space (d=16). Sensitivity to parameters. One of the most compelling aspects of our methods is that each introduces only a single new hyper-parameter to the LMNN framework. During our experiments, ℓwas selected by cross-validation and p was ﬁxed to p=6. We found very little sensitivity in GB-LMNN to regression tree depth, while large margin size was an important but well-behaved parameter for χ2-LMNN. Additional graphs are included in the supplementary material. 7 Conclusion and Future Work In this paper we introduced two non-linear extensions to LMNN, χ2-LMNN and GB-LMNN. Although based on fundamentally different approaches, both algorithms lead to signiﬁcant improvements over the original (linear) LMNN metrics and match or out-perform existing non-linear algorithms. 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ABCC #)%+# ;86 D989=68> ;78D=68@ >;8>=:8? 7:8;=:8@ B6EABCC #)%+#+ #$ D989=68> ;D8?=68< >;8>=98: D>8>=98D FGEABCC #)%+# 68> D:8:=98? 6<8;=;8D >989=986 D>89=98; H6EABCC E E &&#&%"#+ $#&%#) ",#(%#" (#%"#& -./010232456728-39: I9: I6: I>: I@: ;<=32>91?67-1=9 Table 2: kNN classiﬁcation error (in %, ± standard error where applicable) with dimensionality reduction to output dimensionality r. Best results up to one standard error in bold. indicating that convex algorithms (LMNN) as initialization for more expressive non-convex methods can be a winning combination. The strong results obtained with χ2-LMNN show that the incorporation of data-speciﬁc constraints can be highly beneﬁcial—indicating that there is great potential for future research in specialized metric learning algorithms for speciﬁc data types. Further, the ability of χ2-LMNN to reduce the dimensionality of data sampled from probability simplexes is highly encouraging and might lead to interesting applications in computer vision and other ﬁelds, where histogram data is ubiquitous. Here, it might be possible to reduce the running time of time critical algorithms drastically by shrinking the data dimensionality, while strictly maintaining its histogram properties. The high consistency with which GB-LMNN obtains state-of-the-art results across diverse data sets is highly encouraging. In fact, the use of ensembles of CART trees [4] not only inherits all positive aspects of gradient boosting (robustness, speed and insensitivity to hyper-parameters) but is also a natural match for metric learning. Each tree splits the space into different regions and in contrast to prior work [30], this splitting is fully automated, results in new (discriminatively learned) Euclidean representations of the data and gives rise to well-deﬁned pseudo-metrics. 8 Acknowledgements KQW, DK and ST would like to thank NIH for their support through grant U01 1U01NS073457-01 and NSF for grants 1149882 and 1137211. FS would like to thank DARPA for its support with grant D11AP00278 and ONR for grant N00014-12-1-0066. GL was supported in part by the NSF under Grants CCF-0830535 and IIS-1054960, and by the Sloan Foundation. DK would also like to thank the McDonnell International Scholars Academy for their support. References [1] B. Babenko, S. Branson, and S. Belongie. Similarity metrics for categorization: from monolithic to category speciﬁc. In ICCV ’09, pages 293–300. IEEE, 2009. [2] A. Beck and M. Teboulle. Mirror descent and nonlinear projected subgradient methods for convex optimization. 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4,466	Putting Bayes to sleep Wouter M. Koolen∗ Dmitry Adamskiy† Manfred K. Warmuth‡ Abstract We consider sequential prediction algorithms that are given the predictions from a set of models as inputs. If the nature of the data is changing over time in that different models predict well on different segments of the data, then adaptivity is typically achieved by mixing into the weights in each round a bit of the initial prior (kind of like a weak restart). However, what if the favored models in each segment are from a small subset, i.e. the data is likely to be predicted well by models that predicted well before? Curiously, ﬁtting such “sparse composite models” is achieved by mixing in a bit of all the past posteriors. This self-referential updating method is rather peculiar, but it is efﬁcient and gives superior performance on many natural data sets. Also it is important because it introduces a long-term memory: any model that has done well in the past can be recovered quickly. While Bayesian interpretations can be found for mixing in a bit of the initial prior, no Bayesian interpretation is known for mixing in past posteriors. We build atop the “specialist” framework from the online learning literature to give the Mixing Past Posteriors update a proper Bayesian foundation. We apply our method to a well-studied multitask learning problem and obtain a new intriguing efﬁcient update that achieves a signiﬁcantly better bound. 1 Introduction We consider sequential prediction of outcomes y1, y2, . . . using a set of models m = 1, . . . , M for this task. In practice m could range over a mix of human experts, parametric models, or even complex machine learning algorithms. In any case we denote the prediction of model m for outcome yt given past observations y<t = (y1, . . . , yt−1) by P(yt\|y<t, m). The goal is to design a computationally efﬁcient predictor P(yt\|y<t) that maximally leverages the predictive power of these models as measured in log loss. The yardstick in this paper is a notion of regret deﬁned w.r.t. a given comparator class of models or composite models: it is the additional loss of the predictor over the best comparator. For example if the comparator class is the set of base models m = 1, . . . , M, then the regret for a sequence of T outcomes y≤T = (y1, . . . , yT ) is R := T X t=1 −ln P(yt\|y<t) − M min m=1 T X t=1 −ln P(yt\|y<t, m). The Bayesian predictor (detailed below) with uniform model prior has regret at most ln M for all T. Typically the nature of the data is changing with time: in an initial segment one model predicts well, followed by a second segment in which another model has small loss and so forth. For this scenario the natural comparator class is the set of partition models which divide the sequence of T outcomes into B segments and specify the model that predicts in each segment. By running Bayes on all exponentially many partition models comprising the comparator class, we can guarantee regret ln T −1 B−1 +B ln M, which is optimal. The goal then is to ﬁnd efﬁcient algorithms with approximately ∗Supported by NWO Rubicon grant 680-50-1010. †Supported by Veterinary Laboratories Agency of Department for Environment, Food and Rural Affairs. ‡Supported by NSF grant IIS-0917397. 1 the same guarantee as full Bayes. In this case this is achieved by the Fixed Share [HW98] predictor. It assigns a certain prior to all partition models for which the exponentially many posterior weights collapse to M posterior weights that can be maintained efﬁciently. Modiﬁcations of this algorithm achieve essentially the same bound for all T, B and M simultaneously [VW98, KdR08]. In an open problem Yoav Freund [BW02] asked whether there are algorithms that have small regret against sparse partition models where the base models allocated to the segments are from a small subset of N of the M models. The Bayes algorithm when run on all such partition models achieves regret ln M N + ln T −1 B−1 + B ln N, but contrary to the non-sparse case, emulating this algorithm is NP-hard. However in a breakthrough paper, Bousquet and Warmuth in 2001 [BW02] gave the efﬁcient MPP algorithm with only a slightly weaker regret bound. Like Fixed Share, MPP maintains M “posterior” weights, but it instead mixes in a bit of all past posteriors in each update. This causes weights of previously good models to “glow” a little bit, even if they perform bad locally. When the data later favors one of those good models, its weight is pulled up quickly. However the term “posterior” is a misnomer because no Bayesian interpretation for this curious self-referential update was known. Understanding the MPP update is a very important problem because in many practical applications [HLSS00, GWBA02]1 it signiﬁcantly outperforms Fixed Share. Our main philosophical contribution is ﬁnding a Bayesian interpretation for MPP. We employ the specialist framework from online learning [FSSW97, CV09, CKZV10]. So-called specialist models are either awake or asleep. When they are awake, they predict as usual. However when they are asleep, they “go with the rest”, i.e. they predict with the combined prediction of all awake models. T outcomes 3 9 7 3 7 9 7 3 (a) A comparator partition model: segmentation and model assignment z z z z Z Z ZZZ . . . . 3 3 3 7 7 7 9 9 (b) Decomposition into 3 partition specialists, asleep at shaded times Instead of fully coordinated partition models, we construct partition specialists consisting of a base model and a set of segments where this base model is awake. The ﬁgure to the right shows how a comparator partition model is assembled from partition specialists. We can emulate Bayes on all partition specialists; NP-completeness is avoided by forgoing a-priori segment synchronization. By carefully choosing the prior, the exponentially many posterior weights collapse to the small number of weights used by the efﬁcient MPP algorithm. Our analysis technique magically aggregates the contribution of the N partition specialists that constitute the comparator partition, showing that we achieve regret close to the regret of Bayes when run on all full partition models. Actually our new insights into the nature of MPP result in slightly improved regret bounds. We then apply our methods to an online multitask learning problem where a small subset of models from a big set solve a large number of tasks. Again simulating Bayes on all sparse assignments of models to tasks is NP-hard. We split an assignment into subset specialists that assign a single base model to a subset of tasks. With the right prior, Bayes on these subset specialists again gently collapses to an efﬁcient algorithm with a regret bound not much larger than Bayes on all assignments. This considerably improves the previous regret bound of [ABR07]. Our algorithm simply maintains one weight per model/task pair and does not rely on sampling (often used for multitask learning). Why is this line of research important? We found a new intuitive Bayesian method to quickly recover information that was learned before, allowing us to exploit sparse composite models. Moreover, it expressly avoids computational hardness by splitting composite models into smaller constituent “specialists” that are asleep in time steps outside their jurisdiction. This method clearly beats Fixed Share when few base models constitute a partition, i.e. the composite models are sparse. We expect this methodology to become a main tool for making Bayesian prediction adapt to sparse models. The goal is to develop general tools for adding this type of adaptivity to existing Bayesian models without losing efﬁciency. It also lets us look again at the updates used in Nature in a new light, where species/genes cannot dare adapt too quickly to the current environment and must guard themselves against an environment that changes or ﬂuctuates at a large scale. Surprisingly these type of updates might now be amenable to a Bayesian analysis. For example, it might be possible to interpret sex and the double stranded recessive/dominant gene device employed by Nature as a Bayesian update of genes that are either awake or asleep. 1The experiments reported in [HLSS00] are based on precursors of MPP. However MPP outperforms these algorithms in later experiments we have done on natural data for the same problem (not shown). 2 2 Bayes and Specialists We consider sequential prediction of outcomes y1, y2, . . . from a ﬁnite alphabet. Assume that we have access to a collection of models m = 1, . . . , M with data likelihoods P(y1, y2, . . . \|m). We then design a prior P(m) with roughly two goals in mind: the Bayes algorithm should “collapse” (become efﬁcient) and have a good regret bound. After observing past outcomes y<t := (y1, . . . , yt−1), the next outcome yt is predicted by the predictive distribution P(yt\|y<t), which averages the model predictions P(yt\|y<t, m) according to the posterior distribution P(m\|y<t): P(yt\|y<t) = M X m=1 P(yt\|y<t, m)P(m\|y<t), where P(m\|y<t) = P(y<t\|m)P(m) P(y<t) . The latter is conveniently updated step-wise: P(m\|yt, y<t) = P(yt\|y<t, m)P(m\|y<t)/P(yt\|y<t). The log loss of the Bayesian predictor on data y≤T := (y1, . . . , yT ) is the cumulative loss of the predictive distributions and this readily relates to the cumulative loss of any model ˆm: −lnP(y≤T ) \| {z } PT t=1 −lnP (yt\|y<t) − −lnP(y≤T \| ˆm) \| {z } PT t=1 −lnP (yt\|y<t, ˆm) = −ln M X m=1 P(y≤T \|m)P(m) − −lnP(y≤T \| ˆm) ≤−lnP( ˆm). That is, the additional loss (or regret) of Bayes w.r.t. model ˆm is at most −ln P( ˆm). The uniform prior P(m) = 1/M ensures regret at most ln M w.r.t. any model ˆm. This is a so-called individual sequence result, because no probabilistic assumptions were made on the data. Our main results will make essential use of the following fancier weighted notion of regret. Here U(m) is any distribution on the models and △ U(m) P(m) denotes the relative entropy PM m=1 U(m) ln U(m) P (m) between the distributions U(m) and P(m): M X m=1 U(m) −ln P(y≤T )−(−ln P(y≤T \|m)) = △ U(m) P(m) −△ U(m) P(m\|y≤T ) . (1) By dropping the subtracted positive term we get an upper bound. The previous regret bound is now the special case when U is concentrated on model ˆm. However when multiple models are good we achieve tighter regret bounds by letting U be the uniform distribution on all of them. Specialists We now consider a complication of the prediction task, which was introduced in the online learning literature under the name specialists [FSSW97]. The Bayesian algorithm, adapted to this task, will serve as the foundation of our main results. The idea is that in practice the predictions P(yt\|y<t, m) of some models may be unavailable. Human forecasters may be specialized, unreachable or too expensive, algorithms may run out of memory or simply take too long. We call models that may possibly abstain from prediction specialists. The question is how to produce quality predictions from the predictions that are available. We will denote by Wt the set of specialists whose predictions are available at time t, and call them awake and the others asleep. The crucial idea, introduced in [CV09], is to assign to the sleeping specialists the prediction P(yt\|y<t). But wait! That prediction P(yt\|y<t) is deﬁned to average all model predictions, including those of the sleeping specialists, which we just deﬁned to be P(yt\|y<t): P(yt\|y<t) = X m∈Wt P(yt\|y<t, m) P(m\|y<t) + X m/∈Wt P(yt\|y<t) P(m\|y<t). Although this equation is self-referential, it does have a unique solution, namely P(yt\|y<t) := P m∈Wt P(yt\|y<t, m)P(m\|y<t) P(Wt\|y<t) . Thus the sleeping specialists are assigned the average prediction of the awake ones. This completes them to full models to which we can apply the unaltered Bayesian method as before. At ﬁrst this may seem like a kludge, but actually this phenomenon arises naturally wherever concentrations are 3 manipulated. For example, in a democracy abstaining essentially endorses the vote of the participating voters or in Nature unexpressed genes reproduce at rates determined by the active genes of the organism. The effect of abstaining on the update of the posterior weights is also intuitive: weights of asleep specialists are unaffected, whereas weights of awake models are updated with Bayes rule and then renormalised to the original weight of the awake set: P(m\|y≤t) =    P (yt\|y<t,m)P (m\|y<t) P (yt\|y<t) = P (yt\|y<t,m)P (m\|y<t) P m∈Wt P (yt\|y<t,m)P (m\|y<t) P(Wt\|y<t) if m ∈Wt, P (yt\|y<t)P (m\|y<t) P (yt\|y<t) = P(m\|y<t) if m /∈Wt. (2) To obtain regret bounds in the specialist setting, we use the fact that sleeping specialists m /∈Wt are deﬁned to predict P(yt\|y<t, m) := P(yt\|y<t) like the Bayesian aggregate. Now (1) becomes: Theorem 1 ([FSSW97, Theorem 1]). Let U(m) be any distribution on a set of specialists with wake sets W1, W2, . . . Then for any T, Bayes guarantees M X m=1 U(m)  X t≤T : m∈Wt −ln P(yt\|y<t) − X t≤T : m∈Wt −ln P(yt\|y<t, m)  ≤△ U(m) P(m) . 3 Sparse partition learning We design efﬁcient predictors with small regret compared to the best sparse partition model. We do this by constructing partition specialists from the input models and obtain a proper Bayesian predictor by averaging their predictions. We consider two priors. With the ﬁrst prior we obtain the Mixing Past Posteriors (MPP) algorithm, giving it a Bayesian interpretation and slightly improving its regret bound. We then develop a new Markov chain prior. Bayes with this prior collapses to an efﬁcient algorithm for which we prove the best known regret bound compared to sparse partitions. Construction Each partition specialist (χ, m) is parameterized by a model index m and a circadian (wake/sleep pattern) χ = (χ1, χ2, . . .) with χt ∈{w, s}. We use inﬁnite circadians in order to obtain algorithms that do not depend on a time horizon. The wake set Wt includes all partition specialists that are awake at time t, i.e. Wt := {(χ, m) \| χt = w}. An awake specialist (χ, m) in Wt predicts as the base model m, i.e. P(yt\|y<t, (χ, m)) := P(yt\|y<t, m). The Bayesian joint distribution P is completed2 by choosing a prior on partition specialists. In this paper we enforce the independence P(χ, m) := P(χ)P(m) and deﬁne P(m) := 1/M uniform on the base models. We now can apply Theorem 1 to bound the regret w.r.t. any partition model with time horizon T by decomposing it into N partition specialists (χ1 ≤T , ˆm1), . . . , (χN ≤T , ˆmN) and choosing U(·) = 1/N uniform on these specialists: R ≤N ln M N + N X n=1 −ln P(χn ≤T ). (3) The overhead of selecting N reference models from the pool of size M closely approximates the information-theoretic ideal N ln M N ≈ ln M N . This improves previous regret bounds [BW02, ABR07, CBGLS12] by an additive N ln N. Next we consider two choices for P(χ): one for which we retrieve MPP, and a natural one which leads to efﬁcient algorithms and sharper bounds. 3.1 A circadian prior equivalent to Mixing Past Posteriors The Mixing Past Posteriors algorithm is parameterized a so-called mixing scheme, which is a sequence γ1, γ2, . . . of distributions, each γt with support {0, . . . , t −1}. MPP predicts outcome yt with Predt(yt) := PM m=1 P(yt\|y<t, m) vt(m), i.e. by averaging the model predictions with weights vt(m) deﬁned recursively by vt(m) := t−1 X s=0 ˜vs+1(m) γt(s) where ˜v1(m) := 1 M and ˜vt+1(m) := P(yt\|y<t, m)vt(m) Predt(yt) . 2From here on we use the symbol P for the Bayesian joint to avoid a fundamental ambiguity: P(yt\|y<t, m) does not equal the prediction P(yt\|y<t, m) of the input model m, since it averages over both asleep and awake specialists (χ, m). The predictions of base models are now recovered as P(yt\|y<t, Wt, m) = P(yt\|y<t, m). 4 The auxiliary distribution ˜vt+1(m) is formally the (incremental) posterior from prior vt(m). The predictive weights vt(m) are then the pre-speciﬁed γt mixture of all such past posteriors. To make the Bayesian predictor equal to MPP, we deﬁne from the MPP mixing scheme a circadian prior measure P(χ) that puts mass only on sequences with a ﬁnite nonzero number of w’s, by P(χ) := 1 sJ(sJ + 1) J Y j=1 γsj(sj−1) where s≤J are the indices of the w’s in χ and s0 = 0. (4) We built the independence m ⊥χ into the prior P(χ, m) and (4) ensures χ<t ⊥χ>t \| χt = w for all t. Since the outcomes y≤t are a stochastic function of m and χ≤t, the Bayesian joint satisﬁes y≤t, m ⊥χ>t \| χt = w for all t. (5) Theorem 2. Let Predt(yt) be the prediction of MPP for some mixing scheme γ1, γ2, . . . Let P(yt\|y<t) be the prediction of Bayes with prior (4). Then for all outcomes y≤t Predt(yt) = P(yt\|y<t). Proof. Partition the event Wt = {χt = w} into Zt q := {χt = χq = w and χr = s for all q < r < t} for all 0 ≤q < t, with the convention that χ0 = w. We ﬁrst establish that the Bayesian joint with prior (4) satisﬁes y≤t ⊥Wt for all t. Namely, by induction on t, for all q < t P(y<t\|Zt q) = P(y<t\|y≤q)P(y≤q\|Zt q) (5)= P(y<t\|y≤q)P(y≤q\|Wq) Induction = P(y<t), and therefore P(y≤t\|Wt) = P(yt\|y<t) Pt−1 q=0 P(y<t\|Zt q)P(Zt q\|Wt) = P(y≤t), i.e. y≤t ⊥Wt. The theorem will be implied by the stronger claim vt(m) = P(m\|y<t, Wt), which we again prove by induction on t. The case t = 1 is trivial. For t > 1, we expand the right-hand side, apply (5), use the independence we just proved, and the fact that asleep specialist predict with the rest: P(m\|y<t, Wt) = t−1 X q=0 P(m\|y≤q, Wq)P(Zt q\| m, y≤q, Wq)P(Wq\|HH y≤q) P(Wt\|HH y<t) P(y<t\|Zt q, m, y≤q) P(y<t\|y≤q) = t−1 X q=0 P(yq\|y<q, m)P(m\|y<q, Wq) P(yq\|y<q) P(Zt q\|Wt) By (4) P(Zt q\|Wt) = γt(q), and the proof is completed by applying the induction hypothesis. The proof of the theorem provides a Bayesian interpretation of all the MPP weights: vt(m) = P(m\|y<t, Wt) is the predictive distribution, ˜vt+1(m) = P(m\|y≤t, Wt) is the posterior, and γt(q) = P(Zt q\|Wt) is the conditional probability of the previous awake time. 3.2 A simple Markov chain circadian prior In the previous section we recovered circadian priors corresponding to the MPP mixing schemes. Here we design priors afresh from ﬁrst principles. Our goal is efﬁciency and good regret bounds. A simple and intuitive choice for prior P(χ) is a Markov chain on states {w, s} with initial distribution θ(·) and transition probabilities θ(·\|w) and θ(·\|s), that is P(χ≤t) := θ(χ1) tY s=2 θ(χs\|χs−1). (6) By choosing low transition probabilities we obtain a prior that favors temporal locality in that it allocates high probability to circadians that are awake and asleep in contiguous segments. Thus if a good sparse partition model exists for the data, our algorithm will pick up on this and predict well. The resulting Bayesian strategy (aggregating inﬁnitely many specialists) can be executed efﬁciently. Theorem 3. The prediction P(yt\|y<t) of Bayes with Markov prior (6) equals the prediction Predt(yt) of Algorithm 1, which can be computed in O(M) time per outcome using O(M) space. 5 Proof. We prove by induction on t that vt(b, m) = P(χt = b, m\|y<t) for each model m and b ∈{w, s}. The base case t = 1 is automatic. For the induction step we expand P(χt+1 = b, m\|y≤t) (6)= θ(b\|w)P(χt = w, m\|y≤t) + θ(b\|s)P(χt = s, m\|y≤t) (2)= θ(b\|w) P(χt = w, m\|y<t)P(yt\|y<t, m) PM i=1 P(i\|χt = w, y<t)P(yt\|y<t, i) + θ(b\|s)P(χt = s, m\|y<t). By applying the induction hypothesis we obtain the update rule for vt+1(b, m). Algorithm 1 Bayes with Markov circadian prior (6) (for Freund’s problem) Input: Distributions θ(·), θ(·\|w) and θ(·\|s) on {w, s}. Initialize v1(b, m) := θ(b)/M for each model m and b ∈{w, s} for t = 1, 2, . . . do Receive prediction P(yt\|y<t, m) of each model m Predict with Predt(yt) := PM m=1 P(yt\|y<t, m)vt(m\|w) where vt(m\|w) = vt(w,m) PM m′=1 vt(w,m′) Observe outcome yt and suffer loss −ln Predt(yt). Update vt+1(b, m) := θ(b\|w) P (yt\|y<t,m) Predt(yt) vt(w, m) + θ(b\|s)vt(s, m). end for The previous theorem establishes that we can predict fast. Next we show that we predict well. Theorem 4. Let ˆm1, . . . , ˆmT be an N-sparse assignment of M models to T times with B segments. The regret of Bayes (Algorithm 1) with tuning θ(w) = 1/N, θ(s\|w) = B−1 T −1 and θ(w\|s) = B−1 (N−1)(T −1) is at most R ≤N ln M N + N H 1 N + (T −1) H B −1 T −1 + (N −1)(T −1) H B −1 (N −1)(T −1) , where H(p) := −p ln(p) −(1 −p) ln(1 −p) is the binary entropy function. Proof. Without generality assume ˆmt ∈{1, . . . , N}. For each reference model n pick circadian χn ≤T with χn t = w iff ˆmt = n. Expanding the deﬁnition of the prior (6) we ﬁnd N Y n=1 P(χn ≤T ) = θ(w)θ(s)N−1θ(s\|s)(N−1)(T −1)−(B−1)θ(w\|w)T −Bθ(w\|s)B−1θ(s\|w)B−1, which is in fact maximized by the proposed tuning. The theorem follows from (3). The information-theoretic ideal regret is ln M N + ln T −1 B−1 + B ln N. Theorem 4 is very close to this except for a factor of 2 in front of the middle term; since n H(k/n) ≤k ln(n/k) + k we have R ≤N ln M N + 2 (B −1) ln T −1 B −1 + B ln N + 2B. The origin of this factor remained a mystery in [BW02], but becomes clear in our analysis: it is the price of coordination between the specialists that constitute the best partition model. To see this, let us regard a circadian as a sequence of wake/sleep transition times. With this viewpoint (3) bounds the regret by summing the prior costs of all the reference wake/sleep transition times. This means that we incur overhead at each segment boundary of the comparator twice: once as the sleep time of the preceding model, and once more as the wake time of the subsequent model. In practice the comparator parameters T, N and B are unknown. This can be addressed by standard orthogonal techniques. Of particular interest is the method inspired by [SM99, KdR08, Koo11] of changing the Markov transition probabilities as a function of time. It can be shown that by setting θ(w) = 1/2 and increasing θ(w\|w) and θ(s\|s) as exp(− 1 t ln2(t+1)) we keep the update time and space of the algorithm at O(M) and guarantee regret bounded for all T, N and B as R ≤N ln M N + 2N + 2(B −1) ln T + 4(B −1) ln ln(T + 1). At no computational overhead, this bound is remarkably close to the fully tuned bound of the theorem above, especially when the number of segments B is modest as a function of T. 6 4 Sparse multitask learning We transition to an extension of the sequential prediction setup called online multitask learning [ABR07, RAB07, ARB08, LPS09, CCBG10, SRDV11]. The new ingredient is that before predicting outcome yt we are given its task number κt ∈{1, . . . , K}. The goal is to exploit similarities between tasks. As before, we have access to M models that each issue a prediction each round. If a single model predicts well on several tasks we want to ﬁgure this out quickly and exploit it. Simply ignoring the task number would not result in an adaptive algorithm. Applying a separate Bayesian predictor to each task independently would not result in any inter-task synergy. Nevertheless, it would guarantee regret at most K ln M overall. Now suppose each task is predicted well by some model from a small subset of models of size N ≪M. Running Bayes on all N-sparse allocations would achieve regret ln M N +K ln N. However, emulating Bayes in this case is NP-hard [RAB07]. The goal is to design efﬁcient algorithms with approximately the same regret bound. In [ABR07] this multiclass problem is reduced to MPP, giving regret bound N ln M N +B ln N. Here B is the number of same-task segments in the task sequence κ≤T . When all outcomes with the same task number are consecutive, i.e. B = K, then the desired bound is achieved. However the tasks may be interleaved, making the number of segments B much larger than K. We now eliminate the dependence on B, i.e. we solve a key open problem of [ABR07]. We apply the method of specialists to multitask learning, and obtain regret bounds close to the information-theoretic ideal, which in particular do not depend on the task segment count B at all. Construction We create a subset specialist (S, m) for each basic model index m and subset of tasks S ⊆{1, . . . , K}. At time t, specialists with the current task κt in their set S are awake, i.e. Wt := {(S, m) \| κt ∈S}, and issue the prediction P(yt\|y<t, S, m) := P(yt\|y<t, m) of model m. We assign to subset specialist (S, m) prior probability P(S, m) := P(S)P(m) where P(m) := 1/M is uniform, and P(S) includes each task independently with some ﬁxed bias σ(w) P(S) := σ(w)\|S\|σ(s)K−\|S\|. (7) This construction has the property that the product of prior weights of two loners ({κ1}, ˆm) and ({κ2}, ˆm) is dramatically lower than the single pair specialist ({κ1, κ2}, ˆm), especially so when the number of models M is large or when we consider larger task clusters. By strongly favoring it in the prior, any inter-task similarity present will be picked up fast. The resulting Bayesian strategy involving M2K subset specialists can be implemented efﬁciently. Theorem 5. The predictions P(yt\|y<t) of Bayes with the set prior (7) equal the predictions Predt(yt) of Algorithm 2. They can be computed in O(M) time per outcome using O(KM) storage. Of particular interest is Algorithm 2’s update rule for f κ t+1(m). This would be a regular Bayesian posterior calculation if vt(m) in Predt(yt) were replaced by f κ t (m). In fact, vt(m) is the communication channel by which knowledge about the performance of model m in other tasks is received. Proof. The resource analysis follows from inspection, noting that the update is fast because only the weights f κ t (m) associated to the current task κ are changed. We prove by induction on t that P(m\|y<t, Wt) = vt(m). In the base case t = 1 both equal 1/M. For the induction step we expand P(m\|y≤t, Wt+1), which is by deﬁnition proportional to X S∋κt+1 1 M σ(w)\|S\|σ(s)K−\|S\|   Y q≤t : κq∈S P(yq\|y<q, m)     Y q≤t : κq /∈S P(yq\|y<q)  . (8) The product form of both set prior and likelihood allows us to factor this exponential sum of products into a product of binary sums. It follows from the induction hypothesis that f k t (m) = σ(w) σ(s) Y q≤t : κq=k P(yq\|y<q, m) P(yq\|y<q) Then we can divide (8) by P(y≤t)σ(s)K and reorganize to P(m\|y≤t, Wt+1) ∝ 1 M f κt+1 t (m) Y k̸=κt+1 f k t (m) + 1 = 1 M f κt+1 t (m) f κt+1 t (m) + 1 K Y k=1 f k t (m) + 1 7 Since the algorithm maintains πt(m) = QK k=1(f k t (m) + 1) this is proportional to vt+1(m). Algorithm 2 Bayes with set prior (7) (for online multitask learning) Input: Number of tasks K ≥2, distribution σ(·) on {w, s}. Initialize f k 1 (m) := σ(w) σ(s) for each task k and π1(m) := QK k=1(f k 1 (m) + 1). for t = 1, 2, . . . do Observe task index κ = κt. Compute auxiliary vt(m) := f κ t (m) πt(m)/(f κ t (m)+1) PM i=1 f κ t (i) πt(i)/(f κ t (i)+1) . Receive prediction P(yt\|y<t, m) of each model m Issue prediction Predt(yt) := PM m=1 P(yt\|y<t, m)vt(m). Observe outcome yt and suffer loss −ln Predt(yt). Update f κ t+1(m) := P (yt\|y<t,m) Predt(yt) f κ t (m) and keep f k t+1(m) := f k t (m) for all k ̸= κ. Update πt+1(m) := f κ t+1(m)+1 f κ t (m)+1 πt(m). end for The Bayesian strategy is hence emulated fast by Algorithm 2. We now show it predicts well. Theorem 6. Let ˆm1, . . . , ˆmK be an N-sparse allocation of M models to K tasks. With tuned inclusion rate σ(w) = 1/N, the regret of Bayes (Algorithm 2) is bounded by R ≤N ln (M/N ) + KN H(1/N). Proof. Without loss of generality assume that ˆmk ∈{1, . . . , N}. Let Sn := {1 ≤k ≤K \| ˆmk = n}. The sets Sn for n = 1, . . . , N form a partition of the K tasks. By (7) QN n=1 P(Sn) = σ(w)Kσ(s)(N−1)K, which is maximized by the proposed tuning. The theorem now follows from (3). We achieve the desired goal since KN H(1/N) ≈K ln N. In practice N is of course unavailable for tuning, and we may tune σ(w) = 1/K pessimistically to get K ln K + N instead for all N simultaneously. Or alternatively, we may sacriﬁce some time efﬁciency to externally mix over all M possible values with decreasing prior, increasing the tuned regret by just ln N + O(ln ln N). If in addition the number of tasks is unknown or unbounded, we may (as done in Section 3.2) decrease the membership rate σ(w) with each new task encountered and guarantee regret R ≤ N ln(M/N) + K ln K + 4N + 2K ln ln K where now K is the number of tasks actually received. 5 Discussion We showed that Mixing Past Posteriors is not just a heuristic with an unusual regret bound: we gave it a full Bayesian interpretation using specialist models. We then applied our method to a multitask problem. Again an unusual algorithm resulted that exploits sparsity by pulling up the weights of models that have done well before in other tasks. In other words, if all tasks are well predicted by a small subset of base models, then this algorithm improves its prior over models as it learns from previous tasks. Both algorithms closely circumvent NP-hardness. The deep question is whether some of the common updates used in Nature can be brought into the Bayesian fold using the specialist mechanism. There are a large number of more immediate technical open problems (we just discuss a few). We presented our results using probabilities and log loss. However the bounds should easily carry over to the typical pseudo-likelihoods employed in online learning in connection with other loss functions. Next, it would be worthwhile to investigate for which inﬁnite sets of models we can still employ our updates implicitly. It was already shown in [KvE10, Koo11] that MPP can be efﬁciently emulated on all Bernoulli models. However, what about Gaussians, exponential families in general, or even linear regression? Finally, is there a Bayesian method for modeling concurrent multitasking, i.e. can the Bayesian analysis be generalized to the case where a small subset of models solve many tasks in parallel? 8 References [ABR07] Jacob Ducan Abernethy, Peter Bartlett, and Alexander Rakhlin. Multitask learning with expert advice. Technical report, University of California at Berkeley, January 2007. 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4,467	Sparse Approximate Manifolds for Differential Geometric MCMC Ben Calderhead∗ CoMPLEX University College London London, WC1E 6BT, UK b.calderhead@ucl.ac.uk Mátyás A. Sustik Department of Computer Sciences University of Texas at Austin Austin, TX 78712, USA sustik@cs.utexas.edu Abstract One of the enduring challenges in Markov chain Monte Carlo methodology is the development of proposal mechanisms to make moves distant from the current point, that are accepted with high probability and at low computational cost. The recent introduction of locally adaptive MCMC methods based on the natural underlying Riemannian geometry of such models goes some way to alleviating these problems for certain classes of models for which the metric tensor is analytically tractable, however computational efﬁciency is not assured due to the necessity of potentially high-dimensional matrix operations at each iteration. In this paper we ﬁrstly investigate a sampling-based approach for approximating the metric tensor and suggest a valid MCMC algorithm that extends the applicability of Riemannian Manifold MCMC methods to statistical models that do not admit an analytically computable metric tensor. Secondly, we show how the approximation scheme we consider naturally motivates the use of ℓ1 regularisation to improve estimates and obtain a sparse approximate inverse of the metric, which enables stable and sparse approximations of the local geometry to be made. We demonstrate the application of this algorithm for inferring the parameters of a realistic system of ordinary differential equations using a biologically motivated robust Student-t error model, for which the Expected Fisher Information is analytically intractable. 1 Introduction The use of Markov chain Monte Carlo methods can be extremely challenging in many modern day applications. This difﬁculty arises from the more frequent use of complex and nonlinear statistical models that induce strong correlation structures in their often high-dimensional parameter spaces. The exact structure of the target distribution is generally not known in advance and local correlation structure between different parameters may vary across the space, particularly as the chain moves from the transient phase, exploring areas of negligible probability mass, to the stationary phase exploring higher density regions [1]. Constructing a Markov chain that adapts to the target distribution while still drawing samples from the correct stationary distribution is challenging, although much research over the last 15 years has resulted in a variety of approaches and theoretical results. Adaptive MCMC for example, allows for global adaptation based on the partial or full history of a chain; this breaks its Markov property, although it has been shown that subject to some technical conditions [2,3] the resulting chain will still converge to the desired stationary distribution. Most recently, advances in Riemannian Manifold MCMC allow locally changing, position speciﬁc proposals to be made based on the underlying ∗http://www.2020science.net/people/ben-calderhead 1 geometry of the target distribution [1]. This directly takes into account the changing sensitivities of the model for different parameter values and enables very efﬁcient inference over a number of popular statistical models. It is useful for inference over large numbers of strongly covarying parameters, however this methodology is still not suitable for all statistical models; in its current form it is only applicable to models that admit an analytic expression for the metric tensor. In practice, there are many commonly used models for which the Expected Fisher Information is not analytically tractable, such as when a robust Student-t error model is employed to construct the likelihood. In this paper we propose the use of a locally adaptive MCMC algorithm that approximates the local Riemannian geometry at each point in the target space. This extends the applicability of Riemannian Manifold MCMC to a much wider class of statistical models than at present. In particular, we do so by estimating the covariance structure of the tangent vectors at a point on the Riemannian manifold induced by the statistical model. Considering this geometric problem as one of inverse covariance estimation naturally leads us to the use of an ℓ1 regularised maximum likelihood estimator. This approximate inverse approach allows the required geometry to be estimated with few samples, enabling good proposals for the Markov chain while inducing a natural sparsity in the inverse metric tensor that reduces the associated computational cost. We ﬁrst give a brief characterisation of current adaptive approaches to MCMC, making a distinction between locally and globally adaptive methods, since these two approaches have very different requirements in terms of proving convergence to the stationary distribution. We then discuss the use of geometry in MCMC and the interpretation of such methods as being locally adaptive, before giving the necessary background on Riemannian geometry and MCMC algorithms deﬁned on induced Riemannian manifolds. We focus on the manifold MALA sampler, which is derived from a Langevin diffusion process that takes into account local non-Euclidean geometry, and we discuss simpliﬁcations that may be made for computational efﬁciency. Finally we present a valid MCMC algorithm that estimates the Riemannian geometry at each iteration based on covariance estimates of random vectors tangent to the manifold at the chain’s current point. We demonstrate the use of ℓ1 regularisation to calculate sparse approximate inverses of the metric tensor and investigate the sampling properties of the algorithm on an extremely challenging statistical model for which the Expected Fisher Information is analytically intractable. 2 Background We wish to sample from some arbitrary target density π(x) deﬁned on a continuous state space XD, which may be high-dimensional. We may deﬁne a Markov chain that converges to the correct stationary distribution in the usual manner by proposing a new position x∗from the current position xn via some ﬁxed proposal distribution q(x∗\|xn); we accept the new move setting xn+1 = x∗with probability α(x∗\|xn) = min( π(x∗) π(xn) q(xn\|x∗) q(x∗\|xn), 1) and set xn+1 = xn otherwise. In a Bayesian context, we will often have a posterior distribution as our target π(x) = p(θ\|y), where y is the data and θ are the parameters of a statistical model. The choice of proposal distribution is the critical factor in determining how efﬁciently the Markov chain can explore the space and whether new moves will be accepted with high probability and be sufﬁciently far from the current point to keep autocorrelation of the samples to a minimum. There is a lot of ﬂexibility in the choice of proposal distribution, in that it may depend on the current point in a deterministic manner. We note that Adaptive MCMC approaches attempt to change their proposal mechanism throughout the running of the algorithm, and for the purpose of proving convergence to the stationary distribution it is useful to categorise them as follows; locally adaptive MCMC methods make proposals based only on the current position of the chain, whereas globally adaptive MCMC methods use previously collected samples in the chain’s history to generate a new proposal mechanism. This is an important distinction since globally adaptive methods lose their Markov property and convergence to the stationary distribution must be proven in an alternative manner. It has been shown that such chains may still be usefully employed as long as they satisfy some technical conditions, namely diminishing adaptation and bounded convergence [2]. In practice these algorithms represent a step towards MCMC as a “black box” method and may be very useful for sampling from target distributions for which there is no derivative or higher order geometric information available, however there are simple examples of standard Adaptive MCMC methods requiring hundreds of thousands of iterations in higher dimensions before adapting to a suitable proposal distribution [3]. In addi2 tion, if there is more information about the target density available, then there seems little point in trying to guess the geometric structure when it may be calculated directly. In this paper we focus on locally adaptive methods that employ proposals constructed deterministically from information at the current position of the Markov chain. 2.1 Locally Adaptive MCMC Many geometric-based MCMC methods may be categorised as being locally adaptive. When the derivative of the target density is available, MCMC methods such as the Metropolis-adjusted Langevin Algorithm (MALA) [4] allow local adaptation based on the geometry at the current point, but unlike globally adaptive MCMC, they retain their Markovian property and therefore converge to the correct stationary distribution using a standard Metropolis-Hastings step and without the need to satisfy further technical conditions. In general, we can deﬁne position-speciﬁc proposal densities based on deterministic functions that depend only on the current point. This idea has been previously employed to develop approaches for sampling multimodal distributions whereby large initial jumps followed by deterministic optimisation functions were used to create mode-jumping proposal mechanisms [5]. In some instances, the use of ﬁrst order geometric information may drastically speed up the convergence to a stationary distribution, however in other cases such algorithms exhibit very slow convergence, due to the gradients not being isotropic in magnitude [6]; in practice gradients may vary greatly in different directions and the rate of exploration of the target density may in addition be dependent on the problem-speciﬁc choice of parameterisation [1]. Methods using the standard gradient implicitly assume that the slope in each direction is approximately constant over a small distance, when in fact these gradients may rapidly change over short distances. Incorporating higher order geometry often helps although at an increased computational cost. A number of Hessian-based MCMC methods have been proposed as a solution [7]. While such approaches have been shown to work very well for selected problems there are a number of problems with this use of geometry; ad hoc methods are often necessary to deal with the fact that the Hessian might not be everywhere positive-deﬁnite, and second derivatives can be challenging and costly to compute. We can also exploit higher order information through the use of Riemannian geometry. Using a metric tensor instead of a Hessian matrix lends us nice properties such as invariance to reparameterisation of our statistical model, and positive-deﬁniteness is also assured. Riemannian geometry has been useful in a variety of other machine learning and statistical contexts [8] however the limiting factor is usually analytic or computational tractability. 3 Differential Geometric MCMC During the 1940s, Jeffreys and Rao demonstrated that the Expected Fisher Information has the same properties as a metric tensor and indeed induces a natural Riemannian structure for a statistical model [11, 10], providing a fascinating link between statistics and differential geometry. Much work has been done since then elucidating the relationship between statistics and Riemannian geometry, in particular examining geometric concepts such as distance, curvature and geodesics on statistical manifolds, within a ﬁeld that has become known as Information Geometry [6]. We ﬁrst provide an overview of Riemannian geometry and MCMC algorithms deﬁned on Riemannian manifolds. We then describe a sampling scheme that allows the local geometry to be estimated at each iteration for statistical models that do not admit an analytically tractable metric tensor. 3.1 Riemannian Geometry Informally, a manifold is an n-dimensional space that is locally Euclidean; it is locally equivalent to Rn via some smooth transformation. At each point θ ∈Rn on a Riemannian manifold M there exists a tangent space, which we denote as TθM. We can think of this as a linear approximation to the Riemannian manifold at the point θ and this is simply a standard vector space, whose origin is the current point on the manifold and whose vectors are tangent to this point. The vector space TθM is spanned by the differential operators h ∂ ∂θ1 , . . . , ∂ ∂θn i , which act on functions deﬁning paths on the underlying manifold [9]. In the context of MCMC we can consider the target density as the 3 log-likelihood of a statistical model given some data, such that at a particular point θ, the derivatives of the log-likelihood are tangent to the manifold and these are just the score vectors at θ, ∇θL = h ∂ ∂θ1 , . . . , ∂ ∂θn i . The tangent space at each point θ arises when we equip a differentiable manifold with an inner product at each point, which we can use to measure distance and angles between vectors. This inner product is deﬁned in terms of a metric tensor, Gθ, which deﬁnes a basis on each tangent space TθM. The tangent space is therefore a linear approximation of the manifold at a given point and it has the same dimensionality. A natural inner product for this vector space is given by the covariance of the basis score vectors, since the covariance function satisﬁes the same properties as a metric tensor, namely symmetry, bilinearity and positive-deﬁniteness [9]. This inner product then turns out to be equivalent to the Expected Fisher Information, following from the fact that the expectation of the score is zero, with the [i, j]th component of the tensor given by Gi,j = Cov ∂L ∂θi , ∂L ∂θj = Ep(x\|θ) ∂L ∂θi T ∂L ∂θj ! = −Ep(x\|θ) ∂2L ∂θi∂θj (1) Each tangent vector, t1 ∈TθM, at a point on the manifold, θ ∈M, has a length \|\|t1\|\| ∈R+, whose square is given by the inner product, such that \|\|t1\|\|2 Gθ = ⟨t1, t1⟩θ = tT 1 Gθt1. This squared distance is known as the ﬁrst fundamental form in Riemannian geometry [9], is invariant to reparameterisations of the coordinates, and importantly for MCMC provides a local measure of distance that takes into account the local 2nd order sensitivity of the statistical model. We note that when the metric tensor is constant for all values of θ then the Riemannian manifold is equivalent to a vector space with constant inner product; further, if the metric tensor is an identity matrix then the manifold simply becomes a Euclidean space. 3.2 Manifold MCMC We consider the manifold version of the MALA sampling algorithm, which proposes moves based on a stochastic differential equation deﬁning a Langevin diffusion [4]. It turns out we can also deﬁne such a diffusion on a Riemannian manifold [12], and so in a similar manner we can derive a sampling algorithm that takes the underlying geometric structure into account when making proposals. It is based on the Laplace-Beltrami operator, which simply measures the divergence of a vector ﬁeld on a manifold. The stochastic differential equation deﬁning the Langevin diffusion on a Riemannian manifold is dθ(t) = 1 2 ˜∇θL(θ(t))dt + d ˜ b(t), where the natural gradient [6] is the gradient of a function transformed into the tangent space at the current point by a linear transformation using the basis deﬁned by the metric tensor, such that ˜∇θL(θ(t)) = G−1(θ(t))∇θL(θ(t)), and the Brownian motion on the Riemannian manifold is deﬁned as d˜bi(t) = \|G(θ(t))\|−1 2 D X j=1 ∂ ∂θj (G−1(θ(t))ij\|G(θ(t))\| 1 2 )dt + p G−1(θ(t))db(t) i (2) The ﬁrst part of the right hand side of Equation 2 represents the 1st order terms of the LaplaceBeltrami operator and these relate to the local curvature of the manifold, reducing to zero if the metric is everywhere constant. The second term on the right hand side provides a position speciﬁc linear transformation of the Brownian motion b(t) based on the local metric. Employing a ﬁrst order Euler integrator, the discrete form of the Langevin diffusion on a Riemannian manifold follows as θn+1 i = θn i + ϵ2 2 (G−1(θn)∇θL(θn))i −ϵ2 D X j=1 G−1(θn)∂G(θn) ∂θj G−1(θn) ij (3) +ϵ2 2 D X j=1 G−1(θn) ij Tr G−1(θn)∂G(θn) ∂θj + ϵ p G−1(θn)zn i = µ(θn, ϵ)i + ϵ p G−1(θn)zn i 4 which deﬁnes a proposal mechanism with density q(θ∗\|θn) = N(θ∗\|µ(θn, ϵ), ϵ2G−1(θn)) and acceptance probability min{1, p(θ∗)q(θn\|θ∗)/p(θn)q(θ∗\|θn)} to ensure convergence to the invariant density p(θ). We note that this deterministically deﬁnes a position-speciﬁc proposal distribution at each point on the manifold; we may categorise this as another locally adaptive MCMC method and convergence to the invariant density follows from using the standard Metropolis-Hastings ratio. It may be computationally expensive to calculate the 3rd order derivatives needed for working out the rate of change of the metric tensor, and so an obvious approximation is to assume these derivatives are zero for each step. In other words, for each step we can assume that the metric is locally constant. Of course even if the curvature of the manifold is not constant, this simpliﬁed proposal mechanism still deﬁnes a correct MCMC method which converges to the target measure, as we accept or reject moves using a Metropolis-Hastings ratio. This is equivalent to a position-speciﬁc pre-conditioned MALA proposal, where the pre-conditioning is dependent on the current parameter values θn+1 = θn + ϵ2 2 G−1(θn)∇θL(θn) + ϵ p G−1(θn)zn (4) For a manifold whose metric tensor is globally constant, this reduces further to a pre-conditioned MALA proposal, where the pre-conditioning is effectively independent of the current parameter values. In this context, such pre-conditioning no longer needs to be chosen arbitrarily, but rather it may be informed by the geometry of the distribution we are exploring. We point out that any approximations of the metric tensor would be best employed in the simpliﬁed mMALA scheme, deﬁning the covariance of the proposal distribution, or as a ﬂat approximation to a manifold. In the case of full mMALA, or even Hamiltonian Monte Carlo deﬁned on a Riemannian manifold [1], Christoffel symbols are also used, incorporating the derivatives of the metric tensor as it changes across the surface of the manifold - in many cases the extra expense of computing or estimating such higher order information is not sufﬁciently supported by the increase in sampling efﬁciency [1] and for this reason we do not consider such methods further. In the next section we consider the representation of the metric tensor as the covariance of the tangent vectors at each point. We consider a method of estimating this such that convergence is guaranteed by extending the state-space and introducing auxiliary variables that are conditioned on the current point and we demonstrate its potential within a Riemannian geometric context. 4 Approximate Geometry for MCMC Proposals We ﬁrst derive an acceptance ratio on an extended state-space that enables convergence to the stationary distribution before describing the implications for developing new differential geometric MCMC methods. Following [13, 14] we can employ the oft-used trick of deﬁning an extended state space X × D. We may of course choose D to be of any size, however in our particular case we shall choose D to be Rm×s, where m is the dimension of the data and s is the number of samples; the reasons for this shall become clear. We therefore sample from this extended state space, whose joint distribution follows as π∗= π(x)ˆπ(d\|x). Given the current states [xn, dn], we may propose a new state q(x∗\|xn, dn) and the MCMC algorithm will satisfy detailed balance and hence converge to the stationary distribution if we accept joint proposals with Metropolis-Hastings probability ratio, α(x∗, d∗\|xn, dn) = min 1, π∗(x∗, d∗) π∗(xn, dn) q(xn\|x∗, d∗) q(x∗\|xn, dn) ˆπ(dn\|xn) ˆπ(d∗\|x∗) (5) = min 1, π(x∗) π(xn) ˆπ(d∗\|x∗) ˆπ(dn\|xn) q(xn\|x∗, d∗) q(x∗\|xn, dn) ˆπ(dn\|xn) ˆπ(d∗\|x∗) = min 1, π(x∗) π(xn) q(xn\|x∗, d∗) q(x∗\|xn, dn) This is a reversible transition on π(x, d), from which we can sample to obtain π(x) as the marginal distribution. The key point here is that we may deﬁne our proposal distribution q(x∗\|xn, dn) in almost any deterministic manner we wish. In particular, choosing ˆπ(d\|x) to be the same distribution 5 as the log-likelihood for our statistical model, the s samples from the extended state space D may be thought of as pseudo-data, from which we can deterministically calculate an estimate of the Expected Fisher Information to use as the covariance of a proposal distribution. Speciﬁcally, each sampled pseudo-data can be used deterministically to give a sample of ∂L dθ given the current θ, all of which may then be used deterministically to obtain an approximation of the covariance of tangent vectors at the current point. This approximation, unlike the Hessian, will always be positive deﬁnite, and gives us an approximation of the metric tensor deﬁning the local geometry. Further, we may use additional deterministic procedures, given xn and dn, to construct better proposals; we consider a sparsity inducing approach in the next section. 5 Stability and Sparsity via ℓ1 Regularisation We have two motivations for using an ℓ1 regularisation approach for computing the inverse of the metric tensor; ﬁrstly, since the metric is equivalent to the covariance of tangent vectors, we may obtain more stable estimates of the inverse metric tensor using smaller numbers of samples, and secondly, it induces a natural sparsity in the inverse metric, which may be exploited to decrease the computational cost associated with repeated Cholesky factorisations and matrix-vector multiplications. We adopted the graphical lasso [15, 16], in which the maximum likelihood solution results in the matrix optimisation problem, arg min A≻0{−log det(A) + tr(AG) + γ X i̸=j \|Aij\|} (6) where G is an empirical covariance matrix and γ is a regularisation parameter. This convex optimisation problem aims to ﬁnd A, the regularised maximum likelihood estimate for the inverse of the covariance matrix. Importantly, the optimisation algorithm we employ is deterministic given our tangent vectors, and therefore does not affect the validity of our MCMC algorithm; indeed we note that we may use any deterministic sparse matrix inverse estimation approaches within this MCMC algorithm. The use of the ℓ1 regularisation promotes sparsity [23]; larger values for the regularisation parameter matrix Λ results in a solution that is more sparse, on the other hand when Λ approaches zero, the solution converges to the inverse of G (assuming it exists). It is also worth noting that the ℓ1 regularisation helps to recover a sparse structure in a high dimensional setting where the number of samples is less than the number of parameters [17]. In order to achieve sufﬁciently fast computation we carefully implemented the graphical lasso algorithm tailored to this problem. We used no penalisation for the diagonal and uniform regularisation parameter value for the off-diagonal elements. The motivation for not penalising the diagonal is that it has been shown in the covariance estimation setting that the true inverse is approached as the number of samples is increased [18], and the structure is learned more accurately [19]. The simple regularisation structure allowed code simpliﬁcation and reduction in memory use. We refactored the graphical lasso algorithm of [15] and implemented it directly in FORTRAN which we then called from MATLAB, making sure to minimise matrix copying due to MATLAB processing. This code is available as a software package, GLASSOFAST [20]. In the current context, the use of this approach allows us to obtain sparse approximations to the inverse metric tensor, which may then be used in an MCMC proposal. Indeed, even if we have access to an analytic metric tensor we need not use the full inverse for our proposals; we could still obtain an approximate sparse representation, which may be beneﬁcial computationally. The metric tensor varies smoothly across a Riemannian manifold and, theoretically, if we are calculating the inverse of 2 metric tensors that are close to each other, they may be numerically similar enough to be able to use the solution of one to speed up convergence of solution for the other, although in the simulations in this paper we found no beneﬁt in doing so, i.e. the metric tensor varied too much as the MCMC sampler took large steps across the manifold. 6 Simulation Study We consider a challenging class of statistical models that severely tests the sampling capability of MCMC methods; in particular, two examples based on nonlinear differential equations using a 6 (a) Exact full inverse (b) Approximate sparse inverse Figure 1: In this comparison we plotted the exact and the sparse approximate inverses of a typical metric tensor G; we note that only subsets of parameters are typically strongly correlated in the statistical models we consider here and that the sparse approximation still captures the main correlation structure present. Here the dimension is p = 25, and the regularisation parameter γ is 0.05 · \|\|G\|\|∞. Table 1: Summary of results for the Fitzhugh-Nagumo model with 10 runs of each parameter sampling scheme and 5000 posterior samples. Sampling Time (s) Mean ESS Total Time/ Relative Method (a, b, c) (Min mean ESS) Speed Metropolis 14.5 139, 18.2, 23.4 0.80 ×1.1 MALA 24.9 119.3, 28.7, 52.3 0.87 ×1.0 mMALA Simp. 35.9 283.4, 136.6, 173.7 0.26 ×3.4 biologically motivated robust Student-t likelihood, which renders the metric tensor analytically intractable. We examine the efﬁciency of our MCMC method with approximate metric on a well studied toy example, the Fitzhugh-Nagumo model, before examining a realistic, nonlinear and highly challenging example describing enzymatic circadian control in the plant Arabidopsis thaliana [22]. 6.1 Nonlinear Ordinary Differential Equations Statistical modelling using systems of nonlinear ordinary differential equations plays a vital role in unravelling the structure and behaviour of biological processes at a molecular level. The well-used Gaussian error model however is often inappropriate, particularly in molecular biology where limited measurements may not be repeated under exactly the same conditions and are susceptible to bias and systematic errors. The use of a Student-t distribution as a likelihood may help the robustness of the model with respect to possible outliers in the data. This presents a problem for standard manifold MCMC algorithms as it makes the metric tensor analytically intractable. We consider ﬁrst the Fitzhugh-Nagumo model [1]. This synthetic dataset consisted of 200 time points simulated from the model between t = [0, 20] with parameters [a, b, c] = [0.2, 0.2, 3], to which Gaussian distributed noise was added with variance σ2 = 0.25. We employed a Student-t likelihood with scaling parameter v = 3, and compared M-H and MALA (both employing scaled isotropic covariances), and simpliﬁed mMALA with approximate metric. The stepsize for each was automatically adjusted during the burn-in phase to obtain the theoretically optimal acceptance rate. Table 1 shows the results including time-normalised effective sample size (ESS) as a measure of sampling efﬁciency excluding burn-in [1]. The approximate manifold sampler offers a modest improvement on the other two samplers; despite taking longer to run because of the computational cost of estimating the metric, the samples it draws exhibit lower autocorrelation, and as such the approximate manifold sampler offers the highest time-normalised ESS. The toy Fitzhugh-Nagumo model is however rather simple, and despite being a popular example is rather unlike many realistic models used nowadays in the molecular modelling community. As such we consider another larger model that describes the enzymatic control of the circadian networks in Arabidopsis thaliana [21]. This is an extremely challenging, highly nonlinear model. We consider 7 Table 2: Comparison of pseudodata sample size on the quality of metric tensor estimation, and hence on sampling efﬁciency, using the circadian network example model, with 10 runs and 10,000 posterior samples. Number of Time (s) Min Mean ESS Total Time/ Relative Samples (Min mean ESS) Speed 10 155.6 85.1 1.90 ×1.0 20 163.2 171.9 0.95 ×2.0 30 168.9 209.1 0.81 ×2.35 40 175.2 208.3 0.84 ×2.26 Table 3: Summary of results for the circadian network model with 10 runs of each parameter sampling scheme and 10,000 posterior samples. Sampling Time (s) Min Mean ESS Total Time/ Relative Method (Min mean ESS) Speed Metropolis 37.1 6.0 6.2 ×4.4 MALA 101.3 3.7 27.4 ×1.0 Adaptive MCMC 110.4 46.7 2.34 ×11.7 mMALA Simp. 168.9 209.1 0.81 ×33.8 inferring the 6 rate parameters that control production and decay of proteins in the nucleus and cytoplasm (see [22] for the equations and full details of the model), again employing a Student-t likelihood for which the Expected Fisher Information is analytically intractable. We used parameter values from [22] to simulate observations for each of the six species at 48 time points representing 48 hours in the model. Student-t distributed noise was then added to obtain the data for inference. We ﬁrst investigated the effect that the tangent vector sample size for covariance estimation has on the sampling efﬁciency of simpliﬁed mMALA. The results in Table 2 show that there is a threshold above which a more accurate estimate of the metric tensor does not result in additional sampling advantage. The threshold for this particular example model is around 30 pseudodata samples. Table 3 shows the time normalised statistical efﬁciency for each of the sampling methods; this time we also compare an Adaptive MCMC algorithm [2] with M-H, MALA, and simpliﬁed mMALA with approximate geometry. Both the M-H and MALA algorithms fail to explore the target distribution and have severe difﬁculties with the extreme scalings and nonlinear correlation structure present in the manifold. The Adaptive MCMC method works reasonably well after taking 2000 samples to learn the covariance structure, although its performance is still poorer than the simpliﬁed mMALA scheme, which converges almost immediately with no adaptation time required; the approximation mMALA makes of the local geometry allows it to adequately deal with the different scalings and correlations that occur in different parts of the space. 7 Conclusions The use of Riemannian geometry can be very useful for enabling efﬁcient sampling from arbitrary probability densities. The metric tensor may be used for creating position-speciﬁc proposal mechanisms that allow MCMC methods to automatically adapt to the local correlation structure induced by the sensitivities of the parameters of a statistical model. The metric tensor may conveniently be deﬁned as the Expected Fisher Information, however this quantity is often either difﬁcult or impossible to compute analytically. We have presented a sampling scheme that approximates the Expected Fisher Information by estimating the covariance structure of the tangent vectors at each point on the manifold. By considering this problem as one of inverse covariance estimation, this naturally led us to consider the use of ℓ1 regularisation to improve the estimation procedure. This had the added beneﬁt of inducing sparsity into the metric tensor, which may offer computational advantages when proposing MCMC moves across the manifold. For future work it will be exciting to investigate the potential impact of approximate, sparse metric tensors for high dimensional problems. 8 Ben Calderhead gratefully acknowledges his Research Fellowship through the 2020 Science programme, funded by EPSRC grant number EP/I017909/1 and supported by Microsoft Research. References [1] M. Girolami and B. 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4,468	Bayesian active learning with localized priors for fast receptive ﬁeld characterization Mijung Park Electrical and Computer Engineering The University of Texas at Austin mjpark@mail.utexas.edu Jonathan W. Pillow Center For Perceptual Systems The University of Texas at Austin pillow@mail.utexas.edu Abstract Active learning methods can dramatically improve the yield of neurophysiology experiments by adaptively selecting stimuli to probe a neuron’s receptive ﬁeld (RF). Bayesian active learning methods specify a posterior distribution over the RF given the data collected so far in the experiment, and select a stimulus on each time step that maximally reduces posterior uncertainty. However, existing methods tend to employ simple Gaussian priors over the RF and do not exploit uncertainty at the level of hyperparameters. Incorporating this uncertainty can substantially speed up active learning, particularly when RFs are smooth, sparse, or local in space and time. Here we describe a novel framework for active learning under hierarchical, conditionally Gaussian priors. Our algorithm uses sequential Markov Chain Monte Carlo sampling (“particle ﬁltering” with MCMC) to construct a mixture-of-Gaussians representation of the RF posterior, and selects optimal stimuli using an approximate infomax criterion. The core elements of this algorithm are parallelizable, making it computationally efﬁcient for real-time experiments. We apply our algorithm to simulated and real neural data, and show that it can provide highly accurate receptive ﬁeld estimates from very limited data, even with a small number of hyperparameter samples. 1 Introduction Neurophysiology experiments are costly and time-consuming. Data are limited by an animal’s willingness to perform a task (in awake experiments) and the difﬁculty of maintaining stable neural recordings. This motivates the use of active learning, known in statistics as “optimal experimental design”, to improve experiments using adaptive stimulus selection in closed-loop experiments. These methods are especially powerful for models with many parameters, where traditional methods typically require large amounts of data. In Bayesian active learning, the basic idea is to deﬁne a statistical model of the neural response, then carry out experiments to efﬁciently characterize the model parameters [1–6]. (See Fig. 1A). Typically, this begins with a (weakly- or non-informative) prior distribution, which expresses our uncertainty about these parameters before the start of the experiment. Then, recorded data (i.e., stimulus-response pairs) provide likelihood terms that we combine with the prior to obtain a posterior distribution. This posterior reﬂects our beliefs about the parameters given the data collected so far in the experiment. We then select a stimulus for the next trial that maximizes some measure of utility (e.g., expected reduction in entropy, mean-squared error, classiﬁcation error, etc.), integrated with respect to the current posterior. In this paper, we focus on the problem of receptive ﬁeld (RF) characterization from extracellularly recorded spike train data. The receptive ﬁeld is a linear ﬁlter that describes how the neuron integrates its input (e.g., light) over space and time; it can be equated with the linear term in a generalized linear 1 model (GLM) of the neural response [7]. Typically, RFs are high-dimensional (with 10s to 100s of parameters, depending on the choice of input domain), making them an attractive target for active learning methods. Our paper builds on prior work from Lewi et al [6], a seminal paper that describes active learning for RFs under a conditionally Poisson point process model. Here we show that a sophisticated choice of prior distribution can lead to substantial improvements in active learning. Speciﬁcally, we develop a method for learning under a class of hierarchical, conditionally Gaussian priors that have been recently developed for RF estimation [8,9]. These priors ﬂexibly encode a preference for smooth, sparse, and/or localized structure, which are common features of real neural RFs. In ﬁxed datasets (“passive learning”), the associated estimators give substantial improvements over both maximum likelihood and standard lasso/ridge-regression shrinkage estimators, but they have not yet been incorporated into frameworks for active learning. Active learning with a non-Gaussian prior poses several major challenges, however, since the posterior is non-Gaussian, and requisite posterior expectations are much harder to compute. We address these challenges by exploiting a conditionally Gaussian representation of the prior (and posterior) using sampling at the level of the hyperparameters. We demonstrate our method using the Automatic Locality Determination (ALD) prior introduced in [9], where hyperparameters control the locality of the RF in space-time and frequency. The resulting algorithm outperforms previous active learning methods on real and simulated neural data, even under various forms of model mismatch. The paper is organized as follows. In Sec. 2, we formally deﬁne the Bayesian active learning problem and review the algorithm of [6], to which we will compare our results. In Sec. 3, we describe a hierarchical response model, and in Sec. 4 describe the localized RF prior that we will employ for active learning. In Sec. 5, we describe a new active learning method for conditionally Gaussian priors. In Sec. 6, we show results of simulated experiments with simulated and real neural data. 2 Bayesian active learning Bayesian active learning (or “experimental design”) provides a model-based framework for selecting optimal stimuli or experiments. A Bayesian active learning method has three basic ingredients: (1) an observation model (likelihood) p(y\|x, k), specifying the conditional probability of a scalar response y given vector stimulus x and parameter vector k; (2) a prior p(k) over the parameters of interest; and (3) a loss or utility function U, which characterizes the desirability of a stimulusresponse pair (x, y) under the current posterior over k. The optimal stimulus x is the one that maximizes the expected utility Ey\|x[U(x, y)], meaning the utility averaged over the distribution of (as yet) unobserved y\|x. One popular choice of utility function is the mutual information between (x, y) and the parameters k. This is commonly known as information-theoretic or infomax learning [10]. It is equivalent to picking the stimulus on each trial that minimizes the expected posterior entropy. Let Dt = {xi, yi}t i=1 denote the data collected up to time step t in the experiment. Under infomax learning, the optimal stimulus at time step t + 1 is: xt+1 = arg max x Ey\|x,Dt[I(y, k\|x, Dt)] = arg min x Ey\|x,Dt,[H(k\|x, y, Dt)], (1) where H(k\|x, y, Dt) = − R p(k\|x, y, Dt) log p(k\|x, y, Dt)dk denotes the posterior entropy of k, and p(y\|x, Dt) = R p(y\|x, k)p(k\|Dt)dk is the predictive distribution over response y given stimulus x and data Dt. The mutual information provided by (y, x) about k, denoted by I(y, k\|x, Dt), is simply the difference between the prior and posterior entropy. 2.1 Method of Lewi, Butera & Paninski 2009 Lewi et al [6] developed a Bayesian active learning framework for RF characterization in closed-loop neurophysiology experiments, which we henceforth refer to as “Lewi-09”. This method employs a conditionally Poisson generalized linear model (GLM) of the neural spike response: λt = g(k⊤xt) yt ∼ Poiss(λt), (2) 2 C hyperparameters parameters (RF) parameters (RF) hierarchical RF model A select stimulus update posterior experiment B RF model (Lewi et al 09) stimulus spike count Figure 1: (A) Schematic of Bayesian active learning for neurophysiology experiments. For each presented stimulus x and recorded response y (upper right), we update the posterior over receptive ﬁeld k (bottom), then select the stimulus that maximizes expected information gain (upper left). (B) Graphical model for the non-hierarchical RF model used by Lewi-09. It assumes a Gaussian prior p(k) and Poisson likelihood p(yt\|xt, k). (C) Graphical model for the hierarchical RF model used here, with a hyper-prior pθ(θ) over hyper-parameters and conditionally Gaussian prior p(k\|θ) over the RF. For simplicity and speed, we assume a Gaussian likelihood for p(yt\|xt, k), though all examples in the manuscript involved real neural data or simulations from a Poisson GLM. where g is a nonlinear function that ensures non-negative spike rate λt. The Lewi-09 method assumes a Gaussian prior over k, which leads to a (non-Gaussian) posterior given by the product of Poisson likelihood and Gaussian prior. (See Fig. 1B). Neither the predictive distribution p(y\|x, Dt) nor the posterior entropy H(k\|x, y, Dt) can be computed in closed form. However, the log-concavity of the posterior (guaranteed for suitable choice of g [11]) motivates a tractable and accurate Gaussian approximation to the posterior, which provides a concise analytic formula for posterior entropy [12,13]. The key contributions of Lewi-09 include fast methods for updating the Gaussian approximation to the posterior and for selecting the stimulus (subject to a maximum-power constraint) that maximizes expected information gain. The Lewi-09 algorithm yields substantial improvement in characterization performance relative to randomized iid (e.g., “white noise”) stimulus selection. Below, we will benchmark the performance of our method against this algorithm. 3 Hierarchical RF models Here we seek to extend the work of Lewi et al to incorporate non-Gaussian priors in a hierarchical receptive ﬁeld model. (See Fig. 1C). Intuitively, a good prior can improve active learning by reducing the prior entropy, i.e., the effective size of the parameter space to be searched. The drawback of more sophisticated priors is that they may complicate the problem of computing and optimizing the posterior expectations needed for active learning. To focus more straightforwardly on the role of the prior distribution, we employ a simple linearGaussian model of the neural response: yt = k⊤xt + ϵt, ϵt ∼N(0, σ2), (3) where ϵt is iid zero-mean Gaussian noise with variance σ2. We then place a hierarchical, conditionally Gaussian prior on k: k \| θ ∼ N(0, Cθ) (4) θ ∼ pθ, (5) where Cθ is a prior covariance matrix that depends on hyperparameters θ. These hyperparameters in turn have a hyper-prior pθ. We will specify the functional form of Cθ in the next section. In this setup, the effective prior over k is a mixture-of-Gaussians, obtained by marginalizing over θ: p(k) = Z p(k\|θ)p(θ)dθ = Z N(0, Cθ) pθ(θ)dθ. (6) 3 Given data X = (x1, . . . , xt)⊤and Y = (y1, . . . , yt)⊤, the posterior also takes the form of a mixture-of-Gaussians: p(k\|X, Y ) = Z p(k\|X, Y, θ)p(θ\|X, Y )dθ (7) where the conditional posterior given θ is the Gaussian p(k\|X, Y, θ) = N(µθ, Λθ), µθ = 1 σ2 ΛθX⊤Y, Λθ = ( 1 σ2 X⊤X + C−1 θ )−1, (8) and the mixing weights are given by the marginal posterior, p(θ\|X, Y ) ∝p(Y \|X, θ)pθ(θ), (9) which we will only need up to a constant of proportionality. The marginal likelihood or evidence p(Y \|X, θ) is the marginal probability of the data given the hyperparameters, and has a closed form for the linear Gaussian model: p(Y \|X, θ) = \|2πΛθ\| 1 2 \|2πσ2I\| 1 2 \|2πCθ\| 1 2 exp 1 2 µ⊤ θ Λ−1 θ µθ −m⊤L−1m , (10) where L = σ2(X⊤X)−1 and m = 1 σ2 LX⊤Y . Several authors have pointed out that active learning confers no beneﬁt over ﬁxed-design experiments in linear-Gaussian models with Gaussian priors, due to the fact that the posterior covariance is response-independent [1, 6]. That is, an optimal design (one that minimizes the ﬁnal posterior entropy) can be planned out entirely in advance of the experiment. However, this does not hold for linear-Gaussian models with non-Gaussian priors, such as those considered here. The posterior distribution in such models is data-dependent via the marginal posterior’s dependence on Y (eq. 9). Thus, active learning is warranted even for linear-Gaussian responses, as we will demonstrate empirically below. 4 Automatic Locality Determination (ALD) prior In this paper, we employ a ﬂexible RF model underlying the so-called automatic locality determination (ALD) estimator [9].1 The key justiﬁcation for the ALD prior is the observation that most neural RFs tend to be localized in both space-time and spatio-temporal frequency. Locality in space-time refers to the fact that (e.g., visual) neurons integrate input over a limited domain in time and space; locality in frequency refers to the band-pass (or smooth / low pass) character of most neural RFs. The ALD prior encodes these tendencies in the parametric form of the covariance matrix Cθ, where hyperparameters θ control the support of both the RF and its Fourier transform. The hyperparameters for the ALD prior are θ = (ρ, νs, νf, Ms, Mf)⊤, where ρ is a “ridge” parameter that determines the overall amplitude of the covariance; νs and νf are length-D vectors that specify the center of the RF support in space-time and frequency, respectively (where D is the degree of the RF tensor2); and Ms and Mf are D × D positive deﬁnite matrices that describe an elliptical (Gaussian) region of support for the RF in space-time and frequency, respectively. In practice, we will also include the additive noise variance σ2 (eq. 3) as a hyperparameter, since it plays a similar role to C in determining the posterior and evidence. Thus, for the (D = 2) examples considered here, there are 12 hyperparameters, including scalars σ2 and ρ, two hyperparameters each for νs and νf, and three each for symmetric matrices Ms and Mf. Note that although the conditional ALD prior over k\|θ assigns high prior probability to smooth and sparse RFs for some settings of θ, for other settings (i.e., where Ms and Mf describe elliptical regions large enough to cover the entire RF) the conditional prior corresponds to a simple ridge prior and imposes no such structure. We place a ﬂat prior over θ so that no strong prior beliefs about spatial locality or bandpass frequency characteristics are imposed a priori. However, as data from a neuron with a truly localized RF accumulates, the support of the marginal posterior p(θ\|Dt) shrinks down on regions that favor a localized RF, shrinking the posterior entropy over k far more quickly than is achievable with methods based on Gaussian priors. 1“Automatic” refers to the fact that in [9], the model was used for empirical Bayes inference, i.e., MAP inference after maximizing the evidence for θ. Here, we consider perform fully Bayesian inference under the associated model. 2e.g., a space×space×time RF has degree D = 3. 4 5 Bayesian active learning with ALD To perform active learning under the ALD model, we need two basic ingredients: (1) an efﬁcient method for representing and updating the posterior p(k\|Dt) as data come in during the experiment; and (2) an efﬁcient algorithm for computing and maximizing the expected information gain given a stimulus x. We will describe each of these in turn below. 5.1 Posterior updating via sequential Markov Chain Monte Carlo To represent the ALD posterior over k given data, we will rely on the conditionally Gaussian representation of the posterior (eq. 7) using particles {θi}i=1,...,N sampled from the marginal posterior, θi ∼P(θ\|Dt) (eq. 9). The posterior will then be approximated as: p(k\|Dt) ≈1 N X i p(k\|Dt, θi), (11) where each distribution p(k\|Dt, θi) is Gaussian with θi-dependent mean and covariance (eq. 8). Markov Chain Monte Carlo (MCMC) is a popular method for sampling from distributions known only up to a normalizing constant. In cases where the target distribution evolves over time by accumulating more data, however, MCMC samplers are often impractical due to the time required for convergence (i.e., “burning in”). To reduce the computational burden, we use a sequential sampling algorithm to update the samples of the hyperparameters at each time step, based on the samples drawn at the previous time step. The main idea of our algorithm is adopted from the resample-move particle ﬁlter, which involves generating initial particles; resampling particles according to incoming data; then performing MCMC moves to avoid degeneracy in particles [14]. The details are as follows. Initialization: On the ﬁrst time step, generate initial hyperparameter samples {θi} from the hyperprior pθ, which we take to be ﬂat over a broad range in θ. Resampling: Given a new stimulus/response pair {x, y} at time t, resample the existing particles according to the importance weights: p(yt\|θ(t) i , Dt−1, xt) = N(yt\|µi ⊤xt, xt ⊤Λixt + σ2 i ), (12) where (µi, Λi) denote the mean and covariance of the Gaussian component attached to particle θi, This ensures the posterior evolves according to: p(θ(t) i \|Dt) ∝p(yt\|θ(t) i , Dt−1, xt)p(θ(t) i \|Dt−1). (13) MCMC Move: Propagate particles via Metropolis Hastings (MH), with multivariate Gaussian proposals centered on the current particle θi of the Markov chain: θ∗∼N(θi, Γ), where Γ is a diagonal matrix with diagonal entries given by the variance of the particles at the end of time step t−1. Accept the proposal with probability min(1, α), where α = q(θ∗) q(θi) , with q(θi) = p(θi\|Dt). Repeat MCMC moves until computational or time budget has expired. The main bottleneck of this scheme is the updating of conditional posterior mean µi and covariance Λi for each particle θi, since this requires inversion of a d × d matrix. (Note that, unlike Lewi09, these are not rank-one updates due to the fact that Cθi changes after each θi move). This cost is independent of the amount of data, linear in the number of particles, and scales as O(d3) in RF dimensionality d. However, particle updates can be performed efﬁciently in parallel on GPUs or machines with multi-core processors, since the particles do not interact except for stimulus selection, which we describe below. 5.2 Optimal Stimulus Selection Given the posterior over k at time t, represented by a mixture of Gaussians attached to particles {θi} sampled from the marginal posterior, our task is to determine the maximally informative stimulus to present at time t + 1. Although the entropy of a mixture-of-Gaussians has no analytic form, we can 5 true filter 10 20 10 20 200 trials 400 trials Lewi-09 ALD10 ALD100 1 1 B A 1000 trials 0 200 400 600 800 1000 30 40 50 60 70 angle difference in degree # trials Lewi-09 ALD10 ALD100 62.82 51.54 44.94 57.29 40.69 36.65 43.34 35.90 28.98 Passive-ALD Figure 2: Simulated experiment. (A) Angular error in estimates of a simulated RF (20 × 20 pixels, shown in inset) vs. number of stimuli, for Lewi-09 method (blue), the ALD-based active learning method using 10 (pink) or 100 (red) particles, and the ALD-based passive learning method (black). True responses were simulated from a Poisson-GLM neuron. Traces show average over 20 independent repetitions. (B) RF estimates obtained by each method after 200, 400, and 1000 trials. Red numbers below indicate angular error (deg). compute the exact posterior covariance via the formula: ˜Λt = 1 N N X i=1 Λi + µiµi ⊤ −˜µ˜µ⊤, (14) where ˜µt = 1 N P µi is the full posterior mean. This leads to an upper bound on posterior entropy, since a Gaussian is the maximum-entropy distribution for ﬁxed covariance. We then take the next stimulus to be the maximum-variance eigenvector of the posterior covariance, which is the most informative stimulus under a Gaussian posterior and Gaussian noise model, subject to a power constraint on stimuli [6]. Although this selection criterion is heuristic, since it is not guaranteed to maximize mutual information under the true posterior, it is intuitively reasonable since it selects the stimulus direction along which the current posterior is maximally uncertain. Conceptually, directions of large posterior variance can arise in two different ways: (1) directions of large variance for all covariances Λi, meaning that all particles assign high posterior uncertainty over k\|Dt in the direction of x; or (2) directions in which the means µi are highly dispersed, meaning the particles disagree about the mean of k\|Dt in the direction of x. In either scenario, selecting a stimulus proportional to the dominant eigenvector is heuristically justiﬁed by the fact that it will reduce collective uncertainty in particle covariances or cause particle means to converge by narrowing of the marginal posterior. We show that the method performs well in practice for both real and simulated data (Section 6). We summarize the complete method in Algorithm 1. Algorithm 1 Sequential active learning under conditionally Gaussian models Given particles {θi} from p(θ\|Dt), which deﬁne the posterior as P(k\|Dt) = P i N(µi, Λi), 1. Compute the posterior covariance ˜Λt from {(µi, Λi)} (eq. 14). 2. Select optimal stimulus xt+1 as the maximal eigenvector of ˜Λt 3. Measure response yt+1. 4. Resample particles {θi} with the weights {N(yt+1\|µi⊤xt+1, xt+1⊤Λixt+1 + σ2 i )}. 5. Perform MH sampling of p(θ\|Dt+1), starting from resampled particles. repeat 6 Lewi-09 true filter (A) (B) (C) angle difference: 60.68 37.82 62.82 42.57 60.32 50.73 ALD10 Figure 3: Additional simulated examples comparing Lewi-09 and ALDbased active learning. Responses were simulated from a GLM-Poisson model with three different true 400-pixel RFs (left column): (A) a Gabor ﬁlter (shown previously in [6]); (B): a centersurround RF, typical in retinal ganglion cells; (C): a relatively non-localized grid-cell RF. Middle and right columns show RF estimates after 400 trials of active learning under each method, with average angular error (over independent 20 repeats) shown beneath in red. 6 Results Simulated Data: We tested the performance of our algorithm using data simulated from a PoissonGLM neuron with a 20 × 20 pixel Gabor ﬁlter and an exponential nonlinearity (See Fig. 2). This is the response model assumed by the Lewi-09 method, and therefore substantially mismatched to the linear-Gaussian model assumed by our method. For the Lewi-09 method, we used a diagonal prior covariance with amplitude set by maximizing marginal likelihood for a small dataset. We compared two versions of the ALD-based algorithm (with 10 and 100 hyperparameter particles, respectively) to examine the relationship between performance and ﬁdelity of the posterior representation. To quantify the performance, we used the angular difference (in degrees) between the true and estimated RF. Fig 2A shows the angular difference between the true RF and estimates obtained by Lewi-09 and the ALD-based method, as a function of the number of trials. The ALD estimate exhibits more rapid convergence, and performs noticeably better with 100 than with 10 particles (ALD100 vs. ALD10), indicating that accurately preserving uncertainty over the hyperparameters is beneﬁcial to performance. We also show the performance of ALD inference under passive learning (iid random stimulus selection), which indicates that the improvement in our method is not simply due to the use of an improved RF estimator. Fig 2B shows the estimates obtained by each method after 200, 400, and 1000 trials. Note that the estimate with 100 hyperparameter samples is almost indistinguishable from the true ﬁlter after 200 trials, which is substantially lower than the dimensionality of the ﬁlter itself (d = 400). Fig. 3 shows a performance comparison using three additional 2-dimensional receptive ﬁelds, to show that performance improves across a variety of different RF shapes. The ﬁlters included: (A) a gabor ﬁlter similar to that used in [6]; (B) a retina-like center-surround receptive ﬁeld; (C) a grid-cell receptive ﬁeld with multiple modes. As before, noisy responses were simulated from a Poisson-GLM. For the grid-cell example, these ﬁlter is not strongly localized in space, yet the ALDbased estimate substantially outperforms Lewi-09 due to its sensitivity to localized components in frequency. Thus, ALD-based method converges more quickly despite the mismatch between the model used to simulate data and the model assumed for active learning. Neural Data: We also tested our method with an off-line analysis of real neural data from a simple cell recorded in primate V1 (published in [15]). The stimulus consisted of 1D spatiotemporal white noise (“ﬂickering bars”), with 16 spatial bars on each frame, aligned with the cell’s preferred orientation. We took the RF to have 16 time bins, resulting in a 256-dimensional parameter space for the RF. We performed simulated active learning by extracting the raw stimuli from 46 minutes of experimental data. On each trial, we then computed the expected information gain from presenting each of these stimuli (blind to neuron’s actual response to each stimulus). We used ALD-based active learning with 10 hyperparameter particles, and examined performance of both algorithms for 960 trials (selecting from ≈276,000 possible stimuli on each trial). 7 ml (46 min.) 8 16 8 16 Lewi-09 ALD 1 1 B A 0 160 480 960 40 50 60 70 # of stimuli ALD Lewi-09 −140 −100 −60 −20 20 angle: 55.0 42.5 C 0 320 640 960 # of stimuli 160 stimuli 480 stimuli 45.1 47.2 avg angle difference Figure 4: Comparison of active learning methods in a simulated experiment with real neural data from a primate V1 simple cell. (Original data recorded in response to white noise “ﬂickering bars” stimuli, see [15]). (A): Average angular difference between the MLE (inset, computed from an entire 46-minute dataset) and the estimates obtained by active learning, as a function of the amount of data. We simulated active learning via an ofﬂine analysis of the ﬁxed dataset, where methods had access to possible stimuli but not responses. (B): RF estimates after 10 and 30 seconds of data. Note that the ALD-based estimate has smaller error with 10 seconds of data than Lewi-09 with 30 seconds of data. (C): Average entropy of hyperparameter particles as a function of t, showing rapid narrowing of marginal posterior. Fig 4A shows the average angular difference between the maximum likelihood estimate (computed with the entire dataset) and the estimate obtained by each active learning method, as a function of the number of stimuli. The ALD-based method reduces the angular difference by 45 degrees with only 160 stimuli, although the ﬁlter dimensionality of the RF for this example is 256. The Lewi-09 method requires four times more data to achieve the same accuracy. Fig 4B shows estimates after 160 and 480 stimuli. We also examined the average entropy of the hyperparameter particles as a function of the amount of data used. Fig. 4C shows that the entropy of the marginal posterior over hyperparameters falls rapidly during the ﬁrst 150 trials of active learning. The main bottleneck of the algorithm is eigendecomposition of the posterior covariance ˜Λ, which took 30ms for a 256 × 256 matrix on a 2 × 2.66 GHz Quad-Core Intel Xeon Mac Pro. Updating importance weights and resampling 10 particles took 4ms, and a single step of MH resampling for each particle took 5ms. In total, it took <60 ms to compute the optimal stimulus in each trial using a non-optimized implementation of our algorithm, indicating that our methods should be fast enough for use in real-time neurophysiology experiments. 7 Discussion We have developed a Bayesian active learning method for neural RFs under hierarchical response models with conditionally Gaussian priors. To take account of uncertainty at the level of hyperparameters, we developed an approximate information-theoretic criterion for selecting optimal stimuli under a mixture-of-Gaussians posterior. We applied this framework using a prior designed to capture smooth and localized RF structure. The resulting method showed clear advantages over traditional designs that do not exploit structured prior knowledge. We have contrasted our method with that of Lewi et al [6], which employs a more ﬂexible and accurate model of the neural response, but a less ﬂexible model of the RF prior. A natural future direction therefore will be to combine the Poisson-GLM likelihood and ALD prior, which will combine the beneﬁts of a more accurate neural response model and a ﬂexible (low-entropy) prior for neural receptive ﬁelds, while incurring only a small increase in computational cost. 8 Acknowledgments We thank N. C. Rust and J. A. Movshon for V1 data, and several anonymous reviewers for helpful advice on the original manuscript. This work was supported by a Sloan Research Fellowship, McKnight Scholar’s Award, and NSF CAREER Award IIS-1150186 (JP). References [1] D. J. C. MacKay. Information-based objective functions for active data selection. Neural Computation, 4(4):590–604, 1992. [2] K. Chaloner and I. Verdinelli. Bayesian experimental design: a review. Statistical Science, 10:273–304, 1995. [3] D. A. Cohn, Z. Ghahramani, and M. I. Jordan. Active learning with statistical models. J. Artif. Intell. Res. (JAIR), 4:129–145, 1996. [4] A. Watson and D. Pelli. QUEST: a Bayesian adaptive psychophysical method. Perception and Psychophysics, 33:113–120, 1983. [5] L. Paninski. Asymptotic theory of information-theoretic experimental design. Neural Computation, 17(7):1480–1507, 2005. [6] J. Lewi, R. Butera, and L. Paninski. Sequential optimal design of neurophysiology experiments. Neural Computation, 21(3):619–687, 2009. 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4,469	Deep Neural Networks Segment Neuronal Membranes in Electron Microscopy Images Dan C. Cires¸an∗ IDSIA USI-SUPSI Lugano 6900 dan@idsia.ch Alessandro Giusti IDSIA USI-SUPSI Lugano 6900 alessandrog@idsia.ch Luca M. Gambardella IDSIA USI-SUPSI Lugano 6900 luca@idsia.ch J¨urgen Schmidhuber IDSIA USI-SUPSI Lugano 6900 juergen@idsia.ch Abstract We address a central problem of neuroanatomy, namely, the automatic segmentation of neuronal structures depicted in stacks of electron microscopy (EM) images. This is necessary to efﬁciently map 3D brain structure and connectivity. To segment biological neuron membranes, we use a special type of deep artiﬁcial neural network as a pixel classiﬁer. The label of each pixel (membrane or nonmembrane) is predicted from raw pixel values in a square window centered on it. The input layer maps each window pixel to a neuron. It is followed by a succession of convolutional and max-pooling layers which preserve 2D information and extract features with increasing levels of abstraction. The output layer produces a calibrated probability for each class. The classiﬁer is trained by plain gradient descent on a 512 × 512 × 30 stack with known ground truth, and tested on a stack of the same size (ground truth unknown to the authors) by the organizers of the ISBI 2012 EM Segmentation Challenge. Even without problem-speciﬁc postprocessing, our approach outperforms competing techniques by a large margin in all three considered metrics, i.e. rand error, warping error and pixel error. For pixel error, our approach is the only one outperforming a second human observer. 1 Introduction How is the brain structured? The recent ﬁeld of connectomics [2] is developing high-throughput techniques for mapping connections in nervous systems, one of the most important and ambitious goals of neuroanatomy. The main tool for studying connections at the neuron level is serial-section Transmitted Electron Microscopy (ssTEM), resolving individual neurons and their shapes. After preparation, a sample of neural tissue is typically sectioned into 50-nanometer slices; each slice is then recorded as a 2D grayscale image with a pixel size of about 4 × 4 nanometers (see Figure 1). The visual complexity of the resulting stacks makes them hard to handle. Reliable automated segmentation of neuronal structures in ssTEM stacks so far has been infeasible. A solution of this problem, however, is essential for any automated pipeline reconstructing and mapping neural connections in 3D. Recent advances in automated sample preparation and imaging make this increas∗webpage: http://www.idsia.ch/˜ciresan 1 Figure 1: Left: the training stack (one slice shown). Right: corresponding ground truth; black lines denote neuron membranes. Note complexity of image appearance. ingly urgent, as they enable acquisition of huge datasets [6, 21], whose manual analysis is simply unfeasible. Our solution is based on a Deep Neural Network (DNN) [12, 13] used as a pixel classiﬁer. The network computes the probability of a pixel being a membrane, using as input the image intensities in a square window centered on the pixel itself. An image is then segmented by classifying all of its pixels. The DNN is trained on a different stack with similar characteristics, in which membranes were manually annotated. DNN are inspired by convolutional neural networks introduced in 1980 [16], improved in the 1990s [25], reﬁned and simpliﬁed in the 2000s [5, 33], and brought to their full potential by making them both large and deep [12, 13]. Lately, DNN proved their efﬁciency on data sets extending from handwritten digits (MNIST) [10, 12], handwritten characters [11] to 3D toys (NORB) [13] and faces [35]. Training huge nets requires months or even years on CPUs, where high data transfer latency prevented multi-threading code from saving the situation. Our fast GPU implementation [10, 12] overcomes this problem, speeding up single-threaded CPU code by up to two orders of magnitude. Many other types of learning classiﬁers have been applied to segmentation of TEM images, where different structures are not easily characterized by intensity differences, and structure boundaries are not correlated with high image gradients, due to noise and many confounding micro-structures. In most binary segmentation problems, classiﬁers are used to compute one or both of the following probabilities: (a) probability of a pixel belonging to each class; (b) probability of a boundary dividing two adjacent pixels. Segmentation through graph cuts [7] uses (a) as the unary term, and (b) as the binary term. Some use an additional term to account for the expected geometry of neuron membranes[23]. We compute pixel probabilities only (point (a) above), and directly obtain a segmentation by mild smoothing and thresholding, without using graph cuts. Our main contribution lies therefore in the classiﬁer itself. Others have used off-the-shelf random forest classiﬁers to compute unary terms of neuron membranes [22], or SVMs to compute both unary and binary terms for segmenting mitochondria [28, 27]. The former approach uses haar-like features and texture histograms computed on a small region around the pixel of interest, whereas the latter uses sophisticated rotational [17] and ray [34] features computed on superpixels [3]. Feature selection mirrors the researcher’s expectation of which characteristics of the image are relevant for classiﬁcation, and has a large impact on classiﬁcation accuracy. In our approach, we bypass such problems, using raw pixel values as inputs. Due to their convolutional structure, the ﬁrst layers of the network automatically learn to compute meaningful features during training. The main contribution of the paper is a practical state-of-the-art segmentation method for neuron membranes in ssTEM data, described in Section 2. It outperforms existing methods as validated in Section 3. The contribution is particularly meaningful because our approach does not rely on problem-speciﬁc postprocessing: fruitful application to different biomedical segmentation problems is therefore likely. 2 Figure 2: Overview of our approach (see text). 2 Methods For each pixel we consider two possible classes, membrane and non-membrane. The DNN classiﬁer (Section 2.1) computes the probability of a pixel p being of the former class, using as input the raw intensity values of a square window centered on p with an edge of w pixels—w being an odd number to enforce symmetry. When a pixel is close to the image border, its window will include pixels outside the image boundaries; such pixels are synthesized by mirroring the pixels in the actual image across the boundary (see Figure 2). The classiﬁer is ﬁrst trained using the provided training images (Section 2.2). After training, to segment a test image, the classiﬁer is applied to all of its pixels, thus generating a map of membrane probabilities—i.e., a new real-valued image the size of the input image. Binary membrane segmentation is obtained by mild postprocessing techniques discussed in Section 2.3, followed by thresholding. 2.1 DNN architecture A DNN [13] consists of a succession of convolutional, max-pooling and fully connected layers. It is a general, hierarchical feature extractor that maps raw pixel intensities of the input image into a feature vector to be classiﬁed by several fully connected layers. All adjustable parameters are jointly optimized through minimization of the misclassiﬁcation error over the training set. Each convolutional layer performs a 2D convolution of its input maps with a square ﬁlter. The activations of the output maps are obtained by summing the convolutional responses which are passed through a nonlinear activation function. The biggest architectural difference between the our DNN and earlier CNN [25] are max-pooling layers [30, 32, 31] instead of sub-sampling layers. Their outputs are given by the maximum activation over non-overlapping square regions. Max-pooling are ﬁxed, non-trainable layers which select the most promising features. The DNN also have many more maps per layer, and thus many more connections and weights. After 1 to 4 stages of convolutional and max-pooling layers several fully connected layers further combine the outputs into a 1D feature vector. The output layer is always a fully connected layer with one neuron per class (two in our case). Using a softmax activation function for the last layer guarantees that each neuron’s output activation can be interpreted as the probability of a particular input image belonging to that class. 2.2 Training To train the classiﬁer, we use all available slices of the training stack, i.e., 30 images with a 512×512 resolution. For each slice, we use all membrane pixels as positive examples (on average, about 50000), and the same amount of pixels randomly sampled (without repetitions) among all nonmembrane pixels. This amounts to 3 million training examples in total, in which both classes are equally represented. As is often the case in TEM images—but not in other modalities such as phase-contrast microscopy—the appearance of structures is not affected by their orientation. We take advantage of 3 this property, and synthetically augment the training set at the beginning of each epoch by randomly mirroring each training instance, and/or rotating it by ±90◦. 2.3 Postprocessing of network outputs Because each class is equally represented in the training set but not in the testing data, the network outputs cannot be directly interpreted as probability values; instead, they tend to severely overestimate the membrane probability. To ﬁx this issue, a polynomial function post-processor is applied to the network outputs. To compute its coefﬁcients, a network N is trained on 20 slices of the training volume Ttrain and tested on the remaining 10 slices of the same volume (Ttest, for which ground truth is available). We compare all outputs obtained on Ttest (a total of 2.6 million instances) to ground truth, to compute the transformation relating the network output value and the actual probability of being a membrane; for example, we measure that, among all pixels of Ttest which were classiﬁed by N as having a 50% probability of being membrane, only about 18% have in fact such a ground truth label; the reason being the different prevalence of membrane instances in Ttrain (i.e. 50%) and in Ttest (roughly 20%). The resulting function is well approximated by a monotone cubic polynomial, whose coefﬁcients are computed by least-squares ﬁtting. The same function is then used to calibrate the outputs of all trained networks. After calibration (a grayscale transformation in image processing terms), network outputs are spatially smoothed by a 2-pixel-radius median ﬁlter. This results in regularized of membrane boundaries after thresholding. 2.4 Foveation and nonuniform sampling We experimented with two related techniques for improving the network performance by manipulating its input data, namely foveation and nonuniform sampling (see Figure 3). Foveation is inspired by the structure of human photoreceptor topography [14], and has recently been shown to be very effective for improving nonlocal-means denoising algorithms [15]. It imposes a spatially-variant blur on the input window pixels, such that full detail is kept in the central section (fovea), while the peripheral parts are defocused by means of a convolution with a disk kernel, to remove ﬁne details. The network, whose task is to classify the center pixel of the window, is then forced to disregard such peripheral ﬁne details, which are most likely irrelevant, while still retaining the general structure of the window (context). Figure 3: Input windows with w = 65, from the training set. First row shows original window (Plain); other rows show effects of foveation (Fov), nonuniform sampling (Nu), and both (Fov+Nu). Samples on the left and right correspond to instances of class Membrane and Non-membrane, respectively. The leftmost image illustrates how a checkerboard pattern is affected by such transformations. Nonuniform sampling is motivated by the observation that (in this and other applications) larger window sizes w generally result in signiﬁcant performance improvements. However, a large w 4 results in much bigger networks, which take longer to train and, at least in theory, require larger amounts of training data to retain their generalization ability. With nonuniform sampling, image pixels are directly mapped to neurons only in the central part of the window; elsewhere, their source pixels are sampled with decreasing resolution as the distance from the window center increases. As a result, the image in the window is deformed in a ﬁsheye-like fashion, and covers a larger area of the input image with fewer neurons. Simultaneously applying both techniques is a way of exploiting data at multiple resolutions—ﬁne at the center, coarse in the periphery of the window. 2.5 Averaging outputs of multiple networks We observed that large networks with different architectures often exhibit signiﬁcant output differences for many image parts, despite being trained on the same data. This suggests that these powerful and ﬂexible classiﬁers exhibit relatively large variance but low bias. It is therefore reasonable to attempt to reduce such variance by averaging the calibrated outputs of several networks with different architectures. This was experimentally veriﬁed. The submissions obtained by averaging the outputs of multiple large networks scored signiﬁcantly better in all metrics than the single networks. 3 Experimental results All experiments are performed on a computer with a Core i7 950 3.06GHz processor, 24GB of RAM, and four GTX 580 graphics cards. A GPU implementation [12] accelerates the forward propagation and back propagation routines by a factor of 50. We validate our approach on the publicly-available dataset [9] provided by the organizers of the ISBI 2012 EM Segmentation Challenge [1], which represents two portions of the ventral nerve cord of a Drosophila larva. The dataset is composed by two 512 × 512 × 30 stacks, one used for training, one for testing. Each stack covers a 2 × 2 × 1.5 µm volume, with a resolution of 4 × 4 × 50 nm/pixel. For the training stack, a manually annotated ground truth segmentation is provided. For the testing stack, the organizers obtained (but did not distribute) two manual segmentations by different expert neuroanatomists. One is used as ground truth, the other to evaluate the performance of a second human observer and provide a meaningful comparison for the algorithms’ performance. A segmentation of the testing stack is evaluated through an automated online system, which computes three error metrics in relation to the hidden ground truth: Rand error: deﬁned as 1−Frand, where Frand represents the F1 score of the Rand index [29], which measures the accuracy with which pixels are associated to their respective neurons. Warping error: a segmentation metric designed to account for topological disagreements [19]; it accounts for the number of neuron splits and mergers required to obtain the candidate segmentation from ground truth. Pixel error: deﬁned as 1 −Fpixel, where Fpixel represents the F1 score of pixel similarity. The automated system accepts a stack of grayscale images, representing membrane probability values for each pixel; the stack is thresholded using 9 different threshold values, obtaining 9 binary stacks. For each of the stacks, the system computes the error measures above, and returns the minimum error. Pixel error is clearly not a suitable indicator of segmentation quality in this context, and is reported mostly for reference. Rand and Warping error metrics have various strengths and weaknesses, without clear consensus in favor of any. The former tends to provide a more consistent measure but penalizes even slightly misplaced borders, which would not be problematic in most practical applications. The latter has a more intuitive interpretation, but completely disregards non-topological errors. We train four networks N1, N2, N3 and N4, with slightly different architectures, and window sizes w = 65 (for N1, N2, N3) and w = 95 (for N4); all networks use foveation and nonuniform sampling, 5 Figure 4: Above, from left to right: part of a source image from the test set; corresponding calibrated outputs of networks N1, N2, N3 and N4; average of such outputs; average after ﬁltering. Below, the performance of each network, as well as the signiﬁcantly better performance due to averaging their outputs. All results are computed after median ﬁltering (see text). except N3, which uses neither. As the input window size increases, the network depth also increases because we keep the convolutional ﬁlter sizes small. The architecture of N4 is the deepest, and is reported in Table 1. Training time for one epoch varies from approximately 170 minutes for N1 (w = 65) to 340 minutes for N4 (w = 95). All nets are trained for 30 epochs, which leads to a total training time of several days. However, once networks are trained, application to new images is relatively fast: classifying the 8 million pixels comprising the whole testing stack takes 10 to 30 minutes on four GPUs. Such implementation is currently being further optimized (with foreseen speedups of one order of magnitude at least) in view of application to huge, terapixel-class datasets [6, 21]. Table 1: 11-layer architecture for network N4, w = 95. Layer Type Maps and neurons Kernel size 0 input 1 map of 95x95 neurons 1 convolutional 48 maps of 92x92 neurons 4x4 2 max pooling 48 maps of 46x46 neurons 2x2 3 convolutional 48 maps of 42x42 neurons 5x5 4 max pooling 48 maps of 21x21 neurons 2x2 5 convolutional 48 maps of 18x18 neurons 4x4 6 max pooling 48 maps of 9x9 neurons 2x2 7 convolutional 48 maps of 6x6 neurons 4x4 8 max pooling 48 maps of 3x3 neurons 2x2 9 fully connected 200 neurons 1x1 10 fully connected 2 neurons 1x1 The outputs of four such networks are shown in Figure 4, along with their performance after ﬁltering. By averaging the outputs of all networks, results improve signiﬁcantly. The ﬁnal result for one slice of the test stack is shown in Figure 5. Our results are compared to competing methods in Table 2. Since our pure pixel classiﬁer method aims at minimizing pixel error, Rand and warping errors are just minimized as a side-effect, but never explicitly accounted for during segmentation. In contrast, some competing segmentation approaches adopt different post-processing techniques directly opti6 Figure 5: Left: slice 16 of the test stack. Right: corresponding output. Table 2: Results of our approach and competing algorithms. For comparison, the ﬁrst two rows report the performance of the second human observer and of a simple thresholding approach. Group Rand error [·10−3] Warping error [·10−6] Pixel error [·10−3] Second Human Observer 27 344 67 Simple Thresholding 445 15522 222 Our approach 48 434 60 Laptev et al. [24] (1) 65 556 83 Laptev et al. [24] (2) 70 525 79 Sumbul et al. 76 646 65 Liu et al. [26] (1) 84 1602 134 Kaynig et al. [23] 84 1124 157 Liu et al. [26] (2) 89 1134 78 Kamentsky et al. [20] 90 1512 100 Burget et al. [8] 139 2641 102 Tan et al. [36] 153 685 88 Bas et al. [4] 162 1613 109 Iftikhar et al. [18] 230 16156 150 mizing the rand error. Nevertheless, their results are inferior. But such post-processing techniques— which unlike our general classiﬁer are speciﬁc to this particular problem—could be successfully applied to ﬁnetune our outputs, further improving results. Preliminary results in this direction are encouraging: the problem-speciﬁc postprocessing techniques in [20] and [24], operating on our segmentation, reduce the Rand error to measure to 36·10−3 and 32·10−3, respectively. Further research along these lines is planned for the near future. 4 Discussion and conclusions The main strength of our approach to neuronal membrane segmentation in EM images lies in a deep and wide neural network trained by online back-propagation to become a very powerful pixel classiﬁer with superhuman pixel-error rate, made possible by an optimized GPU implementation more than 50 times faster than equivalent code on standard microprocessors. 7 Our approach outperforms all other approaches in the competition, despite not even being tailored to this particular segmentation task. Instead, the DNN acts as a generic image classiﬁer, using raw pixel intensities as inputs, without ad-hoc post-processing. This opens interesting perspectives on applying similar techniques to other biomedical image segmentation tasks. Acknowledgments This work was partially supported by the Supervised Deep / Recurrent Nets SNF grant, Project Code 140399. 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4,470	Learning from the Wisdom of Crowds by Minimax Entropy Dengyong Zhou, John C. Platt, Sumit Basu, and Yi Mao Microsoft Research 1 Microsoft Way, Redmond, WA 98052 {denzho,jplatt,sumitb,yimao}@microsoft.com Abstract An important way to make large training sets is to gather noisy labels from crowds of nonexperts. We propose a minimax entropy principle to improve the quality of these labels. Our method assumes that labels are generated by a probability distribution over workers, items, and labels. By maximizing the entropy of this distribution, the method naturally infers item confusability and worker expertise. We infer the ground truth by minimizing the entropy of this distribution, which we show minimizes the Kullback-Leibler (KL) divergence between the probability distribution and the unknown truth. We show that a simple coordinate descent scheme can optimize minimax entropy. Empirically, our results are substantially better than previously published methods for the same problem. 1 Introduction There is an increasing interest in using crowdsourcing to collect labels for machine learning [19, 6, 21, 17, 20, 10, 13, 12]. Currently, many companies provide crowdsourcing services. Amazon Mechanical Turk (MTurk) [2] and CrowdFlower [4] are perhaps the most well-known ones. An advantage of crowdsourcing is that we can obtain a large number of labels at the low cost of pennies per label. However, these workers are not experts, so the labels collected from them are often fairly noisy. A fundamental challenge in crowdsourcing is inferring ground truth from noisy labels by a crowd of nonexperts. When each item is labeled several times by different workers, a straightforward approach is to use the most common label as the true label. From reported experimental results on real crowdsourcing data [19] and our own experience, majority voting performs signiﬁcantly better on average than individual workers. However, majority voting considers each item independently. When many items are simultaneously labeled, it is reasonable to assume that the performance of a worker is consistent across different items. This assumption underlies the work of Dawid and Skene [5, 18, 19, 11, 17], where each worker is associated with a probabilistic confusion matrix that generates her labels. Each entry of the matrix indicates the probability that items in one class are labeled as another. Given the observed responses, the true labels for each item and the confusion matrices for each worker can be jointly estimated by a maximum likelihood method. The optimization can be implemented by the expectation-maximization (EM) algorithm [7]. Dawid and Skene’s method works well in practice. However, their method only contains a perworker probabilistic confusion model of generating labels. In this paper, we assume a separate probabilistic distribution for each worker-item pair. We propose a novel minimax entropy principle to jointly estimate the distributions and the ground truth given the observed labels by workers in Section 2. The theoretical justiﬁcation of minimum entropy is given in Section 2.1. To prevent overﬁtting, we relax the minimax entropy optimization in Section 3. We describe an easy-to-implement technique to carry out the minimax program in Section 4 and link minimax entropy to a principle of 1 item 1 item 2 ... item n worker 1 z11 z12 ... z1n worker 2 z21 z22 ... z2n ... ... ... ... ... worker m zm1 zm2 ... zmn item 1 item 2 ... item n worker 1 π11 π12 ... π1n worker 2 π21 π22 ... π2n ... ... ... ... ... worker m πm1 πm2 ... πmn Figure 1: Left: observed labels. Right: underlying distributions. Highlights on both tables indicate that rows and columns of the distributions are constrained by sums over observations. objective measurements in Section 5. Finally, we present superior experimental results on real-world crowdsourcing data in Section 6. 2 Minimax Entropy Principle We propose a model illustrated in Figure 1. Each row corresponds to a crowdsourced worker indexed by i (from 1 to m). Each column corresponds to an item to be labeled, indexed by j (from 1 to n). Each item has an unobserved label represented as a vector yjl, which is 1 when item j is in class l (from 1 to c), and 0 otherwise. More generally, we can treat yjl as the probability that item j is in class l. We observe a matrix of labels zij by workers. The label matrix can also be represented as a tensor zijk, which is 1 when worker i labels item j as class k , and 0 otherwise. We assume that zij are drawn from πij, which is the distribution for worker i to generate a label for item j. Again, πij can also be represented as a tensor πijk, which is the probability that worker i labels item j as class k. Our method will estimate yjl from the observed zij. We specify the form of πij through the maximum entropy principle, where the constraints on the maximum entropy combine the best ideas from previous work. Majority voting suggests that we should be constraining the πij per column, with the empirical observation of the number of votes per class per item P i zijk should match P i πijk. Dawid and Skene’s method suggests that we should be constraining the πij per row, with the empirical confusion matrix per worker P j yjlzijk should match P j yjlπijk. We thus have the following maximum entropy model for πij given yjl: max π − m X i=1 n X j=1 c X k=1 πijk ln πijk s.t. m X i=1 πijk = m X i=1 zijk, ∀j, k, n X j=1 yjlπijk = n X j=1 yjlzijk, ∀i, k, l, (1a) c X k=1 πijk = 1, ∀i, j, πijk ≥0, ∀i, j, k. (1b) We propose that, to infer yjl, we should choose yjl to minimize the entropy in Equation (1). Intuitively, making πij “peaky” means that zij is the least random given yjl. We make this intuition rigorous in Section 2.1. Thus, the inference for yjl can be expressed by a minimax entropy program: min y max π − m X i=1 n X j=1 c X k=1 πijk ln πijk s.t. m X i=1 πijk = m X i=1 zijk, ∀j, k, n X j=1 yjlπijk = n X j=1 yjlzijk, ∀i, k, l, (2a) c X k=1 πijk = 1, ∀i, j, πijk ≥0, ∀i, j, k, c X l=1 yjl = 1, ∀j, yjl ≥0, ∀j, l. (2b) 2 2.1 Justiﬁcation for Minimum Entropy Now we justify the principle of choosing yjl by minimizing entropy. Think of yjl as a set of parameters to the worker-item label models πij. The goal in choosing the yjl is to select πij that are as close as possible to the true distributions π∗ ij. To ﬁnd a principle to choose the yjl, assume that we have access to the row and column measurements on the true distributions π∗ ij. That is, assume that we know the true values of the column measurements φjk = P i π∗ ijk and row measurements ϕikl = P j yjlπ∗ ijk, for a chosen set of yjl values. Knowing these true row and column measurements, we can apply the maximum entropy principle to generate distributions πij: max π − m X i=1 n X j=1 c X k=1 πijk ln πijk s.t. m X i=1 πijk = φjk, ∀j, k, n X j=1 yjlπijk = ϕikl, ∀i, k, l. (3) Let DKL(· ∥·) denote the KL divergence between two distributions. We can choose yjl to minimize a loss of πij with respect to π∗ ij given by ℓ(π∗, π) = m X i=1 n X j=1 DKL(π∗ ij ∥πij). (4) The minimum loss can be attained by choosing yjl to minimize the entropy of the maximum distributions πij. This can be shown by writing the Lagrangian of program (3): L = − m X i=1 n X j=1 c X k=1 πijk ln πijk + m X i=1 n X j=1 λij c X k=1 πijk −1 + n X j=1 c X k=1 τjk m X i=1 (πijk −π∗ ijk) + m X i=1 c X k=1 c X l=1 σikl n X j=1 yjl(πijk −π∗ ijk), where the newly introduced variables τjk and σikl are the Lagrange multipliers. For a solution to be optimal, the Karush-Kuhn-Tucker (KKT) conditions must be satisﬁed [3]. Thus, ∂L ∂πijk = −ln πijk −1 + λij + c X l=1 yjl(τjk + σikl) = 0, ∀i, j, k, which can be rearranged as πijk = exp c X l=1 yjl(τjk + σikl) + λij −1 , ∀i, j, k. (5) For being a probability measure, the variables πijk have to satisfy c X k=1 πijk = c X k=1 exp c X l=1 yjl(τjk + σikl) + λij −1 = 1, ∀i, j. (6) Eliminating λij by jointly considering Equations (5) and (6), we obtain a labeling model in the exponential family: πijk = exp Pc l=1 yjl (τjk + σikl) Pc s=1 exp Pc l=1 yjl (τjs + σisl), ∀i, j, k. (7) Plugging Equation (7) into (4) and performing some algebraic manipulations, we prove Theorem 2.1 Let πij be the maximum entropy distributions in (3). Then, ℓ(π∗, π) = m X i=1 n X j=1 c X k=1 (π∗ ijk ln π∗ ijk −πijk ln πijk). 3 The second term is the only term that depends on yjl. Therefore, we should choose yjl to minimize the entropy of the maximum entropy distributions. The labeling model expressed by Equation (7) has a natural interpretation. For each worker i, the multiplier set {σikl} is a measure of her expertise, while for each item j, the multiplier set {τjk} is a measure of its confusability. A worker correctly labels an item either because she has good expertise or because the item is not that confusing. When the item or worker parameters are shifted by an arbitrary constant, the probability given by Equation (7) does not change. The redundancy of the constraints in (2a) causes the redundancy of the parameters. 3 Constraint Relaxation In real crowdsourcing applications, each item is usually labeled only a few times. Moreover, a worker usually only labels a small subset of items rather than all of them. In such cases, it is unreasonable to expect that the constraints in (2a) hold for the true underlying distributions πij. As in the literature of regularized maximum entropy [14, 1, 9], we relax the optimization problem to prevent overﬁtting: min y max π,ξ,ζ − m X i=1 n X j=1 c X k=1 πijk ln πijk − n X j=1 c X k=1 ξ2 jk 2αj − m X i=1 c X k=1 c X l=1 ζ2 ikl 2βi s.t. m X i=1 (πijk −zijk) = ξjk, ∀j, k, n X j=1 yjl(πijk −zijk) = ζikl, ∀i, k, l, (8a) c X k=1 πijk = 1, ∀i, j, πijk ≥0, ∀i, j, k, c X l=1 yjl = 1, ∀j, yjl ≥0, ∀j, l, (8b) where αj and βi are regularization parameters. It is obvious that program (8) is reduced to program (2) when the slack variables ξjk and ζikl are set to zero. The two ℓ2-norm based regularization terms in the objective function force the slack variables to be not far away from zero. Other vector or matrix norms, such as the ℓ1-norm and the trace norm, can be applied as well [14, 1, 9]. We choose the ℓ2-norm only for the sake of simplicity in computation. The justiﬁcation for minimum entropy in Section 2.1 can be extended to the regularized minimax entropy formulation (8) with minor modiﬁcations. Instead of knowing the exact marginals, we need to choose πij based on noisy marginals: φjk = m X i=1 π∗ ijk + ξ∗ jk, ∀j, k, ϕikl = n X j=1 yjlπ∗ ijk + ζ∗ ikl, ∀i, k, l. We thus maximize the regularized entropy subject to the relaxed constraints: m X i=1 πijk + ξjk = φjk, ∀j, k, n X j=1 yjlπijk + ζikl = ϕikl, ∀i, k, l. (9) Lemma 3.1 To be the regularized maximum entropy distributions subject to (9), πij must be represented as in Equation (7). Moreover, we should have ξjk = αjτjk, ζikl = βiσikl. Proof The ﬁrst part of the result can be veriﬁed as before. By using the labeling model in Equation (7), the Lagrangian of the regularized maximum entropy program can be written as L = − m X i=1 n X j=1 ln c X s=1 exp c X l=1 yjl (τjs + σisl) − n X j=1 c X k=1 ξ2 jk 2αj − m X i=1 c X k=1 c X l=1 ζ2 ikl 2βi + n X j=1 c X k=1 τjk − m X i=1 π∗ ijk + (ξjk −ξ∗ jk) + m X i=1 c X k=1 c X l=1 σikl − n X j=1 yjlπ∗ ijk + (ζikl −ζ∗ ikl) . For ﬁxed τjk and σikl, maximizing the Lagrange dual over ξjk and ζikl provides the proof. 4 By Lemma 3.1 and some algebraic manipulations, we obtain Theorem 3.2 Let πij be the regularized maximum entropy distributions subject to (9). Then, ℓ(π∗, π) = m X i=1 n X j=1 c X k=1 π∗ ijk ln π∗ ijk − m X i=1 n X j=1 c X k=1 πijk ln πijk − n X j=1 c X k=1 ξ2 jk αj − m X i=1 c X k=1 c X l=1 ζ2 ikl βi + n X j=1 c X k=1 ξ∗ jkξjk αj + m X i=1 c X k=1 c X l=1 ζ∗ iklζikl βi . (10) We cannot minimize the loss by minimizing the right side of Equation (10) since the random noise is unknown. However, we can consider minimizing an upper bound instead. Note that ξ∗ jkξjk ≤(ξ∗2 jk + ξ2 jk)/2, ∀j, k, ζ∗ iklζikl ≤(ζ∗2 ikl + ζ2 ikl)/2, ∀i, k, l. (11) Denote by Ω(π, ξ, ζ) the objective function of the regularized minimax entropy program (8). Substituting the inequalities in (11) into Equation (10), we have ℓ(π∗, π) ≤Ω(π, ξ, ζ) −Ω(π∗, ξ∗, ζ∗). (12) So minimizing the regularized maximum entropy leads to minimizing an upper bound of the loss. 4 Optimization Algorithm A typical approach to constrained optimization is to covert the primal problem to its dual form. By Lemma 3.1, the Lagrangian of program (8) can be written as L = − n X j=1 ln m Y i=1 exp Pc k=1 zijk Pc l=1 yjl(τjk + σikl) Pc s=1 exp Pc l=1 yjl (τjs + σisl) + n X j=1 c X k=1 αjτ 2 jk 2 + m X i=1 c X k=1 c X l=1 βiσ2 ikl 2 . The dual problem minimizes L subject to the constraints ∆= {yjl\| Pc l=1 yjl = 1, ∀j, yjl ≥ 0, ∀j, l}. It can be solved by coordinate descent with the variables being split into two groups: {yjl} and {τjk, σikl}. It is easy to check that, when the variables in one group are ﬁxed, the optimization problem on the variables in the other group is convex. When the yjl are restricted to be {0, 1}, that is, deterministic labels, the coordinate descent procedure can be simpliﬁed. Let pjl = m Y i=1 exp Pc k=1 zijk(τjk + σikl) Pc s=1 exp (τjs + σisl) . For any set of real-valued numbers {µjl\| Pc l=1 µjl = 1, ∀j, µjl > 0, ∀j, l}, we have the inequality n X j=1 ln m Y i=1 exp Pc k=1 zijk Pc l=1 yjl(τjk + σikl) Pc s=1 exp Pc l=1 yjl (τjs + σisl) = n X j=1 ln c X l=1 yjlpjl deterministic labels = n X j=1 ln c X l=1 µjl yjlpjl µjl ≥ n X j=1 c X l=1 µjl ln yjlpjl µjl Jensen’s inequality = n X j=1 c X l=1 µjl ln(yjlpjl) − n X j=1 c X l=1 µjl ln µjl. Plugging the last line into the Lagrangian L, we obtain an upper bound of L, called F. It can be shown that we must have yjl = µjl at any stationary point of F. Our optimization algorithm is a coordinate descent minimization of this F [15, 7]. We initialize yjl with majority vote in Equation (13). In each iteration step, we ﬁrst optimize over τjk and σikl in (14a), which can be solved by any convex optimization procedure, and next optimize over yjl using a simple closed form in (14b). The optimization over yjl is the same as applying Bayes’ theorem where the result from the last iteration is considered as a prior. This algorithm can be shown to produce only deterministic labels. 5 Algorithm 1 Minimax Entropy Learning from Crowds input: {zijk} ∈{0, 1}m×n×c, {αj} ∈Rn +, {βi} ∈Rm + initialization: y0 jl = Pm i=1 zijl Pm i=1 Pc k=1 zijk , ∀j, l (13) for t = 1, 2, . . . {τ t jk, σt ikl} = arg min τ,σ m X i=1 n X j=1 c X l=1 yt−1 jl log c X s=1 exp(τjs + σisl) − c X k=1 zijk(τjk + σikl) + n X j=1 c X k=1 αjτ 2 jk 2 + m X i=1 c X k=1 c X l=1 βiσ2 ikl 2 (14a) yt jl ∝yt−1 jl m Y i=1 exp Pc k=1 zijk(τ t jk + σt ikl) Pc s=1 exp τ t js + σt isl , ∀j, l (14b) output: {yt jl} 5 Measurement Objectivity Principle The measurement objectivity principle can be roughly stated as follows: (1) a comparison of labeling confusability between two items should be independent of which particular workers are included for the comparison; (2) symmetrically, a comparison of labeling expertise between two workers should be independent of which particular items are included for the comparison. The ﬁrst statement is about the objectivity of item confusability. The second statement is about the objectivity of worker expertise. In what follows, we mathematically deﬁne the measurement objectivity principle. For deterministic labels, we show that the labeling model in Equation (7) can be recovered from the measurement objectivity principle. From Equation (7), given item j in class l, the probability that worker i labels it as class k is πijkl = exp (τjk + σikl) Pc s=1 exp (τjs + σisl). (15) Assume that a worker i has labeled two items j and j′ both of which are from the same class l. With respect to the given worker i, for each item, we measure the confusability for class k by ρijk = πijkl πijll , ρij′k = πij′kl πij′ll . (16) For comparing the item confusabilities, we compute a ratio between them. To maintain the objectivity of confusability, the ratio should not depend on whichever worker is involved in the comparison. Hence, given another worker i′, we should have πijkl πijll πij′kl πij′ll = πi′jkl πi′jll πi′j′kl πi′j′ll . (17) It is straightforward to verify that the labeling model in Equation (15) indeed satisﬁes the objectivity requirement given by Equation (17). We can further show that a labeling model which satisﬁes Equation (17) has to be expressed by Equation (15). Let us rewrite Equation (17) as πijkl πijll = πij′kl πij′ll πi′jkl πi′jll πi′j′ll πi′j′kl . Without loss of generality, choose i′ = 0 and j′ = 0 as the ﬁxed references such that πijkl πijll = πi0kl πi0ll π0jkl π0jll π00ll π00kl . (18) Assume that the referenced worker 0 chooses a class uniformly at random for the referenced item 0. So we have π00ll = π00kl = 1/c. Equation (18) implies πijkl ∝πi0klπ0jkl. Reparameterizing with 6 (a) Norfolk Terrier (b) Norwich Terrier (c) Irish Wolfhound (d) Scottish Deerhound Figure 2: Sample images of four breeds of dogs from the Stanford dogs dataset πi0kl = exp(σikl) and π0jkl = exp(τjk) (note that l is dropped since it is determined by j), we have πijkl ∝exp(τjk + σikl). The labeling model in Equation (15) has been recovered. Symmetrically, we can also start from the objectivity of worker expertise to recover the labeling model in (15). Assume that two workers i and i′ have labeled a common item j which is from class l. With respect to the given item j, for each worker, we measure the confusion from class l to k by ρijk = πijkl πijll , ρi′jk = πi′jkl πi′jll . (19) For comparing the worker expertises, we compute a ratio between them. To maintain the objectivity of expertise, the ratio should not depend on whichever item is involved in the comparison. Hence, given another item j′ in class l, we should have πijkl πijll πi′jkl πi′jll = πij′kl πij′ll πi′j′kl πi′j′ll . (20) We can see that Equation (20) is actually just a rearrangement of Equation (17). 6 Experimental Validation We compare our method with majority voting and Dawid & Skene’s method [5] using real crowdsourcing data. One is multiclass image labeling, and the other is web search relevance judging. 6.1 Image Labeling We chose the images of 4 breeds of dogs from the Stanford dogs dataset [8]: Norfolk Terrier (172), Norwich Terrier (185), Irish Wolfhound (218), and Scottish Deerhound (232) (see Figure 2). The numbers of the images for each breed are in the parentheses. There are 807 images in total. We submitted them to MTurk, and received the labels from 109 MTurk workers. A worker labeled an image at most once, and each image was labeled 10 times. It is difﬁcult to evaluate a worker if she only labeled few images. We thus only consider the workers who labeled at least 40 images, which yields a label set that contains 7354 labels by 52 workers. Each image has at least 4 labels and around 95% of the images have at least 8 labels. The average accuracy of the workers is 70.60%. The best worker achieved an accuracy of 88.24% while only labeled 68 images. The worker who labeled the most labeled 345 images and achieved an accuracy of 68.99%. The average worker confusion matrix between breeds is shown in Table 2. As expected, it consists of two blocks. One block contains Norfolk Terrier and Norwich Terrier, and the other block contains Irish Wolfhound and Scottish Deerhound. For our method, the regularization parameters are set as αj = 100/(number of labels for item j), βi = 100/(number of labels by worker i). The performance of various methods on this image labeling task is summarized in Table 1. For this problem, our method is somewhat better than Dawid and Skene’s method. 6.2 Web Search Relevance Judging In another experiment, we asked workers to rate a set of 2665 query-URL pairs on a relevance rating scale from 1 to 5. The larger the rating, the more relevant the URL. The true labels were derived by 7 Method Dogs Web Minimax Entropy 84.63 88.05 Dawid & Skene 84.14 83.98 Majority Voting 82.09 73.07 Average Worker 70.60 37.05 Table 1: Accuracy of methods (%) Norfolk Norwich Irish Scottish Norfolk 71.04 27.35 1.03 0.58 Norwich 31.99 66.71 1.13 0.18 Irish 1.19 0.55 69.35 28.91 Scottish 1.20 0.38 26.77 71.65 Table 2: Average worker confusion (%) using consensus from 9 experts. The noisy labels were provided by 177 nonexpert workers. Each pair was judged by around 6 workers, and each worker judged a subset of the pairs. The average accuracy of workers is 37.05%. Seventeen workers have an accuracy of 0 and they judged at most 7 pairs. The worker who judged the most judged 1225 pairs and achieved an accuracy of 76.73%. For our method, the regularization parameters are set as αj = 200/(number of labels for item j), βi = 200/(number of labels by worker i). The performance of various methods on this relevance judging task is summarized in Table 1. In this case, our method is substantially better. 7 Related Work This paper can be regarded as a natural extension to Dawid and Skene’s work [5], discussed in Section 1. Our approach can be reduced to Dawid and Skene’s by setting the regularization parameters to be αj = ∞, βi = 0. The essential difference between our work and Dawid and Skene’s work is that, in addition to worker expertise, we also take item confusability into account. In computer vision, a minimax entropy method was proposed for estimating the probability density of certain visual patterns such as textures [22]. The authors compute empirical marginal distributions through various features, then construct a density model that can reproduce all empirical marginal distributions. Among all models satisfying the constraints, the one with maximum entropy is preferred. However, one wants to select the features which are most informative: the constructed model should approximate the underlying density by minimizing a KL divergence. The authors formulate the combined density estimation and feature selection as a minimax entropy problem. The measurement objectivity principle is inspired by the Rasch model [16], used to design and analyze psychological and educational measurements. In the Rasch model, given an examinee and a test item, the probability of a correct response is modeled as a logistic function of the difference between the examinee ability and the item difﬁculty. Rasch deﬁned “speciﬁc objectivity”: the comparison of any two subjects can be carried out in such a way that no other parameters are involved than those of the two subjects. The speciﬁc objectivity property of the Rasch model comes from the algebraic separation of examinee and item parameters. If the probability of a correct response is modeled with other forms, such as a logistic function of the ratio between the examinee ability and the item difﬁculty [21], objective measurements cannot be achieved. The most fundamental difference between the Rasch model and our work is that we must infer ground truth, rather than take them as given. 8 Conclusion We have proposed a minimax entropy principle for estimating the true labels from the judgements of a crowd of nonexperts. We have also shown that the labeling model derived from the minimax entropy principle uniquely satisﬁes an objectivity principle for measuring worker expertise and item confusability. Experimental results on real-world crowdsourcing data demonstrate that the proposed method estimates ground truth more accurately than previously proposed methods. The presented framework can be easily extended. For example, in the web search experiment, the multilevel relevance scale is treated as multiclass. By taking the ordinal property of ratings into account, the accuracy may be further improved. The framework could be extended to real-valued labels. A detailed discussion on those topics is beyond the scope of this paper. Acknowledgments We thank Daniel Hsu, Xi Chen, Chris Burges and Chris Meek for helpful discussions, and Gabriella Kazai for generating the web search dataset. 8 References [1] Y. Altun and A. Smola. Unifying divergence minimization and statistical inference via convex duality. In Proceedings of the 19th Annual Conference on Learning Theory, 2006. [2] Amazon Mechanical Turk. https://www.mturk.com/mturk. [3] S. Boyd and L. Vandenberghe. Convex Optimization. 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4,471	Parametric Local Metric Learning for Nearest Neighbor Classiﬁcation Jun Wang Department of Computer Science University of Geneva Switzerland Jun.Wang@unige.ch Adam Woznica Department of Computer Science University of Geneva Switzerland Adam.Woznica@unige.ch Alexandros Kalousis Department of Business Informatics University of Applied Sciences Western Switzerland Alexandros.Kalousis@hesge.ch Abstract We study the problem of learning local metrics for nearest neighbor classiﬁcation. Most previous works on local metric learning learn a number of local unrelated metrics. While this ”independence” approach delivers an increased ﬂexibility its downside is the considerable risk of overﬁtting. We present a new parametric local metric learning method in which we learn a smooth metric matrix function over the data manifold. Using an approximation error bound of the metric matrix function we learn local metrics as linear combinations of basis metrics deﬁned on anchor points over different regions of the instance space. We constrain the metric matrix function by imposing on the linear combinations manifold regularization which makes the learned metric matrix function vary smoothly along the geodesics of the data manifold. Our metric learning method has excellent performance both in terms of predictive power and scalability. We experimented with several largescale classiﬁcation problems, tens of thousands of instances, and compared it with several state of the art metric learning methods, both global and local, as well as to SVM with automatic kernel selection, all of which it outperforms in a signiﬁcant manner. 1 Introduction The nearest neighbor (NN) classiﬁer is one of the simplest and most classical non-linear classiﬁcation algorithms. It is guaranteed to yield an error no worse than twice the Bayes error as the number of instances approaches inﬁnity. With ﬁnite learning instances, its performance strongly depends on the use of an appropriate distance measure. Mahalanobis metric learning [4, 15, 9, 10, 17, 14] improves the performance of the NN classiﬁer if used instead of the Euclidean metric. It learns a global distance metric which determines the importance of the different input features and their correlations. However, since the discriminatory power of the input features might vary between different neighborhoods, learning a global metric cannot ﬁt well the distance over the data manifold. Thus a more appropriate way is to learn a metric on each neighborhood and local metric learning [8, 3, 15, 7] does exactly that. It increases the expressive power of standard Mahalanobis metric learning by learning a number of local metrics (e.g. one per each instance). 1 Local metric learning has been shown to be effective for different learning scenarios. One of the ﬁrst local metric learning works, Discriminant Adaptive Nearest Neighbor classiﬁcation [8], DANN, learns local metrics by shrinking neighborhoods in directions orthogonal to the local decision boundaries and enlarging the neighborhoods parallel to the boundaries. It learns the local metrics independently with no regularization between them which makes it prone to overﬁtting. The authors of LMNN-Multiple Metric (LMNN-MM) [15] signiﬁcantly limited the number of learned metrics and constrained all instances in a given region to share the same metric in an effort to combat overﬁtting. In the supervised setting they ﬁxed the number of metrics to the number of classes; a similar idea has been also considered in [3]. However, they too learn the metrics independently for each region making them also prone to overﬁtting since the local metrics will be overly speciﬁc to their respective regions. The authors of [16] learn local metrics using a least-squares approach by minimizing a weighted sum of the distances of each instance to apriori deﬁned target positions and constraining the instances in the projected space to preserve the original geometric structure of the data in an effort to alleviate overﬁtting. However, the method learns the local metrics using a learning-order-sensitive propagation strategy, and depends heavily on the appropriate deﬁnition of the target positions for each instance, a task far from obvious. In another effort to overcome the overﬁtting problem of the discriminative methods [8, 15], Generative Local Metric Learning, GLML, [11], propose to learn local metrics by minimizing the NN expected classiﬁcation error under strong model assumptions. They use the Gaussian distribution to model the learning instances of each class. However, the strong model assumptions might easily be very inﬂexible for many learning problems. In this paper we propose the Parametric Local Metric Learning method (PLML) which learns a smooth metric matrix function over the data manifold. More precisely, we parametrize the metric matrix of each instance as a linear combination of basis metric matrices of a small set of anchor points; this parametrization is naturally derived from an error bound on local metric approximation. Additionally we incorporate a manifold regularization on the linear combinations, forcing the linear combinations to vary smoothly over the data manifold. We develop an efﬁcient two stage algorithm that ﬁrst learns the linear combinations of each instance and then the metric matrices of the anchor points. To improve scalability and efﬁciency we employ a fast ﬁrst-order optimization algorithm, FISTA [2], to learn the linear combinations as well as the basis metrics of the anchor points. We experiment with the PLML method on a number of large scale classiﬁcation problems with tens of thousands of learning instances. The experimental results clearly demonstrate that PLML signiﬁcantly improves the predictive performance over the current state-of-the-art metric learning methods, as well as over multi-class SVM with automatic kernel selection. 2 Preliminaries We denote by X the n×d matrix of learning instances, the i-th row of which is the xT i ∈Rd instance, and by y = (y1, . . . , yn)T , yi ∈{1, . . . , c} the vector of class labels. The squared Mahalanobis distance between two instances in the input space is given by: d2 M(xi, xj) = (xi −xj)T M(xi −xj) where M is a PSD metric matrix (M ⪰0). A linear metric learning method learns a Mahalanobis metric M by optimizing some cost function under the PSD constraints for M and a set of additional constraints on the pairwise instance distances. Depending on the actual metric learning method, different kinds of constraints on pairwise distances are used. The most successful ones are the large margin triplet constraints. A triplet constraint denoted by c(xi, xj, xk), indicates that in the projected space induced by M the distance between xi and xj should be smaller than the distance between xi and xk. Very often a single metric M can not model adequately the complexity of a given learning problem in which discriminative features vary between different neighborhoods. To address this limitation in local metric learning we learn a set of local metrics. In most cases we learn a local metric for each learning instance [8, 11], however we can also learn a local metric for some part of the instance space in which case the number of learned metrics can be considerably smaller than n, e.g. [15]. We follow the former approach and learn one local metric per instance. In principle, distances should then be deﬁned as geodesic distances using the local metric on a Riemannian manifold. However, this is computationally difﬁcult, thus we deﬁne the distance between instances xi and xj as: d2 Mi(xi, xj) = (xi −xj)T Mi(xi −xj) 2 where Mi is the local metric of instance xi. Note that most often the local metric Mi of instance xi is different from that of xj. As a result, the distance d2 Mi(xi, xj) does not satisfy the symmetric property, i.e. it is not a proper metric. Nevertheless, in accordance to the standard practice we will continue to use the term local metric learning following [15, 11]. 3 Parametric Local Metric Learning We assume that there exists a Lipschitz smooth vector-valued function f(x), the output of which is the vectorized local metric matrix of instance x. Learning the local metric of each instance is essentially learning the value of this function at different points over the data manifold. In order to signiﬁcantly reduce the computational complexity we will approximate the metric function instead of directly learning it. Deﬁnition 1 A vector-valued function f(x) on Rd is a (α, β, p)-Lipschitz smooth function with respect to a vector norm ∥·∥ if ∥f(x) −f(x′)∥ ≤ α ∥x −x′∥ and f(x) −f(x′) −∇f(x′)T (x −x′) ≤ β ∥x −x′∥1+p, where ∇f(x′)T is the derivative of the f function at x′. We assume α, β > 0 and p ∈(0, 1]. [18] have shown that any Lipschitz smooth real function f(x) deﬁned on a lower dimensional manifold can be approximated by a linear combination of function values f(u), u ∈U, of a set U of anchor points. Based on this result we have the following lemma that gives the respective error bound for learning a Lipschitz smooth vector-valued function. Lemma 1 Let (γ, U) be a nonnegative weighting on anchor points U in Rd. Let f be an (α, β, p)Lipschitz smooth vector function. We have for all x ∈Rd: f(x) − X u∈U γu(x)f(u) ≤α x − X u∈U γu(x)u + β X u∈U γu(x) ∥x −u∥1+p (1) The proof of the above Lemma 1 is similar to the proof of Lemma 2.1 in [18]; for lack of space we omit its presentation. By the nonnegative weighting strategy (γ, U), the PSD constraints on the approximated local metric is automatically satisﬁed if the local metrics of anchor points are PSD matrices. Lemma 1 suggests a natural way to approximate the local metric function by parameterizing the metric Mi of each instance xi as a weighted linear combination, Wi ∈Rm, of a small set of metric basis, {Mb1, . . . , Mbm}, each one associated with an anchor point deﬁned in some region of the instance space. This parametrization will also provide us with a global way to regularize the ﬂexibility of the metric function. We will ﬁrst learn the vector of weights Wi for each instance xi, and then the basis metric matrices; these two together, will give us the Mi metric for the instance xi. More formally, we deﬁne a m × d matrix U of anchor points, the i-th row of which is the anchor point ui, where uT i ∈Rd. We denote by Mbi the Mahalanobis metric matrix associated with ui. The anchor points can be deﬁned using some clustering algorithm, we have chosen to deﬁne them as the means of clusters constructed by the k-means algorithm. The local metric Mi of an instance xi is parametrized by: Mi = X bk WibkMbk, Wibk ≥0, X bk Wibk = 1 (2) where W is a n × m weight matrix, and its Wibk entry is the weight of the basis metric Mbk for the instance xi. The constraint P bk Wibk = 1 removes the scaling problem between different local metrics. Using the parametrization of equation (2), the squared distance of xi to xj under the metric Mi is: d2 Mi(xi, xj) = X bk Wibkd2 Mbk (xi, xj) (3) where d2 Mbk (xi, xj) is the squared Mahalanobis distance between xi and xj under the basis metric Mbk. We will show in the next section how to learn the weights of the basis metrics for each instance and in section 3.2 how to learn the basis metrics. 3 Algorithm 1 Smoothl Local Linear Weight Learning Input: W0, X, U, G, L, λ1, and λ2 Output: matrix W deﬁne egβ,Y(W) = g(Y) + tr(∇g(Y)T (W −Y)) + β 2 ∥W −Y∥2 F initialize: t1 = 1, β = 1,Y1 = W0, and i = 0 repeat i = i + 1, Wi = Proj((Yi −1 β ∇g(Yi))) while g(Wi) > egβ,Yi(Wi) do β = 2β, Wi = Proj((Yi −1 β ∇g(Yi))) end while ti+1 = 1+√ 1+4t2 i 2 , Yi+1 = Wi + ti−1 ti+1 (Wi −Wi−1) until converges; 3.1 Smooth Local Linear Weighting Lemma 1 bounds the approximation error by two terms. The ﬁrst term states that x should be close to its linear approximation, and the second that the weighting should be local. In addition we want the local metrics to vary smoothly over the data manifold. To achieve this smoothness we rely on manifold regularization and constrain the weight vectors of neighboring instances to be similar. Following this reasoning we will learn Smooth Local Linear Weights for the basis metrics by minimizing the error bound of (1) together with a regularization term that controls the weight variation of similar instances. To simplify the objective function, we use the term x −P u∈U γu(x)u 2 instead of x −P u∈U γu(x)u . By including the constraints on the W weight matrix in (2), the optimization problem is given by: min W g(W) = ∥X −WU∥2 F + λ1tr(WG) + λ2tr(WTLW) (4) s.t. Wibk ≥0, X bk Wibk = 1, ∀i, bk where tr(·) and ∥·∥F denote respectively the trace norm of a square matrix and the Frobenius norm of a matrix. The m × n matrix G is the squared distance matrix between each anchor point ui and each instance xj, obtained for p = 1 in (1), i.e. its (i, j) entry is the squared Euclidean distance between ui and xj. L is the n × n Laplacian matrix constructed by D −S, where S is the n × n symmetric pairwise similarity matrix of learning instances and D is a diagonal matrix with Dii = P k Sik. Thus the minimization of the tr(WTLW) term constrains similar instances to have similar weight coefﬁcients. The minimization of the tr(WG) term forces the weights of the instances to reﬂect their local properties. Most often the similarity matrix S is constructed using k-nearest neighbors graph [19]. The λ1 and λ2 parameters control the importance of the different terms. Since the cost function g(W) is convex quadratic with W and the constraint is simply linear, (4) is a convex optimization problem with a unique optimal solution. The constraints on W in (4) can be seen as n simplex constraints on each row of W; we will use the projected gradient method to solve the optimization problem. At each iteration t, the learned weight matrix W is updated by: Wt+1 = Proj(Wt −η∇g(Wt)) (5) where η > 0 is the step size and ∇g(Wt) is the gradient of the cost function g(W) at Wt. The Proj(·) denotes the simplex projection operator on each row of W. Such a projection operator can be efﬁciently implemented with a complexity of O(nm log(m)) [6]. To speed up the optimization procedure we employ a fast ﬁrst-order optimization method FISTA, [2]. The detailed algorithm is described in Algorithm 1. The Lipschitz constant β required by this algorithm is estimated by using the condition of g(Wi) ≤egβ,Yi(Wi) [1]. At each iteration, the main computations are in the gradient and the objective value with complexity O(nmd + n2m). To set the weights of the basis metrics for a testing instance we can optimize (4) given the weight of the basis metrics for the training instances. Alternatively we can simply set them as the weights of its nearest neighbor in the training instances. In the experiments we used the latter approach. 4 3.2 Large Margin Basis Metric Learning In this section we deﬁne a large margin based algorithm to learn the basis metrics Mb1, . . . , Mbm. Given the W weight matrix of basis metrics obtained using Algorithm 1, the local metric Mi of an instance xi deﬁned in (2) is linear with respect to the basis metrics Mb1, . . . , Mbm. We deﬁne the relative comparison distance of instances xi, xj and xk as: d2 Mi(xi, xk) −d2 Mi(xi, xj). In a large margin constraint c(xi, xj, xk), the squared distance d2 Mi(xi, xk) is required to be larger than d2 Mi(xi, xj) + 1, otherwise an error ξijk ≥0 is generated. Note that, this relative comparison deﬁnition is different from that deﬁned in LMNN-MM [15]. In LMNN-MM to avoid over-ﬁtting, different local metrics Mj and Mk are used to compute the squared distance d2 Mj(xi, xj) and d2 Mk(xi, xk) respectively, as no smoothness constraint is added between metrics of different local regions. Given a set of triplet constraints, we learn the basis metrics Mb1, . . . , Mbm with the following optimization problem: min Mb1,...,Mbm,ξ α1 X bl \|\|Mbl\|\|2 F + X ijk ξijk + α2 X ij X bl Wibld2 Mbl (xi, xj) (6) s.t. X bl Wibl(d2 Mbl (xi, xk) −d2 Mbl (xi, xj)) ≥1 −ξijk ∀i, j, k ξijk ≥0; ∀i, j, k Mbl ⪰0; ∀bl where α1 and α2 are parameters that balance the importance of the different terms. The large margin triplet constraints for each instance are generated using its k1 same class nearest neighbors and k2 different class nearest neighbors by requiring its distances to the k2 different class instances to be larger than those to its k1 same class instances. In the objective function of (6) the basis metrics are learned by minimizing the sum of large margin errors and the sum of squared pairwise distances of each instance to its k1 nearest neighbors computed using the local metric. Unlike LMNN we add the squared Frobenius norm on each basis metrics in the objective function. We do this for two reasons. First we exploit the connection between LMNN and SVM shown in [5] under which the squared Frobenius norm of the metric matrix is related to the SVM margin. Second because adding this term leads to an easy-to-optimize dual formulation of (6) [12]. Unlike many special solvers which optimize the primal form of the metric learning problem [15, 13], we follow [12] and optimize the Lagrangian dual problem of (6). The dual formulation leads to an efﬁcient basis metric learning algorithm. Introducing the Lagrangian dual multipliers γijk, pijk and the PSD matrices Zbl to respectively associate with every large margin triplet constraints, ξijk ≥0 and the PSD constraints Mbl ⪰0 in (6), we can easily derive the following Lagrangian dual form max Zb1,...,Zbm,γ X ijk γijk − X bl 1 4α1 · ∥Zbl + X ijk γijkWiblCijk −α2 X ij WiblAij∥2 F (7) s.t. 1 ≥γijk ≥0; ∀i,j,k Zbl ⪰0; ∀bl and the corresponding optimality conditions: M∗ bl = (Z∗ bl+P ijk γ∗ ijkWiblCijk−α2 P ij WiblAij) 2α1 and 1 ≥γijk ≥0, where the matrices Aij and Cijk are given by xT ijxij and xT ikxik−xT ijxij respectively, where xij = xi −xj. Compared to the primal form, the main advantage of the dual formulation is that the second term in the objective function of (7) has a closed-form solution for Zbl given a ﬁxed γ. To drive the optimal solution of Zbl, let Kbl = α2 P ij WiblAij −P ijk γijkWiblCijk. Then, given a ﬁxed γ, the optimal solution of Zbl is Z∗ bl = (Kbl)+, where (Kbl)+ projects the matrix Kbl onto the PSD cone, i.e. (Kbl)+ = U[max(diag(Σ)), 0)]UT with Kbl = UΣUT. Now, (7) is rewritten as: min γ g(γ) = − X ijk γijk + X bl 1 4α1 ∥(Kbl)+ −Kbl∥2 F (8) s.t. 1 ≥γijk ≥0; ∀i, j, k 5 And the optimal condition for Mbl is M∗ bl = 1 2α1 ((K∗ bl)+ −K∗ bl). The gradient of the objective function in (8), ∇g(γijk), is given by: ∇g(γijk) = −1 + P bl 1 2α1 ⟨(Kbl)+ −Kbl, WiblCijk⟩. At each iteration, γ is updated by: γi+1 = BoxProj(γi −η∇g(γi)) where η > 0 is the step size. The BoxProj(·) denotes the simple box projection operator on γ as speciﬁed in the constraints of (8). At each iteration, the main computational complexity lies in the computation of the eigendecomposition with a complexity of O(md3) and the computation of the gradient with a complexity of O(m(nd2 + cd)), where m is the number of basis metrics and c is the number of large margin triplet constraints. As in the weight learning problem the FISTA algorithm is employed to accelerate the optimization process; for lack of space we omit the algorithm presentation. 4 Experiments In this section we will evaluate the performance of PLML and compare it with a number of relevant baseline methods on six datasets with large number of instances, ranging from 5K to 70K instances; these datasets are Letter, USPS, Pendigits, Optdigits, Isolet and MNIST. We want to determine whether the addition of manifold regularization on the local metrics improves the predictive performance of local metric learning, and whether the local metric learning improves over learning with single global metric. We will compare PLML against six baseline methods. The ﬁrst, SML, is a variant of PLML where a single global metric is learned, i.e. we set the number of basis in (6) to one. The second, Cluster-Based LML (CBLML), is also a variant of PLML without weight learning. Here we learn one local metric for each cluster and we assign a weight of one for a basis metric Mbi if the corresponding cluster of Mbi contains the instance, and zero otherwise. Finally, we also compare against four state of the art metric learning methods LMNN [15], BoostMetric [13]1, GLML [11] and LMNN-MM [15]2. The former two learn a single global metric and the latter two a number of local metrics. In addition to the different metric learning methods, we also compare PLML against multi-class SVMs in which we use the one-against-all strategy to determine the class label for multi-class problems and select the best kernel with inner cross validation. Since metric learning is computationally expensive for datasets with large number of features we followed [15] and reduced the dimensionality of the USPS, Isolet and MINIST datasets by applying PCA. In these datasets the retained PCA components explain 95% of their total variances. We preprocessed all datasets by ﬁrst standardizing the input features, and then normalizing the instances to so that their L2-norm is one. PLML has a number of hyper-parameters. To reduce the computational time we do not tune λ1 and λ2 of the weight learning optimization problem (4), and we set them to their default values of λ1 = 1 and λ2 = 100. The Laplacian matrix L is constructed using the six nearest neighbors graph following [19]. The anchor points U are the means of clusters constructed with k-means clustering. The number m of anchor points, i.e. the number of basis metrics, depends on the complexity of the learning problem. More complex problems will often require a larger number of anchor points to better model the complexity of the data. As the number of classes in the examined datasets is 10 or 26, we simply set m = 20 for all datasets. In the basis metric learning problem (6), the number of the dual parameters γ is the same as the number of triplet constraints. To speedup the learning process, the triplet constraints are constructed only using the three same-class and the three different-class nearest neighbors for each learning instance. The parameter α2 is set to 1, while the parameter α1 is the only parameter that we select from the set {0.01, 0.1, 1, 10, 100} using 2-fold inner cross-validation. The above setting of basis metric learning for PLML is also used with the SML and CBLML methods. For LMNN and LMNN-MM we use their default settings, [15], in which the triplet constraints are constructed by the three nearest same-class neighbors and all different-class samples. As a result, the number of triplet constraints optimized in LMNN and LMNN-MM is much larger than those of PLML, SML, BoostMetric and CBLML. The local metrics are initialized by identity matrices. As in [11], GLML uses the Gaussian distribution to model the learning instances from the same class. Finally, we use the 1-NN rule to evaluate the performance of the different metric learning methods. In addition as we already mentioned we also compare against multi-class SVM. Since the performance of the latter depends heavily on the kernel with which it is coupled we do automatic kernel selection with inner cross validation to select the best 1http://code.google.com/p/boosting 2http://www.cse.wustl.edu/∼kilian/code/code.html. 6 (a) LMNN-MM (b) CBLML (c) GLML (d) PLML Figure 1: The visualization of learned local metrics of LMNN-MM, CBLML, GLML and PLML. Table 1: Accuracy results. The superscripts +−= next to the accuracies of PLML indicate the result of the McNemar’s statistical test with LMNN, BoostMetric, SML, CBLML, LMNN-MM, GMLM and SVM. They denote respectively a signiﬁcant win, loss or no difference for PLML. The number in the parenthesis indicates the score of the respective algorithm for the given dataset based on the pairwise comparisons of the McNemar’s statistical test. Single Metric Learning Baselines Local Metric Learning Baselines Datasets PLML LMNN BoostMetric SML CBLML LMNN-MM GLML SVM Letter 97.22+++\|+++\|+(7.0) 96.08(2.5) 96.49(4.5) 96.71(5.5) 95.82(2.5) 95.02(1.0) 93.86(0.0) 96.64(5.0) Pendigits 98.34+++\|+++\|+(7.0) 97.43(2.0) 97.43(2.5) 97.80(4.5) 97.94(5.0) 97.43(2.0) 96.88(0.0) 97.91(5.0) Optdigits 97.72===\|+++\|=(5.0) 97.55(5.0) 97.61(5.0) 97.22(5.0) 95.94(1.5) 95.94(1.5) 94.82(0.0) 97.33(5.0) Isolet 95.25=+=\|+++\|=(5.5) 95.51(5.5) 89.16(2.5) 94.68(5.5) 89.03(2.5) 84.61(0.5) 84.03(0.5) 95.19(5.5) USPS 98.26+++\|+++\|=(6.5) 97.92(4.5) 97.65(2.5) 97.94(4.0) 96.22(0.5) 97.90(4.0) 96.05(0.5) 98.19(5.5) MNIST 97.30=++\|+++\|=(6.0) 97.30(6.0) 96.03(2.5) 96.57(4.0) 95.77(2.5) 93.24(1.0) 84.02(0.0) 97.62(6.0) Total Score 37 25.5 19.5 28.5 14.5 10 1 32.5 kernel and parameter setting. The kernels were chosen from the set of linear, polynomial (degree 2,3 and 4), and Gaussian kernels; the width of the Gaussian kernel was set to the average of all pairwise distances. Its C parameter of the hinge loss term was selected from {0.1, 1, 10, 100}. To estimate the classiﬁcation accuracy for Pendigits, Optdigits, Isolet and MNIST we used the default train and test split, for the other datasets we used 10-fold cross-validation. The statistical signiﬁcance of the differences were tested with McNemar’s test with a p-value of 0.05. In order to get a better understanding of the relative performance of the different algorithms for a given dataset we used a simple ranking schema in which an algorithm A was assigned one point if it was found to have a statistically signiﬁcantly better accuracy than another algorithm B, 0.5 points if the two algorithms did not have a signiﬁcant difference, and zero points if A was found to be signiﬁcantly worse than B. 4.1 Results In Table 1 we report the experimental results. PLML consistently outperforms the single global metric learning methods LMNN, BoostMetric and SML, for all datasets except Isolet on which its accuracy is slightly lower than that of LMNN. Depending on the single global metric learning method with which we compare it, it is signiﬁcantly better in three, four, and ﬁve datasets ( for LMNN, SML, and BoostMetric respectively), out of the six and never singiﬁcantly worse. When we compare PLML with CBLML and LMNN-MM, the two baseline methods which learn one local metric for each cluster and each class respectively with no smoothness constraints, we see that it is statistically signiﬁcantly better in all the datasets. GLML fails to learn appropriate metrics on all datasets because its fundamental generative model assumption is often not valid. Finally, we see that PLML is signiﬁcantly better than SVM in two out of the six datasets and it is never signiﬁcantly worse; remember here that with SVM we also do inner fold kernel selection to automatically select the appropriate feature speace. Overall PLML is the best performing methods scoring 37 points over the different datasets, followed by SVM with automatic kernel selection and SML which score 32.5 and 28.5 points respectively. The other metric learning methods perform rather poorly. Examining more closely the performance of the baseline local metric learning methods CBLML and LMNN-MM we observe that they tend to overﬁt the learning problems. This can be seen by their considerably worse performance with respect to that of SML and LMNN which rely on a single global model. On the other hand PLML even though it also learns local metrics it does not suffer from the overﬁtting problem due to the manifold regularization. The poor performance of LMNN7 (a) Letter (b) Pendigits (c) Optdigits (d) USPS (e) Isolet (f) MNIST Figure 2: Accuracy results of PLML and CBLML with varying number of basis metrics. MM is not in agreement with the results reported in [15]. The main reason for the difference is the experimental setting. In [15], 30% of the training instance of each dataset were used as a validation set to avoid overﬁtting. To provide a better understanding of the behavior of the learned metrics, we applied PLML LMNNMM, CBLML and GLML, on an image dataset containing instances of four different handwritten digits, zero, one, two, and four, from the MNIST dataset. As in [15], we use the two main principal components to learn. Figure 1 shows the learned local metrics by plotting the axis of their corresponding ellipses(black line). The direction of the longer axis is the more discriminative. Clearly PLML ﬁts the data much better than LMNN-MM and as expected its local metrics vary smoothly. In terms of the predictive performance, PLML has the best with 82.76% accuracy. The CBLML, LMNN-MM and GLML have an almost identical performance with respective accuracies of 82.59%, 82.56% and 82.51%. Finally we investigated the sensitivity of PLML and CBLML to the number of basis metrics, we experimented with m ∈{5, 10, 15, 20, 25, 30, 35, 40}. The results are given in Figure 2. We see that the predictive performance of PLML often improves as we increase the number of the basis metrics. Its performance saturates when the number of basis metrics becomes sufﬁcient to model the underlying training data. As expected different learning problems require different number of basis metrics. PLML does not overﬁt on any of the datasets. In contrast, the performance of CBLML gets worse when the number of basis metrics is large which provides further evidence that CBLML does indeed overﬁt the learning problems, demonstrating clearly the utility of the manifold regularization. 5 Conclusions Local metric learning provides a more ﬂexible way to learn the distance function. However they are prone to overﬁtting since the number of parameters they learn can be very large. In this paper we presented PLML, a local metric learning method which regularizes local metrics to vary smoothly over the data manifold. Using an approximation error bound of the metric matrix function, we parametrize the local metrics by a weighted linear combinations of local metrics of anchor points. Our method scales to learning problems with tens of thousands of instances and avoids the overﬁtting problems that plague the other local metric learning methods. The experimental results show that PLML outperforms signiﬁcantly the state of the art metric learning methods and it has a performance which is signiﬁcantly better or equivalent to that of SVM with automatic kernel selection. Acknowledgments This work was funded by the Swiss NSF (Grant 200021-137949). The support of EU projects DebugIT (FP7-217139) and e-LICO (FP7-231519), as well as that of COST Action BM072 (’Urine and Kidney Proteomics’) is also gratefully acknowledged. 8 References [1] F. Bach, R. Jenatton, J. Mairal, and G. Obozinski. Convex optimization with sparsity-inducing norms. Optimization for Machine Learning. [2] A. Beck and M. Teboulle. 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4,472	The topographic unsupervised learning of natural sounds in the auditory cortex Hiroki Terashima The University of Tokyo / JSPS Tokyo, Japan teratti@teratti.jp Masato Okada The University of Tokyo / RIKEN BSI Tokyo, Japan okada@k.u-tokyo.ac.jp Abstract The computational modelling of the primary auditory cortex (A1) has been less fruitful than that of the primary visual cortex (V1) due to the less organized properties of A1. Greater disorder has recently been demonstrated for the tonotopy of A1 that has traditionally been considered to be as ordered as the retinotopy of V1. This disorder appears to be incongruous, given the uniformity of the neocortex; however, we hypothesized that both A1 and V1 would adopt an efﬁcient coding strategy and that the disorder in A1 reﬂects natural sound statistics. To provide a computational model of the tonotopic disorder in A1, we used a model that was originally proposed for the smooth V1 map. In contrast to natural images, natural sounds exhibit distant correlations, which were learned and reﬂected in the disordered map. The auditory model predicted harmonic relationships among neighbouring A1 cells; furthermore, the same mechanism used to model V1 complex cells reproduced nonlinear responses similar to the pitch selectivity. These results contribute to the understanding of the sensory cortices of different modalities in a novel and integrated manner. 1 Introduction Despite the anatomical and functional similarities between the primary auditory cortex (A1) and the primary visual cortex (V1), the computational modelling of A1 has proven to be less fruitful than V1, primarily because the responses of A1 cells are more disorganized. For instance, the receptive ﬁelds of V1 cells are localized within a small portion of the ﬁeld of view [1], whereas certain A1 cells have receptive ﬁelds that are not localized, as these A1 cells demonstrate signiﬁcant responses to multiple distant frequencies [2, 3]. An additional discrepancy that has recently been discovered between these two regions relates to their topographic structures, i.e., the retinotopy of V1 and the tonotopy of A1; these structures had long been considered to be quite similar, but studies on a microscopic scale have demonstrated that in mice, the tonotopy of A1 is much more disordered [4, 5] than the retinotopy of V1 [6, 7]. This result is consistent with previous investigations involving other species [8, 9], suggesting that the discrepancy in question constitutes a general tendency among mammals. This disorderliness appears to pose signiﬁcant difﬁculties for the development of computational models of A1. A number of computational modelling studies have emphasized the close associations between V1 cells and natural image statistics, which suggests that the V1 adopts an unsupervised, efﬁcient coding strategy [10]. For instance, the receptive ﬁelds of V1 simple cells were reproduced by either sparse coding [11] or the independent component analysis [12] of natural images. This line of research yields explanations for the two-dimensional topography, the orientation and retinotopic maps of V1 [13, 14, 15]. Similar efforts to address A1 have been attempted by only a few studies, which demonstrated that the efﬁcient coding of natural, harmonic sounds, such as human voices or piano 1 recordings, can explain the basic receptive ﬁelds of A1 cells [16, 17] and their harmony-related responses [18, 19]. However, these studies have not yet addressed the topography of A1. In an integrated and computational manner, the present paper attempts to explain why the tonotopy of A1 is more disordered than the retinotopy of V1. We hypothesized that V1 and A1 still share an efﬁcient coding strategy, and we therefore proposed that the distant correlations in natural sounds would be responsible for the relative disorder in A1. To test this hypothesis, we ﬁrst demonstrated the signiﬁcant differences between natural images and natural sounds. Natural images and natural sounds were then each used as inputs for topographic independent component analysis, a model that had previously been proposed for the smooth topography of V1, and maps were generated for these images and sounds. Due to the distant correlations of natural sounds, greater disorder was observed in the learned map that had been adapted to natural sounds than in the analogous map that had been adapted to images. For natural sounds, this model not only predicted harmonic relationships between neighbouring cells but also demonstrated nonlinear responses that appeared similar to the responses of the pitch-selective cells that were recently found in A1. These results suggest that the apparently dissimilar topographies of V1 and A1 may reﬂect statistical differences between natural images and natural sounds; however, these two regions may employ a common adaptive strategy. 2 Methods 2.1 Topographic independent component analysis Herein, we discuss an unsupervised learning model termed topographic independent component analysis (TICA), which was originally proposed for the study of V1 topography [13, 14]. This model comprises two layers: the ﬁrst layer of N units models the linear responses of V1 simple cells, whereas the second layer of N units models the nonlinear responses of V1 complex cells, and the connections between the layers deﬁne a topography. Given a whitened input vector I(x) ∈Rd (here, d = N), the input is reconstructed by the linear superposition of a basis ai ∈Rd, each of which corresponds to the ﬁrst-layer units I = ∑ i siai (1) where si ∈R are activity levels of the units or model neurons. Inverse ﬁlters wi to determine si can typically be obtained, and thus si = IT wi (inner product). Using the activities of the ﬁrst layer, the activities of the second-layer units ci ∈R can be deﬁned as follows: ci = ∑ j h(i, j)s2 j (2) where h(i, j) is the neighbourhood function that takes the value of 1 if i and j are neighbours and is 0 otherwise. The neighbourhood is deﬁned by a square window (e.g., 5 × 5) in cases of two-dimensional topography. The learning of wi is accomplished through the minimization of the energy function E or the negative log likelihood: E = −log L(I; {wi}) = − ∑ i G(ci) (3) ∆wi ∝ ⟨ Isi  ∑ j h(i, j)g(cj)   ⟩ (4) where G(ci) = −√ϵ + ci imposes sparseness on the second-layer activities (ϵ = 0.005 for the stability), and g(ci) is the derivative of G(ci). The operator ⟨· · · ⟩is the mean over the iterations. 2.1.1 An extension for overcomplete representation Ma and Zhang [15] extended the TICA model to account for overcomplete representations (d < N), which are observed in the V1 of primates. In this extension, inverse ﬁlters cannot be uniquely deﬁned; therefore, a set of ﬁrst-layer responses si to an input is computed by minimizing the following extended energy function: 2 45 1812 285 4 0 -4 1 0 -1 Frequency [Hz] Correlation A B Angle of view [°] Figure 1: Local correlations in natural images and distant correlations in natural sounds. (A) The correlation matrix of image strips (right) demonstrated only local correlations (∼6◦) in the ﬁeld of view (∼120◦). (B) The correlation matrix of the human voice spectra (right) demonstrated not only local correlation but also off-diagonal distant correlations produced by harmonics. E = −log L(I; {ai}, {si}) = I − ∑ i siai 2 −λ ∑ i G(ci) (5) ∆si ∝aT i  I − ∑ j sjaj  −λsi  ∑ j h(i, j)g(cj)   (6) where λ is the relative weight of the activity sparseness, in accordance with sparse coding [11]. The initial value of si is set equal to the inner product of I and ai. Every 256 inputs, the basis is updated using the following gradient. In this study, we used the learning rate η = 0.08. ∆ai = η ⟨ si  I − ∑ j sjaj   ⟩ (7) 2.2 The discontinuity index for topographic representation To compare the degrees of disorder in topographies of different modalities, we deﬁned a discontinuity index (DI) for each point i of the maps. Features deﬁning a topography f(i) (e.g., a retinotopic position or a frequency) were normalized to the range of [0, 1]. Features f(j) within the neighbourhood of the ith unit deﬁned by h(i, j) were linearly ﬁtted using the least squares method, and the DI value at i was then determined using the following equation: DI(i) = √∑ j h(i, j)r(j)2 NNB (8) where r(j) is the residual error of linear regression at j and NNB is the number of units within a neighbourhood window. If the input space is a torus (see Section 3.3), another DI value is computed using modiﬁed f values that are increased by 1 if they were initially within [0, 1 2), and the smaller of the calculated DI values is used. 3 Results 3.1 Correlations of natural images and natural sounds Given that V1 is supposed to adapt to natural images and that A1 is supposed to adapt to natural sounds, the ﬁrst analysis in this study simply compared statistics for natural images and natural sounds. The natural images were taken from the van Hateren database [20] and were reduced four times from their original size. Vertical arrays of 120 pixels each were extracted from the reduced 3 C A B CF [Hz] E Tonotopy (smoothed) 90 3.6k CF [Hz] F Tonotopy 90 3.6k D Retinotopy 1 25 Gabor position 3.6k 200 [ms] [Hz] 90 0 0.1 0.2 0.3 0 50 100 Retinotopy Tonotopy Discontinuity index (DI) Number of units Figure 2: The ordered retinotopy and disordered tonotopy. (A) The topography of units adapted to natural images. A small square indicates a unit ai (grey: 0; white: max value). (B) The distributions of DI for the two topographies. (C) The topography of spectro-temporal units that have been adapted to natural sounds. (D-F) The retinotopy of the visual map (D) is smooth, whereas the tonotopy of the auditory map (F) is more disordered, although global tonotopy still exists (E). images, each of which covered approximately 8◦( 1 15 of the vertical range of the human ﬁeld of view). Figure 1A (right) illustrates the correlation matrix for these images, which is a simple structure that contains local correlations that span approximately 6◦. This result was not surprising, as distant pixels typically depict different objects. For natural sounds, we used human narratives from the Handbook of the International Phonetic Association [21], as efﬁcient representations of human voices have been successful in facilitating studies of various components of the auditory system [22, 23], including A1 [16, 17]. After these sounds were downsampled to 4 kHz, their spectrograms were generated using the NSL toolbox [24] to approximate peripheral auditory processing. Short-time spectra were extracted from the spectrograms, each of which were 128 pixels wide on a logarithmic scale (24 pixels = 1 octave). Note that the frequency range (> 5 octaves) spans approximately half of a typical mammalian hearing range (∼10 octaves [25]), whereas the image pixel array spans only 1 15 of the ﬁeld of view. Figure 1B illustrates the correlation matrix for these sounds, which is a complex structure that incorporates distant, off-diagonal correlations. The most prominent off-diagonal correlation, which was just 1 octave away from the main diagonal, corresponded to the second harmonic of a sound, i.e., frequencies at a ratio of 1:2. Similarly, other off-diagonal peaks indicated correlations due to higher harmonics, i.e., frequencies that were related to each other by simple integral ratios. These distant correlations represent relatively typical results for natural sounds and differ greatly from the strictly local correlations observed for natural images. 3.2 Greater disorder for the tonotopy than the retinotopy To test the hypothesis that V1 and A1 share a learning strategy, the TICA model was applied to natural images and natural sounds, which exhibit different statistical proﬁles, as discussed above. 4 0.2 0.2 0.1 0 0.4 0.2 Discontinuity index (DI) Strength of distant correlation Vision-like Audition-like Figure 3: The correlation between discontinuity and input “auditoriness”. When inputs only correlated locally (pa ∼0: vision-like inputs), DI was low, and DI increased with the input “auditoriness” pa. Three lines: the quartiles (25, 50 (bold), and 75%) obtained from 100 iterations. Learning with natural images was accomplished in accordance with the original TICA study [13, 14]. Images from the van Hateren database were reduced four times from their original size, and 25 × 25 pixel image patches were randomly extracted (n = 50,000). The patches were whitened and bandpassed by applying principal component analysis, whereby we selected 400 components and rejected certain components with low variances and the three components with the largest variances [13]. The topography was a 20 × 20 torus, and the neighbourhood window was 5 × 5. Figure 2A illustrates the visual topographic map obtained from this analysis, a small square of which constitutes a basis vector ai. As previously observed in the original TICA study [13, 14], each unit was localized, oriented, and bandpassed; thus, these units appeared to be organized similarly to the receptive ﬁelds of V1 simple cells. The orientation and position of the units changed smoothly with the coordinates that were examined, which suggested that this map evinces an ordered topography. To quantify the retinotopic discontinuity, each unit was ﬁtted using a two-dimensional Gabor function, and DI was calculated using the y values of the centre coordinates of the resulting Gabor functions as the features. Figure 2B graphically indicates that the obtained DI values were quite low, which is consistent with the smooth retinotopy illustrated in Figure 2D. Next, another TICA model was applied to natural sounds to create an auditory topographic map that could be compared to the visual topography. As detailed in the previous section, spectrograms of human voices (sampled at 8 kHz) were generated using the NSL toolbox to approximate peripheral auditory processing. Spectrogram patches of 200 ms (25 pixels) in width were randomly extracted (n = 50,000) and vertically reduced from 128 to 25 pixels, which enabled these spectrogram patches to be directly compared with the image patches. The sound patches were whitened, bandpassed, and adapted using the model in the same manner as was described for the image patches. Figure 2C shows the resulting auditory topographic map, which is composed of spectro-temporal units of ai that are represented by small squares. The units were localized temporally and spectrally, and some units demonstrated multiple, harmonic peaks; thus, these units appeared to reasonably represent the typical spectro-temporal receptive ﬁelds of A1 cells [16, 3]. The frequency to which an auditory neuron responds most signiﬁcantly is called its characteristic frequency (CF) [2]. In this analysis, the CF of a unit was deﬁned as the frequency that demonstrated the largest absolute value for the unit in question. Figure 2F illustrates the spatial distribution of CFs, i.e., the tonotopic map. Within local regions, the tonotopy was not necessarily smooth, i.e., neighbouring units displayed distant CFs. However, at a global level, a smooth tonotopy was observed (Figure 2E). Both of these ﬁndings are consistent with established experimental results [4, 5]. The distribution of tonotopic DI values is shown in Figure 2B, which clearly demonstrates that the tonotopy was more disordered than the retinotopy (p < 0.0001; Wilcoxon rank test). 3.3 The topographic disorder due to distant input correlations The previous section demonstrated that natural sounds could induce greater topographic disorder than natural images, and this section discusses the attempts to elucidate the disorder resulting from a speciﬁc characteristic of natural sounds, namely, distant correlations. For this purpose, we generated artiﬁcial inputs (d = 16) with a parameter pa ∈[0, 1] that regulates the degree of distant correlations. 5 1 1.2 1.4 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Distance of two units 1:3 1:2 2:3 CF difference [octave] B 2 4 6 8 10 0 1.0 2.0 3.0 4.0 5.0 A Distance of two units CF difference [octave] Figure 4: The harmonic relationships between CFs of neighbouring units. (A) The full distribution of distance and CF difference between two units. (B) The distribution of CF differences within neighbourhoods (the red-dotted rectangle in (A)). There were three peaks that indicate harmonic relationships between neighbouring units. The distances were jittered to obtain the visualization. After the inputs were initially generated from a standard normal distribution, a constant value of 4 was added at k points of each input, where k was from a uniform distribution over {3, 4, 5, 6} and the points’ coordinates x were from a normal distribution with a random centre and σ = 2. After adding this constant value at x, we also added another at xdist = x + 5 with a probability pa that deﬁnes its “auditoriness”, i.e., its degree of distant correlations. For greater simplicity and to avoid border effects, the input space was deﬁned to be a one-dimensional torus. The topography was also set as a one-dimensional torus of 16 units with a neighbourhood window size of 5. Figure 3 shows the positive correlation between the input “auditoriness” pa and the DI of the learned topographies. In computations of DI, the feature f of a unit was considered to be its peak coordinate with the largest absolute value, and a toric input space was used (Section 2.2). If the input only demonstrated local correlations like visual stimuli (pa ∼0), then its learned topography was smooth (i.e., its DI was low). The DI values generally increased as distant correlations appeared more frequently, i.e., more “auditoriness” of the inputs grew. Thus, the topographic disorder of auditory maps results from distant correlations presented by natural auditory signals. 3.4 The harmonic relationship among neighbouring units Several experiments [4, 5] have reported that the CFs of neighbouring cells can differ by up to 4 octaves, although these studies have failed to provide additional detail regarding the local spatial patterns of the CF distributions. However, if the auditory topography is representative of natural stimulus statistics, the topographic map is likely to possess certain additional spatial features that reﬂect the statistical characteristics of natural sounds. To enable a detailed investigation of the CF distribution, we employed a model that had been adapted to ﬁner frequency spectra of natural sounds, and this model was then used throughout the remainder of the study. As the temporal structure of the auditory receptive ﬁelds was less dominant than their spectral structure (Figure 2C), we focused solely on the spectral domain and did not attempt to address temporal information. Therefore, the inputs for the new model (n = 100,000) were short-time frequency spectra of 128 pixels each (24 pixels = 1 octave). The data for these spectra were ﬁrst obtained from the spectrograms of human voices (8 kHz) using the method detailed in Section 3.1, and these data were then whitened, bandpassed, and reduced to 100 dimensions prior to input into the model. To illustrate patterns more clearly, the results shown below were obtained using the overcomplete extension of TICA described in Section 2.1.1, which included a 14 × 14 torus (approximately 2× overcomplete) and 3×3 windows. The CF of a unit was determined using pure-tone inputs of 128 frequencies. Figure 4A illustrates the full distribution of the distance and CF difference between two units in a learned topography. The CFs of even neighbouring units differed by up to ∼4 octaves, which is consistent with recent experimental ﬁndings [4, 5]. A closer inspection of the red-dotted rectangular region of Figure 4A is shown in Figure 4B. The histogram in Figure 4B demonstrates several peaks 6 A C B Frequency [Hz] Pitch selective units Harmonic composition of MFs 12f0 f0 10-12 2-4 1-3 0 4f0 8*f0 CF 90 [Hz] 8k f0 2 4 6 Lowest harmonic present Normalized activity 8 10 0 0.2 0.4 0.6 0.8 1.0 n = 66 units (6 simulations) Figure 5: Nonlinear responses similar to pitch selectivity. (A) The spectra of MFs that share a f0, all of which are perceived similarly. (B) The responses of pitch-selective units to MFs. (C) The distribution of pitch-selective units on the smoothed tonotopy in a single session. at harmony-related CF differences, such as 0.59 (= log2 1.5), 1.0 (= log2 2; the largest peak), and 1.59 (= log2 3). These examples indicate that CFs of neighbouring units did not differ randomly, but tended to be harmonically related. A careful inspection of published data (Figure 5d from [5]) suggests that this relationship may be discernible in those published results; however, the magnitude of non-harmonic relationships cannot be clearly established from the inspection of this previously published study, as the stimuli used by the relevant experiment [5] were separated by an interval of 0.25 octaves and were therefore biased towards being harmonic. Thus, this prediction of a harmonic relationship in neighbouring CFs will need to be examined in more detailed investigations. 3.5 Nonlinear responses similar to pitch-selectivity Psychoacoustics have long demonstrated interesting phenomena related to harmony, namely, the perception of pitch, which represents a subjective attribute of perceived sounds. Forming a rigid deﬁnition for the notion of pitch is difﬁcult; however, if a tone consists of a stack of harmonics (f0, 2f0, 3f0, . . .), then its pitch is the frequency of the lowest harmonic, which is called the fundamental frequency f0. The perception of pitch is known to remain constant even if the sound lacks power at lower harmonics; in fact, pitch at f0 can be perceived from a sound that lacks f0, a phenomenon known as “missing fundamental” [26]. Nonlinear pitch-selective responses similar to this perception have recently been demonstrated in certain A1 neurons [27] that localize in the low-frequency area of the global tonotopy. To investigate pitch-related responses, previously described complex tones [27] that consisted of harmonics were selected as inputs for the model described in Section 3.4. For each unit, responses were calculated to complex tones termed missing fundamental complex tones (MFs) [27]. The MFs were composed of three consecutive harmonics sharing a single f0; the lowest frequency for these consecutive harmonics varied from the fundamental frequency (f0) to the tenth harmonic (10f0), as shown in Figure 5A. For each unit, ﬁve patterns of f0 around its CF (∼0.2 octave) were tested, resulting in a total of 10 × 5 = 50 variations of MFs. The activity of a unit was normalized to its maximum response to the MFs. Pitch-selective units were deﬁned as those that signiﬁcantly responded (normalized activity > 0.4) to all of the MFs sharing a single f0 with a lowest harmonic from 1 to 4. We found certain pitch-selective units in the second layer (n = 66; 6 simulations), whereas none were found in the ﬁrst layer. Figure 5B illustrates the response proﬁles of the pitch-selective units, which demonstrated sustained activity for MFs with a lowest harmonic below the sixth harmonic (6f0), and this result is similar to previously published data [27]. Additionally, these units were located in a low-frequency region of the global tonotopy, as shown in Figure 5C, and this feature of pitch-selective units is also consistent with previous ﬁndings [27]. The second layer of the TICA model, which contained the pitch-selective units, was originally designed to represent the layer of V1 complex cells, which have nonlinear responses that can be modelled by a summation of “energies” of neighbouring simple cells [13, 14, 15]. Our result suggests that the mechanism underlying V1 complex cells may be similar to the organizational mechanism for A1 pitch-selective cells. 7 Cortical position Retinal position Smooth V1 retinotopy Cortical position CF Disordered A1 tonotopy Natural images Natural sounds 0 2:3 1:2 1:3 1.0 Δ Frequency Correlation [octave] 0 Retinal distance Correlation No correlation (different objects) A B Figure 6: The suggested relationships between natural stimulus statistics and topographies. 4 Discussion Using a single model, we have provided a computational account explaining why the tonotopy of A1 is more disordered than the retinotopy of V1. First, we demonstrated that there are signiﬁcant differences between natural images and natural sounds; in particular, the latter evince distant correlations, whereas the former do not. The topographic independent component analysis therefore generated a disordered tonotopy for these sounds, whereas the retinotopy adapted to natural images was locally organized throughout. Detailed analyses of the TICA model predicted harmonic relationships among neighbouring neurons; furthermore, these analyses successfully replicated pitch selectivity, a nonlinear response of actual cells, using a mechanism that was designed to model V1 complex cells. The results suggest that A1 and V1 may share an adaptive strategy, and the dissimilar topographies of visual and auditory maps may therefore reﬂect signiﬁcant differences in the natural stimuli. Figure 6 summarizes the ways in which the organizations of V1 and A1 reﬂect these input differences. Natural images correlate only locally, which produces a smooth retinotopy through an efﬁcient coding strategy (Figure 6A). By contrast, natural sounds exhibit additional distant correlations (primarily correlations among harmonics), which produce the topographic disorganization observed for A1 (Figure 6B). To extract the features of natural sounds in the auditory pathway, A1 must integrate multiple channels of distant frequencies [2]; for this purpose, the disordered tonotopy can be beneﬁcial because a neuron can easily collect information regarding distant (and often harmonically related) frequencies from other cells within its neighbourhood. Our result suggests the existence of a common adaptive strategy underlying V1 and A1, which would be consistent with experimental studies that exchanged the peripheral inputs of the visual and auditory systems and suggested the sensory experiences had a dominant effect on cortical organization [28, 29, 30]. Our ﬁnal result suggested that a common mechanism may underlie the complex cells of V1 and the pitch-selective cells of A1. Additional support for this notion was provided by recent evidence indicating that the pitch-selective cells are most commonly found in the supragranular layer [27], and V1 complex cells display a similar tendency. It has been hypothesized that V1 complex cells collect information from neighbouring cells that are selective to different phases of similar orientations; in an analogous way, A1 pitch-selective cells could collect information from the activities of neighbouring cells, which in this case could be selective to different frequencies sharing a single f0. To the best of our knowledge, no previous studies in the literature have attempted to use this analogy of V1 complex cells to explain A1 pitch-selective cells (however, other potential analogues have been mentioned [31, 32]). Our results and further investigations should help us to understand these pitch-selective cells from an integrated, computational viewpoint. 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4,473	Efﬁcient Sampling for Bipartite Matching Problems Maksims N. Volkovs University of Toronto mvolkovs@cs.toronto.edu Richard S. Zemel University of Toronto zemel@cs.toronto.edu Abstract Bipartite matching problems characterize many situations, ranging from ranking in information retrieval to correspondence in vision. Exact inference in realworld applications of these problems is intractable, making efﬁcient approximation methods essential for learning and inference. In this paper we propose a novel sequential matching sampler based on a generalization of the PlackettLuce model, which can effectively make large moves in the space of matchings. This allows the sampler to match the difﬁcult target distributions common in these problems: highly multimodal distributions with well separated modes. We present experimental results with bipartite matching problems—ranking and image correspondence—which show that the sequential matching sampler efﬁciently approximates the target distribution, signiﬁcantly outperforming other sampling approaches. 1 Introduction Bipartite matching problems (BMPs), which involve mapping one set of items to another, are ubiquitous, with applications ranging from computational biology to information retrieval to computer vision. Many problems in these domains can be expressed as a bipartite graph, with one node for each of the items, and edges representing the compatibility between pairs. In a typical BMP a set of labeled instances with target matches is provided together with feature descriptions of the items. The features for any two items do not provide a natural measure of compatibility between the items, i.e., should they be matched or not. Consequently the goal of learning is to create a mapping from the item features to the target matches such that when an unlabeled instance is presented the same mapping can be applied to accurately infer the matches. Probabilistic formulations of this problem, which involve specifying a distribution over possible matches, have become increasingly popular, e.g., [23, 26, 1], and these models have been applied to problems ranging from preference aggregation in social choice and information retrieval [7, 13] to multiple sequence protein alignment in computational biology [24, 27]. However, exact learning and inference in real-world applications of these problems quickly become intractable because the state space is typically factorial in the number of items. Approximate inference methods are also problematic in this domain. Variational approaches, in which aspects of the joint distribution are treated independently, may miss important contingencies in the joint. On the other hand sampling is hard, plagued by the multimodality and strict constraints inherent in discrete combinatorial spaces. Recently there has been a ﬂurry of new methods for sampling for bipartite matching problems. Some of these have strong theoretical properties [10, 9], while others are appealingly simple [6, 13]. However, to the best of our knowledge, even for simple versions of bipartite matching problems, no efﬁcient sampler exists. In this paper we propose a novel Markov Chain Monte Carlo (MCMC) method applicable to a wide subclass of BMPs. We compare the efﬁciency and performance of our sampler to others on two applications. 1 2 Problem Formulation A standard BMP consists of the two sets of N items U = {u1, ..., uN} and V = {v1, ..., vN}. The goal is to ﬁnd an assignment of the items so that every item in U is matched to exactly one item in V and no two items share the same match. In this problem an assignment corresponds to a permutation π where π is a bijection {1, ..., N} →{1, ..., N}, mapping each item in U to its match in V ; we use the terms assignment and permutation interchangeably. We deﬁne π(i) = j to denote the index of a match vπ(i) = vj for item ui in π and use π−1(j) = i to denote the reverse. Permutations have a useful property that any subset of the permutation also constitutes a valid permutation with respect to the items in the subset. We will utilize this property in later sections; here we introduce the notation. Given a full permutation π we deﬁne π1:t (π1:0 = ∅) as a partial permutation of only the ﬁrst t items in U. To express uncertainty over assignments, we use the standard Gibbs form to deﬁne the probability of a permutation π: P(π\|θ) = 1 Z(θ) exp(−E(π, θ)) Z(θ) = X π exp(−E(π, θ)) (1) where θ is the set of model parameters and E(π, θ) is the energy. We assume, without loss of generality, that the energy E(π, θ) is given by a sum of single and/or higher order potentials. Many important problems can be formulated in this form. For example, in information retrieval the crucial problem of learning a ranking function can be modeled as a BMP [12, 26]. In this domain U corresponds to a set of documents and V to a set of ranks. The energy of a given assignment is typically formulated as a combination of ranks and the model’s output from the query-document features. For example in [12] the energy is deﬁned as: E(π, θ) = −1 N N X i=1 θi(N −π(i) + 1) (2) where θi is a score assigned by the model to ui. Similarly, in computer vision the problem of ﬁnding a correspondence between sets of images can be expressed as a BMP [5, 3, 17]. Here U and V are typically sets of points in the two images and the energy is deﬁned on the feature descriptors of these points. For example in [17] the energy is given by: E(π, θ) = 1 \|ψ\| N X i=1 θ, (ψu i −ψv π(i))2 (3) where ψu i and ψv π(i) are feature descriptors for points ui and vπ(i). Finally, some clustering problems can also be expressed in the form of a BMP [8]. It is important to note here that for all models where the energy is additive we can compute the energy E(π1:t, θ) for any partial permutation π1:t by summing the potentials only over the t assignments in π1:t. For instance for the energy in Equation 3, E(π1:t, θ) = 1 \|ψ\| Pt i=1 D θ, (ψu i −ψv π1:t(i))2E with E(π1:0, θ) = 0. Learning in these models typically involves maximizing the log probability of the correct match as a function of θ. To do this one generally needs to ﬁnd the gradient of the log probability with respect to θ: ∂log(P (π\|θ)) ∂θ = −∂E(π,θ) ∂θ −∂log(Z(θ)) ∂θ . Unfortunately, computing the gradient with respect to the partition function requires a summation over N! valid assignments, which very quickly becomes intractable. For example for N = 20 ﬁnding ∂log(Z(θ)) ∂θ requires over 1017 summations. Thus effective approximation techniques are necessary to learn such models. A particular instance of BMP that has been studied extensively is the maximum weight bipartite matching problem (WBMP). In WBMP the energy is reduced to only the single potential φ: Eunary(π, θ) = X i φ(ui, vπ(i), θ) (4) Equations 2 and 3 are both examples of WBMP energies. Finding the assignment with the maximum energy is tractable and can be solved in O(N 3) [16]. Determining the partition function in a WBMP is equivalent to ﬁnding the permanent of the edge weight matrix (deﬁned by the unary potential), a well-known #P problem [25]. The majority of the proposed samplers are designed for 2 WBMPs and cannot be applied to the more general BMPs where the energy includes higher order potentials. However, distributions based on higher order potentials allow greater ﬂexibility and have been actively used in problems ranging from computer vision and robotics [20, 2] to information retrieval [19, 26]. There is thus an evident need to develop an effective sampler applicable to any BMP distribution. 3 Related Approaches In this section we brieﬂy describe existing sampling approaches, some of which have been developed speciﬁcally for bipartite matching problems while others come from matrix permanent research. 3.1 Gibbs Sampling Gibbs and block-Gibbs sampling can be applied straightforwardly to sample from distributions deﬁned by Equation 1. To do that we start with some initial assignment π and consider a subset of items in U; for illustration purposes we will use two items ui and uj. Given the selected subset of items the Gibbs sampler considers all possible assignment swaps within this subset. In our example there are only two possibilities: leave π unchanged or swap π(i) with π(j) to produce a new permutation π′. Conditioned on the assignment of all the other items in U that were not selected, the probability of each permutation is: p(π′\|π\{i,j}) = exp(−E(π′, θ)) exp(−E(π, θ)) + exp(−E(π′, θ)) p(π\|π\{i,j}) = 1 −p(π′\|π\{i,j}) where π\{i,j} is permutation π with ui and uj removed. We sample using these probabilities to either stay at π or move to π′, and repeat the process. Gibbs sampling has been applied to a wide range of energy-based probabilistic models. It is often found to mix very slowly and to get trapped in local modes [22]. The main reason for this is that the path from one probable assignment to another using only pairwise swaps is likely to go through regions that have very low probability [5]. This makes it very unlikely that those moves will be accepted, which typically traps the sampler in one mode. Thus, the local structure of the Gibbs sampler is likely to be inadequate for problems of the type considered here, in which several probable assignments will produce well-separated modes. 3.2 Chain-Based Approaches Chain-based methods extend the assignment swap idea behind the Gibbs sampler to generate samples more efﬁciently from WBMP distributions. Instead of randomly choosing subsets of items to swap, chain-based method generate a sequence (chain) of interdependent swaps. Given a (random) starting permutation π, an item ui (currently matched with vπ(i)) is selected at random and a new match vj is proposed with probability p(ui, vj\|θ) where p depends on the unary potential φ(ui, vj, θ) in the WBMP energy (see Equation 4). Now, assuming that the match {ui, vj}, is selected, matches {ui, vπ(i)} and {uπ−1(j), vj} are removed from π and {ui, vj} is added to make π′. After this change uπ−1(j) and vπ(i) are no longer matched to any item so π′ is a partial assignment. The procedure then ﬁnds a new match for uπ−1(j) using p. This chain-like match sampling is repeated either until π′ is a complete assignment or a termination criteria is reached. Several chain-based methods have been proposed including the chain ﬂipping approach [5] and the Markov Chain approach [11]. Dellaert et al., [5] empirically demonstrated that the chain ﬂipping sampler can mix better than the Gibbs sampler when applied to multimodal distributions. However, chain-based methods also have several drawbacks that signiﬁcantly affect their performance. First, unlike the Gibbs sampler which always maintains a valid assignment, the intermediate assignments π′ in chain-based methods are incomplete. This means that the chain either has to be run until a valid assignment is generated [5] or terminated early and produce an incomplete assignment [11]. In the ﬁrst case the sampler has a non-deterministic run-time whereas in the second case the incomplete assignment can not be taken as a valid sample from the model. Finally, to the best of our knowledge no chain-based method can be applied to general BMPs because they are speciﬁcally designed for Eunary (see Equation 4). 3 (a) t = 0 (b) t = 1 (c) t = 2 (d) t = 3 Figure 1: Top row: Plackett-Luce generative process viewed as rank matching. Bottom row: sequential matching procedure. Items are U = {u1, u2, u3} and V = {v1, v2, v3}; the reference permutation is σ = {2, 3, 1}. The proposed matches are shown in red dotted arrows and accepted matches in black arrows. 3.3 Recursive Partitioning Algorithm The recursive partitioning [10] algorithm was developed to obtain exact samples from the distribution for WBMP. This method is considered to be the state-of-the-art in matrix permanent research and to the beset of our knowledge has the lowest expected run time. Recursive partitioning proceeds by splitting the space of all valid assignments Ωinto K subsets Ω1, ..., ΩK with corresponding partition functions Z1, ..., ZK. It then samples one of these subsets and repeats the partitioning procedure recursively, generating exact samples from a WBMP distribution. Despite strong theoretical guarantees the recursive partitioning procedure has a number of limitations that signiﬁcantly affect its applicability. First, the running time of the sampler is nondeterministic as the algorithm has to be restarted every time a sample falls outside of Ω. The probability of restart increases with N which is an undesirable property especially for training large models where one typically needs to have precise control over the time spent in each training phase. Moreover, this algorithm is also speciﬁc to WBMP and cannot be generalized to sample from arbitrary BMP distributions with higher order potentials. 3.4 Plackett-Luce Model Our proposed sampling approach is based on a generalization of the well-established Plackett-Luce model [18, 14], which is a generative model for permutations. Given a set of items V = {v1, ..., vN}, a Plackett-Luce model is parametrized by a set of weights (one per item) W = {w1, ..., wN}. Under this model a permutation π is generated by ﬁrst selecting item vπ(1) from the set of N items and placing it in the ﬁrst position, then selecting vπ(2) from the remaining N −1 items and placing it the second position, and so on until all N items are placed. The probability of π under this model is given by: Q(π) = exp(wπ(1)) PN i=1 exp(wπ(i)) × exp(wπ(2)) PN i=2 exp(wπ(i)) × ... × exp(wπ(N)) exp(wπ(N)) (5) Here exp(wπ(t)) exp(PN i=t wπ(i)) is the probability of choosing the item vπ(t) out of the N −t + 1 remaining items. It can be shown that Q is a valid distribution with P π Q(π) = 1. Moreover, note that it is very easy to draw samples from Q by applying the sequential procedure described above. In the next section we show how this model can be generalized to draw samples from any BMP distribution. 4 Sampling by Sequentially Matching Vertices In this section we introduce a class of proposal distributions that can be effectively used in conjunction with the Metropolis-Hastings algorithm to obtain samples from a BMP distribution. Our approach is based on the observation that the sequential procedure behind the Plackett-Luce model can also be extended to generate matches between item sets. Instead of placing items into ranked positions we can think of the Plackett-Luce generative process as sequentially matching ranks to the items in V , as illustrated in the top row of Figure 1. To generate the permutation π = {3, 1, 2} the Plackett-Luce model ﬁrst matches rank 1 with vπ(1) = v2 then rank 2 with vπ(2) = v3 and ﬁnally rank 3 with vπ(3) = v1. Taking this one step further we can replace ranks with a general item set 4 U and repeat the same process. Unlike ranks, items in U do not have a natural order so we use a reference permutation σ, which speciﬁes the order in which items in U are matched. We refer to this procedure as sequential matching. The bottom row of Figure 1 illustrates this process. Formally the sequential matching process proceeds as follows: given some reference permutation σ, we start with an empty assignment π1:0 = ∅. Then at each iteration t = 1, ..., N the corresponding item uσ(t) gets matched with one of the V \ π1:t−1 items, where V \ π1:t−1 = {vjt, ..., vjN } denotes the set of items not matched in π1:t−1. Note that similarly to the Plackett-Luce model, \|V \ π1:t−1\| = N −t + 1 so at each iteration, uσ(t) will have N −t + 1 left over items in V \ π1:t−1 to match with. We deﬁne the conditional probability of each such match to be p(vj\|uσ(t), π1:t−1), P vj∈V \π1:t−1 p(vj\|uσ(t), π1:t−1) = 1. After N iterations the permutation π1:N = π is produced with probability: Q(π\|σ) = N Y t=1 p(vπ(σ(t))\|uσ(t), π1:t−1) (6) where vπ(σ(t)) is a match for uσ(t) in π. The conditional match probabilities depend on both the current item uσ(t) and on the partial assignment π1:t−1. Introducing this dependency generalizes the Plackett-Luce model which only takes into account that the items in π1:t−1 are already matched but does not take into account how these items are matched. This dependency becomes very important when the energy contains pairwise and/or higher order potentials as it allows us to compute the change in energy for each new match, in turn allowing for close approximations to the target BMP distribution. We can show that the distribution Q deﬁned by the p’s is a valid distribution over assignments: Proposition 1 For any reference permutation σ and any choice of matching probabilities that satisfy P vj∈V \π1:t−1 p(vj\|uσ(t), π1:t−1) = 1, the distribution given by: Q(π\|σ) = QN t=1 p(vπ(σ(t))\|uσ(t), π1:t−1) is a valid probability distribution over assignments.1 The important consequence of this proposition is that it allows us to work with a very rich class of matching probabilities with arbitrary dependencies and still obtain a valid distribution over assignments with a simple way to generate exact samples from it. This opens many avenues for tailoring proposal distributions for MCMC applications to speciﬁc BMPs. In the next section we propose one such approach. 4.1 Proposal Distribution Given the general matching probabilities the goal is to deﬁne them so that the resulting proposal distribution Q matches the target distribution as closely as possible. One potential way of achieving this is through the partial energy E(π1:t, θ) (see Section 2). The partial energy ignores all the items that are not matched in π1:t and thus provides an estimate of the ”current” energy at each iteration t. Using partial energies we can also ﬁnd the changes in energy when a given item is matched. Given that our goal is to explore low-energy (high-probability) modes we deﬁne the matching probabilities as: p(vj\|uσ(t), π1:t−1) = exp(−E(H(vj, uσ(t), π1:t−1), θ)) Zt(uσ(t), π1:t−1) Zt(uσ(t), π1:t−1) = X vj∈V \π1:t−1 exp(−E(H(vj, uσ(t), π1:t−1), θ)) (7) where H(vj, uσ(t), π1:t−1) is the resulting partial assignment after match {uσ(t), vj} is added to π1:t−1. The normalizing constant Zt ensures that the probabilities sum to 1, which is the necessary condition for Proposition 1 to apply. It is useful to rewrite the matching probabilities as: p(vj\|uσ(t), π1:t−1) = exp(−E(H(vj, uσ(t), π1:t−1), θ) + E(π1:t−1, θ)) Z∗ t (uσ(t), π1:t−1) Z∗ t (uσ(t), π1:t−1) = X vj∈V \π1:t−1 exp(−E(H(vj, uσ(t), π1:t−1), θ) + E(π1:t−1, θ)) Adding E(π1:t−1, θ) to each item’s energy does not change the probabilities because this term cancels out during normalization (but it does change the partition function, denoted by Z∗ t here). However, in this form we see that p(vj\|uσ(t), π1:t−1) is directly related to the change in the partial energy 1The proof is in the supplementary material. 5 from π1:t−1 to H(vj, uσ(t), π1:t−1) – the larger the change the bigger the resulting probability will be. Thus, the matching choices will be made solely based on the changes in the partial energy. Reorganizing the terms yields the proposal distribution: Q(π\|σ) = exp(−E(π1:1, θ) + E(π1:0, θ)) Z∗ 1(uσ(1), π1:0) × ... × exp(−E(π1:N, θ) + E(π1:N−1, θ)) Z∗ N(uσ(N), π1:N−1) = exp(−E(π, θ)) Z∗(π, σ) Here Z∗(π, σ) is the normalization factor which depends both on the reference permutation σ and the generated assignment π. The resulting proposal distribution is essentially a renormalized version of the target distribution. The numerator remains the exponent of the energy but the denominator is no longer a constant; rather it is a function which depends on the generated assignment and the reference permutation. Note that the proposal distribution deﬁned above can be used to generate samples for any target distribution with arbitrary energy consisting of single and/or higher order potentials. To the best of our knowledge aside from the Gibbs sampler this is the only sampling procedure that can be applied to arbitrary BMP distributions. 4.2 Temperature and Chain Properties Acceptance rate, a key property of any sampler, is typically controlled by a parameter which either shrinks or expands the proposal distribution. To achieve this effect with the sequential matching model we introduce an additional parameter ρ which we refer to as temperature: p(vj\|uσ(t), π1:t−1, ρ) ∝exp(−E(H(vj, uσ(t), π1:t−1), θ)/ρ). Decreasing ρ leads to sharp proposal distributions typically highly skewed towards one speciﬁc assignment, while increasing ρ makes the proposal distribution approach the uniform distribution. By adjusting ρ we can control the range of the proposed moves therefore controlling the acceptance rate. To ensure that the SM sampler converges to the required distribution we demonstrate that it satisﬁes the three requisite properties: detailed balance, ergodicity, and aperiodicity [15]. The detailed balance condition is satisﬁed because every Metropolis-Hastings algorithm satisﬁes detailed balance. Ergodicity follows from the fact that the insertion probabilities are always strictly greater than 0. Therefore any π is reachable from any σ in one proposal cycle. Finally, aperiodicity follows from the fact that the chain allows self-transitions. 4.3 Reference Permutation Algorithm 1 Sequential Matching (SM) Input: σ, M, ρ for m = 1 to M do Initialize π1:0 = ∅ for t = 1 to N do {generate sample from Q(·\|σ)} Find a match vj for uσ(t) using: p(vj\|uσ(t), π1:t−1, ρ) Add {uσ(t), vj} to π1:t−1 to get π1:t end for Calculate forward probability: Q(π\|σ) = QN t=1 p(vπ(σ(t))\|uσ(t), π1:t−1, ρ) Calculate backward probability: Q(σ\|π) = QN t=1 p(vσ(π(t))\|uπ(t), σ1:t−1, ρ) if Uniform(0, 1) < exp(−E(π,θ))Q(σ\|π) exp(−E(σ,θ))Q(π\|σ) then σ ←π end if end for Return: σ Fixing the reference permutation σ yields a state independent sampler. Empirically we found that setting σ to the MAP permutation gives good performance for WBMP problems. However, for the general energy based distributions considered here ﬁnding the MAP state can be very expensive and in many cases intractable. Moreover, even if MAP can be found efﬁciently there is still no guarantee that using it as the reference permutation will lead to a good sampler. To avoid these problems we use a state dependent sampler where the reference permutation σ is updated every time a sample gets accepted. In the matching example (bottom row of Figure 1) if the new match at t = 3 is accepted then σ would be updated to {3, 1, 2}. Empirically we found the state dependent sampler to be more stable, with consistent performance across different random initializations of the reference permutation. Algorithm 1 summarizes the Metropolis-Hastings procedure for the state dependent sequential matching sampler. 5 Experiments To test the sequential matching sampling approach we conducted extensive experiments. We considered document ranking and image matching, two popular applications of BMP; and for the sake of 6 Table 1: Average Hellinger distances for learning to rank (left half) and image matching (right half) problems. Statistically signiﬁcant results are underlined. Note that Hellinger distances for N = 8 are not directly comparable to those for N = 25, 50 since approximate normalization is used for N > 8. For N = 50 we were unable to get a single sample from the RP sampler for any c in the allocated time limit (over 5 minutes). Learning to Rank Image Matching c = 20 c = 40 c = 60 c = 80 c = 100 c = 0.2 c = 0.4 c = 0.6 c = 0.8 c = 1 N = 8: GB 0.7948 0.6211 0.4635 0.4218 0.3737 0.9108 0.8868 0.8320 0.7616 0.6533 CF 0.9012 0.8987 0.8887 0.8714 0.8748 0.9112 0.8882 0.8336 0.7672 0.6623 RP 0.7945 0.6209 0.4629 0.4986 0.3734 0.9110 0.8870 0.8312 0.7623 0.6548 SM 0.7902 0.6188 0.4636 0.4474 0.3725 0.9109 0.8866 0.8307 0.7621 0.6557 N = 25: GB 0.9533 0.9728 0.9646 0.9449 0.9486 0.7246 0.8669 0.9902 0.9960 0.9976 CF 0.9767 0.9990 0.9937 0.9953 0.9781 0.7243 0.8675 0.9904 0.9950 0.9807 RP 0.9533 0.9728 0.9694 0.9462 0.9673 0.7279 0.9788 0.9896 0.9988 0.9969 SM 0.1970 0.1937 0.2899 0.4166 0.3858 0.7234 0.8471 0.8472 0.6350 0.5576 N = 50: GB 0.9983 0.9991 0.9988 0.9974 0.9985 0.6949 0.9646 1.0000 1.0000 1.0000 CF 0.9841 0.9995 0.9993 0.9906 0.9305 0.6960 0.9635 1.0000 1.0000 0.9992 SM 0.1617 0.2335 0.3462 0.4931 0.4895 0.6941 0.9243 0.7016 0.3550 0.1677 comparison we concentrated on WBMP, as most of the methods cannot be applied to general BMP problems. When comparing the samplers we concentrated on evaluating how well the Monte Carlo estimates of probabilities produced by the samplers approximate the true distribution P. When target probabilities are known this method of evaluation provides a good estimate of performance since the ultimate goal of any sampler is to approximate P as closely as possible. For all experiments the Hellinger distance was used to compare the true distributions with the approximations produced by samplers. We chose this metric because it is symmetric and bounded. Furthermore it avoids the log(0) problems that arise in cross entropy measures. For any two distributions P and Q the Hellinger distance is given by D = (1 −(P π P(π)Q(π))1/2)1/2. Note that 0 ≤D ≤1 where 0 indicates that P = Q. Computing D exactly quickly becomes intractable as the number of items grows. To overcome this problem we note that if a given permutation π is not generated by any of the samplers then the term P(π)Q(π) is 0 and does not affect the resulting estimate of D for any sampler. Hence we can locally approximate D up to a constant for all samplers by changing Equation 1 to: P(π\|θ) ≈ exp(−E(π,θ)) P π′∈Ω∗exp(−E(π′,θ), where Ω∗is the union of all distinct permutations produced by the samplers. The Hellinger distance is then estimated with respect to Ω∗. For all experiments we ran the samplers on small (N = 8), medium (N = 25) and large (N = 50) scale problems. The sampling chains for each method were run in parallel using 4 cores; the use of multiprocessor boards such as GPUs allows our method to scale to large problems. We compare the SM approach with Gibbs (GB), chain ﬂipping (CF) and recursive partitioning (RP) samplers. To run RP we used the code available from the author’s webpage. These methods cover all of the primary leading approaches in WBMP and matrix permanent research. Since any valid sampler will eventually produce samples from the target distribution, we tested the methods with short chain lengths. This regime also simulates real applications of the methods where, due to computational time limits, the user is typically unable to run long chains. Note that this is especially relevant if the distributions are being sampled as an inner loop during parameter optimization. Furthermore, to make comparisons fair we used the block GB sampler with the block size of 7 (the largest computationally feasible size) as the reference point. We used 2N swaps for each GB chain, setting the number of iterations for other methods to match the total run time for GB (for all experiments the difference in running times between GB and SM did not exceed 10%). The run times of the CF and RP methods are difﬁcult to control as they are non-deterministic. To deal with this we set an upper-bound on the running time (consistent with the other methods) after which CF and RP were terminated. Finally, the temperature for SM was chosen in the [0.1, 1] interval to keep the acceptance rate approximately between 20% and 60%. 5.1 Learning to Rank For a learning to rank problem we used the Yahoo! Learning To Rank dataset [4]. For each query the distribution over assignments was parametrized by the energy given in Equation 2. Here θi 7 is the output of the neural network scoring function trained on query-document features. After pretraining the network on the full dataset we randomly selected 50 queries with N = 8, 25, 50 documents and used GB, CF, RP and SM methods to generate 1000 samples for each query. To gain insight into sampling accuracy we experimented with different distribution shapes by introducing an additional scaling constant c so that P(π\|θ, c) ∝exp(−c × E(π, θ)). In this form c controls the ”peakiness” of the distribution with large values resulting in highly peaked distributions; we used c ∈{20, 40, 60, 80, 100}. The left half of Table 1 shows Hellinger distances for N = 8, 25, 50, averaged across the 50 queries.2 From the table it is seen that all the samplers perform equally well when the number of items is small (N = 8). However, as the number of items increases SM signiﬁcantly outperforms all other samplers. Throughout experiments we found that the CF and RP samplers often reached the allocated time limit and had to be forced to terminate early. For N = 50 we were unable to get a single sample from the RP sampler after running it for over 5 minutes. This is likely due to the fact that at each matching step t = 1, ..., N the RP sampler has a non-zero probability of failing (rejecting). Consequently the total rejection probability increases linearly with the number of items N. Even for N = 25 we found the RP sampler to reject over 95% of the time. This further suggests that approaches with non-deterministic run times are not suitable for this problem because their worst-case performance can be extremely slow. Overall, the results indicate that SM can produce higher quality samples more rapidly, a crucial property for learning large-scale models. 5.2 Image Matching For an image matching task we followed the framework of Petterson et al. [17]. Here, we used the Giraffe dataset [21] which is a video sequence of a walking giraffe. From this data we randomly selected 50 pairs of images that were at least 20 frames apart. Using the available set of 61 hand labeled points we then randomly selected three sets of correspondence points for each image pair, containing 8, 25 and 50 points respectively, and extracted SIFT feature descriptors at each point. The target distribution over matchings was parametrized by the energy given by Equation 3 where ψ’s are the SIFT feature descriptors. We also experimented with different scale settings: c ∈{0.2, 0.4, 0.6, 0.8, 1}. Figure 2 shows an example pair of images with 25 labeled points and the inferred MAP assignment. Figure 2: Example image pair with N = 25. Green lines show the inferred MAP assignment. The results for N = 8, 25, 50 are shown in in the right half of Table 1. We see that when the distributions are relatively ﬂat (c < 0.6) all samplers have comparable performance. However, as the distributions become sharper with several well deﬁned modes (c ≥0.6), the SM sampler signiﬁcantly outperforms all other samplers. As mentioned above, when the distribution has well deﬁned modes the path from one mode to the other using only local swaps is likely to go through low probability modes. This is the likely cause of the poor performance of the GB and CF samplers as both samplers propose new assignments through local moves. As in the learning to rank experiments, we found the rejection rate for the RP sampler to increase signiﬁcantly for N ≥25. We were unable to obtain any samples in the allocated time (over 5 mins) from the RP sampler for N = 50. Overall, the results further show that the SM method is able to generate higher quality samples faster than the other methods. 6 Conclusion In this paper we introduced a new sampling approach for bipartite matching problems based on a generalization of the Plackett-Luce model. In this approach the matching probabilities at each stage are conditioned on the partial assignment made to that point. This global dependency allows us to deﬁne a rich class of proposal distributions that accurately approximate the target distribution. Empirically we found that our method is able to generate good quality samples faster and is less prone to getting stuck in local modes. Future work involves applying the sampler during inference while learning BMP models. We also plan to investigate the relationship between the proposal distribution produced by sequential matching and the target one. 2Trace and Hellinger distance plots (for both experiments) are in the supplementary material. 8 References [1] A. Bouchard-Cote and M. I. Jordan. Variational inference over combinatorial spaces. In NIPS, 2010. [2] C. Cadena, D. Galvez-Lopez, F. Ramos, J. D. Tardos, and J. Neira. Robust place recognition with stereo cameras. In IROS, 2010. [3] T. S. Caetano, L. Cheng, Q. V. Le, and A. J. Smola. Learning graph matching. In ICML, 2009. [4] O. Chapelle, Y. Chang, and T.-Y. Liu. The Yahoo! Learning to Rank Challenge. 2010. [5] F. Dellaert, S. M. Seitz, C. E. Thorpe, and S. Thrun. EM, MCMC, and chain ﬂipping for structure from motion with unknown correspondence. Machine Learning, 50, 2003. [6] J.-P. Doignon, A. Pekec, and M. Regenwetter. The repeated insertion model for rankings: Missing link between two subset choice models. Psychometrika, 69, 2004. [7] C. Dwork, R. Kumar, M. Naor, and D. Sivakumar. Rank aggregation methods for the web. In WWW, 2001. [8] B. Huang and T. Jebara. Loopy belief propagation for bipartite maximum weight b-matching. In AISTATS, 2007. [9] J. Huang, C. Guestrin, and L. Guibas. Fourier theoretic probabilistic inference over permutations. Machine Learning Research, 10, 2009. [10] M. Huber and J. Law. Fast approximation of the permanent for very dense problems. In SODA, 2008. [11] M. Jerrum, A. Sinclair, and E. Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. 2004. [12] Q. V. Le and A. Smola. Direct optimization of ranking measures. In arxiv: 0704.3359, 2007. [13] T. Lu and C. Boutilier. Learning Mallows models with pairwise preferences. In ICML, 2011. [14] R. D. Luce. Individual choice behavior: A theoretical analysis. Wiley, 1959. [15] R. M. Neal. Probabilistic inference using Markov Chain Monte Carlo methods. Technical report, University of Toronto, 1993. [16] C. H. Papadimitriou and K. Steiglitz. Combinatorial optimization: Algorithms and complexity. Prentice-Hall, 1982. [17] J. Petterson, T. S. Caetano, J. J. McAuley, and J. Yu. Exponential family graph matching and ranking. In NIPS, 2009. [18] R. Plackett. The analysis of permutations. Applied Statistics, 24, 1975. [19] T. Qin, T.-Y. Liu, X.-D. Zhang, D.-S. Wang, and H. Li. Global ranking using continuous conditional random ﬁelds. In NIPS, 2008. [20] F. Ramos, D. Fox, and H. Durrant-Whyte. CRF-Matching: Conditional random ﬁelds for feature-based scan matching. In Robotics: Science and Systems, 2007. [21] D. A. Ross, D. Tarlow, and R. S. Zemel. Learning articulated structure and motion. International Journal on Computer Vision, 88, 2010. [22] R. Salakhutdinov. Learning deep Boltzmann machines using adaptive MCMC. In ICML, 2010. [23] M. Taylor, J. Guiver, S. Robertson, and T. Minka. Softrank: Optimizing non-smooth rank metrics. In WSDM, 2008. [24] W. R. Taylor. Protein structure comparison using bipartite graph matching and its application to protein structure classiﬁcation. In Molecular Cell Proteomics, 2002. [25] L. G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8, 1979. [26] M. N. Volkovs and R. S. Zemel. Boltzrank: Learning to maximize expected ranking gain. In ICML, 2009. [27] Y. Wang, F. Makedon, and J. Ford. A bipartite graph matching framework for ﬁnding correspondences between structural elements in two proteins. In IEBMS, 2004. 9	2012	119
4,474	Factoring nonnegative matrices with linear programs Victor Bittorf bittorf@cs.wisc.edu Benjamin Recht brecht@cs.wisc.edu Computer Sciences University of Wisconsin Christopher R´e chrisre@cs.wisc.edu Joel A. Tropp Computing and Mathematical Sciences California Institute of Technology tropp@cms.caltech.edu Abstract This paper describes a new approach, based on linear programming, for computing nonnegative matrix factorizations (NMFs). The key idea is a data-driven model for the factorization where the most salient features in the data are used to express the remaining features. More precisely, given a data matrix X, the algorithm identiﬁes a matrix C that satisﬁes X ≈CX and some linear constraints. The constraints are chosen to ensure that the matrix C selects features; these features can then be used to ﬁnd a low-rank NMF of X. A theoretical analysis demonstrates that this approach has guarantees similar to those of the recent NMF algorithm of Arora et al. (2012). In contrast with this earlier work, the proposed method extends to more general noise models and leads to efﬁcient, scalable algorithms. Experiments with synthetic and real datasets provide evidence that the new approach is also superior in practice. An optimized C++ implementation can factor a multigigabyte matrix in a matter of minutes. 1 Introduction Nonnegative matrix factorization (NMF) is a popular approach for selecting features in data [16–18, 23]. Many machine-learning and data-mining software packages (including Matlab [3], R [12], and Oracle Data Mining [1]) now include heuristic computational methods for NMF. Nevertheless, we still have limited theoretical understanding of when these heuristics are correct. The difﬁculty in developing rigorous methods for NMF stems from the fact that the problem is computationally challenging. Indeed, Vavasis has shown that NMF is NP-Hard [27]; see [4] for further worst-case hardness results. As a consequence, we must instate additional assumptions on the data if we hope to compute nonnegative matrix factorizations in practice. In this spirit, Arora, Ge, Kannan, and Moitra (AGKM) have exhibited a polynomial-time algorithm for NMF that is provably correct—provided that the data is drawn from an appropriate model, based on ideas from [8]. The AGKM result describes one circumstance where we can be sure that NMF algorithms are capable of producing meaningful answers. This work has the potential to make an impact in machine learning because proper feature selection is an important preprocessing step for many other techniques. Even so, the actual impact is damped by the fact that the AGKM algorithm is too computationally expensive for large-scale problems and is not tolerant to departures from the modeling assumptions. Thus, for NMF, there remains a gap between the theoretical exercise and the actual practice of machine learning. 1 The present work presents a scalable, robust algorithm that can successfully solve the NMF problem under appropriate hypotheses. Our ﬁrst contribution is a new formulation of the nonnegative feature selection problem that only requires the solution of a single linear program. Second, we provide a theoretical analysis of this algorithm. This argument shows that our method succeeds under the same modeling assumptions as the AGKM algorithm with an additional margin constraint that is common in machine learning. We prove that if there exists a unique, well-deﬁned model, then we can recover this model accurately; our error bound improves substantially on the error bound for the AGKM algorithm in the high SNR regime. One may argue that NMF only “makes sense” (i.e., is well posed) when a unique solution exists, and so we believe our result has independent interest. Furthermore, our algorithm can be adapted for a wide class of noise models. In addition to these theoretical contributions, our work also includes a major algorithmic and experimental component. Our formulation of NMF allows us to exploit methods from operations research and database systems to design solvers that scale to extremely large datasets. We develop an efﬁcient stochastic gradient descent (SGD) algorithm that is (at least) two orders of magnitude faster than the approach of AGKM when both are implemented in Matlab. We describe a parallel implementation of our SGD algorithm that can robustly factor matrices with 105 features and 106 examples in a few minutes on a multicore workstation. Our formulation of NMF uses a data-driven modeling approach to simplify the factorization problem. More precisely, we search for a small collection of rows from the data matrix that can be used to express the other rows. This type of approach appears in a number of other factorization problems, including rank-revealing QR [15], interpolative decomposition [20], subspace clustering [10, 24], dictionary learning [11], and others. Our computational techniques can be adapted to address large-scale instances of these problems as well. 2 Separable Nonnegative Matrix Factorizations and Hott Topics Notation. For a matrix M and indices i and j, we write Mi· for the ith row of M and M·j for the jth column of M. We write Mij for the (i, j) entry. Let Y be a nonnegative f × n data matrix with columns indexing examples and rows indexing features. Exact NMF seeks a factorization Y = F W where the feature matrix F is f × r, where the weight matrix W is r × n, and both factors are nonnegative. Typically, r ≪min{f, n}. Unless stated otherwise, we assume that each row of the data matrix Y is normalized so it sums to one. Under this hypothesis, we may also assume that each row of F and of W also sums to one [4]. It is notoriously difﬁcult to solve the NMF problem. Vavasis showed that it is NP-complete to decide whether a matrix admits a rank-r nonnegative factorization [27]. AGKM proved that an exact NMF algorithm can be used to solve 3-SAT in subexponential time [4]. The literature contains some mathematical analysis of NMF that can be used to motivate algorithmic development. Thomas [25] developed a necessary and sufﬁcient condition for the existence of a rank-r NMF. More recently, Donoho and Stodden [8] obtained a related sufﬁcient condition for uniqueness. AGKM exhibited an algorithm that can produce a nonnegative matrix factorization under a weaker sufﬁcient condition. To state their results, we need a deﬁnition. Deﬁnition 2.1 A set of vectors {v1, . . . , vr} ⊂Rd is simplicial if no vector vi lies in the convex hull of {vj : j ̸= i}. The set of vectors is α-robust simplicial if, for each i, the ℓ1 distance from vi to the convex hull of {vj : j ̸= i} is at least α. Figure 1 illustrates these concepts. These ideas support the uniqueness results of Donoho and Stodden and the AGKM algorithm. Indeed, we can ﬁnd an NMF of Y efﬁciently if Y contains a set of r rows that is simplicial and whose convex hull contains the remaining rows. Deﬁnition 2.2 An NMF Y = F W is called separable if the rows of W are simplicial and there is a permutation matrix Π such that ΠF = Ir M . (1) 2 Algorithm 1: AGKM: Approximably Separable Nonnegative Matrix Factorization [4] 1: Initialize R = ∅. 2: Compute the f × f matrix D with Dij = ∥Xi· −Xj·∥1. 3: for k = 1, . . . f do 4: Find the set Nk of rows that are at least 5ϵ/α + 2ϵ away from Xk·. 5: Compute the distance δk of Xk· from conv({Xj· : j ∈Nk}). 6: if δk > 2ϵ, add k to the set R. 7: end for 8: Cluster the rows in R as follows: j and k are in the same cluster if Djk ≤10ϵ/α + 6ϵ. 9: Choose one element from each cluster to yield W . 10: F = arg minZ∈Rf×r ∥X −ZW ∥∞,1 2 1 3 2 1 3 d3 d2 d1 d1 Figure 1: Numbered circles are hott topics. Their convex hull (orange) contains the other topics (small circles), so the data admits a separable NMF. The arrow d1 marks the ℓ1 distance from hott topic (1) to the convex hull of the other two hott topics; deﬁnitions of d2 and d3 are similar. The hott topics are α-robustly simplicial when each di ≥α. To compute a separable factorization of Y , we must ﬁrst identify a simplicial set of rows from Y . Afterward, we compute weights that express the remaining rows as convex combinations of this distinguished set. We call the simplicial rows hott and the corresponding features hott topics. This model allows us to express all the features for a particular instance if we know the values of the instance at the simplicial rows. This assumption can be justiﬁed in a variety of applications. For example, in text, knowledge of a few keywords may be sufﬁcient to reconstruct counts of the other words in a document. In vision, localized features can be used to predict gestures. In audio data, a few bins of the spectrogram may allow us to reconstruct the remaining bins. While a nonnegative matrix one encounters in practice might not admit a separable factorization, it may be well-approximated by a nonnnegative matrix with separable factorization. AGKM derived an algorithm for nonnegative matrix factorization of a matrix that is well-approximated by a separable factorization. To state their result, we introduce a norm on f × n matrices: ∥∆∥∞,1 := max 1≤i≤f n X j=1 \|∆ij\| . Theorem 2.3 (AGKM [4]) Let ϵ and α be nonnegative constants satisfying ϵ ≤ α2 20+13α. Let X be a nonnegative data matrix. Assume X = Y + ∆where Y is a nonnegative matrix whose rows have unit ℓ1 norm, where Y = F W is a rank-r separable factorization in which the rows of W are α-robust simplicial, and where ∥∆∥∞,1 ≤ϵ. Then Algorithm 1 ﬁnds a rank-r nonnegative factorization ˆF ˆ W that satisﬁes the error bound X −ˆF ˆ W ∞,1 ≤10ϵ/α + 7ϵ. In particular, the AGKM algorithm computes the factorization exactly when ϵ = 0. Although this method is guaranteed to run in polynomial time, it has many undesirable features. First, the algorithm requires a priori knowledge of the parameters α and ϵ. It may be possible to calculate ϵ, but we can only estimate α if we know which rows are hott. Second, the algorithm computes all ℓ1 distances between rows at a cost of O(f 2n). Third, for every row in the matrix, we must determine its distance to the convex hull of the rows that lie at a sufﬁcient distance; this step requires us to solve a linear program for each row of the matrix at a cost of Ω(fn). Finally, this method is intimately linked to the choice of the error norm ∥·∥∞,1. It is not obvious how to adapt the algorithm for other noise models. We present a new approach, based on linear programming, that overcomes these drawbacks. 3 Main Theoretical Results: NMF by Linear Programming This paper shows that we can factor an approximately separable nonnegative matrix by solving a linear program. A major advantage of this formulation is that it scales to very large data sets. 3 Algorithm 2 Separable Nonnegative Matrix Factorization by Linear Programming Require: An f × n nonnegative matrix Y with a rank-r separable NMF. Ensure: An f × r matrix F and r × n matrix W with F ≥0, W ≥0, and Y = F W . 1: Find the unique C ∈Φ(Y ) to minimize pT diag(C) where p is any vector with distinct values. 2: Let I = {i : Cii = 1} and set W = YI· and F = C·I. Here is the key observation: Suppose that Y is any f × n nonnegative matrix that admits a rank-r separable factorization Y = F W . If we pad F with zeros to form an f × f matrix, we have Y = ΠT Ir 0 M 0 ΠY =: CY We call the matrix C factorization localizing. Note that any factorization localizing matrix C is an element of the polyhedral set Φ(Y ) := {C ≥0 : CY = Y , Tr(C) = r, Cjj ≤1 ∀j, Cij ≤Cjj ∀i, j}. Thus, to ﬁnd an exact NMF of Y , it sufﬁces to ﬁnd a feasible element of C ∈Φ(Y ) whose diagonal is integral. This task can be accomplished by linear programming. Once we have such a C, we construct W by extracting the rows of X that correspond to the indices i where Cii = 1. We construct the feature matrix F by extracting the nonzero columns of C. This approach is summarized in Algorithm 2. In turn, we can prove the following result. Theorem 3.1 Suppose Y is a nonnegative matrix with a rank-r separable factorization Y = F W . Then Algorithm 2 constructs a rank-r nonnegative matrix factorization of Y . As the theorem suggests, we can isolate the rows of Y that yield a simplicial factorization by solving a single linear program. The factor F can be found by extracting columns of C. 3.1 Robustness to Noise Suppose we observe a nonnegative matrix X whose rows sum to one. Assume that X = Y + ∆ where Y is a nonnegative matrix whose rows sum to one, which has a rank-r separable factorization Y = F W such that the rows of W are α-robust simplicial, and where ∥∆∥∞,1 ≤ϵ. Deﬁne the polyhedral set Φτ(X) := n C ≥0 : ∥CX −X∥∞,1 ≤τ, Tr(C) = r, Cjj ≤1 ∀j, Cij ≤Cjj ∀i, j o The set Φ(X) consists of matrices C that approximately locate a factorization of X. We can prove the following result. Theorem 3.2 Suppose that X satisﬁes the assumptions stated in the previous paragraph. Furthermore, assume that for every row Yj,· that is not hott, we have the margin constraint ∥Yj,·−Yi,·∥≥d0 for all hott rows i. Then we can ﬁnd a nonnegative factorization satisfying X −ˆF ˆ W ∞,1 ≤2ϵ provided that ϵ < min{αd0,α2} 9(r+1) . Furthermore, this factorization correctly identiﬁes the hott topics appearing in the separable factorization of Y . Algorithm 3 requires the solution of two linear programs. The ﬁrst minimizes a cost vector over Φ2ϵ(X). This lets us ﬁnd ˆ W . Afterward, the matrix ˆF can be found by setting ˆF = arg min Z≥0 X −Z ˆ W ∞,1 . (2) Our robustness result requires a margin-type constraint assuming that the original conﬁguration consists either of duplicate hott topics, or topics that are reasonably far away from the hott topics. On the other hand, under such a margin constraint, we can construct a considerably better approximation that guaranteed by the AGKM algorithm. Moreover, unlike AGKM, our algorithm does not need to know the parameter α. 4 Algorithm 3 Approximably Separable Nonnegative Matrix Factorization by Linear Programming Require: An f × n nonnegative matrix X that satisﬁes the hypotheses of Theorem 3.2. Ensure: An f ×r matrix F and r ×n matrix W with F ≥0, W ≥0, and ∥X −F W ∥∞,1 ≤2ϵ. 1: Find C ∈Φ2ϵ(X) that minimizes pT diag C where p is any vector with distinct values. 2: Let I = {i : Cii = 1} and set W = XI·. 3: Set F = arg minZ∈Rf×r ∥X −ZW ∥∞,1 The proofs of Theorems 3.1 and 3.2 can be found in the b version of this paper [6]. The main idea is to show that we can only represent a hott topic efﬁciently using the hott topic itself. Some earlier versions of this paper contained incomplete arguments, which we have remedied. For a signifcantly stronger robustness analysis of Algorithm 3, see the recent paper [13]. Having established these theoretical guarantees, it now remains to develop an algorithm to solve the LP. Off-the-shelf LP solvers may sufﬁce for moderate-size problems, but for large-scale matrix factorization problems, their running time is prohibitive, as we show in Section 5. In Section 4, we turn to describe how to solve Algorithm 3 efﬁciently for large data sets. 3.2 Related Work Localizing factorizations via column or row subset selection is a popular alternative to direct factorization methods such as the SVD. Interpolative decomposition such as Rank-Revealing QR [15] and CUR [20] have favorable efﬁciency properties as compared to factorizations (such as SVD) that are not based on exemplars. Factorization localization has been used in subspace clustering and has been shown to be robust to outliers [10,24]. In recent work on dictionary learning, Esser et al. and Elhamifar et al. have proposed a factorization localization solution to nonnegative matrix factorization using group sparsity techniques [9, 11]. Esser et al. prove asymptotic exact recovery in a restricted noise model, but this result requires preprocessing to remove duplicate or near-duplicate rows. Elhamifar shows exact representative recovery in the noiseless setting assuming no hott topics are duplicated. Our work here improves upon this work in several aspects, enabling ﬁnite sample error bounds, the elimination of any need to preprocess the data, and algorithmic implementations that scale to very large data sets. 4 Incremental Gradient Algorithms for NMF The rudiments of our fast implementation rely on two standard optimization techniques: dual decomposition and incremental gradient descent. Both techniques are described in depth in Chapters 3.4 and 7.8 of Bertsekas and Tstisklis [5]. We aim to minimize pT diag(C) subject to C ∈Φτ(X). To proceed, form the Lagrangian L(C, β, w) = pT diag(C) + β(Tr(C) −r) + f X i=1 wi (∥Xi· −[CX]i·∥1 −τ) with multipliers β and w ≥0. Note that we do not dualize out all of the constraints. The remaining ones appear in the constraint set Φ0 = {C : C ≥0, diag(C) ≤1, and Cij ≤Cjj for all i, j}. Dual subgradient ascent solves this problem by alternating between minimizing the Lagrangian over the constraint set Φ0, and then taking a subgradient step with respect to the dual variables wi ←wi + s (∥Xi· −[C⋆X]i·∥1 −τ) and β ←β + s(Tr(C⋆) −r) where C⋆is the minimizer of the Lagrangian over Φ0. The update of wi makes very little difference in the solution quality, so we typically only update β. We minimize the Lagrangian using projected incremental gradient descent. Note that we can rewrite the Lagrangian as L(C, β, w) = −τ1T w −βr + n X k=1   X j∈supp(X·k) wj∥Xjk −[CX]jk∥1 + µj(pj + β)Cjj  . 5 Algorithm 4 HOTTOPIXX: Approximate Separable NMF by Incremental Gradient Descent Require: An f × n nonnegative matrix X. Primal and dual stepsizes sp and sd. Ensure: An f ×r matrix F and r ×n matrix W with F ≥0, W ≥0, and ∥X −F W ∥∞,1 ≤2ϵ. 1: Pick a cost p with distinct entries. 2: Initialize C = 0, β = 0 3: for t = 1, . . . , Nepochs do 4: for i = 1, . . . n do 5: Choose k uniformly at random from [n]. 6: C ←C + sp · sign(X·k −CX·k)XT ·k −sp diag(µ ◦(β1 −p)). 7: end for 8: Project C onto Φ0. 9: β ←β + sd(Tr(C) −r) 10: end for 11: Let I = {i : Cii = 1} and set W = XI·. 12: Set F = arg minZ∈Rf×r ∥X −ZW ∥∞,1 Here, supp(x) is the set indexing the entries where x is nonzero, and µj is the number of nonzeros in row j divided by n. The incremental gradient method chooses one of the n summands at random and follows its subgradient. We then project the iterate onto the constraint set Φ0. The projection onto Φ0 can be performed in the time required to sort the individual columns of C plus a linear-time operation. The full procedure is described in the extended version of this paper [6]. In the case where we expect a unique solution, we can drop the constraint Cij ≤Cjj, resulting in a simple clipping procedure: set all negative items to zero and set any diagonal entry exceeding one to one. In practice, we perform a tradeoff. Since the constraint Cij ≤Cjj is used solely for symmetry breaking, we have found empirically that we only need to project onto Φ0 every n iterations or so. This incremental iteration is repeated n times in a phase called an epoch. After each epoch, we update the dual variables and quit after we believe we have identiﬁed the large elements of the diagonal of C. Just as before, once we have identiﬁed the hott rows, we can form W by selecting these rows of X. We can ﬁnd F just as before, by solving (2). Note that this minimization can also be computed by incremental subgradient descent. The full procedure, called HOTTOPIXX, is described in Algorithm 4. 4.1 Sparsity and Computational Enhancements for Large Scale. For small-scale problems, HOTTOPIXX can be implemented in a few lines of Matlab code. But for the very large data sets studied in Section 5, we take advantage of natural parallelism and a host of low-level optimizations that are also enabled by our formulation. As in any numerical program, memory layout and cache behavior can be critical factors for performance. We use standard techniques: in-memory clustering to increase prefetching opportunities, padded data structures for better cache alignment, and compiler directives to allow the Intel compiler to apply vectorization. Note that the incremental gradient step (step 6 in Algorithm 4) only modiﬁes the entries of C where X·k is nonzero. Thus, we can parallelize the algorithm with respect to updating either the rows or the columns of C. We store X in large contiguous blocks of memory to encourage hardware prefetching. In contrast, we choose a dense representation of our localizing matrix C; this choice trades space for runtime performance. Each worker thread is assigned a number of rows of C so that all rows ﬁt in the shared L3 cache. Then, each worker thread repeatedly scans X while marking updates to multiple rows of C. We repeat this process until all rows of C are scanned, similar to the classical block-nested loop join in relational databases [22]. 5 Experiments Except for the speedup curves, all of the experiments were run on an identical conﬁguration: a dual Xeon X650 (6 cores each) machine with 128GB of RAM. The kernel is Linux 2.6.32-131. 6 0 20 40 60 0 0.2 0.4 0.6 0.8 1 τ Pr(error≤τ errormin) (a) hott hott (fast) hott (lp) AGKM 0 20 40 60 0 0.2 0.4 0.6 0.8 1 τ Pr(error≤τ errormin) (b) hott hott (fast) AGKM 0 100 200 300 0 0.2 0.4 0.6 0.8 1 τ Pr(time≤τ timemin) (c) hott hott (fast) AGKM 0 20 40 60 0 0.2 0.4 0.6 0.8 1 τ Pr(RMSE≤τ RMSEmin) (d) hott hott (fast) hott (lp) AGKM 0 20 40 60 0 0.2 0.4 0.6 0.8 1 τ Pr(RMSE≤τ RMSEmin) (e) hott hott (fast) AGKM 0 20 40 60 0 0.2 0.4 0.6 0.8 1 τ Pr(error≤τ errormin) (f) hott hott (fast) AGKM Figure 2: Performance proﬁles for synthetic data. (a) (∞, 1)-norm error for 40 × 400 sized instances and (b) all instances. (c) is the performance proﬁle for running time on all instances. RMSE performance proﬁles for the (d) small scale and (e) medium scale experiments. (f) (∞, 1)-norm error for the η ≥1. In the noisy examples, even 4 epochs of HOTTOPIXX is sufﬁcient to obtain competitive reconstruction error. In small-scale, synthetic experiments, we compared HOTTOPIXX to the AGKM algorithm and the linear programming formulation of Algorithm 3 implemented in Matlab. Both AGKM and Algorithm 3 were run using CVX [14] coupled to the SDPT3 solver [26]. We ran HOTTOPIXX for 50 epochs with primal stepsize 1e-1 and dual stepsize 1e-2. Once the hott topics were identiﬁed, we ﬁt F using two cleaning epochs of incremental gradient descent for all three algorithms. To generate our instances, we sampled r hott topics uniformly from the unit simplex in Rn. These topics were duplicated d times. We generated the remaining f −r(d + 1) rows to be random convex combinations of the hott topics, with the combinations selected uniformly at random. We then added noise with (∞, 1)-norm error bounded by η · α2 20+13α. Recall that AGKM algorithm is only guaranteed to work for η < 1. We ran with f ∈{40, 80, 160}, n ∈{400, 800, 1600}, r ∈{3, 5, 10}, d ∈{0, 1, 2}, and η ∈{0.25, 0.95, 4, 10, 100}. Each experiment was repeated 5 times. Because we ran over 2000 experiments with 405 different parameter settings, it is convenient to use the performance proﬁles to compare the performance of the different algorithms [7]. Let P be the set of experiments and A denote the set of different algorithms we are comparing. Let Qa(p) be the value of some performance metric of the experiment p ∈P for algorithm a ∈A. Then the performance proﬁle at τ for a particular algorithm is the fraction of the experiments where the value of Qa(p) lies within a factor of τ of the minimal value of minb∈A Qb(p). That is, Pa(τ) = # {p ∈P : Qa(p) ≤τ mina′∈A Qa′(p)} #(P) . In a performance proﬁle, the higher a curve corresponding to an algorithm, the more often it outperforms the other algorithms. This gives a convenient way to contrast algorithms visually. Our performance proﬁles are shown in Figure 2. The ﬁrst two ﬁgures correspond to experiments with f = 40 and n = 400. The third ﬁgure is for the synthetic experiments with all other values of f and n. In terms of (∞, 1)-norm error, the linear programming solver typically achieves the lowest error. However, using SDPT3, it is prohibitively slow to factor larger matrices. On the other hand, HOTTOPIXX achieves better noise performance than the AGKM algorithm in much less time. Moreover, the AGKM algorithm must be fed the values of ϵ and α in order to run. HOTTOPIXX does not require this information and still achieves about the same error performance. We also display a graph for running only four epochs (hott (fast)). This algorithm is by far the fastest algorithm, but does not achieve as optimal a noise performance. For very high levels of noise, however, it achieves a lower reconstruction error than the AGKM algorithm, whose performance 7 data set features documents nonzeros size (GB) time (s) jumbo 1600 64000 1.02e8 2.7 338 clueweb 44739 351849 1.94e7 0.27 478 RCV1 47153 781265 5.92e7 1.14 430 Table 1: Description of the large data sets. Time is to ﬁnd 100 hott topics on the 12 core machines. 0 10 20 30 40 0 10 20 30 40 threads speedup jumbo clueweb 0 500 1000 1500 2000 2500 4 6 8 10 12 14 16 number of topics RMSE 0 1000 2000 3000 4000 5000 5 10 15 20 25 30 number of topics class error Figure 3: (left) The speedup over a serial implementation for HOTTOPIXX on the jumbo and clueweb data sets. Note the superlinear speedup for up to 20 threads. (middle) The RMSE for the clueweb data set. (right) The test error on RCV1 CCAT class versus the number of hott topics. The horizontal line indicates the test error achieved using all of the features. degrades once η approaches or exceeds 1 (Figure 2(f)). We also provide performance proﬁles for the root-mean-square error of the nonnegative matrix factorizations (Figure 2 (d) and (e)). The performance is qualitatively similar to that for the (∞, 1)-norm. We also coded HOTTOPIXX in C++, using the design principles described in Section 4.1, and ran on three large data sets. We generated a large synthetic example (jumbo) as above with r = 100. We generated a co-occurrence matrix of people and places from the ClueWeb09 Dataset [2], normalized by TFIDF. We also used HOTTOPIXX to select features from the RCV1 data set to recognize the class CCAT [19]. The statistics for these data sets can be found in Table 1. In Figure 3 (left), we plot the speed-up over a serial implementation. In contrast to other parallel methods that exhibit memory contention [21], we see superlinear speed-ups for up to 20 threads due to hardware prefetching and cache effects. All three of our large data sets can be trained in minutes, showing that we can scale HOTTOPIXX on both synthetic and real data. Our algorithm is able to correctly identify the hott topics on the jumbo set. For clueweb, we plot the RMSE Figure 3 (middle). This curve rolls off quickly for the ﬁrst few hundred topics, demonstrating that our algorithm may be useful for dimensionality reduction in Natural Language Processing applications. For RCV1, we trained an SVM on the set of features extracted by HOTTOPIXX and plot the misclassiﬁcation error versus the number of topics in Figure 3 (right). With 1500 hott topics, we achieve 7% misclassiﬁcation error as compared to 5.5% with the entire set of features. 6 Discussion This paper provides an algorithmic and theoretical framework for analyzing and deploying any factorization problem that can be posed as a linear (or convex) factorization localizing program. Future work should investigate the applicability of HOTTOPIXX to other factorization localizing algorithms, such as subspace clustering, and should revisit earlier theoretical bounds on such prior art. Acknowledgments The authors would like to thank Sanjeev Arora, Michael Ferris, Rong Ge, Nicolas Gillis, Ankur Moitra, and Stephen Wright for helpful suggestions. BR is generously supported by ONR award N00014-11-1-0723, NSF award CCF-1139953, and a Sloan Research Fellowship. CR is generously supported by NSF CAREER award under IIS-1054009, ONR award N000141210041, and gifts or research awards from American Family Insurance, Google, Greenplum, and Oracle. JAT is generously supported by ONR award N00014-11-1002, AFOSR award FA9550-09-1-0643, and a Sloan Research Fellowship. 8 References [1] docs.oracle.com/cd/B28359_01/datamine.111/b28129/algo_nmf.htm. [2] lemurproject.org/clueweb09/. [3] www.mathworks.com/help/toolbox/stats/nnmf.html. [4] S. Arora, R. Ge, R. Kannan, and A. Moitra. 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Join processing in database systems with large main memories. ACM Transactions on Database Systems, 11(3):239–264, 1986. [23] P. Smaragdis. Non-negative matrix factorization for polyphonic music transcription. In IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, pages 177–180, 2003. [24] M. Soltanolkotabi and E. J. Cand`es. A geometric analysis of subspace clustering with outliers. Preprint available at arxiv.org/abs/1112.4258, 2011. [25] L. B. Thomas. Problem 73-14, rank factorization of nonnegative matrices. SIAM Review, 16(3):393–394, 1974. [26] K. C. Toh, M. Todd, and R. H. T¨ut¨unc¨u. SDPT3: A MATLAB software package for semideﬁnite-quadratic-linear programming. Available from http://www.math.nus.edu.sg/˜mattohkc/sdpt3.html. [27] S. A. Vavasis. On the complexity of nonnegative matrix factorization. SIAM Joural on Optimization, 20(3):1364–1377, 2009. 9	2012	12
4,475	Volume Regularization for Binary Classiﬁcation Koby Crammer Department of Electrical Enginering The Technion - Israel Institute of Technology Haifa, 32000 Israel koby@ee.technion.ac.il Tal Wagner∗ Faculty of Mathematics and Computer Science Weizmann Institute of Science Rehovot, 76100, Israel tal.wagner@gmail.com Abstract We introduce a large-volume box classiﬁcation for binary prediction, which maintains a subset of weight vectors, and speciﬁcally axis-aligned boxes. Our learning algorithm seeks for a box of large volume that contains “simple” weight vectors which most of are accurate on the training set. Two versions of the learning process are cast as convex optimization problems, and it is shown how to solve them efﬁciently. The formulation yields a natural PAC-Bayesian performance bound and it is shown to minimize a quantity directly aligned with it. The algorithm outperforms SVM and the recently proposed AROW algorithm on a majority of 30 NLP datasets and binarized USPS optical character recognition datasets. 1 Introduction Linear models are widely used for a variety of tasks including classiﬁcation and regression. Support vectors machines [3, 22] (SVMs) are considered a primary method to efﬁciently build linear classiﬁers from data, yielding state-of-the-art performance. SVMs and many other methods are often easy to implement and efﬁcient, yet return only a single weight-vector with no additional information about alternative models nor about conﬁdence in prediction. An alternative approach is taken by Bayesian methods [21, 13]. The primary object is a (posterior) distribution over models that is updated using Bayes rule. Unfortunately, the posterior is very complicated even for simple models, such as Bayesian logistic regression [15], and it is not known how to perform the update analytically, and approximations are required. In this work we integrate the advantages of both approaches. We propose to model uncertainty over weight-vectors by maintaining a (simple) set of possible weight-vectors, rather than a single weight-vector. Learning is motivated from principles of discriminative learning rather than Bayes’ rule, and it is optimizing a combination of an hand-crafted regularization term and the empirical loss. Speciﬁcally, our algorithm maintains an axis-aligned box, which only requires double number of parameters than maintaing a single weight-vector, a dominating model for many tasks. We use a similar conceptual reasoning as used in Bayes point machines (BPM) [13]. Both approaches maintain a set of possible weights, which can be thought of as a posterior. BPMs use the version space, the set of all consistent weight vectors, which is a convex polyhedron. Since the size of the polyhedron’s representation grows with the number of training examples, BPMs approximate the polyhedron with a single weight-vector, the Bayes point. Our algorithms model the set as a box, with a representation that is ﬁxed in the size of the input, and ﬁnd an optimal prediction box. We cast learning as a convex optimization problem and propose methods to solve it efﬁciently. We further provide generalization bounds using PAC-Bayesian theory, and show that our algorithm is ∗The research was performed while TW was a student at the Technion. 1 minimizing a quantity directly related to the generalization bound. We give two formulations or versions of the algorithm, one that is closely related to the bound, while the other is smooth. We experiment with 30 binary text categorization datasets from various tasks: sentiment classiﬁcation, predicting domain of product-review, assigning topics to news items, tagging spam emails, and classifying posts to news-groups. The results indicate that our algorithms outperform SVM and the recently proposed AROW [4] algorithm, which was shown to be the state-of-the-art in numerous NLP tasks. Additional support for the superiority and robustness of our algorithms, especially in high-noise setting, is provided using experiments with 45 pairs of binarized USPS OCR problems. Notation: Given a vector x ∈Rd, we denote its kth element by xk ∈R, and by \|x\| ∈Rd the vector with component-wise absolute value of its elements, \|x\| = (\|x1\|, . . . , \|xd\|). 2 Large-Volume Box Classiﬁers Standard linear classiﬁcation learning algorithms maintain and return a single weight vector w⋆∈ Rd used to predict the label of any test point. We study a generalization of these algorithms where hypotheses are uncertainty (sub)sets of weight vectors w. Such a hypothesis can be seen as a randomized linear classiﬁer or a voting process. To classify an instance x, a parameter vector w is drawn according to the hypothesis and predicts the label sign(w · x). Herbrich et.al. [13, 12], argued in a similar context that such a randomization yields a more robust solution. PAC-Bayesian analysis and its generalization bounds give additional justiﬁcation to this approach (see Sec 5). The uncertainty subsets we study are axis aligned boxes parametrized with two vectors u, v ∈ Rd where we assume, uk ≤vk for all k = 1 . . . d. In words, u is the vertex with the lowest coordinates, and v is the vertex with the largest coordinates. The projection of the box onto the k-axis yields the interval [uk, vk]. The set of weight vectors contained in the box is denoted by, Q = {w : uk ≤wk ≤vk for k = 1 . . . d} . Given an instance x to be classiﬁed, a Gibbs classiﬁer samples a weight vector uniformly in random from the box w ∈Q and returns sign(w · x). A deterministic alternative we use in practice is to employ the center of mass deﬁned by µ = 1 2 (u + v) and return sign(µ · x). For linear classiﬁers, the majority prediction with Gibbs sampling coincides with predicting using the center of mass. We also deﬁne the uncertainty intervals σ = 1 2 (v −u). Intuitively, the uncertainty in the weight associated with the kth feature is σk. Clearly, v = µ + σ and u = µ −σ. 3 Learning as Optimization Given a labeled sample S = {(xi, yi)}n i=1, a common practice in learning linear models w is to perform structural risk minimization (SRM) [25] that picks a weight-vector that is both “simple” (eg small norm) and performs well on the training set. Learning is cast as an optimization problem, w⋆= arg min w 1 n X i ℓ(w, (xi, yi)) + D R(w) . (1) The ﬁrst term is the empirical loss evaluated on the training set with some loss function ℓ(w, (x, y)), and the second term is a regularization that penalizes weight-vectors according to their complexity. The parameter D > 0 is a tradeoff parameter. Learning with uncertainty sets invites us to balance three desires rather than two as when learning a single weight-vector. The ﬁrst two desires are generalizations of the structural risk minimization principle [25] to boxes: we prefer boxes containing weight-vectors that attain both low loss ℓ(w, (xi, yi)) and are “simple” (eg small norm). This alone though is not enough, as if the loss and regularization functions are strictly convex then the optimal box would in fact be a single weightvector. The third desire is thus to prefer boxes with large volume. Intuitively, if during training an algorithm ﬁnds a box with large volume, such that all weight-vectors belonging to it attain low training error and are simple, we expect the classiﬁer based on the center of mass to be robust to noise or ﬂuctuations. This will be formally stated in the analysis described in Sec 5. We formalize this requirement by adding a term that is inversely proportional to the volume of the box Q. We take a worst-case approach, and deﬁne the loss of the box Q given an example (x, y) denoted by ℓ(Q, (x, y)) to be the loss of the worst member w ∈Q. Similarly, we deﬁne the complexity of the box Q to be the complexity of the most complex member of the box w ∈Q, formally, ℓ(Q, (x, y)) = supw∈Q ℓ(w, (x, y)) and R(Q) = supw∈Q R(w). 2 Putting it all together, we replace (1) with, Q⋆= arg min Q∈Q sup w∈Q 1 m X i ℓ(w, (xi, yi)) + DR(w) ! , (2) where Q is a set of boxes with some minimal volume. In other words, the algorithm is seeking for a set of alternative weight-vectors all of which are performing well on the training data. We expect this formulation to be robust, as a box is evaluated with its worst performing member. We modify the problem by removing the constraint Q ∈Q and adding an equivalent penalty term to the objective, namely the log-volume of the box. We use the log-volume function for three reasons. First, it is a common barrier function in optimization [26], and in our case it keeps the box from actually shrinking to a zero volume box. Second, this choice is supported by the analysis below, and third, it is additive in the dimension of the data d, like all other quantities of the objective. Additionally, we bound the supremum over w with a sum of supremum operators. To conclude, we cast the learning problem as the following optimization problem over boxes, arg min Q 1 m X i sup w∈Q ℓ(w, (xi, yi)) −C log volQ + D sup w∈Q R(w) , (3) where C, D > 0 are two trade-off parameters used to balance the three goals. (In the analysis below it will be shown that D can be also interpreted as a Langrange multiplier of a constrained optimization problem.) We further develop the last equation by making additional assumptions over the loss function and the regularization. We assume that the loss is a monotonically decreasing function of the product y(x⊤w), often called the margin (or the signed margin). This is a property of many popular loss functions for binary classiﬁcation, including the hinge-loss and its square used by SVMs [3, 22], exp-loss used by boosting [9], logistic-regression [11] and the Huber-loss [14]. Under this assumption we compute analytically the ﬁrst term of the objective (3). Lemma 1 If the loss function is monotonically decreasing in the margin, ℓ(w, (x, y)) = ℓ(y x⊤w ) then supw∈Q ℓ(w, (xi, yi)) = ℓ(y(x⊤µ) −\|x\|σ). Proof: From the monotonicity of ℓ(·) we have supw∈Q ℓ y(x⊤w) = ℓ infw∈Q y(x⊤w) . Computing the inﬁmum we get, inf w∈Q y(x⊤w) = inf wk∈[uk,vk] for k=1...d d X k=1 (yxk)wk = d X k=1 inf wk∈[uk,vk](yxk)wk = d X k=1 (yxk) uk (yxk) ≥0 vk (yxk) < 0 = d X k=1 (yxk) (µk −sign(yxk)σk) = y(x⊤µ) −\|x\|σ , using u = µ −σ and v = u + σ as stated above. The lemma highlights the need to constrain the volume to be strictly larger than zero: due to monotonicity and the fact that σ ≥0 (component wise) we have ℓ(y(x⊤µ) −\|x\|σ) ≥ℓ(y(x⊤µ)), so the loss is always minimized when we set σ = 0. We next turn to analyse the third term of (3) with the following lemma. Lemma 2 (1) Assuming R(w) is convex, then supw∈Q R(w) is attained on vertices of the box Q. (2) Additionally, if R(w) is strictly convex then the supremum is attained only on vertices. Proof: We use the fact that every point in the box can be represented as a convex combination of the vertices. Formally, given a point in the box w ∈Q, there exists a vector α ∈R2d with non-negative elements and P t αt = 1 such that w = P t αtzt where zt are the vertices of the box. Convexity of R(·) yields, R(w) ≤P t αtR(zt) ≤maxt {R(zt)} . Thus, if w attains the supremum supw∈Q R(w) then so does at least one vertex. Additionally, if R(w) is a strictly convex function, then the ﬁrst inequality in the last equation is a strict inequality, and thus a non-vertex cannot attain the supremum. Common regularization functions are deﬁned as sums over individual features, that is R(w) = P k r(wk). In this case the supremum is attained on each coordinate independently as follows. 3 Corollary 3 Assuming R(w) is a sum of scalar-convex functions P k r(wk), we have, sup w∈Q R(w) = X k max {r(uk), r(vk)} = X k max {r(µk −σk), r(µk + σk)} . The corollary follows from the lemma since a supremum of a scalar-function over a box is equivalent to taking the supremum over the box projected to a single coordinate. Finally, the volume of a box is given by a product of the length of its axes, that is, vol (Q) = Q k (vk −uk) = Q k (2σk) = 2d Q k σk . To summarize, the learning problem of the large-volume box algorithm is cast by solving the following minimization problem, in terms of the center µ and the size (or dimensions) σ, min σ≥0,µ 1 m m X i=1 ℓ yi(x⊤ i µ) −\|xi\|σ −C X k log σk +D X k max {r(µk −σk), r(µk + σk)} , (4) where ℓ(·) is a monotonically decreasing function, r(·) is a convex function, and C, D > 0 are two trade-off parameters used to balance our three desires. We denote by zi,+ = yixi + \|xi\| ∈Rd , zi,−= yixi −\|xi\| ∈Rd . (5) The kth element of zi,+ (zi,−) is twice the kth element of \|xi\| if the sign of the kth element of xi agrees (disagrees) with yi, and zero otherwise. This problem can equivalently be written in terms of the two “extreme” vertices u and v as follows, min v≥u 1 m m X i=1 ℓ 1 2 v⊤(zi,−) + u⊤(zi,+) −C X k log (vk −uk)+D X k max {r(vk), r(uk)} , (6) by using the relation yix⊤ i (v+u)−\|xi\|(v−u)=v⊤(zi,−)+u⊤(zi,+) . Note, if the loss function ℓ(·) is convex, then both formulations (4) and (6) of the learning problem are convex in their arguments, as each is a sum of convex functions of linear combination of the arguments, and a maximum of convex functions is convex. We conclude this section with an additional alternative formulation, which for convenience, we present in the notation of (6). Although the above problem is convex, the regularization term P k max {r(vk), r(uk)} is not smooth because of the max operator. In this alternative, we replace it with a smooth term, by changing the max to a sum, yielding P k r(vk) + r(uk) = R(u) + R(v). The problem then becomes, min v≥u 1 m m X i=1 ℓ 1 2 v⊤(zi,−) + u⊤(zi,+) −C X k log (vk −uk) + D (R(u) + R(v)) . (7) The two alternatives are related via the following chain of inequalities, 0.5 max {r(vk), r(uk)} ≤ 0.5 (r(vk) + r(uk)) ≤max {r(vk), r(uk)} ≤r(vk) + r(uk) . In other words, given either one of the problems (6) or (7), we can lower and upper bound it with the other problem with a proper choice of trade-off parameter D. We call the two versions BoW for box-of-weights algorithm, and refer to them as BoW-M(ax) and BoW-S(um), respectively. 4 Optimization Algorithm We now present an algorithm to solve (6) for the special case r(x) = x2. The algorithm is a based on COMID [8] and its convergence analysis follows directly from the analysis of COMID, which is omitted due to lack of space. The algorithm works in iterations. On each iteration a (stochastic) gradient decent step is performed, followed by a regularization-optimization step. Formally, the algorithm picks a random example i and updates, (˜u, ˜v) ←(u, v) −αη 2 zi,+ , zi,− for α = ℓ′ 1 2 v⊤(zi,−) + u⊤(zi,+) . 4 The algorithm then solves the following regularization-oriented optimization problem, min u,v 1 2∥u −˜u∥2 + 1 2∥v −˜v∥2 −C X k log (vk −uk) + D X k max v2 k, u2 k . The objective of the last problem decomposes over individual pairs uk, vk so we reduce the optimization to d independent problems, each deﬁned over 2 scalars u and v (omitting index k), min u,v F(u, v) = 1 2 (u −˜u)2 + 1 2 (v −˜v)2 −C log (v −u) + D max v2, u2 . (8) We denote the half-plane H = {(u, v) ∈R2 : v > u} and partition it into three subsets: G1 = {(u, v) ∈H : v > −u}, G2 = {(u, v) ∈H : v < −u}, and the line L = {(u, v) ∈R2 : v = −u}. The following lemma describes the optimal solution of (8). Lemma 4 Exactly one of the items below holds and describe the optimal solution of (8). 1. If there exists (u, v) ∈G1 such that v is a root of f(v) = αv2 + βv + γ and u = ˜u −2Dv + (˜v −v) where α = 2(1 + D)(1 + 2D), β = −˜u(1 + 2D) −˜v(3 + 4D), and γ = ˜v + ˜u˜v −C, then it is a global minimum of F. 2. If there exists (u, v) ∈G2 such that u is a root of f(u) = αu2 + βu + γ and v = ˜v −2Du + (˜u −u) where α = 2(1 + D)(1 + 2D), β = −˜v(1 + 2D) −˜u(3 + 4D) and γ = ˜u + ˜v˜u −C, then it is a global minimum of F. Furthermore, such point and a point described in 1 cannot exist simultaneously. 3. If no points as described in 1 nor 2 exist, then the global minimum of F is (u, −u) such that u is a root of f(u) = αu2 + βu + C where α = 2 + 2D, β = ˜v −˜u, γ = −C. Proof sketch: By deﬁnition, the function F is smooth and convex on G1. The condition in 1 is equivalent to satisfying ∇F(u, v) = 0, and therefore any point that satisﬁes it, is a minimum of F G1. A similar argument applies to G2 with 2. The convexity of F on the entire set H yields that any such point is also a global minimum of F, and that if no such point exists then F attains a global minimum on L (which is derived in 3). The latter is sure to exist since limv→0 F\|L = limv→∞F\|L = ∞. The algebraic derivation is omitted due to lack of space. Similarly, we develop the update for solving (7). Here after the gradient step we need to solve the following problem per coordinate k, minu,v F(u, v) = 1 2 (u −˜u)2 + 1 2 (v −˜v)2 −C log (v −u)+ D v2 + u2 . The following lemma characterizes the optimal solution. Lemma 5 The optimal solution (u, v) ∈{(u, v) ∈R2 : v −u > 0} of the last problem is such that u is a root of the polynomial f(u) = αu2 + βu + γ where α = 2 + 2D + 6D + 8D2,β = −(˜v +2D˜v + ˜u+6D˜u)−2˜u,γ = ˜u2 + ˜u˜v −4C −2CD and v = (˜v + ˜u −u(1 + 2D)) /(1 + 2D). Its proof is similar to the proof of Lemma 4, but simpler and omitted due to lack of space. 5 Analysis PAC-Bayesian bounds were introduced by McAllester [19], were further reﬁned later (e.g. [17, 23]), and applied to analyze SVMs [18]. They often have been shown to be quite tight. We ﬁrst introduce some notation needed for the discussion of these bounds. Let ¯ℓ(w, (x, y)) denote the zero-one loss, that is ¯ℓ(w, (x, y)) = 1 if sign(w ·x) ̸= y and ¯ℓ(w, (x, y)) = 0 otherwise. Let D be a distribution over the labeled examples (x, y), and denote by ¯ℓ(w, D) the expected zero-one loss of a linear classiﬁer characterized by its weight vector w: ¯ℓ(w, D) = Pr(x,y)∼D[sign(w·x) ̸= y] = E(x,y)∼D[¯ℓ(w, (x, y))] . We abuse notation, and denote by ¯ℓ(w, S) the expected loss ¯ℓ(w, DS) for the empirical distribution DS of a sample S. PAC-Bayesian analysis states generalization bounds in terms of two distributions - prior and posterior - over all hypotheses (i.e. over weight-vectors w). Below, we identify a compact set with a uniform distribution over the set, and in particular we identify a box Q with a uniform distribution 5 over all weight vectors it contains (and zero mass otherwise). Similarly, we identify any compact body P with a uniform distribution over its elements. In other words, we refer to the prior P and the posterior Q both as two uniform distributions and as their support (which are subsets). We also denote by ℓ(Q, D) the expectation of ℓ(w, D) over weight vectors w drawn according to the distribution Q. We quote Cor. 2.2 of Germain et.al. [10], Corollary 6 ([10]) : For any distribution D, for any set H of weight-vectors, for any distribution P of support H, for any δ ∈(0, 1], and any positive number γ the following statement holds with probability ≥1 −δ over samples S of size n, ¯ℓ(Q, D) ≤ 1 1 −e−γ ( 1 −exp " − γ · ¯ℓ(Q, S) + 1 nDKL (Q∥P) + 1 n ln 1 δ !#) . (9) The corollary states that the expected number of mistakes over examples drawn according to some ﬁxed and unknown distribution D over inputs, and over weight-vectors drawn from the box Q uniformly, is bounded by the right term, which is a monotonic function of the following sum, ¯ℓ(Q, S) + 1 nγ DKL (Q∥P) . (10) For uniform distributions we have the following, DKL (Q∥P) = log vol(P) vol(Q) Q ⊆P ∞ otherwise . (11) Additionally, we bound the empirical training error, ¯ℓ(Q, S) = 1 n n X i 1 volQ Z w∈Q ¯ℓ(w, (xi, yi)) dw ≤1 n X i ℓ inf w∈Q yi(x⊤ i w) , (12) where the equality is the deﬁnition of ¯ℓ(Q, S), and the inequality follows by choosing a loss function ℓ(·) which upper bounds the zero-one loss (e.g. Hinge loss), by bounding an expectation with the supremum value, and from Lemma 1. We get that to minimize the generalization bound of (9) we can minimize a bound on (10) which is obtained by substituting (11) and (12) in (10). Omitting constants we get, min Q 1 n X i ℓ inf w∈Q yiw⊤xi −1 nγ log volQ s.t. Q ⊆P . (13) Next, we set P to be a ball of radius R about the origin, and, as in Sec 2, we set Q as a box parametrized with the vectors u and v. We use the following lemma, of which proof is omitted due to lack of space, Lemma 7 If P is a ball of radius R about the origin and Q is a box parametrized using u and v, we have Q ⊆P ⇔P k max{v2 k, u2 k} ≤R2 . Finally, plugging Lemma 7 and Lemma 1 in (13) we get the following problem, which is monotonically related to a bound of the generalization loss, minv≥u 1 n Pm i=1 ℓ 1 2 v⊤(zi,−) + u⊤(zi,+) − 1 nγ P k log (vk −uk) subject to P k max {r(vk), r(uk)} ≤R2 . To solve the last problem we write its Lagrangian, max η min v≥u 1 n m X i=1 ℓ 1 2 v⊤(zi,−) + u⊤(zi,+) −1 nγ X k log (vk −uk) + η X k max {r(vk), r(uk)} −ηR2 , (14) where η is the Lagrange multiplier ensuring the constraint. Comparing (14), whose objective is used in the PAC-Bayesian bound, and our learning algorithm in (6), we observe that the three terms in both objectives are the same by setting C = 1 nγ and identifying the optimal value of the Lagrange 6 10 −1 10 0 10 1 10 −1 10 0 10 1 error (%) BoW−M error (%) SVM 10 −1 10 0 10 1 10 −1 10 0 10 1 error (%) BoW−S error (%) SVM 10 −1 10 0 10 1 10 −1 10 0 10 1 error (%) BoW−M error (%) AROW 10 −1 10 0 10 1 10 −1 10 0 10 1 error (%) BoW−S error (%) AROW Figure 1: Fraction of error on text classiﬁcation datasets of BoW-M and BoW-S vs SVM (two left plots); and BoW-M and BoW-S vs AROW (two right plots). Markers above the line indicate superior BoW performance. multipler with the trade-off constant η = D. In fact, each value of the radius R yields a unique optimal value of the Lagrange multiplier η. Thus, we can interpret the role of the constant D as setting implicitly the effective radius of the prior ball P. Few comments are in order. First, the KL-divergence between distributions is minimized more effectively if both P and Q are of the same form, e.g. both P and Q are boxes. However, we chose Q to be a box, as it has a nice interpretation of uncertainty over features, and P to be a ball, as it decomposes (as opposed to an ℓ∞ball), which allows simpler optimization algorithms. Second, as noted above, BoW-S is indeed smoother than BoW-M, yet, from (14) it follows that the latter is better motivated from the PAC-Bayesian bound, as we want Q ⊆P. Third, the bound is small if the volume of the box Q is large, which motivates seeking for large-volume boxes, whose members perform well. 6 Empirical Evaluation We evaluated BoW-M and BoW-S on NLP tasks experimenting with all the 12 datasets used by Dredze et al [6] (sentiment classiﬁcation in 6 Amazon domains, 3 pairs of 20 newsgroups and 3 pairs of Reuters (RCV1)). We deﬁned an additional task from the 6 Amazon domains (book, dvd, music, video, electronics, kitchen). Given reviews from two domains, the goal is to identify the domain identity. We used all 6×5/2=15 unordered pairs of domains. Additionally, we selected 3 users from task A of the 2006 ECML/PKDD Discovery Challenge spam data set. The goal is to classify an email as either a spam or a not-spam. This yielded a total of 30 datasets. For each problem we selected about 2, 000 instances and represented them with vectors of uni/bi-grams counts. Feature extraction followed a previous protocol [6, 2]. Each dataset was randomly divided for 10-fold cross validation. We also experimented with USPS OCR data which we binarized into 45 all-pairs problems, maintaing the standard split into training and test sets. Given an image of one of two digits, the goal is to detect which of the two digits is shown in the image. We implemented BoW-M and BoW-S both with Hinge loss and Huber loss. The performance of the latter was slightly worse than the former, thus we report results only for the Hinge loss. We also tried AdaGrad [7] but surprisingly it did not work as well as COMID. We compared BoW with support vector machines (SVM) [3] and AROW [4] which was shown to outperform many algorithms on NLP tasks. (Other algorithms we evaluated, including maximum-entropy and SGD with Huberloss, performed worse than either of these two algorithms and thus are omitted.) It is not clear at this point how to incorporate Mercer kernels into BoW, and thus we are restricted to evaluate all algorithms on data that can be classiﬁed well with linear models. Classiﬁers parameters (C for SVM, r for AROW and C, D for BoW) were tuned for each task on a single additional randomized run over the data splitting it into 80%, used for training, and the remaining 20% of examples were used to choose the parameters. Results are reported for NLP tasks as the average error over the 10 folds per problem , while for USPS the standard test sets are used. The mean error for 30 NLP tasks over 10 folds of BoW-M and BoW-S vs SVM is summarized in the two left panels of Fig. 1. Markers above the line indicate superior BoW performance. Clearly, both BoW versions outperform SVM obtaining lower test error on most (26) datasets and higher only on few (at most 3). The right two panels compare the performance of both BoW versions with AROW. Here the trend remains yet with a smaller gap, BoW-M outperforms AROW in 20 datasets, and is outperformed in 9, while BoW-S outperforms AROW in 19 datasets and outperformed in 12. Note, AROW was previously shown [4] to have superior performance on text data over other algorithms. 7 0% 10% 20% 30% 0 10 20 30 40 No wins Training Label Error BoW−M SVM TIE 0% 10% 20% 30% 0 10 20 30 40 No wins Training Label Error BoW−S SVM TIE 0% 10% 20% 30% 0 10 20 30 40 No wins Training Label Error BoW−M AROW TIE 0% 10% 20% 30% 0 10 20 30 40 No wins Training Label Error BoW−S AROW TIE Figure 2: No. of USPS 1-vs-1 datasets (out of 45) for which one algorithm is better than the other (see legend) shown for four levels of label noise during training: 0%, 10%, 20% and 30% (left to right). Higher values indicate better performance. The results of the experiments with USPS are summarized in Fig. 2. Each panel shows the number of datasets (out of 45) for which one algorithm outperforms another algorithm, for four levels of label noise (i.e. probability of ﬂipping the correct label) during training: 0%, 10%, 20% and 30%.The four pairs compared are BoW vs SVM (two left panels, BoW-M most left) and BoW vs AROW (two right panels, BoW-S most right). A left bar higher than a middle bar (in each group in each panel) indicates superior BoW performance. With no label noise (left group in each panel) SVM outperforms both BoW algorithms (e.g. SVM attains lower test error than BoW-S on 20 datasets and higher on 12 datasets, with a tie in 13 datasets). The average test error of SVM is 1.81, AROW is 1.98 and BoW-S is 1.97. When the level of noise increases both BoW algorithms outperform AROW and SVM. With maximal level of 30% label noise, the average test error is 16.1% for SVM, 14.8 for AROW, and 6.1% for BoW-S. BoW-M achieves lower test error on 27 datasets (compared both with SVM and AROW), while BoW-S achieves lower test error than SVM on 38 datasets and than AROW on 40 datasets. Interestingly, while, in general, BoW-M achieved lower test error than BoW-S on the NLP problems, the situation is reversed in the USPS data where BoW-S achieves in general lower test error. 7 Related Work There is much previous work on a related topic of incorporating additional constraints, using prior knowledge of the problem. Shivaswamy and Jebara [24] use a geometric motivation to modify SVMs. Their effort and other related works, ﬁrst deduce some additional knowledge about the problem [16, 20, 1], and keep it ﬁxed while learning. In contrast, our method learns together the classiﬁer and some additional information. Another line of research is about algorithms that are maintaining a Gaussian distribution over weights, as opposed to uniform distribution as in our case, either AROW [4] in the online setting and its predecessors, or Gaussian Margin Machines (GMMs) [5] in the batch setting. Our motivation is similar to the motivation behind GMMs, yet it is different in few important aspects. (1) BoW maintains only 2d parameters, while GMM employs d+d(d+1)/2 as it maintains a full covariance matrix. (2) As a consequence, GMMs are not feasible to run on data with more than hundreds of features, which is further supported by the fact that GMMs were evaluated only on data of dimension 64 [5]. (3) We use directly a specialized PAC-Bayes bound for convex loss functions [10] while the analysis of GMMs uses a bound designed for the 0 −1 loss which is then further bounded. (4) The optimization problem of both versions of BoW is convex, while the optimization problem of GMMs is not convex, and it is only approximated with a convex problem. (5) Therefore, we can and do employ COMID [8] which is theoretically justiﬁed and fast in practice, while GMMs are trained using another technique with no convergence (to local minima) guarantees. (6) Conceptually, BoW maintains a compact set (box) while the set of possible weights for GMM is not compact. This allows us to extend our work to other types of sets (in progress), while its not clear how to extend the GMMs approach from Gaussian distributions to other objects. 8 Conclusion We extend the commonly used linear classiﬁers to subsets of the class of possible classiﬁers, or in other words uniform distributions over weight vectors. Our learning algorithm is based on a worst-case margin minimization principle, and it beneﬁts from strong theoretical guarantees based on tight PAC-Bayesian bounds. The empirical evaluation presented shows that our method performs favourably with respect to SVMs and AROW, and is more robust in the presence of label noise. We 8 plan to study the integration of kernels, extend our framework for various shapes and problems, and develop specialized large scale algorithms. Acknowledgments: The paper was partially supported by an Israeli Science Foundation grant ISF1567/10 and by a Google research award. References [1] J. Bi and T. Zhang. Support vector classiﬁcation with input data uncertainty. In NIPS, 2004. [2] J. Blitzer, M. Dredze, and F. Pereira. Biographies, bollywood, boom-boxes and blenders: Domain adaptation for sentiment classiﬁcation. In ACL, 2007. [3] C. Cortes and V. Vapnik. Support-vector networks. Machine Learning, 20(3):273–297, September 1995. [4] K. Crammer, A. Kulesza, and M. Dredze. Adaptive regularization of weighted vectors. In NIPS, 2009. [5] K. Crammer, M. Mohri, and F. Pereira. Gaussian margin machines. In AISTATS, 2009. [6] M. Dredze, K. Crammer, and F. Pereira. Conﬁdence-weighted linear classiﬁcation. In ICML, 2008. [7] J. Duchi, E. Hazan, and Y. Singer. Adaptive subgradient methods for online learning and stochastic optimization. In COLT, 2010. [8] J. Duchi, S. Shalev-Shwartz, Y. Singer, and A. Tewari. Composite objective mirror descent. In COLT, pages 250–264, 2010. [9] Y. Freund and R.E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. In Euro-COLT, pages 23–37, 1995. [10] P. Germain, A. Lacasse, F. Laviolette, and M. Marchand. Pac-bayesian learning of linear classiﬁers. In ICML, 2009. [11] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, 2001. [12] R. Herbrich, T. Graepel, and C. Campbell. Robust Bayes point machines. In ESANN 2000, pages 49–54, 2000. [13] R. Herbrich, T. Graepel, and C. Campbell. Bayes point machines. JMLR, 1:245–279, 2001. [14] P.J. Huber. 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4,476	Causal discovery with scale-mixture model for spatiotemporal variance dependencies Zhitang Chen, Kun Zhang†, and Laiwan Chan *Department of Computer Science and Engineering, Chinese University of Hong Kong, Hong Kong {ztchen,lwchan}@cse.cuhk.edu.hk †Max Planck Institute for Intelligent Systems, T¨ubingen, Germany kzhang@tuebingen.mpg.de Abstract In conventional causal discovery, structural equation models (SEM) are directly applied to the observed variables, meaning that the causal effect can be represented as a function of the direct causes themselves. However, in many real world problems, there are signiﬁcant dependencies in the variances or energies, which indicates that causality may possibly take place at the level of variances or energies. In this paper, we propose a probabilistic causal scale-mixture model with spatiotemporal variance dependencies to represent a speciﬁc type of generating mechanism of the observations. In particular, the causal mechanism including contemporaneous and temporal causal relations in variances or energies is represented by a Structural Vector AutoRegressive model (SVAR). We prove the identiﬁability of this model under the non-Gaussian assumption on the innovation processes. We also propose algorithms to estimate the involved parameters and discover the contemporaneous causal structure. Experiments on synthetic and real world data are conducted to show the applicability of the proposed model and algorithms. 1 Introduction Causal discovery aims to discover the underlying generating mechanism of the observed data, and consequently, the causal relations allow us to predict the effects of interventions on the system [15, 19]. For example, if we know the causal structure of a stock market, we are able to predict the reactions of other stocks against the sudden collapse of one share price in the market. A traditional way to infer the causal structure is by controlled experiments. However, controlled experiments are in general expensive, time consuming, technically infeasible and/or ethically prohibited. Thus, causal discovery from non-experimental data is of great importance and has drawn considerable attention in the past decades [15, 19, 16, 17, 12, 22, 2]. Probabilistic models such as Bayesian Networks (BNs) and Linear Non-Gaussian Acyclic Models (LiNGAM) have been proposed and applied to many real world problems [18, 13, 14, 21]. Conventional models such as LiNGAM assume that the causal relations are of a linear form, i.e., if the observed variable x is the cause of another observed variable y, we model the causal relation as y = αx+e, where α is a constant coefﬁcient and e is the additive noise independent of x. However, in many types of natural signals or time series such as MEG/EEG data [23] and ﬁnancial data [20], a common form of nonlinear dependency, as seen from the correlation in variances or energies, is found [5]. This observation indicates that causality may take place at the level of variances or energies instead of the observed variables themselves. Generally speaking, traditional methods cannot detect this type of causal relations. Another restriction of conventional causal models is that these models assume constant variances of the observations; this assumption is unrealistic for those data with strong heteroscedasticity [1]. 1 In this paper, we propose a new probabilistic model called Causal Scale-Mixture model with SpatioTemporal Variance Dependencies (CSM-STVD) incorporating the spatial and temporal variance or energy dependencies among the observed data. The main feature of the new model is that we model the spatiotemporal variance dependencies based on the Structural Vector AutoRegressive (SVAR) model, in particular the Non-Gaussian SVAR [11]. The contributions of this study are two-fold. First, we provide an alternative way to model the causal relations among the observations, i.e., causality in variances or energies. In this model, causality takes place at the level of variances or energies, i.e., the variance or energy of one observed series at time instant t0 is inﬂuenced by the variances or energies of other variables at time instants t ≤t0 and its past values at time instants t < t0. Thus, both contemporaneous and temporal causal relations in variances are considered. Secondly, we prove the identiﬁability of this model and more speciﬁcally, we show that Non-Gaussianity makes the model fully identiﬁable. Furthermore, we propose a method which directly estimates such causal structures without explicitly estimating the variances. 2 Related work To model the variance or energy dependencies of the observations, a classic method is to use a scalemixture model [5, 23, 9, 8]. Mathematically, we can represent a signal as si = uiσi, where ui is a signal with zero mean and constant variance, and σi is a positive factor which is independent of ui and modulates the variance or energy of si [5]. For multivariate case, we have s = u ⊙σ, (1) where ⊙means element-wise multiplication. In basic scale-mixture model, u and σ are statistically independent and the components ui are spatiatemporally independent, i.e. ui,tτ1 ⊥⊥uj,tτ2, ∀tτ1, tτ2. However, σi, the standard deviations of the observations, are dependent across i. The observation x, in many situations, is assumed to be a linear mixture of the source s, i.e., x = As, where A is a mixing matrix. In [5], Hirayama and Hyv¨arinen proposed a two-stage model. The ﬁrst stage is a classic ICA model [3, 10], where the observation x is a linear mixture of the hidden source s, i.e., x = As. On the second stage, the variance dependencies are modeled by applying a linear Non-Gaussian (LiN) SEM to the log-energies of the sources. yi = ∑ j hijyj + hi0 + ri, i = 1, 2, · · · , d, where yi = log ϕ(σi) are the log-energies of sources si and the nonlinear function ϕ is any appropriate measure of energy; ri are non-Gaussian distributed and independent of yj. To make the problem tractable, they assumed that ui are binary, i.e., ui ∈{−1, 1} and uniformly distributed. The parameters of this two-stage model including A and hij are estimated by maximum likelihood without approximation due to the uniform binary distribution assumption of u. However, this assumption is restrictive and thus may not ﬁt real world observations well. Furthermore, they assumed that σi are spatially dependent but temporally white. However, many time series show strong heterosecadasticity and temporal variance dependencies such as ﬁnancial time series and brain signals. Taking into account of temporal variance dependencies would improve the quality of the estimated underlying structure of the observed data. Another two-stage model for magnetoencephalography (MEG) or electroencephalography (EEG) data was propsoed earlier in [23]. The ﬁrst stage also performs linear separation; they proposed a blind source separation algorithm by exploiting the autocorrelations and time-varying variances of the sources. In the second stage, si(t) are modeled by autoregressive processes with L lags (AR(L)) driven by innovations ei(t). The innovation processes ei(t) are mutually uncorrelated and temporally white. However, ei(t) are not necessarily independent. They modeled ei(t) as follows: ei(t) = σitzi(t), where zi(t) ∼N(0, 1). (2) Two different methods are used to model the dependencies among the variances of the innovations. The ﬁrst method is causal-in-variance GARCH (CausalVar-GARCH). Speciﬁcally σ2 it are modeled by a multivariate GARCH model. The advantage of this model is that we are able to estimate the temporal causal structure in variances. However, this model provides no information about the 2 contemporaneous causal relations among the sources if there indeed exist such causal relations. The second method to model the variance dependencies is applying a factor model to the log-energies (log σ2 it) of the sources. The disadvantage of this method is that we cannot model the causal relations among the sources which are more interesting to us. In many real world observations, there are causal inﬂuences in variances among the observed variables. For instance, there are signiﬁcant mutual inﬂuences among the volatilities of the observed stock prices. We are more interested in investigating the underlying causal structure among the variances of the observed data. Consequently, in this paper, we consider the situation where the correlation in the variances of the observed data is interesting. That is, the ﬁrst stage of [5, 23] is not needed, and we focus on the second stage, i.e., modeling the spatiotemporal variance dependencies and causal mechanism among the observations. In the following sections, we propose our probabilistic model based on SVAR to describe the spatiotemporal variance dependencies among the observations. Our model is, as shown in later sections, closely related to the models introduced in [5, 23], but has signiﬁcant advantages: (1) both contemporaneous and temporal causal relations can be modeled; (2) this model is fully identiﬁable under certain assumptions. 3 Causal scale-mixture model with spatiotemporal variances dependencies We propose the causal scale-mixture model with spatiotemporal variance dependencies as follows. Let z(t) be the m × 1 observed vector with components zi(t), which are assumed to be generated according to the scale-mixture model: zi(t) = ui(t)σi(t). (3) Here we assume that ui(t) are temporally independent processes, i.e., ui(tτ1) ⊥⊥uj(tτ2), ∀tτ1 ̸= tτ2 but unlike basic scale-mixture model, here ui(t) may be contemporarily dependent, i.e., ui(t) ̸⊥⊥ uj(t), ∀i ̸= j. σ(t) is spatially and temporally independent of u(t). Using vector notation, zt = ut ⊙σt. (4) Here σit > 0 are related to the variances or energies of the observations zt and are assumed to be spatiotemporally dependent. As in [5, 23], let yt = log σt. In this paper, we model the spatiotemporal variance dependencies by a Structural Vector AutoRegressive model (SVAR), i.e., yt = A0yt + L ∑ τ=1 Bτyt−τ + ϵt, (5) where A0 contains the contemporaneous causal strengths among the variances of the observations, i.e., if [A0]ij ̸= 0, we say that yit is contemporaneously affected by yjt; Bτ contains the temporal (time-lag) causal relations, i.e., if [Bτ]ij ̸= 0, we say that yi,t is affected by yj,t−τ. Here, ϵt are i.i.d. mutually independent innovations. Let xt = log \|zt\| (In this model, we assume that ui(t) do not take value zero) and ηt = log \|ut\|.Take log of the absolute values of both sides of equation (4), then we have the following model: xt = yt + ηt, yt = A0yt + L ∑ τ=1 Bτyt−τ + ϵt. (6) We make the following assumptions on the model: A1 Both ηt and ϵt are temporally white with zero means. The components of ηt are not necessarily independent, and we assume that the covariance matrix of ηt is Ση. The components of ϵt are independent and Σϵ = I1. A2 The contemporaneous causal structure is acyclic, i.e., by simultaneous row and column permutations, A0 can be permuted to a strictly lower triangular matrix. BL is of full rank. 1Note that Σϵ = I is assumed just for convenience. A0 and Bτ can also be correctly estimated if Σϵ is a general diagonal covariance matrix. The explanation why the scaling indeterminacy can be eliminated is the same as LiNGAM given in [16]. 3 A3 The innovations ϵt are non-Gaussian, and ηt are either Gaussian or non-Gaussian. Inspired by the identiﬁability results of the Non-Gaussian state-space model in [24], we show that our model is identiﬁable. Note that our new model and the state-space model proposed in [24] are two different models, while interestingly by simple re-parameterization we can prove the following Lemma 3.1 and Theorem 3.1 following [24]. Lemma 3.1 Given the log-transformed observation xt = log \|zt\| generated by Equations (6), if the assumptions A1 ∼A2 hold, by solving simple linear equations involving the autocovariances of xt, the covariance Ση and ABτ can be uniquely determined, where A = (I −A0)−1; furthermore, A and Bτ can be identiﬁed up to some rotation transformations. That is, suppose that two models with parameters (A, {Bτ}L τ=1, Ση) and ( ˜A, { ˜Bτ}L τ=1, ˜Σ ˜η) generate the same observation xt, then we have Ση = ˜Σ ˜η, ˜A = AU, ˜Bτ = UT Bτ, where U is an orthogonal matrix. Non-Gaussianity of the innovations ϵt makes the model fully identiﬁable, as seen in the following theorem. Theorem 3.1 Given the log-transformed observation xt = log \|zt\| generated by Equations (6) and given L, if assumptions A1 ∼A3 hold, then the model is identiﬁable. In other words, suppose that two models with parameters (A, {Bτ}L τ=1, Ση) and ( ˜A, { ˜Bτ}L τ=1, ˜Σ ˜η) generate the same observation xt; then these two models are identical, i.e., we have ˜Σ ˜η = Ση, ˜A = A, ˜Bτ = Bτ, and ˜yt = yt. 4 Parameter learning and causal discovery In this section, we propose an effective algorithm to estimate the contemporaneous causal structure matrix A0 and temporal causal structure matrices Bτ, τ = 1, · · · , L (see (6)). 4.1 Estimation of ABτ We have shown that ABτ can be uniquely determined, where A = (I −A0)−1. The proof of Lemma 3.1 also suggests a way to estimate ABτ, as given below. Readers can refer to the appendix for the detailed mathematical derivation. Although we are aware that this method might not be statistically efﬁcient, we adopt this estimation method due to its great computational efﬁciency. Given the log-transformed observations xt = log \|zt\|, denoted by Rx(k) the autocovariance function of xt at lag k, we have Rx(k) = E(xtxT t+k). Based on the model assumptions A1 and A2, we have the following linear equations of the autocovarainces of xt.   Rx(L + 1) Rx(L + 2) ... Rx(2L)   =   Rx(L) Rx(L −1) · · · Rx(1) Rx(L + 1) Rx(L) · · · Rx(2) ... ... ... ... Rx(2L −1) Rx(2L −2) · · · Rx(L)   \| {z } ≜H   CT 1 CT 2 ... CT L   , (7) where Cτ = ABτ(τ = 1, · · · , L). As shown in the proof of Lemma 3.1, H is invertible. We can easily estimate ABτ by solving the linear Equations (7). 4.2 Estimation of A0 The estimations of ABτ(τ = 1, · · · , L) still contain the mixing information of the causal structures A0 and Bτ. In order to further obtain the contemporaneous and temporal causal relations, we need to estimate both A0 and Bτ(τ = 1, · · · , L). Here, we show that the estimation of A0 can be reduced to solving a Linear Non-Gaussian Acyclic Models with latent confounders. Substituting yt = xt −ηt into Equations (6), we have xt −ηt = L ∑ τ=1 ABτ(xt−τ −ηt−τ) + Aϵt. (8) 4 Since ABτ can be uniquely determined according to Lemma 3.1 or more speciﬁcally Equations (7), we can easily obtain ξt = xt −∑L τ=1 ABτxt−τ, then we have: ξt = Aϵt + ηt − L ∑ τ=1 ABτηt−τ. (9) This is exactly a Linear Non-Gaussian Acyclic Model with latent confounders and the estimation of A is a very challenging problem [6, 2]. To make to problem tractable, we further have the following two assumptions on the model: • A4 If the components of ηt are not independent, we assume that ηt follows a factor model: ηt = Dft, where the components of ft are spatially and temporally independent Gaussian factors and D is the factor loading matrix (not necessarily square). • A5 The components of ϵt are simultaneously super-Gaussian or sub-Gaussian. By replacing ηt with Dft , we have: ξt = Aϵt + Dft − L ∑ τ=1 ABτDft−τ \| {z } confounding effects . (10) To identify the matrix A which contains the contemporaneous causal information of the observed variables, we treat ft and ft−τ as latent confounders and the interpretation of assumption A4 is that we can treat the independent factors ft as some external factors outside the system. The Gaussian assumption of ft can be interpreted hierarchically as the result of central limit theorem because these factors themselves represent the ensemble effects of numerous factors from the whole environment. On the contrary, the disturbances ϵit are local factors that describe the intrinsic behaviors of the observed variables [4]. Since they are local and thus not regarded as the ensembles of large amount of factors. In this case, the disturbances ϵit are assumed to be non-Gaussian. The LiNGAM-GC model [2] takes into the consideration of latent confounders. In that model, the confounders are assumed to follow Gaussian distribution, which was interpreted as the result of central limit theorem. It mainly focuses on the following cause-effect pair: x = e1 + αc, y = ρx + e2 + βc, (11) where e1 and e2 are non-Gaussian and mutually independent, and c is the latent Gaussian confounder independent of the disturbances e1 and e2. To tackle the causal discovery problem of LiNGAMGC, it was ﬁrstly shown that if x and y are standardized to unit absolute kurtosis then \|ρ\| < 1 based on the assumption that e1 and e2 are simultaneously super-Gaussian or sub-Gaussian. Note that assumption A5 is a natural extension of this assumption. It holds in many practical problems, especially for ﬁnancial data. After the standardization, the following cumulant-based measure ˜Rxy was proposed [2]: ˜Rxy = (Cxy + Cyx)(Cxy −Cyx), where Cxy = ˆE{x3y} −3ˆE{xy}ˆE{x2}, Cyx = ˆE{xy3} −3ˆE{xy}ˆE{y2}, (12) and ˆE means sample average. It was shown that the causal direction can be identiﬁed simply by examining the sign of ˜Rxy, i.e., if ˜Rxy > 0, x →y is concluded; otherwise if ˜Rxy < 0, y → x is concluded. Once the causal direction has been identiﬁed, the estimation of causal strength is straightforward. The work can be extended to multivariate causal network discovery following DirectLiNGAM framework [17]. Here we adopt LiNGAM-GC-UK, the algorithm proposed in [2], to ﬁnd the contemporaneous casual structure matrix A0. Once A0 has been estimated, Bτ can be easily obtained by ˆBτ = (I−ˆA0) ˆCτ, where ˆA0 and ˆCτ are the estimations of A0 and ABτ, respectively. The algorithm for learning the model is summarized in the following algorithm. 5 Algorithm 1 Causal discovery with scale-mixture model for spatiotemporal variance dependencies 1: Given the observations zt, compute xt = log \|zt\|. 2: Subtract the mean ¯xt from xt, i.e., xt = xt −¯xt 3: Choose an appropriate lag L for the SVAR and then estimate ABτ where A = (I −A0)−1 and τ = 1, · · · , L, using Equations(7). 4: Obtain the residues by ξt = xt −∑L τ=1 ABτxt−τ. 5: Apply LiNGAM-GC algorithms to ξt and obtain the estimation of A0 and Bτ(τ = 1, · · · , L) and the corresponding comtemporaneous and temporal causal orderings. 5 Experiment We conduct experiments using synthetic data and real world data to investigate the effectiveness of our proposed model and algorithms. 5.1 Synthetic data We generate the observations according to the following model: zt = r ⊙ut ⊙σt, r is a m×1 scale vector of which the elements are randomly selected from interval [1.0, 6.0]; ut > 0 and ηt = log ut follows a factor model: ηt = Dft, where D is m × m and the elements of D are randomly selected from [0.2, 0.4] . fit are i.i.d. and fit ∼N(0, 0.5). Denoted by yt = log σt, we model the spatiotemporal variance dependencies of the observations xt by an SVAR(1): yt = A0yt + B1yt−1 + ϵt, where A0 is a m × m strictly lower triangular matrix of which the elements are randomly selected from [0.1, 0.2] or [−0.2, −0.1]; B1 is a m × m matrix of which the diagonal elements [B1]ii are randomly selected from [0.7, 0.8], 80% of the off-diagonal elements [B1]i̸=j are zero and the remaining 20% are randomly selected from [−0.1, 0.1]; ϵit are i.i.d. super-Gaussian generated by ϵit = sign(nit)\|nit\|2(nit ∼N(0, 1)) and normalized to unit variance. The generated observations are permuted to a random order. The task of this experiment is to investigate the performance of our algorithms in estimating the coefﬁcient matrix (I −A0)−1B1 and also the contemporaneous causal ordering induced by A0. We estimate the matrix (I −A0)−1B1 using Lemma 3.1 or speciﬁcally Equations (7). We use different algorithms: LiNGAM-GC-UK proposed in [2], C-M proposed in [7] and LiNGAM [16] to estimate the contemporaneous causal structure. We investigate the performances of different algorithms in the scenarios of m = 4 with sample size from 500 to 4000 and m = 8 with sample size from 1000 to 10000. For each scenario, we randomly conduct 100 independent trials and discard those trials where the SVAR processes are not stable. We calculate the accuracies of LiNGAM-GC-UK, C-M and LiNGAM in ﬁnding (1) whole causal ordering (2) exogenous variable (root) of the causal network. We also calculate the sum square error Err of estimated causal strength matrix of different algorithms with respect to the true one. The average SNR deﬁned as SNR = 10 log ∑ i V ar(ϵi) ∑ i V ar(fi) is about 13.85 dB. The experimental results are shown in Figure 1 and Table 1. Figure 1 shows the plots of the estimated entries of (I−A0)−1B1 versus the true ones when the dimension of the observations m = 8. From Figure 1, we can see that the matrix (I −A0)−1B1 is estimated well enough when the sample size is only 1000. This conﬁrms the correctness of our theoretical analysis of the proposed model. From Table 1, we can see that when the dimension of the observations is small (m = 4), all algorithms have acceptable performances. The performance of LiNGAM is the best when the sample size is small. This is because C-M and LiNGAM-GC-UK are cumulant-based methods which need sufﬁciently large sample size. When the dimension of the observations m increases to 8, we can see that the performances of C-M and LiNGAM degrade dramatically. While LiNGAM-GC-UK still successfully ﬁnds the exogenous variable (root) or even the whole contemporaneous causal ordering among the variances of the observations if the sample size is sufﬁciently large enough. This is mainly due to the fact that when the dimension increases, 6 −1 0 1 −1 −0.5 0 0.5 1 sample size: 2000 true parameters −1 0 1 −1 −0.5 0 0.5 1 estimated parameters sample size: 1000 −1 0 1 −1 −0.5 0 0.5 1 sample size: 4000 −1 0 1 −1 −0.5 0 0.5 1 estimated parameters sample size: 6000 −1 0 1 −1 −0.5 0 0.5 1 true parameters sample size: 8000 −1 0 1 −1 −0.5 0 0.5 1 sample size: 10000 Figure 1: Estimated entries causal strength matrix (I − A0)−1B1 vs the true ones (m = 8) FTSE FCHI GDAXI DJI NDX 1.005 0.8427 0.7404 -0.624 0.9833 0.4798 Figure 2: Contemporaneous causal network of the selected stock indices Table 1: Accuracy of ﬁnding the causal ordering sample size whole causal ordering ﬁrst variable found Err C-M LiNGAM LiNGAM-GC-UK C-M LiNGAM LiNGAM-GC-UK C-M LiNGAM LiNGAM-GC-UK m = 4 500 37% 70% 28% 61% 85% 60% 0.1101 0.0326 0.0938 1000 47% 75% 25% 25% 92% 72% 0.0865 0.024 0.0444 2000 74% 86% 81% 82% 90% 92% 0.0679 0.02 0.0199 3000 67% 78% 90% 79% 88% 96% 0.0716 0.0201 0.0126 4000 63% 83% 90% 81% 92% 94% 0.0669 0.0193 0.0109 m = 8 1000 0% 23.08% 8.79% 20.88% 75.82% 65.93% 0.8516 0.2318 0.3017 2000 1.14% 26.14% 25% 25% 70.45% 75% 0.7866 0.2082 0.1396 4000 0% 31.87% 58.24% 19.78% 82.41% 86.81% 0.7537 0.1916 0.0634 6000 0% 25.29% 83.91% 25.29% 75.86% 96.55% 0.7638 0.1843 0.0341 8000 2.20% 30.77% 80.22% 17.58% 79.12% 91.21% 0.7735 0.1824 0.029 10000 0% 23.53% 91.76% 12.94% 68.24% 97.64% 0.7794 0.194 0.0199 the confounding effects of Dft −(I −A)−1B1Dft−1 become more problematic such that the performances of C-M and LiNGAM are strongly affected by confounding effect. Table 1 also shows the estimation accuracies of the compared methods. Among them, LiNGAM-GC-UK signiﬁcantly outperforms other methods given sufﬁciently large sample size. In order to investigate the robustness of our methods against the Gaussian assumption on the external factors ft, we conduct the following experiment. The experimental setting is the same as that in the above experiment but here the external factors ft are non-Gaussian, and more speciﬁcally fit = sign(nit)\|nit\|p, where nit ∼N(0, 0.5). When p > 1, the factor is super-Gaussian and when p < 1 the factor is sub-Gaussian. We investigate the performances of LiNGAMGC-UK, LiNGAM and C-M in ﬁnding the whole causal ordering in difference scenarios where p = {0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6} with sample size of 6000. The results in Figure 3 show that LiNGAM-GC-UK achieved satisfying results compared to LiNGAM and C-M. This suggests that although LiNGAM-GC is developed based on the assumption that the latent confounders are Gaussian distributed, it is still robust in the scenarios where the latent confounders are mildly non-Gaussian with mild causal strengths. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 20 40 60 80 100 p accuracy(%) whole causal ordering LiNGAM−GC−UK C−M LiNGAM 0.5 1 1.5 −0.4 −0.2 0 0.2 0.4 p kurtosis kurtosis of ft Figure 3: Robustness against Gaussianity of ft 7 5.2 Real world data In this section, we use our new model to discover the causal relations among ﬁve major world stocks indices: (1) Dow Jones Industrial Average (DJI) (2) FTSE 100 (FTSE) (3) Nasdaq-100 (NDX) (4) CAC 40 (FCHI) (5) DAX (GDAXI), where DJI and NDX are stock indices in US, and FTSE, FCHI and GDAXI are indices in Europe. Note that because of the time difference, we believe that the causal relations among these stock indices are mainly acyclic, as we assumed in this paper. We collect the adjusted close prices of these selected indices from May 2nd, 2006 to April 12th, 2012, and use linear interpolation to estimate the prices on those dates when the data are not available. We apply our proposed model with SVAR(1) to model the spatiotemporal variance dependencies of the data. For the contemporaneous causal structure discovery, we use LiNGAM-GC-UK, C-M, LiNGAM2 and Direct-LiNGAM3 to estimate the causal ordering. The discovered causal orderings of different algorithms are shown in Table 2. From Table 2, we see that in the causal ordering Table 2: Contemporaneous causal ordering of the selected stock indices algorithm causal ordering LiNGAM-GC-UK {2} →{4} →{5} →{1} →{3} C-M {1} →{2} →{4} →{5} {1} →{3} LiNGAM {2} →{5} →{3} →{1} {2} →{4} Direct-LiNGAM {3} →{1} →{5} →{4} →{2} discovered by LiNGAM-GC-UK and LiNGAM, the stock indices in US, i.e., DJI and NDX are contemporaneously affected by the indices in Europe. Note that each stock index is given in local time. Because of the time difference between Europe and America and the efﬁcient market hypothesis (the market is quick to absorb new information and adjust stock prices relative to that), the contemporaneous causal relations should be from Europe to America, if they exist. This is consistent with the results our method and LiNGAM produced. Another interesting ﬁnding is that in the graphs obtained by LiNGAM-GC-UK and LiNGAM, we can see that FTSE is the root, which is consistent with the fact that London is the ﬁnancial centre of Europe and FTSE is regarded as Europe’s most important index. However, in results by C-M and DirectLiNGAM, we have the opposite direction, i.e., the stock indices in US is contemporaneously the cause of the indices in Europe, which is difﬁcult to interpret. The contemporaneous causal network of the stock indices are shown in Figure 2. Further interpretation on the discovered causal strengths needs expertise knowledge. 6 Conclusion In this paper, we investigate the causal discovery problem where causality takes place at the level of variances or energies instead of the observed variables themselves. We propose a causal scalemixture model with spatiotemporal variance dependencies to describe this type of causal mechanism. We show that the model is fully identiﬁable under the non-Gaussian assumption of the innovations. In addition, we propose algorithms to estimate the parameters, especially the contemporaneous causal structure of this model. Experimental results on synthetic data verify the practical usefulness of our model and the effectiveness of our algorithms. Results using real world data further suggest that our new model can possibly explain the underlying interaction mechanism of major world stock markets. Acknowledgments The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administration Region, China. 2LiNGAM converges to several local optima. We only show one of the discovered causal ordering here. The code is available at:http://www.cs.helsinki.ﬁ/group/neuroinf/lingam/ 3http://www.ar.sanken.osaka-u.ac.jp/∼inazumi/dlingam.html 8 References [1] T. Bollerslev. Generalized autoregressive conditional heteroskedasticity. Journal of econometrics, 31(3):307–327, 1986. [2] Z. Chen and L. Chan. Causal discovery for linear non-gaussian acyclic models in the presence of latent gaussian confounders. In Proceedings of the 10th international conference on Latent Variable Analysis and Signal Separation, pages 17–24. Springer-Verlag, 2012. [3] P. Comon. Independent component analysis, a new concept? Signal processing, 36(3):287–314, 1994. [4] R. Henao and O. Winther. Sparse linear identiﬁable multivariate modeling. 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4,477	Neurally Plausible Reinforcement Learning of Working Memory Tasks Jaldert O. Rombouts, Sander M. Bohte CWI, Life Sciences Amsterdam, The Netherlands {j.o.rombouts, s.m.bohte}@cwi.nl Pieter R. Roelfsema Netherlands Institute for Neuroscience Amsterdam, The Netherlands p.r.roelfsema@nin.knaw.nl Abstract A key function of brains is undoubtedly the abstraction and maintenance of information from the environment for later use. Neurons in association cortex play an important role in this process: by learning these neurons become tuned to relevant features and represent the information that is required later as a persistent elevation of their activity [1]. It is however not well known how such neurons acquire these task-relevant working memories. Here we introduce a biologically plausible learning scheme grounded in Reinforcement Learning (RL) theory [2] that explains how neurons become selective for relevant information by trial and error learning. The model has memory units which learn useful internal state representations to solve working memory tasks by transforming partially observable Markov decision problems (POMDP) into MDPs. We propose that synaptic plasticity is guided by a combination of attentional feedback signals from the action selection stage to earlier processing levels and a globally released neuromodulatory signal. Feedback signals interact with feedforward signals to form synaptic tags at those connections that are responsible for the stimulus-response mapping. The neuromodulatory signal interacts with tagged synapses to determine the sign and strength of plasticity. The learning scheme is generic because it can train networks in different tasks, simply by varying inputs and rewards. It explains how neurons in association cortex learn to 1) temporarily store task-relevant information in non-linear stimulus-response mapping tasks [1, 3, 4] and 2) learn to optimally integrate probabilistic evidence for perceptual decision making [5, 6]. 1 Introduction By giving reward at the right times, animals like monkeys can be trained to perform complex tasks that require the mapping of sensory stimuli onto responses, the storage of information in working memory and the integration of uncertain sensory evidence. While signiﬁcant progress has been made in reinforcement learning theory [2, 7, 8, 9], a generic learning rule for neural networks that is biologically plausible and also accounts for the versatility of animal learning has yet to be described. We propose a simple biologically plausible neural network model that can solve a variety of working memory tasks. The network predicts action-values (Q-values) for different possible actions [2], and it learns to minimize SARSA [10, 2] temporal difference (TD) prediction errors by stochastic gradient descent. The model has memory units inspired by neurons in lateral intraparietal (LIP) cortex and prefrontal cortex. Such neurons exhibit persistent activations for task related cues in visual working memory tasks [1, 11, 4]. Memory units learn to represent an internal state that allows the network to solve working memory tasks by transforming POMDPs into MDPs [25]. The updates for synaptic weights have two components. The ﬁrst is a synaptic tag [12] that arises from an interaction between feedforward and feedback activations. Tags form on those synapses that are responsible for the chosen actions by an attentional feedback process [13]. The second factor is a 1 Instant Off On Feedforward Feedback Feedforward Q-values Action Selection Sensory Association Action Figure 1: Model and learning (see section 2). Pentagons represent synaptic tags. global neuromodulatory signal δ that reﬂects the TD error, and this signal interacts with the tags to yield synaptic plasticity. TD-errors are represented by dopamine neurons in the ventral tegmental area and substantia nigra [9, 14]. The persistence of tags permits learning if time passes between synaptic activity and the animal’s choice, for example if information is stored in working memory or evidence accumulates before a decision is made. The learning rules are biologically plausible because the information required for computing the synaptic updates is available at the synapse. We call the new learning scheme AuGMEnT (Attention-Gated MEmory Tagging). We ﬁrst discuss the model and then show that it explains how neurons in association cortex learn to 1) temporarily store task-relevant information in non-linear stimulus-response mapping tasks [1, 3, 4] and 2) learn to optimally integrate probabilistic evidence for perceptual decision making [5, 6]. 2 Model AuGMEnT is modeled as a three layer neural network (Fig. 1). Units in the motor (output) layer predict Q-values [2] for their associated actions. Predictions are learned by stochastic gradient descent on prediction errors. The sensory layer contains two types of units; instantaneous and transient on(+)/off(-) units. Instantaneous units xi encode sensory inputs si(t), and + and - units encode positive and negative changes in sensory inputs with respect to the previous time step t −1: x+ i (t) = [si(t) −si(t −1)]+ ; x− i (t) = [si(t −1) −si(t)]+ , (1) where [.]+ is a threshold operator that returns 0 for all negative inputs but leaves positive inputs unchanged. Each sensory variable si is thus represented by three units xi, x+ i , x− i (we only explicitly write the time dependence if it is ambiguous). We denote the set of differentiating units as x′. The hidden layer models the association cortex and it contains regular units and memory units. The regular units j (Fig. 1, circles) are fully connected to the instantaneous units i in the sensory layer by connections vR ij; vR 0j is a bias weight. Regular unit activations yR j are computed as: yR j = σ(aR j ) = 1 1 + exp (θ −aR j ) with aR j = X i vR ijxi . (2) Memory units m (Fig. 1, diamonds) are fully connected to the +/- units in the sensory layer by connections vM lm and they derive their activations yM j (t) by integrating their inputs: yM m = σ(aM m ) with aM m = aM m (t −1) + X l vM lmx′ l , (3) with σ as deﬁned in eqn. (2). Output layer units k are fully connected to the hidden layer by connections wR jk (for regular hiddens, wR 0k is a bias weight) and wM mk (for memory hiddens). Activations are computed as: qk = X j yR j wR jk + X m yM m wM mk . (4) 2 A Winner Takes All (WTA) competition now selects an action based on the estimated Q-values. We used a max-Boltzmann [15] controller which executes the action with the highest estimated Qvalue with probability 1 −ϵ and otherwise it chooses an action with probabilities according to the Boltzmann distribution: Pr(zk = 1) = exp qk P k′ exp qk′ . (5) The WTA mechanism then sets the activation of the winning unit to 1 and the activation of all other units to 0; zk = δkK where δkK is the Kronecker delta function. The winning unit sends feedback signals to the earlier processing layers, informing the rest of the network about the action that was taken. This feedback signal interacts with the feedforward activations to give rise to synaptic tags on those synapses that were involved in taking the decision. The tags then interact with a neuromodulatory signal δ, which codes a TD error, to modify synaptic strengths. 2.1 Learning After executing an action, the environment returns a new observation s′, a scalar reward r, and possibly a signal indicating the end of a trial. The network computes a SARSA TD error [10, 2]: δ = r + γqK′ −qK , (6) where qK′ is the predicted value of the winning action for the new observation, and γ ∈[0, 1] is the temporal discount parameter [2]. AuGMEnT learns by minimizing the squared prediction error E: E = 1 2(δ)2 = 1 2(r + γqK′ −qK)2 , (7) The synaptic updates have two factors. The ﬁrst is a synaptic tag (Fig. 2, pentagons; equivalent to an eligibility trace in RL [2]) that arises from an interaction between feedforward and feedback activations. The second is a global neuromodulatory signal δ which interacts with these tags to yield synaptic plasticity. The updates can be derived by the chain rule for derivatives [16]. The update for synapses wR jk is: ∆wR jk = −β ∂E ∂qK TagR jk = βδ(t)TagR jk , (8) ∆TagR jk = (λγ −1)TagR jk + ∂qK ∂wR jk = (λγ −1)TagR jk + yR j zk , (9) where β is a learning rate, TagR jk are the synaptic tags on synapses between regular hidden units and the motor layer, and λ is a decay parameter [2]. Note that ∆wR jk ∝−β ∂E ∂qK ∂qK ∂wR jk = −β ∂E ∂wR jk , holding with equality if λγ = 0. If λγ > 0, tags decay exponentially so that synapses that were responsible for previous actions are also assigned credit for the currently observed error. Equivalently, updates for synapses between memory units and motor units are: ∆wM mk = βδ(t)TagM mk , (10) ∆TagM mk = (λγ −1)TagM mk + yM m zk . (11) The updates for synapses between instantaneous sensory units and regular association units are: ∆vR ij = −β ∂E ∂qK TagR ij = βδTagR ij , (12) ∆TagR ij = (λγ −1)TagR ij + ∂qK ∂yR j ∂yR j ∂aR j ∂aR j ∂vR ij , (13) = (λγ −1)TagR ij + w ′R KjyR j (1 −yR j )xi , (14) where w ′R Kj are feedback weights from the motor layer back to the association layer. The intuition for the last equation is that the winning output unit K provides feedback to the units in the association layer that were responsible for its activation. Association units with a strong feedforward connection also have a strong feedback connection. As a result, synapses onto association units that 3 provided strong input to the winning unit will have the strongest plasticity. This ‘attentional feedback’ mechanism was introduced in [13]. For convenience, we have assumed that feedforward and feedback weights are symmetrical, but they can also be trained as in [13]. For the updates for the synapses between +/- sensory units and memory units we ﬁrst approximate the activation aM m (see eqn. (3)) as: aM m = aM m (t −1) + X l vM lmx′ l ≈vM lm t X t′=0 x′ l(t′) , (15) which is a good approximation if the synapses vM lm change slowly. We can then write the updates as: ∆vM lm = −β ∂E ∂qK TagM lm = βδTagM lm , (16) ∆TagM lm = −TagM lm + ∂qK ∂yM m ∂yM m ∂aM m ∂aM m ∂vM lm , (17) = −TagM lm + w ′M KjyM m (t)(1 −yM m (t)) " t X t′=0 x′ l(t′) # . (18) Note that one can interpret a memory unit as a regular one that receives all sensory input in a trial simultaneously. For synapses onto memory units, we set λ = 0 to arrive at the last equation. The intuition behind the last equation is that because the activity of a memory unit does not decay, the inﬂuence of its inputs x′ l on the activity in the motor layer does not decay either (λγ = 0). A special condition occurs when the environment returns the end-trial signal. In this case, the estimate qK in eqn. (6) is set to 0 (see [2]) and after the synaptic updates we reset the memory units and synaptic tags, so that there is no confounding between different trials. AuGMEnT is biologically plausible because the information required for the synaptic updates is locally available by the interaction of feedforward and feedback signals and a globally released neuromodulator coding TD errors. As we will show, this mechanism is powerful enough to learn non-linear transformations and to create relevant working memories. 3 Experiments We tested AuGMEnT on a set of memory tasks that have been used to investigate the effects of training on neuronal activity in area LIP. Across all of our simulations, we ﬁxed the conﬁguration of the association layer (three regular units, four memory units) and Q-layer (three output units, for directing gaze to the left, center or right of a virtual screen). The input layer was tailored to the speciﬁc task (see below). In all tasks, we trained the network by trial and error to ﬁxate on a ﬁxation mark and to respond to task-related cues. As is usual in training animals for complex tasks, we used a small shaping reward rfix (arbitrary units) to facilitate learning to ﬁxate [17]. At the end of trials the model had to make an eye-movement to the left or right. The full task reward rfin was given if this saccade was accurate, while we aborted trials and gave no reward if the model made the wrong eye-movement or broke ﬁxation before the go signal. We used a single set of parameters for the network; β = 0.15; λ = 0.20; γ = 0.90; ϵ = 0.025 and θ = 2.5, which shifts the sigmoidal activation function for association units so that that units with little input have almost zero output. Initial synaptic weights were drawn from a uniform distribution U ∼[−0.25, 0.25]. For all tasks we used rfix = 0.2 and rfin = 1.5. 3.1 Saccade/Antisaccade The memory saccade/anti-saccade task (Fig. 2A) is based on [3]. This task requires a non-linear transformation and cannot be solved by a direct mapping from sensory units to Q-value units. Trials started with an empty screen, shown for one time step. Then either a black or white ﬁxation mark was shown indicating a pro-saccade or anti-saccade trial, respectively. The model had to ﬁxate on the ﬁxation mark within ten time-steps, or the trial was terminated. After ﬁxating for two timesteps, a cue was presented on the left or right and a small shaping reward rfix was given. The 4 Pro Anti Fixation Cue Delay Go R L R L R R L F 0 0.5 Assoc. 0 0.5 Assoc. 0 0.5 Assoc. F C G 0 1 Q F C G F C G F C G R R R R L L L L F F F F F F F F F F F F F F F F F F F F D D D D Left Cue Right Cue Left Cue Right Cue Pro-Saccade Anti-Saccade 0 0.65 0 0.65 SI Cue Location SI Trial Type A B C D Figure 2: A Memory saccade/antisaccade task. B Model network. In the association layer, a regular unit and two memory units are color coded gray, green and orange, respectively. Output units L,F,R are colored green, blue and red, respectively. C Unit activation traces for a sample trained network. Symbols in bottom graph indicate highest valued action. F, ﬁxation onset; C, cue onset; D, delay; G, ﬁxation offset (‘Go’ signal). Thick blue: ﬁxate, dashed green: left, red: right. D Selectivity indices of memory units in saccade/antisaccade task (black) and in pro-saccade only task (red). cue was shown for one time-step, and then only the ﬁxation mark was visible for two time-steps before turning off. In the pro-saccade condition, the offset of the ﬁxation mark indicated that the model should make an eye-movement towards the cue location to collect rfin. In the anti-saccade condition, the model had to make an eye-movement away from the cue location. The model had to make the correct eye-movement within eight time steps. The input to the model (Fig. 2B) consisted of four binary variables representing the information on the virtual screen; two for the ﬁxation marks and two for the cue location. Due to the +/−cells, the input layer thus had 12 binary units. We trained the models for at most 25, 000 trials, or until convergence. We measured convergence as the proportion of correct trials for the last 50 examples of all trial-types (N = 4). When this proportion reached 0.9 or higher for all trial-types, learning in the network was stopped and we evaluated accuracy on all trial types without stochastic exploration of actions. We considered learning successful if the model performed all trial-types accurately. We trained 10, 000 randomly initialized networks with and without a shaping reward (rfix = 0). Of the networks that received ﬁxation rewards, 9, 945 learned the task versus 7, 641 that did not receive ﬁxation rewards; χ2(1, N = 10, 000) = 2, 498, P < 10−6. The 10, 000 models trained with shaping learned the complete task in a median of 4, 117 trials. This is at least an order of magnitude faster than monkeys that typically learn such a task after months of training with more than 1, 000 trials per day, e. g. [6]. The activity of a trained network is illustrated in Fig. 2C. The Q-unit for ﬁxating at the center had strongest activity at ﬁxation onset and throughout the ﬁxation and memory delays, whereas the Qunit for the appropriate eye movement became more active after the go-signal. Interestingly, the activity of the Q-cells also depended on cue-location during the memory delay, as is observed, for example, in the frontal eye ﬁelds [18]. This activity derives from memory units in the association layer that maintain a trace of the cue as persistent elevation of their activity and are also tuned to the difference between pro- and antisaccade trials. To illustrate this, we deﬁned selectivity indices (SIs) to characterize the tuning of memory units to the difference between pro- or antisaccade trials and to the difference in cue location. The sensitivity of units to differences in trial types, SItype was \|0.5((RP L + RP R) −(RAL + RAR))\|, with R representing a units’ activation level (at ‘Go’ time) in pro (P) and anti-saccade trials (A) with a left (L) or right (R) cue. A unit has an SI of 0 if it does not distinguish between pro- and antisaccade trials, and an SI of 1 if it is fully active for one trial type and inactive for the other. The sensitivity to cue location, SIcue, was deﬁned \|0.5((RP L + RAL) −(RP R + RAR))\|. We trained 100 networks and found that units tuned to cue-location also tended to be selective for trial-type (black data points in Fig. 2D; SI correlation 0.79, (N = 400, P < 10−6)). To show that the association layer only learns to represent relevant features, we trained the same 100 networks using the same stimuli, but now only required pro5 S1 Delay Go Fixation S4 C Response (sp/s) 0.6 0 40 30 20 0.6 0 0.6 0 0.6 0 Time (s) 0.1 Activation 0.5 0.3 S1 S2 S3 S4 Model Data Symbols presented A +∞ 0.9 0.7 0.5 0.3 –0.3 –0.5 –0.7 –0.9 –∞ Assigned weights Favouring green Favouring red Shapes L F R G R G R B subjective WOE +∞ -∞ 0 1 -1 1 -1 0 Average Weights True symbol weights 0 +∞ -∞ 1 -1 3 -3 0 + – LogLR D Prop. of Units -1 -0.5 1 0.5 Spearman Correlation 0.4 0.2 E Data Model Figure 3: A Probabilistic classiﬁcation task (redrawn from [6]). B Model network C Population averages, conditional on LogLR-quintile (inset) for LIP neurons (redrawn from [6]) (top) and model memory units over 100, 000 trials after learning had converged (bottom). D Subjective weights inferred for a trained monkey (redrawn from [6]) (left) and average synaptic weights to an example memory unit (right) versus true symbol weights (A, right). E Histogram of weight correlations for 400 memory units from 100 trained networks. saccades, rendering the color of the ﬁxation point irrelevant. Memory units in the 97 converged networks now became tuned to cue-location but not to ﬁxation point color (Fig. 2D, red data points. SI Correlation 0.04, (N = 388, P > 0.48)), indicating that the association layer indeed only learns to represent relevant features. 3.2 Probabilistic Classiﬁcation Neurons in area LIP also play a role in perceptual decision making [5]. We hypothesized that memory units could learn to integrate probabilistic evidence for a decision. Yang and Shadlen [6] investigated how monkeys learn to combine information about four brieﬂy presented symbols, which provided probabilistic cues whether a red or green eye movement target was baited with reward (Fig. 3A). A previous model with only one layer of modiﬁable synapses could learn a simpliﬁed, linear version of this task [19]. We tested if AuGMEnT could train the network to adapt to the full complexity of the task that demands a non-linear combination of information about the four symbols with the position of the red and green eye-movement targets. Trials followed the same structure as described in section 3.1, but now four cues were subsequently added to the display. Cues were drawn with replacement from a set of ten (Fig. 3A, right), each with a different associated weight. The sum of these weights, W, determined the probability that rfin was assigned to the red target (R) as follows: P(R\|W) = 10W /(1 + 10W ). For the green target G, P(G\|W) = 1 −P(R\|W). At ﬁxation mark offset, the model had to make a saccade to the target with the highest reward probability. The sensory layer of the model (Fig. 3B) had four retinotopic ﬁelds with binary units for all possible symbols, a binary unit for the ﬁxation mark and four binary units coding the locations of the colored targets on the virtual screen. Due to the +/- units, this made 45 × 3 units in total. As in [6], we increased the difﬁculty of the task gradually (i. e. we used a shaping strategy) by increasing the set of input symbols (2, 4, . . . , 10) and sequence length (1−4) in eight steps. Training started with the ‘trump’ shapes which guarantee reward for the correct decision (Fig. 3A, right; see [6]) and then added the symbols with the next absolute highest weights. We determined that the task had been learned when the proportion of trials on which the correct decision was taken over the last n trials reached 0.85, where n was increased with the difﬁculty level l of the task. For the ﬁrst 5 levels, n(l) = 500 + 500l and for l = 6, 7, 8 n was 10, 000; 10, 000 and 20, 000, respectively. Networks were trained for at most 500, 000 trials. The behavior of a trained network is shown in ﬁgure 3C (bottom). Memory units integrated information for one of the choices over the symbol sequence and maintained information about the value of this choice as persistent activity during the memory delay. Their activation was correlated to the log likelihood that the targets were baited, just like LIP neurons [6] (Fig. 3C). The graphs show average activations of populations of real and model neurons in the four cue presentation epochs. Each pos6 Convergence Rate 0 1 A B 25 0 2 4 Median Convergence Speed Convergence Rate 0 1 25 0 2 4 Median Convergence Speed 6+8 384+512 192+256 96+128 48+64 24+32 12+16 3+4 Association layer units (reg. + mem.) 6+8 384+512 192+256 96+128 48+64 24+32 12+16 3+4 Association layer units (reg. + mem.) 6+8 384+512 192+256 96+128 48+64 24+32 12+16 3+4 Association layer units (reg. + mem.) 6+8 384+512 192+256 96+128 48+64 24+32 12+16 3+4 Association layer units (reg. + mem.) Figure 4: Association layer scaling behavior for A default learning parameters and, B optimized learning parameters. Error bars are 95% conﬁdence intervals. Parameters used are indicated by shading (see inset) sible sub-sequence of cues was assigned to a log-likelihood ratio (logLR) quintile, which correlates with the probability that the neurons’ preferred eye-movement is rewarded. Note that sub-sequences from the same trial might be assigned to different quintiles. We computed LogLR quintiles by enumerating all combinations of four symbols and then computing the probabilities of reward for saccades to red and green targets. Given these probabilities, we computed reward probability for all sub-sequences by marginalizing over the unknown symbols, i. e. to compute the probability that the red target was baited given only a ﬁrst symbol si, P(R\|si), we summed the probabilities for full sequences starting with si and divided by the number of such full sequences. We then computed the logLR for the sub-sequences and divided those into quintiles. For model units we rearranged the quintiles so that they were aligned in the last epoch to compute the population average. Synaptic weights from input neurons to memory cells became strongly correlated to the true weights of the symbols (Fig. 3D, right; Spearman correlation, ρ = 1, P < 10−6). Thus, the training of synaptic weights to memory neurons in parietal cortex can explain how the monkeys valuate the symbols [19]. We trained 100 networks on the same task and computed Spearman correlations for the memory unit weights with the true weights and found that in general they learn to represent the symbols (Fig. 3E). The learning scheme thus offers a biologically realistic explanation of how neurons in LIP learn to integrate relevant information in a probabilistic classiﬁcation task. 3.3 Scaling behavior To show that the learning scheme scales well, we ran a series of simulations with increasing numbers of association units. We scaled the number of association units by powers of two, from 21 = 2 (yielding 6 regular units and 8 memory units) to 27 = 128 (yielding 384 regular and 512 memory units). For each scale, we trained 100 networks on the saccade/antisaccade task, as described in section 3.1. We ﬁrst evaluated these scaled networks with the standard set of learning parameters and found that these yielded stable results within a wide range but that performance deteriorated for the largest networks (from 26 = 64; 192 regular units and 256 memory units) (Fig. 4A). In a second experiment (Fig. 4B), we also varied the learning rate (β) and trace decay (λ) parameters. We jointly scaled these parameters by 1 2, 1 4 and 1 8 and selected the parameter combination which resulted in the highest convergence rate and the fastest median convergence speed. It can be seen that the performance of the larger networks was at least as good as that of the default network, provided the learning parameters were scaled. Furthermore, we ran extensive grid-searches over the λ, β parameter space using default networks (not shown) and found that the model robustly learns both tasks with a wide range of parameters. 4 Discussion We have shown that AuGMEnT can train networks to solve working memory tasks that require nonlinear stimulus-response mappings and the integration of sensory evidence in a biologically plausible way. All the information required for the synaptic updates is available locally, at the synapses. The network is trained by a form of SARSA(λ) [10, 2], and synaptic updates minimize TD errors by stochastic gradient descent. Although there is an ongoing debate whether SARSA or Q-learning 7 [20] like algorithms are used by the brain [21, 22], we used SARSA because this has stronger convergence guarantees than Q-learning when used to train neural networks [23]. Although stability is considered a problem for neural networks implementing reinforcement learning methods [24], AuGMEnT robustly trained networks on our tasks for a wide range of model parameters. Technically, working memory tasks are Partially Observable Markov Decision Processes (POMDPs), because current observations do not contain the information to make optimal decisions [25]. Although AuGMEnT is not a solution for all POMDPs, as these are in general intractable [25], its simple learning mechanism is well able to learn challenging working memory tasks. The problem of learning new working memory representations by reinforcement learning is not well-studied. Some early work used the biologically implausible backpropagation-through-time algorithm to learn memory representations [26, 27]. Most other work pre-wires some aspects of working memory and only has a single layer of plastic weights (e. g. [19]), so that the learning mechanism is not general. To our knowledge, the model by O’Reilly and Frank [7] is most closely related to AuGMEnT. This model is able to learn a variety of working memory tasks, but it requires a teaching signal that provides the correct actions on each time-step and the architecture and learning rules are elaborate. AuGMEnT only requires scalar rewards and the learning rules are simple and well-grounded in RL theory [2]. AuGMEnT explains how neurons become tuned to relevant sensory stimuli in sequential decision tasks that animals learn by trial and error. The scheme uses units with properties that resemble cortical and subcortical neurons: transient and sustained neurons in sensory cortices [28], action-value coding neurons in frontal cortex and basal ganglia [29, 30] and neurons which integrate input and therefore carry traces of previously presented stimuli in association cortex. The persistent activity of these memory cells could derive from intracellular processes, local circuit reverberations or recurrent activity in larger networks spanning cortex, thalamus and basal ganglia [31]. The learning scheme adopts previously proposed ideas that globally released neuromodulatory signals code deviations from reward expectancy and gate synaptic plasticity [8, 9, 14]. In addition to this neuromodulatory signal, plasticity in AuGMEnT is gated by an attentional feedback signal that tags synapses responsible for the chosen action. Such a feedback signal exists in the brain because neurons at the motor stage that code a selected action enhance the activity of upstream neurons that provided input for this action [32], a signal that explains a corresponding shift of visual attention [33]. AuGMEnT trains networks to direct feedback (i.e. selective attention) to features that are critical for the stimulus-response mapping and are associated with reward. 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4,478	Simultaneously Leveraging Output and Task Structures for Multiple-Output Regression Piyush Rai† Dept. of Computer Science University of Texas at Austin Austin, TX piyush@cs.utexas.edu Abhishek Kumar† Dept. of Computer Science University of Maryland College Park, MD abhishek@cs.umd.edu Hal Daum´e III Dept. of Computer Science University of Maryland College Park, MD hal@umiacs.umd.edu Abstract Multiple-output regression models require estimating multiple parameters, one for each output. Structural regularization is usually employed to improve parameter estimation in such models. In this paper, we present a multiple-output regression model that leverages the covariance structure of the latent model parameters as well as the conditional covariance structure of the observed outputs. This is in contrast with existing methods that usually take into account only one of these structures. More importantly, unlike some of the other existing methods, none of these structures need be known a priori in our model, and are learned from the data. Several previously proposed structural regularization based multiple-output regression models turn out to be special cases of our model. Moreover, in addition to being a rich model for multiple-output regression, our model can also be used in estimating the graphical model structure of a set of variables (multivariate outputs) conditioned on another set of variables (inputs). Experimental results on both synthetic and real datasets demonstrate the effectiveness of our method. 1 Introduction Multivariate response prediction, also known as multiple-output regression [3] when the responses are real-valued vectors, is an important problem in machine learning and statistics. The goal in multiple-output regression is to learn a model for predicting K > 1 real-valued responses (the output) from D predictors or features (the input), given a training dataset consisting of N inputoutput pairs. Multiple-output prediction is also an instance of the problem of multitask learning [5, 10] where predicting each output is a task and all the tasks share the same input data. Multipleoutput regression problems are encountered frequently in various application domains. For example, in computational biology [11], we often want to predict the gene-expression levels of multiple genes based on a set of single nucleotide polymorphisms (SNPs); in econometrics [17], we often want to predict the stock prices in the future using relevant macro-economic variables and stock prices in the past as inputs; in geostatistics, we are often interested in jointly predicting the concentration levels of different heavy metal pollutants [9]; and so on. One distinguishing aspect of multiple-output regression is that the outputs are often related to each other via some underlying (and often a priori unknown) structure. A part of this can be captured by the imposing a relatedness structure among the regression coefﬁcients (e.g., the weight vectors in a linear regression model) of all the outputs. We refer to the relatedness structure among the regression coefﬁcients as task structure. However, there can still be some structure left in the outputs that is not explained by the regression coefﬁcients alone. This can be due to a limited expressive power of our chosen hypothesis class (e.g., linear predictors considered in this paper). The residual structure that is left out when conditioned on inputs will be referred to as output structure here. This can be also be seen as the covariance structure in the output noise. It is therefore desirable to simultaneously learn †Contributed equally 1 and leverage both the output structure and the task structure in multiple-output regression models for improved parameter estimation and prediction accuracy. Although some of the existing multiple-output regression models have attempted to incorporate such structures [17, 11, 13], most of these models are restrictive in the sense that (1) they usually exploit only one of the two structures (output structure or task structure, but not both), and (2) they assume availability of prior information about such structures which may not always be available. For example, Multivariate Regression with Covariance Estimation [17] (MRCE) is a recently proposed method which learns the output structure (in form of the covariance matrix for correlated noise across multiple outputs) along with the regression coefﬁcients (i.e., the weight vector) for predicting each output. However MRCE does not explicitly model the relationships among the regression coefﬁcients of the multiple tasks and therefore fails to account for the task structure. More recently, [14] proposed an extension of the MRCE model by allowing weighting the individual entries of the regression coefﬁcients and the entries of the output (inverse) covariance matrix, but otherwise this model has essentially the same properties as MRCE. Among other works, Graph-guided Fused Lasso [11] (GFlasso) incorporates task structure to some degree by assuming that the regression coefﬁcients of all the outputs have similar sparsity patterns. This amounts to assuming that all the outputs share almost same set of relevant features. However, GFlasso assumes that output graph structure is known which is rarely true in practice. Some other methods such as[13] take into account the task structure by imposing structural sparsity on the regression coefﬁcients of the multiple tasks but again assume that output structure is known a priori and/or is of a speciﬁc form. In [22], the authors proposed a multitask learning model by explicitly modeling the task structures as the task covariance matrix but this model does not take into account the output structure which is important in multiple-output regression problems. In this paper, we present a multiple-output regression model that allows leveraging both output structure and task structure without assuming an a priori knowledge of either. In our model, both output structure and task structure are learned from the data, along with the regression coefﬁcients for each task. Speciﬁcally, we model the output structure using the (inverse) covariance matrix of the correlated noise across the multiple outputs, and the task structure using the (inverse) covariance matrix of the regression coefﬁcients of the multiple tasks being learned in the model. By explicitly modeling and learning the output structure and task structure, our model also addresses the limitations of the existing models that typically assume certain speciﬁc type of output structures (e.g., tree [13]) or task structures (e.g., shared sparsity [11]). In particular, a model with task relatedness structure based on shared sparsity on the task weight vectors may not be appropriate in many real applications where all the features are important for prediction and the true task structure is at a more higher level (e.g., weight vectors for some tasks are closer to each other compared to others). Apart from providing a ﬂexible way of learning multiple-output regression, our model can also be used for the problem of conditional inverse covariance estimation of the (multivariate) outputs that depend on another set of inputs variables, an important problem that has been gaining signiﬁcant attention recently [23, 15, 20, 4, 7, 6]. 2 Multiple-Output Regression In multiple-output regression, each input is associated with a vector of responses and the goal is the learn the input-output relationship given some training data consisting of input-output pairs. Formally, given an N × D input matrix X = [x1, . . . , xN]⊤and an N × K output matrix Y = [y1, . . . , yN]⊤, the goal in multiple-output regression is to learn the functional relationship between the inputs xn ∈RD and the outputs yn ∈RK. For a linear regression model, we write: yn = W⊤xn + b + ǫn ∀n = 1, . . . , N (1) Here W = [w1, . . . , wK] denotes the D × K matrix where wk denotes the regression coefﬁcient of the k-th output, b = [b1, . . . , bK]⊤∈RK is a vector of bias terms for the K outputs, and ǫn = [ǫn1, . . . , ǫnK]⊤∈RK is a vector consisting of the noise for each of the K outputs. The noise is typically assumed to be Gaussian with a zero mean and uncorrelated across the K outputs. Standard parameter estimation for Equation 1 involves maximizing the (penalized) log-likelihood of the model, or equivalently minimizing the (regularized) loss function over the training data: arg min W,b tr((Y −XW −1b⊤)(Y −XW −1b⊤)⊤) + λR(W) (2) 2 where tr(.) denotes matrix trace, 1 an N ×1 vector of all 1s and R(W) the regularizer on the weight matrix W consisting of the regression weight vectors of all the outputs. For a choice of R(W) = tr(W⊤W) (the ℓ2-squared norm, equivalent to assuming independent, zero-mean Gaussian priors on the weight vectors), solving Equation 2 amounts to solving K independent regression problems and this solution ignores any correlations among the outputs or among the weight vectors. 3 Multiple-Output Regression with Output and Task Structures To take into account both conditional output covariance and the covariance among the weight vectors W = [w1, . . . , wK], we assume a full covariance matrix Ωof size K × K on the output noise distribution to capture conditional output covariance, and a structured prior distribution on the weight vector matrix W that induces structural regularization of W. We place the following prior distribution on W p(W) ∝ K Y k=1 Nor(wk\|0, ID)MN D×K(W\|0D×K, ID ⊗Σ) (3) where MN D×K(M, A ⊗B) denotes the matrix-variate normal distribution with M ∈RD×K being its mean, A ∈RD×D its row-covariance matrix and B ∈RK×K its column-covariance matrix. Here ⊗denotes the Kronecker product. In this prior distribution, the Nor(wk\|0, ID) factors regularize the weight vectors wk individually, and the MN D×K(W\|0D×K, ID ⊗Σ) term couples the K weight vectors, allowing them to share statistical strength. To derive our objective function, we start by writing down the likelihood of the model, for a set of N i.i.d. observations: N Y n=1 p(yn\|xn, W, b) = N Y n=1 Nor(yn\|W⊤xn + b, Ω) (4) In the above, a diagonal Ωwould imply that the K outputs are all conditionally independent of each other. In this paper, we assume a full Ωwhich will allow us to capture the conditional output correlations. Combining the prior on W and the likelihood, we can write down the posterior distribution of W: p(W\|X, Y, b, Ω, Σ) ∝ p(W) QN n=1 p(yn\|xn, W, b) = QK k=1 Nor(wk\|0, ID) MN D×K(W\|0D×K, ID ⊗Σ) QN n=1 Nor(yn\|W⊤xn + b, Ω) Taking the log of the above and simplifying the resulting expression, we can then write the negative log-posterior of W as (ignoring the constants): tr((Y −XW −1b⊤)Ω−1(Y −XW −1b⊤)⊤) + N log \|Ω\| + tr(WW⊤) + tr(WΣ−1W⊤) + D log \|Σ\| where 1 denotes a N × 1 vector of all 1s. Note that in the term tr(WΣ−1W⊤), the inverse covariance matrix Σ−1 plays the role of coupling pairs of weight vectors, and therefore controls the amount of sharing between any pair of tasks. The task covariance matrix Σ as well as the conditional output covariance matrix Ωwill be learned from the data. For reasons that will become apparent later, we parameterize our model in terms of the inverse covariance matrices Ω−1 and Σ−1 instead of covariance matrices. With this parameterization, the negative log-posterior becomes: tr((Y −XW −1b⊤)Ω−1(Y −XW −1b⊤)⊤) −N log \|Ω−1\| + tr(WW⊤) + tr(WΣ−1W⊤) −D log \|Σ−1\| (5) The objective function in Equation 5 naturally imposes positive-deﬁnite constraints on the inverse covariance matrices Ω−1 and Σ−1. In addition, we will impose sparsity constraints (via an ℓ1 penalty) on Ω−1 and Σ−1. Sparsity on these parameters is appealing in this context for two reasons: (1) Sparsity leads to improved robust estimates [19, 8] of Ω−1 and Σ−1, and (2) Sparsity supports the notion that the output correlations and the task correlations tend to be sparse [21, 4, 8] 3 – not all pairs of outputs are related (given the inputs and other outputs), and likewise not all task pairs (and therefore the corresponding weight vectors) are related. Finally, we will also introduce regularization hyperparameters to control the trade-off between data-ﬁt and model complexity. Parameter estimation in the model involves minimizing the negative log-posterior which is equivalent to minimizing the (regularized) loss function. The minimization problem is given as arg min W,b,Σ−1,Ω−1 tr((Y −XW −1b⊤)Ω−1(Y −XW −1b⊤)⊤) −N log \|Ω−1\| + λ tr(WW⊤) +λ1 tr(WΣ−1W⊤) −D log \|Σ−1\| + λ2\|\|Ω−1\|\|1 + λ3\|\|Σ−1\|\|1 (6) where \|\|A\|\|1 denotes the sum of absolute values of the matrix A. Note that by replacing the regularizer tr(WW⊤) with a sparsity inducing regularizer on the individual weight vectors w1, . . . , wK, one can also learn Lasso-like sparsity [19] in the regression weights. In this exposition, however, we consider ℓ2 regularization on the regression weights and let the tr(WΣ−1W⊤) term capture the similarity between the weights of two tasks by learning the task inverse covariance matrix Σ−1. The above cost function is not jointly convex in the variables but is individually convex in each variable when others are ﬁxed. We adopt an alternating optimization strategy that was empirically observed to converge in all our experiments. More details are provided in the experiments section. Finally, although it is not the main goal of this paper, since our model provides an estimate of the inverse covariance structure Ω−1 of the outputs conditioned on the inputs, it can also be used for the more general problem of estimating the conditional inverse covariance [23, 15, 20, 4, 7] of a set of variables y = {y1, . . . , yK} conditioned on another set of variables x = {x1, . . . , xD}, given paired samples of the form {(x1, y1), . . . , (xN, yN)}. 3.1 Special Cases In this section, we show that our model subsumes/generalizes some previously proposed models for multiple-output regression. Some of these include: • Multivariate Regression with Covariance Estimation (MRCE-ℓ2): With the task inverse covariance matrix Σ−1 = IK and the bias term set to zero, our model results in the ℓ2 regularized weights variant of the MRCE model [17] which would be equivalent to minimizing the following objective: arg min W,Ω−1 tr((Y −XW)Ω−1(Y −XW)⊤)+λ tr(WW⊤)−N log \|Ω−1\|+λ2\|\|Ω−1\|\|1 • Multitask Relationship Learning for Regression (MTRL): With the output inverse covariance matrix Ω−1 = IK and the sparsity constraint on Σ−1 dropped, our model results in the regression version of the multitask relationship learning model proposed in [22]. Speciﬁcally, the corresponding objective function would be: arg min W,Σ−1 tr((Y−XW)(Y−XW)⊤)+λ tr(WW⊤)+λ1 tr(WΣ−1W⊤)−D log \|Σ−1\| In [22], the −log \|Σ−1\| term is dropped since the authors solve their cost function in terms of Σ and this term is concave in Σ. A constraint of tr(Σ) = 1 was introduced in its place to restrict the complexity of the model. We keep the log \| · \| constraint in our cost function since we parameterize our model in terms of Σ−1, and −log \|Σ−1\| is convex in Σ−1. 3.2 Optimization We take an alternating optimization approach to solve the optimization problem given by Equation 6. Each sub-problem in the alternating optimization steps is convex. The matrices Σ and Ωare initialized to I in the beginning. The bias vector b is initialized to 1 N Y⊤1. Optimization w.r.t. W when Ω−1, Σ−1 and b are ﬁxed: Given Ω−1, Σ−1, b, the matrix W consisting of the regression weight vectors of all the tasks can be obtained by solving the following optimization problem: arg min W tr((Y−XW−1b⊤)Ω−1(Y−XW−1b⊤)⊤)+λ tr(WW⊤)+λ1 tr(WΣ−1W⊤) (7) 4 The estimate ˆ W is given by solving the following system of linear equations w.r.t. W: Ω−1 ⊗X′X + λ1Σ−1 + λIK ⊗ID vec(W) = vec(X′(Y −1b⊤)Ω−1) (8) It is easy to see that with Ωand Σ set to identity, the model becomes equivalent to solving K regularized independent linear regression problems. Optimization w.r.t. b when Ω−1, Σ−1 and W are ﬁxed: Given Ω−1, Σ−1, W, the bias vector b for all the K outputs can be obtained by solving the following optimization problem: arg min b tr((Y −XW −1b⊤)Ω−1(Y −XW −1b⊤)⊤) (9) The estimate ˆb is given by ˆb = 1 N PN n=1(Y −XW)⊤1 Optimization w.r.t. Σ−1 when Ω−1, W and b are ﬁxed: Given Ω−1, W, b, the task inverse covariance matrix Σ−1 can be estimated by solving the following optimization problem: arg min Σ−1 λ1 tr(WΣ−1W⊤) −D log \|Σ−1\| + λ3\|\|Σ−1\|\|1 (10) It is easy to see that the above is an instance of the standard inverse covariance estimation problem with sample covariance λ1 D W⊤W, and can be solve using standard tools for inverse covariance estimation. We use the graphical Lasso procedure [8] to solve Equation 10 to estimate Σ−1: ˆΣ−1 = gLasso(λ1 D W⊤W, λ3) (11) If we assume Σ−1 to be non-sparse, we can drop the ℓ1 penalty on Σ−1 from Equation 10. However, the solution to Σ−1 will not be deﬁned (when K > D) or will overﬁt (when K is of the same order as D). To avoid this, we add a regularizer of the form λ tr(Σ−1) to Equation 10. This can be seen as imposing a matrix variate Gaussian prior on Σ−1/2 with both row and column covariance matrices equal to I to make the solution well deﬁned. In the previous case of sparse Σ−1, the solution was well deﬁned because of the sparsity prior on Σ−1. The optimization problem for Σ−1 is then given as arg min Σ−1 λ1 tr(WΣ−1W⊤) −D log \|Σ−1\| + λ tr Σ−1 . (12) Equation 12 admits a closed form solution which is given by λ1W⊤W+λI D −1 . For the non-sparse Σ−1 case, we keep the parameter λ same as the hyperparameter for the term tr(WW⊤) in Equation 6. Optimization w.r.t. Ω−1 when Σ−1, W and b are ﬁxed: Given Σ−1, W, b, the task inverse covariance matrix Ω−1 can be estimated by solving the following optimization problem: arg min Ω−1 tr((Y −XW −1b⊤)Ω−1(Y −XW −1b⊤)⊤) −N log \|Ω−1\| + λ2\|\|Ω−1\|\|1 (13) It is again easy to see that the above problem is an instance of the standard inverse covariance estimation problem with sample covariance 1 N (Y−XW−1b⊤)′(Y−XW−1b⊤), and can be solved using standard tools for inverse covariance estimation. We use the graphical Lasso procedure [8] to solve Equation 10 to estimate Σ−1: ˆΩ−1 = gLasso( 1 N (Y −XW −cb⊤)⊤(Y −XW −cb⊤), λ2) (14) 4 Experiments In this section, we evaluate our model by comparing it with several relevant baselines on both synthetic and real-world datasets. Our main set of results are on multiple-output regression problems on which we report mean-squared errors averaged across all the outputs. However, since our model also provides an estimate of the conditional inverse covariance structure Ω−1 of the outputs, in Section 4.3 we provide experimental results on the structure recovery task as well. We compare our method with following baselines: 5 • Independent regressions (RLS): This baseline learns regularized least squares (RLS) regression model for each output, without assuming any structure among the weight vectors or among the outputs. This corresponds to our model with Σ = IK and Ω= IK. The weight vector of each individual problem is ℓ2 regularized with a hyperparameter λ. • Curds and Whey (C&W): The predictor in Curds and Whey [3] takes the form Wcw = WrlsUΛU−, where Wrls denotes the regularized least squares predictor, the columns of matrix U are the projection directions for the responses Y obtained from canonical correlation analysis (CCA) of X and Y, and U−denotes Moore-Penrose pseudoinverse of U. The diagonal matrix Λ contains the shrinkage factors for each CCA projection direction. • Multi-task Relationship Learning (MTRL): This method leverages task relationships by assuming a matrix-variate prior on the weight matrix W [22]. We chose this baseline because of its ﬂexibility in modeling the task relationships by “discovering” how the weight vectors are related (via Σ−1), rather than assuming a speciﬁc structure on them such as shared sparsity [16], low-rank assumption [2], etc. However MTRL in the multiple-output regression setting cannot take into account the output structure. It is therefor a special case of our model if we assume the output inverse covariance matrix Ω−1 = I. The MTRL approach proposed in [22] does not have sparse penalty on Σ−1. We experimented with both sparse and non-sparse variants of MTRL and report the better of the two results here. • Multivariate Regression with Covariance Estimation (MRCE-ℓ2): This baseline is the ℓ2 regularized variant of the MRCE model [17]. MRCE leverages output structure by assuming a full noise covariance in multiple-output regression and learning it along with the weight matrix W from the data. MRCE however cannot take into account the task structure because it cannot capture the relationships among the columns of W. It is therefore a special case of our model if we assume the task inverse covariance matrix Σ−1 = I. We do not compare with the original ℓ1 regularized MRCE [17] to ensure a fair comparison by keeping all the models non-sparse in weight vectors. In the experiments, we refer to our model as MROTS (Multiple-output Regression with Output and Task Structures). We experiment with two variants of our proposed model, one without a sparsity inducing penalty on the task coupling matrix Σ−1 (called MROTS-I), and the other with the sparse penalty on Σ−1 (called MROTS-II). The hyperparameters are selected using four-fold cross-validation. Both MTRL and MRCE-ℓ2 have two hyperparameters each and these are selected by searching on a two-dimensional grid. For the proposed model with non-sparse Σ−1, we ﬁx the hyperparameter λ in Equations 6 and 12 as 0.001 for all the experiments. This is used to ensure that the task inverse covariance matrix estimate ˆ Σ−1 exists and is robust when number of response variables K is of the same order or larger than the input dimension D. The other two parameters λ1 and λ2 are selected using cross-validation. For sparse Σ−1 case, we use the same values of λ1 and λ2 that were selected for non-sparse case, and only the third parameter λ3 is selected by crossvalidation. This procedure avoids a potentially expensive search over a three dimensional grid. The hyperparameter λ in Equation 6 is again ﬁxed at 0.001. 4.1 Synthetic data We describe the process for synthetic data generation here. First, we generate a random positive deﬁnite matrix Σ−1 which will act as the task inverse covariance matrix. Next, a matrix V of size D×K is generated with each entry sampled from a zero mean and 1/D variance normal distribution. We compute the square-root S of Σ (= SS, where S is also a symmetric positive deﬁnite matrix), and S is used to generate the ﬁnal weight matrix W as W = VS. It is clear that for a W generated in this fashion, we will have E[WT W] = SS = Σ. This process generates W such that its columns (and therefore the weight vectors for different outputs) are correlated. A bias vector b of size K is generated randomly from a zero mean unit variance normal distribution. Then we generate a sparse random positive deﬁnite matrix Ω−1 that acts as the conditional inverse covariance matrix on output noise making the outputs correlated (given the inputs). Next, input samples are generated i.i.d. from a normal distribution and the corresponding multivariate output variables are generated as yi = Wxi +b+ǫi, ∀i = 1, 2, . . . , N, where ǫi is the correlated noise vector randomly sampled from a zero mean normal distribution with covariance matrix Ω. We generate three sets of synthetic data using the above process to gauge the effectiveness of the proposed model under varying circumstances: (i) D = 20, K = 10 and non-sparse Σ−1, (ii) 6 Method Synth data I Synth data II Synth data III Paper I Paper II Gene data RLS 37.29 3.22 3.94 1.08 1.04 1.92 C&W 37.14 21.88 7.06 1.08 1.08 1.51 MTRL 34.45 3.12 3.86 1.07 1.03 1.24 MRCE-ℓ2 29.84 3.08 3.92 1.36 1.03 1.55 MROTS-I 26.65 2.61 3.75 0.90 1.03 1.18 MROTS-II 25.90 2.60 3.55 0.90 1.03 1.20 Table 1: Prediction error (MSE) on synthetic and real datasets. RLS: Independent regression, C&W: Curds and Whey model [3], MTRL: Multi-task relationship learning [22], MRCE-ℓ2: The ℓ2-regularized version of MRCE [17], MROTS-I: our model without sparse penalty on Σ−1, MROTS-II: our model with sparse penalty on Σ−1. Best results are highlighted in bold fonts. D = 10, K = 20 and non-sparse Σ−1, and (iii) D = 10, K = 20 and sparse Σ−1. We also experiment with varying number of training samples (N = 20, 30, 40 and 50). 10 20 30 40 50 60 0 50 100 150 200 250 300 Number of training samples Mean square error RLS C&W MTRL MRCE−l2 MROTS−I MROTS−II (a) Synthetic data I 10 20 30 40 50 60 2.5 3 3.5 4 4.5 5 Number of training samples Mean square error RLS MTRL MRCE−l2 MROTS−I MROTS−II (b) Synthetic data II 0 10 20 30 20 30 40 50 60 Iterations MSE or Obj. value MSE OBJ. VALUE (c) Synthetic data I 0 10 20 30 0.8 1 1.2 1.4 Iterations MSE or Obj. value MSE OBJ. VALUE (d) Paper data I Figure 1: (a) and (b): Mean Square Error with varying number of training samples, (c) and (d): Mean Square Error and the value of the Objective function with increasing iterations for the proposed method. 4.2 Real data We also evaluate our model on the following real-world multiple-output regression datasets: • Paper datasets: These are two multivariate multiple-response regression datasets from paper industry [1]. The ﬁrst dataset has 30 samples with each sample having 9 features and 32 outputs. The second dataset has 29 samples (after ignoring one sample with missing response variables), each having 9 features and 13 outputs. We take 15 samples for training and the remaining samples for test. • Genotype dataset: This dataset has genotypes as input variables and phenotypes or observed traits as output variables [12]. The number of genotypes (features) is 25 and the number of phenotypes (outputs) is 30. We have a total of 100 samples in this dataset and we split it equally into training and test data. The results on synthetic and real-world datasets are shown in Table 1. For synthetic datasets, the reported results are with 50 training samples. Independent linear regression performs the worst on all synthetic datasets. MRCE-ℓ2 performs better than MTRL on ﬁrst and second synthetic data while MTRL is better on the third dataset. This mixed behavior of MRCE-ℓ2 and MTRL supports our motivation that both task structure (i.e., relationships among weight vectors) and output structure are important in multiple-output regression. Both MTRL and MRCE-ℓ2 are special cases of our model with former ignoring the output structure (captured by Ω−1) and the latter ignoring the weight vector relationships (captured by Σ−1). Both variants of our model (MROTS-I and MROTS-II) perform signiﬁcantly better than the compared baselines. The improvement with sparse Σ−1 variant is more prominent on the third dataset which is generated with sparse Σ−1 (5.33% relative reduction in MSE), than on the ﬁrst two datasets (2.81% and 0.3% relative reduction in MSE). However, in our experiments, the sparse Σ−1 variant (MROTS-II) always performed better or as good as the nonsparse variant on all synthetic and real datasets, which suggests that explicitly encouraging zero entries in Σ−1 leads to better estimates of task relationships (by avoiding spurious correlations between weight vectors). This can potentially improve the prediction performance. Finally, we also note that the Curds & Whey method [3] performs signiﬁcantly worse than RLS for Synthetic data II and III. C&W uses CCA to project the response matrix Y to a lower min(D, K)-dimensional space learning min(D, K) predictors there and then projecting them back to the original K-dimensional 7 space. This procedure may end up throwing away relevant information from responses if K is much higher than D. These empirical results suggest that C&W may adversely affect the prediction performance when the number of response variables K is higher than the number of explanatory variables D (D = 2K in these cases). On the real-world datasets too, our model performs better than or on par with the compared baselines. Both MROTS-I and MROTS-II perform signiﬁcantly better than the other baselines on the ﬁrst Paper dataset (9 features and 32 outputs per sample). All models perform almost similarly on the second Paper dataset (9 features and 13 outputs per sample), which could be due to the absence of a strong task or output structure in this data. C&W does not preform well on both Paper datasets which might be due to the reason discussed earlier. On the genotype-phenotype prediction task too, both our models achieve better average mean squared errors than the other baselines, with both variants performing roughly comparably. We also evaluate our model’s performance with varying number of training examples and compare with the other baselines. Figures 1(a) and 1(b) show the plots of mean square error vs. number of training examples for ﬁrst two synthetic datasets. We do not plot C&W for Synthetic data II since it performs worse than RLS. On the ﬁrst synthetic data, the performance gain of our model is more pronounced when number of training examples is small. For the second synthetic data, we retain similar performance gain over other models when number of training examples are increased from 20. The MSE numbers for the ﬁrst synthetic data are higher than the ones obtained for the second synthetic data because of a difference in the magnitude of error covariances used in the generation of datasets. We also experiment with the convergence properties of our method. Figures 1(c) and 1(d) show that plots of average MSE and the value of the objective function (given by Equation 6) with increasing number of iterations on the ﬁrst synthetic dataset and the ﬁrst Paper dataset. The plots show that our alternating optimization procedure converges in roughly 10–15 iterations. 4.3 Covariance structure recovery Although not the main goal of the paper, we experiment with learned inverse covariance matrix of the outputs (given the inputs) as a sanity check on the proposed model. To better visualize, we generate a dataset with 5 responses and 3 predictors using the same process as described in Sec. 4.1. Figure on the right shows the true conditional inverse covariance matrix Ω−1 (Top), the matrix learned with MROTS ˆΩ−1 (Middle), and the precision matrix learned with graphical lasso ignoring the predictors (Bottom). Taking into account the regression weights results in better estimate of the true covariance matrix. We got similar results for MRCE-ℓ2 which also takes into account the predictors while learning the inverse covariance, although MROTS estimates were closer to the ground truth in terms of the Frobenius norm. 5 Related Work Apart from the prior works discussed in Section 1, our work has connections to some other works which we discuss in this section. Recently, Sohn & Kim [18] proposed a model for jointly estimating the weight vector for each output and the covariance structure of the outputs. However, they assume a shared sparsity structure on the weight vectors. This assumption may be restrictive in some problems. Some other works on conditional graphical model estimation [20, 4] are based on similar structural sparsity assumptions. In contrast, our model does not assume any speciﬁc structure on the weight vectors, and by explicitly modeling the covariance structure of the weight vectors, learns the appropriate underlying structure from the data. 6 Future Work and Conclusion We have presented a ﬂexible model for multiple-output regression taking into account the covariance structure of the outputs and the covariance structure of the underlying prediction tasks. Our model does not require a priori knowledge of these structures and learns these from the data. Our model leads to improved accuracies on multiple-output regression tasks. Our model can be extended in several ways. For example, one possibility is to model nonlinear input-output relationships by kernelizing the model along the lines of [22]. 8 References [1] M. Aldrin. Moderate projection pursuit regression for multivariate response data. Computational Statistics and Data Analysis, 21, 1996. 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4,479	Feature Clustering for Accelerating Parallel Coordinate Descent Chad Scherrer Independent Consultant Yakima, WA chad.scherrer@gmail.com Ambuj Tewari Department of Statistics University of Michigan Ann Arbor, MI tewaria@umich.edu Mahantesh Halappanavar Paciﬁc Northwest National Laboratory Richland, WA mahantesh.halappanavar@pnnl.gov David J Haglin Paciﬁc Northwest National Laboratory Richland, WA david.haglin@pnnl.gov Abstract Large-scale `1-regularized loss minimization problems arise in high-dimensional applications such as compressed sensing and high-dimensional supervised learning, including classiﬁcation and regression problems. High-performance algorithms and implementations are critical to efﬁciently solving these problems. Building upon previous work on coordinate descent algorithms for `1-regularized problems, we introduce a novel family of algorithms called block-greedy coordinate descent that includes, as special cases, several existing algorithms such as SCD, Greedy CD, Shotgun, and Thread-Greedy. We give a uniﬁed convergence analysis for the family of block-greedy algorithms. The analysis suggests that block-greedy coordinate descent can better exploit parallelism if features are clustered so that the maximum inner product between features in different blocks is small. Our theoretical convergence analysis is supported with experimental results using data from diverse real-world applications. We hope that algorithmic approaches and convergence analysis we provide will not only advance the ﬁeld, but will also encourage researchers to systematically explore the design space of algorithms for solving large-scale `1-regularization problems. 1 Introduction Consider the `1-regularized loss minimization problem min w 1 n n X i=1 `(yi, (Xw)i) + λkwk1 , (1) where X 2 IRn⇥p is the design matrix, w 2 IRp is a weight vector to be estimated, and the loss function ` is such that `(y, ·) is a convex differentiable function for each y. This formulation includes `1-regularized least squares (Lasso) (when `(y, t) = 1 2(y −t)2) and `1-regularized logistic regression (when `(y, t) = log(1+exp(−yt))). In recent years, coordinate descent (CD) algorithms have been shown to be efﬁcient for this class of problems [Friedman et al., 2007; Wu and Lange, 2008; Shalev-Shwartz and Tewari, 2011; Bradley et al., 2011]. Motivated by the need to solve large scale `1 regularized problems, researchers have begun to explore parallel algorithms. For instance, Bradley et al. [2011] developed the Shotgun algorithm. More recently, Scherrer et al. [2012] have developed “GenCD”, a generic framework for expressing 1 parallel coordinate descent algorithms. Special cases of GenCD include Greedy CD [Li and Osher, 2009; Dhillon et al., 2011], the Shotgun algorithm of [Bradley et al., 2011], and Thread-Greedy CD [Scherrer et al., 2012]. In fact, the connection between these three special cases of GenCD is much deeper, and more fundamental, than is obvious under the GenCD abstraction. As our ﬁrst contribution, we describe a general randomized block-greedy that includes all three as special cases. The block-greedy algorithm has two parameter: B, the total number of feature blocks and P, the size of the random subset of the B blocks that is chosen at every time step. For each of these P blocks, we greedily choose, in parallel, a single feature weight to be updated. Second, we present a non-asymptotic convergence rate analysis for the randomized block-greedy coordinate descent algorithms for general values of B 2 {1, . . . , p} (as the number of blocks cannot exceed the number of features) and P 2 {1, . . . , B}. This result therefore applies to stochastic CD, greedy CD, Shotgun, and thread-greedy. Indeed, we build on the analysis and insights in all of these previous works. Our general convergence result, and in particular its instantiation to thread-greedy CD, is novel. Third, based on the convergence rate analysis for block-greedy, we optimize a certain “block spectral radius” associated with the design matrix. This parameter is a direct generalization of a similar spectral parameter that appears in the analysis of Shotgun. We show that the block spectral radius can be upper bounded by the maximum inner product (or correlation if features are mean zero) between features in distinct blocks. This motivates the use of correlation-based feature clustering to accelerate the convergence of the thread-greedy algorithm. Finally, we conduct an experimental study using a simple clustering heuristic. We observe dramatic acceleration due to clustering for smaller values of the regularization parameter, and show characteristics that must be paid particularly close attention for heavily regularized problems, and that can be improved upon in future work. 2 Block-Greedy Coordinate Descent Scherrer et al. [2012] describe “GenCD”, a generic framework for parallel coordinate descent algorithms, in which a parallel coordinate descent algorithm can be determined by specifying a select step and an accept step. At each iteration, features chosen by select are evaluated, and a proposed increment is generated for each corresponding feature weight. Using this, the accept step then determines which proposals are to be updated. Figure 1: The design space of block-greedy coordinate descent. In these terms, we consider the block-greedy algorithm that takes as part of the input a partition of the features into B blocks. Given this, each iteration selects features corresponding to a set of P randomly selected blocks, and accepts a single feature from each block, based on an estimate of the resulting reduction in the objective function. The pseudocode for the randomized block-greedy coordinate descent is given by Algorithm 1. The algorithm can be applied to any function of the form F + R where F is smooth and convex, and R is convex and separable across coordinates. Our objective function (1) satisﬁes these conditions. The greedy step chooses a feature within a block for which the guaranteed descent in the objective function (if that feature alone were updated) is maximized. This descent is quantiﬁed by \|⌘j\|, which is deﬁned precisely in the next section. To arrive at an heuristic understanding, it is best to think of \|⌘j\| as being proportional to the absolute value \|rjF(w)\| of the jth entry in the gradient of the smooth part F. In fact, if R is zero (no regularization) then this heuristic is exact. The two parameters, B and P, of the block-greedy CD algorithm have the ranges B 2 {1, . . . , p} and P 2 {1, . . . , B}. Setting these to speciﬁc values gives many existing algorithms. For instance when B = p, each feature is a block on its own. Then, setting P = 1 amounts to randomly choosing a single coordinate and updating it which gives us the stochastic CD algorithm of Shalev-Shwartz and Tewari [2011]. Shotgun [Bradley et al., 2011] is obtained when B is still p but P ≥1. Another 2 Algorithm 1 Block-Greedy Coordinate Descent Parameters: B (no. of blocks) and P B (degree of parallelism) while not converged do Select a random subset of size P from the B available blocks Set J to be the features in the selected blocks Propose increment ⌘j, j 2 J // parallel Accept J0 = {j : ⌘j has maximal absolute value in its block} Update weight wj wj −⌘j for all j 2 J0 // parallel extreme is the case when all the features constitute a single block. That is, B = 1. Then blockgreedy CD is a deterministic algorithm and becomes the greedy CD algorithm of Li and Osher [2009]; Dhillon et al. [2011]. Finally, we can choose non-trivial values of B that lie strictly between 1 and p. When this is the case, and we choose to update all blocks in parallel each time (P = B), we arrive at the thread-greedy algorithm of Scherrer et al. [2012]. Figure 1 shows a schematic representation of the parameterization of these special cases. 3 Convergence Analysis Of course, there is no reason to expect block-greedy CD to converge for all values of B and P. In this section, we give a sufﬁcient condition for convergence and derive a convergence rate assuming this condition. Bradley et al. express the convergence criteria for Shotgun algorithm in terms of the spectral radius (maximal eigenvalue) ⇢(XT X). For block-greedy, the corresponding quantity is a bit more complicated. We deﬁne ⇢block = max M2M ⇢(M) where M is the set of all B ⇥B submatrices that we can obtain from XT X by selecting exactly one index from each of the B blocks. The intuition is that if features from different blocks are almost orthogonal then the matrices M in M will be close to identity and will therefore have small ⇢(M). Highly correlated features within a block do not increase ⇢block. As we said above, we will assume that we are minimizing a “smooth plus separable” convex function F + R where the convex differentiable function F : Rp ! R satisﬁes a second order upper bound F(w + ∆) F(w) + rF(w)T ∆+ β 2 ∆T XT X∆ In our case, this inequality will hold as soon as `00(y, t) β for any y, t (where differentiation is w.r.t. t). The function R is separable across coordinates: R(w) = Pp j=1 r(wj). The function λkwk1 is clearly separable. The quantity ⌘j appearing in Algorithm 1 serves to quantify the guaranteed descent (based on second order upper bound) if feature j alone is updated and is obtained as a solution of the one-dimensional minimization problem ⌘j = argmin ⌘ rjF(w)⌘+ β 2 ⌘2 + r(wj + ⌘) −r(wj) . Note that if there is no regularization, then ⌘j is simply −rjF(w)/β = −gj/β (if we denote rjF(w) by gj for brevity). In the general case, by ﬁrst order optimality conditions for the above one-dimensional convex optimization problem, we have gj +β⌘j +⌫j = 0 where ⌫j is a subgradient of r at wj +⌘j. That is, ⌫j 2 @r(wj +⌘j). This implies that r(wj +⌘j)−r(w0) ⌫j(wj +⌘j −w0) for any w0. Theorem 1. Let P be chosen so that ✏= (P −1)(⇢block −1) (B −1) 3 is less than 1. Suppose the randomized block-greedy coordinate algorithm is run on a smooth plus separable convex function f = F +R to produce the iterates {wk}k≥1. Then the expected accuracy after k steps is bounded as E [f(wk) −f(w?)] C B R2 1 (1 −✏)P · 1 k . Here the constant C only depends on (Lipschitz and smoothness constants of) the function F, R1 is an upper bound on the norms {kwk −w?k1}k≥1, and w? is any minimizer of f. Proof. We ﬁrst calculate the expected change in objective function following the Shotgun analysis. We will use wb to denote wjb (similarly for ⌫b, gb etc.) E [f(w0) −f(w)] = PEb  ⌘bgb + β 2 (⌘b)2 + r(wb + ⌘b) −r(wb) $ + β 2 P(P −1)Eb6=b0 ⇥ ⌘b · ⌘b0 · AT jbAjb0 ⇤ Deﬁne the B ⇥B matrix M (that depends on the current iterate w) with entries Mb,b0 = AT jbAjb. Then, using r(wb + ⌘b) −r(wb) ⌫b⌘b, we continue P B  ⌘T g + β 2 ⌘T ⌘+ ⌫T ⌘ $ + βP(P −1) 2B(B −1) ⇥ ⌘>M⌘−⌘T ⌘ ⇤ Above (with some abuse of notation), ⌘, ⌫and g are B length vectors with components ⌘b, ⌫b and gb respectively. By deﬁnition of ⇢block, we have ⌘>M⌘⇢block⌘T ⌘. So, we continue P B  ⌘T g + β 2 ⌘T ⌘−gT ⌘−β⌘T ⌘ $ + βP(P −1) 2B(B −1) (⇢block −1)⌘T ⌘ where we used ⌫= −g −β⌘. Simplifying we get E [f(w0) −f(w)] Pβ 2B [−1 + ✏] k⌘k2 2 where ✏= (P −1)(⇢block −1) (B −1) should be less than 1. Now note that k⌘k2 2 = P b \|⌘jb\|2 = k⌘k2 1,2 where the “inﬁnity-2” norm k · k1,2 of a p-vector is, by deﬁnition, as follows: take the `1 norm within a block and take the `2 of the resulting values. Note that in the second step above, we moved from a B-length ⌘to a p length ⌘. This gives us E [f(w0) −f(w)] −(1 −✏)Pβ 2B k⌘k2 1,2 . For the rest of the proof, assume λ = 0. In that case ⌘= −g/β. Thus, convexity and the fact that the dual norm of the “inﬁnity-2” norm is the “1-2” norm, give f(w) −f(w?) rf(w)T (w −w?) krf(w)k1,2 · kw −w?k1,2 Putting the last two inequalities together gives (for any upper bound R1 on kw −w?k1 ≥kw − w?k1,2) E [f(w0) −f(w)] −(1 −✏)P 2βBR2 1 (f(w) −f(w?))2 . Deﬁning the accuracy ↵k = f(wk) −f(w?), we translate the above into the recurrence E [↵k+1 −↵k] −(1 −✏)P 2βBR2 1 E ⇥ ↵2 k ⇤ 4 and by Jensen’s we have (E [↵k])2 E ⇥ ↵2 k ⇤ and therefore E [↵k+1] −E [↵k] −(1 −✏)P 2βBR2 1 (E [↵k])2 which solves to (up to a universal constant factor) E [↵k] 2βBR2 1 (1 −✏)P · 1 k Even when λ > 0, we can still relate k⌘k1,2 to f(w) −f(w?) but the argument is a little more involved. We refer the reader to the supplementary for more details. In particular, consider the case where all blocks are updated in parallel as in the thread-greedy coordinate descent algorithm of Scherrer et al. [2012]. Then P = B and there is no randomness in the algorithm, yielding the following corollary. Corollary 2. Suppose the block-greedy coordinate algorithm with B = P (thready-greedy) is run on a smooth plus separable convex function f = F + R to produce the iterates {wk}k≥1. If ⇢block < 2, then f(wk) −f(w?) = O ✓ 1 (2 −⇢block)k ◆ . 4 Feature Clustering The convergence analysis of section 3 shows that we need to minimize the block spectral radius. Directly ﬁnding a clustering that minimizes ⇢block is a computationally daunting task. Even with equal-sized blocks, the number of possible partitions is p!/ ) p B *B. In the absence of an efﬁcient search strategy for this enormous space, we ﬁnd it convenient to work instead in terms of the inner product of features from distinct blocks. The following proposition makes the connection between these approaches precise. Proposition 3. Let S 2 RB⇥B be positive semideﬁnite, with Sii = 1, and \|Sij\| < " for i 6= j. Then the spectral radius of S has the upper bound ⇢(S) 1 + (B −1) " . Proof. Let x be the eigenvector corresponding to the largest eigenvalue of S, scaled so that kxk1 = 1. Then ⇢(S) = kSxk1 = X i ++++++ xi + Sij X j6=i xj ++++++  X i 0 @\|xi\| + " X j6=i \|xj\| 1 A = 1 + (B −1) " Proposition 3 tells us that we can partition the features into clusters using a heuristic approach that strives to minimize the maximum absolute inner product between the features (columns of the design matrix) Xi and Xj where i and j are features in different blocks. 4.1 Clustering Heuristic Given p features and B blocks, we wish to distribute the features evenly among the blocks, attempting to minimize the absolute inner product between features across blocks. Moreover, we require an approach that is efﬁcient, since any time spent clustering could instead have been used for iterations of the main algorithm. We describe a simple heuristic that builds uniform-sized clusters of features. To construct a given block, we select a feature as a “seed”, and assign the nearest features to the seed (in terms of absolute inner product) to be in the same block. Because inner products with very sparse features result in a large number of zeros, we choose at each step the most dense unassigned feature as the seed. Algorithm 2 provides a detailed description. This heuristic requires computation of O(Bp) inner products. In practice it is very fast—less than three seconds for even the large KDDA dataset. 5 Algorithm 2 A heuristic for clustering p features into B blocks, based on correlation U {1, · · · , p} for b = 1 to B −1 do s arg maxj2U NNZ(Xj) for j 2 U do // parallel cj \|hXs, Xji\| Jb {j yielding the dp/Be largest values of cj} U U\Jb JB U return {Jb\|b = 1, · · · , B} Name # Features # Samples # Nonzeros Source NEWS20 1, 355, 191 19, 996 9, 097, 916 Keerthi and DeCoste [2005] REUTERS 47, 237 23, 865 1, 757, 800 Lewis et al. [2004] REALSIM 20, 958 72, 309 3, 709, 083 RealSim KDDA 20, 216, 830 8, 407, 752 305, 613, 510 Lo et al. [2011] Table 1: Summary of input characteristics. 5 Experimental Setup Platform All our experiments are conducted on a 48-core system comprising of 4 sockets and 8 banks of memory. Each socket is an AMD Opteron processor codenamed Magny-Cours, which is a multichip processor with two 6-core chips on a single die. Each 6-core processor is equipped with a three-level memory hierarchy as follows: (i) 64 KB of L1 cache for data and 512 KB of L2 cache that are private to each core, and (ii) 12 MB of L3 cache that is shared among the 6 cores. Each 6-core processor is linked to a 32 GB memory bank with independent memory controllers leading to a total system memory of 256 GB (32 ⇥8) that can be globally addressed from each core. The four sockets are interconnected using HyperTransport-3 technology1. Datasets A variety of datasets were chosen2 for experimentation; these are summarized in Table 1. We consider four datasets: (i) NEWS20 contains about 20, 000 UseNet postings from 20 newsgroups. The data was gathered by Ken Lang at Carnegie Mellon University circa 1995. (ii) REUTERS is the RCV1-v2/LYRL2004 Reuters text data described by Lewis et al. [2004]. In this term-document matrix, each example is a training document, and each feature is a term. Nonzero values of the matrix correspond to term frequencies that are transformed using a standard tf-idf normalization. (iii) REALSIM consists of about 73, 000 UseNet articles from four discussion groups: simulated auto racing, simulated aviation, real auto racing, and real aviation. The data was gathered by Andrew McCallum while at Just Research circa 1997. We consider classifying real vs simulated data, irrespective of auto/aviation. (iv) KDDA represents data from the KDD Cup 2010 challenge on educational data mining. The data represents a processed version of the training set of the ﬁrst problem, algebra 2008 2009, provided by the winner from the National Taiwan University. These four inputs cover a broad spectrum of sizes and structural properties. Implementation For the current work, our empirical results focus on thread-greedy coordinate descent with 32 blocks. At each iteration, a given thread must step through the nonzeros of each of its features to compute the proposed increment (the ⌘j of Section 3) and the estimated beneﬁt of choosing that feature. Once this is complete, the thread (without waiting) enters the line search phase, where it remains until all threads are being updated by less than the speciﬁed tolerance. Finally, all updates are performed concurrently. We use OpenMP’s atomic directive to maintain consistency. Testing framework We compare the effect of clustering to randomization (i.e. features are randomly assigned to blocks), for a variety of values for the regularization parameter λ. To test the effect of clustering for very 1Further details on AMD Opteron can be found at http://www.amd.com/us/products/ embedded/processors/opteron/Pages/opteron-6100-series.aspx. 2from http://www.csie.ntu.edu.tw/˜cjlin/libsvmtools/datasets/ 6 10 × REL 0 5 10 15 0 1 2 3 4 5 6 7 Time (min) NNZ 0 5 10 15 10 102 103 104 (a) NEWS20, λ0 = 10−4 10 × REL 0 5 10 15 1 2 3 4 5 6 7 Time (min) NNZ 0 5 10 15 10 102 103 104 (b) REUTERS, λ0 = 10−4 10 × REL 0 5 10 15 2 3 4 5 6 7 Time (min) NNZ 0 5 10 15 10 102 103 104 (c) REALSIM, λ0 = 10−4 10 × REL 0 50 100 150 6.920 6.925 6.930 Time (min) NNZ 0 50 100 150 1 10 102 103 104 (d) KDDA, λ0 = 10−6 Figure 2: Convergence results. For each dataset, we show the regularized expected loss (top) and number of nonzeros (bottom), using powers of ten as regularization parameters. Results for randomized features are shown in black, and those for clustered features are shown in red. Note that the allowed running time for KDDA was ten times that of other datasets. λ = 10−4 λ = 10−5 λ = 10−6 Randomized Clustered Randomized Clustered Randomized Clustered Active blocks 32 6 32 32 32 32 Iterations per second 153 12.9 152 12.3 136 12.3 NNZ @ 1K sec 184 215 797 8592 1248 19473 Objective @ 1K sec 0.472 0.591 0.264 0.321 0.206 0.136 NNZ @ 10K iter 74 203 82 8812 110 19919 Objective @ 10K iter 0.570 0.593 0.515 0.328 0.472 0.141 Table 2: The effect of feature clustering, for REUTERS. sparse weights, we ﬁrst let λ0 be the largest power of ten that leads to any nonzero weight estimates. This is followed by the next three consecutive powers of ten. For each run, we measure the regularized expected loss and the number of nonzeros at one-second intervals. Times required for clustering and randomization are negligible, and we do not report them here. 6 Results Figure 2 shows the regularized expected loss (top) and number of nonzeros (bottom), for several values of the regularization parameter λ. Black and red curves indicate randomly-permuted features and clustered features, respectively. The starting value of λ was 10−4 for all data except KDDA, which required λ = 10−6 in order to yield any nonzero weights. In the upper plots, within a color, the order of the 4 curves, top to bottom, corresponds to successively decreasing values of λ. Note that a larger value of λ results in a sparser solution with greater regularized expected loss and a smaller number of nonzeros. Thus, for each subﬁgure of Figure 2, the order of the curves in the lower plot is reversed from that of the upper plot. Overall, results across datasets are very consistent. For large values of λ, the simple clustering heuristic results in slower convergence, while for smaller values of λ we see considerable beneﬁt. Due to space limitations, we choose a single dataset for which to explore results in greater detail. Of the datasets we tested, REUTERS might reasonably lead to the greatest concern. Like the other datasets, clustered features lead to slow convergence for large λ and fast convergence for small λ. However, REUTERS is particularly interesting because for λ = 10−5, clustered features seem to provide an initial beneﬁt that does not last; after about 250 seconds it is overtaken by the run with randomized features. 7 ● ● ●●● ● ●●●●●●●●●●●●●● ●●●●●●●●●●●● Block NNZ ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 104 105 106 (a) Block density Iterations 10 × REL 1 10 102 103 104 105 1 2 3 4 5 6 7 (b) Regularized expected loss Iterations NNZ 1 10 102 103 104 105 10 102 103 104 (c) Number of nonzeros Figure 3: A closer look at performance characteristics for REUTERS. Table 2 gives a more detailed summary of the results for REUTERS, for the three largest values of λ. The ﬁrst row of this table gives the number of active blocks, by which we mean the number of blocks containing any nonzeros. For an inactive block, the corresponding thread repeatedly conﬁrms that all weights remain zero without contributing to convergence. In the most regularized case λ = 10−4, clustered data results in only six active blocks, while for other cases every block is active. Thus in this case features corresponding to nonzero weights are colocated within these few blocks, severely limiting the advantage of parallel updates. In the second row, we see that for randomized features, the algorithm is able to get through over ten times as many iterations per second. To see why, note that the amount of work for a given thread is a linear function of the number of nonzeros over all of the features in its block. Thus, the block with the greatest number of nonzeros serves as a bottleneck. The middle two rows of Figure 2 summarize the state of each run after 1000 seconds. Note that for this test, randomized features result in faster convergence for the two largest values of λ. For comparison, the ﬁnal two rows of Figure 2 provide a similar summary based instead on the number of iterations. In these terms, clustering is advantageous for all but the largest value of λ. Figure 3 shows the source of this problem. First, Figure 3a shows the number of nonzeros in all features for each of the 32 blocks. Clearly the simple heuristic results in poor load-balancing. For comparison, Figures 3b and 3c show convergence rates as a function of the number of iterations. 7 Conclusion We have presented convergence results for a family of randomized coordinate descent algorithms that we call block-greedy coordinate descent. This family includes Greedy CD, Thread-Greedy CD, Shotgun, and Stochastic CD. We have shown that convergence depends on ⇢block, the maximal spectral radius over submatrices of XT X resulting from the choice of one feature for each block. Even though a simple clustering heuristic helps for smaller values of the regularization parameter, our results also show the importance of considering issues of load-balancing and the distribution of weights for heavily-regularized problems. A clear next goal in this work is the development of a clustering heuristic that is relatively well load-balanced and distributes weights for heavily-regularized problems evenly across blocks, while maintaining good computational efﬁciency. Acknowledgments The authors are grateful for the helpful suggestions of Ken Jarman, Joseph Manzano, and our anonymous reviewers. Funding for this work was provided by the Center for Adaptive Super Computing Software - MultiThreaded Architectures (CASS-MT) at the U.S. Department of Energy’s Paciﬁc Northwest National Laboratory. 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Hung-Yi Lo, Kai-Wei Chang, Shang-Tse Chen, Tsung-Hsien Chiang, Chun-Sung Ferng, Cho-Jui Hsieh, Yi-Kuang Ko, Tsung-Ting Kuo, Hung-Che Lai, Ken-Yi Lin, Chia-Hsuan Wang, Hsiang-Fu Yu, Chih-Jen Lin, Hsuan-Tien Lin, and Shou de Lin. Feature engineering and classiﬁer ensemble for KDD Cup 2010, 2011. To appear in JMLR Workshop and Conference Proceedings. 9	2012	124
4,480	Phoneme Classiﬁcation using Constrained Variational Gaussian Process Dynamical System Hyunsin Park Department of EE, KAIST Daejeon, South Korea hs.park@kaist.ac.kr Sungrack Yun Qualcomm Korea Seoul, South Korea sungrack@qualcomm.com Sanghyuk Park Department of EE, KAIST Daejeon, South Korea shine0624@kaist.ac.kr Jongmin Kim Department of EE, KAIST Daejeon, South Korea kimjm0309@gmail.com Chang D. Yoo Department of EE, KAIST Daejeon, South Korea cdyoo@ee.kaist.ac.kr Abstract For phoneme classiﬁcation, this paper describes an acoustic model based on the variational Gaussian process dynamical system (VGPDS). The nonlinear and nonparametric acoustic model is adopted to overcome the limitations of classical hidden Markov models (HMMs) in modeling speech. The Gaussian process prior on the dynamics and emission functions respectively enable the complex dynamic structure and long-range dependency of speech to be better represented than that by an HMM. In addition, a variance constraint in the VGPDS is introduced to eliminate the sparse approximation error in the kernel matrix. The effectiveness of the proposed model is demonstrated with three experimental results, including parameter estimation and classiﬁcation performance, on the synthetic and benchmark datasets. 1 Introduction Automatic speech recognition (ASR), the process of automatically translating spoken words into text, has been an important research topic for several decades owing to its wide array of potential applications in the area of human-computer interaction (HCI). The state-of-the-art ASR systems typically use hidden Markov models (HMMs) [1] to model the sequential articulator structure of speech signals. There are various issues to consider in designing a successful ASR and certainly the following two limitations of an HMM need to be overcome. 1) An HMM with a ﬁrst-order Markovian structure is suitable for capturing short-range dependency in observations and speech requires a more ﬂexible model that can capture long-range dependency in speech. 2) Discrete latent state variables and sudden state transitions in an HMM have limited capacity when used to represent the continuous and complex dynamic structure of speech. These limitations must be considered when seeking to improve the performance of an ASR. To overcome these limitations, various models have been considered to model the complex structure of speech. For example, the stochastic segment model [2] is a well-known generalization of the HMM that represents long-range dependency over observations using a time-dependent emission function. And the hidden dynamical model [3] is used for modeling the complex nonlinear dynamics of a physiological articulator. Another promising research direction is to consider a nonparametric Bayesian model for nonlinear probabilistic modeling of speech. Owing to the fact that nonparametric models do not assume any 1 ﬁxed model structure, they are generally more ﬂexible than parametric models and can allow dependency among observations naturally. The Gaussian process (GP) [4], a stochastic process over a real-valued function, has been a key ingredient in solving such problems as nonlinear regression and classiﬁcation. As a standard supervised learning task using the GP, Gaussian process regression (GPR) offers a nonparametric Bayesian framework to infer the nonlinear latent function relating the input and the output data. Recently, researchers have begun focusing on applying the GP to unsupervised learning tasks with high-dimensional data, such as the Gaussian process latent variable model (GP-LVM) for reduction of dimensionality [5-6]. In [7], a variational inference framework was proposed for training the GP-LVM. The variational approach is one of the sparse approximation approaches [8]. The framework was extended to the variational Gaussian process dynamical system (VGPDS) in [9] by augmenting latent dynamics for modeling high-dimensional time series data. High-dimensional time series have been incorporated in many applications of machine learning such as robotics (sensor data), computational biology (gene expression data), computer vision (video sequences), and graphics (motion capture data). However, no previous work has considered the GP-based approach for speech recognition tasks that involve high-dimensional time series data. In this paper, we propose a GP-based acoustic model for phoneme classiﬁcation. The proposed model is based on the assumption that the continuous dynamics and nonlinearity of the VGPDS can be better represent the statistical characteristic of real speech than an HMM. The GP prior over the emission function allows the model to represent long-range dependency over the observations of speech, while the HMM does not. Furthermore, the GP prior over the dynamics function enables the model to capture the nonlinear dynamics of a physiological articulator. Our contributions are as follows: 1) we introduce a GP-based model for phoneme classiﬁcation tasks for the ﬁrst time, showing that the model has the potential of describing the underlying characteristics of speech in a nonparametric way; 2) we propose a prior for hyperparameters and a variance constraint that are specially designed for ASR; and 3) we provide extensive experimental results and analyses to reveal clearly the strength of our proposed model. The remainder of the paper is structured as follows: Section 2 introduces the proposed model after a brief description of the VGPDS. Section 3 provides extensive experimental evaluations to prove the effectiveness of our model, and Section 4 concludes the paper with a discussion and plans for future work. 2 Acoustic modeling using Gaussian Processes 2.1 Variational Gaussian Process Dynamical System The VGPDS [9] models time series data by assuming that there exist latent states that govern the data. Let Y = [[y11, · · · yN1]T , · · · , [y1D, · · · yND]T ] ∈RN×D, t = [t1, · · · , tN]T ∈RN +, and X = [[x11, · · · xN1]T , · · · , [x1Q, · · · xNQ]T ] ∈RN×Q be observed data, time, and corresponding latent state, where N, D, and Q(< D) are the number of samples, the dimension of the observation space, and the dimension of the latent space, respectively. In the VGPDS, these variables are related as follows: xnj = gj(tn) + ηnj, ηnj ∼N(0, 1/βx j ), yni = fi(xn) + ϵni, ϵni ∼N(0, 1/βy i ), (1) where fi(x) ∼GP(µf i (x), kf i (x, x′)) and gj(t) ∼GP(µg j(t), kg j (t, t′)) are the emission function from the latent space to the i-th dimension of the observation space and the dynamics function from the time space to the j-th dimension of the latent space, respectively. Here, n ∈{1, · · · , N}, i ∈{1, · · · , D}, and j ∈{1, · · · , Q}. In this paper, a zero-mean function is used for all GPs. Fig. 1 shows graphical representations of HMM and VGPDS. Although the Gaussian process dynamical model (GPDM) [10], which involves an auto-regressive dynamics function, is also a GP-based model for time-series, it is not considered in this paper. The marginal likelihood of the VGPDS is given as p(Y\|t) = Z p(Y\|X)p(X\|t)dX. (2) 2 Figure 1: Graphical representations of (left) the left-to-right HMM and (right) the VGPDS: In the left ﬁgure, yn ∈RD and xn ∈{1, · · · , C} are observations and discrete latent states. In the right ﬁgure, yni, fni, xnj, gnj, and tn are observations, emission function points, latent states, dynamics function points, and times, respectively. All function points in the same plate are fully connected. Since the integral in Eq. (2) is not tractable, a variational method is used by introducing a variational distribution q(X). A variational lower bound on the logarithm of the marginal likelihood is log p(Y\|t) ≥ Z q(X) log p(Y\|X)p(X\|t) q(X) dX = Z q(X) log p(Y\|X)dX − Z q(X) log q(X) p(X\|t)dX = L −KL(q(X)\|\|p(X\|t)). (3) By the assumption of independence over the observation dimension, the ﬁrst term in Eq. (3) is given as L = D X i=1 Z q(X) log p(yi\|X)dX = D X i=1 Li. (4) In [9], a variational approach which involves sparse approximation of the covariance matrix obtained from GP is proposed. The variational lower bound on Li is given as Li ≥ log " (βy i )N/2\| ˜Ki\|1/2 (2π)N/2\|βy i Ψ2i + ˜Ki\|1/2 e(−1 2 yT i Wiyi) # −βy i 2 (ψ0i −Tr( ˜K−1 i Ψ2i)), (5) where Wi = βy i IN −(βy i )2Ψ1i(βy i Ψ2i + ˜Ki)−1ΨT 1i. Here, ˜Ki ∈RM×M is a kernel matrix calculated using the i-th kernel function and inducing input variables ˜X ∈RM×Q that are used for sparse approximation of the full kernel matrix Ki. The closed-form of the statistics {ψ0i, Ψ1i, Ψ2i}D i=1, which are functions of variational parameters and inducing points, can be found in [9]. In the second term of Eq. (3), p(X\|t) = QQ j=1 p(xj) and q(X) = QN n QQ j=1 N(µnj, snj) are the prior for the latent state and the variational distribution that is used for approximating the posterior of the latent state, respectively. The parameter set Θ, which consists of the hyperparameters {θf, θg} of the kernel functions, the noise variances {βy, βx}, the variational parameters {[µn1, · · · , µnQ], [sn1, · · · , snQ]}N n=1 of q(X), and the inducing input points ˜X, is estimated by maximizing the lower bound on log p(Y\|t) in Eq. (3) using a scaled conjugate gradient (SCG) algorithm. 2.2 Acoustic modeling using VGPDS For several decades, HMM has been the predominant model for acoustic speech modeling. However, as we mentioned in Section 1, the model suffers from two major limitations: discrete state variables and ﬁrst-order Markovian structure which can model short-range dependency over the observations. 3 To overcome such limitations of the HMM, we propose an acoustic speech model based on the VGPDS, which is a nonlinear and nonparametric model that can be used to represent the complex dynamic structure of speech and long-range dependency over observations of speech. In addition, to ﬁt the model to large-scale speech data, we describe various implementation issues. 2.2.1 Time scale modiﬁcation The time length of each phoneme segment in an utterance varies with various conditions such as position of the phoneme segment in the utterance, emotion, gender, and other speaker and environment conditions. To incorporate this fact into the proposed acoustic model, the time points tn are modiﬁed as follows: tn = n −1 N −1, (6) where n and N are the observation index and the number of observations in a phoneme segment, respectively. This time scale modiﬁcation makes all phoneme signals have unit time length. 2.2.2 Hyperparameters To compute the kernel matrices in Eq. (5), the kernel function must be deﬁned. We use the radial basis function (RBF) kernel for the emission function f as follows: kf(x, x′) = αf exp  − Q X j=1 ωf j (xj −x′ j)2  , (7) where αf and ωf j are the RBF kernel variance and the j-th inverse length scale, respectively. The RBF kernel function is adopted for representing smoothness of speech. For the dynamics function g, the following kernel function is used: kg(t, t′) = αg exp −ωg(t −t′)2 + λtt′ + b, (8) where λ and b are linear kernel variance and bias, respectively. The above dynamics kernel, which consists of both linear and nonlinear components, is used for representing the complex dynamics of the articulator. All hyperparameters are assumed to be independent in this paper. In [11], same kernel function parameters are shared over all dimensions of human-motion capture data and high-dimensional raw video data. However, this extensive sharing of the hyperparameters is unsuitable for speech modeling. Even though each dimension of observations is normalized in advance to have unit variance, the signal-to-noise ratio (SNR) is not consistent over all dimensions. To handle this problem, this paper considers each dimension to be modeled independently using different kernel function parameters. Therefore, the hyperparameter sets are deﬁned as θf = {αf i , {ωf 1i, · · · , ωf Qi}}D i=1 and θg = {αg j, ωg j , λj, bj}Q j=1. 2.2.3 Priors on the hyperparameters In the parameter estimation of the VGPDS, the SCG algorithm does not guarantee the optimal solution. To overcome this problem, we place the following prior on the hyperparameters of the kernel functions as given below p(γ) ∝exp(−γ2/¯γ), (9) where γ ∈{θf, θg} and ¯γ are the hyper-parameter and the model parameter of the prior, respectively. In this paper, ¯γ is set to the sample variance for the hyperparameters of the emission kernel functions, and ¯γ is set to 1 for the hyperparameters of the dynamics kernel functions. Uniform priors are adopted for other hyperparameters, then the parameters of the VGPDS are estimated by maximizing the joint distribution p(Y, Θ\|t) = p(Y\|t, Θ)p(Θ). 2.2.4 Variance constraint In the lower bound of Eq. (5), the second term on the right-hand side is the regularization term that represents the sparse approximation error of the full kernel matrix Ki. Note that with more inducing 4 input points, approximation error becomes smaller. However, only a small number of inducing input points can be used owing to the limited availability of computational power, which increases the effect of the regularization term. To mitigate this problem, we introduce the following constraint on the diagonal terms of the covariance matrix as given below: Tr(⟨Ki⟩q(X)) N + 1/βy i = σ2 i , (10) where ⟨Ki⟩q(X) and σ2 i are the expectation of the full kernel matrix Ki and the sample variance of the i-th dimension of the observation, respectively. This constraint is designed so that the variance of each observation calculated from the estimated model is equal to the sample variance. By using ψ0i = Tr(⟨Ki⟩q(X)), the inverse noise variance parameter is obtained directly by βy i = (σ2 i − ψ0i/N)−1 without separate gradient-based optimization. Then, the partial derivative ∂log βy i ∂ψ0i = 1 Nσ2−ψ0i is used for SCG-based optimization. In Section 3.1, the effectiveness of the variance constraint is demonstrated empirically. 2.3 Classiﬁcation For classiﬁcation with trained VGPDSs, maximum-likelihood (ML) decoding is used. Let D(l) = {Y(l), t(l)} and Θ(l) be the observation and parameter sets of the l-th VGPDS, respectively. Given the test data D∗= {Y∗, t∗}, the classiﬁcation result ˆl ∈{1, · · · , L} can be obtained by ˆl = arg max l log p(Y∗\|t∗, Y(l), t(l), Θ(l)) = arg max l log p(Y(l), Y∗\|t(l), t∗, Θ(l)) p(Y(l)\|t(l), Θ(l)) . (11) 3 Experiments To evaluate the effectiveness of the proposed model, three different kinds of experiments have been designed: 1. Parameter estimation: validating the effectiveness of the proposed variance constraint (Section 2.2.4) on model parameter estimation 2. Two-class classiﬁcation using synthetic data: demonstrating explicitly the advantages of the proposed model over the HMM with respect to the degree of dependency over the observations 3. Phoneme classiﬁcation: evaluating the performance of the proposed model on real speech data Each experiment is described in detail in the following subsections. In this paper, the proposed model is referred to as the constrained-VGPDS (CVGPDS). 3.1 Parameter estimation In this subsection, the experiments of parameter estimation on synthetic data are described. Synthetic data are generated by using a phoneme model that is selected from the trained models in Section 3.3 and then modiﬁed. The RBF kernel variances of the emission functions and the emission noise variances are modiﬁed from the selected model. In this experiment, the emission noise variances and inducing input points are estimated, while all other parameters are ﬁxed to the true values used in generating the data. Fig. 2 shows the parameter estimation results. The estimates of the 39-dimensional noise variance of the emission functions are shown with the true noise variances, the true RBF kernel variances, and the sample variances of the synthetic data. The top row denotes the estimation results without the variance constraint, and the bottom row with the variance constraint. By comparing the two ﬁgures 5 Figure 2: Results of parameter estimation: (top-left) VGPDS with M = 5, (top-right) VGPDS with M = 30, and (bottom) CVGPDS with M = 5 on the top row, we can conﬁrm that the estimation result of the noise variance with M = 30 inducing input points is better than that with M = 5 inducing input points. This result is obvious in the sense that smaller values of M produce more errors in the sparse approximation of the covariance matrix. However, both noise variance estimates are still different from the true values. By comparing the top and bottom rows, we can see that the proposed CVGPDS outperforms the VGPDS in terms of parameter estimation. Remarkably, the estimation result of the CVGPDS with M = 5 inducing input points is much better than the result of the VGPDS with M = 30. Based on these observations, we can conclude that the proposed CVGPDS is considerably more robust to the sparse approximation error compared to the VGPDS, as we claimed in Section 2.2.4. 3.2 Two-class classiﬁcation using synthetic data This section aims to show that when there is strong dependency over the observations, the proposed CVGPDS is a more appropriate model than the HMM for the classiﬁcation task. To this end, we ﬁrst generated several sets of two-class classiﬁcation datasets with different degrees of dependency over the observations. The considered classiﬁcation task is to map each input segment to one of two class labels. Using s ∈{1, ..., S} as the segment index, the synthetic dataset D = {Ys, ts, ls}S s=1 consists of S segments, where the s-th segment has Ns samples. Here, Ys ∈RNs×D, ts ∈RNs, and ls are the observation data, time, and class label of the s-th segment, respectively. The synthetic dataset is generated as follows: • Mean and kernel functions of two GPs gj(t) and fi(x) are deﬁned as gj(t) : µg j(t) = ajt + bj, kg j (t, t′) = 1t=t′ fi(x) : µf i (x) = PZi z=1 wzN(x; mz i Λz i ), kf i (x, x′) = αi exp(−ωi\|\|x −x′\|\|) (12) where {aj, bj}, {wz, mz i , Λz i }, and {αi, ωi} are respectively the parameters of the linear, Gaussian mixture, and RBF kernel functions. The superscript z denotes the component index of the Gaussian mixture, and Zi is the number of components in fi(x). 6 • For the s-th segment, {Ys, ts, ls}, 1. ls is selected as either class 1 or 2. 2. Ns is randomly selected from interval [20, 30], and ts is obtained by using Eq. (6). 3. From ts, the mean vector µg j(ts) and covariance matrix Kg j are obtained for j = 1, ..., Q. Let Xs ∈RNs×Q be the latent state of the s-th segment. Then, the j-th column of Xs is generated by the Ns-dimensional Gaussian distribution N(µg j(ts), Kg j). 4. From Xs, the mean vector µf i (Xs) and covariance matrix Kf i are obtained for i = 1, ..., D. Then, the i-th column of Ys is generated by the Ns-dimensional Gaussian distribution, N(µf i (Xs), Kf i ). Note that parameter ωi controls the degree of dependency over the observations. For instance, if ωi decreases, the off-diagonal terms of the emission kernel matrix Kf i increase, which means stronger correlations over the observations. The experimental setups are as follows. The synthesized dataset consists of 200 segments in total (100 segments per class). The dimensions of the latent space and observation space are set to Q = 2 and D = 5, respectively. We use 6(= Zi) components for the mean function of the emission kernel function. In this experiment, three datasets are synthesized and used to compare the CVGPDS and the HMM. When generating each dataset, we use two different ωi values, one for each class, while all other parameters in Eq. (12) are shared between the two classes. As a result, the degree of correlation between the observations is the only factor that distinguishes the two classes. The three generated datasets have different degrees of correlation over the observations, as a result of setting different ωi values for generating each dataset. In particular, the third dataset is constructed with two limitations of HMM such that it is well represented by an HMM. This could be achieved simply by changing the form of the mean function µg j(t) from a linear to a step function, and setting ωi = ∞so that each data sample is generated independently of the others. In the third dataset, the two classes are set to have different αi values. The classiﬁcation experiments are conducted using an HMM and CVGPDS. Table 1: Classiﬁcation accuracy for the two-class synthetic datasets (10-fold CV average [%]): All parameters except ωi are set to be equal for classes 1 and 2. In the case of ωi = ∞, αi are set to be different. ωi (class 1 : class 2) 0.1 : 0.5 1.0 : 2.0 ∞: ∞ HMM 61.0 68.5 88.5 CVGPDS 78.0 79.0 92.0 Table 1 summarizes the classiﬁcation performance of the HMM and CVGPDS for the three synthetic datasets. Remarkably, in all cases, the proposed CVGPDS outperforms the HMM, even in the case of ωi = ∞(the fourth column), where we assumed the dataset follows HMM-like characteristics. Comparing the second and the third columns of Table 1, we can see that the performance of the HMM degrades by 6.5% as ωi becomes smaller, while the proposed CVGPDS almost maintains its performance with only 1.0% reduction. This result demonstrates the superiority of the proposed CVGPDS in modeling data with strong correlations over the observations. Apparently, the HMM failed to distinguish the two classes with different degree of dependency over the observations. In contrast, the proposed CVGPDS distinguishes the two classes more effectively by capturing the different degrees of inter-dependencies over the observations incorporated in each class. 3.3 Phoneme classiﬁcation In this section, phoneme classiﬁcation experiments is described on real speech data from the TIMIT database. The TIMIT database contains a total of 6300 phonetically rich utterances, each of which is manually segmented based on 61 phoneme transcriptions. Following the standard regrouping of phoneme labels [11], 61 phonemes are reduced to 48 phonemes selected for modeling. As observations, 39-dimensional Mel-frequency cepstral coefﬁcients (MFCCs) (13 static coefﬁcients, ∆, and 7 ∆∆) extracted from the speech signals with standard 25 ms frame size, and 10 ms frame shifts are used. The dimension of the latent space is set to Q = 2. For the ﬁrst phoneme classiﬁcation experiment, 100 segments per phoneme are randomly selected using the phoneme boundary provided information in the TIMIT database. The number of inducing input points is set to M = 10. A 10-fold cross-validation test was conducted to evaluate the proposed model in comparison with an HMM that has three states and a single Gaussian distribution with a full covariance matrix per state. Parameters of the HMMs are estimated by using the conventional expectation-maximization (EM) algorithm with a maximum likelihood criterion. Table 2: Classiﬁcation accuracy on the 48-phoneme dataset (10-fold CV average [%]): 100 segments are used for training and testing each phoneme model HMM VGPDS CVGPDS 49.19 48.17 49.36 Table 2 shows the experimental results of a 48-phoneme classiﬁcation. Compared to the HMM and VGPDS, the proposed CVGPDS performs more effectively. For the second phoneme classiﬁcation experiment, the TIMIT core test set consisting of 192 sentences is used for evaluation. We use the same 100 segments for training the phoneme models as in the ﬁrst phoneme classiﬁcation experiment. The size of the training dataset is smaller than that of conventional approaches due to our limited computational ability. When evaluating the models, we merge the labels of 48 phonemes into the commonly used 39 phonemes [11]. Given speech observations with boundary information, a sequence of log-likelihoods is obtained, and then a bigram is constructed to incorporate linguistic information into the classiﬁcation score. In this experiment, the number of inducing input points is set to M = 5. Table 3: Classiﬁcation accuracy on the TIMIT core test set [%]: 100 segments are used for training each phoneme model HMM VGPDS CVGPDS 57.83 61.44 61.54 Table 3 shows the experimental results of phoneme classiﬁcation for the TIMIT core test set. As shown by the results in Table 2, the proposed CVGPDS performed better than the HMM and VGPDS. However, the classiﬁcation accuracies in Table 3 are lower than the state-of-the-art phoneme classiﬁcation results [12-13]. The reasons for low accuracy are as follows: 1) insufﬁcient amount of data is used for training the model owing to limited availability of computational power; 2) a mixture model for the emission is not considered. These remaining issues need to be addressed for improved performance. 4 Conclusion In this paper, a VGPDS-based acoustic model for phoneme classiﬁcation was considered. The proposed acoustic model can represent the nonlinear latent dynamics and dependency among observations by GP priors. In addition, we introduced a variance constraint on the VGPDS. Although the proposed model could not achieve the state-of-the-art performance of phoneme classiﬁcation, the experimental results showed that the proposed acoustic model has potential for speech modeling. For future works, extension to phonetic recognition and mixture of the VGPDS will be considered. Acknowledgments This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No.2012-0005378 and No.2012-0000985) 8 References [1] F. Jelinek, “Continuous speech recognition by statistical methods,” Proceedings of the IEEE, Vol.64, pp.532556, 1976. [2] M. Ostendorf, V. Digalakis, and J. Rohlicek, “From HMMs to segment models: A uniﬁed view of stochastic modeling for speech recognition,” IEEE Trans. on Speech and Audio Processing, Vol.4, pp.360-378, 1996. [3] L. Deng, D. Yu, and A. Acero, “Structured Speech Modeling,” IEEE Trans. on Audio, Speech, and Language Processing, Vol.14, pp.1492-1504, 2006. [4] C. E. Rasmussen and C. K. I. Williams, “Gaussian Process for Machine Learning,” MIT Press, Cambridge, MA, 2006. [5] N. D. Lawrence, “Probabilistic non-linear principal component analysis with Gaussian process latent variable models,” Journal of Machine Learning Research (JMLR), Vol.6, pp.1783-1816, 2005. [6] N. D. Lawrence, “Learning for larger datasets with the Gaussian process latent variable model,” International Conference on Artiﬁcial Intelligence and Statistics (AISTATS), pp.243-250, 2007. [7] M. K. Titsias and N. D. Lawrence, “Bayesian Gaussian Process Latent Variable Model,” International Conference on Artiﬁcial Intelligence and Statistics (AISTATS), pp.844-851, 2010. [8] J. Qui˜nonero-Candela and C. E. Rasmussen, “A Unifying View of Sparse Approximate Gaussian Process Regression,” Journal of Machine Learning Research (JMLR), Vol.6, pp.1939-1959, 2005. [9] A. C. Damianou, M. K. Titsias, and N. D. Lawrence, “Variational Gaussian Process Dynamical Systems,” Advances in Neural Information Processing Systems (NIPS), 2011. [10] J. M. Wang, D. J. Fleet, and A. Hertzmann, “Gaussian Process Dynamical Models for Human Motion,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol.30, pp.283-298, 2008. [11] K. F. Lee and H. W. Hon, “Speaker-independent phone recognition using hidden Markov models,” IEEE Trans. on Acoustics, Speech and Signal Processing, vol.37, pp.1641-1648, 1989. [12] A. Mohamed, G. Dahl, and G. Hinton, “Acoustic modeling using deep belief networks,” IEEE Trans. on Audio, Speech, and Language Processing, Vol.20, no.1, pp. 14-22, 2012. [13] F. Sha and L. K. Saul, “Large margin hidden markov models for automatic speech recognition,” Advances in Neural Information Processing Systems (NIPS), 2007. 9	2012	125
4,481	Fast Variational Inference in the Conjugate Exponential Family James Hensman∗ Department of Computer Science The University of Shefﬁeld james.hensman@sheffield.ac.uk Magnus Rattray Faculty of Life Science The University of Manchester magnus.rattray@manchester.ac.uk Neil D. Lawrence∗ Department of Computer Science The University of Shefﬁeld n.lawrence@sheffield.ac.uk Abstract We present a general method for deriving collapsed variational inference algorithms for probabilistic models in the conjugate exponential family. Our method uniﬁes many existing approaches to collapsed variational inference. Our collapsed variational inference leads to a new lower bound on the marginal likelihood. We exploit the information geometry of the bound to derive much faster optimization methods based on conjugate gradients for these models. Our approach is very general and is easily applied to any model where the mean ﬁeld update equations have been derived. Empirically we show signiﬁcant speed-ups for probabilistic inference using our bound. 1 Introduction Variational bounds provide a convenient approach to approximate inference in a range of intractable models [Ghahramani and Beal, 2001]. Classical variational optimization is achieved through coordinate ascent which can be slow to converge. A popular solution [King and Lawrence, 2006, Teh et al., 2007, Kurihara et al., 2007, Sung et al., 2008, L´azaro-Gredilla and Titsias, 2011, L´azaro-Gredilla et al., 2011] is to marginalize analytically a portion of the variational approximating distribution, removing this from the optimization. In this paper we provide a unifying framework for collapsed inference in the general class of models composed of conjugate-exponential graphs (CEGs). First we review the body of earlier work with a succinct and unifying derivation of the collapsed bounds. We describe how the applicability of the collapsed bound to any particular CEG can be determined with a simple d-separation test. Standard variational inference via coordinate ascent turns out to be steepest ascent with a unit step length on our unifying bound. This motivates us to consider natural gradients and conjugate gradients for fast optimization of these models. We apply our unifying approach to a range of models from the literature obtaining, often, an order of magnitude or more increase in convergence speed. Our unifying view allows collapsed variational methods to be integrated into general inference tools like infer.net [Minka et al., 2010]. ∗also at Shefﬁeld Institute for Translational Neuroscience, SITraN 1 2 The Marginalised Variational Bound The advantages to marginalising analytically a subset of variables in variational bounds seem to be well understood: several different approaches have been suggested in the context of speciﬁc models. In Dirichlet process mixture models Kurihara et al. [2007] proposed a collapsed approach using both truncated stick-breaking and symmetric priors. Sung et al. [2008] proposed ‘latent space variational Bayes’ where both the cluster-parameters and mixing weights were marginalised, again with some approximations. Teh et al. [2007] proposed a collapsed inference procedure for latent Dirichlet allocation (LDA). In this paper we unify all these results from the perspective of the ‘KL corrected bound’ [King and Lawrence, 2006]. This lower bound on the model evidence is also an upper bound on the original variational bound, the difference between the two bounds is given by a Kullback Leibler divergence. The approach has also been referred to as the marginalised variational bound by L´azaro-Gredilla et al. [2011], L´azaro-Gredilla and Titsias [2011]. The connection between the KL corrected bound and the collapsed bounds is not immediately obvious. The key difference between the frameworks is the order in which the marginalisation and variational approximation are applied. However, for CEGs this order turns out to be irrelevant. Our framework leads to a more succinct derivation of the collapsed approximations. The resulting bound can then be optimised without recourse to approximations in either the bound’s evaluation or its optimization. 2.1 Variational Inference Assume we have a probabilistic model for data, D, given parameters (and/or latent variables), X, Z, of the form p(D, X, Z) = p(D \| Z, X)p(Z \| X)p(X). In variational Bayes (see e.g. Bishop [2006]) we approximate the posterior p(Z, X\|D) by a distribution q(Z, X). We use Jensen’s inequality to derive a lower bound on the model evidence L, which serves as an objective function in the variational optimisation: p(D) ≥L = Z q(Z, X) ln p(D, Z, X) q(Z, X) dZ dX. (1) For tractability the mean ﬁeld (MF) approach assumes q factorises across its variables, q(Z, X) = q(Z)q(X). It is then possible to implement an optimisation scheme which analytically optimises each factor alternately, with the optimal distribution given by q⋆(X) ∝exp Z q(Z) ln p(D, X\|Z) dZ , (2) and similarly for Z: these are often referred to as VBE and VBM steps. King and Lawrence [2006] substituted the expression for the optimal distribution (for example q⋆(X)) back into the bound (1), eliminating one set of parameters from the optimisation, an approach that has been reused by L´azaroGredilla et al. [2011], L´azaro-Gredilla and Titsias [2011]. The resulting bound is not dependent on q(X). King and Lawrence [2006] referred to this new bound as ‘the KL corrected bound’. The difference between the bound, which we denote LKL, and a standard mean ﬁeld approximation LMF, is the Kullback Leibler divergence between the optimal form of q∗(X) and the current q(X). We re-derive their bound by ﬁrst using Jensen’s inequality to construct the variational lower bound on the conditional distribution, ln p(D\|X) ≥ Z q(Z) ln p(D, Z\|X) q(Z) dZ ≜L1. (3) This object turns out to be of central importance in computing the ﬁnal KL-corrected bound and also in computing gradients, curvatures and the distribution of the collapsed variables q⋆(X). It is easy to see that it is a function of X which lower-bounds the log likelihood p(D \| X), and indeed our derivation treats it as such. We now marginalize the conditioned variable from this expression, ln p(D) ≥ln Z p(X) exp{L1} dX ≜LKL, (4) giving us the bound of King and Lawrence [2006] & L´azaro-Gredilla et al. [2011]. Note that one set of parameters was marginalised after the variational approximation was made. Using (2), this expression also provides the approximate posterior for the marginalised variables X: q⋆(X) = p(X)eL1−LKL (5) and eLKL appears as the constant of proportionality in the mean-ﬁeld update equation (2). 2 3 Partial Equivalence of the Bounds We can recover LMF from LKL by again applying Jensen’s inequality, LKL = ln Z q(X)p(X) q(X) exp{L1} dX ≥ Z q(X) ln p(X) q(X) exp{L1} dX, (6) which can be re-arranged to give the mean-ﬁeld bound, LKL ≥ Z q(X)q(Z) ln p(D\|Z, X)p(Z)p(X) q(Z)q(X) dX dZ, (7) and it follows that LKL = LMF + KL(q∗(X)\|\|q(X)) and1 LKL ≥LMF. For a given q(Z), the bounds are equal after q(X) is updated via the mean ﬁeld method: the approximations are ultimately the same. The advantage of the new bound is to reduce the number of parameters in the optimisation. It is particularly useful when variational parameters are optimised by gradient methods. Since VBEM is equivalent to a steepest descent gradient method with a ﬁxed step size, there appears to be a lot to gain by combining the KLC bound with more sophisticated optimization techniques. 3.1 Gradients Consider the gradient of the KL corrected bound with respect to the parameters of q(Z): ∂LKL ∂θz = exp{−LKL} ∂ ∂θz Z exp{L1}p(X) dX = Eq⋆(X) h∂L1 ∂θz i , (8) where we have used the relation (5). To ﬁnd the gradient of the mean-ﬁeld bound we note that it can be written in terms of our conditional bound (3) as LMF = Eq(X) h L1 + ln p(X) −ln q(X) i giving ∂LMF ∂θz = Eq(X) h∂L1 ∂θz i (9) thus setting q(X) = q⋆(X) not only makes the bounds equal, LMF = LKL, but also their gradients with respect to θZ. Sato [2001] has shown that the variational update equation can be interpreted as a gradient method, where each update is also a step in the steepest direction in the canonical parameters of q(Z). We can combine this important insight with the above result to realize that we have a simple method for computing the gradients of the KL corrected bound: we only need to look at the update expressions for the mean-ﬁeld method. This result also reveals the weakness of standard variational Bayesian expectation maximization (VBEM): it is a steepest ascent algorithm. Honkela et al. [2010] looked to rectify this weakness by applying a conjugate gradient algorithm to the mean ﬁeld bound. However, they didn’t obtain a signiﬁcant improvement in convergence speed. Our suggestion is to apply conjugate gradients to the KLC bound. Whilst the value and gradient of the MF bound matches that of the KLC bound after an update of the collapsed variables, the curvature is always greater. In practise this means that much larger steps (which we compute using conjugate gradient methods) can be taken when optimizing the KLC bound than for the MF bound leading to more rapid convergence. 3.2 Curvature of the Bounds King and Lawrence [2006] showed empirically that the KLC bound could lead to faster convergence because the bounds differ in their curvature: the curvature of the KLC bound enables larger steps to be taken by an optimizer. We now derive analytical expressions for the curvature of both bounds. For the mean ﬁeld bound we have ∂2LMF ∂θ2z = Eq(X) h∂2L1 ∂θ2z i , (10) 1We use KL(·\|\|·) to denote the Kullback Leibler divergence between two distributions. 3 and for the KLC bound, with some manipulation of (4) and using (5): ∂2LKL ∂θ[i] z ∂θ[j] z = e−LKL ∂2eLKL ∂θ[i] z ∂θ[j] z −e−2LKLn∂eLKL ∂θ[i] z on∂eLKL ∂θ[j] z o = Eq⋆(X) h ∂2L1 ∂θ[i] z ∂θ[j] z i + Eq⋆(X) h ∂L1 ∂θ[i] z ∂L1 ∂θ[j] z i − n Eq⋆(X) h ∂L1 ∂θ[i] z ion Eq⋆(X) h ∂L1 ∂θ[j] z io . (11) In this result the ﬁrst term is equal to (10), and the second two terms combine to be always positive semi-deﬁnite, proving King and Lawrence [2006]’s intuition about the curvature of the bound. When curvature is negative deﬁnite (e.g. near a maximum), the KLC bound’s curvature is less negative deﬁnite, enabling larger steps to be taken in optimization. Figure 1(b) illustrates the effect of this as well as the bound’s similarities. 3.3 Relationship to Collapsed VB In collapsed inference some parameters are marginalized before applying the variational bound. For example, Sung et al. [2008] proposed a latent variable model where the model parameters were marginalised, and Teh et al. [2007] proposed a non-parametric topic model where the document proportions were collapsed. These procedures lead to improved inference, or faster convergence. The KLC bound derivation we have provided also marginalises parameters, but after a variational approximation is made. The difference between the two approaches is distilled in these expressions: ln Ep(X) exp Eq(Z) ln p(D\|X, Z) Eq(Z) ln Ep(X) p(D\|X, Z) (12) where the left expression appears in the KLC bound, and the right expression appears in the bound for collapsed variational Bayes, with the remainder of the bounds being equal. Whilst appropriately conjugate formulation of the model will always ensure that the KLC expression is analytically tractable, the expectation in the collapsed VB expression is not. Sung et al. [2008] propose a ﬁrst order approximation to the expectation of the form Eq(Z) h f(Z) i ≈f(Eq(Z) h Z i ), which reduces the right expression to the that on the left. Under this approximation2 the KL corrected approach is equivalent to the collapsed variational approach. 3.4 Applicability To apply the KLC bound we need to specify a subset, X, of variables to marginalize. We select the variables that break the dependency structure of the graph to enable the analytic computation of the integral in (4). Assuming the appropriate conjugate exponential structure for the model we are left with the requirement to select a sub-set that induces the appropriate factorisation. These induced factorisations are discussed in some detail in Bishop [2006]. They are factorisations in the approximate posterior which arise from the form of the variational approximation and from the structure of the model. These factorisations allow application of KLC bound, and can be identiﬁed using a simple d-separation test as Bishop discusses. The d-separation test involves checking for independence amongst the marginalised variables (X in the above) conditioned on the observed data D and the approximated variables (Z in the above). The requirement is to select a sufﬁcient set of variables, Z, such that the effective likelihood for X, given by (3) becomes conjugate to the prior. Figure 1(a) illustrates the d-separation test with application to the KLC bound. For latent variable models, it is often sufﬁcient to select the latent variables for X whilst collapsing the model variables. For example, in the speciﬁc case of mixture models and topic models, approximating the component labels allows for the marginalisation of the cluster parameters (topics 2Kurihara et al. [2007] and Teh et al. [2007] suggest a further second order correction and assume that that q(Z) is Gaussian to obtain tractability. This leads to additional correction terms that augment KLC bound. The form of these corrections would need to be determined on a case by case basis, and has in fact been shown to be less effective than those methods uniﬁed here [Asuncion et al., 2012]. 4 D E F A B C (a) (b) Figure 1: (a) An example directed graphical model on which we could use the KLC bound. Given the observed node C, the nodes A,F d-separate given nodes B,D,E. Thus we could make an explicit variational approximation for A,F, whilst marginalising B,D,E. Alternatively, we could select B,D,E for a parameterised approximate distribution, whilst marginalising A,F. (b) A sketch of the KLC and MF bounds. At the point where the mean ﬁeld method has q(X) = q⋆(X), the bounds are equal in value as well as in gradient. Away from the this point, the different between the bounds is the Kullback Leibler divergence between the current MF approximation for X and the implicit distribution q⋆(X) of the KLC bound. allocations) and mixing proportions. This allowed Sung et al. [2008] to derive a general form for latent variable models, though our formulation is general to any conjugate exponential graph. 4 Riemannian Gradient Based Optimisation Sato [2001] and Hoffman et al. [2012] showed that the VBEM procedure performs gradient ascent in the space of the natural parameters. Using the KLC bound to collapse the problem, gradient methods seem a natural choice for optimisation, since there are fewer parameters to deal with, and we have shown that computation of the gradients is straightforward (the variational update equations contain the model gradients). It turns out that the KLC bound is particularly amenable to Riemannian or natural gradient methods, because the information geometry of the exponential family distrubution(s), over which we are optimising, leads to a simple expression for the natural gradient. Previous investigations of natural gradients for variational Bayes [Honkela et al., 2010, Kuusela et al., 2009] required the inversion of the Fisher information at every step (ours does not), and also used VBEM steps for some parameters and Riemannian optimisation for other variables. The collapsed nature of the KLC bound means that these VBEM steps are unnecessary: the bound can be computed by parameterizing the distribution of only one set of variables (q(Z)) whilst the implicit distribution of the other variables is given in terms of the ﬁrst distribution and the data by equation (5). We optimize the lower bound LKL with respect to the parameters of the approximating distribution of the non-collapsed variables. We showed in section 2 that the gradient of the KLC bound is given by the gradient of the standard MF variational bound, after an update of the collapsed variables. It is clear from their deﬁnition that the same is true of the natural gradients. 4.1 Variable Transformations We can compute the natural gradient of our collapsed bound by considering the update equations of the non-collapsed problem as described above. However, if we wish to make use of more powerful optimisation methods like conjugate gradient ascent, it is helpful to re-parameterize the natural parameters in an unconstrained fashion. The natural gradient is given by [Amari and Nagaoka, 2007]: eg(θ) = G(θ)−1 ∂LKL ∂θ (13) where G(θ) is the Fisher information matrix whose i,jth element is given by G(θ)[i,j] = −Eq(X \| θ) h∂2 ln q(X \| θ) ∂θ[i]∂θ[j] i . (14) 5 For exponential family distributions, this reduces to ∇2 θψ(θ), where ψ is the log-normaliser. Further, for exponential family distributions, the Fisher information in the canonical parameters (θ) and that in the expectation parameters (η) are reciprocal, and we also have G(θ) = ∂η/∂θ. This means that the natural gradient in θ is given by eg(θ) = G(θ)−1 ∂η ∂θ ∂LKL ∂η = ∂LKL ∂η and eg(η) = ∂LKL ∂θ . (15) The gradient in one set of parameters provides the natural gradient in the other. Thus when our approximating distribution q is exponential family, we can compute the natural gradient without the expensive matrix inverse. 4.2 Steepest Ascent is Coordinate Ascent Sato [2001] showed that the VBEM algorithm was a gradient based algorithm. In fact, VBEM consists of taking unit steps in the direction of the natural gradient of the canonical parameters. From equation (9) and the work of Sato [2001], we see that the gradient of the KLC bound can be obtained by considering the standard mean-ﬁeld update for the non-collapsed parameter Z. We conﬁrm these relationships for the models studied in the next section in the supplementary material. Having conﬁrmed that the VB-E step is equivalent to steepest-gradient ascent we now explore whether the procedure could be improved by the use of conjugate gradients. 4.3 Conjugate Gradient Optimization One idea for solving some of the problems associated with steepest ascent is to ensure each gradient step is conjugate (geometrically) to the previous. Honkela et al. [2010] applied conjugate gradients to the standard mean ﬁeld bound, we expect much faster convergence for the KLC bound due to its differing curvature. Since VBEM uses a step length of 1 to optimize,3 we also used this step length in conjugate gradients. In the natural conjugate gradient method, the search direction at the ith iteration is given by si = −egi + βsi−1. Empirically the Fletcher-Reeves method for estimating β worked well for us: βF R = ⟨egi, egi⟩i ⟨egi−1, egi−1⟩i−1 (16) where ⟨·, ·⟩i denotes the inner product in Riemannian geometry, which is given by eg⊤G(ρ)eg. We note from Kuusela et al. [2009] that this can be simpliﬁed since eg⊤Geg = eg⊤GG−1g = eg⊤g, and other conjugate methods, deﬁned in the supplementary material, can be applied similarly. 5 Experiments For empirical investigation of the potential speed ups we selected a range of probabilistic models. We provide derivations of the bound and fuller explanations of the models in the supplementary material. In each experiment, the algorithm was considered to have converged when the change in the bound or the Riemannian gradient reached below 10−6. Comparisons between optimisation procedures always used the same initial conditions (or set of initial conditions) for each method. First we recreate the mixture of Gaussians example described by Honkela et al. [2010]. 5.1 Mixtures of Gaussians For a mixture of Gaussians, using the d-separation rule, we select for X the cluster allocation (latent) variables. These are parameterised through the softmax function for unconstrained optimisation. Our model includes a fully Bayesian treatment of the cluster parameters and the mixing proportions, whose approximate posterior distributions appear as (5). Full details of the algorithm derivation are given in the supplementary material. A neat feature is that we can make use of the discussion above to derive an expression for the natural gradient without a matrix inverse. 3We empirically evaluated a line-search procedure, but found that in most cases that Wolfe-Powell conditions were met after a single step of unit length. 6 Table 1: Iterations to convergence for the mixture of Gaussians problem, with varying overlap (R). This table reports the average number of iterations taken to reach (within 10 nats of) the best known solution. For the more difﬁcult scenarios (with more overlap in the clusters) the VBEM method failed to reach the optimum solution within 500 restarts CG. method R = 1 R = 2 R = 3 R = 4 R = 5 Polack-Ribi´ere 3, 100.37 15, 698.57 5, 767.12 1, 613.09 3, 046.25 Hestenes-Stiefel 1, 371.55 5, 501.25 5, 922.4 358.03 172.39 Fletcher-Reeves 416.18 1, 161.35 5, 091.0 792.10 494.24 VBEM ∞ ∞ ∞ 992.07 429.57 Table 2: Time and iterations taken to run LDA on the NIPS 2011 corpus, ± one standard deviation, for two conjugate methods and VBEM. The Fletcher-Reeves conjugate algorithm is almost ten times as fast as VBEM. The value of the bound at the optimum was largely the same: deviations are likely just due to the choice of initialisations, of which we used 12. Method Time (minutes) Iterations Bound Hestenes-Steifel 56.4 ± 18.5 644.3 ± 214.5 −1, 998, 780 ± 201 Fletcher-Reeves 38.5 ± 8.7 447.8 ± 100.5 −1, 998, 743 ± 194 VBEM 370 ± 105 4, 459 ± 1, 296 −1, 998, 732 ± 241 In Honkela et al. [2010] data are drawn from a mixture of ﬁve two-dimensional Gaussians with equal weights, each with unit spherical covariance. The centers of the components are at (0, 0) and (±R, ±R). R is varied from 1 (almost completely overlapping) to 5 (completely separate). The model is initialised with eight components with an uninformative prior over the mixing proportions: the optimisation procedure is left to select an appropriate number of components. Sung et al. [2008] reported that their collapsed method led to improved convergence over VBEM. Since our objective is identical, though our optimisation procedure different, we devised a metric for measuring the efﬁcacy of our algorithms which also accounts for their propensity to fall into local minima. Using many randomised restarts, we measured the average number of iterations taken to reach the best-known optimum. If the algorithm converged at a lesser optimum, those iterations were included in the denomiator, but we didn’t increment the numerator when computing the average. We compared three different conjugate gradient approaches and standard VBEM (which is also steepest ascent on the KLC bound) using 500 restarts. Table 1 shows the number of iterations required (on average) to come within 10 nats of the best known solution for three different conjugate-gradient methods and VBEM. VBEM sometimes failed to ﬁnd the optimum in any of the 500 restarts. Even relaxing the stringency of our selection to 100 nats, the VBEM method was always at least twice as slow as the best conjugate method. 5.2 Topic Models Latent Dirichlet allocation (LDA) [Blei et al., 2003] is a popular approach for extracting topics from documents. To demonstrate the KLC bound we applied it to 200 papers from the 2011 NIPS conference. The PDFs were preprocessed with pdftotext, removing non-alphabetical characters and coarsely ﬁltering words by popularity to form a vocabulary size of 2000.4 We selected the latent topic-assignment variables for parameterisation, collapsing the topics and the document proportions. Conjugate gradient optimization was compared to the standard VBEM approach. We used twelve random initializations, starting each algorithm from each initial condition. Topic and document distributions where treated with ﬁxed, uninformative priors. On average, the HestenesSteifel algorithm was almost ten times as fast as standard VB, as shown in Table 2, whilst the ﬁnal bound varied little between approaches. 4Some extracted topics are presented in the supplementary material. 7 5.3 RNA-seq alignment An emerging problem in computational biology is inference of transcript structure and expression levels using next-generation sequencing technology (RNA-Seq). Several models have been proposed. The BitSeq method [Glaus et al., 2012] is based on a probabilistic model and uses Gibbs sampling for approximate inference. The sampler can suffer from particularly slow convergence due to the large size of the problem, which has six million latent variables for the data considered here. We implemented a variational version of their model and optimised it using VBEM and our collapsed Riemannian method. We applied the model to data described in Xu et al. [2010], a study of human microRNA. The model was initialised using four random initial conditions, and optimised using standard VBEM and the conjugate gradient versions of the algorithm. The Polack-Ribi´ere conjugate method performed very poorly for this problem, often giving negative conjugation: we omit it here. The solutions found for the other algorithms were all fairly close, with bounds coming within 60 nats. The VBEM method was dramatically outperformed by the Fletcher-Reeves and Hestenes-Steifel methods: it took 4600 ± 20 iterations to converge, whilst the conjugate methods took only 268±4 and 265±1 iterations to converge. At about 8 seconds per iteration, our collapsed Riemannian method requires around forty minutes, whilst VBEM takes almost eleven hours. All the variational approaches represent an improvement over a Gibbs sampler, which takes approximately one week to run for this data [Glaus et al., 2012]. 6 Discussion Under very general conditions (conjugate exponential family) we have shown the equivalence of collapsed variational bounds and marginalized variational bounds using the KL corrected perspective of King and Lawrence [2006]. We have provided a succinct derivation of these bounds, unifying several strands of work and laying the foundations for much wider application of this approach. When the collapsed variables are updated in the standard MF bound the KLC bound is identical to the MF bound in value and gradient. Sato [2001] has shown that coordinate ascent of the MF bound (as proscribed by VBEM updates) is equivalent to steepest ascent of the MF bound using natural gradients. This implies that standard variational inference is also performing steepest ascent on the KLC bound. This equivalence between natural gradients and the VBEM update equations means our method is quickly implementable for any model where the mean ﬁeld update equations have been computed. It is only necessary to determine which variables to collapse using a d-separation test. Importantly this implies our approach can readily be incorporated in automated inference engines such as that provided by infer.net [Minka et al., 2010]. We’d like to emphasise the ease with which the method can be applied: we have provided derivations of equivalencies of the bounds and gradients which should enable collapsed conjugate optimisation of any existing mean ﬁeld algorithm, with minimal changes to the software. Indeed our own implementations (see supplementary material) use just a few lines of code to switch between the VBEM and conjugate methods. The improved performance arises from the curvature of the KLC bound. We have shown that it is always less negative than that of the original variational bound allowing much larger steps in the variational parameters as King and Lawrence [2006] suggested. This also provides a gateway to second-order optimisation, which could prove even faster. We provided empirical evidence of the performance increases that are possible using our method in three models. 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4,482	Identiﬁability and Unmixing of Latent Parse Trees Daniel Hsu Microsoft Research Sham M. Kakade Microsoft Research Percy Liang Stanford University Abstract This paper explores unsupervised learning of parsing models along two directions. First, which models are identiﬁable from inﬁnite data? We use a general technique for numerically checking identiﬁability based on the rank of a Jacobian matrix, and apply it to several standard constituency and dependency parsing models. Second, for identiﬁable models, how do we estimate the parameters efﬁciently? EM suffers from local optima, while recent work using spectral methods [1] cannot be directly applied since the topology of the parse tree varies across sentences. We develop a strategy, unmixing, which deals with this additional complexity for restricted classes of parsing models. 1 Introduction Generative parsing models, which deﬁne joint distributions over sentences and their parse trees, are one of the core techniques in computational linguistics. We are interested in the unsupervised learning of these models [2–6], where the goal is to estimate the model parameters given only examples of sentences. Unsupervised learning can fail for a number of reasons [7]: model misspeciﬁcation, non-identiﬁability, estimation error, and computation error. In this paper, we delve into two of these issues: identiﬁability and computation. In doing so, we confront a central challenge of parsing models—that the topology of the parse tree is unobserved and varies across sentences. This is in contrast to standard phylogenetic models [8] and other latent tree models for which there is a single ﬁxed global tree across all examples [9]. A model is identiﬁable if there is enough information in the data to pinpoint the parameters (up to some trivial equivalence class); establishing the identiﬁability of a model is often a highly nontrivial task. A classic result of Kruskal [10] has been employed to prove the identiﬁability of a wide class of latent variable models, including hidden Markov models and certain restricted mixtures of latent tree models [11–13]. However, these techniques cannot be directly applied to parsing models since the tree topology varies over an exponential set of possible topologies. Instead, we turn to techniques from algebraic geometry [14–17]; we show that a simple numerical procedure can be used to check identiﬁability for a wide class of models in NLP. Using this tool, we discover that probabilistic context-free grammars (PCFGs) are non-identiﬁable, but that simpler PCFG variants and dependency models are identiﬁable. The most common way to estimate unsupervised parsing models is by using local techniques such as EM [18] or MCMC sampling [19], but these methods can suffer from local optima and slow mixing. Meanwhile, recent work [1,20–23] has shown that spectral methods can be used to estimate mixture models and HMMs with provable guarantees. These techniques express low-order moments of the observable distribution as a product of matrix parameters and use eigenvalue decomposition to recover these matrices. However, these methods are not directly applicable to parsing models because the tree topology again varies non-trivially. To address this, we propose a new technique, unmixing. The main idea is to express moments of the observable distribution as a mixture over the possible topologies. For restricted parsing models, the moments for a ﬁxed tree structure can E-mail: dahsu@microsoft.com, skakade@microsoft.com, pliang@cs.stanford.edu 1 T T T T O x1 z01 O x2 z12 z02 O x3 z23 z03 π T T O x1 z01 T T O x2 z12 O x3 z23 z13 z03 π Topology(z) = 1 Topology(z) = 2 x1 x2 x3 π A A Topology(z) = 1 x1 x2 x3 π A A Topology(z) = 2 x1 x2 x3 π A A Topology(z) = 3 x1 x2 x3 A π A Topology(z) = 4 x1 x2 x3 A A π Topology(z) = 5 x1 x2 x3 A A π Topology(z) = 6 x1 x2 x3 A A π Topology(z) = 7 (a) Constituency (PCFG-IE) (b) Dependency (DEP-IE) Figure 1: The two constituency trees and seven dependency trees over L = 3 words, x1, x2, x3. (a) A constituency tree consists of a hierarchical grouping of the words with a latent state zv for each node v. (b) A dependency tree consists of a collection of directed edges between the words. In both cases, we have labeled each edge from i to j with the parameters used to generate the state of node j given i. be “unmixed”, thereby reducing the problem to one with a ﬁxed topology, which can be tackled using standard techniques [1]. Importantly, our unmixing technique does not require the training sentences be annotated with the tree topologies a priori, in contrast to recent extensions of [21] to learning PCFGs [24] and dependency trees [25,26], which work on a ﬁxed topology. 2 Notation For a positive integer n, deﬁne [n] def = {1, . . . , n} and ⟨n⟩= {e1, . . . , en}, where ei is the vector which is 1 in component i and 0 elsewhere. For integers a, b ∈[n], let a⊗nb = (a−1)n+b ∈[n2] be the integer encoding of the pair (a, b). For a pair of matrices, A, B ∈Rm×n, deﬁne the columnwise tensor product A ⊗C B ∈Rm2×n to be such that (A ⊗C B)(i1⊗mi2)j = Ai1jBi2j. For a matrix A ∈Rm×n, let A† denote the Moore-Penrose pseudoinverse. 3 Parsing models A sentence is a sequence of L words, x = (x1, . . . , xL), where each word xi ∈⟨d⟩is one of d possible word types. A (generative) parsing model deﬁnes a joint distribution Pθ(x, z) over a sentence x and its parse tree z (to be made precise later), where θ are the model parameters (a collection of multinomials). Each parse tree z has a topology Topology(z) ∈Topologies, which is both unobserved and varying across sentences. The learning problem is to recover θ given only samples of x. Two important classes of models of natural language syntax are constituency models, which represent a hierarchical grouping and labeling of the phrases of a sentence (e.g., Figure 1(a)), and dependency models, which represent pairwise relationships between the words of a sentence (e.g., Figure 1(b)). 2 3.1 Constituency models A constituency tree z = (V, s) consists of a set of nodes V and a collection of hidden states s = {sv}v∈V . Each state sv ∈⟨k⟩represents one of k possible syntactic categories. Each node v has the form [i : j] for 0 ≤i < j ≤L corresponding to the phrase between positions i and j of the sentence. These nodes form a binary tree as follows: the root node is [0 : L] ∈V , and for each node [i : j] ∈V with j −i > 1, there exists a unique m with i < m < j deﬁning the two children nodes [i : m] ∈V and [m : j] ∈V . Let Topology(z) be an integer encoding of V . PCFG. Perhaps the most well-known constituency parsing model is the probabilistic context-free grammar (PCFG). The parameters of a PCFG are θ = (π, B, O), where π ∈Rk speciﬁes the initial state distribution, B ∈Rk2×k speciﬁes the binary production distributions, and O ∈Rd×k speciﬁes the emission distributions. A PCFG corresponds to the following generative process (see Figure 1(a) for an example): choose a topology Topology(z) uniformly at random; generate the state of the root node using π; recursively generate pairs of children states given their parents using B; and ﬁnally generate words xi given their parents using O. This generative process deﬁnes a joint probability over a sentence x and a parse tree z: Pθ(x, z) = \| Topologies \|−1π⊤s[0:L] Y [i:m],[m:j]∈V (s[i:m] ⊗k s[m:j])⊤Bs[i:j] L Y i=1 x⊤ i Os[i−1:i], (1) We will also consider two variants of the PCFG with additional restrictions: PCFG-I. The left and right children states are generated independently—that is, we have the following factorization: B = T1 ⊗C T2 for some T1, T2 ∈Rk×k. PCFG-IE. The left and the right productions are independent and equal: B = T ⊗C T. 3.2 Dependency tree models In contrast to constituency trees, which posit internal nodes with latent states, dependency trees connect the words directly. A dependency tree z is a set of directed edges (i, j), where i, j ∈[L] are distinct positions in the sentence. Let Root(z) denote the position of the root node of z. We consider only projective dependency trees [27]: z is projective if for every path from i to j to k in z, we have that j and k are on the same side of i (that is, j −i and k −i have the same sign). Let Topology(z) be an integer encoding of z. DEP-I. We consider the simple dependency model of [4]. The parameters of this model are θ = (π, A↙, A↘), where π ∈Rd is the initial word distribution and A↙, A↘∈Rd×d are the left and right argument distributions. The generative process is as follows: choose a topology Topology(z) uniformly at random, generate the root word using π, and recursively generate argument words to the left to the right given the parent word using A↙and A↘, respectively. The corresponding joint probability distribution is as follows: Pθ(x, z) = \| Topologies \|−1π⊤xRoot(z) Y (i,j)∈z x⊤ j Adir(i,j)xi, (2) where dir(i, j) = ↙if j < i and ↘if j > i. We also consider the following two variants: DEP-IE. The left and right argument distributions are equal: A = A↙= A↘. DEP-IES. A = A↙= A↘and π is the stationary distribution of A (that is, π = Aπ). Usually a PCFG induces a topology via a state-dependent probability of choosing a binary production versus an emission. Our model is a restriction which corresponds to a state-independent probability. 3 4 Identiﬁability Our goal is to estimate model parameters θ0 ∈Θ given only access to sentences x ∼Pθ0. Speciﬁcally, suppose we have an observation function φ(x) ∈Rm, which is the only lens through which an algorithm can view the data. We ask a basic question: in the limit of inﬁnite data, is it informationtheoretically possible to identify θ0 from the observed moments µ(θ0) def = Eθ0[φ(x)]? To be more precise, deﬁne the equivalence class of θ0 to be the set of parameters θ that yield the same observed moments: SΘ(θ0) = {θ ∈Θ : µ(θ) = µ(θ0)}. (3) It is impossible for an algorithm to distinguish among the elements of SΘ(θ0). Therefore, one might want to ensure that \|SΘ(θ0)\| = 1 for all θ0 ∈Θ. However, this requirement is too strong for two reasons. First, models often have natural symmetries—e.g., the k states of any PCFG can be permuted without changing µ(θ), so \|SΘ(θ0)\| ≥k!. Second, \|SΘ(θ0)\| = ∞for some pathological θ0’s—e.g., PCFGs where all states have the same emission distribution O are indistinguishable regardless of the production distributions B. The following deﬁnition of identiﬁability accommodates these two exceptional cases: Deﬁnition 1 (Identiﬁability). A model family with parameter space Θ is (globally) identiﬁable from φ if there exists a measure zero set E such that \|SΘ(θ0)\| is ﬁnite for every θ0 ∈Θ\E. It is locally identiﬁable from φ if there exists a measure zero set E such that, for every θ0 ∈Θ\E, there exists an open neighborhood N(θ0) around θ0 such that SΘ(θ0) ∩N(θ0) = {θ0}. Example of non-identiﬁability. Consider the DEP-IE model with L = 2 with the full observation function φ(x) = x1 ⊗x2. The corresponding observed moments are µ(θ) = 0.5A diag(π) + 0.5 diag(π)A⊤. Note that A diag(π) is an arbitrary d × d matrix whose entries sum to 1, which has d2 −1 degrees of freedom, whereas µ(θ) is a symmetric matrix whose entries sum to 1, which has d+1 2 −1 degrees of freedom. Therefore, SΘ(θ) has dimension d 2 and therefore the model is non-identiﬁable. Parameter counting. It is important to compute the degrees of freedom correctly—simple parameter counting is insufﬁcient. For example, consider the PCFG-IE model with L = 2. The observed moments with respect to φ(x) = x1 ⊗x2 is a d × d matrix, which places d2 constraints on the k2 +(d−1)k parameters. When d ≥2k, there are more constraints than parameters, but the PCFGIE model with L = 2 is actually non-identiﬁable (as we will see later). The issue here is that the number of constraints does not reveal the fact that some of these constraints are redundant. 4.1 Observation functions An observation function φ(x) and its associated observed moments µ(θ0) = Eθ0[φ(x)] reveals aspects of the distribution Pθ0(x). For example, φ(x) = x1 would only reveal the marginal distribution of the ﬁrst word, whereas φ(x) = x1 ⊗· · · ⊗xL reveals the entire distribution of x. There is a tradeoff: Higher-order moments provide more information, but are harder to estimate reliably given ﬁnite data, and are also computationally more expensive. In this paper, we consider the following intermediate moments: φ12(x) def = x1 ⊗x2 φ∗∗(x) def = xi ⊗xj : i, j ∈[L] φ123(x) def = x1 ⊗x2 ⊗x3 φ∗∗∗(x) def = xi ⊗xj ⊗xk : i, j, k ∈[L] φ123η(x) def = (x1 ⊗x2)(η⊤x3) φ∗∗∗η(x) def = (xi ⊗xj)(η⊤xk) : i, j, k ∈[L] φall(x) def = x1 ⊗· · · ⊗xL Above, η ∈Rd denotes a unit vector in Rd (e.g., e1) which picks out a linear combination of matrix slices from a third-order d × d × d tensor. 4.2 Automatically checking identiﬁability One immediate goal is to determine which models in Section 3 are identiﬁable from which of the observed moments (Section 4.1). A powerful analytic tool that has been succesfully applied in 4 previous work is Kruskal’s theorem [10,11], but (i) it is does not immediately apply to models with random topologies, and (ii) only gives sufﬁcient conditions for identiﬁability, and cannot be used to determine non-identiﬁability. Furthermore, since it is common practice to explore many different models for a given problem in rapid succession, we would like to check identiﬁability quickly and reliably. In this section, we develop an automatic procedure to do this. To establish identiﬁability, let us examine the algebraic structure of SΘ(θ0) for θ0 ∈Θ, where we assume that the parameter space Θ is an open subset of [0, 1]n. Recall that SΘ(θ0) is deﬁned by the moment constraints µ(θ) = µ(θ0). We can write these constraints as hθ0(θ) = 0, where hθ0(θ) def = µ(θ) −µ(θ0) is a vector of m polynomials in θ. Let us now compute the number of degrees of freedom of hθ0 around θ0. The key quantity is J(θ) ∈Rm×n, the Jacobian of hθ0 at θ (note that the Jacobian of hθ0 does not depend on θ0; it is precisely the Jacobian of µ). This Jacobian criterion is well-established in algebraic geometry, and has been adopted in the statistical literature for testing model identiﬁability and other related properties [14–17]. Intuitively, each row of J(θ0) corresponds to a direction of a constraint violation, and thus the row space of J(θ0) corresponds to all directions that would take us outside the equivalence class SΘ(θ0). If J(θ0) has less than rank n, then there is a direction orthogonal to all the rows along which we can move and still satisfy all the constraints—in other words, \|SΘ(θ0)\| is inﬁnite, and therefore the model is non-identiﬁable. This intuition leads to the following algorithm: CHECKIDENTIFIABILITY: −1. Choose a point ˜θ ∈Θ uniformly at random. −2. Compute the Jacobian matrix J(˜θ). −3. Return “yes” if the rank of J(˜θ) = n and “no” otherwise. The following theorem asserts the correctness of CHECKIDENTIFIABILITY. It is largely based on techniques in [16], although we have not seen it explicitly stated in this form. Theorem 1 (Correctness of CHECKIDENTIFIABILITY). Assume the parameter space Θ is a nonempty open connected subset of [0, 1]n; and the observed moments µ: Rn →Rm, with respect to observation function φ, is a polynomial map. Then with probability 1, CHECKIDENTIFIABILITY returns “yes” iff the model family is locally identiﬁable from φ. Moreover, if it returns “yes”, then there exists E ⊂Θ of measure zero such that the model family with parameter space Θ \ E is identiﬁable from φ. The proof of Theorem 1 is given in Appendix A. 4.3 Implementation of CHECKIDENTIFIABILITY Computing the Jacobian. The rows of J correspond to ∂Eθ[φj(x)]/∂θ and can be computed efﬁciently by adapting dynamic programs used in the E-step of an EM algorithm for parsing models. There are two main differences: (i) we must sum over possible values of x in addition to z, and (ii) we are not computing moments, but rather gradients thereof. Speciﬁcally, we adapt the CKY algorithm for constituency models and the algorithm of [27] for dependency models. See Appendix C.1 for more details. Numerical issues. Because we implemented CHECKIDENTIFIABILITY on a ﬁnite precision machine, the results are subject to numerical precision errors. However, we veriﬁed that our numerical results are consistent with various analytically-derived identiﬁability results (e.g., from [11]). While we initially deﬁned θ to be a tuple of conditional probability matrices, we will now use its nonredundant vectorized form θ ∈Rn. 5 Model \ Observation function φ12 φ∗∗ φ123e1 φ123 φ∗∗∗e1 φ∗∗∗ PCFG No, even from φall for L ∈{3, 4, 5} PCFG-I / PCFG-IE No Yes iff L ≥4 Yes iff L ≥3 DEP-I No Yes iff L ≥3 DEP-IE / DEP-IES Yes iff L ≥3 Figure 2: Local identiﬁability of parsing models. These ﬁndings are given by CHECKIDENTIFIABILITY have the semantics from Theorem 1. These were checked for d ∈ {2, 3, . . . , 8}, k ∈{2, . . . , d} (applies only for PCFG models), L ∈{2, 3, . . . , 9}. 4.4 Identiﬁability of constituency and dependency tree models We checked the identiﬁability status of various constituency and dependency tree models using our implementation of CHECKIDENTIFIABILITY. We focus on the regime where d ≥k for PCFGs; additional results for d < k are given in Appendix B. The results are reported in Figure 2. First, we found that the PCFG is not identiﬁable from φall (and therefore not identiﬁable from any φ) for L ∈{3, 4, 5}; we believe that the same holds for all L. This negative result motivates exploring restricted subclasses of PCFGs, such as PCFG-I and PCFG-IE, which factorize the binary productions. For these classes, we found that the sentence length L and choice of observation function can inﬂuence identiﬁability: Both models are identiﬁable for large enough L (e.g., L ≥3) and with a sufﬁciently rich observation function (e.g., φ123η). The dependency models, DEP-I and DEP-IE, were all found to be identiﬁable for L ≥3 from second-order moments φ∗∗. The conditions for identiﬁability are less stringent than their constituency counterparts (PCFG-I and PCFG-IE), which is natural since dependency models are simpler without the latent states. Note that in all identiﬁable models, second-order moments sufﬁce to determine the distribution—this is good news because low-order moments are easier to estimate. 5 Unmixing algorithms Having established which parsing models are identiﬁable, we now turn to parameter estimation for these models. We will consider algorithms based on moment matching—those that try to ﬁnd a θ satisfying µ(θ) = u for some u. Typically, u is an empirical estimate of µ(θ0) = Eθ0[φ(x)] based on samples x ∼Pθ0. In general, solving µ(θ) = u corresponds to ﬁnding solutions to systems of multivariate polynomials, which is NP-hard [28]. However, µ(θ) often has additional structure which we can exploit. For instance, for an HMM, the sliced third-order moments µ123η(θ) can be written as a product of parameter matrices in θ, and each matrix can be recovered by decomposing the product [1]. For parsing models, the challenge is that the topology is random, so the moments is not a single product, but a mixture over products. To deal with this complication, we propose a new technique, which we call unmixing: We “unmix” the products from the mixtures, essentially reducing the problem to one with a ﬁxed topology. We will ﬁrst present the general idea of unmixing (Section 5.1) and then apply it to the PCFG-IE model (Section 5.2) and the DEP-IES model (Section 5.3). 5.1 General case We assume the observation function φ(x) consists of a collection of observation matrices {φo(x)}o∈O (e.g., for o = (i, j), φo(x) = xi ⊗xj). Given an observation matrix φo(x) and a topology t ∈Topologies, consider the mapping that computes the observed moment conditioned on Note that these subclasses occupy measure zero subsets of the PCFG parameter space, which is expected given the non-identiﬁability of the general PCFG. We will develop our algorithms assuming true moments (u = µ(θ0)). The empirical moments converge to the true moments at Op(n−1 2 ), and matrix perturbation arguments (e.g., [1]) can be used derive sample complexity arguments for the parameter error. 6 that topology: Ψo,t(θ) = Eθ[φo(x) \| Topology = t]. As we range o over O and t over Topologies, we will enounter a ﬁnite number of such mappings. We call these mappings compound parameters, denoted {Ψp}p∈P. Now write the observed moments as a weighted sum: µo(θ) = X p∈P P(Ψo,Topology = Ψp) \| {z } def = Mop Ψp for all o ∈O, (4) where we have deﬁned Mop to be the probability mass over tree topologies that yield compound parameter Ψp. We let {Mop}o∈O,p∈P be the mixing matrix. Note that (4) deﬁnes a system of equations µ = MΨ, where the variables are the compound parameters and the constraints are the observed moments. In a sense, we have replaced the original system of polynomial equations (in θ) with a system of linear equations (in Ψ). The key to the utility of this technique is that the number of compound parameters can be polynomial in L even when the number of possible topologies is exponential in L. Previous analytic techniques [13] based on Kruskal’s theorem [10] cannot be applied here because the possible topologies are too many and too varied. Note that the mixing equation µ = MΨ holds for each sentence length L, but many compound parameters p appear in the equations of multiple L. Therefore, we can combine the equations across all observed sentence lengths, yielding a more constrained system than if we considered the equations of each L separately. The following proposition shows how we can recover θ by unmixing the observed moments µ: Proposition 1 (Unmixing). Suppose that there exists an efﬁcient base algorithm to recover θ from some subset of compound parameters {Ψp(θ) : p ∈P0}, and that e⊤ p is in the row space of M for each p ∈P0. Then we can recover θ as follows: UNMIX(µ): −1. Compute the mixing matrix M (4). −2. Retrieve the compound parameters Ψp(θ) = (M †µ)p for each p ∈P0. −3. Call the base algorithm on {Ψp(θ) : p ∈P0} to obtain θ. For all our parsing models, M can be computed efﬁciently using dynamic programming (Appendix C.2). Note that M is data-independent, so this computation can be done once in advance. 5.2 Application to the PCFG-IE model As a concrete example, consider the PCFG-IE model over L = 3 words. Write A = OT. For any η ∈Rd, we can express the observed moments as a sum over the two possible topologies in Figure 1(a): µ123η def = E[x1 ⊗x2(η⊤x3)] = 0.5Ψ1;η + 0.5Ψ2;η, Ψ1;η def = A diag(T diag(π)A⊤η)A⊤, µ132η def = E[x1 ⊗x3(η⊤x2)] = 0.5Ψ3;η + 0.5Ψ2;η, Ψ2;η def = A diag(π)T ⊤diag(A⊤η)A⊤, µ231η def = E[x2 ⊗x3(η⊤x1)] = 0.5Ψ3;η + 0.5Ψ1;η, Ψ3;η def = A diag(A⊤η)T diag(π)A⊤, or compactly in matrix form: µ123η µ132η µ231η ! \| {z } observed moments µη = 0.5I 0.5I 0 0 0.5I 0.5I 0.5I 0 0.5I ! \| {z } mixing matrix M Ψ1;η Ψ2;η Ψ3;η ! \| {z } compound parameters Ψη . Let us observe µη at two different values of η, say at η = 1 and η = τ for some random τ. Since the mixing matrix M is invertible, we can obtain the compound parameters Ψ2;1 = (M −1µ1)2 and Ψ2;τ = (M −1µτ)2. 7 Now we will recover θ from Ψ2;1 and Ψ2;τ by ﬁrst extracting A = OT via an eigenvalue decomposition, and then recovering π, T, and O in turn (all up to the same unknown permutation) via elementary matrix operations. For the ﬁrst step, we will use the following tool (adapted from Algorithm A of [1]), which allow us to decompose two related matrix products: Lemma 1 (Spectral decomposition). Let M1, M2 ∈Rd×k have full column rank and D be a diagonal matrix with distinct diagonal entries. Suppose we observe X = M1M ⊤ 2 and Y = M1DM ⊤ 2 . Then DECOMPOSE(X, Y ) recovers M1 up to a permutation and scaling of the columns. DECOMPOSE(X, Y ): −1. Find U1, U2 ∈Rd×k such that range(U1) = range(X) and range(U2) = range(X⊤). −2. Perform an eigenvalue decomposition of (U ⊤ 1 Y U2)(U ⊤ 1 XU2)−1 = V SV −1. −3. Return (U ⊤ 1 )†V . First, run DECOMPOSE(X = Ψ⊤ 2;1, Y = Ψ⊤ 2;τ) (Lemma 1), which corresponds to M1 = A and M2 = A diag(π)T ⊤. This produces AΠS for some permutation matrix Π and diagonal scaling S. Since we know that the columns of A sum to one, we can identify AΠ. To recover the initial distribution π (up to permutation), take Ψ2;11 = Aπ and left-multiply by (AΠ)† to get Π−1π. For T, put the entries of π in a diagonal matrix: Π−1 diag(π)Π. Take Ψ⊤ 2;1 = AT diag(π)A⊤and multiply by (AΠ)† on the left and ((AΠ)⊤)†(Π−1 diag(π)Π)−1 on the right, which yields Π−1TΠ. (Note that Π is orthogonal, so Π−1 = Π⊤.) Finally, multiply AΠ = OTΠ and (Π−1TΠ)−1, which yields OΠ. The above algorithm identiﬁes the PCFG-IE from only length 3 sentences. To exploit sentences of different lengths, we can compute a mixing matrix M which includes constraints from sentences of length 1 ≤L ≤Lmax up to some upper bound Lmax. For example, Lmax = 10 results in a 990 × 2376 mixing matrix. We can retrieve the same compound parameters (Ψ2;1 and Ψ2;τ) from the pseudoinverse of M and as proceed as before. 5.3 Application to the DEP-IES model We now turn to the DEP-IES model over L = 3 words. Our goal is to recover the parameters θ = (π, A). Let D = diag(π) = diag(Aπ), where the second equality is due to stationarity of π. µ1 def = E[x1] = π, µ12 def = E[x1 ⊗x2] = 7−1(DA⊤+ DA⊤+ DA⊤A⊤+ AD + ADA⊤+ AD + DA⊤), µ13 def = E[x1 ⊗x3] = 7−1(DA⊤+ DA⊤A⊤+ DA⊤+ ADA⊤+ AD + AAD + AD), ˜µ12 def = ˜E[x1 ⊗x2] = 2−1(DA⊤+ AD), where ˜E[·] is taken with respect to length 2 sentences. Having recovered π from µ1, it remains to recover A. By selectively combining the moments above, we can compute AA + A = [7(µ13 − µ12) + 2˜µ12] diag(µ1)−1. Assuming A is generic position, it is diagonalizable: A = QΛQ−1 for some diagonal matrix Λ = diag(λ1, . . . , λd), possibly with complex entries. Therefore, we can recover Λ2 + Λ = Q−1(AA + A)Q. 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4,483	On the (Non-)existence of Convex, Calibrated Surrogate Losses for Ranking Cl´ement Calauz`enes, Nicolas Usunier, Patrick Gallinari LIP6 - UPMC 4 place Jussieu, 75005 Paris, France firstname.lastname@lip6.fr Abstract We study surrogate losses for learning to rank, in a framework where the rankings are induced by scores and the task is to learn the scoring function. We focus on the calibration of surrogate losses with respect to a ranking evaluation metric, where the calibration is equivalent to the guarantee that near-optimal values of the surrogate risk imply near-optimal values of the risk deﬁned by the evaluation metric. We prove that if a surrogate loss is a convex function of the scores, then it is not calibrated with respect to two evaluation metrics widely used for search engine evaluation, namely the Average Precision and the Expected Reciprocal Rank. We also show that such convex surrogate losses cannot be calibrated with respect to the Pairwise Disagreement, an evaluation metric used when learning from pairwise preferences. Our results cast lights on the intrinsic difﬁculty of some ranking problems, as well as on the limitations of learning-to-rank algorithms based on the minimization of a convex surrogate risk. 1 Introduction A surrogate loss is a loss function used as a substitute for the true quality measure during training in order to ease the optimization of the empirical risk. The hinge loss or the exponential loss, which are used in Support Vector Machines or AdaBoost as convex upper bounds of the classiﬁcation error, are well-known examples of surrogate losses for binary classiﬁcation. In this paper, we study surrogate losses for learning to rank, in a context where a set of items should be ranked given an input query and where the ranking is obtained by sorting the items according to predicted numerical scores. This work is motivated by the intensive research that has recently been carried out on machine learning approaches to improve the quality of search engine results, and more speciﬁcally on the design of surrogate losses that lead to high quality rankings (see [16] for a review). Considering algorithms for learning to rank on the axis of scalability, there are ﬁrst algorithms that are designed for small-scale datasets only and that directly solve the NP-hard problem [5] without using any surrogate loss; after them come algorithms that use a surrogate loss chosen as a non-convex but continuous and (almost everywhere) differentiable approximation of the evaluation metric [3, 21, 10], and ﬁnally algorithms that use a convex surrogate loss. Most algorithms for learning to rank fall into the latter category, including the reference algorithms RankBoost [12] and Ranking SVMs [14, 4] or the regression approach of [8], because convex surrogate losses lead to optimization problems that can be solved efﬁciently while non-convex approaches may require intensive computations to ﬁnd a good local optimum. The disadvantage of convex surrogate losses is that they cannot closely approximate the evaluation metrics on the whole prediction space. However, as more examples are available and smaller values of the surrogate risk are achieved, the only region of interest becomes that of near-optimal predictions. It is thus possible that the minimization of the surrogate risk provably leads to optimal predictions according to the risk deﬁned by the evaluation measure. In that case, the surrogate loss is said to be calibrated with respect to the evaluation metric. 1 The calibration of surrogate losses has been extensively studied for various classiﬁcation settings [1, 26, 27, 18, 19] and for AUC optimization [7, 15]. For each of these tasks, many usual convex losses are calibrated with respect to the natural evaluation metric. In the context of learning to rank for search engines, several families of convex losses are calibrated with respect to the Discounted Cumulative Gain (DCG) and its variants [8, 2, 17]. However, other metrics than the DCG are often used as reference for the evaluation of ranked results, such as the Average Precision (AP), used in past TREC competitions [22], the Expected Reciprocal Rank (ERR), used the Yahoo! Learning to Rank Challenge [6], or the Pairwise Disagreement (PD), used when learning from pairwise preferences. And despite the multiplicity of convex losses that have been proposed for ranking, none of them was proved to be calibrated with respect to any of these three metrics. This lead us to the question of whether convex losses can be calibrated with respect to the AP, the ERR, or the PD. Our main contribution is a deﬁnitive and negative answer to that question. We prove that if a surrogate loss is convex, then it cannot be calibrated with respect to any of the AP, the ERR or the PD. Thus, if one of these metrics should be optimized, the price to pay for the computational advantage of convex losses is an inconsistent learning procedure, which may converge to non-optimal predictions as the number of examples increases. Our result generalizes previous works on non-calibration. First, Duchi et al. [11] showed that many convex losses based on pairwise comparisons, such as those of RankBoost [12] or Ranking SVMs [14, 4], are not calibrated with respect to the PD. Secondly, Buffoni et al. [2] showed that speciﬁc convex losses, called order-preserving, are not calibrated with respect to the AP or the ERR, even though these losses are calibrated with respect to (any variant of) the DCG. Our result is stronger than those because we do not make any assumption on the exact structure of the loss; our approach as a whole is also more general because it directly applies to the three evaluation metrics (AP, ERR and PD). Finally, Duchi et al. conjectured that no convex loss can be calibrated with the PD in general [11, Section 2.1] because it would provide a polynomial algorithm to solve an NP-hard problem. Our approach thus leads to a direct proof of this conjecture. In the next section, we describe our framework for learning to rank. We then present in Section 3 the general framework of calibration of [20], and give a new characterization of calibration for the evaluation metrics we consider (Theorem 2), and the implications of the convexity of a surrogate loss. Our main result is proved in Section 4. Section 5 concludes the paper, and Section 6 is a technical part containing the full proof of Theorem 2. Notation Let V, W be two sets. A set-valued function g from V to W maps all v ∈V to a subset of W (set-valued functions appear in the paper as the result of arg min operations). Given a subset V of V, the image of V by g, denoted by g(V ), is the union of the images by g of all members of V , i.e. g(V ) = S v∈V g(v). If n is a positive integer, [n] is the set {1, ..., n}, and Sn is the set of permutations of [n]. Boldface characters are used for vectors of Rn. If x ∈Rn, the i-th component of x is denoted by xi (normal font and subscript). The cardinal of a ﬁnite set V is denoted by \|V\|. 2 Ranking Framework We describe in this section the formal framework of ranking we consider. We ﬁrst present the prediction problem we address, and then deﬁne the two main objects of our study: evaluation metrics for ranking and surrogate losses. We end the section with an outline of our technical contributions. 2.1 Framework and Deﬁnitions We consider a framework similar to label ranking [9] or subset ranking [8]. Let X be a measurable space (the instance space). An instance x ∈X represents a query and its associated n items to rank, for an integer n ≥3. The items are indexed from 1 to n, and the goal is to order the set of item indexes [n] = {1, ..., n} given x. The ordering (or ranking) is predicted by a scoring function, which is a measurable function f : X →Rn. For any input instance x, the scoring function f predicts a vector of n relevance scores (one score for each item) and the ranking is predicted by sorting the item indexes by decreasing scores. We use permutations over [n] to represent rankings, with the following conventions. First, given a permutation σ in Sn, k in [n] is the rank of the item σ(k); second, low ranks are better, so that σ(1) is the top-ranked item. 2 Table 1: Formulas of r(y, σ) for some common ranking evaluation metrics TYPE OF FEEDBACK METRIC FORMULA y ∈Y = {0, ..., Y }n , Y ∈N, Y ≥1 Discounted Cumulative Gain (higher values mean better performances) nP k=1 2 yσ(k)−1 log(1+k) Expected Reciprocal Rank (higher values mean better performances) nP k=1 Rk k k−1 Q q=1 (1 −Rq), Rk = 2 yσ(k)–1 2Y y ∈Y = {0, 1}n Average Precision (higher values mean better performances) 1 \|{i:yi=1}\| P i:yi=1 σ-1(i) P k=1 yσ(k) σ-1(i) y ∈Y = all DAGs over [n] Pairwise Disagreement (lower values mean better performances) P i→j∈y I σ-1(i) > σ-1(j) The quality of a ranking is measured by a ranking evaluation metric, relatively to a feedback. The feedback space, denoted by Y, is a ﬁnite set, and an evaluation metric is a function r : Y ×Sn →R. We use the convention that lower values of r are preferable, and thus when we discuss existing metrics for which higher values are better (e.g. the DCG, the AP or the ERR), we implicitly consider their opposite. Table 1 gives the formula and feedback spaces of the evaluation metrics that we discuss in the paper. The ﬁrst three metrics – the DCG, the ERR and the AP – are commonly used for search engine evaluation. The feedback they consider is a vector of relevance judgments (one judgment per item). The last measure we consider is the PD, which is widely used when learning from pairwise preferences. For the feedback space of the PD, we follow [11] and take Y as the set of all directed acyclic graph (DAG) over [n]. For a DAG y ∈Y, there is an edge from item i to j (denoted i →j ∈y) when i is preferred to j, or, equivalently when i should have better rank than j. In general, using a sorting algorithm, any ranking evaluation metric r induces a quality measure on vectors of scores instead of rankings, considering that the sorting algorithm break ties randomly. Thus, using the following set-valued function from Rn to Sn, called arg sort, which gives the set of rankings induced by a vector of scores: ∀s = (s1, ..., sn) ∈Rn, arg sort(s) = σ ∈Sn\|∀k ∈[n −1] , sσ(k) ≥sσ(k+1) , the evaluation metric on vectors of scores induced by r is deﬁned by: ∀y ∈Y, ∀s ∈Rn, r′(y, s) = X σ∈arg sort(s) r (y, σ) \| arg sort(s)\| . For a ﬁxed, but unknown, probability measure D on X × Y, the objective of a learning algorithm is to ﬁnd a scoring function f with low ranking risk R′(D, f) = R X×Y r′(y, f(x))dD(x, y) using a training set of (instance, feedback) pairs (e.g. drawn i.i.d. according to D). The optimization of the empirical ranking risk is usually intractable because the ranking loss is discontinuous. To address this issue, algorithms optimize the empirical risk associated to a surrogate loss instead. Throughout the paper, we assume that this loss is bounded below, so that all the inﬁma we take are well-deﬁned. Without loss of generality, we assume that the surrogate loss has nonnegative values, and we deﬁne a surrogate loss as a measurable function ℓ: Y × Rn →R+. The surrogate risk of a scoring function f is then deﬁned by L(D, f) = R X×Y ℓ(y, f(x))dD(x, y). 2.2 Outline of the Analysis Any learning algorithm that performs empirical or structural risk minimization on the surrogate risk can, at most, be expected to reach low values of the surrogate risk. The question we address in this paper is whether such an algorithm provably solves the real learning task, which is to achieve low values of the ranking risk. More formally, the criterion under study is whether the following implication holds for every sequence of scoring functions (fk)k≥0 and every data distribution D: L(D, fk) −→ k→∞inf f L(D, f) ⇒ R′(D, fk) −→ k→∞inf f R′(D, f) (1) where the inﬁma are taken over all scoring functions. In particular, we show that if a surrogate loss is convex in the sense that ℓ(y, .) is convex for every y ∈Y, and if the evaluation metric is the AP, 3 the ERR or the PD, then there are distributions and sequences of scoring functions for which (1) does not hold. In other words, we show that learning-to-rank algorithms that deﬁne their objective through a convex surrogate loss cannot provably optimize any of these evaluation metrics. In order to perform a general analysis for all the three evaluation metrics, we consider Assumption (A) below, which formalizes the common property of these metrics that is relevant to our study. Intuitively, it means that for any given item, there is a feedback for which the performance only depends on the rank of this item, with a strict improvement of performances when one improves the rank of the item: (A) ∃β1 <β2 <...<βn such that ∀i ∈[n] , ∃y ∈Y : ∀σ ∈Sn, r(y, σ) = βσ-1(i). Note that in the assumption, the values of βk (i.e. the performance when item i is predicted at rank k) are the same for all items. This is not a strong requirement because the metrics we consider do not depend on how we index the elements. The DCG, the AP and the ERR satisfy (A): for each i, we take the vector of relevance with a 1 for item i and 0 for all other items so that the values of the metrics only depends on the rank of i (which should be ranked ﬁrst). The PD satisﬁes Assumption (A) as well: for each i, take y as the DAG containing the edges i →j, ∀j ∈[n] \ {i} and only those edges. For this feedback, i is preferred to all other items (and no preference is speciﬁed regarding the other items) and thus the quality of a ranking only depends on the rank of i. Our analysis is organized as follows. In the next section, we introduce the notion of a calibrated surrogate loss deﬁned by Steinwart [20], which is a criterion equivalent to (1). We then obtain a new condition that is equivalent to calibration when (A) holds, and ﬁnally we restrict our attention to evaluation metrics satisfying (A) and to convex surrogate losses. In that context, using our new condition for calibration, we show that evaluation metrics with a calibrated surrogate loss necessarily satisfy a speciﬁc property. Then, in Section 4, we prove that the AP, the ERR and the PD do not satisfy this property. Since Assumption (A) holds for these three metrics, this latter result implies that they do not have any convex and calibrated surrogate loss. Equivalently, it implies that (1) does not hold in general for these metrics if the surrogate loss is convex. 3 A New Characterization of Calibration We present in this section the notion of calibration as studied in [20], which is the basis of our work. Then, we provide a characterization of calibration more speciﬁc to the evaluation metrics we consider, that relates more closely calibrated surrogate losses and evaluation metrics. This more speciﬁc characterization of calibration is the starting point of the analysis of convex and calibrated surrogate losses carried out in the last subsection and that allows us to state the results of Section 4. 3.1 The Framework of Calibration Applying the general results of [20] to our setting, the criterion deﬁned by (1) can be studied by restricting our attention to the contributions of a single instance to the surrogate and ranking risk. These contributions are called the inner surrogate risk and the inner ranking risk respectively. Denoting the set of probability distributions over Y by P = p : Y →[0, 1]\| P y∈Y p(y) = 1 , the inner risks are respectively deﬁned for all p ∈P and all s ∈Rn by: L (p, s) = X y∈Y p(y)ℓ(y, s) and R′(p, s) = X σ∈arg sort(s) R (p, σ) \| arg sort(s)\| , where ∀σ ∈Sn, R (p, σ) = X y∈Y p(y)r (y, σ) . Their optimal values are denoted by L (p) = inf s∈Rn L (p, s) and R′(p) = R (p) = min σ∈Sn R (p, σ). More precisely, [20, Theorem 2.8] shows that (1) holds for any distribution D and any sequence of scoring functions if and only if the surrogate loss is r-calibrated according to the deﬁnition below. Similarly to (1), the calibration is an implication of two limits, but it involves the inner risks L and R′ instead of the risks L and R′. For convenience in the rest of the work, we write the implication 4 between the two limits of L and R′as an inclusion of the sets of near-optimal vectors of scores. For any ε > 0 and δ > 0, the latter sets are respectively denoted by Mℓ(p, δ) = {s ∈Rn\|L (p, s) −L (p) < δ} and Mr(p, ε) = {s ∈Rn\|R′(p, s) −R′(p) < ε} , so that the deﬁnition of an r-calibrated loss is the following: Deﬁnition 1. [20, Deﬁnition 2.7] The surrogate loss ℓis r-calibrated if ∀p ∈P, ∀ε > 0, ∃δ > 0 : Mℓ(p, δ) ⊆Mr(p, ε) . 3.2 Calibration through Optimal Rankings Deﬁnition 1 is the starting point of our analysis, and our goal is to show that if the evaluation metric is the AP, the ERR or the PD, then no convex surrogate loss can satisfy it. The goal of this subsection is to give a stronger characterization of r-calibrated surrogate losses when Assumption (A) holds. The starting point of this characterization is to rewrite Deﬁnition 1 in terms of rankings induced by the sets of near-optimal scores, from which we can deduce that ℓis r-calibrated if and only if1: ∀p ∈P, ∀ε > 0, ∃δ > 0 : arg sort(Mℓ(p, δ)) ⊆arg sort(Mr(p, ε)) . In contrast to this characterization of calibration, our result (Theorem 2 below), which is speciﬁc to metrics that satisfy (A), replaces the inclusion (which can be strict in general) of sets of ranking by an equality when ε tends to 0. More speciﬁcally, we deﬁne the set of optimal rankings for the inner ranking risk with the following set-valued function from P to Sn: ∀p ∈P, Ar(p) = arg min σ∈Sn R (p, σ) , so that when Assumption (A) holds, the set of optimal rankings is equal to a set of rankings induced by near-optimal scores of the inner surrogate risk: Theorem 2. If Assumption (A) holds, then ℓis r-calibrated if and only if ∀p ∈P, ∃δ > 0 s.t. arg sort(Mℓ(p, δ)) = Ar(p) . The proof of Theorem 2 is deferred to Section 6 at the end of the paper. This theorem enables us to relate the surrogate loss and the evaluation metric so that the convexity of ℓinduces some constraints on r that are not satisﬁed by all evaluation metrics. 3.3 The implication of Convexity on Sets of Optimal Rankings If ℓ(y, .) is convex for all y ∈P, then the inner risk L(p, .) is also convex for every distribution p ∈P. This implies that Mℓ(p, δ) is a convex subset of Rn. Thus, if ℓis r-calibrated, then Theorem 2 implies that Ar(p) = arg sort(Mℓ(p, δ)) is a set of rankings induced by a convex set of Rn. The following theorem presents a condition that the set Ar(p) must satisfy if it is generated by a convex set of scores: if there exists at least one pair of items (i, j) which are inverted in two rankings of Ar(p), then i and j are “indifferent” in Ar(p): Theorem 3. Assume that for all y ∈Y, the function s 7→ℓ(y, s) is convex. If Assumption (A) holds and ℓis r-calibrated, then r satisﬁes: ∀p ∈P, ∀i, j ∈[n] , ∀σ, σ′ ∈Ar(p), σ-1(i) < σ-1(j) and σ′-1(i) > σ′-1(j) ⇒∃s ∈Rn : si = sj and arg sort(s) ⊆Ar(p) . (2) Proof of Theorem 3. Assume that the conditions of the theorem are satisﬁed. From now on, we ﬁx some p ∈P and two i and j in [n]. Take σ and σ′ in Ar(p) and assume that σ-1(i) < σ-1(j) and σ′-1(i) > σ′-1(j). Since Assumption (A) holds, there is a δ > 0 such that Ar(p) = arg sort Mℓ(p, δ) by Theorem 2. Thus, there are two score vectors u and v in Mℓ(p, δ) such that ui ≥uj (u induces the ranking σ) and vi ≤vj (v induces the ranking σ′). Moreover, since ℓis convex, the function L (p, .) is convex for every p ∈P, and thus Mℓ(p, δ) is convex. Consequently, for all t ∈[0, 1], the vector γ(t) = (1 −t)u + tv belongs to Mℓ(p, δ). We deﬁne g : t 7→γi(t) −γj(t) for t ∈[0, 1]. Then, g is continuous, with g(0) = ui −uj ≥0 and g(1) = vi −vj ≤0. By the intermediate value theorem, there is t0 ∈[0, 1] such that g(t0) = 0. The consequence is that the score vector s, deﬁned by s = γ(t0), satisﬁes s ∈Mℓ(p, δ) and si = sj. 1We remind to the reader the notation arg sort(Mℓ(p, δ)) = S s∈Mℓ(p,δ) arg sort(s). 5 Table 2: Examples for Corollary 4. There are three elements to rank. i ≻j ≻k represents the permutation that ranks item i ﬁrst, j second and k last. For the ERR and the AP, we consider binary relevance judgments. p110 denotes a Dirac distribution at the feedback vector y = [1, 1, 0]. p001 is deﬁned similarly. For the Pairwise Disagreement, p1≻2≻3 is the Dirac distribution at the DAG containing the edges 1 →2, 2 →3 and 1 →3, i.e. the DAG corresponding to 1 ≻2 ≻3. The Dirac distribution at the DAG containing only the edge 3→1 is denoted by p3≻1. In all cases, ˜p(α) is a mixture between two Dirac distributions. The sets Ar(˜p(α)) are obtained by direct calculations. The set Ar(˜p(α)) is the same for all αs in the range given in the third column. DISTRIBUTION ˜p(α) METRIC RANGE OF α Ar(˜p(α)) (1 −α)p110 + αp001 ERR α ∈ 1 3, 1 2 {(1 ≻3 ≻2), (2 ≻3 ≻1)} AP α = 5 13 {(1 ≻2 ≻3), (3 ≻1 ≻2), (2 ≻1 ≻3), (3 ≻2 ≻1)} (1 −α)p1≻2≻3 + αp3≻1 PD α ∈ 2 3, 1 {(2 ≻3 ≻1), (3 ≻1 ≻2)} The contrapositive of Theorem 3 is our technical tool to prove the nonexistence of convex and calibrated losses. Indeed, for a given evaluation metric r, if we are able to exhibit a distribution p ∈P such that (2) is not satisﬁed, this evaluation metric cannot have a surrogate loss both convex and calibrated. In the next subsection, we apply this argument to the AP, the ERR and the PD. Remark 1. It has been proved by several authors that there exist convex surrogate losses that are DCG-calibrated [8, 2, 17]. Thus, the DCG satisﬁes (2). It can be seen by observing that the optimal rankings for the DCG are exactly those generated by sorting the items according to the vector of score s∗(p) deﬁned by s∗ i (p) = P y∈Y p(y)2yi, i.e. Ar(p) = arg sort(s∗(p)). 4 Nonexistence Results We now present the main result of the nonexistence of convex, calibrated surrogate losses: Corollary 4. No convex surrogate loss is calibrated with respect to the AP, the ERR or the PD. Proof. We consider the case where there are three elements to rank, and we use the examples and the notations of Table 2. Since all three metrics satisfy (A), Theorem 3 implies that if r (taken as either the AP, the ERR or the PD) has a calibrated, convex surrogate loss, then, for any distribution ˜p(α), we have: if item i is preferred to j according to a ranking in Ar(˜p(α)), and j is preferred to i according to another ranking in Ar(˜p(α)), then one of the two assertions below must hold: (a) (i ≻j ≻k), (j ≻i ≻k) ⊆Ar(˜p(α)) , (b) (k ≻i ≻j), (k ≻j ≻i) ⊆Ar(˜p(α)) because there exists s ∈R3 such that arg sort(s) ⊆Ar(˜p(α)) for which either si = sj ≤sk or si = sj ≥sk. Now, let us consider the case of the ERR. Taking an arbitrary α ∈ 1 3, 1 2 , we see on the last column of Table 2 that Ar(˜p(α)) contains two rankings: one of them ranks item 1 before item 2, and the other one ranks 2 before 1. If the ERR had a convex calibrated surrogate loss, then either (a) or (b) should hold. However, we see that neither (a) nor (b) holds. Thus their is no convex, ERR-calibrated surrogate loss. For the AP, a similar argument with items 1 and 3 leads to the conclusion. For the PD, taking any two items leads to the result. A ﬁrst consequence of Corollary 4 is that for ranking problems evaluated in terms of AP, ERR or PD, surrogate losses deﬁned as convex upper bounds on an evaluation metric as discussed in [24], as well as convex surrogate losses proposed in the structured output framework such as SVMmap [25] are not calibrated with respect to the evaluation metric they are designed for. The convex surrogate losses used by most participants of the recent Yahoo! Learning to Rank Challenge [6] are also not calibrated with respect to the ERR, the ofﬁcial evaluation metric of the challenge. The fact that the minimization of a non-calibrated surrogate risk leads to suboptimal prediction functions on some data distributions suggests that convex losses are not a deﬁnitive solution to learning to rank. Signiﬁcant improvements in performances may then be obtained by switching to other approaches than the optimization of a convex risk. 6 5 Conclusion We proved that convex surrogate losses cannot be calibrated with three major ranking evaluation metrics. The result cast light on the intrinsic limitations of all algorithms based on (empirical) convex risk minimization for ranking, even though most existing algorithms for learning to rank follow this approach. A possible direction for future work is to study whether the calibration of convex losses can be obtained under low noise conditions. Such studies was carried out for the PD [11], and calibrated, convex surrogate losses were found for special cases of practical interest. Nonetheless, in order to obtain algorithms that do not rely on low noise assumptions, our results suggest to explore whether alternatives to convex surrogate approaches can lead to improvements in terms of performances. A ﬁrst possibility is to turn to non-convex losses for ranking as in [10, 3], and to study the calibration of such losses. Another alternative is to use another surrogate approach than scoring, such as directly learning pairwise preferences [13], even though the reconstruction of an optimal ranking, given the pairwise predictions, that is optimal for evaluation metrics such as the AP, the ERR or the PD is still mostly an open issue. 6 Proof of Theorem 2 We remind the statement of Theorem 2: if r satisﬁes (A), then ℓis r-calibrated if and only if for all p ∈P, there exists δ > 0 such that Ar(p) = arg sort(Mℓ(p, δ) . We prove the result using the following set-valued function which deﬁnes the set of optimal rankings for the inner surrogate risk: Aℓ(p) = arg min σ∈Sn eL (p, σ) where eL (p, σ) = inf L (p, s) \| s ∈Rn s.t. σ ∈arg sort(s) . Then, Theorem 2 is a direct implication of the two following claims that we prove in this section: (a) the assertion ∀p ∈P, ∃δ > 0, arg sort(Mℓ(p, δ) = Aℓ(p) is true in general; (b) if Assumption (A) holds, then ℓis r-calibrated if and only if ∀p ∈P, Aℓ(p) = Ar(p). The proof of these two claims is based on three lemmas that we present before the ﬁnal proof. The ﬁrst lemma, which does not need any assumption on the evaluation metric, both proves equality (a) and provides a general characterization of calibration in terms of optimal rankings. The second lemma concerns the surrogate loss; it states that a slight perturbation in p does not affect “too much” Aℓ(p). The third lemma concerns evaluation metrics and gives a simple consequence of Assumption (A). The ﬁnal proof of Theorem 2 connects all these pieces together to prove (b). Lemma 5. The following claims are true: (i) ∀p ∈P, ∀δ > 0, Aℓ(p) ⊆arg sort Mℓ(p, δ) . (ii) ∀p ∈P, ∃δ0 > 0 : Aℓ(p) = arg sort Mℓ(p, δ0) . (iii) ℓis r-calibrated if and only if: ∀p ∈P, Aℓ(p) ⊆Ar(p). Proof. (i) Fix p ∈P and δ > 0. Let σ ∈Aℓ(p). By the deﬁnition of eL, there is an s ∈Rn such that σ ∈arg sort(s) and L (p, s) −eL (p, σ) < δ. Since eL (p, σ) = minσ′∈Sn eL (p, σ′) = L (p), we have L (p, s) −L (p) < δ . This proves s ∈Mℓ(p, δ) and thus σ ∈arg sort Mℓ(p, δ) . (ii) Fix p ∈P and take δ0 = minσ̸∈Aℓ(p) eL (p, σ) −L (p) > 0, with the convention min ∅= +∞. The choice of δ0 guarantees that ∀s ∈Rn, L (p, s) −L (p) < δ0 ⇒arg sort(s) ⊆Aℓ(p), which is equivalent to arg sort Mℓ(p, δ0) ⊆Aℓ(p). The reverse inclusion is given by the ﬁrst point. (iii) Since r can only take a ﬁnite set of values, we can prove that ℓis r-calibrated if and only if: ∀p ∈P, ∃δ > 0 : ∀s ∈Rn, L (p, s) −L (p) < δ ⇒R′(p, s) = R′(p). Moreover, we have R′(p, s) = R′(p) ⇔arg sort(s) ⊆Ar(p) since R′(p, s) is the mean of R (p, σ) for σ ∈arg sort(s). Thus, ℓis r-calibrated if and only if for every p ∈P, there exists δ > 0 such that arg sort Mℓ(p, δ) ⊆Ar(p). This characterization and the ﬁrst two points give the result. 7 We now present a more technical result on Aℓ, which shows the set of optimal rankings cannot dramatically change under a slight perturbation in the distribution over the feedback space. From now on, for any p ∈P and any η > 0, we denote by B(p, η) the open ball of P (with respect to ∥.∥1) of radius η centered at p, i.e. B(p, η) = {p′ ∈P\|∥p −p′∥1 < η}. Lemma 6. ∀p ∈P, ∃η > 0 such that Aℓ(B(p, η)) = Aℓ(p). Proof. Note that Aℓ(p) ⊆Aℓ(B(p, η)) since p ∈B(p, η). We now prove Aℓ(B(p, η)) ⊆Aℓ(p); the main argument is that eL (., σ) is continuous for every σ because Y is ﬁnite [23, Theorem 2]. Indeed, let us ﬁx p ∈P and deﬁne ε = 1 2 minσ′̸∈Aℓ(p) eL (p, σ′) −L (p) . For each σ ∈Sn, since eL (., σ) is continuous, there exists ησ > 0 such that ∀p′ ∈B(p, ησ), \|eL (p′, σ) −eL (p, σ)\| < ε. Let η = minσ∈Sn ησ, and let p′ be an arbitrary member of B(p, η). By the deﬁnition of ε, we have: ∀σ′ ̸∈Aℓ(p) , eL (p′, σ′) = eL (p′, σ′) −eL (p, σ′) + eL (p, σ′) −L (p) + L (p) > −ε + 2ε + L (p) . Thus, ∀σ′ ̸∈Aℓ(p) , eL (p′, σ′) > L (p) + ε. Additionally, the deﬁnition of η gives ∀σ ∈ Aℓ(p) , eL (p′, σ) < L (p) + ε. Thus, we have minσ′̸∈Aℓ(p) eL (p′, σ′) > minσ∈Aℓ(p) eL (p′, σ). This proves that a ranking that is not optimal for eL (p, .) cannot be optimal for eL (p′, .). Thus Aℓ(p′) ⊆Aℓ(p) from which we conclude Aℓ(B(p, η)) ⊆Aℓ(p). Now that we have studied the properties of Aℓ, we analyze in more depth the evaluation metrics. We prove the following consequence of Assumption (A): for each possible ranking there is a distribution over the feedback space for which this ranking is the unique optimal ranking. Lemma 7. If Assumption (A) holds, then ∀σ ∈Sn, ∃pσ ∈P such that Ar(pσ) = {σ}. Proof. Assume (A) holds, and, for each item k, let us denote by yk the feedback corresponding to item k in Assumption (A). Now, let us take some σ ∈Sn and deﬁne pσ as pσ(yk) = ασ-1(k) with α1 > ... > αn > 0 and Pn k=1 αk = 1. Then, for any σ′ ∈Sn, we have the equality R (pσ, σ′) = Pn k=1 ασ-1(k)r yk, σ′ = Pn k=1 ασ-1(k)βσ′-1(k). Since the αs are non-negative, and since there are ties neither the αs nor in the βs, the rearrangement inequality implies that the minimum value of R (pσ, σ′) is obtained for the single permutation σ′ for which the βσ′-1(k) are in reverse order relatively to the ασ-1(k) (i.e. smaller values βσ′-1(k) should be associated to greater values of ασ-1(k)). Since the αks are decreasing with k and the βks are increasing, the minimum value of σ′ 7→R (pσ, σ′) = Pn k=1 ασ-1(k)βσ′-1(k) is obtained if and only if σ-1 = σ′-1 (i.e. σ′ = σ). Proof of Theorem 2. We remind to the reader that by the second point of Lemma 5, for any p ∈P, there is δ > 0 such that Aℓ(p) = arg sort(Mℓ(p, δ) . What remains to show is that if Assumption (A) holds, then ℓis r-calibrated if and only if ∀p ∈P, Aℓ(p) = Ar(p). (“if” direction) If ∀p ∈P, Aℓ(p) = Ar(p) then ℓis r-calibrated by Lemma 5. (“only if” direction) Assume that (A) holds and that ℓis r-calibrated. Let p ∈P. By Point (iii) of Lemma 5, we know that Aℓ(p) ⊆Ar(p). We now prove the reverse inclusion Aℓ(p′) ⊆Ar(p′). By Lemma 6, there exists some η > 0 such that Aℓ(B(p, η)) = Aℓ(p). Let σ ∈Ar(p). The idea is to use Lemma 7 to ﬁnd some p′ ∈B(p, η) such that Aℓ(p′) = {σ} which would prove σ ∈Aℓ(p) and thus the result. The rest of the proof consists in building p′. Using Lemma 7, let pσ ∈P such that Ar(pσ) = {σ}. Now, let p′ = (1−η 4)p+ η 4pσ. Then, we have ∥p −p′∥1 = η 4∥p −pσ∥1 ≤η/2 and thus p′ ∈B(p, η). Moreover, Ar(p′) = {σ} since σ is optimal for both p and pσ, and any other permutation is suboptimal for pσ. We also have Aℓ(p′) = {σ} because Aℓhas non-empty values and calibration implies that Aℓ(p′) ⊆Ar(p′) by Lemma 5. 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4,484	Learning with Partially Absorbing Random Walks Xiao-Ming Wu1, Zhenguo Li1, Anthony Man-Cho So3, John Wright1 and Shih-Fu Chang1,2 1Department of Electrical Engineering, Columbia University 2Department of Computer Science, Columbia University 3Department of SEEM, The Chinese University of Hong Kong {xmwu, zgli, johnwright, sfchang}@ee.columbia.edu, manchoso@se.cuhk.edu.hk Abstract We propose a novel stochastic process that is with probability αi being absorbed at current state i, and with probability 1 −αi follows a random edge out of it. We analyze its properties and show its potential for exploring graph structures. We prove that under proper absorption rates, a random walk starting from a set S of low conductance will be mostly absorbed in S. Moreover, the absorption probabilities vary slowly inside S, while dropping sharply outside, thus implementing the desirable cluster assumption for graph-based learning. Remarkably, the partially absorbing process uniﬁes many popular models arising in a variety of contexts, provides new insights into them, and makes it possible for transferring ﬁndings from one paradigm to another. Simulation results demonstrate its promising applications in retrieval and classiﬁcation. 1 Introduction Random walks have been widely used for graph-based learning, leading to a variety of models including PageRank [14] for web page ranking, hitting and commute times [8] for similarity measure between vertices, harmonic functions [20] for semi-supervised learning, diffusion maps [7] for dimensionality reduction, and normalized cuts [12] for clustering. In graph-based learning one often adopts the cluster assumption, which states that the semantics usually vary smoothly for vertices within regions of high density [17], and suggests to place the prediction boundary in regions of low density [5]. It is thus interesting to ask how the cluster assumption can be realized in terms of random walks. Although a random walk appears to explore the graph globally, it converges to a stationary distribution determined solely by vertex degrees regardless of the starting points, a phenomenon well known as the mixing of random walks [11]. This causes some random walk approaches intended to capture non-local graph structures to fail, especially when the underlying graph is well connected, i.e., the random walk has a large mixing rate. For example, it was recently proven in [16] that under some mild conditions the hitting and commute times on large graphs do not take into account the global structure of the graph at all, despite the fact that they have integrated all the relevant paths on the graph. It is also shown in [13] that the “harmonic” walks [20] in high-dimensional spaces converge to a constant distribution as the data size approaches inﬁnity, which is undesirable for classiﬁcation and regression. These ﬁndings show that intuitions regarding random walks can sometimes be misleading, and should be taken with caution. A natural question is: can we design a random walk which implements the cluster assumption with some guarantees? In this paper, we propose partially absorbing random walks (PARWs), a novel random walk model whose properties can be analyzed theoretically. In PARWs, a random walk is with probability αi being absorbed at current state i, and with probability 1 −αi follows a random edge out of it. PARWs are guaranteed to implement the cluster assumption in the sense that under proper absorp1 0 t = 1 t = 2 t = i j k ii p jj p kk p i 'i j k 'k 'j (a) (b) (c) Figure 1: A partially absorbing random walk. (a) A ﬂow perspective (see text). (b) A second-order Markov chain. (c) An equivalent standard Markov chain with additional sinks. tion rates, a random walk starting from a set S of low conductance will be mostly absorbed in S. Furthermore, we show that by setting the absorption rates, the absorption probabilities can vary slowly inside S, while dropping sharply outside S. This approximately piecewise constant property makes PARWs highly desirable and robust for a variety of learning tasks including ranking, clustering, and classiﬁcation, as demonstrated in Section 4. More interestingly, it turns out that many existing models including PageRank, hitting and commute times, and label propagation algorithms in semi-supervised learning, can be uniﬁed or related in PARWs, which brings at least two beneﬁts. On one hand, our theoretical analysis sheds some light on the understanding of existing models; on the other hand, it enables transferring ﬁndings among different paradigms. We present our model in Section 2, analyze a special case of it in Section 3, and show simulation results in Section 4. Section 5 concludes the paper. Most of our proofs are included in supplementary material. 2 Partially Absorbing Random Walks Let us consider a simple diffusion process illustrated in Fig. 1(a). At the beginning, a unit ﬂow (blue) is injected to the graph at a selected vertex. After one step, some of the ﬂow (red) is “stored” at the vertex while the rest (blue) propagates to its neighbors. Whenever the ﬂow passes a vertex, some fraction of it is retained at that vertex. As this process continues, the amount of ﬂow stored in each vertex will accumulate and there will be less and less ﬂow left running on the graph. After a certain number of steps, there will be almost no ﬂow left running and the ﬂow stored will nearly sum up to 1. The above diffusion process can be made precise in terms of random walks, as shown below. Consider a discrete-time stochastic process X = {Xt : t ≥0} on the state space N = {1, 2, . . ., n}, where the initial state X0 is given, say X0 = i, the next state X1 is determined by the transition probability P(X1 = j\|X0 = i) = pij, and the subsequent states are determined by the transition probabilities P(Xt+2 = j\|Xt+1 = i, Xt = k) = ( 1, i = j, i = k, 0, i ̸= j, i = k, P(Xt+2 = j\|Xt+1 = i) = pij, i ̸= k, (1) where t ≥0. Note that the process X is time homogeneous, i.e., the transition probabilities in (1) are independent of t. In other words, if the previous and current states are the same, the process will remain in the current state forever. Otherwise, the next state is conditionally independent of the previous state given the current state, i.e., the process behaves like a usual random walk. To illustrate the above construction, consider Fig. 1(b). Starting from state i, there is some probability pii that the process will stay at i in the next step; and once it stays, the process will be absorbed into state i. Hence, we shall call the above process a partially absorbing random walk (PARW), where pii is the absorption rate of state i. If 0 < pii < 1, then we say that i is a partially absorbing state. If pii = 1, then we say that i is a fully absorbing state. Finally, if pii = 0, then we say that i is a transient state. Note that if pii ∈{0, 1} for every state i ∈N, then the above process reduces to a standard Markov chain [9]. A PARW is a second-order Markov chain completely speciﬁed by its ﬁrst order transition probabilities {pij}. One can observe that any PARW can be realized as a standard Markov chain by adding a sink (fully absorbing state) to each vertex in the graph, as illustrated in Fig. 1(c). The transition 2 probability from i to its sink i′ equals the absorption rate pii in PARWs. One may also notice that the construction of PARWs can be generalized to the m-th order, i.e., the process is absorbed at a state only after it has stayed at that state for m-consecutive steps. However, it can be shown that any m-th order PARW can be realized by a second-order PARW. We will not elaborate on this due to space constraints. 2.1 PARWs on Graphs Let G = (V, W) be an undirected weighted graph, where V is a set of n vertices and W = [wij] ∈ Rn×n is a symmetric non-negative matrix of pairwise afﬁnities among vertices. We assume G is connected. Let D = diag(d1, d2, . . . , dn) with di = P j wij as the degree of vertex i, and deﬁne the Laplacian of G by L = D −W [6]. Denote by d(S) := P i∈S di the volume of a subset S ⊆V of vertices. Let λ1, λ2, . . . , λn ≥0 be arbitrary, and set Λ = diag(λ1, λ2, . . . , λn). Suppose that we deﬁne the ﬁrst order transition probabilities of a PARW by pij = ( λi λi+di , i = j, wij λi+di , i ̸= j. (2) Then, we see that state i is an absorbing state (either partially or fully) when λi > 0, and is a transient state when λi = 0. In particular, the matrix Λ acts like a regularizer that controls the absorption rate of each state, i.e., the larger λi, the larger pii. In the sequel, we refer to Λ as the regularizer matrix. Absorption Probabilities. We are interested in the probability aij that a random walk starting from state i, is absorbed at state j in any ﬁnite number of steps. Let A = [aij] ∈Rn×n be the matrix of absorption probabilities. The following theorem shows that A has a closed-form. Theorem 2.1. Suppose λi > 0 for some i. Then A = (Λ + L)−1Λ. Proof. Since λi > 0 for some i, the matrix Λ + L is positive deﬁnite and hence non-singular. Moreover, the matrix Λ + D is non-singular, since D is non-singular. Thus, the matrix I −(Λ + D)−1W = (Λ + D)−1(Λ + L) is also non-singular. Now, observe that the absorbing probabilities {aij} satisfy the following equations: aii = λi λi + di × 1 + X j̸=i wij λi + di aji, (3) aij = X k̸=i wik λi + di akj, i ̸= j. (4) Upon writing equations (3) and (4) in matrix form, we have (I −(Λ + D)−1W)A = (Λ + D)−1Λ, whence A = (I −(Λ + D)−1W)−1(Λ + D)−1Λ = (Λ + D −W)−1Λ = (Λ + L)−1Λ. The following result conﬁrms that A is indeed a probability matrix. Proposition 2.1. Suppose λi > 0 for some i. Then A is a non-negative matrix with each row summing up to 1. By Proposition 2.1, P k ajk = 1 for any j. This means that a PARW starting from any vertex will eventually be absorbed, provided that there is at least one absorbing state in the state space. 2.2 Limits of Absorption Probabilities By Theorem 2.1, we see that the absorption probabilities (A) are governed by both the structure of the graph (L) and the regularizer matrix (Λ). It would be interesting to see how A varies with Λ, particularly when λi’s become small which allows the ﬂow to propagate sufﬁciently (Fig. 1(a)). The following result shows that as Λ (λi’s) vanishes, each row of A converges to a distribution proportional to (λ1, λ2, . . . , λn), regardless of graph structure. Theorem 2.2. Suppose λi > 0 for all i. Then lim α→0+(αΛ + L)−1αΛ = 1¯λ ⊤, (5) where (¯λ)i = λi/(Pn j=1 λj). In particular, limα→0+(αI + L)−1αI = 1 n11⊤. 3 Theorem 2.2 tells us that with Λ = αI and as α →0 a PARW will converge to the constant distribution 1/n, regardless of the starting vertex. At ﬁrst glance, this limit seems meaningless. However, the following lemma will show that it actually has interesting connections with L+, the pseudo-inverse of the graph Laplacian, a matrix that is widely studied and proven useful for many learning tasks including recommendation and clustering [8]. Proposition 2.2. Suppose Λ = αI and denote Aα := (Λ + L)−1Λ = (αI + L)−1α. Then, lim α→0 Aα −1 n11⊤ α = L+. (6) Proposition 2.2 gives a novel probabilistic interpretation of L+. Note that by Theorem 2.2, A0 := limα→0 Aα = 1 n11⊤. Thus L+ is the derivative of Aα w.r.t. α at α = 0, implying that L+ reﬂects the variation of absorption probabilities when the absorption rate is very small. By (6), we see that ranking by L+ is essentially the same as ranking by Aα, when α is sufﬁciently small. 2.3 Relations with Popular Ranking and Classiﬁcation Models Relations with PageRank Vectors. Suppose λj > 0 for all j. Let a be the absorption probability vector of a PARW starting from vertex i. Denote by s the indicator vector of i, i.e., s(i) = 1 and s(j) = 0 for j ̸= i. Then a⊤= s⊤(Λ + L)−1Λ, which can be rewritten as a⊤= s⊤(Λ + D)−1Λ + a⊤Λ−1W(Λ + D)−1Λ. (7) By letting Λ = β 1−βD, we have a⊤= βs⊤+ (1 −β)a⊤D−1W, which is exactly the equilibrium equation for personalized PageRank [14]. Note that β is often referred to as the “teleportation” probability in PageRank. This shows that personalized PageRank is a special case of PARWs with absorption rates pii = λi λi+di = β. Relations with Hitting and Commute Times. The hitting time Hij is the expected time that it takes a random walk starting from i to ﬁrst arrive at j, and the commute time Cij is the expected time it takes a random walk starting from i to travel to j and back to i, which can be computed as Hij = d(G)(L+ jj −L+ ij), Cij = Hij + Hji = d(G)(L+ ii + L+ jj −2L+ ij), (8) where d(G) := P i di denotes the volume of the graph. By (6), when Λ = αI and α is sufﬁciently small, ranking with Hij or Cij (say, with respect to i) is the same as ranking by Aα jj −Aα ij or Aα ii + Aα jj −2Aα ij respectively. This appears to be not particularly meaningful because the term Aα jj is the self-absorption probability that does not contain any essential information with the starting vertex i. Accordingly, it should not be included as part of the ranking function with respect to i. This argument is also supported in a recent study by [16], where the hitting and commute times are shown to be dominated by the inverse of degrees of vertices. In other words, they do not take into account the graph structure at all. A remedy they propose is to throw away the diagonal terms of L+ and only use the off-diagonal terms. This happens to suggest using absorption probabilities for ranking and as similarity measure, because when α is sufﬁciently small, ranking by the off-diagonal terms of L+ is essentially the same as ranking by Aα ij, i.e., the absorption probability of starting from i and being absorbed at j. Our theoretical analysis in Section 3 and the simulation results in Section 4 further conﬁrm this argument. Relations with Semi-supervised Learning. Interestingly, many label propagation algorithms in semi-supervised learning can be cast in PARWs. The harmonic function method [20] is a PARW when setting λi = ∞(absorption rate 1) for the labeled vertices while λi = 0 (absorption rate 0) for the unlabeled. In [19] the authors have made this interpretation in terms of absorbing random walks, where a random walk arriving at an absorbing state will stay there forever. PARWs can be viewed as an extension of absorbing random walks. The regularized harmonic function method [5] is also a PARW when setting λi = α for the labeled vertices while λi = 0 for the unlabeled. The consistency method [17], if using un-normalized Laplacian instead of normalized Laplacian, is a PARW with Λ = αI. Our analysis in this paper reveals several nice properties of this case (Section 3). A variant of this method is a PARW with Λ = αD, which is the same as PageRank as shown above. If we add an additional sink to the graph, a variant of harmonic function method [10] and a variant of the regularized harmonic function method [3] can all be included as instances of PARWs. We omit the details here due to space constraints. 4 Beneﬁts of a Unifying View. We have shown that PARWs can unify or relate many models from different contexts. This brings at least two beneﬁts. First, it sheds some light on existing models. For instance, hitting and commute times are not suitable for ranking given its interpretation in absorption probabilities, as discussed above. In the next section, we will show that a special case of PARWs is better suited for implementing the cluster assumption for graph-based learning. Second, a unifying view builds bridges between different paradigms thus making it easier to transfer ﬁndings between them. For example, it has been shown in [2,4] that approximate personalized PageRank vectors can be computed in O(1/ǫ) iterations, where ǫ is a precision tolerance parameter. We indicate here that such a technique is also applicable to PARWs due to PARWs’s generalizing nature. Consequently, most models included in PARWs can be substantially accelerated using the same technique. 3 PARWs with Graph Conductance In this section, we present results on the properties of the absorption probability vector ai obtained by a PARW starting from vertex i (i.e., a⊤ i is the row i of A). We show that properties of ai relate closely to the connectivity between i and the rest of graph, which can be captured by the conductance of the cluster S where i belongs. We also ﬁnd that properties of ai depend on the setting of absorption rates. Our key results can be summarized as follows. In general, the probability mass of ai is mostly absorbed by S. Under proper absorption rates, ai can vary slowly within S while dropping sharply outside S. Such properties are highly desirable for learning tasks such as ranking, clustering, and classiﬁcation. The conductance of a subset S ⊂V of vertices is deﬁned as Φ(S) = w(S, ¯ S) min(d(S),d( ¯S)), where w(S, ¯S) := P (i,j)∈e(S, ¯ S) wij is the cut between S and its complement ¯S [6]. We denote the indicator vector of S by χS such that χS(i) = 1 if i ∈S and χS(i) = 0 otherwise; and denote the stationary distribution w.r.t. S by πS such that πS(i) = di/d(S) if i ∈S and πS(i) = 0 otherwise. In terms of the conductance of S, the following theorem gives an upper bound on the expected probability mass escaped from S if the distribution of the starting vertex is πS. Theorem 3.1. Let S be any set of vertices satisfying d(S) ≤ 1 2d(G). Let γ1 = mini∈S λi di and γ2 = maxi∈¯ S λi di . Then, π⊤ S Aχ ¯ S ≤ γ2 1 + γ2 1 + γ1 γ2 1 Φ(S). (9) Theorem 3.1 shows that most of the probability mass will be absorbed in S, provided that S is of small conductance and the random walk starts from S according to πS. In other words, a PARW will be trapped inside the cluster1 from where it starts, as desired. To identify the entire cluster, what is more desirable would be that the absorption probabilities vary slowly within the cluster while dropping sharply outside. As such, the cluster can be identiﬁed by detecting the sharp drop. We show below that such property can be achieved by setting appropriate absorption rates at vertices. 3.1 PARWs with Λ = αI We will prove that the choice of Λ = αI can fulﬁll the above goal. Before presenting theoretical analysis, let us discuss the intuition behind it from both ﬂow (Fig. 1(a)) and random walk perspectives. To vary slowly within the cluster, the ﬂow needs to be distributed evenly within it; while to drop sharply outside, the ﬂow must be prevented from escaping. This means that the absorption rates should be small in the interior but large near the boundary area of the cluster. Setting Λ = αI achieves this. It corresponds to the absorption rates pii = λi λi+di = α α+di , which decrease monotonically with di. Since the degrees of vertices are usually relatively large in the interior of the cluster due to denser connections, and small near its boundary area (Fig. 2(a)), the absorption rates are therefore much larger at its boundary than in its interior (Fig. 2(b)). State differently, a random walk may move freely inside the cluster, but it will get absorbed with high probability when traveling near the cluster’s boundary. In this way, the absorption rates set up a bounding “wall” around the cluster to prevent the random walk from escaping, leading to an absorption probability vector that 1A cluster is understood as a subset of vertices of small conductance. 5 0 300 600 900 0 2 4x 10 −3 0 300 600 900 0 2 4x 10 −3 (a) (b) (c) (d) Figure 2: Absorption rates and absorption probabilities. (a) A data set of three Gaussians with the degrees of vertices in the underlying graph shown (see Section 4 for the descriptions of the data and graph construction). A starting vertex is denoted in black circle. (b–c) Absorption rates and absorption probabilities for Λ = αI (α = 10−3). (d) Sorted absorption probabilities of (c). For illustration purpose, in (a–b), the degrees of vertices and the absorption rates have been properly scaled, and in (c), the data are arranged such that points within each Gaussian appear consecutively. varies slowly within the cluster while dropping sharply outside (Figs. 2(c–d)), thus implementing the cluster assumption. We make these arguments precise below. It is worth pointing out that a PARW with Λ = αI is symmetric, i.e., the absorption probability of starting from i and absorbed at j is equal to the probability of starting from j and absorbed at i. For simplicity, we use the abbreviated notation a to denote ai, the absorption probability vector for the PARW starting from vertex i. By (3) and the symmetry property, we immediately see that a has the following “harmonic” property: a(i) = λi λi + di + X k̸=i wik λi + di a(k), a(j) = X k̸=j wjk λj + dj a(k), j ̸= i. (10) We will use this property to prove some interesting results. Another desirable property one should notice for this PARW is that the starting vertex always has the largest absorption probability, as shown by the following lemma. Lemma 3.2. Given Λ = αI, then aii > aij for any i ̸= j. By Lemma 3.2 and without loss of generality, we assume the vertices are sorted so that a(1) > a(2) ≥· · · ≥a(n), where vertex 1 is the starting vertex. Let Sk be the set of vertices {1, . . . , k}. Denote e(Si, Sj) as the set of edges between Si and Sj. The following theorem quantiﬁes the drop of the absorption probabilities between Sk and ¯Sk. Theorem 3.3. For every S ∈{Sk \| k = 1, 2, . . ., n}, X (u,v)∈e(S, ¯ S) wuv (a(u) −a(v)) = α 1 − X k∈S a(k) ! . (11) Theorem 3.3 shows that the weighted difference in absorption probabilities between Sk and ¯Sk is α 1 −Pk j=1 a(j) , implying that it drops slowly when α is small and as k increases, as expected. Next we show the variation of absorption probabilities with graph conductance. Without loss of generality, we consider sets Sj where d(Sj) ≤1 2d(G). The following theorem says that a(j+1) will drop little from a(j) if the set Sj has high conductance or if the vertex j is far away from the starting vertex 1 (i.e., j ≫1). Lemma 3.4. If Φ(Sj) = φ, then a(j + 1) ≥a(j) − α 1 −Pj k=1 a(k) φd(Sj) . (12) The above result can be extended to describe the drop in a much longer range, as stated in the following theorem. 6 0 300 600 900 0 0.1 0.2 0.3 0.4 0 300 600 900 0 0.005 0.01 0 300 600 900 0 1 2x 10 −3 0 300 600 900 1.06 1.08 1.1 1.12x 10 −3 0 300 600 900 1.1108 1.111 1.1112 1.1114x 10 −3 (a) (b) (c) (d) (e) 0 300 600 900 0 0.2 0.4 0.6 0.8 0 300 600 900 0 0.01 0.02 0.03 0 300 600 900 0 2 4 6x 10 −3 0 300 600 900 0 1 2 3x 10 −3 0 300 600 900 0 1 2 3x 10 −3 (f) (g) (h) (i) (j) Figure 3: Absorption probabilities on the three Gaussians in Fig. 2(a) with the starting vertex denoted in black circle. (a–e) Λ = αI, α = 100, 10−2, 10−4, 10−6, 10−8; (f–j) Λ = αD, α = 100, 10−2, 10−4, 10−6, 10−8. For illustration purpose, the data are arranged such that points within each Gaussian appear consecutively, as in Fig. 2(c). Table 1: Ranking results (MAP) on USPS Digits 0 1 2 3 4 5 6 7 8 9 All Λ = αI .981 .988 .876 .893 .646 .778 .940 .919 .746 .730 .850 PageRank .886 .972 .608 .764 .488 .568 .837 .825 .626 .702 .728 Manifold Ranking .957 .987 .827 .827 .467 .630 .917 .822 .675 .719 .783 Euclidean Distance .640 .980 .318 .499 .337 .294 .548 .620 .368 .480 .508 Theorem 3.5. If Φ(Sj) ≥2φ, then there exists a k > j such that d(Sk) ≥(1 + φ)d(Sj) and a(k) ≥a(j) − α 1 −Pj k=1 a(k) φd(Sj) . Theorem 3.5 tells us that if the set Sj has high conductance, then there will be a set Sk much larger than Sj where the absorption probability a(k) remains large. In other words, a(k) will not drop much if Sj is closely connected with the rest of graph. Combining Theorems 3.3, 3.5, and 3.1, we see that the absorption probability vector of the PARW with Λ = αI has the nice property of varying slowly within the cluster while dropping sharply outside. We remark that similar analyses have been conducted in [1, 2] on personalized PageRank, for the local clustering problem [15] whose goal is to ﬁnd a local cut of low conductance near a speciﬁed starting vertex. As shown in Section 2, personalized PageRank is a special case of PARWs with Λ = αD = β 1−β D, which corresponds to setting the same absorption rate pii = β at each vertex. This setting does not take advantage of the cluster assumption. Indeed, despite the signiﬁcant cluster structure in the three Gaussians (Fig. 2), no clear drop emerges by varying β (Section 4). This explains the “heuristic” used in [1, 2] where the personalized PageRank vector is divided by the degrees of vertices to generate a sharp drop. In contrast, our choice of Λ = αI appears to be more justiﬁed, without the need of such post-processing while retaining a probabilistic foundation. 4 Simulation In this section, we demonstrate our theoretical results on both synthetic and real data. For each data set, a weighted k-NN graph is constructed with k = 20. The similarity between vertices i and j is computed as wij = exp(−d2 ij/σ) if i is within j’s k nearest neighbors or vice versa, and wij = 0 otherwise (wii = 0), where σ = 0.2 × r and r denotes the average square distance between each point to its 20th nearest neighbor. The ﬁrst experiment is to examine the absorption probabilities when varying absorption rates. We use the synthetic three Gaussians in Fig. 2(a), which consists of 900 points from three Gaussians, with 300 in each. Fig. 3 compares the cases of Λ = αI and Λ = αD (PageRank). We can 7 Table 2: Classiﬁcation accuracy on USPS HMN LGC Λ = αD Λ = αI .782 ± .068 .792 ± .062 .787 ± .048 .881 ± .039 draw several observations. For Λ = αI, when α is large, most probability mass is absorbed in the cluster of the starting vertex (Fig. 3(a)). As it becomes appropriately small, the probability mass distributes evenly within the cluster, and a sharp drop emerges (Fig. 3(b)). As α →0, the probability mass distributes more evenly within each cluster and also on the entire graph (Figs. 3(c–e)), but the drops between clusters are still quite signiﬁcant. In contrast, for Λ = αD, no signiﬁcant drops show for all α’s (Figs. 3(f–j)). This is due to the uniform absorption rates on the graph, which makes the ﬂow favor vertices with denser connections (i.e., of large degrees). These observations support the theoretical arguments in Section 3 for PARWs with Λ = αI and suggest its robustness in distinguishing between different clusters. The second experiment is to test the potential of PARWs for information retrieval. We compare PARWs with Λ = αI to PageRank (i.e., PARWs with Λ = αD), Manifold Ranking [18], and the baseline using Euclidean distance. For parameter selection, we use α = 10−6 for Λ = αI and β = 0.15 for PageRank (see Section 2.3) as suggested in [14]. The regularization parameter in Manifold Ranking is set to 0.99, following [18]. The image benchmark USPS2 is used for this experiment, which contains 9298 images of handwritten digits from 0 to 9 of size 16 × 16, with 1553, 1269, 929, 824, 852, 716, 834, 792, 708, and 821 instances of each digit respectively. Each instance is used as a query and the mean average precision (MAP) is reported. The results are shown in Table 1. We see that the PARW with Λ = αI consistently gives best results for individual digits as well as the entire data set. In the last experiment, we test PARWs on classiﬁcation/semi-supervised learning, also on USPS with all 9298 images. We randomly sample 20 instances as labeled data and make sure there is at least one label for each class. For PARWs, we classify each unlabeled instance u to the class of the labeled vertex v where u is most likely to be absorbed, i.e., v = arg maxi∈L aui where L denotes the labeled data and aui is the absorption probability. We compare PARWs with Λ = αI (α = 10−6) and Λ = αD (β = 0.15) to the harmonic function method (HMN) [20] coupled with class mass normalization (CMN) and the local and global consistency (LGC) method [17]. No parameter in HMN is required, and the regularization parameter in LGC is set to 0.99 following [17]. The classiﬁcation accuracy averaged over 1000 runs is shown in Table 2. Again, it conﬁrms the superior performance of the PARW with Λ = αI. In the second and third experiments, we also tried other parameter settings for methods where appropriate. We found that the performance of PARWs with Λ = αI is quite stable with small α, and varying parameters in other methods did not lead to signiﬁcantly better results, which validates our previous arguments. 5 Conclusions We have presented partially absorbing random walks (PARWs), a novel stochastic process generalizing ordinary random walks. Surprisingly, it has been shown to unify or relate many popular existing models and provide new insights. Moreover, a new algorithm developed from PARWs has been theoretically shown to be able to reveal cluster structure under the cluster assumption. Simulation results have conﬁrmed our theoretical analysis and suggested its potential for a variety of learning tasks including retrieval, clustering, and classiﬁcation. In future work, we plan to apply our model to real applications. Acknowledgements This work is supported in part by Ofﬁce of Naval Research (ONR) grant #N00014-10-1-0242. 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4,485	Privacy Aware Learning John C. Duchi1 Michael I. Jordan1,2 Martin J. Wainwright1,2 1Department of Electrical Engineering and Computer Science, 2Department of Statistics University of California, Berkeley Berkeley, CA USA 94720 {jduchi,jordan,wainwrig}@eecs.berkeley.edu Abstract We study statistical risk minimization problems under a version of privacy in which the data is kept conﬁdential even from the learner. In this local privacy framework, we establish sharp upper and lower bounds on the convergence rates of statistical estimation procedures. As a consequence, we exhibit a precise tradeoff between the amount of privacy the data preserves and the utility, measured by convergence rate, of any statistical estimator. 1 Introduction There are natural tensions between learning and privacy that arise whenever a learner must aggregate data across multiple individuals. The learner wishes to make optimal use of each data point, but the providers of the data may wish to limit detailed exposure, either to the learner or to other individuals. It is of great interest to characterize such tensions in the form of quantitative tradeoffs that can be both part of the public discourse surrounding the design of systems that learn from data and can be employed as controllable degrees of freedom whenever such a system is deployed. We approach this problem from the point of view of statistical decision theory. The decisiontheoretic perspective offers a number of advantages. First, the use of loss functions and risk functions provides a compelling formal foundation for deﬁning “learning,” one that dates back to Wald [28] in the 1930’s, and which has seen continued development in the context of research on machine learning over the past two decades. Second, by formulating the goals of a learning system in terms of loss functions, we make it possible for individuals to assess whether the goals of a learning system align with their own personal utility, and thereby determine the extent to which they are willing to sacriﬁce some privacy. Third, an appeal to decision theory permits abstraction over the details of speciﬁc learning procedures, providing (under certain conditions) minimax lower bounds that apply to any speciﬁc procedure. Finally, the use of loss functions, in particular convex loss functions, in the design of a learning system allows powerful tools of optimization theory to be brought to bear. In more formal detail, our framework is as follows. Given a compact convex set Θ ⊂Rd, we wish to ﬁnd a parameter value θ ∈Θ achieving good average performance under a loss function ℓ: X × Rd →R+. Here the value ℓ(X, θ) measures the performance of the parameter vector θ ∈Θ on the sample X ∈X, and ℓ(x, ·) : Rd →R+ is convex for x ∈X. We measure the expected performance of θ ∈Θ via the risk function R(θ) := E[ℓ(X, θ)]. (1) In the standard formulation of statistical risk minimization, a method M is given n samples X1, . . . , Xn, and outputs an estimate θn approximately minimizing R(θ). Instead of allowing M access to the samples Xi, however, we study the effect of giving only a perturbed view Zi of each datum Xi, quantifying the rate of convergence of R(θn) to infθ∈Θ R(θ) as a function of both the number of samples n and the amount of privacy Zi provides for Xi. 1 There is a long history of research at the intersection of privacy and statistics, where there is a natural competition between maintaining the privacy of elements in a dataset {X1, . . . , Xn} and the output of statistical procedures. Study of this issue goes back at least to the 1960s, when Warner [29] suggested privacy-preserving methods for survey sampling. Recently, there has been substantial work on privacy—focusing on a measure known as differential privacy [12]—in statistics, computer science, and other ﬁelds. We cannot hope to do justice to the large body of related work, referring the reader to the survey by Dwork [10] and the statistical framework studied by Wasserman and Zhou [30] for background and references. In this paper, we study local privacy [13, 17], in which each datum Xi is kept private from the method M. The goal of many types of privacy is to guarantee that the output !θn of the method M based on the data cannot be used to discover information about the individual samples X1, . . . , Xn, but locally private algorithms access only disguised views of each datum Xi. Local algorithms are among the most classical approaches to privacy, tracing back to Warner’s work on randomized response [29], and rely on communication only of some disguised view Zi of each true sample Xi. Locally private algorithms are natural when the providers of the data—the population sampled to give X1, . . . , Xn—do not trust even the statistician or statistical method M, but the providers are interested in the parameters θ∗minimizing R(θ). For example, in medical applications, a participant may be embarrassed about his use of drugs, but if the loss ℓis able to measure the likelihood of developing cancer, the participant has high utility for access to the optimal parameters θ∗. In essence, we would like the statistical procedure M to learn from the data X1, . . . , Xn but not about it. Our goal is to understand the fundamental tradeoffs between maintaining privacy while still retaining the utility of the statistical inference method M. Though intuitively there must be some tradeoff, quantifying it precisely has been difﬁcult. In the machine learning literature, Chaudhuri et al. [7] develop differentially private empirical risk minimization algorithms, and Dwork and Lei [11] and Smith [26] analyze similar statistical procedures, but do not show that there must be negative effects of privacy. Rubinstein et al. [24] are able to show that it is impossible to obtain a useful parameter vector θ that is substantially differentially private; it is unclear whether their guarantees are improvable. Recent work by Hall et al. [15] gives sharp minimax rates of convergence for differentially private histogram estimation. Blum et al. [5] also give lower bounds on the closeness of certain statistical quantities computed from the dataset, though their upper and lower bounds do not match. Sankar et al. [25] provide rate-distortion theorems for utility models involving information-theoretic quantities, which has some similarity to our risk-based framework, but it appears challenging to map their setting onto ours. The work most related to ours is probably that of Kasiviswanathan et al. [17], who show that that locally private algorithms coincide with concepts that can be learned with polynomial sample complexity in Kearns’s statistical query (SQ) model. In contrast, our analysis addresses sharp rates of convergence, and applies to estimators for a broad class of convex risks (1). 2 Main results and approach Our approach to local privacy is based on a worst-case measure of mutual information, where we view privacy preservation as a game between the providers of the data—who wish to preserve privacy—and nature. Recalling that the method sees only the perturbed version Zi of Xi, we adopt a uniform variant of the mutual information I(Zi; Xi) between the random variables Xi and Zi as our measure for privacy. This use of mutual information is by no means original [13, 25], but because standard mutual information has deﬁciencies as a measure of privacy [e.g. 13], we say the distribution Q generating Z from X is private only if I(X; Z) is small for all possible distributions P on X (possibly subject to constraints). This is similar to the worst-case information approach of Evﬁmievski et al. [13], which limits privacy breaches. (In the long version of this paper [9] we also consider differentially private algorithms.) The central consequences of our main results are, under standard conditions on the loss functions ℓ, sharp upper and lower bounds on the possible convergence rates for estimation procedures when we wish to guarantee a level of privacy I(Xi; Zi) ≤I∗. We show there are problem dependent constants a(Θ, ℓ) and b(Θ, ℓ) such that the rates of convergence of all possible procedures are lower bounded by a(Θ, ℓ)/ √ nI∗and that there exist procedures achieving convergence rates of b(Θ, ℓ)/ √ nI∗, where the ratio b(Θ, ℓ)/a(Θ, ℓ) is upper bounded by a universal constant. Thus, we establish and quantify explicitly the tradeoff between statistical estimation and the amount of privacy. 2 We show that stochastic gradient descent is one procedure that achieves the optimal convergence rates, which means additionally that our upper bounds apply in streaming and online settings, requiring only a ﬁxed-size memory footprint. Our subsequent analysis builds on this favorable property of gradient-based methods, whence we focus on statistical estimation procedures that access data through the subgradients of the loss functions ∂ℓ(X, θ). This is a natural restriction. Gradients of the loss ℓare asymptotically sufﬁcient [18] (in an asymptotic sense, gradients contain all of the statistical information for risk minimization problems), stochastic gradient-based estimation procedures are (sample) minimax optimal and Bahadur efﬁcient [23, 1, 27, Chapter 8], many estimation procedures are gradient-based [20, 6], and distributed optimization procedures that send gradient information across a network to a centralized procedure M are natural [e.g. 3]. Our mechanism gives M access to a vector Zi that is a stochastic (sub)gradient of the loss evaluated on the sample Xi at a parameter θ of the method’s choosing: E[Zi \| Xi, θ] ∈∂ℓ(Xi, θ), (2) where ∂ℓ(Xi, θ) denotes the subgradient set of the function θ '→ℓ(Xi, θ). In a sense, the unbiasedness of the subgradient inclusion (2) is information-theoretically necessary [1]. To obtain upper and lower bound on the convergence rate of estimation procedures, we provide a two-part analysis. One part requires studying saddle points of the mutual information I(X; Z) (as a function of the distributions P of X and Q(· \| X) of Z) under natural constraints that allow inference of the optimal parameters θ∗for the risk R. We show that for certain classes of loss functions ℓand constraints on the communicated version Zi of the data Xi, there is a unique distribution Q(· \| Xi) that attains the smallest possible mutual information I(X; Z) for all distributions on X. Using this unique distribution, we can adapt information-theoretic techniques for obtaining lower bounds on estimation [31, 1] to derive our lower bounds. The uniqueness results for the conditional distribution Q show that no algorithm guaranteeing privacy between M and the samples Xi can do better. We can obtain matching upper bounds by application of known convergence rates for stochastic gradient and mirror descent algorithms [20, 21], which are computationally efﬁcient. 3 Optimal learning rates and tradeoffs Having outlined our general approach, we turn in this section to providing statements of our main results. Before doing so, we require some formalization of our notions of privacy and error measures, which we now provide. 3.1 Optimal Local Privacy We begin by describing in slightly more detail the communication protocol by which information about the random variables X is communicated to the procedure M. We assume throughout that there exist two d-dimensional compact sets C, D, where C ⊂int D ⊂Rd, and we have that ∂ℓ(x, θ) ⊂C for all θ ∈Θ and x ∈X. We wish to maximally “disguise” the random variable X with the random variable Z satisfying Z ∈D. Such a setting is natural; indeed, many online optimization and stochastic approximation algorithms [34, 21, 1] assume that for any x ∈X and θ ∈Θ, if g ∈∂ℓ(x, θ) then ∥g∥≤L for some norm ∥·∥. We may obtain privacy by allowing a perturbation to the subgradient g so long as the perturbation lives in a (larger) norm ball of radius M ≥L, so that C = {g ∈Rd : ∥g∥≤L} ⊂D = {g ∈Rd : ∥g∥≤M}. Now let X have distribution P, and for each x ∈X, let Q(· \| x) denote the regular conditional probability measure of Z given that X = x. Let Q(·) denote the marginal probability deﬁned by Q(A) = EP [Q(A \| X)]. The mutual information between X and Z is the expected KullbackLeibler (KL) divergence between Q(· \| X) and Q(·): I(P, Q) = I(X; Z) := EP [Dkl (Q(· \| X)\|\|Q(·))] . (3) We view the problem of privacy as a game between the adversary controlling P and the data owners, who use Q to obscure the samples X. In particular, we say a distribution Q guarantees a level of privacy I∗if and only if supP I(P, Q) ≤I∗. (Evﬁmievski et al. [13, Deﬁnition 6] present a similar condition.) Thus we seek a saddle point P ∗, Q∗such that sup P I(P, Q∗) ≤I(P ∗, Q∗) ≤inf Q I(P ∗, Q), (4) 3 where the ﬁrst supremum is taken over all distributions P on X such that ∇ℓ(X, θ) ∈C with P-probability 1, and the inﬁmum is taken over all regular conditional distributions Q such that if Z ∼Q(· \| X), then Z ∈D and EQ[Z \| X, θ] = ∇ℓ(X, θ). Indeed, if we can ﬁnd P ∗and Q∗ satisfying the saddle point (4), then the trivial direction of the max-min inequality yields sup P inf Q I(P, Q) = I(P ∗, Q∗) = inf Q sup P I(P, Q). To fully formalize this idea and our notions of privacy, we deﬁne two collections of probability measures and associated losses. For sets C ⊂D ⊂Rd, we deﬁne the source set P (C) := {Distributions P such that supp P ⊂C} (5a) and the set of regular conditional distributions (r.c.d.’s), or communicating distributions, Q (C, D) := " r.c.d.’s Q s.t. supp Q(· \| c) ⊂D and # D zdQ(z \| c) = c for c ∈C $ . (5b) The deﬁnitions (5a) and (5b) formally deﬁne the sets over which we may take inﬁma and suprema in the saddle point calculations, and they capture what may be communicated. The conditional distributions Q ∈Q (C, D) are deﬁned so that if ∇ℓ(x, θ) ∈C then EQ[Z \| X, θ] := % D zdQ (z \| ∇ℓ(x, θ)) = ∇ℓ(x, θ). We now make the following key deﬁnition: Deﬁnition 1. The conditional distribution Q∗satisﬁes optimal local privacy for the sets C ⊂D ⊂Rd at level I∗if sup P I(P, Q∗) = inf Q sup P I(P, Q) = I∗, where the supremum is taken over distributions P ∈P (C) and the inﬁmum is taken over regular conditional distributions Q ∈Q (C, D). If a distribution Q∗satisﬁes optimal local privacy, then it guarantees that even for the worst possible distribution on X, the information communicated about X is limited. In a sense, Deﬁnition 1 captures the natural competition between privacy and learnability. The method M speciﬁes the set D to which the data Z it receives must belong; the “teachers,” or owners of the data X, choose the distribution Q to guarantee as much privacy as possible subject to this constraint. Using this mechanism, if we can characterize a unique distribution Q∗attaining the inﬁmum (4) for P ∗(and by extension, for any P), then we may study the effects of privacy between the method M and X. 3.2 Minimax error and loss functions Having deﬁned our privacy metric, we now turn to our original goal: quantiﬁcation of the effect privacy has on statistical estimation rates. Let M denote any statistical procedure or method (that uses n stochastic gradient samples) and let θn denote the output of M after receiving n such samples. Let P denote the distribution according to which samples X are drawn. We deﬁne the (random) error of the method M on the risk R(θ) = E[ℓ(X, θ)] after receiving n sample gradients as ϵn(M, ℓ, Θ, P) := R(θn) −inf θ∈Θ R(θ) = EP [ℓ(X, θn)] −inf θ∈Θ EP [ℓ(X, θ)]. (6) In our settings, in addition to the randomness in the sampling distribution P, there is additional randomness from the perturbation applied to stochastic gradients of the objective ℓ(X, ·) to mask X from the statistitician. Let Q denote the regular conditional probability—the channel distribution— whose conditional part is deﬁned on the range of the subgradient mapping ∂ℓ(X, ·). As the output θn of the statistical procedure M is a random function of both P and Q, we measure the expected sub-optimality of the risk according to both P and Q. Now, let L be a collection of loss functions, where L(P) denotes the losses ℓ: supp P × Θ →R belonging to L. We deﬁne the minimax error ϵ∗ n(L, Θ) := inf M sup ℓ∈L(P ),P EP,Q[ϵn(M, ℓ, Θ, P)], (7) where the expectation is taken over the random samples X ∼P and Z ∼Q(· \| X). We characterize the minimax error (7) for several classes of loss functions L(P), giving sharp results when the privacy distribution Q satisﬁes optimal local privacy. We assume that our collection of loss functions obey certain natural smoothness conditions, which are often (as we see presently) satisﬁed. We deﬁne the class of losses as follows. 4 Deﬁnition 2. Let L > 0 and p ≥1. The set of (L, p)-loss functions are those measurable functions ℓ: X × Θ →R such that x ∈X, the function θ '→ℓ(x, θ) is convex and \|ℓ(x, θ) −ℓ(x, θ′)\| ≤L ∥θ −θ′∥q (8) for any θ, θ′ ∈Θ, where q is the conjugate of p: 1/p + 1/q = 1. A loss ℓsatisﬁes the condition (8) if and only if for all θ ∈Θ, we have the inequality ∥g∥p ≤L for any subgradient g ∈∂ℓ(x, θ) (e.g. [16]). We give a few standard examples of such loss functions. First, we consider ﬁnding a multi-dimensional median, in which case the data x ∈Rd and ℓ(x, θ) = L ∥θ −x∥1. This loss is L-Lipschitz with respect to the ℓ1 norm, so it belongs to the class of (L, ∞) losses. A second example includes classiﬁcation problems, using either the hinge loss or logistic regression loss. In these cases, the data comes in pairs x = (a, b), where a ∈Rd is the set of regressors and b ∈{−1, 1} is the label; the losses are ℓ(x, θ) = [1 −b ⟨a, θ⟩]+ or ℓ(x, θ) = log (1 + exp(−b ⟨a, θ⟩)) By computing (sub)gradients, we may verify that each of these belong to the class of (L, p)-losses if and only if the data a satisﬁes ∥a∥p ≤L, which is a common assumption [7, 24]. The privacy-guaranteeing channel distributions Q∗we construct in Section 4 are motivated by our concern with the (L, p) families of loss functions. In our model of computation, the learning method M queries the loss ℓ(Xi, ·) at the point θ; the owner of the datum Xi then computes the subgradient ∂ℓ(Xi, θ) and returns a masked version Zi with the property that E[Zi \| Xi, θ] ∈∂ℓ(Xi, θ). In the following two theorems, we give lower bounds on ϵ∗ n for the (L, ∞) and (L, 1) families of loss functions under the constraint that the channel distribution Q must guarantee that a limited amount of information I(Xi; Zi) is communicated: the channel distribution Q satisﬁes our Deﬁnition 1 of optimal local privacy. 3.3 Main theorems We now state our two main theorems, deferring proofs to Appendix B. Our ﬁrst theorem applies to the class of (L, ∞) loss functions (recall Deﬁnition 2). We assume that the set to which the perturbed data Z must belong is [−M∞, M∞]d, where M∞≥L. We state two variants of the theorem, as one gives sharper results for an important special case. Theorem 1. Let L be the collection of (L, ∞) loss functions and assume the conditions of the preceding paragraph. Let Q satisfy be optimally private for the collection L. Then (a) If Θ contains the ℓ∞ball of radius r, ϵ∗ n(L, Θ) ≥ 1 163 · M∞rd √n . (b) If Θ = {θ ∈Rd : ∥θ∥1 ≤r}, ϵ∗ n(L, Θ) ≥rM∞ & log(2d) 17√n . For our second theorem, we assume that the loss functions L consist of (L, 1) losses, and that the perturbed data must belong to the ℓ1 ball of radius M1, i.e., Z ∈{z ∈Rd \| ∥z∥1 ≤M1}. Setting M = M1/L, we deﬁne (these constants relate to the optimal local privacy distribution for ℓ1-balls) γ := log ' 2d −2 + & (2d −2)2 + 4(M 2 −1) 2(M −1) ( , and ∆(γ) := eγ −e−γ eγ + e−γ + 2(d −1). (9) Theorem 2. Let L be the collection of (L, 1) loss functions and assume the conditions of the preceding paragraph. Let Q be optimally locally private for the collection L. Then ϵ∗ n(L, Θ) ≥ 1 163 · rL √ d √n∆(γ). 5 Remarks We make two main remarks about Theorems 1 and 2. First, we note that each result yields a minimax rate for stochastic optimization problems when there is no random distribution Q. Indeed, in Theorem 1, we may take M∞= L, in which case (focusing on the second statement of the theorem) we obtain the lower bound rL & log(2d)/17√n when Θ = {θ ∈Rd : ∥θ∥1 ≤r}. Mirror descent algorithms [20, 21] attain a matching upper bound (see the long version of this paper [9, Sec. 3.3] for more substantial explanation). Moreover, our analysis is sharper than previous analyses [1, 20], as none (to our knowledge) recover the logarithmic dependence on the dimension d, which is evidently necessary. Theorem 2 provides a similar result when we take M1 ↓L, though in this case stochastic gradient descent attains the matching upper bound. Our second set of remarks are somewhat more striking. In these, we show that the lower bounds in Theorems 1 and 2 give sharp tradeoffs between the statistical rate of convergence for any statistical procedure and the desired privacy of a user. We present two corollaries establishing this tradeoff. In each corollary, we look ahead to Section 4 and use one of Propositions 1 or 2 to derive a bijection between the size M∞or M1 of the perturbation set and the amount of privacy—as measured by the worst case mutual information I∗—provided. We then combine Theorems 1 and 2 with results on stochastic approximation to demonstrate the tradeoffs. Corollary 1. Let the conditions of Theorem 1(b) hold, and assume that M∞≥2L. Assume Q∗ satisﬁes optimal local privacy at information level I∗. For universal constants c ≤C, c · rL√d log d √ nI∗ ≤ϵ∗ n(L, Θ) ≤C · rL√d log d √ nI∗ . Proof Since Θ ⊆{θ ∈Rd : ∥θ∥1 ≤r}, mirror descent [2, 21, 20, Chapter 5], using n unbiased stochastic gradient samples whose ℓ∞norms are bounded by M∞, obtains convergence rate O(M∞r√log d/√n). This matches the second statement of Theorem 1. Now ﬁx our desired amount of mutual information I∗. From the remarks following Proposition 1, if we must guarantee that I∗≥supP I(P, Q) for any distribution P and loss function ℓwhose gradients are bounded in ℓ∞-norm by L, we must (by the remarks following Proposition 1) have I∗≍dL2 M 2∞ . Up to higher-order terms, to guarantee a level of privacy with mutual information I∗, we must allow gradient noise up to M∞= L & d/I∗. Using the bijection between M∞and the maximal allowed mutual information I∗under local privacy that we have shown, we substitute M∞= L √ d/ √ I∗ into the upper and lower bounds that we have already attained. Similar upper and lower bounds can be obtained under the conditions of part (a) of Theorem 1, where we need not assume Θ is an ℓ1-ball, but we lose a factor of √log d in the lower bound. Now we turn to a parallel result, but applying Theorem 2 and Proposition 2. Corollary 2. Let the conditions of Theorem 2 hold and assume that M1 ≥2L. Assume that Q∗ satisﬁes optimal local privacy at information level I∗. For universal constants c ≤C, c · rLd √ nI∗≤ϵ∗ n(L, Θ) ≤C · rLd √ nI∗. Proof By the conditions of optimal local privacy (Proposition 2 and Corollary 3), to have I∗≥ supP I(P, Q) for any loss ℓwhose gradients are bounded in ℓ1-norm by L, we must have I∗≍dL2 2M 2 1 , using Corollary 3. Rewriting this, we see that we must have M1 = L & d/2I∗(to higher-order terms) to be able to guarantee an amount of privacy I∗. As in the ℓ∞case, we have a bijection between the multiplier M1 and the amount of information I∗and can apply similar techniques. Indeed, stochastic gradient descent (SGD) enjoys the following convergence guarantees (e.g. [21]). Let Θ ⊆Rd be contained in the ℓ∞ball of radius r and the gradients of the loss ℓbelong to the ℓ1-ball of radius M1. Then SGD has ϵ∗ n(L, Θ) ≤CM1r √ d/√n. Now apply the lower bound provided by Theorem 2 and substitute for M1. 6 4 Saddle points, optimal privacy, and mutual information In this section, we explore conditions for a distribution Q∗to satisfy optimal local privacy, as given by Deﬁnition 1. We give characterizations of necessary and sufﬁcient conditions based on the compact sets C ⊂D for distributions P ∗and Q∗to achieve the saddle point (4). Our results can be viewed as rate distortion theorems [14, 8] (with source P and channel Q) for certain compact alphabets, though as far as we know, they are all new. Thus we sometimes refer to the conditional distribution Q, which is designed to maintain the privacy of the data X by communication of Z, as the channel distribution. Since we wish to bound I(X; Z) for arbitrary losses ℓ, we must address the case when ℓ(X, θ) = ⟨θ, X⟩, in which case ∇ℓ(X, θ) = X; by the data-processing inequality [14, Chapter 5] it is thus no loss of generality to assume that X ∈C and that E[Z \| X] = X. We begin by deﬁning the types of sets C and D that we use in our characterization of privacy. As we see in Section 3, such sets are reasonable for many applications. We focus on the case when the compact sets C and D are (suitably symmetric) norm balls: Deﬁnition 3. Let C ⊂Rd be a compact convex set with extreme points ui ∈Rd, i ∈I for some index set I. Then C is rotationally invariant through its extreme points if ∥ui∥2 = ∥uj∥2 for each i, j, and for any unitary matrix U such that Uui = uj for some i ̸= j, then UC = C. Some examples of convex sets rotationally invariant through their extreme points include ℓp-norm balls for p = 1, 2, ∞, though ℓp-balls for p ̸∈{1, 2, ∞} are not. The following theorem gives a general characterization of the minimax mutual information for rotationally invariant norm balls with ﬁnite numbers of extreme points by providing saddle point distributions P ∗and Q∗. We provide the proof of Theorem 3 in Section A.1. Theorem 3. Let C be a compact, convex, polytope rotationally invariant through its extreme points {ui}m i=1 and D = (1 + α)C for some α > 0. Let Q∗be the conditional distribution on Z \| X that maximizes the entropy H(Z \| X = x) subject to the constraints that EQ[Z \| X = x] = x for x ∈C and that Z is supported on (1 + α)ui for i = 1, . . . , m. Then Q∗satisﬁes Deﬁnition 1, optimal local privacy, and Q∗is (up to measure zero sets) unique. Moreover, the distribution P ∗ uniform on {ui}m i=1 uniquely attains the saddle point (4). Remarks: While in the theorem we assume that Q∗(· \| X = x) maximizes the entropy for each x ∈C, this is not in fact essential. In fact, we may introduce a random variable X′ between X and Z: let X′ be distributed among the extreme points {ui}m i=1 of C in any way such that E[X′ \| X] = X, then use the maximum entropy distribution Q∗(· \| ui) deﬁned in the theorem when X ∈{ui}m i=1 to sample Z from X′. The information processing inequality [14, Chapter 5] guarantees the Markov chain X →X′ →Z satisﬁes the minimax bound I(X; Z) ≤infQ supP I(P, Q). With Theorem 3 in place, we can explicitly characterize the distributions achieving optimal local privacy (recall Deﬁnition 1) for ℓ1 and ℓ∞balls. We present the propositions in turn, providing some discussion here and deferring proofs to Appendices A.2 and A.3. First, consider the case where X ∈[−1, 1]d and Z ∈[−M, M]d. For notational convenience, we deﬁne the binary entropy h(p) = −p log p −(1 −p) log(1 −p). We have Proposition 1. Let X ∈[−1, 1]d and Z ∈[−M, M]d be random variables with M ≥1 and E[Z \| X] = X almost surely. Deﬁne Q∗to be the conditional distribution on Z \| X such that the coordinates of Z are independent, have range {−M, M}, and Q∗(Zi = M \| X) = 1 2 + Xi 2M and Q∗(Zi = −M \| X) = 1 2 −Xi 2M . Then Q∗satisﬁes Deﬁnition 1, optimal local privacy, and moreover, sup P I(P, Q∗) = d −d · h )1 2 + 1 2M * . Before continuing, we give a more intuitive understanding of Proposition 1. Concavity implies that for a, b > 0, log(a) ≤log b + b−1(a −b), or −log(a) ≥−log(b) + b−1(b −a), so in particular h )1 2 + 1 2M * ≥− )1 2 + 1 2M * ) −log 2 −1 M * − )1 2 − 1 2M * ) −log 2 + 1 M * = log 2−1 M 2 . 7 That is, we have for any distribution P on X ∈[−1, 1]d that (in natural logarithms) I(P, Q∗) ≤ d M 2 and I(P, Q∗) = d M 2 + O(M −3). We now consider the case when X ∈ + x ∈Rd \| ∥x∥1 ≤1 , and Z ∈ + z ∈Rd \| ∥z∥1 ≤M , . Here the arguments are slightly more complicated, as the coordinates of the random variables are no longer independent, but Theorem 3 still allows us to explicitly characterize the saddle point of the mutual information. Proposition 2. Let X ∈{x ∈Rd \| ∥x∥1 ≤1} and Z ∈{z ∈Rd \| ∥z∥1 ≤M} be random variables with M > 1. Deﬁne the parameter γ as in Eq. (9), and let Q∗be the distribution on Z \| X such that Z is supported on {±Mei}d i=1, and Q∗(Z = Mei \| X = ei) = eγ eγ + e−γ + (2d −2), (10a) Q∗(Z = −Mei \| X = ei) = e−γ eγ + e−γ + (2d −2), (10b) Q∗(Z = ±Mej \| X = ei, j ̸= i) = 1 eγ + e−γ + (2d −2). (10c) (For X ̸∈{±ei}, deﬁne X′ to be randomly selected in any way from among {±ei} such that E[X′ \| X] = X, then sample Z conditioned on X′ according to (10a)–(10c).) Then Q∗satisﬁes Deﬁnition 1, optimal local privacy, and sup P I(P, Q∗) = log(2d)−log eγ + e−γ + 2d −2 . +γ eγ eγ + e−γ + 2d −2−γ e−γ eγ + e−γ + 2d −2. We remark that the additional sampling to guarantee that X′ ∈{±ei} (where the conditional distribution Q∗is deﬁned) can be accomplished simply: deﬁne the random variable X′ so that X′ = ei sign(xi) with probability \|xi\|/ ∥x∥1. Evidently E[X′ \| X] = x, and X →X′ →Z for Z distributed according to Q∗deﬁnes a Markov chain as in our remarks following Theorem 3. Additionally, an asymptotic expansion allows us to gain a somewhat clearer picture of the values of the mutual information, though we do not derive upper bounds as we did for Proposition 1. We have the following corollary, proved in Appendix E.1. Corollary 3. Let Q∗denote the conditional distribution in Proposition 2. Then sup P I(P, Q∗) = d 2M 2 + Θ ) min / d3 M 4 , log4(d) d 0* . 5 Discussion and open questions This study leaves a number open issues and areas for future work. We study procedures that access each datum only once and through a perturbed view Zi of the subgradient ∂ℓ(Xi, θ), which allows us to use (essentially) any convex loss. A natural question is whether there are restrictions on the loss function so that a transformed version (Z1, . . . , Zn) of the data are sufﬁcient for inference. Zhou et al. [33] study one such procedure, and nonparametric data releases, such as those Hall et al. [15] study, may also provide insights. Unfortunately, these (and other) current approaches require the data be aggregated by a trusted curator. Our constraints on the privacy-inducing channel distribution Q require that its support lie in some compact set. We ﬁnd this restriction useful, but perhaps it possible to achieve faster estimation rates under other conditions. A better understanding of general privacy-preserving channels Q for alternative constraints to those we have proposed is also desirable. 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4,486	Relax and Randomize: From Value to Algorithms Alexander Rakhlin University of Pennsylvania Ohad Shamir Microsoft Research Karthik Sridharan University of Pennsylvania Abstract We show a principled way of deriving online learning algorithms from a minimax analysis. Various upper bounds on the minimax value, previously thought to be non-constructive, are shown to yield algorithms. This allows us to seamlessly recover known methods and to derive new ones, also capturing such “unorthodox” methods as Follow the Perturbed Leader and the R2 forecaster. Understanding the inherent complexity of the learning problem thus leads to the development of algorithms. To illustrate our approach, we present several new algorithms, including a family of randomized methods that use the idea of a “random playout”. New versions of the Follow-the-Perturbed-Leader algorithms are presented, as well as methods based on the Littlestone’s dimension, efﬁcient methods for matrix completion with trace norm, and algorithms for the problems of transductive learning and prediction with static experts. 1 Introduction This paper studies the online learning framework, where the goal of the player is to incur small regret while observing a sequence of data on which we place no distributional assumptions. Within this framework, many algorithms have been developed over the past two decades [6]. More recently, a non-algorithmic minimax approach has been developed to study the inherent complexities of sequential problems [2, 1, 15, 20]. It was shown that a theory in parallel to Statistical Learning can be developed, with random averages, combinatorial parameters, covering numbers, and other measures of complexity. Just as the classical learning theory is concerned with the study of the supremum of empirical or Rademacher process, online learning is concerned with the study of the supremum of martingale processes. While the tools introduced in [15, 17, 16] provide ways of studying the minimax value, no algorithms have been exhibited to achieve these non-constructive bounds in general. In this paper, we show that algorithms can, in fact, be extracted from the minimax analysis. This observation leads to a unifying view of many of the methods known in the literature, and also gives a general recipe for developing new algorithms. We show that the potential method, which has been studied in various forms, naturally arises from the study of the minimax value as a certain relaxation. We further show that the sequential complexity tools introduced in [15] are, in fact, relaxations and can be used for constructing algorithms that enjoy the corresponding bounds. By choosing appropriate relaxations, we recover many known methods, improved variants of some known methods, and new algorithms. One can view our framework as one for converting a nonconstructive proof of an upper bound on the value of the game into an algorithm. Surprisingly, this allows us to also study such “unorthodox” methods as Follow the Perturbed Leader [10], and the recent method of [7] under the same umbrella with others. We show that the idea of a random playout has a solid theoretical basis, and that Follow the Perturbed Leader algorithm is an example of such a method. Based on these developments, we exhibit an efﬁcient method for the trace norm matrix completion problem, novel Follow the Perturbed Leader algorithms, and efﬁcient methods for the problems of online transductive learning. The framework of this paper gives a recipe for developing algorithms. Throughout the paper, we stress that the notion of a relaxation, introduced below, is not appearing out of thin air but rather as an upper bound on the sequential Rademacher complexity. The understanding of inherent complexity thus leads to the development of algorithms. 1 Let us introduce some notation. The sequence x1,...,xt is often denoted by x1∶t , and the set of all distributions on some set A by ∆(A). Unless speciﬁed otherwise, ✏denotes a vector (✏1,...,✏T ) of i.i.d. Rademacher random variables. An X-valued tree x of depth d is deﬁned as a sequence (x1,...,xd) of mappings xt ∶{±1}t−1 X (see [15]). We often write xt(✏) instead of xt(✏1∶t−1). 2 Value, The Minimax Algorithm, and Relaxations Let F be the set of learner’s moves and X the set of moves of Nature. The online protocol dictates that on every round t = 1,...,T the learner and Nature simultaneously choose ft ∈F, xt ∈X, and observe each other’s actions. The learner aims to minimize regret RegT ∑T t=1 `(ft,xt) − inff∈F ∑T t=1 `(f,xt) where ` ∶F × X →R is a known loss function. Our aim is to study this online learning problem at an abstract level without assuming convexity or other such properties of `, F and X. We do assume, however, that `, F, and X are such that the minimax theorem in the space of distributions over F and X holds. By studying the abstract setting, we are able to develop general algorithmic and non-algorithmic ideas that are common across various application areas. The starting point of our development is the minimax value of the associated online learning game: VT (F) = inf q1∈∆(F) sup x1∈X E f1∼q1 ... inf qT ∈∆(F) sup xT ∈X E fT ∼qT T t=1 `(ft,xt) −inf f∈F T t=1 `(f,xt) (1) where ∆(F) is the set of distributions on F. The minimax formulation immediately gives rise to the optimal algorithm that solves the minimax expression at every round t and returns argmin q∈∆(F) sup xt E ft∼q `(ft,xt) + inf qt+1 sup xt+1 E ft+1 ...inf qT sup xT E fT T i=t+1 `(fi,xi) −inf f∈F T i=1 `(f,xi) Henceforth, if the quantiﬁcation in inf and sup is omitted, it will be understood that xt, ft, pt, qt range over X, F, ∆(X), ∆(F), respectively. Moreover, Ext is with respect to pt while Eft is with respect to qt. We now notice a recursive form for the value of the game. Deﬁne for any t ∈[T −1] and any given preﬁx x1,...,xt ∈X the conditional value VT (Fx1,...,xt) inf q∈∆(F) sup x∈X E f∼q [`(f,x)] + VT (Fx1,...,xt,x) with VT (Fx1,...,xT ) −inff∈F ∑T t=1 `(f,xt) and VT (F) = VT (F{}). The minimax optimal algorithm specifying the mixed strategy of the player can be written succinctly as qt = argmin q∈∆(F) sup x∈X E f∼q [`(f,x)] + VT (Fx1,...,xt−1,x). (2) Similar recursive formulations have appeared in the literature [8, 13, 19, 3], but now we have tools to study the conditional value of the game. We will show that various upper bounds on VT (Fx1,...,xt−1,x) yield an array of algorithms. In this way, the non-constructive approaches of [15, 16, 17] to upper bound the value of the game directly translate into algorithms. We note that the minimax algorithm in (2) can be interpreted as choosing the best decision that takes into account the present loss and the worst-case future. The ﬁrst step in our analysis is to appeal to the minimax theorem and perform the same manipulation as in [1, 15], but only on the conditional values: VT (Fx1,...,xt) = sup pt+1 E xt+1 ...sup pT E xT T i=t+1 inf fi∈F E xi∼pi `(fi,xi) −inf f∈F T i=1 `(f,xi). (3) The idea now is to come up with more manageable, yet tight, upper bounds on the conditional value. A relaxation RelT is a sequence of real-valued functions RelT (Fx1,...,xt) for each t ∈[T]. We call a relaxation admissible if for any x1,...,xT ∈X, RelT (Fx1,...,xt) ≥ inf q∈∆(F) sup x∈X E f∼q [`(f,x)] + RelT (Fx1,...,xt,x) (4) for all t ∈[T −1], and RelT (Fx1,...,xT ) ≥−inff∈F ∑T t=1 `(f,xt). We use the notation RelT (F) for RelT (F{}). A strategy q that minimizes the expression in (4) deﬁnes an optimal Meta-Algorithm for an admissible relaxation RelT : on round t, compute qt = arg min q∈∆(F)sup x∈X E f∼q [`(f,x)] + RelT (Fx1,...,xt−1,x), (5) 2 play ft ∼qt and receive xt from the opponent. Importantly, minimization need not be exact: any qt that satisﬁes the admissibility condition (4) is a valid method, and we will say that such an algorithm is admissible with respect to the relaxation RelT . Proposition 1. Let RelT be an admissible relaxation. For any admissible algorithm with respect to RelT , (including the Meta-Algorithm), irrespective of the strategy of the adversary, T t=1 Eft∼qt`(ft,xt) −inf f∈F T t=1 `(f,xt) ≤RelT (F) , (6) and therefore, E[RegT ] ≤RelT (F). If `(⋅,⋅) is bounded, the Hoeffding-Azuma inequality yields a high-probability bound on RegT . We also have that VT (F) ≤RelT (F). Further, if for all t ∈[T], the admissible strategies qt are deterministic, RegT ≤RelT (F). The reader might recognize RelT as a potential function. It is known that one can derive regret bounds by coming up with a potential such that the current loss of the player is related to the difference in the potentials at successive steps, and that the regret can be extracted from the ﬁnal potential. The origin/recipe for “good” potential functions has always been a mystery (at least to the authors). One of the key contributions of this paper is to show that they naturally arise as relaxations on the conditional value, and the conditional value is itself the tightest possible relaxation. In particular, for many problems a tight relaxation is achieved through symmetrization applied to the expression in (3). Deﬁne the conditional Sequential Rademacher complexity RT (Fx1,...,xt) = sup x E✏t+1∶T sup f∈F 2 T s=t+1 ✏s`(f,xs−t(✏t+1∶s−1)) − t s=1 `(f,xs). (7) Here the supremum is over all X-valued binary trees of depth T −t. One may view this complexity as a partially symmetrized version of the sequential Rademacher complexity RT (F) RT (F {}) = sup x E✏1∶T sup f∈F 2 T s=1 ✏s`(f,xs(✏1∶s−1)) (8) deﬁned in [15]. We shall refer to the term involving the tree x as the “future” and the term being subtracted off in (7) – as the “past”. This indeed corresponds to the fact that the quantity is conditioned on the already observed x1,...,xt, while for the future we have the worst possible binary tree.1 Proposition 2. The conditional Sequential Rademacher complexity is admissible. We now show that several well-known methods arise as further relaxations on RT . Exponential Weights [12, 21] Suppose F is a ﬁnite class and `(f,x)≤1. In this case, a (tight) upper bound on sequential Rademacher complexity leads to the following relaxation: RelT (Fx1, . . . , xt) = inf λ>0 1 λ log f∈F exp −λ t i=1 `(f, xi) + 2λ(T −t) (9) Proposition 3. The relaxation (9) is admissible and RT (Fx1,...,xt) ≤RelT (Fx1,...,xt). Furthermore, it leads to a parameter-free version of the Exponential Weights algorithm, deﬁned on round t + 1 by the mixed strategy qt+1(f) ∝exp−λ∗ t ∑t s=1 `(f,xs)with λ∗ t the optimal value in (9). The algorithm’s regret is bounded by RelT (F) ≤2 2T log F. We point out that the exponential-weights algorithm arising from the relaxation (9) is a parameterfree algorithm. The learning rate λ∗can be optimized (via 1D line search) at each iteration. Mirror Descent [4, 14] In the setting of online linear optimization [22], the loss is `(f,x) = f,x. Suppose F is a unit ball in some Banach space and X is the dual. Let ⋅be some (2,C)-smooth norm on X (in the Euclidean case, C = 2). Using the notation ˜xt−1 = ∑t−1 s=1 xs, a straightforward upper bound on sequential Rademacher complexity is the following relaxation: RelT (Fx1, . . . , xt) = ˜xt−12 + ∇˜xt−12 , xt+ C(T −t + 1) (10) 1It is cumbersome to write out the indices on xs−t(✏t+1∶s−1) in (7), so we will instead use xs(✏) whenever this doesn’t cause confusion. 3 Proposition 4. The relaxation (10) is admissible and RT (Fx1,...,xt) ≤RelT (Fx1,...,xt) . It yields the update ft = −∇˜xt−12 2 ˜xt−12+C(T −t+1) with regret bound RelT (F) ≤ √ 2CT. We would like to remark that the traditional mirror descent update can be shown to arise out of an appropriate relaxation. The algorithms proposed are parameter free as the step size is tuned automatically. We chose the popular methods of Exponential Weights and Mirror Descent for illustration. In the remainder of the paper, we develop new algorithms to show universality of our approach. 3 Classiﬁcation We start by considering the problem of supervised learning, where X is the space of instances and Y the space of responses (labels). There are two closely related protocols for the online interaction between the learner and Nature, so let us outline them. The “proper” version of supervised learning follows the protocol presented in Section 2: at time t, the learner selects ft ∈F, Nature simultaneously selects (xt,yt) ∈X × Y, and the learner suffers the loss `(f(xt),yt). The “improper” version is as follows: at time t, Nature chooses xt ∈X and presents it to the learner as “side information”, the learner then picks ˆyt ∈Y and Nature simultaneously chooses yt ∈Y. In the improper version, the loss of the learner is `(ˆyt,yt), and it is easy to see that we may equivalently state this protocol as the learner choosing any function ft ∈YX (not necessarily in F), and Nature simultaneously choosing (xt,yt). We mostly focus on the “improper” version of supervised learning in this section. For the improper version, we may write the value in (1) as VT (F) = sup x1∈X inf q1∈∆(Y) sup y1∈X E ˆy1∼q1 . . . sup xT ∈X inf qT ∈∆(Y) sup yT ∈X E ˆyT ∼qT T t=1 `(ˆyt, yt) −inf f∈F T t=1 `(f(xt), yt) and a relaxation RelT is admissible if for any (x1,y1)...,(xT ,yT ) ∈X × Y, sup x∈X inf q∈∆(Y) sup y∈Y E ˆy∼q`(ˆy, y) + RelT F{(xi, yi)}t i=1, (x, y)≤RelT F{(xi, yi)}t i=1 (11) Let us now focus on binary prediction, i.e. Y = {±1}. In this case, the supremum over y in (11) becomes a maximum over two values. Let us now take the absolute loss `(ˆy,y) = ˆy −y= 1 −ˆyy. We can see2 that the optimal randomized strategy, given the side information x, is given by (11) as qt = argmin q∈∆(Y) max 1 −q + RelT F{(xi, yi)}t i=1, (x, 1), 1 + q + RelT F{(xi, yi)}t i=1, (x, −1) or equivalently as : qt = 1 2 RelT F{(xi, yi)}t i=1, (x, 1)−RelT F{(xi, yi)}t i=1, (x, −1) (12) We now assume that F has a ﬁnite Littlestone’s dimension Ldim(F) [11, 5]. Suppose the loss function is `(ˆy,y) = ˆy −y, and consider the “mixed” conditional Rademacher complexity sup x E✏sup f∈F 2 T −t i=1 ✏if(xi(✏)) − t i=1 f(xi) −yi (13) as a possible relaxation. The admissibility condition (11) with the conditional sequential Rademacher (13) as a relaxation would require us to upper bound sup xt inf qt∈[−1,1] max yt∈{±1} E ˆyt∼qt ˆyt −yt+ sup x E✏sup f∈F 2 T −t i=1 ✏if(xi(✏)) − t i=1 f(xi) −yi (14) However, the supremum over x is preventing us from obtaining a concise algorithm. We need to further “relax” this supremum, and the idea is to pass to a ﬁnite cover of F on the given tree x and then proceed as in the Exponential Weights example for a ﬁnite collection of experts. This leads to an upper bound on (13) and gives rise to algorithms similar in spirit to those developed in [5], but with more attractive computational properties and deﬁned more concisely. Deﬁne the function g(d,t) = ∑d i=0 t i, which is shown in [15] to be the maximum size of an exact (zero) cover for a function class with the Littlestone’s dimension Ldim = d. Given {(x1,yt),...,(xt,yt)} and σ = (σ1,...,σt) ∈{±1}t, let Ft(σ) = {f ∈F ∶f(xi) = σi ∀i ≤ t}, the subset of functions that agree with the signs given by σ on the “past” data and let Fx1,...,xt Fxt {(f(x1),...,f(xt)) ∶f ∈F} be the projection of F onto x1,...,xt. Denote Lt(f) = ∑t i=1 f(xi) −yiand Lt(σ) = ∑t i=1 σi −yifor σ ∈{±1}t. The following proposition gives a relaxation and an algorithm which achieves the O( Ldim(F)T log T) regret bound. Unlike the algorithm of [5], we do not need to run an exponential number of experts in parallel and only require access to an oracle that computes the Littlestone’s dimension. 2The extension to k-class prediction is immediate. 4 Proposition 5. The relaxation RelT F(xt, yt)= 1 λ log σ∈Fxt g(Ldim(Ft(σ)), T −t) exp {−λLt(σ)} + 2λ(T −t) . is admissible and leads to an admissible algorithm which uses weights qt(−1) = 1 −qt(+1) and qt(+1) = ∑(σ,+1)∈Fxt g(Ldim(Ft(σ, +1)), T −t) exp {−λLt−1(σ)} ∑(σ,σt)∈Fxt g(Ldim(Ft(σ, σt)), T −t) exp {−λLt−1(σ)} , (15) There is a very close correspondence between the proof of Proposition 5 and the proof of the combinatorial lemma of [15], the analogue of the Vapnik-Chervonenkis-Sauer-Shelah result. 4 Randomized Algorithms and Follow the Perturbed Leader We now develop a class of admissible randomized methods that arise through sampling. Consider the objective (5) given by a relaxation RelT . If RelT is the sequential (or classical) Rademacher complexity, it involves an expectation over sequences of coin ﬂips, and this computation (coupled with optimization for each sequence) can be prohibitively expensive. More generally, RelT might involve an expectation over possible ways in which the future might be realized. In such cases, we may consider a rather simple “random playout” strategy: draw the random sequence and solve only one optimization problem for that random sequence. The ideas of random playout have been discussed in previous literature for estimating the utility of a move in a game (see also [3]). We show that random playout strategy has a solid basis: for the examples we consider, it satisﬁes admissibility. In many learning problems the sequential and the classical Rademacher complexities are within a constant factor of each other. This holds true, for instance, for linear functions in ﬁnite-dimensional spaces. In such cases, the relaxation RelT does not involve the supremum over a tree, and the randomized method only needs to draw a sequence of coin ﬂips and compute a solution to an optimization problem slightly more complicated than ERM. We show that Follow the Perturbed Leader (FPL) algorithms [10] arise in this way. We note that FPL has been previously considered as a rather unorthodox algorithm providing some kind of regularization via randomization. Our analysis shows that it arises through a natural relaxation based on the sequential (and thus the classical) Rademacher complexity, coupled with the random playout idea. As a new algorithmic contribution, we provide a version of the FPL algorithm for the case of the decision sets being `2 balls, with a regret bound that is independent of the dimension. We also provide an FPL-style method for the combination of `1 and `∞balls. To the best of our knowledge, these results are novel. The assumption below implies that the sequential and classical Rademacher complexities are within constant factor C of each other. We later verify that it holds in the examples we consider. Assumption 1. There exists a distribution D ∈∆(X) and constant C ≥2 such that for any t ∈[T] and given any x1,...,xt−1,xt+1,...,xT ∈X and any ✏t+1,...,✏T ∈{±1}, sup p∈∆(X) E xt∼p sup f∈F CAt+1(f) −Lt−1(f) + E x∼p [`(f, x)] −`(f, xt)≤ E ✏t,xt∼D sup f∈F [ CAt(f) −Lt−1(f)] where ✏t’s are i.i.d. Rademacher, Lt−1(f) = ∑t−1 i=1 `(f,xi), and At(f) = ∑T i=t ✏i`(f,xi). Under the above assumption one can use the following relaxation RelT (Fx1, . . . , xt) = E xt+1,...xT ∼D E✏sup f∈F C T i=t+1 ✏i`(f, xi) − t i=1 `(f, xi) (16) which is a partially symmetrized version of the classical Rademacher averages. The proof of admissibility for the randomized methods is quite curious – the forecaster can be seen as mimicking the sequential Rademacher complexity by sampling from the “equivalently bad” classical Rademacher complexity under the speciﬁc distribution D speciﬁed by the above assumption. Lemma 6. Under Assumption 1, the relaxation in Eq. (16) is admissible and a randomized strategy that ensures admissibility is given by: at time t, draw xt+1,...,xT ∼D and ✏t+1,...,✏T and then: (a) In the case the loss ` is convex in its ﬁrst argument and set F is convex and compact, deﬁne 5 ft = argmin g∈F sup x∈X `(g, x) + sup f∈F C T i=t+1 ✏i`(f, xi) − t−1 i=1 `(f, xi) −`(f, x) (17) (b) In the case of non-convex loss, sample ft from the distribution ˆqt = argmin ˆq∈∆(F) sup x∈X E f∼ˆq [`(f, x)] + sup f∈F C T i=t+1 ✏i`(f, xi) − t−1 i=1 `(f, xi) −`(f, x) (18) The expected regret for the method is bounded by the classical Rademacher complexity: E [RegT ] ≤C Ex1∶T ∼D E ✏sup f∈F T t=1 ✏t`(f, xt), Of particular interest are the settings of static experts and transductive learning, which we consider in Section 5. In the transductive case, the xt’s are pre-speciﬁed before the game, and in the static expert case – effectively absent. In these cases, as we show below, there is no explicit distribution D and we only need to sample the random signs ✏’s. We easily see that in these cases, the expected regret bound is simply two times the transductive Rademacher complexity. The idea of sampling from a ﬁxed distribution is particularly appealing in the case of linear loss, `(f,x) = f,x. Suppose X is a unit ball in some norm ⋅in a vector space B, and F is a unit ball in the dual norm ⋅∗. A sufﬁcient condition implying Assumption 1 is then Assumption 2. There exists a distribution D ∈∆(X) and constant C ≥2 such that for any w ∈B, sup x∈X E xt∼p w + 2✏txt≤ E xt∼D E ✏t w + C✏txt (19) At round t, the generic algorithm speciﬁed by Lemma 18 draws fresh Rademacher random variables ✏and xt+1,...,xT ∼D and picks ft = argmin f∈F sup x∈X f, x+ C T i=t+1 ✏ixi − t−1 i=1 xi −x (20) We now look at `2`2 and `1`∞cases and provide corresponding randomized algorithms. Example : `1`∞Follow the Perturbed Leader Here, we consider the setting similar to that in [10]. Let F ⊂RN be the `1 unit ball and X the (dual) `∞unit ball in RN. In [10], F is the probability simplex and X = [0,1]N but these are subsumed by the `1`∞case. Next we show that any symmetric distribution satisﬁes Assumption 2. Lemma 7. If D is any symmetric distribution over R, then Assumption 2 is satisﬁed by using the product distribution DN and any C ≥6Ex∼Dx. In particular, Assumption 2 is satisﬁed with a distribution D that is uniform on the vertices of the cube {±1}N and C = 6. The above lemma is especially attractive with Gaussian perturbations as sum of normal random variables is again normal. Hence, instead of drawing xt+1,...,xT ∼N(0,1) on round t, one can simply draw one vector Xt ∼N(0,T −t) as the perturbation. In this case, C ≤8. The form of update in Equation (20), however, is not in a convenient form, and the following lemma shows a simple Follow the Perturbed Leader type algorithm with the associated regret bound. Lemma 8. Suppose F is the `N 1 unit ball and X is the dual `N ∞unit ball, and let D be any symmetric distribution. Consider the randomized algorithm that at each round t, freshly draws Rademacher random variables ✏t+1,...,✏T and xt+1,...,xT ∼DN and picks ft = argmin f∈F f,∑t−1 i=1 xi −C ∑T i=t+1 ✏ixiwhere C = 6Ex∼Dx. The expected regret is bounded as : E [RegT ] ≤C E x1∶T ∼DN E✏ T t=1 ✏txt ∞ + 4 T t=1 Pyt+1∶T ∼D C T i=t+1 yi≤4 For instance, for the case of coin ﬂips (with C = 6) or the Gaussian distribution (with C = 3 √ 2⇡) the bound above is 4C√T log N, as the second term is bounded by a constant. Example : `2`2 Follow the Perturbed Leader We now consider the case when F and X are both the unit `2 ball. We can use as perturbation the uniform distribution on the surface of unit sphere, as the following lemma shows. This result was hinted at in [2], as in high dimensional case, the random draw from the unit sphere is likely to produce orthogonal directions. However, we do not require dimensionality to be high for our result. Lemma 9. Let X and F be unit balls in Euclidean norm. Then Assumption 2 is satisﬁed with a uniform distribution D on the surface of the unit sphere with constant C = 4 √ 2. 6 As in the previous example the update in (20) is not in a convenient form and this is addressed below. Lemma 10. Let X and F be unit balls in Euclidean norm, and D be the uniform distribution on the surface of the unit sphere. Consider the randomized algorithm that at each round (say round t) freshly draws xt+1,...,xT ∼D and picks ft = −∑t−1 i=1 xi + C ∑T i=t+1 xiL where C = 4 √ 2 and scaling factor L = −∑t−1 i=1 xi + C ∑T i=t+1 ✏ixi 2 2 + 1 12 . The randomized algorithm enjoys a bound on the expected regret given by E[RegT ] ≤C Ex1,...,xT ∼D ∑T t=1 xt2 ≤4 √ 2T . Importantly, the bound does not depend on the dimensionality of the space. To the best of our knowledge, this is the ﬁrst such result for Follow the Perturbed Leader style algorithms. Further, unlike [10, 6], we directly deal with the adaptive adversary. 5 Static Experts with Convex Losses and Transductive Online Learning We show how to recover a variant of the R2 forecaster of [7], for static experts and transductive online learning. At each round, the learner makes a prediction qt ∈[−1,1], observes the outcome yt ∈[−1,1], and suffers convex L-Lipschitz loss `(qt,yt). Regret is deﬁned as the difference between learner’s cumulative loss and inff∈F ∑T t=1 `(f[t],yt), where F ⊂[−1,1]T can be seen as a set of static experts. The transductive setting is equivalent to this: the sequence of xt’s is known before the game starts, and hence the effective function class is once again a subset of [−1,1]T . It turns out that in these cases, sequential Rademacher complexity becomes the classical Rademacher complexity (see [17]), which can thus be taken as a relaxation. This is also the reason that an efﬁcient implementation by sampling is possible. For general convex loss, one possible admissible relaxation is just a conditional version of the classical Rademacher averages: RelT (Fy1, . . . , yt) = E✏t+1∶T sup f∈F 2L T s=t+1 ✏sf[s] −Lt(f) (21) where Lt(f) = ∑t s=1 `(f[s],ys). If (21) is used as a relaxation, the calculation of prediction ˆyt involves a supremum over f ∈F with (potentially nonlinear) loss functions of instances seen so far. In some cases this optimization might be hard and it might be preferable if the supremum only involves terms linear in f. To this end we start by noting that by convexity T t=1 `(ˆyt, yt) −inf f∈F T t=1 `(f(xt), yt) ≤ T t=1 @`(ˆyt, yt) ⋅ˆyt −inf f∈F T t=1 @`(ˆyt, yt) ⋅f[t] (22) One can now consider an alternative online learning problem which, if we solve, also solves the original problem. More precisely, the new loss is `′(ˆy,r) = r ⋅ˆy; we ﬁrst pick prediction ˆyt (deterministically) and the adversary picks rt (corresponding to rt = @`(ˆyt,yt) for choice of yt picked by adversary). Now note that `′ is indeed convex in its ﬁrst argument and is L Lipschitz because @`(ˆyt,yt)≤L. This is a one dimensional convex learning game where we pick ˆyt and regret is given by the right hand side of (22). Hence, we can consider the relaxation RelT (F@`(ˆy1, y1), . . . , @`(ˆyt, yt)) = E✏t+1∶T sup f∈F 2L T i=t+1 ✏if[t] − t i=1 @`(ˆyi, yi) ⋅f[i] (23) as a linearized form of (21). At round t, the prediction of the algorithm is then ˆyt = E ✏sup f∈F T i=t+1 ✏if[i] − 1 2L t−1 i=1 @`(ˆyi, yi)f[i] + 1 2f[t]−sup f∈F T i=t+1 ✏if[i] − 1 2L t−1 i=1 @`(ˆyi, yi)f[i] −1 2f[t] (24) Lemma 11. The relaxation in Eq. (23) is admissible w.r.t. the prediction strategy speciﬁed in Equation (24). Further the regret of the strategy is bounded as RegT ≤2L E✏supf∈F ∑T t=1 ✏tf[t]. This algorithm is similar to R2, with the main difference that R2 computes the inﬁma over a sum of absolute losses, while here we have a more manageable linearized objective. While we need to evaluate the expectation over ✏’s on each round, we can estimate ˆyt by sampling ✏’s and using McDiarmid’s inequality argue that the estimate is close to ˆyt with high probability. The randomized prediction is now given simply as: on round t, draw ✏t+1,...,✏T and predict ˆyt(✏) = inf f∈F − T i=t+1 ✏if[i] + 1 2L t−1 i=1 `(f[i], yi) + 1 2f[t]−inf f∈F − T i=t+1 ✏if[i] + 1 2L t−1 i=1 `(f[i], yi) −1 2f[t] (25) We now show that this predictor enjoys regret bound of the transductive Rademacher complexity : 7 Lemma 12. The relaxation speciﬁed in Equation (21) is admissible w.r.t. the randomized prediction strategy speciﬁed in Equation (25), and enjoys bound E[RegT ] ≤2L E✏supf∈F ∑T t=1 ✏tf[t]. 6 Matrix Completion Consider the problem of predicting unknown entries in a matrix, in an online fashion. At each round t the adversary picks an entry in an m × n matrix and a value yt for that entry. The learner then chooses a predicted value ˆyt, and suffers loss `(yt, ˆyt), assumed to be ⇢-Lipschitz. We deﬁne our regret with respect to the class F of all matrices whose trace-norm is at most B (namely, we can use any such matrix to predict just by returning its relevant entry at each round). Usually, one has B = ⇥(√mn). Consider a transductive version, where we know in advance the location of all entries we need to predict. We show how to develop an algorithm whose regret is bounded by the (transductive) Rademacher complexity of F, which by Theorem 6 of [18], is O(B√n) independent of T. Moreover, in [7], it was shown how one can convert algorithms with such guarantees to obtain the same regret even in a “fully” online case, where the set of entry locations is unknown in advance. In this section we use the two alternatives provided for transductive learning problem in the previous subsection, and provide two alternatives for the matrix completion problem. Both variants proposed here improve on the one provided by the R2 forecaster in [7], since that algorithm competes against the smaller class F′ of matrices with bounded trace-norm and bounded individual entries, and our variants are also computationally more efﬁcient. Our ﬁrst variant also improves on the recently proposed method in [9] in terms of memory requirements, and each iteration is simpler: Whereas that method requires storing and optimizing full m × n matrices every iteration, our algorithm only requires computing spectral norms of sparse matrices (assuming T mn, which is usually the case). This can be done very efﬁciently, e.g. with power iterations or the Lanczos method. Our ﬁrst algorithm follows from Eq. (24), which for our setting gives the following prediction rule: ˆyt = B E ✏ T i=t+1 ✏ixi − 1 2⇢ t−1 i=1 @`(ˆyi, yi)xi + 1 2xt σ − T i=t+1 ✏ixi − 1 2⇢ t−1 i=1 @`(ˆyi, yi)xi −1 2xt σ (26) In the above ⋅σ stands for the spectral norm and each xi is a matrix with a 1 at some speciﬁc position and 0 elsewhere. Notice that the algorithm only involves calculation of spectral norms on each round, which can be done efﬁciently. As mentioned in previous subsection, one can approximately evaluate the expectation by sampling several ✏’s on each round and averaging. The second algorithm follows (25), and is given by ﬁrst drawing ✏at random and then predicting ˆyt(✏)= sup f⌃≤B T i=t+1 ✏if[xi]− 1 2⇢ t−1 i=1 `(f[xi], yi)+ 1 2f[xt]−sup f⌃≤B T i=t+1 ✏if[xi]− 1 2⇢ t−1 i=1 `(f[xi], yi)−1 2f[xt] where f⌃is the trace norm of the m×n f, and f[xi] is the entry of the matrix f at the position xi. Notice that the above involves solving two trace norm constrained convex optimization problems per round. As a simple corollary of Lemma 12, together with the bound on the Rademacher complexity mentioned earlier, we get that the expected regret of either variant is O B ⇢(√m + √n). 7 Conclusion In [2, 1, 15, 20] the minimax value of the online learning game has been analyzed and nonconstructive bounds on the value have been provided. In this paper, we provide a general constructive recipe for deriving new (and old) online learning algorithms, using techniques from the apparently non-constructive minimax analysis. The recipe is rather simple: we start with the notion of conditional sequential Rademacher complexity, and ﬁnd an “admissible” relaxation which upper bounds it. This relaxation immediately leads to an online learning algorithm, as well as to an associated regret guarantee. In addition to the development of a uniﬁed algorithmic framework, our contributions include (1) a new algorithm for online binary classiﬁcation whenever the Littlestone dimension of the class is ﬁnite; (2) a family of randomized online learning algorithms based on the idea of a random playout, with new Follow the Perturbed Leader style algorithms arising as special cases; and (3) efﬁcient algorithms for trace norm based online matrix completion problem which improve over currently known methods. Acknowledgements: We gratefully acknowledge the support of NSF under grants CAREER DMS0954737 and CCF-1116928. 8 References [1] J. Abernethy, A. Agarwal, P. L. Bartlett, and A. Rakhlin. A stochastic view of optimal regret through minimax duality. In COLT, 2009. [2] J. Abernethy, P. L. Bartlett, A. Rakhlin, and A. Tewari. Optimal strategies and minimax lower bounds for online convex games. In COLT, 2008. [3] J. Abernethy, M.K. Warmuth, and J. Yellin. Optimal strategies from random walks. 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4,487	Inverse Reinforcement Learning through Structured Classiﬁcation Edouard Klein1,2 1LORIA – team ABC Nancy, France edouard.klein@supelec.fr Matthieu Geist2 2Supélec – IMS-MaLIS Research Group Metz, France matthieu.geist@supelec.fr Bilal Piot2,3, Olivier Pietquin2,3 3 UMI 2958 (GeorgiaTech-CNRS) Metz, France {bilal.piot,olivier.pietquin}@supelec.fr Abstract This paper adresses the inverse reinforcement learning (IRL) problem, that is inferring a reward for which a demonstrated expert behavior is optimal. We introduce a new algorithm, SCIRL, whose principle is to use the so-called feature expectation of the expert as the parameterization of the score function of a multiclass classiﬁer. This approach produces a reward function for which the expert policy is provably near-optimal. Contrary to most of existing IRL algorithms, SCIRL does not require solving the direct RL problem. Moreover, with an appropriate heuristic, it can succeed with only trajectories sampled according to the expert behavior. This is illustrated on a car driving simulator. 1 Introduction Inverse reinforcement learning (IRL) [14] consists in ﬁnding a reward function such that a demonstrated expert behavior is optimal. Many IRL algorithms (to be brieﬂy reviewed in Sec. 5) search for a reward function such that the associated optimal policy induces a distribution over trajectories (or some measure of this distribution) which matches the one induced by the expert. Often, this distribution is characterized by the so-called feature expectation (see Sec. 2.1): given a reward function linearly parameterized by some feature vector, it is the expected discounted cumulative feature vector for starting in a given state, applying a given action and following the related policy. In this paper, we take a different route. The expert behavior could be mimicked by a supervised learning algorithm generalizing the mapping from states to actions. Here, we consider generally multi-class classiﬁers which compute from a training set the parameters of a linearly parameterized score function; the decision rule for a given state is the argument (the action) which maximizes the score function for this state (see Sec. 2.2). The basic idea of our SCIRL (Structured Classiﬁcationbased IRL) algorithm is simply to take an estimate of the expert feature expectation as the parameterization of the score function (see Sec. 3.1). The computed parameter vector actually deﬁnes a reward function for which we show the expert policy to be near-optimal (Sec. 3.2). Contrary to most existing IRL algorithms, a clear advantage of SCIRL is that it does not require solving repeatedly the direct reinforcement learning (RL) problem. It requires estimating the expert feature expectation, but this is roughly a policy evaluation problem (for an observed policy, so is less involved than repeated policy optimization problems), see Sec. 4. Moreover, up to the use of some heuristic, SCIRL may be trained solely from transitions sampled from the expert policy (no need to sample the whole dynamic). We illustrate this on a car driving simulator in Sec. 6. 1 2 Background and Notations 2.1 (Inverse) Reinforcement Learning A Markov Decision process (MDP) [12] is a tuple {S, A, P, R, γ} where S is the ﬁnite state space1, A the ﬁnite actions space, P = {Pa = (p(s′\|s, a))1≤s,s′≤\|S\|, a ∈A} the set of Markovian transition probabilities, R ∈RS the state-dependent reward function and γ the discount factor. A deterministic policy π ∈SA deﬁnes the behavior of an agent. The quality of this control is quantiﬁed by the value function vπ R ∈RS, associating to each state the cumulative discounted reward for starting in this state and following the policy π afterwards: vπ R(s) = E[P t≥0 γtR(St)\|S0 = s, π]. An optimal policy π∗ R (according to the reward function R) is a policy of associated value function v∗ R satisfying v∗ R ≥vπ R, for any policy π and componentwise. Let Pπ be the stochastic matrix Pπ = (p(s′\|s, π(s)))1≤s,s′≤\|S\|. With a slight abuse of notation, we may write a the policy which associates the action a to each state s. The Bellman evaluation (resp. optimality) operators T π R (resp. T ∗ R) : RS →RS are deﬁned as T π Rv = R + γPπv and T ∗ Rv = maxπ T π Rv. These operators are contractions and vπ R and v∗ R are their respective ﬁxedpoints: vπ R = T π Rvπ R and v∗ R = T ∗ Rv∗ R. The action-value function Qπ ∈RS×A adds a degree of freedom on the choice of the ﬁrst action, it is formally deﬁned as Qπ R(s, a) = [T a Rvπ R](s). We also write ρπ the stationary distribution of the policy π (satisfying ρ⊤ π Pπ = ρ⊤ π ). Reinforcement learning and approximate dynamic programming aim at estimating the optimal control policy π∗ R when the model (transition probabilities and the reward function) is unknown (but observed through interactions with the system to be controlled) and when the state space is too large to allow exact representations of the objects of interest (as value functions or policies) [2, 15, 17]. We refer to this as the direct problem. On the contrary, (approximate) inverse reinforcement learning [11] aim at estimating a reward function for which an observed policy is (nearly) optimal. Let us call this policy the expert policy, denoted πE. We may assume that it optimizes some unknown reward function RE. The aim of IRL is to compute some reward ˆR such that the expert policy is (close to be) optimal, that is such that v∗ ˆ R ≈vπE ˆ R . We refer to this as the inverse problem. Similarly to the direct problem, the state space may be too large for the reward function to admit a practical exact representation. Therefore, we restrict our search of a good reward among linearly parameterized functions. Let φ(s) = (φ1(s) . . . φp(s))⊤be a feature vector composed of p basis function φi ∈RS, we deﬁne the parameterized reward functions as Rθ(s) = θ⊤φ(s) = Pp i=1 θiφi(s). Searching a good reward thus reduces to searching a good parameter vector θ ∈Rp. Notice that we will use interchangeably Rθ and θ as subscripts (e.g., vπ θ for vπ Rθ). Parameterizing the reward this way implies a related parameterization for the action-value function: Qπ θ (s, a) = θ⊤µπ(s, a) with µπ(s, a) = E[ X t≥0 γtφ(St)\|S0 = s, A0 = a, π]. (1) Therefore, the action-value function shares the parameter vector of the reward function, with an associated feature vector µπ called the feature expectation. This notion will be of primary importance for the contribution of this paper. Notice that each component µπ i of this feature vector is actually the action-value function of the policy π assuming the reward is φi: µπ i (s, a) = Qπ φi(s, a). Therefore, any algorithm designed for estimating an action-value function may be used to estimate the feature expectation, such as Monte-Carlo rollouts or temporal difference learning [7]. 2.2 Classiﬁcation with Linearly Parameterized Score Functions Let X be a compact or a ﬁnite set (of inputs to be classiﬁed) and let Y be a ﬁnite set (of labels). Assume that inputs x ∈X are drawn according to some unknown distribution P(x) and that there exists some oracle which associates to each of these inputs a label y ∈Y drawn according to the unknown conditional distribution P(y\|x). Generally speaking, the goal of multi-class classiﬁcation is, given a training set {(xi, yi)1≤i≤N} drawn according to P(x, y), to produce a decision rule g ∈YX which aims at minimizing the classiﬁcation error E[χ{g(x)̸=y}] = P(g(x) ̸= y), where χ denotes the indicator function. 1This work can be extended to compact state spaces, up to some technical aspects. 2 Here, we consider a more restrictive set of classiﬁcation algorithms. We assume that the decision rule associates to an input the argument which maximizes a related score function, this score function being linearly parameterized and the associated parameters being learnt by the algorithm. More formally, let ψ(s, a) = (ψ1(x, y) . . . ψd(x, y))⊤∈Rd be a feature vector whose components are d basis functions ψi ∈RX×Y. The linearly parameterized score function sw ∈RX×Y of parameter vector w ∈Rd is deﬁned as sw(x, y) = w⊤ψ(x, y). The associated decision rule gw ∈YX is deﬁned as gw(x) ∈argmaxy∈Y sw(x, y). Using a training set {(xi, yi)1≤i≤N}, a linearly parameterized score function-based multi-class classiﬁcation (MC2 for short) algorithm computes a parameter vector θc. The quality of the solution is quantiﬁed by the classiﬁcation error ϵc = P(gθc(x) ̸= y). We do not consider a speciﬁc MC2 algorithm, as long as it classiﬁes inputs by maximizing the argument of a linearly parameterized score function. For example, one may choose a multi-class support vector machine [6] (taking the kernel induced by the feature vector) or a structured large margin approach [18]. Other choices may be possible, one can choose its preferred algorithm. 3 Structured Classiﬁcation for Inverse Reinforcement Learning 3.1 General Algorithm Consider the classiﬁcation framework of Sec. 2.2. The input x may be seen as a state and the label y as an action. Then, the decision rule gw(x) can be interpreted as a policy which is greedy according to the score function w⊤ψ(x, y), which may itself be seen as an action-value function. Making the parallel with Eq. (1), if ψ(x, y) is the feature expectation of some policy π which produces labels of the training set, and if the classiﬁcation error is small, then w will be the parameter vector of a reward function for which we may hope the policy π to be near optimal. Based on these remarks, we’re ready to present the proposed Structured Classiﬁcation-based IRL (SCIRL) algorithm. Let πE be the expert policy from which we would like to recover a reward function. Assume that we have a training set D = {(si, ai = πE(si))1≤i≤N} where states are sampled according to the expert stationary distribution2 ρE = ρπE. Assume also that we have an estimate ˆµπE of the expert feature expectation µπE deﬁned in Eq. (1). How to practically estimate this quantity is postponed to Sec. 4.1; however, recall that estimating µπE is simply a policy evaluation problem (estimating the action-value function of a given policy), as noted in Sec. 2.1. Assume also that an MC2 algorithm has been chosen. The proposed algorithm simply consists in choosing θ⊤ˆµπE(s, a) as the linearly parameterized score function, training the classiﬁer on D which produces a parameter vector θc, and outputting the reward function Rθc(s) = θ⊤ c φ(s). Algorithm 1: SCIRL algorithm Given a training set D = {(si, ai = πE(si))1≤i≤N}, an estimate ˆµπE of the expert feature expectation µπE and an MC2 algorithm; Compute the parameter vector θc using the MC2 algorithm fed with the training set D and considering the parameterized score function θ⊤ˆµπE(s, a); Output the reward function Rθc(s) = θ⊤ c φ(s) ; The proposed approach is summarized in Alg. 1. We call this Structured Classiﬁcation-based IRL because using the (estimated) expert feature expectation as the feature vector for the classiﬁer somehow implies taking into account the MDP structure into the classiﬁcation problem and allows outputting a reward vector. Notice that contrary to most of existing IRL algorithms, SCIRL does not require solving the direct problem. If it possibly requires estimating the expert feature expectation, it is just a policy evaluation problem, less difﬁcult than the policy optimization issue involved by the direct problem. This is further discussed in Sec. 5. 2For example, if the Markov chain induced by the expert policy is fast-mixing, sampling a trajectory will quickly lead to sample states according to this distribution. 3 3.2 Analysis In this section, we show that the expert policy πE is close to be optimal according to the reward function Rθc, more precisely that Es∼ρE[v∗ θc(s) −vπE θc (s)] is small. Before stating our main result, we need to introduce some notations and to deﬁne some objects. We will use the ﬁrst order discounted future state distribution concentration coefﬁcient Cf [9]: Cf = (1 −γ) X t≥0 γtc(t) with c(t) = max π1,...,πt,s∈S (ρ⊤ EPπ1 . . . Pπt)(s) ρE(s) . We note πc the decision rule of the classiﬁer: πc(s) ∈argmaxa∈A θ⊤ c ˆµπE(s, a). The classiﬁcation error is therefore ϵc = Es∼ρE[χ{πc(s)̸=πE(s)}] ∈[0, 1]. We write ˆQπE θc = θ⊤ c ˆµπE the score function computed from the training set D (which can be interpreted as an approximate action-value function). Let also ϵµ = ˆµπE −µπE : S × A →Rp be the feature expectation error. Consequently, we deﬁne the action-value function error as ϵQ = ˆQπE θc −QπE θc = θ⊤ c (ˆµπE −µπE) = θ⊤ c ϵµ : S × A →R. We ﬁnally deﬁne the mean delta-max action-value function error as ¯ϵQ = Es∼ρE[maxa∈A ϵQ(s, a) −mina∈A ϵQ(s, a)] ≥0. Theorem 1. Let Rθc be the reward function outputted by Alg. 1. Let also the quantities Cf, ϵc and ¯ϵQ be deﬁned as above. We have 0 ≤Es∼ρE[v∗ Rθc −vπE Rθc ] ≤ Cf 1 −γ ¯ϵQ + ϵc 2γ∥Rθc∥∞ 1 −γ . Proof. As the proof only relies on the reward Rθc, we omit the related subscripts to keep the notations simple (e.g., vπ for vπ θc = vπ Rθc or R for Rθc). First, we link the error Es∼ρE[v∗(s) −vπE(s)] to the Bellman residual Es∼ρE[[T ∗vπE](s) −vπE(s)]. Componentwise, we have that: v∗−vπE = T ∗v∗−T π∗vπE + T π∗vπE −T ∗vπE + T ∗vπE −vπE (a) ≤γPπ∗(v∗−vπE) + T ∗vπE −vπE (b) ≤(I −γPπ∗)−1(T ∗vπE −vπE). Inequality (a) holds because T π∗vπE ≤T ∗vπE and inequality (b) holds thanks to [9, Lemma 4.2]. Moreover, v∗being optimal we have that v∗−vπE ≥0 and T ∗being the Bellman optimality operator, we have T ∗vπE ≥T πEvπE = vπE. Additionally, remark that (I −γPπ∗)−1 = P t≥0 γtP t π∗. Therefore, using the deﬁnition of the concentration coefﬁcient Cf, we have that: 0 ≤Es∼ρE[v∗(s) −vπE(s)] ≤ Cf 1 −γ Es∼ρE [[T ∗vπE](s) −vπE(s)] . (2) This results actually follows closely the one of [9, Theorem 4.2]. There remains to bound the Bellman residual Es∼ρE[[T ∗vπE](s) −vπE(s)]. Considering the following decomposition, T ∗vπE −vπE = T ∗vπE −T πcvπE + T πcvπE −vπE, we will bound Es∼ρE[[T ∗vπE](s) −[T πcvπE](s)] and Es∼ρE[[T πcvπE](s) −vπE(s)]. The policy πc (the decision rule of the classiﬁer) is greedy with respect to ˆQπE = θ⊤ c ˆµπE. Therefore, for any state-action couple (s, a) ∈S × A we have: ˆQπE(s, πc(s)) ≥ˆQπE(s, a) ⇔QπE(s, a) ≤QπE(s, πc(s)) + ϵQ(s, πc(s)) −ϵQ(s, a). By deﬁnition, QπE(s, a) = [T avπE](s) and QπE(s, πc(s)) = [T πcvπE](s). Therefore, for s ∈S: ∀a ∈A, [T avπE](s) ≤[T πcvπE](s) + ϵQ(s, πc(s)) −ϵQ(s, a) ⇒[T ∗vπE](s) ≤[T πcvπE](s) + max a∈A ϵQ(s, a) −min a∈A ϵQ(s, a). Taking the expectation according to ρE and noticing that T ∗vπE ≥vπE, we bound the ﬁrst term: 0 ≤Es∼ρE [[T ∗vπE](s) −[T πcvπE](s)] ≤¯ϵQ. (3) There ﬁnally remains to bound the term Es∼ρE[[T πcvπE](s) −vπE(s)]. 4 Let us write M ∈R\|S\|×\|S\| the diagonal matrix deﬁned as M = diag(χ{πc(s)̸=πE(s)}). Using this, the Bellman operator T πc may be written as, for any v ∈RS: T πcv = R + γMPπcv + γ(I −M)PπEv = R + γPπEv + γM(Pπc −PπE)v. Applying this operator to vπE and recalling that R + γPπEvπE = T πEvπE = vπE, we get: T πcvπE −vπE = γM(Pπc −PπE)vπE ⇒\|ρ⊤ E(T πcvπE −vπE)\| = γ\|ρ⊤ EM(Pπc −PπE)vπE\|. One can easily see that ∥(Pπc −PπE)vπE∥∞≤ 2 1−γ ∥R∥∞, which allows bounding the last term: \|Es∼ρE[[T πcvπE](s) −vπE(s)]\| ≤ϵc 2γ 1 −γ ∥R∥∞. (4) Injecting bounds of Eqs. (3) and (4) into Eq. (2) gives the stated result. This result shows that if the expert feature expectation is well estimated (in the sense that the estimation error ϵµ is small for states sampled according to the expert stationary policy and for all actions) and if the classiﬁcation error ϵc is small, then the proposed generic algorithm outputs a reward function Rθc for which the expert policy will be near optimal. A direct corollary of Th. 1 is that given the true expert feature expectation µπE and a perfect classiﬁer (ϵc = 0), πE is the unique optimal policy for Rθc. One may argue that this bounds trivially holds for the null reward function (a reward often exhibited to show that IRL is an ill-posed problem), obtained if θc = 0. However, recall that the parameter vector θc is computed by the classiﬁer. With θc = 0, the decision rule would be a random policy and we would have ϵc = \|A\|−1 \|A\| , the worst possible classiﬁcation error. This case is really unlikely. Therefore, we advocate that the proposed approach somehow allows disambiguating the IRL problem (at least, it does not output trivial reward functions such as the null vector). Also, this bound is scale-invariant: one could impose ∥θc∥= 1 or normalize (action-) value functions by ∥Rθc∥−1 ∞. One should notice that there is a hidden dependency of the classiﬁcation error ϵc to the estimated expert feature expectation ˆµπE. Indeed, the minimum classiﬁcation error depends on the hypothesis space spanned by the chosen score function basis functions for the MC2 algorithm (here ˆµπE). Nevertheless, provided a good representation for the reward function (that is a good choice of basis functions φi) and a small estimation error, this should not be a practical problem. Finally, if our bound relies on the generalization errors ϵc and ¯ϵQ, the classiﬁer will only use (ˆµπE(si, a))1≤i≤N,a∈A in the training phase, where si are the states from the set D. It outputs θc, seen as a reward function, thus the estimated feature expectation ˆµπE is no longer required. Therefore, practically it should be sufﬁcient to estimate well ˆµπE on state-action couples (si, a)1≤i≤N,a∈A, which allows envisioning Monte-Carlo rollouts for example. 4 A Practical Approach 4.1 Estimating the Expert Feature Expectation SCIRL relies on an estimate ˆµπE of the expert feature expectation. Basically, this is a policy evaluation problem. An already made key observation is that each component of µπE is the action-value function of πE for a reward function φi: µπE i (s, a) = QπE φi (s, a) = [T a φivπE φi ](s). We brieﬂy review its exact computation and possible estimation approaches, and consider possible heuristics. If the model is known, the feature expectation can be computed explicitly. Let Φ ∈R\|S\|×p be the feature matrix whose rows contain the feature vectors φ(s)⊤for all s ∈S. For a ﬁxed a ∈A, let µπE a ∈R\|S\|×p be the feature expectation matrix whose rows are the expert feature vectors, that is (µπE(s, a))⊤for any s ∈S. With these notations, we have µπE a = Φ + γPa(I −γPπE)−1Φ. Moreover, the related computational cost is the same order of magnitude as evaluating a single policy (as the costly part, computing (I −γPπE)−1, is shared by all components). If the model is unknown, any temporal difference learning algorithm can be used to estimate the expert feature expectation [7], as LSTD (Least-Squares Temporal Differences) [4]. Let ψ : S×A → Rd be a feature vector composed of d basis functions ψi ∈RS×A. Each component µπE i of the 5 expert feature expectation is parameterized by a vector ξi ∈Rd: µπE i (s, a) ≈ξ⊤ i ψ(s, a). Assume that we have a training set {(si, ai, s′ i, a′ i = πE(s′ i))1≤i≤M} with actions ai not necessarily sampled according to policy πE (e.g., this may be obtained by sampling trajectories according to an expertbased ϵ-greedy policy), the aim being to have a better variability of tuples (non-expert actions should be tried). Let ˜Ψ ∈RM×d (resp. ˜Ψ′) be the feature matrix whose rows are the feature vectors ψ(si, ai)⊤(resp. ψ(s′ i, a′ i)⊤). Let also ˜Φ ∈RM×p be the feature matrix whose rows are the reward’s feature vectors φ(si)⊤. Finally, let Ξ = [ξ1 . . . ξp] ∈Rd×p be the matrix of all parameter vectors. Applying LSTD to each component of the feature expectation gives the LSTD-µ algorithm [7]: Ξ = (˜Ψ⊤(˜Ψ −γ ˜Ψ′))−1 ˜Ψ⊤˜Φ and ˆµπE(s, a) = Ξ⊤ψ(s, a). As for the exact case, the costly part (computing the inverse matrix) is shared by all feature expectation components, the computational cost is reasonable (same order as LSTD). Provided a simulator and the ability to sample according to the expert policy, the expert feature expectation may also be estimated using Monte-Carlo rollouts for a given state-action pair (as noted in Sec. 3.2, ˆµπE need only be known on (si, a)1≤i≤N,a∈A). Assuming that K trajectories are sampled for each required state-action pair, this method would require KN\|A\| rollouts. In order to have a small error ¯ϵQ, one may learn using transitions whose starting state is sampled according to ρE and whose actions are uniformly distributed. However, it may happen that only transitions of the expert are available: T = {(si, ai = πE(si), s′ i)1≤i≤N}. If the state-action couples (si, ai) may be used to feed the classiﬁer, the transitions (si, ai, s′ i) are not enough to provide an accurate estimate of the feature expectation. In this case, we can still expect an accurate estimate of µπE(s, πE(s)), but there is little hope for µπE(s, a ̸= πE(s)). However, one can still rely on some heuristic; this does not ﬁt the analysis of Sec. 3.2, but it can still provide good experimental results, as illustrated in Sec. 6. We propose such a heuristic. Assume that only data T is available and that we use it to provide an (accurate) estimate ˆµπE(s, πE(s)) (this basically means estimating a value function instead of an action-value function as described above). We may adopt an optimistic point of view by assuming that applying a non-expert action just delays the effect of the expert action. More formally, we associate to each state s a virtual state sv for which p(.\|sv, a) = p(.\|s, πE(s)) for any action a and for which the reward feature expectation is the null vector, φ(sv) = 0. In this case, we have µπE(s, a ̸= πE(s)) = γµπE(s, πE(s)). Applying this idea to the available estimate (recalling that the classiﬁers only requires evaluating ˆµπE on (si, a)1≤i≤N,a∈A) provides the proposed heuristic: for 1 ≤i ≤N, ˆµπE(si, a ̸= ai) = γˆµπE(si, ai). We may even push this idea further, to get the simpler estimate of the expert feature expectation (but with the weakest guarantees). Assume that the set T consists of one long trajectory, that is s′ i = si+1 (thus T = {s1, a1, s2, . . . , sN−1, aN−1, sN, aN}). We may estimate µπE(si, ai) using the single rollout available in the training set and use the proposed heuristic for other actions: ∀1 ≤i ≤N, ˆµπE(si, ai) = N X j=i γj−iφ(sj) and ˆµπE(si, a ̸= ai) = γˆµπE(si, ai). (5) To sum up, the expert feature expectation may be seen as a vector of action-value functions (for the same policy πE and different reward functions φi). Consequently, any action-value function evaluation algorithm may be used to estimate µπ(s, a). Depending on the available data, one may have to rely on some heuristic to assess the feature expectation for a unexperienced (non-expert) action. Also, this expert feature expectation estimate is only required for training the classiﬁer, so it is sufﬁcient to estimate on state-action couples (si, a)1≤i≤N,a∈A. In any case, estimating µπE is not harder than estimating the action-value function of a given policy in the on-policy case, which is much easier than computing an optimal policy for an arbitrary reward function (as required by most of existing IRL algorithms, see Sec. 5). 4.2 An Instantiation As stated before, any MC2 algorithm may be used. Here, we choose the structured large margin approach [18]. Let L : S × A →R+ be a user-deﬁned margin function satisfying L(s, πE(s)) ≤ 6 L(s, a) (here, L(si, ai) = 0 and L(si, a ̸= ai) = 1). The MC2 algorithm solves: min θ,ζ 1 2∥θ∥2 + η N N X i=1 ζi s.t. ∀i, θ⊤ˆµπE(si, ai) + ζi ≥max a θ⊤ˆµπE(si, a) + L(si, a). Following [13], we express the equivalent hinge-loss form (noting that the slack variables ζi are tight, which allows moving the constraints in the objective function): J(θ) = 1 N N X i=1 max a θ⊤ˆµπE(si, a) + L(si, a) −θ⊤ˆµπE(si, ai) + λ 2 ∥θ∥2. This objective function is minimized using a subgradient descent. The expert feature expectation is estimated using the scheme described in Eq. (5). 5 Related Works The notion of IRL has ﬁrst been introduced in [14] and ﬁrst been formalized in [11]. A classic approach to IRL, initiated in [1], consists in ﬁnding a policy (through some reward function) such that its feature expectation (or more generally some measure of the underlying trajectories’ distribution) matches the one of the expert policy. See [10] for a review. Notice that related algorithms are not always able to output a reward function, even if they may make use of IRL as an intermediate step. In such case, they are usually refereed to as apprenticeship learning algorithms. Closer to our contribution, some approaches also somehow introduce a structure in a classiﬁcation procedure [8][13]. In [8], a metric induced by the MDP is used to build a kernel which is used in a classiﬁcation algorithm, showing improvements compared to a non-structured kernel. However, this approach is not an IRL algorithm, and more important assessing the metric of an MDP is a quite involved problem. In [13], a classiﬁcation algorithm is also used to produce a reward function. However, instead of associating actions to states, as we do, it associates optimal policies (labels) to MDPs (inputs), which is how the structure is incorporated. This involves solving many MDPs. As far as we know, all IRL algorithms require solving the direct RL problem repeatedly, except [5, 3]. [5] applies to linearly-solvable MDPs (where the control is done by imposing any dynamic to the system). In [3], based on a relative entropy argument, some utility function is maximized using a subgradient ascent. Estimating the subgradient requires sampling trajectories according to the policy being optimal for the current estimated reward. This is avoided thanks to the use of importance sampling. Still, this requires sampling trajectories according to a non-expert policy and the direct problem remains at the core of the approach (even if solving it is avoided). SCIRL does not require solving the direct problem, just estimating the feature expectation of the expert policy. In other words, instead of solving multiple policy optimization problems, we only solve one policy evaluation problem. This comes with theoretical guarantees (which is not the case of all IRL algorithms, e.g. [3]). Moreover, using heuristics which go beyond our analysis, SCIRL may rely solely on data provided by expert trajectories. We demonstrate this empirically in the next section. To the best of our knowledge, no other IRL algorithm can work in such a restrictive case. 6 Experiments We illustrate the proposed approach on a car driving simulator, similar to [1, 16]. The goal si to drive a car on a busy three-lane highway with randomly generated trafﬁc (driving off-road is allowed on both sides). The car can move left and right, accelerate, decelerate and keep a constant speed. The expert optimizes a handcrafted reward RE which favours speed, punish off-road, punish collisions even more and is neutral otherwise. We compare SCIRL as instantiated in Sec. 4.2 to the unstructured classiﬁer (using the same classiﬁcation algorithm) and to the algorithm of [1] (called here PIRL for Projection IRL). We also consider the optimal behavior according to a randomly sampled reward function as a baseline (using the same reward feature vector as SCIRL and PIRL, the associated parameter vector is randomly sampled). For SCIRL and PIRL we use a discretization of the state space as the reward feature vector, φ ∈ R729: 9 horizontal positions for the user’s car, 3 horizontal and 9 vertical positions for the closest 7 50 100 150 200 250 300 350 400 Number of samples from the expert −4 −2 0 2 4 6 8 10 Es∼U[V π RE(s)] 50 100 150 200 250 300 350 400 Number of samples from the expert −4 −2 0 2 4 6 8 10 Es∼U[V π RE(s)] Figure 1: Highway problem. The highest line is the expert value. For each curves, we show the mean (plain line), the standard deviation (dark color) and the min-max values (light color). The policy corresponding to the random reward is in blue, the policy outputted by the classiﬁer is in yellow and the optimal policy according the SCIRL’s reward is in red. PIRL is the dark blue line. trafﬁc’s car and 3 speeds. Notice that these features are much less informative than the ones used in [1, 16]. Actually, in [16] features are so informative that sampling a random positive parameter vector θ already gives an acceptable behavior. The discount factor is γ = 0.9. The classiﬁer uses the same feature vector reproduced for each action. SCIRL is fed with n trajectories of length n (started in a random state) with n varying from 3 to 20 (so fed with 9 to 400 transitions). Each experiment is repeated 50 times. The classiﬁer uses the same data. PIRL is an iterative algorithm, each iteration requiring to solve the MDP for some reward function. It is run for 70 iterations, all required objects (a feature expectations for a non-expert policy and an optimal policy according to some reward function at each iteration) are computed exactly using the model. We measure the performance of each approach with Es∼U[vπ RE(s)], where U is the uniform distribution (this allows measuring the generalization capability of each approach for states infrequently encountered), RE is the expert reward and π is one of the following polices: the optimal policy for RE (upper baseline), the optimal policy for a random reward (lower baseline), the optimal policy for Rθc (SCIRL), the policy produced by PIRL and the classiﬁer decision rule. Fig. 1 shows the performance of each approach as a number of used expert transitions (except PIRL which uses the model). We can see that the classiﬁer does not work well on this example. Increasing the number of samples would improve its performance, but after 400 transitions it does not work as well as SCIRL with only a ten of transitions. SCIRL works pretty well here: after only a hundred of transitions it reaches the performance of PIRL, both being close to the expert value. We do not report exact computational times, but running SCIRL one time with 400 transitions is approximately hundred time faster than running PIRL for 70 iteration. 7 Conclusion We have introduced a new way to perform IRL by structuring a linearly parameterized score function-based multi-class classiﬁcation algorithm with an estimate of the expert feature expectation. This outputs a reward function for which we have shown the expert to be near optimal, provided a small classiﬁcation error and a good expert feature expectation estimate. How to practically estimate this quantity has been discussed and we have introduced a heuristic for the case where only transitions from the expert are available, along with a speciﬁc instantiation of the SCIRL algorithm. We have shown on a car driving simulator benchmark that the proposed approach works well (even combined with the introduced heuristic), much better than the unstructured classiﬁer and as well as a state-of-the-art algorithm making use of the model (and with a much lower computational time). In the future, we plan to deepen the theoretical properties of SCIRL (notably regarding possible heuristics) and to apply it to real-world robotic problems. Acknowledgments. This research was partly funded by the EU FP7 project ILHAIRE (grant n◦270780), by the EU INTERREG IVa project ALLEGRO and by the Région Lorraine (France). 8 References [1] Pieter Abbeel and Andrew Y. Ng. Apprenticeship learning via inverse reinforcement learning. In Proceedings of the 21st International Conference on Machine learning (ICML), 2004. [2] Dimitri P. Bertsekas and John N. Tsitsiklis. Neuro-Dynamic Programming (Optimization and Neural Computation Series, 3). Athena Scientiﬁc, 1996. 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4,488	To appear in: Neural Information Processing Systems (NIPS), Lake Tahoe, Nevada. December 3-6, 2012. Efﬁcient and direct estimation of a neural subunit model for sensory coding Brett Vintch Andrew D. Zaharia J. Anthony Movshon Eero P. Simoncelli † Center for Neural Science, and †Howard Hughes Medical Institute New York University New York, NY 10003 vintch@cns.nyu.edu Abstract Many visual and auditory neurons have response properties that are well explained by pooling the rectiﬁed responses of a set of spatially shifted linear ﬁlters. These ﬁlters cannot be estimated using spike-triggered averaging (STA). Subspace methods such as spike-triggered covariance (STC) can recover multiple ﬁlters, but require substantial amounts of data, and recover an orthogonal basis for the subspace in which the ﬁlters reside rather than the ﬁlters themselves. Here, we assume a linear-nonlinear–linear-nonlinear (LN-LN) cascade model in which the ﬁrst linear stage is a set of shifted (‘convolutional’) copies of a common ﬁlter, and the ﬁrst nonlinear stage consists of rectifying scalar nonlinearities that are identical for all ﬁlter outputs. We refer to these initial LN elements as the ‘subunits’ of the receptive ﬁeld. The second linear stage then computes a weighted sum of the responses of the rectiﬁed subunits. We present a method for directly ﬁtting this model to spike data, and apply it to both simulated and real neuronal data from primate V1. The subunit model signiﬁcantly outperforms STA and STC in terms of cross-validated accuracy and efﬁciency. 1 Introduction Advances in sensory neuroscience rely on the development of testable functional models for the encoding of sensory stimuli in neural responses. Such models require procedures for ﬁtting their parameters to data, and should be interpretable in terms both of sensory function and of the biological elements from which they are made. The most common models in the visual and auditory literature are based on linear-nonlinear (LN) cascades, in which a linear stage serves to project the highdimensional stimulus down to a one-dimensional signal, where it is then nonlinearly transformed to drive spiking. LN models are readily ﬁt to data, and their linear operators specify the stimulus selectivity and invariance of the cell. The weights of the linear stage may be loosely interpreted as representing the efﬁcacy of synapses, and the nonlinearity as a transformation from membrane potential to ﬁring rate. For many visual and auditory neurons, responses are not well described by projection onto a single linear ﬁlter, but instead reﬂect a combination of several ﬁlters. In the cat retina, the responses of Y cells have been described by linear pooling of shifted rectiﬁed linear ﬁlters, dubbed “subunits” [1, 2]. Similar behaviors are seen in guinea pig [3] and monkey retina [4]. In the auditory nerve, responses are described as computing the envelope of the temporally ﬁltered sound waveform, which can be computed via summation of squared quadrature ﬁlter responses [5]. In primary visual cortex (V1), simple cells are well described using LN models [6, 7], but complex cell responses are more like a 1 superposition of multiple spatially shifted simple cells [8], each with the same orientation and spatial frequency preference [9]. Although the description of complex cells is often reduced to a sum of two squared ﬁlters in quadrature [10], more recent experiments indicate that these cells (and indeed most ’simple’ cells) require multiple shifted ﬁlters to fully capture their responses [11, 12, 13]. Intermediate nonlinearities are also required to describing the response properties of some neurons in V2 to stimuli (e.g., angles [14] and depth edges [15]). Each of these examples is consistent with a canonical but constrained LN-LN model, in which the ﬁrst linear stage consists of convolution with one (or a few) ﬁlters, and the ﬁrst nonlinear stage is point-wise and rectifying. The second linear stage then pools the responses of these “subunits” using a weighted sum, and the ﬁnal nonlinearity converts this to a ﬁring rate. Hierarchical stacks of this type of “generalized complex cell” model have also been proposed for machine vision [16, 17]. What is lacking is a method for validating this model by ﬁtting it directly to spike data. A widely used procedure for ﬁtting a simple LN model to neural data is reverse correlation [18, 19]. The spike-triggered average of a set of Gaussian white noise stimuli provides an unbiased estimate of the linear kernel. In a subunit model, the initial linear stage projects the stimulus into a multidimensional subspace, which can be estimated using spike-triggered covariance (STC) [20, 21]. This has been used successfully for ﬂy motion neurons [22], vertebrate retina [23], and primary visual cortex [24, 11]. But this method relies on a Gaussian stimulus ensemble, requires a substantial amount of data, and recovers only a set of orthogonal axes for the response subspace—not the underlying biological ﬁlters. More general methods based on information maximization alleviate some of the stimulus restrictions [25] but strongly limit the dimensionality of the recoverable subspace and still produce only a basis for the subspace. Here, we develop a speciﬁc subunit model and a maximum likelihood procedure to estimate its parameters from spiking data. We ﬁt the model to both simulated and real V1 neuronal data, demonstrating that it is substantially more accurate for a given amount of data than the current state-of-theart V1 model which is based on STC [11], and that it produces biologically interpretable ﬁlters. 2 Subunit model We assume that neural responses arise from a weighted sum of the responses of a set of nonlinear subunits. Each subunit applies a linear ﬁlter to its input (which can be either the raw stimulus, or the responses arising from a previous stage in a hierarchical cascade), and transforms the ﬁltered response using a memoryless rectifying nonlinearity. A critical simpliﬁcation is that the subunit ﬁlters are related by a ﬁxed transformation; here, we assume they are spatially translated copies of a common ﬁlter, and thus the population of subunits can be viewed as computing a convolution. For example, the subunits of a V1 complex cell could be simple cells in V1 that share the same orientation and spatial frequency preference, but differ in spatial location, as originally proposed by Hubel & Wiesel [8, 9]. We also assume that all subunits use the same rectifying nonlinearity. The response to input deﬁned over two discrete spatial dimensions and time, x(i, j, t), is written as: ˆr(t) = X m,n wm,n f⇥ 0 @X i,j,⌧ k(m, n, ⌧)· x(i −m, j −n, t −⌧) 1 A + . . . + b, (1) where k is the subunit ﬁlter, f⇥is a point-wise function parameterized by vector ⇥, wn,m are the spatial weights, and b is an additive baseline. The ellipsis indicates that we allow for multiple subunit channels, each with its own ﬁlter, nonlinearity, and pooling weights. We interpret ˆr(t) as a ‘generator potential’, (e.g., time-varying membrane voltage) which is converted to a ﬁring rate by another rectifying nonlinearity. The subunit model of Eq. (1) may be seen as a speciﬁc instance of a subspace model, in which the input is initially projected onto a linear subspace. Bialek and colleagues introduced spike-triggered covariance as a means of recovering such subspaces [20, 22]. Speciﬁcally, eigenvector analysis of the covariance matrix of the spike-triggered input ensemble exposes orthogonal axes for which the spike-triggered ensemble has a variance that differs signiﬁcantly from that of the raw input ensemble. These axes may be separated into those along which variance is greater (excitatory) or less than (suppressive) that of the input. Figure 1 demonstrates what happens when STC is applied to a simulated complex cell with 15 spatially shifted subunits. The response of this model cell is 2 a) b) c) // 1 0 // 1 0 λ1-2 eigenvalues eigenvectors 10 4 data points 10 6 data points shifted filter manifold STC plane position envelope λ1-4 Figure 1: Spike-triggered covariance analysis of a simulated V1 complex cell. (a) The model output is formed by summing the rectiﬁed responses of multiple linear ﬁlter kernels which are shifted and scaled copies of a canonical form. (b) The shifted ﬁlters lie along a manifold in stimulus space (four shown), and are not mutually orthogonal in general. STC recovers an orthogonal basis for a low-dimensional subspace that contains this manifold by ﬁnding the directions in stimulus space along which spikes are elicited or suppressed. (c) STC analysis of this model cell returns a variable number of ﬁlters dependent upon the amount of acquired data. A modest amount of data typically reveals two strong STC eigenvalues (top), whose eigenvectors form a quadrature (90-degree phaseshifted) pair and span the best-ﬁtting plane for the set of shifted model ﬁlters. These will generally have tuning properties (orientation, spatial frequency) similar to the true model ﬁlters. However, the manifold does not generally lie in a two-dimensional subspace [26], and a larger data set reveals additional eigenvectors (bottom) that serve to capture the deviations from the ~e1,2 plane. Due to the constraint of mutual orthogonality, these ﬁlters are usually not localized and they have tuning properties that differ from true model ﬁlters. ˆr(t) = P i wib(~ki· ~x(t))c2, where the ~k’s are shifted ﬁlters, w weights ﬁlters by position, and ~x is the stimulus vector. The recovered STC axes span the same subspace as the shifted model ﬁlters, but there are fewer of them, and the enforced orthogonality of eigenvectors means that they are generally not a direct match to any of the model ﬁlters. This has also been observed in ﬁlters extracted from physiological data [11, 12]. Although one may follow the STC analysis by indirectly identifying a localized ﬁlter whose shifted copies span the recovered subspace [11, 13], the reliance on STC still imposes the stimulus limitations and data requirements mentioned above. 3 Direct subunit model estimation A generic subspace method like STC does not exploit the speciﬁc structure of the subunit model. We therefore developed an estimation procedure explicitly tailored for this type of computation. We ﬁrst introduce a piecewise-linear parameterization of the subunit nonlinearity: f(s) = X l ↵lTl(s), (2) where the ↵’s scale a small set of overlapping ‘tent’ functions, Tl(·), that represent localized portions of f(·) (we ﬁnd that a dozen or so basis functions are typically sufﬁcient to provide the needed ﬂexibility). Incorporating this into the model response of Eq. (1) allows us to fold the second linear pooling stage and the subunit nonlinearity into a single sum: ˆr(t) = X m,n,l wm,n↵l Tl 0 @X i,j,⌧ k(m, n, ⌧)· x(i −m, j −n, t −⌧) 1 A + ... + b. (3) The model is now partitioned into two linear stages, separated by the ﬁxed nonlinear functions Tl(·). In the ﬁrst, the stimulus is convolved with k, and in the second, the nonlinear responses are summed with a set of weights that are separable in the indices l and n, m. The partition motivates the use of an iterative coordinate descent scheme: the linear weights of each portion are optimized in alternation, 3 while the other portion is held constant. For each step, we minimized the mean square error between the observed ﬁring rate of a cell and the ﬁring rate predicted by the model. For models that include two subunit channels we optimize over both channels simultaneously (see section 3.3 for comments regarding two-channel initialization). 3.1 Estimating the convolutional subunit kernel The ﬁrst coordinate descent leg optimizes the convolutional subunit kernel, k, using gradient descent while ﬁxing the subunit nonlinearity and the ﬁnal linear pooling. Because the tent basis functions are ﬁxed and piecewise linear, the gradient is easily determined. This property also ensures that the descent is locally convex: assuming that updating k does not cause any of the the linear subunit responses to jump between the localized tent functions representing f, then the optimization is linear and the objective function is quadratic. In practice, the full gradient descent path causes the linear subunit responses to move slowly across bins of the piecewise nonlinearity. However, we include a regularization term to impose smoothness on the nonlinearity (see below) and this yields a wellbehaved minimization problem for k. 3.2 Estimating the subunit nonlinearities and linear subunit pooling The second leg of coordinate descent optimizes the subunit nonlinearity (more speciﬁcally, the weights on the tent functions, ↵l), and the subunit pooling, wn,m. As described above, the objective is bilinear in ↵l and wn,m when k is ﬁxed. Estimating both ↵l and wn,m can be accomplished with alternating least-squares, which assures convergence to a (local) minimum [27]. We also include two regularization terms in the objective function. The ﬁrst ensures smoothness in the nonlinearity f, by penalizing the square of the second derivative of the function in the least-squares ﬁt. This smooth nonlinearity helps to guarantee that the optimization of k is well behaved, even where ﬁnite data sets leave the function poorly constrained. We also include a cross-validated ridge prior for the pooling weights to bias wn,m toward zero. The ﬁlter kernel k can also be regularized to ensure smoothness, but for the examples shown here we did not ﬁnd the need to include such a term. 3.3 Model initialization Our objective function is non-convex and contains local minima, so the selection of initial parameter values may affect the solution. We found that initializing our two-channel subunit model to have a positive pooling function for one channel and a negative pooling function for the second channel allowed the optimization of the second channel to proceed much more quickly. This is probably due in part to a suppressive channel that is much weaker than the excitatory channel in general. We initialized the nonlinearity to halfwave-rectiﬁcation for the excitatory channel and fullwaverectiﬁcation for the suppressive channel. To initialize the convolutional ﬁlter we use a novel technique that we term ‘convolutional STC’. The subunit model describes a receptive ﬁeld as the linear combination of nonlinear kernel responses that spatially tile the stimulus. Thus, the contribution of each localized patch of stimulus (of a size equal to the subunit kernel) is the same, up to a scale factor set by the weighting used in the subsequent pooling stage. As such, we compute an STC analysis on the union of all localized patches of stimuli. For each subunit location, {m, n}, we extract the local stimulus values in a window, gm,n(i, j), the size of the convolutional kernel and append them vertically in a ’local’ stimulus matrix. As an initial guess for the pooling weights, we weight each of these blocks by a Gaussian spatial proﬁle, chosen to roughly match the size of the receptive ﬁeld. We also generate a vector containing the vertical concatenation of copies of the measured spike train, ~r (one copy for each subunit location). 0 B @ w1,1Xg1,1(i,j) w1,2Xg1,2(i,j) ... 1 C A ! Xloc ; 0 B @ ~r ~r ... 1 C A ! ~rloc. (4) After performing STC analysis on the localized stimulus matrix, we use the ﬁrst (largest variance) eigenvector to initialize the subunit kernel of the excitatory channel, and the last (lowest variance) eigenvector to initialize the kernel of the suppressive channel. In practice, we ﬁnd that this initialization greatly reduces the number of iterations, and thus the run time, of the optimization procedure. 4 5/60 10/60 1 5 10 20 40 0 0.25 0.5 0.75 1 5/60 10/60 1 5 10 20 40 0 0.25 0.5 0.75 1 a) b) Minutes of simulated data Minutes of simulated data Model performance (r) Simulated simple cell Simulated complex cell subunit model Rust-STC model train test Figure 2: Model ﬁtting performance for simulated V1 neurons. Shown are correlation coefﬁcients for the subunit model (black circles) and the Rust-STC model (blue squares) [11], computed on both the training data (open), and on a holdout test set (closed). Spike counts for each presented stimulus frame are drawn from a Poisson distribution. Shaded regions indicate ± 1 s.d. for 5 simulation runs. (a) ‘Simple’ cell, with spike rate determined by the halfwave-rectiﬁed and squared response of a single oriented linear ﬁlter. (b) ‘Complex’ cell, with rate determined by a sum of squared Gabor ﬁlters arranged in spatial quadrature. Insets show estimated ﬁlters for the subunit (top) and RustSTC (bottom) models with ten seconds (400 frames; left) and 20 minutes (48,000 frames; right) of data. 4 Experiments We ﬁt the subunit model to physiological data sets in 3 different primate cortical areas: V1, V2, and MT. The model is able to explain a signiﬁcant amount of variance for each of these areas, but for illustrative purposes we show here only data for V1. Initially, we use simulated V1 cells to compare the performance of the subunit model to that of the Rust-STC model [11], which is based upon STC analysis. 4.1 Simulated V1 data We simulated the responses of canonical V1 simple cells and complex cells in response to white noise stimuli. Stimuli consisted of a 16x16 spatial array of pixels whose luminance values were set to independent ternary white noise sequences, updated every 25 ms (or 40 Hz). The simulated cells use spatiotemporally oriented Gabor ﬁlters: The simple cell has one even-phase ﬁlter and a half-squaring output nonlinearity while the complex cell has two ﬁlters (one even and one odd) whose squared responses are combined to give a ﬁring rate. Spike counts are drawn from a Poisson distribution, and overall rates are scaled so as to yield an average of 40 ips (i.e. one spike per time bin). For consistency with the analysis of the physiological data, we ﬁt the simulated data using a subunit model with two subunit channels (even though the simulated cells only possess an excitatory channel). When ﬁtting the Rust-STC model, we followed the procedure described in [11]. Brieﬂy, after the STA and STC ﬁlters are estimated, they are weighted according to their predictive power and combined in excitatory and suppressive pools, E and S (we use cross-validation to determine the number of ﬁlters to use for each pool). These two pooled responses are then combined using a joint output nonlinearity: ˆr(t)Rust = ↵+ (βE⇢−δS⇢)/(γE⇢+ ✏S⇢+ 1). Parameters {↵, β, δ, γ, ✏, ⇢} are optimized to minimizing mean squared error between observed spike counts and the model rate. Model performances, measured as the correlation between the model rate and spike count, are shown in Figure 2. In low data regimes, both models perform nearly perfectly on the training data, but poorly on separate test data not used for ﬁtting, a clear indication of over-ﬁtting. But as the data set increases in size, the subunit model rapidly improves, reaching near-perfect performance for modest spike counts. The Rust-STC model also improves, but much more slowly; It requires more than an order of magnitude more data to achieve the same performance as the subunit model. This 5 Excitatory Suppressive Convolutional subunit filters Nonlinearity Position map 0 90 180 270 0 50 100 150 200 Orientation Firing rate (ips) measured subunit Rust 1s Trials spikes 5 0 a) b) c) 0 ms 25 50 75 100 125 150 175 0.96º + Figure 3: Two-channel subunit model ﬁt to a physiological data from a macaque V1 cell. (a) Fitted parameters for the excitatory (top row) and suppressive (bottom row) channels, including the spacetime subunit ﬁlters (8 grayscale images, corresponding to different time frames), the nonlinearity, and the spatial weighting function wn,m that is used to combine the subunit responses. (b) A raster showing spiking responses to 20 repeated presentations of an identical stimulus with the average spike count (black) and model prediction (blue) plotted above. (c) Simulated models (subunit model: blue, Rust-STC model: purple) and measured (black) responses to drifting sinusoidal gratings. inefﬁciency is more pronounced for the complex cell, because the simple cell is fully explained by the STA ﬁlter, which can be estimated much more reliably than the STC ﬁlters for small amounts of data. We conclude that directly ﬁtting the subunit model is much more efﬁcient in the use of data than using STC to estimate a subspace model. 4.2 Physiological data from macaque V1 We presented spatio-temporal pixel noise to 38 cells recorded from V1 in anesthetized macaques (see [11] for details of experimental design). The stimulus was a 16x16 grid with luminance values set by independent ternary white noise sequences refreshed at 40 Hz. For 21 neurons we also presented 20 repeats of a sequence of 1000 stimulus frames as a validation set. The model ﬁlters were assumed to respond over a 200 ms (8 frame) causal time window in which the stimulus most strongly affected the ﬁring of the neurons, and thus, model responses were derived from a stimulus vector with 2048 dimensions (16x16x8). Figure 3 shows the ﬁt of a 2-channel subunit model to data from a typical V1 cell. Figure 3a illustrates the subunit kernels and their associated nonlinearities and spatial pooling maps, for both the excitatory channel (top row) and the suppressive channel (bottom row). The two channels show clear but opposing direction selectivity, starting at a latency of 50 ms. The fact that this cell is complex is reﬂected in two aspects of the model parameters. First, the model shows a symmetric, full-wave rectifying nonlinearity for the excitatory channel. Second, the ﬁnal linear pooling for this channel is diffuse over space, eliciting a response that is invariant to the exact spatial position and phase of the stimulus. For this particular example the model ﬁts well. For the cross-validated set of repeated stimuli (which have the same structure as for the ﬁtting data), on average the model correlates with each trial’s ﬁring rate with an r-value of 0.54. A raster of spiking responses to twenty repetitions of a 5 s stimulus are depicted in Fig. 3b, along with the average ﬁring rate and the model prediction, which are well matched. The model can also capture the direction selectivity of this cell’s response to moving sinusoidal gratings (whose spatial and temporal frequency are chosen to best drive the cell) (Fig. 3c). The subunit model acceptably ﬁts most of the cells we recorded in V1. Moreover, ﬁt quality is not correlated with modulation index (r = −0.08; n.s.), suggesting that the model captures the behavior of both simple and complex cells equally well. The ﬁtted subunit model also signiﬁcantly outperforms the Rust-STC model in terms of predicting responses to novel data. Figure 4a shows the performance of the Rust-STC and subunit models for 21 V1 neurons, for both training data and test data on single trials. For the training data, the Rust6 a) b) c) 0 0.25 0.5 0.75 1 0 25,000 50,000 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 n = 21 training test Subunit model accuracy (r) Rust-STC model accuracy (r) ‘Oracle’ accuracy (r) Total number of recorded spikes 75% 50% 25% 100% Oracle Figure 4: Model performance comparisons on physiological data. (a) Subunit model performance vs. Rust-STC model for V1 data. Training accuracy is computed for a single variable-length sequence extracted from the ﬁtting data. Test accuracy is computed on the average response to 20 repeats of a 25 s stimulus. (b) Subunit model performance vs. an ‘Oracle’ model for V1 data (see text). Each point represents the average accuracy in predicting responses to each of 20 repeated stimuli. The oracle model uses the average spike count over the other 19 repeats as a prediction. Inset: Ratio of subunit-to-oracle performance. Error bars indicate 1 s.d. (c) Subunit model performance on test data, as a function of the total number of recorded spikes. STC model performs signiﬁcantly better than the subunit model (Figure 4a; < rRust >= 0.81, < rsubunit >= 0.33; p ⌧0.005). However, this is primarily due to over-ﬁtting: Visual inspection of the STC kernels for most cells reveals very little structure. For test data (that was not included in the data used to ﬁt the models), the subunit model exhibits signiﬁcantly better performance than the Rust-STC model (< rRust >= 0.16, < rsubunit >= 0.27; p ⌧0.005). This is primarily due to over-ﬁtting in the STC analysis. For a stimulus composed of a 16x16 pixel grid with 8 frames, the spike-triggered covariance matrix contains over 2 million parameters. For the same stimulus, a subunit model with two channels and an 8x8x8 subunit kernel has only about 1200 parameters. The subunit model performs well when compared to the Rust-STC model, but we were interested in obtaining a more absolute measure of performance. Speciﬁcally, no purely stimulus-driven model can be expected to explain the response variability seen across repeated presentations of the same stimulus. We can estimate an upper bound on stimulus-driven model performance by implementing an empirical ‘oracle’ model that uses the average response over all but one of a set of repeated stimulus trials to predict the response on the remaining trial. Over the 21 neurons with repeated stimulus data, we found that the subunit model achieved, on average, 76% the performance of the oracle model (Figure 4b). Moreover, the cells that were least well ﬁt by the subunit model were also the cells that responded only weakly to the stimulus (Figure 4c). We conclude that, for most cells, the ﬁtted subunit model explains a signiﬁcant fraction of the response that can be explained by any stimulus-driven model. 5 Discussion Subunits have been proposed as a qualitative description of many types of receptive ﬁelds in sensory systems [2, 28, 8, 11, 12], and have enjoyed a recent renewal of interest by the modeling community [13, 29]. Here we have described a new parameterized canonical subunit model that can be applied to an arbitrary set of inputs (either a sensory stimulus, or a population of afferents from a previous stage of processing), and we have developed a method for directly estimating the parameters of this model from measured spiking data. Compared with STA or STC, the model ﬁts are more accurate for a given amount of data, less sensitive to the choice of stimulus ensemble, and more interpretable in terms of biological mechanism. For V1, we have applied this model directly to the visual stimuli, adopting the simplifying assumption that subcortical pathways faithfully relay the image data to V1. Higher visual areas build their responses on the afferent inputs arriving from lower visual areas, and we have applied this subunit 7 model to such neurons by ﬁrst simulating the responses of a population of the afferent V1 neurons, and then optimizing a subunit model that best maps these afferent responses to the spiking responses observed in the data. Speciﬁcally, for neurons in area V2, we model the afferent V1 population as a collection of simple cells that tile visual space. The V1 ﬁlters are chosen to uniformly cover the space of orientations, scales, and positions [30]. We also include four different phases. For neurons in area MT (V5), we use an afferent V1 population that also includes direction selective subunits, because the projections from V1 to MT are known to be sensitive to the direction of visual motion [31]. Speciﬁcally, the V1 ﬁlters are a rotation-invariant set of 3-dimensional, space-space-time steerable ﬁlters [32]. We ﬁt these models to neural responses to textured stimuli that varied in contrast and local orientation content (for MT, the local elements also drift over time). Our preliminary results show that the subunit model outperforms standard models for these higher order areas as well. We are currently working to reﬁne and generalize the subunit model in a number of ways. The mean squared error objective function, while computationally appealing, does not accurately reﬂect the noise properties of real neurons, whose variance changes with their mean rate. A likelihood objective function, based on a Poisson or similar spiking model, can improve the accuracy of the ﬁtted model, but it does so at a cost to the simplicity of model estimation (e.g. Alternating Least Squares can no longer be used to solve the bilinear problem). Real neurons also possess other forms of nonlinearities, such as local gain control that is been observed in neurons through the visual and auditory systems [33]. We are exploring means by which this functionality can be included directly in the model framework (e.g. [11]), while retaining the tractability of the parameter estimation. Acknowledgments This work was supported by the Howard Hughes Medical Institute, and by NIH grant EY04440. References [1] H. B. Barlow and W. R. Levick. The mechanism of directionally selective units in rabbit’s retina. The Journal of Physiology, 178(3):477, June 1965. [2] S. Hochstein and R. M. Shapley. Linear and nonlinear spatial subunits in Y cat retinal ganglion cells, 1976. [3] J. 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4,489	Discriminative Learning of Sum-Product Networks Robert Gens Pedro Domingos Department of Computer Science and Engineering University of Washington Seattle, WA 98195-2350, U.S.A. {rcg,pedrod}@cs.washington.edu Abstract Sum-product networks are a new deep architecture that can perform fast, exact inference on high-treewidth models. Only generative methods for training SPNs have been proposed to date. In this paper, we present the ﬁrst discriminative training algorithms for SPNs, combining the high accuracy of the former with the representational power and tractability of the latter. We show that the class of tractable discriminative SPNs is broader than the class of tractable generative ones, and propose an efﬁcient backpropagation-style algorithm for computing the gradient of the conditional log likelihood. Standard gradient descent suffers from the diffusion problem, but networks with many layers can be learned reliably using “hard” gradient descent, where marginal inference is replaced by MPE inference (i.e., inferring the most probable state of the non-evidence variables). The resulting updates have a simple and intuitive form. We test discriminative SPNs on standard image classiﬁcation tasks. We obtain the best results to date on the CIFAR-10 dataset, using fewer features than prior methods with an SPN architecture that learns local image structure discriminatively. We also report the highest published test accuracy on STL-10 even though we only use the labeled portion of the dataset. 1 Introduction Probabilistic models play a crucial role in many scientiﬁc disciplines and real world applications. Graphical models compactly represent the joint distribution of a set of variables as a product of factors normalized by the partition function. Unfortunately, inference in graphical models is generally intractable. Low treewidth ensures tractability, but is a very restrictive condition, particularly since the highest practical treewidth is usually 2 or 3 [2, 9]. Sum-product networks (SPNs) [23] overcome this by exploiting context-speciﬁc independence [7] and determinism [8]. They can be viewed as a new type of deep architecture, where sum layers alternate with product layers. Deep networks have many layers of hidden variables, which greatly increases their representational power, but inference with even a single layer is generally intractable, and adding layers compounds the problem [3]. SPNs are a deep architecture with full probabilistic semantics where inference is guaranteed to be tractable, under general conditions derived by Poon and Domingos [23]. Despite their tractability, SPNs are quite expressive [16], and have been used to solve difﬁcult problems in vision [23, 1]. Poon and Domingos introduced an algorithm for generatively training SPNs, yet it is generally observed that discriminative training fares better. By optimizing P(Y\|X) instead of P(X, Y) conditional random ﬁelds retain joint inference over dependent label variables Y while allowing for ﬂexible features over given inputs X [22]. Unfortunately, the conditional partition function Z(X) is just as prone to intractability as with generative training. For this reason, low treewidth models (e.g. chains and trees) of Y are commonly used. Research suggests that approximate inference can make it harder to learn rich structured models [21]. In this paper, discriminatively training SPNs will allow us to combine ﬂexible features with fast, exact inference over high treewidth models. 1 With inference and learning that easily scales to many layers, SPNs can be viewed as a type of deep network. Existing deep networks employ discriminative training with backpropagation through softmax layers or support vector machines over network variables. Most networks that are not purely feed-forward require approximate inference. Poon and Domingos showed that deep SPNs could be learned faster and more accurately than deep belief networks and deep Boltzmann machines on a generative image completion task [23]. This paper contributes a discriminative training algorithm that could be used on its own or with generative pre-training. For the ﬁrst time we combine the advantages of SPNs with those of discriminative models. In this paper we will review SPNs and describe the conditions under which an SPN can represent the conditional partition function. We then provide a training algorithm, demonstrate how to compute the gradient of the conditional log-likelihood of an SPN using backpropagation, and explore variations of inference. Finally, we show state-of-the-art results where a discriminatively-trained SPN achieves higher accuracy than SVMs and deep models on image classiﬁcation tasks. 2 Sum-Product Networks SPNs were introduced with the aim of identifying the most expressive tractable representation possible. The foundation for their work lies in Darwiche’s network polynomial [14]. We deﬁne an unnormalized probability distribution Φ(x) ≥0 over a vector of Boolean variables X. The indicator function [.] is one when its argument is true and zero otherwise; we abbreviate [Xi] and [¯Xi] as xi and ¯xi. To distinguish random variables from indicator variables, we use roman font for the former and italic for the latter. Vectors of variables are denoted by bold roman and bold italic font, respectively. The network polynomial of Φ(x) is deﬁned as P x Φ(x) Q(x), where Q(x) is the product of indicators that are one in state x. For example, the network polynomial of the Bayesian network X1 →X2 is P(x1)P(x2\|x1)x1x2 + P(x1)P(¯x2\|x1)x1¯x2 + P(¯x1)P(x2\|¯x1)¯x1x2 + P(¯x1)P(¯x2\|¯x1)¯x1¯x2. To compute P(X1 = true, X2 = false), we access the corresponding term of the network polynomial by setting indicators x1 and ¯x2 to one and the rest to zero. To ﬁnd P(X2 = true), we ﬁx evidence on X2 by setting x2 to one and ¯x2 to zero and marginalize X1 by setting both x1 and ¯x1 to one. Notice that there are two reasons we might set an indicator xi = 1: (1) evidence {Xi = true}, in which case we set ¯xi = 0 and (2) marginalization of Xi, where ¯xi = 1 as well. In general the role of an indicator xi is to determine whether terms compatible with variable state Xi = true are included in the summation, and similarly for ¯xi. With this notation, the partition function Z can be computed by setting all indicators of all variables to one. The network polynomial has size exponential in the number of variables, but in many cases it can be represented more compactly using a sum-product network [23, 14]. Deﬁnition 1. (Poon & Domingos, 2011) A sum-product network (SPN) over variables X1, . . . , Xd is a rooted directed acyclic graph whose leaves are the indicators x1, . . . , xd and ¯x1, . . . , ¯xd and whose internal nodes are sums and products. Each edge (i, j) emanating from a sum node i has a non-negative weight wij. The value of a product node is the product of the values of its children. The value of a sum node is P j∈Ch(i) wijvj, where Ch(i) are the children of i and vj is the value of node j. The value of an SPN S[x1, ¯x1, . . . , xd, ¯xd] is the value of its root. + + + + + + + x2 x2 x3 x3 x1 x1 0.8 0.2 0.3 0.5 0.1 0.7 0.6 0.4 0.5 0.9 Figure 1: SPN over Boolean variables X1, X2, X3 If we could replace the exponential sum over variable states in the partition function with the linear evaluation of the network, inference would be tractable. For example, the SPN in Figure 1 represents the joint probability of three Boolean variables P(X1, X2, X3) in the Bayesian network X2 ←X1 →X3 using six indicators S[x1, ¯x1, x2, ¯x2, x3, ¯x3]. To compute P(X1 = true), we could sum over the joint states of X2 and X3, evaluating the network a total of four times S[1, 0, 0, 1, 0, 1]+. . .+ S[1, 0, 1, 0, 1, 0]. Instead, we set the indicators so that the network sums out both X2 and X3. An indicator setting of S[1,0,1,1,1,1] computes 2 the sum over all states compatible with our evidence e = {X1 = true} and requires only one evaluation. However, not every SPN will have this property. If a linear evaluation of an SPN with indicators set to represent evidence equals the exponential sum over all variable states consistent with that evidence, the SPN is valid. Deﬁnition 2. (Poon & Domingos, 2011) A sum-product network S is valid iff S(e) = ΦS(e) for all evidence e. In their paper, Poon and Domingos prove that there are two conditions sufﬁcient for validity: completeness and consistency. Deﬁnition 3. (Poon & Domingos, 2011) A sum-product network is complete iff all children of the same sum node have the same scope. Deﬁnition 4. (Poon & Domingos, 2011) A sum-product network is consistent iff no variable appears negated in one child of a product node and non-negated in another. Theorem 1. (Poon & Domingos, 2011) A sum-product network is valid if it is complete and consistent. The scope of a node is deﬁned as the set of variables that have indicators among the node’s descendants. To “appear in a child” means to be among that child’s descendants. If a sum node is incomplete, the SPN will undercount the true marginals. Since an incomplete sum node has scope larger than a child, that child will be non-zero for more than one state of the sum (e.g. if S[x1, ¯x1, x2, ¯x2] = (x1 +x2), S[1, 0, 1, 1] < S[1, 0, 1, 0]+S[1, 0, 0, 1]). If a product node is inconsistent, the SPN will overcount the marginals as it will incorporate impossible states (e.g. x1 × ¯x1) into its computation. Poon and Domingos show how to generatively train the parameters of an SPN. One method is to compute the likelihood gradient and optimize with gradient descent (GD). They also show how to use expectation maximization (EM) by considering each sum node as the marginalization of a hidden variable [17]. They found that online EM using most probable explanation (MPE or “hard”) inference worked the best for their image completion task. Gradient diffusion is a key issue in training deep models. It is commonly observed in neural networks that when the gradient is propagated to lower layers it becomes less informative [3]. When every node in the network takes fractional responsibility for the errors of a top level node, it becomes difﬁcult to steer parameters out of local minima. Poon and Domingos also saw this effect when using gradient descent and EM to train SPNs. They found that online hard EM could provide a sparse but strong learning signal to synchronize the efforts of upper and lower nodes. Note that hard training is not exclusive to EM. In the next section we show how to discriminatively train SPNs with hard gradient descent. 3 Discriminative Learning of SPNs We deﬁne an SPN S[y, h\|x] that takes as input three disjoint sets of variables H, Y, and X (hidden, query, and given). We denote the setting of all h indicator functions to 1 as S[y, 1\|x], where the bold 1 is a vector. We do not sum over states of given variables X when discriminatively training SPNs. Given an instance, we treat X as constants. This means that one ignores X variables in the scope of a node when considering completeness and consistency. Since adding a constant as a child to a product node cannot make that product inconsistent, a variable x can be the child of any product node in a valid SPN. To maintain completeness, x can only be the child of a sum node that has scope outside of Y or H. Algorithm 1: LearnSPN Input: Set D of instances over variables X and label variables Y, a valid SPN S with initialized parameters. Output: An SPN with learned weights repeat forall the d ∈D do UpdateWeights(S, Inference(S,xd,yd)) until convergence or early stopping condition; 3 The parameters of an SPN can be learned using an online procedure as in Algorithm 1 as proposed by Poon and Domingos. The three dimensions of the algorithm are generative vs. discriminative, the inference procedure, and the weight update. Poon and Domingos discussed generative gradient descent with marginal inference as well as EM with marginal and MPE inference. In this section we will derive discriminative gradient descent with marginal and MPE inference, where hard gradient descent can also be used for generative training. EM is not typically used for discriminative training as it requires modiﬁcation to lower bound the conditional likelihood [25] and there may not be a closed form for the M-step. 3.1 Discriminative Training with Marginal Inference A component of the gradient of the conditional log likelihood takes the form ∂ ∂w log P(y\|x) = ∂ ∂w log X h Φ(Y = y, H = h\|x) −∂ ∂w log X y′,h Φ(Y = y′, H = h\|x) = 1 S[y, 1\|x] ∂S[y, 1\|x] ∂w − 1 S[1, 1\|x] ∂S[1, 1\|x] ∂w where the two summations are separate bottom-up evaluations of the SPN with indicators set as S[y, 1\|x] and S[1, 1\|x], respectively. The partial derivatives of the SPN with respect to all weights can be computed with backpropagation, detailed in Algorithm 2. After performing a bottom-up evaluation of the SPN, partial derivatives are passed from parent to child as follows from the chain rule and described in [15]. The form of backpropagation presented takes time linear in the number of nodes in the SPN if product nodes have a bounded number of children. Our gradient descent update then follows the direction of the partial derivative of the conditional log likelihood with learning rate η: ∆w = η ∂ ∂w log P(y\|x). After each gradient step we optionally renormalize the weights of a sum node so they sum to one. Empirically we have found this to produce the best results. The second SPN evaluation that marginalizes H and Y can reuse computation from the ﬁrst, for example, when Y is modeled by a root sum node. In this case the values of all non-root nodes are equivalent between the two evaluations. For any architecture, one can memoize values of nodes that do not have a query variable indicator as a descendant. Algorithm 2: BackpropSPN Input: A valid SPN S, where Sn denotes the value of node n after bottom-up evaluation. Output: Partial derivatives of the SPN with respect to every node ∂S ∂Sn and weight ∂S ∂wi,j Initialize all ∂S ∂Sn = 0 except ∂S ∂S = 1 forall the n ∈S in top-down order do if n is a sum node then forall the j ∈Ch(n) do ∂S ∂Sj ← ∂S ∂Sj + wn,j ∂S ∂Sn ∂S ∂wn,j ←Sj ∂S ∂Sn else forall the j ∈Ch(n) do ∂S ∂Sj ← ∂S ∂Sj + ∂S ∂Sn Q k∈Ch(n)\{j} Sk 3.2 Discriminative Training with MPE Inference There are several reasons why MPE inference is appealing for discriminatively training SPNs. As discussed above, hard inference was crucial for overcoming gradient diffusion when generatively training SPNs. For many applications the goal is to predict the most probable structure, and therefore it makes sense to use this also during training. Finally, it is common to approximate summations with maximizations for reasons of speed or tractability. Though summation in SPNs is fast and exact, MPE inference is still faster. We derive discriminative gradient descent using MPE inference. 4 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f Figure 2: Positive and negative terms in the hard gradient. The root node sums out the variable Y, the two sum nodes on the left sum out the hidden variable H1, the two sum nodes on the right sum out H2, and a circled ‘f’ denotes an input variable Xi. Dashed lines indicate negative elements in the gradient. We deﬁne a max-product network (MPN) M[y, h\|x] based on the max-product semiring. This network compactly represents the maximizer polynomial maxx Φ(x) Q(x), which computes the MPE [15]. To convert an SPN to an MPN, we replace each sum node by a max node, where weights on children are retained. The gradient of the conditional log likelihood with MPE inference is then ∂ ∂w log ˜P(y\|x) = ∂ ∂w log max h Φ(Y = y, H = h\|x) −∂ ∂w log max y′,h Φ(Y = y′, H = h\|x) where the two maximizations are computed by M[y, 1\|x] and M[1, 1\|x]. MPE inference also consists of a bottom-up evaluation followed by a top-down pass. Inference yields a branching path through the SPN called a complete subcircuit that includes an indicator (and therefore assignment) for every variable [15]. Analogous to Viterbi decoding, the path starts at the root node and at each max (formerly sum) node it only travels to the max-valued child. At product nodes, the path branches to all children. We deﬁne W as the multiset of weights traversed by this path1. The value of the MPN takes the form of a product Q wi∈W wci i , where ci is the number of times wi appears in W. The partial derivatives of the MPN with respect to all nodes and weights is computed by Algorithm 2 modiﬁed to accommodate MPNs: (1) S becomes M, (2) when n is a sum node, the body of the forall loop is run once for j as the max-valued child. The partial derivative of the logarithm of an MPN with respect to a weight takes the form ∂log M ∂wi = ∂log M ∂M ∂M ∂wi = 1 M ∂M ∂wi = ci · wci−1 i Q wj∈W \{wi} w cj j Q wj∈W w cj j = ci wi The gradient of the conditional log likelihood with MPE inference is therefore ∆ci/wi, where ∆ci = c′ i −c′′ i is the difference between the number of times wi is traversed by the two MPE inference paths in M[y, 1\|x] and M[1, 1\|x], respectively. The hard gradient update is then ∆wi = η ∂ ∂wi log ˜P(y\|x) = η ∆ci wi . The hard gradient for a training instance (xd, yd) is illustrated in Figure 2. In the ﬁrst two expressions, the complete subcircuit traveled by each MPE inference is shown in bold. Product nodes do not have weighted children, so they do not appear in the gradient, depicted in the last expression We can also easily add regularization to SPN training. An L2 weight penalty takes the familiar form of −λ\|\|w\|\|2 and partial derivatives −2λwi can be added to the gradient. With an appropriate optimization method, an L1 penalty could also be used for learning with marginal inference on dense SPN architectures. However, sparsity is not as important for SPNs as it is for Markov random ﬁelds, where a non-zero weight can have outsize impact on inference time; with SPNs inference is always linear with respect to model size. A summary of the variations of Algorithm 1 is provided in Tables 1 and 2. The generative hard gradient can be used in place of online EM for datasets where it would be prohibitive to store inference results from past epoch. For architectures that have high fan-in sum nodes, soft inference may be able to separate groups of modes faster than hard inference, which can only alter one child of a sum node at a time. We observe the similarity between the updates of hard EM and hard gradient descent. In particular, if we reparameterize the SPN so that each child of a sum node is weighted by wi = ew′ i, the form of 1A consistent SPN allows for MPE inference to reach the same indicator more than once in the same branching path 5 Table 1: Inference procedures Node Soft Inference Hard Inference Sum ∂S ∂Sn = P k∈P a(n) ∂S ∂Sk Q l∈Ch(k)\{n} Sl ∂M ∂Mn = P k∈P a(n) ∂M ∂Mk Q l∈Ch(k)\{n} Ml Product ∂S ∂Sn = P k∈P a(n) wkn ∂S ∂Sk ∂M ∂Mn = P k∈P a(n) wkn ∂M ∂Mk : wkn ∈W 0 : otherwise Weight ∂S ∂wki = ∂S ∂Sk Si ∂M ∂wki = ∂M ∂Mk Mi Table 2: Weight updates Update Soft Inference Hard Inference Gen. GD ∆w = η ∂S[x,y] ∂w ∆wi = η ci wi Gen. EM P(Hk = i\|x, y) ∝wki ∂S[x,y] ∂Sk P(Hk = i\|x, y) = 1 : wki ∈W 0 : otherwise Disc. GD ∆w = η 1 S[y,1\|x] ∂S[y,1\|x] ∂w − ∆wi = η ∆ci wi 1 S[1,1\|x] ∂S[1,1\|x] ∂w the partial derivative of the log MPN becomes ∂log M ∂w′ i = 1 M ∂M ∂w′ i = ci Q w′ j∈W ′ ecj·w′ j Q w′ j∈W ′ ecj·w′ j = ci This means that the hard gradient update for weights in logspace is ∆w′ i = ∆ci, which resembles structured perceptron [13]. 4 Experiments We have applied discriminative training of SPNs to image classiﬁcation benchmarks. CIFAR-10 and STL-10 are standard datasets for deep networks and unsupervised feature learning. Both are 10-class small image datasets. We achieve the best results to date on both tasks. We follow the feature extraction pipeline of Coates et al. [10], which was also used recently to learn pooling functions [20]. The procedure consists of extracting 4 × 105 6x6 pixel patches from the training set images, ZCA whitening those patches [19], running k-means for 50 rounds, and then normalizing the dictionary to have zero mean and unit variance. We then use the dictionary to extract K features at every 6x6 pixel site in the image (unit stride) with the “triangle” encoding fk(x) = max{0, ¯z −zk}, where zk = \|\|x −ck\|\|2, ck is the k-th item in the dictionary, and ¯z is the average zk. For each image of CIFAR-10, for example, this yields a 27 × 27 × K feature vector that is ﬁnally downsampled by max-pooling to a G × G × K feature vector. GxGxK Mixture x Parts + + + Classes Location WxWxK exij·f111 Figure 3: SPN architecture for experiments. Hidden variable indicators omitted for legibility. We experiment with a simple architecture that allows for discriminative learning of local structure. This architecture cannot be generatively trained as it violates consistency over X. Inspired by the successful star models in Felzenszwalb et al. [18], we construct a network with C classes, P parts per class, and T mixture components per part. A part is a pattern of image patch features that can occur anywhere in the image (e.g. an arrangement of patches that deﬁnes a curve). Each part ﬁlter ⃗fcpt is of dimension W × W × K and is initialized to ⃗0. The root of the SPN is a sum node with a child Sc for each class c in the dataset multiplied by the indicator for that state of the label variable Y. Sc is a product over P nodes Scp, where each Scp is a sum node over T nodes 6 Scpt. The hidden variables H represent the choice of cluster in the mixture over a part and its position (Scp and Scpt, respectively). Finally, Scpt sums over positions i, j in the image of the logistic function e⃗xij·⃗fcpt where the given variable ⃗xij is the same dimension as f and parts can overlap. Notice that the mixture Scp models an additional level of spatial structure on top of the image patch features learned by k-means. Coates and Ng [12] also learn higher-order structure, but whereas our method learns structure discriminatively in the context of a parts-based model, their unsupervised algorithm greedily groups features based on correlation and is unable to learn mixtures. Compared with the pooling functions in Jia et al. [20] that model independent translation of patch features, our architecture models how nearby features move together. Other deep probabilistic architectures should be able to model high-level structure, but considering the difﬁculty in training these models with approximate inference, it is hard to make full use of their representational power. Unlike the star model of Felzenswalb et al. [18] that learns ﬁlters over predeﬁned HOG image features, our SPN learns on top of learned image features that can model color and detailed patterns. Generative SPN architectures on the same features produce unsatisfactory results as generative training is led astray by the large number of features, very few of which differentiate labels. In the generative SPN paper [23], continuous variables are modeled with univariate Gaussians at the leaves (viewed as a sum node with inﬁnite children but ﬁnite weight sum). With discriminative training, X can be continuous because we always condition on it, which effectively folds it into the weights. All networks are learned with stochastic gradient descent regularized by early stopping. We found that using marginal inference for the root node and MPE inference for the rest of the network worked best. This allows the SPN to continue learning the difference between classes even when it correctly classiﬁes a training instance. The fraction of the training set reserved for validation with CIFAR10 and STL-10 were 10% and 20%, respectively. Learning rates, P, and T were chosen based on validation set performance. 4.1 Results on CIFAR-10 CIFAR-10 consists of 32x32 pixel images: 5×104 for training and 104 for testing. We ﬁrst compare discriminative SPNs with other methods as we vary the size of the dictionary K. The results are seen in Figure 4. To fairly compare with recent work [10, 20] we also set G = 4. In general, we observe that SPNs can achieve higher performance using half as many features as the next best approach, the learned pooling function. We hypothesize that this is because the SPN architecture allows us to discriminatively train large moveable parts, image structure that cannot be captured by larger dictionaries. In Jia et al. [20] the pooling functions blur individual features (i.e. a 6x6 pixel dictionary item), from which the classiﬁer may have trouble inferring the coordination of image parts. We then experimented with a ﬁner grid and fewer dictionary items (G = 7, K = 400). Pooling functions destroy information, so it is better if less is done before learning. Finer grids are less feasible for the method in Jia et al. [20] as the number of rectangular pooling functions grows O(G4). Our best test accuracy of 83.96% was achieved with W = 3, P = 200, and T = 2, chosen 200 400 800 1600 4000 Dictionary Size 64 68 72 76 80 84 Accuracy Performance on CIFAR-10 Discriminative SPN Learned Pooling, Jia et al. K-means (tri.), white, Coates et al. Auto-encoder, raw, Coates et al. RBM, whitened, Coates et al. Figure 4: Impact of dictionary size K with a 4x4 pooling grid (W=3) on CIFAR-10 test accuracy 7 Table 3: Test accuracies on CIFAR-10. Method Dictionary Accuracy Logistic Regression [24] 36.0% SVM [5] 39.5% SIFT [5] 65.6% mcRBM [24] 68.3% mcRBM-DBN [24] 71.0% Convolutional RBM [10] 78.9% K-means (Triangle) [10] 4000, 4x4 grid 79.6 % HKDES [4] 80.0% 3-Layer Learned RF [12] 1600, 9x9 grid 82.0% Learned Pooling [20] 6000, 4x4 grid 83.11% Discriminative SPN 400, 7x7 grid 83.96% Table 4: Comparison of average test accuracies on all folds of STL-10. Method Accuracy (±σ) 1-layer Vector Quantization [11] 54.9% (± 0.4%) 1-layer Sparse Coding [11] 59.0% (± 0.8%) 3-layer Learned Receptive Field [12] 60.1% (± 1.0%) Discriminative SPN 62.3% (± 1.0%) by validation set performance. This architecture achieves the highest published test accuracy on the CIFAR-10 dataset, remarkably using one ﬁfth the number of features of the next best approach. We compare top CIFAR-10 results in Table 3, highlighting the dictionary size of systems that use the feature extraction from Coates et al. [10]. 4.2 Results on STL-10 STL-10 has larger 96x96 pixel images and less labeled data (5,000 training and 8,000 test) than CIFAR-10 [10]. The training set is mapped to ten predeﬁned folds of 1,000 images. We experimented on the STL-10 dataset in a manner similar to CIFAR-10, ignoring the 105 items of unlabeled data. Ten models were trained on the pre-speciﬁed folds, and test accuracy is reported as an average. With K=1600, G=8, W=4, P=10, and T=3 we achieved 62.3% (± 1.0% standard deviation among folds), the highest published test accuracy as of writing. Notably, this includes approaches that make use of the unlabeled training images. Like Coates and Ng [12], our architecture learns local relations among different feature maps. However, the SPN is able to discriminatively learn latent mixtures, which can encode a more nuanced decision boundary than the linear classiﬁer used in their work. After we carried out our experiments, Bo et al. [6] reported a higher accuracy with their unsupervised features and a linear SVM. Just as with the features of Coates et al. [10], we anticipate that using an SPN instead of the SVM would be beneﬁcial by learning spatial structure that the SVM cannot model. 5 Conclusion Sum-product networks are a new class of probabilistic model where inference remains tractable despite high treewidth and many hidden layers. This paper introduced the ﬁrst algorithms for learning SPNs discriminatively, using a form of backpropagation to compute gradients. Discriminative training allows for a wider variety of SPN architectures than generative training, because completeness and consistency do not have to be maintained over evidence variables. We proposed both “soft” and “hard” gradient algorithms, using marginal inference in the “soft” case and MPE inference in the “hard” case. The latter successfully combats the diffusion problem, allowing deep networks to be learned. Experiments on image classiﬁcation benchmarks illustrate the power of discriminative SPNs. Future research directions include applying other discriminative learning paradigms to SPNs (e.g. max-margin methods), automatically learning SPN structure, and applying discriminative SPNs to a variety of structured prediction problems. Acknowledgments: This research was partly funded by ARO grant W911NF-08-1-0242, AFRL contract FA8750-09-C-0181, NSF grant IIS-0803481, and ONR grant N00014-12-1-0312. 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, AFRL, NSF, ONR, or the United States Government. 8 References [1] M. Amer and S. Todorovic. Sum-product networks for modeling activities with stochastic structure. CVPR, 2012. [2] F. Bach and M.I. Jordan. Thin junction trees. Advances in Neural Information Processing Systems, 14:569–576, 2002. [3] Y. 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4,490	Stochastic optimization and sparse statistical recovery: Optimal algorithms for high dimensions Alekh Agarwal Microsoft Research New York NY alekha@microsoft.com Sahand N. Negahban Dept. of EECS MIT sahandn@mit.edu Martin J. Wainwright Dept. of EECS and Statistics UC Berkeley wainwrig@stat.berkeley.edu Abstract We develop and analyze stochastic optimization algorithms for problems in which the expected loss is strongly convex, and the optimum is (approximately) sparse. Previous approaches are able to exploit only one of these two structures, yielding a O(d/T ) convergence rate for strongly convex objectives in d dimensions and O( p s(log d)/T) convergence rate when the optimum is s-sparse. Our algorithm is based on successively solving a series of ℓ1-regularized optimization problems using Nesterov’s dual averaging algorithm. We establish that the error of our solution after T iterations is at most O(s(log d)/T ), with natural extensions to approximate sparsity. Our results apply to locally Lipschitz losses including the logistic, exponential, hinge and least-squares losses. By recourse to statistical minimax results, we show that our convergence rates are optimal up to constants. The effectiveness of our approach is also conﬁrmed in numerical simulations where we compare to several baselines on a least-squares regression problem. 1 Introduction Stochastic optimization algorithms have many desirable features for large-scale machine learning, and have been studied intensively in the last few years (e.g., [18, 4, 8, 22]). The empirical efﬁciency of these methods is backed with strong theoretical guarantees on their convergence rates, which depend on various structural properties of the objective function. More precisely, for an objective function that is strongly convex, stochastic gradient descent enjoys a convergence rate ranging from O(1/T ), when features vectors are extremely sparse, to O(d/T ), when feature vectors are dense [9, 14, 10]. This strong convexity condition is satisﬁed for many common machine learning problems, including boosting, least squares regression, SVMs and generalized linear models among others. A complementary condition is that of (approximate) sparsity in the optimal solution. Sparse models have proven useful in many applications (see e.g., [6, 5] and references therein), and many statistical procedures seek to exploit such sparsity. It has been shown [15, 19] that when the optimal solution θ∗ is s-sparse, appropriate versions of the mirror descent algorithm converge at a rate O(s p (log d)/T). Srebro et al. [20] exploit the smoothness of common loss functions, and obtain improved rates of the form O(η p (s log d)/T), where η is the noise variance. While the √log d scaling makes these methods attractive in high dimensions, their scaling with respect to the iterations T is relatively slow—namely, O(1/ √ T) as opposed to O(1/T ) for strongly convex problems. Many optimization problems encountered in practice exhibit both features: the objective function is strongly convex, and the optimum is (approximately) sparse. This fact leads to the natural question: is it possible to design algorithms for stochastic optimization that enjoy the best features of both types of structure? More speciﬁcally, an algorithm should have a O(1/T ) convergence rate, as well as a logarithmic dependence on dimension. The main contribution of this paper is to answer this question in the afﬁrmative, and to analyze a new algorithm that has convergence rate O((s log d)/T ) 1 for a strongly convex problem with an s-sparse optimum in d dimensions. This rate is unimprovable (up to constants) in our setting, meaning that no algorithm can converge at a substantially faster rate. Our analysis also yields optimal rates when the optimum is only approximately sparse. The algorithm proposed in this paper builds off recent work on multi-step methods for strongly convex problems [11, 10, 12], but involves some new ingredients so as to obtain optimal rates for statistical problems with sparse optima. In particular, we form a sequence of objective functions by decreasing the amount of regularization as the optimization algorithm proceeds which is quite natural from a statistical viewpoint. Each step of our algorithm can be computed efﬁciently, with a closed form update rule in many common examples. In summary, the outcome of our development is an optimal one-pass algorithm for many structured statistical problems in high dimensions, and with computational complexity linear in the sample size. Numerical simulations conﬁrm our theoretical predictions regarding the convergence rate of the algorithm, and also establish its superiority compared to regularized dual averaging [22] and stochastic gradient descent algorithms. They also conﬁrm that a direct application of the multi-step method of Juditsky and Nesterov [11] is inferior to our algorithm, meaning that our gradual decrease of regularization is quite critical. More details on our results and their proofs can be found in the full-length version of this paper [2]. 2 Problem set-up and algorithm description Given a subset Ω⊆Rd and a random variable Z taking values in a space Z, we consider an optimization problem of the form θ∗∈arg min θ∈ΩE[L(θ; Z)], (1) where L : Ω× Z →R is a given loss function. As is standard in stochastic optimization, we do not have direct access to the expected loss function L(θ) := E[L(θ; Z)], nor to its subgradients. Rather, for a given query point θ ∈Ω, we observe a stochastic subgradient, meaning a random vector g(θ) ∈Rd such that E[g(θ)] ∈∂L(θ). The goal of this paper is to design algorithms that are suitable for solving the problem (1) when the optimum θ∗is (approximately) sparse. Algorithm description: In order to solve a sparse version of the problem (1), our strategy is to consider a sequence of regularized problems of the form min θ∈Ω′ L(θ) + λ∥θ∥1 . (2) Our algorithm involves a sequence of KT different epochs, where the regularization parameter λ > 0 and the constraint set Ω′ ⊂Ωchange from epoch to epoch. The epochs are speciﬁed by: • a sequence of natural numbers {Ti}KT i=1, where Ti speciﬁes the length of the ith epoch, • a sequence of positive regularization weights {λi}KT i=1, and • a sequence of positive radii {Ri}KT i=1 and d-dimensional vectors {yi}KT i=1, which specify the constraint set, Ω(Ri) := θ ∈Ω\| ∥θ −yi∥p ≤Ri , that is used throughout the ith epoch. We initialize the algorithm in the ﬁrst epoch with y1 = 0, and with any radius R1 that is an upper bound on ∥θ∗∥1. The norm ∥· ∥p used in deﬁning the constraint set Ω(Ri) is speciﬁed by p = 2 log d/(2 log d −1), a choice that will be clariﬁed momentarily. The goal of the ith epoch is to update yi 7→yi+1, in such a way that we are guaranteed that ∥yi+1 −θ∗∥2 1 ≤R2 i+1 for each i = 1, 2, . . .. We choose the radii such that R2 i+1 = R2 i /2, so that upon termination, ∥yKT −θ∗∥2 1 ≤R2 1/2KT −1. In order to update yi 7→yi+1, we run Ti rounds of the stochastic dual averaging algorithm [17] (henceforth DA) on the regularized objective min θ∈Ω(Ri) L(θ) + λi∥θ∥1 . (3) The DA method generates two sequences of vectors {µt}Ti t=0 and {θt}Ti t=0 initialized as µ0 = 0 and θ0 = yi, using a sequence of step sizes {αt}Ti t=0. At iteration t = 0, 1, . . ., Ti, we let gt be a stochastic subgradient of L at θt, and we let νt be any element of the subdifferential of the ℓ1-norm ∥· ∥1 at θt. The DA update at time t maps (µt, θt) 7→(µt+1, θt+1) via the recursions µt+1 = µt + gt + λiνt, and θt+1 = arg min θ∈Ω(Ri) αt+1⟨µt+1, θ⟩+ ψyi,Ri(θ) , (4) 2 where the prox function ψ is speciﬁed below (5). The pseudocode describing the overall procedure is given in Algorithm 1. In the stochastic dual averaging updates (4), we use the prox function ψyi,Ri(θ) = 1 2R2 i (p −1) ∥θ −yi∥2 p, where p = 2 log d 2 log d −1. (5) This particular choice of the prox-function and the speciﬁc value of p ensure that the function ψ is strongly convex with respect to the ℓ1-norm, and has been previously used for sparse stochastic optimization (see e.g. [15, 19, 7]). In most of our examples, Ω= Rd and owing to our choice of the prox-function and the feasible set in the update (4), we can compute θt+1 from µt+1 in closed form. Some algebra yields that the update (4) with Ω= Rd is equivalent to θt+1 = yi + R2 i αt+1 (p −1)(1 + ξ) \|µt+1\|(q−1)sign(µt+1) ∥µt+1∥(q−2) q , where ξ = max 0, αt+1∥µt+1∥qRi p −1 −1 . Here \|µt+1\|(q−1) refers to elementwise operations and q = p/(p −1) is the conjugate exponent to p. We observe that our update (4) computes a subgradient of the ℓ1-norm rather than computing an exact prox-mapping as in some previous methods [16, 7, 22]. Computing such a prox-mapping for yi ̸= 0 requires O(d2) computation, which is why we adopt the update (4) with a complexity O(d). Algorithm 1 Regularization Annealed epoch Dual AveRaging (RADAR) Require: Epoch length schedule {Ti}KT i=1, initial radius R1, step-size multiplier α, prox-function ψ, initial prox-center y1, regularization parameters λi. for Epoch i = 1, 2, . . ., KT do Initialize µ0 = 0 and θ0 = yi. for Iteration t = 0, 1, . . . , Ti −1 do Update (µt, θt) 7→(µt+1, θt+1) according to rule (4) with step size αt = α/ √ t. end for Set yi+1 = PTi t=1 θt Ti . Update R2 i+1 = R2 i /2. end for Return yKT +1 Conditions: Having deﬁned our algorithm, we now discuss the conditions on the objective function L(θ) and stochastic gradients that underlie our analysis. Assumption 1 (Locally Lipschitz). For each R > 0, there is a constant G = G(R) such that \|L(θ) −L(˜θ)\| ≤G ∥θ −˜θ∥1 (6) for all pairs θ, ˜θ ∈Ωsuch that ∥θ −θ∗∥1 ≤R and ∥˜θ −θ∗∥1 ≤R. We note that it sufﬁces to have ∥∇L(θ)∥∞≤G(R) for the above condition. As mentioned, our goal is to obtain fast rates for objectives satisfying a local strong convexity condition, deﬁned below. Assumption 2 (Local strong convexity (LSC)). The function L : Ω→R satisﬁes a R-local form of strong convexity (LSC) if there is a non-negative constant γ = γ(R) such that L(˜θ) ≥L(θ) + ⟨∇L(θ), ˜θ −θ⟩+ γ 2 ∥θ −˜θ∥2 2 ∀θ, ˜θ ∈Ωwith ∥θ∥1 ≤R and ∥˜θ∥1 ≤R. (7) Some of our results regarding stochastic optimization from a ﬁnite sample will use a weaker form of the assumption, called local RSC, exploited in our recent work on statistics and optimization [1, 13]. Our ﬁnal assumption is a tail condition on the error in stochastic gradients: e(θ) := g(θ) −E[g(θ)]. Assumption 3 (Sub-Gaussian stochastic gradients). There is a constant σ = σ(R) such that E exp(∥e(θ)∥2 ∞/σ2) ≤exp(1) for all θ such that ∥θ −θ∗∥1 ≤R. (8) Clearly, this condition holds whenever the error vector e(θ) has bounded components. More generally, the bound (8) holds whenever each component of the error vector has sub-Gaussian tails. 3 Some illustrative examples: We now describe some examples that satisfy the above conditions to illustrate how the various parameters of interest might be obtained in different scenarios. Example 1 (Classiﬁcation under Lipschitz losses). In binary classiﬁcation, the samples consist of pairs z = (x, y) ∈Rd × {−1, 1}. Common choices for the loss function L(θ; z) are the hinge loss max(0, 1−y⟨θ, x⟩) or the logistic loss log(1+exp(−y⟨θ, x⟩). Given a distribution P over Z (either the population or the empirical distribution), a common strategy is to draw (xt, yt) ∼P at iteration t and use gt = ∇L(θ; (xt, yt)). We now illustrate how our conditions are satisﬁed in this setting. • Locally Lipschitz: Both the above examples actually satisfy a stronger global Lipschitz condition since we have the bound G ≤∥∇L(θ)∥∞≤E∥x∥∞. Often, the data satisﬁes the normalization ∥x∥∞≤B, in which case we get G ≤B. More generally, tail conditions on the marginal distribution of each coordinate of x ensure G = O(√log d)) is valid with high probability. • LSC: When the expectation in the objective (1) is under the population distribution, the above examples satisfy LSC. Here we focus on the example of the logistic loss, where we deﬁne the link function ψ(α) = exp(α)/(1+exp(α))2. We also deﬁne Σ = E[xxT ] to be the covariance matrix and let σmin(Σ) denote its minimum singular value. Then a second-order Taylor expansion yields L(˜θ) −L(θ) −⟨∇L(θ), ˜θ −θ⟩= ψ(⟨eθ, x⟩) 2 ∥Σ1/2(θ −˜θ)∥2 2 ≥ψ(BR)σmin(Σ) 2 ∥θ −˜θ∥2 2, where eθ = aθ + (1 −a)˜θ for some a ∈(0, 1). Hence γ ≥ψ(BR)σmin(Σ) in this example. • Sub-Gaussian gradients: Assuming the bound E∥x∥∞≤B, this condition is easily veriﬁed. A simple calculation yields σ = 2B, since ∥e(θ)∥∞= ∥∇L(θ; (x, y)) −∇L(θ)∥∞≤∥∇L(θ; (x, y))∥∞+ ∥∇L(θ)∥∞≤2B. Example 2 (Least-squares regression). In the regression setup, we are given samples of the form z = (x, y) ∈Rd × R. The loss function of interest is L(θ; (x, y)) = (y −⟨θ, x⟩)2/2. To illustrate the conditions more clearly, we assume that our samples are generated as y = ⟨x, θ∗⟩+ w, where w ∼N(0, η2) and ExxT = Σ so that EL(θ; (x, y)) = ∥Σ1/2(θ −θ∗)∥2 2/2. • Locally Lipschitz: For this example, the Lipschitz parameter G(R) depends on the bound R. If we deﬁne ρ(Σ) = maxi Σii to be the largest variance of a coordinate of x, then a direct calculation yields the bound G(R) ≤ρ(Σ)R. • LSC: Again we focus on the case where the expectation is taken under the population distribution, where we have γ = σmin(Σ). • Sub-Gaussian gradients: Once again we assume that ∥x∥∞≤B. It can be shown with some work that Assumption 3 is satisﬁed with σ2(R) = 8ρ(Σ)2R2 + 4B4R2 + 10B2η2. 3 Main results and their consequences In this section we state our main results, regarding the convergence of Algorithm 1. We focus on the cases where Assumptions 1 and 3 hold over the entire set Ω, and RSC holds uniformly for all ∥θ∥1 ≤R1; key examples being the hinge and logistic losses from Example 1. Extensions to examples such as least-squares loss, which are not Lipschitz on all of Ωrequire a more delicate treatment and these results as well the proofs of our results can be found in the long version [2]. Formally, we assume that G(R) ≡G and σ(R) ≡σ in Assumptions 1 and 3. We also use γ to denote γ(R1) in Assumption 2. For a constant ω > 0 governing the error probability in our results, we also deﬁne ω2 i = ω2 + 24 log i at epoch i. Our results assume that we run Algorithm 1 with Ti ≥c1 s2 γ2R2 i (G2 + σ2) log d + ω2 i σ2 + log d , (9) where c1 is a universal constant. For a total of T iterations in Algorithm 1, we state our results for the parameter bθT = y(KT +1) where KT is the last epoch completed in T iterations. 3.1 Main theorem and some remarks We start with our main result which shows an overall convergence rate of O(1/T ) after T iterations. This O(1/T ) convergence is analogous to earlier work on multi-step methods for strongly convex 4 objectives [11, 12, 10]. For each subset S ⊆{1, 2, . . ., d} of cardinality s, we deﬁne ε2(θ∗; S) := ∥θ∗ Sc∥2 1/s. (10) This quantity captures the degree of sparsity in the optimum θ∗; for instance, ε2(θ∗; S) = 0 if and only if θ∗is supported on S. Given the probability parameter ω > 0, we also deﬁne the shorthand κT = log2 γ2R2 1T s2((G2 + σ2) log d + ω2σ2) log d. (11) Theorem 1. Suppose the expected loss L satisﬁes Assumptions 1— 3 with parameters G(R) ≡G, γ and σ(R) ≡σ, and we perform updates (4) with epoch lengths (9) and parameters λ2 i = Riγ s√Ti q (G2 + σ2) log d + ω2 i σ2 and α(t) = 5Ri s log d (G2 + λ2 i + σ2)t. (12) Then for any subset S ⊆{1, . . . , d} of cardinality s and any T ≥2κT, there is a universal constant c0 such that with probability at least 1 −6 exp(−ω2/12) we have ∥bθT −θ∗∥2 2 ≤c3 s γ2T ((G2 + σ2) log d + σ2(ω2 + log κT log d)) + ε2(θ∗; S) . (13) Consequently, the theorem predicts a convergencerate of O(1/γ2T ) which is the best possible under our assumptions. Under the setup of Example 1, the error bound of Theorem 1 further simpliﬁes to ∥bθT −θ∗∥2 2 = O sB2 γ2T (log d + ω2) + ε2(θ∗; S) . (14) We note that for an approximately sparse θ∗, Theorem 1 guarantees convergence only to a tolerance ε2(θ∗; S) due to the error terms arising out of the approximate sparsity. Overall, the theorem provides a family of upper bounds, one for each choice of S. The best bound can be obtained by optimizing this choice, trading off the competing contributions of s and ∥θ∗ Sc∥1. At this point, we can compare the result of Theorem 1 to some of the previous work. One approach to minimize the objective (1) is to perform stochastic gradient descent on the objective, which has a convergence rate of O(( eG2 + eσ2)/(γ2T )) [10, 14], where ∥∇L(θ)∥2 ≤eG and E exp ∥e(θ)∥2 2 eσ2 ≤ exp(1). In the setup of Example 1, eG2 = Bd and similarly for eσ; giving an exponentially worse scaling in the dimension d. An alternative is to perform mirror descent [15, 19] or regularized dual averaging [22] using the same prox-function as Algorithm 1 but without breaking it up into epochs. As mentioned in the introduction, this single-step method fails to exploit the strong convexity of our problem and obtains inferior convergence rates of O(s p log d/T) [19, 22, 7]. A proposal closer to our approach is to minimize the regularized objective (3), but with a ﬁxed value of λ instead of the decreasing schedule of λi used in Theorem 1. This amounts to using the method of Juditsky and Nesterov [11] on the regularized problem, and by using the proof techniques developed in this paper, it can be shown that setting λ = σ p log d/T leads to an overall convergence rate of eO sB2 γ2T (log d + ω2) , which exhibits the same scaling as Theorem 1. However, with this ﬁxed setting of λ, the initial epochs tend to be much longer than needed for halving the error. Indeed, our setting of λi is based on minimizing the upper bound at each epoch, and leads to an improved performance in our numerical simulations. The beneﬁts of slowly decreasing the regularization in the context of deterministic optimization were also noted in the recent work of Xiao and Zhang [23]. 3.2 Some illustrative corollaries We now present some consequences of Theorem 1 by making speciﬁc assumptions regarding the sparsity of θ∗. The simplest situation is when θ∗is supported on some subset S of size s. More generally, Theorem 1 also applies to the case when the optimum θ∗is only approximately sparse. One natural form of approximate sparsity is to assume that θ∗∈Bq(Rq) for 0 < q ≤1, where Bq(Rq) := ( θ ∈Rd \| d X i=1 \|θi\|q ≤Rq ) . 5 For 0 < q ≤1, membership in the set Bq(Rq) enforces a decay rate on the components of the vector θ. We now present a corollary of Theorem 1 under such an approximate sparsity condition. To facilitate comparison with minimax lower bounds, we set ω2 = δ log d in the corollaries. Corollary 1. Under the conditions of Theorem 1, for all T > 2κT with probability at least 1 − 6 exp(−δ log d/12), there is a universal constant c0 such that ∥bθT −θ∗∥2 2 ≤      c0 h G2+σ2(1+δ) γ2 s log d T + sσ2 γ2T log κT log d i θ∗is s-sparse, c0Rq n (G2+σ2(1+δ)) log d γ2T o 2−q 2 + σ2 γ2T 2−q 2 log κT log d ((1+δ) log d) q 2 θ∗∈Bq(Rq). The ﬁrst part of the corollary follows directly from Theorem 1 by noting that ε2(θ∗; S) = 0 under our assumptions. Note that as q ranges over the interval [0, 1], reﬂecting the degree of sparsity, the convergencerate ranges from from e O(1/T ) (for q = 0 corresponding to exact sparsity) to eO(1/ √ T) (for q = 1). This is a rather interesting trade-off, showing in a precise sense how convergence rates vary quantitatively as a function of the underlying sparsity. It is useful to note that the results on recovery for generalized linear models presented here exactly match those that have been developed in the statistics literature [13, 21], which are optimal under our assumptions on the design vectors. Concretely, ignoring factors of O(log T ), we get a parameter bθT having error at most O(s log d/(γ2T ) with an error probability decyaing to zero with d. Moreover, in doing so our algorithm only goes over at most T data samples, as each stochastic gradient can be evaluated with one fresh data sample drawn from the underlying distribution. Since the statistical minimax lower bounds [13, 21] demonstrate that this is the smallest possible error that any method can attain from T samples, our method is statistically optimal in the scaling of the estimation error with the number of samples. We also observe that it is easy to instead set the error probability to δ = ω2 log T , if an error probability decaying with T is desired, incurring at most additional log T factors in the error bound. Finally, we also remark that our techniques extend to handle examples such as the least-squares loss that are not uniformly Lipschitz. The details of this extension are deferred to the long version of this paper [2]. Stochastic optimization over ﬁnite pools: A common setting for the application of stochastic optimization methods in machine learning is when one has a ﬁnite pool of examples, say {Z1, . . . , Zn}, and the objective (1) takes the form θ∗= arg min θ∈Ω 1 n n X i=1 L(θ; Zi) (15) In this setting, a stochastic gradient g(θ) can be obtained by drawing a sample Zj at random with replacement from the pool {Z1, . . . , Zn}, and returning the gradient ∇L(θ; Zj). In high-dimensional problems where d ≫n, the sample loss is not strongly convex. However, it has been shown by many researchers [3, 13, 1] that under suitable conditions, this objective does satisfy restricted forms of the LSC assumption, allowing us to appeal to a generalized form of Theorem 1. We will present this corollary only for settings where θ∗is exactly sparse and also specialize to the logistic loss, L(θ; (x, y)) = log(1 + exp(−y⟨θ, x⟩)) to illustrate the key aspects of the result. We recall the deﬁnition of the link function ψ(α) = exp(α)/(1 + exp(α))2. We will state the result for sub-Gaussian data design with parameters (Σ, η2 x), meaning that the E[xixT i ] = Σ and ⟨u, xi⟩is ηx-sub-Gaussian for any unit norm vector u ∈Rd. Corollary 2. Consider the ﬁnite-pool loss (15), based on n i.i.d. samples from a sub-Gaussian design with parameters (Σ, η2 x). Suppose that Assumptions 1-3 are satisﬁed and the optimum θ∗ of (15) is s-sparse. Then there are universal constants (c0, c1, c2, c3) such that for all T ≥2κT and n ≥c3 log d σ2 min(Σ) max(σ2 min(Σ), η4 x), we have ∥bθT −θ∗∥2 2 ≤ c0 σ2 min(Σ) s log d T n 1 ψ2(2BR1) B2(1 + δ) o + c0 sσ2 σ2 min(Σ)ψ2(2BR1)T log κT log d. with probability at least 1 −2 exp(−c1n min(σ2 min(Σ)/η4 x, 1)) −6 exp(−δ log d/12). 6 We observe that the bound only holds when the number of samples n in the objective (15) is large enough, which is necessary for the restricted form of the LSC condition to hold with non-trivial parameters in the ﬁnite sample setting. A modiﬁed method with constant epoch lengths: Algorithm 1 as described is efﬁcient and simple to implement. However, the convergence results critically rely on the epoch length Ti to be set appropriately in a doubling manner. This could be problematic in practice, where it might be tricky to know when an epoch should be terminated. Following Juditsky and Nesterov [11], we next demonstrate how a variant of our algorithm with constant epoch lengths enjoys similar rates of convergence. The key challenge here is that unlike the previous set-up [11], our objective function changes at each epoch which leads to signiﬁcant technical difﬁculties. At a very coarse level, if we have a total budget of T iterations, then this version of our algorithm allows us to set the epoch lengths to O(log T ), and guarantees convergence rates that are O((log T )/T ). Theorem 2. Suppose the expected loss satisﬁes Assumptions 1- 3 with parameters G, γ, and σ resp. Let S be any subset of {1, . . ., d} of cardinality s. Suppose we run Algorithm 1 for a total of T iterations with epoch length Ti ≡T log d/κT and with parameters as in Equation 12. Assuming that this setting ensures Ti = O(log d), for any set S, with probability at least 1 −3 exp(ω2/12) ∥bθT −θ∗∥2 2 = O s (G2 + σ2) log d + (ω2 + log(κ/ log d))σ2 T log d κ . The theorem shows that up to logarithmic factors in T , setting the epoch lengths optimally is not critical. A similar result can also be proved for the case of least-squares regression. 4 Simulations In this section we will present numerical simulations that back our theoretical convergence results. We focus on least-squares regression, discussed in Example 2. Speciﬁcally, we generate samples (xt, yt) with each coordinate of xt distributed as Unif[−B, B] and yt = ⟨θ∗, xt⟩+ wt. We pick θ∗ to be s-sparse vector with s = ⌈log d⌉, and wt ∼N(0, η2) with η2 = 0.5. Given an iterate θt, we generate a stochastic gradient of the expected loss (1) at (xt, yt). For the ℓ1-norm, we pick the sign vector of θt, with 0 for any component that is zero, a member of the ℓ1-sub-differential. Our ﬁrst set of results evaluate Algorithm 1 against other stochastic optimization baselines assuming a complete knowledge of problem parameters. Speciﬁcally, we epoch i is terminated once ∥yi+1 −θ∗∥2 p ≤∥yi −θ∗∥2 p/2. This ensures that θ∗remains feasible throughout, and tests the performance of Algorithm 1 in the most favorable scenario. We compare the algorithm against two baselines. The ﬁrst baseline is the regularized dual averaging (RDA) algorithm [22], applied to the regularized objective (3) with λ = 4η p log d/T, which is the statistically optimal regularization parameter with T samples. We use the same prox-function ψ(θ) = ∥θ∥2 p 2(p−1), so that the theory for RDA predicts a convergence rate of O(s p log d/T) [22]. Our second baseline is the stochastic gradient (SGD) algorithm which exploits the strong convexity but not the sparsity of the problem (1). Since the squared loss is not uniformly Lipschitz, we impose an additional constraint ∥θ∥1 ≤R1, without which the algorithm does not converge. The results of this comparison are shown in Figure 1(a), where we present the error ∥θt −θ∗∥2 2 averaged over 5 random trials. We observe that RADAR comprehensively outperforms both the baselines, conﬁrming the predictions of our theory. The second set of results focuses on evaluating algorithms better tailored for our assumptions. Our ﬁrst baseline here is the approach that we described in our remarks following Theorem 1. In this approach we use the same multi-step strategy as Algorithm 1 but keep λ ﬁxed. We refer to this as Epoch Dual Averaging (henceforth EDA), and again employ λ = 4η p log d/T with this strategy. Our epochs are again determined by halving of the squared ℓp-error measured relative to θ∗. Finally, we also evaluate the version of our algorithm with constant epoch lengths that we analyzed in Theorem 2 (henceforth RADAR-CONST), using epochs of length log(T ). As shown in Figure 1(b), the RADAR-CONST has relatively large error during the initial epochs, before converging quite 7 rapidly, a phenomenon consistent with our theory.1 Even though the RADAR-CONST method does not use the knowledge of θ∗to set epochs, all three methods exhibit the same eventual convergence rates, with RADAR (set with optimal epoch lengths) performing the best, as expected. Although RADAR-CONST is very slow in initial iterations, its convergence rate remains competitive with EDA (even though EDA does exploit knowledge of θ∗), but is worse than RADAR as expected. Overall, our experiments demonstrate that RADAR and RADAR-CONST have practical performance consistent with our theoretical predictions. Although optimal epoch length setting is not too critical for our approach, better data-dependent empirical rules for determining epoch lengths remains an interesting question for future research. The relatively poorer performance of EDA demonstrates the importance of our decreasing regularization schedule. 0 0.5 1 1.5 2 x 10 4 0 1 2 3 4 5 6 Iterations ∥θt −θ∗∥2 2 Error vs. iterations RADAR SGD RDA 0 0.5 1 1.5 2 x 10 4 0 1 2 3 4 5 Iterations ∥θt −θ∗∥2 2 Error vs. iterations RADAR EDA RADAR-CONST (a) (b) Figure 1. A comparison of RADAR with other stochastic optimization algorithms for d = 40000 and s = ⌈log d⌉. The left plot compares RADAR with the RDA and SGD algorithms, neither of which exploits both the sparsity and the strong convexity structures simultaneously. The right one compares RADAR with the EDA and RADAR-CONST algorithms, all of which exploit the problem structure but with varying degrees of effectiveness. We plot ∥θt −θ∗∥2 2 averaged over 5 random trials versus the number of iterations. 5 Discussion In this paper we present an algorithm that is able to take advantage of the strong convexity and sparsity conditions that are satsiﬁed by many common problems in machine learning. Our algorithm is simple and efﬁcient to implement, and for a d-dimensional objective with an s-sparse optima, it achieves the minimax-optimal convergence rate O(s log d/T ). We also demonstrate optimal convergence rates for problems that have weakly sparse optima, with implications for problems such as sparse linear regression and sparse logistic regression. While we focus our attention exclusively on sparse vector recovery due to space constraints, the ideas naturally extend to other structures such as group sparse vectors and low-rank matrices. It would be interesting to study similar developments for other algorithms such as mirror descent or Nesterov’s accelerated gradient methods, leading to multi-step variants of those methods with optimal convergence rates in our setting. Acknowledgements The work of all three authors was partially supported by ONR MURI grant N00014-11-1-0688 to MJW. In addition, AA was partially supported by a Google Fellowship, and SNN was partially supported by the Yahoo KSC award. 1 To clarify, the epoch lengths in RADAR-CONST are set large enough to guarantee that we can attain an overall error bound of O(1/T ), meaning that the initial epochs for RADAR-CONST are much longer than for RADAR. Thus, after roughly 500 iterations, RADAR-CONST has done only 2 epochs and operates with a crude constraint set Ω(R1/4). During epoch i, the step size scales proportionally to Ri/ √ t, where t is the iteration number within the epoch; hence the relatively large initial steps in an epoch can take us to a bad solution even when we start with a good solution yi when Ri is large. As Ri decreases further with more epochs, this effect is mitigated and the error of RADAR-CONST does rapidly decrease like our theory predicts. 8 References [1] A. Agarwal, S. N. Negahban, and M. J. Wainwright. Fast global convergence rates of gradient methods for high-dimensional statistical recovery. To appear in The Annals of Statistics, 2012. Full-length version http://arxiv.org/pdf/1104.4824v2. [2] A. Agarwal, S. N. Negahban, and M. J. Wainwright. Stochastic optimization and sparse statistical recovery: An optimal algorithm for high dimensions. 2012. URL http://arxiv.org/abs/1207.4421. [3] P. J. Bickel, Y. Ritov, and A. B. Tsybakov. Simultaneous analysis of Lasso and Dantzig selector. Ann. Stat., 37(4):1705–1732, 2009. [4] L. 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4,491	Bandit Algorithms boost motor-task selection for Brain Computer Interfaces Joan Fruitet INRIA, Sophia Antipolis 2004 Route des Lucioles 06560 Sophia Antipolis, France joan.fruitet@inria.fr Alexandra Carpentier Statistical Laboratory, CMS Wilberforce Road, Cambridge CB3 0WB UK a.carpentier@statslab.cam.ac.uk R´emi Munos INRIA Lille - Nord Europe 40, avenue Halley 59000 Villeneuve d’ascq, France remi.munos@inria.fr Maureen Clerc INRIA, Sophia Antipolis 2004 Route des Lucioles 06560 Sophia Antipolis, France Maureen.Clerc@inria.fr Abstract Brain-computer interfaces (BCI) allow users to “communicate” with a computer without using their muscles. BCI based on sensori-motor rhythms use imaginary motor tasks, such as moving the right or left hand, to send control signals. The performances of a BCI can vary greatly across users but also depend on the tasks used, making the problem of appropriate task selection an important issue. This study presents a new procedure to automatically select as fast as possible a discriminant motor task for a brain-controlled button. We develop for this purpose an adaptive algorithm, UCB-classif, based on the stochastic bandit theory. This shortens the training stage, thereby allowing the exploration of a greater variety of tasks. By not wasting time on inefﬁcient tasks, and focusing on the most promising ones, this algorithm results in a faster task selection and a more efﬁcient use of the BCI training session. Comparing the proposed method to the standard practice in task selection, for a ﬁxed time budget, UCB-classif leads to an improved classiﬁcation rate, and for a ﬁxed classiﬁcation rate, to a reduction of the time spent in training by 50%. 1 Introduction Scalp recorded electroencephalography (EEG) can be used for non-muscular control and communication systems, commonly called brain-computer interfaces (BCI). BCI allow users to “communicate” with a computer without using their muscles. The communication is made directly through the electrical activity from the brain, collected by EEG in real time. This is a particularly interesting prospect for severely handicapped people, but it can also be of use in other circumstances, for instance for enhanced video games. A possible way of communicating through the BCI is by using sensori-motor rhythms (SMR), which are modulated in the course of movement execution or movement imagination. The SMR corresponding to movement imagination can be detected after pre-processing the EEG, which is corrupted by important noise, and after training (see [1, 2, 3]). A well-trained classiﬁer can then use features of the SMR in order to discriminate periods of imagined movement from resting periods, when the user is idle. The detected mental states can be used as buttons in a Brain Computer Interface, mimicking traditional interfaces such as keyboard or mouse button. This paper deals with training a BCI corresponding to a single brain-controlled button (see [2, 4]), in which a button is pressed (and instantaneously released) when a certain imagined movement is detected. The important steps are thus to ﬁnd a suitable imaginary motor task, and to train a 1 classiﬁer. This is far from trivial, because appropriate tasks which can be well classiﬁed from the background resting state are highly variable among subjects; moreover, the classiﬁer requires to be trained on a large set of labeled data. The setting up of such a brain-controlled button can be very time consuming, given that many training examples need to be acquired for each of the imaginary motor task to be tested. The usual training protocol for a brain-controlled button is to display sequentially to the user a set of images, that serve as prompts to perform the corresponding imaginary movements. The collected data are used to train the classiﬁer, and to select the imaginary movement that seems to provide the highest classiﬁcation rate (compared to the background resting state). We refer to this imaginary movement as the “best imaginary movement”. In this paper, we focus on the part of the training phase that consists in efﬁciently ﬁnding this best imaginary movement. This is an important problem, since the SMR collected by the EEG are heterogeneously noisy: some imaginary motor tasks will provide higher classiﬁcation rates than others. In the literature, ﬁnding such imaginary motor tasks is deemed an essential issue (see [5, 6, 7]), but, to the best of our knowledge, no automatized protocol has yet been proposed to deal with it. We believe that enhancing the efﬁciency of the training phase is made even more essential by the facts that (i) the best imaginary movement differs from one user to another, e.g. the best imaginary movement for one user could be to imagine moving the right hand, and for the next, to imagine moving both feet (see [8]) and (ii) using a BCI requires much concentration, and a long training phase exhausts the user. If an “oracle” were able to state what the best imaginary movement is, then the training phase would consist only in requiring the user to perform this imaginary movement. The training set for the classiﬁer on this imaginary movement would be large, and no training time would be wasted in asking the user to perform sub-optimal and thus useless imaginary movements. The best imaginary movement is however not known in advance, and so the commonly used strategy (which we will refer to as uniform) consists in asking the user to perform all the movements a ﬁxed number of times. An alternative strategy is to learn while building the training set what imaginary movements seem the most promising, and ask the classiﬁer to perform these more often. This problem is quite archetypal to a ﬁeld of Machine Learning called Bandit Theory (initiated in [9]). Indeed, the main idea in Bandit Theory is to mix the Exploration of the possible actions1, and their Exploitation to perform the empirical best action. Contributions This paper builds on ideas of Bandit Theory, in order to propose an efﬁcient method to select the best imaginary movement for the activation of a brain-controlled button. To the best of our knowledge, this is the ﬁrst contribution to the automation and optimization of this task selection. • We design a BCI experiment for imaginary motor task selection, and collect data on several subjects, for different imaginary motor tasks, in the aim of testing our methods. • We provide a bandit algorithm (which is strongly inspired by the Upper Conﬁdence Bound Algorithm of [10]) adapted to this classiﬁcation problem. In addition, we propose several variants of this algorithm that are intended to deal with other slightly different scenarios that the practitioner might face. We believe that this bandit-based classiﬁcation technique is of independent interest and could be applied to other task selection procedures under constraints on the samples. • We provide empirical evidence that using such an algorithm considerably speeds up the training phase for the BCI. We gain up to 18% in terms of classiﬁcation rate, and up to 50% in training time, when compared to the uniform strategy traditionally used in the literature. The rest of the paper is organized as follows: in Section 2, we describe the EEG experiment we built in order to acquire data and simulate the training of a brain-controlled button. In Section 3, we model the task selection as a bandit problem, which is solved using an Upper Conﬁdence Bound algorithm. We motivate the choice of this algorithm by providing a performance analysis. Section 4, which is the main focus of this paper, presents results on simulated experiments, and proves empirically the gain brought forth by adaptive algorithms in this setting. We then conclude this paper with further perspectives. 1Here, the actions are images displayed to the BCI user as prompts to perform the corresponding imaginary tasks. 2 2 Material and protocol BCI systems based on SMR rely on the users’ ability to control their SMR in the mu (8-13Hz) and/or beta (16-24Hz) frequency bands [1, 2, 3]. Indeed, these rhythms are naturally modulated during real and imagined motor action. More precisely, real and imagined movements similarly activate neural structures located in the sensori-motor cortex, which can be detected in EEG recordings through increases in power (event related synchronization or ERS) and/or decreases in power (event related de-synchronization or ERD) in the mu and beta frequency bands [11, 12]. Because of the homuncular organization of the sensori-motor cortex [13], different limb movements may be distinguished according to the spatial layout of the ERD/ERS. BCI based on the control of SMR generally use movements lasting several seconds, that enable continuous control of multidimensional interfaces [1]. On the contrary this work targets a braincontrolled button that can be rapidly triggered by a short motor task [2, 4]. A vast variety of motor tasks can be used in this context, like imagining rapidly moving the hand, grasping an object, or kicking an imaginary ball. We remind that the best imaginary movement differs from one user to another (see [8]). As explained in the Introduction, the use of a BCI must always be preceded by a training phase. In the case of a BCI managing a brain-controlled button through SMR, this training phase consists in displaying to the user a sequence of images corresponding to movements, that he/she must imagine performing. By processing the EEG, the SMR associated to the imaginary movements and to idle periods can be extracted. Collecting these labeled data results in a training set, which serves to train the classiﬁer between the movements, and the idle periods. The imaginary movement with highest classiﬁcation rate is then selected to activate the button in the actual use of the BCI. The rest of this Section explains in more detail the BCI material and protocol used to acquire the EEG, and to extract the features from the signal. 2.1 The EEG experiment The EEG experiment was similar to the training of a brain-controlled button: we presented, at random timing, cue images during which the subjects were asked to perform 2 second long motor tasks (intended to activate the button). Six right-handed subjects, aged 24 to 39, with no disabilities, were sitting at 1.5m of a 23’ LCD screen. EEG was recorded dat a sampling rate of 512Hz via 11 scalp electrodes of a 64-channel cap and ampliﬁed with a TMSI ampliﬁer (see Figure 1). The OpenViBE platform [14] was used to run the experiment. The signal was ﬁltered in time through a band-pass ﬁlter, and in space through a surface Laplacian to increase the signal to noise ratio. The experiment was composed of 5 to 12 blocks of approximately 5 minutes. During each block, 4 cue images were presented for 2 seconds in a random order, 10 times each. The time between two image presentations varied between 1.5s and 10s. Each cue image was a prompt for the subject to perform or imagine the corresponding motor action during 2 seconds, namely moving the right or left hand, the feet or the tongue. 2.2 Feature extraction In the case of short motor tasks, the movement (real or imagined) produces an ERD in the mu and beta bands during the task, and is followed by a strong ERS [4] (sometimes called beta rebound as it is most easily seen in the beta frequency band). We extracted features of the mu and beta bands during the 2-second windows of the motor action and in the subsequent 1.5 seconds of signal in order to use the bursts of mu and beta power (ERS or rebound) that follow the indicated movement. Figure 1 shows a time-frequency map on which the movement and rebound windows are indicated. One may observe that, during the movement, the power in the mu and beta bands decreases (ERD) and that, approximately 1 second after the movement, it increases to reach a higher level than in the resting state (ERS). More precisely, the features were chosen as the power around 12Hz and 18Hz extracted at 3 electrodes over the sensori-motor cortex (C3, C4 and Cz). Thus, 6 features are extracted during the movement and 6 during the rebound. The lengths and positions of the windows and the frequency bands were chosen according to a preliminary study with one of the subjects and were deliberately kept ﬁxed for the other subjects. 3 One of the goals of our algorithm is to be able to select the best task among a large number of tasks. However, in our experiment, only a limited number of tasks were used (four), because we limited the length of the sessions in order not to tire the subjects. To demonstrate the usefulness of our method for a larger number of tasks, we decided to create artiﬁcial (degraded) tasks by mixing the features of one of the real tasks (the feet) with different proportions of the features extracted during the resting period. Figure 1: A: Layout of the 64 EEG cap, with (in black) the 3 electrodes from which the features are extracted. The electrodes marked in blue/grey are used for the Laplacian. B: Time-frequency map of the signal recorded on electrode C3, for a right hand movement lasting 2 seconds (subject 1). Four features (red windows) are extracted for each of the 3 electrodes. 2.3 Evaluation of performances For each task k, we can classify between when the subject is inactive and when he/she is performing task k. Consider a sample (X, Y ) ∼Dk where Dk is the distribution of the data restricted to task k and the idle task (task 0), X is the feature set, and Y is the label (1 if the sample corresponds to task k and 0 otherwise). We consider a compact set of classiﬁers H. Deﬁne the best classiﬁer in H for task k as h∗ k = arg minh∈H E(X,Y )∼Dk[1{h(X) = Y }]. Deﬁne the theoretical loss r∗ k of a task k as the probability of labeling incorrectly a new data drawn from Dk with the best classiﬁer h∗ k, that is to say r∗ k = 1 −P(X,Y )∼D(h∗ k(X) = Y ). At time t, there are Tk,t +T0,t samples (Xi, Yi)i≤Tk,t+T0,t (where Tk,t is the number of samples for task k, and T0,t is the number of samples for the idle task) that are available. With these data, we build the empirical minimizer of the loss ˆhk,t = arg minh∈H Tk,t+T0,t i=1 1{h(Xi) = Yi} . We deﬁne the empirical loss of this classiﬁer ˆrk,t = 1 −minh∈H Tk,t+T0,t i=1 1{h(Xi) = Yi} . Since during our experiments we collect, between each imaginary task, a sample of idle condition, we have T0,t Tk,t. From Vapnik-Chervonenkis theory (see [15] and also the Supplementary Material), we obtain with probability 1 −δ, that the error in generalization of classiﬁer ˆhk,t is not larger than r∗ k + O d log(1/δ) Tk,n , where d is the VC dimension of the domain of X. This implies that the performance of the optimal empirical classiﬁer for task k is close to the performance of the optimal classiﬁer for task k. Also with probability 1 −δ, \|ˆrk,t −r∗ k\| = O d log(1/δ) Tk,n . (1) We consider in this paper linear classiﬁers. In this case, the VC dimension d is the dimension of X, i.e. the number of features. The loss we considered ((0, 1) loss) is difﬁcult to minimize in practice because it is not convex. This is why we consider in this work the classiﬁer ˆhk,t provided by linear SVM. We also estimate the performance ˆrk,t of this classiﬁer by cross-validation: we use the leaveone-out technique when less than 8 samples of the task are available, and a 8-fold validation when more repetitions of the task have been recorded. As explained in [15], results similar to Equation 1 hold for this classiﬁer. We will use in the next Section the results of Equation 1, in order to select as fast as possible the task with highest r∗ k and collect as many samples from it as possible. 4 3 A bandit algorithms for optimal task selection In order to improve the efﬁciency of the training phase, it is important to ﬁnd out as fast as possible what are the most promising imaginary tasks (i.e. tasks with large r∗ k). Indeed, it is important to collect as many samples as possible from the best imaginary movement, so that the classiﬁer built for this task is as precise as possible. In this Section, we propose the UCB-Classif algorithm, inspired by the Upper Conﬁdence Bound algorithm in Bandit Theory (see [10]). 3.1 Modeling the problem by a multi-armed bandit Let K denote the number of different tasks2 and N the total number of rounds (the budget) of the training stage. Our goal is to ﬁnd a presentation strategy for the images (i.e. that choose at each timestep t ∈{1, . . . , N} an image kt ∈{1, . . . , K} to show), which allows to determine the “best”, i.e. most discriminative imaginary movement, with highest classiﬁcation rate in generalization). Note that, in order to learn an efﬁcient classiﬁer, we need as many training data as possible, so our presentation strategy should rapidly focus on the most promising tasks in order to obtain more samples from these rather than from the ones with small classiﬁcation rate. This issue is relatively close to the stochastic bandit problem [9]. The classical stochastic bandit problem is deﬁned by a set of K actions (pulling different arms of bandit machines) and to each action is assigned a reward distribution, initially unknown to the learner. At time t ∈{1, . . . , N}, if we choose an action kt ∈{1, . . . , K}, we receive a reward sample drawn independently from the distribution of the corresponding action kt. The goal is to ﬁnd a sampling strategy which maximizes the sum of obtained rewards. We model the K different images to be displayed as the K possible actions, and we deﬁne the reward as the classiﬁcation rate of the corresponding motor action. In the bandit problem, pulling a bandit arm directly gives a stochastic reward which is used to estimate the distribution of this arm. In our case, when we display a new image, we obtain a new data sample for the selected imaginary movement, which provides one more data sample to train or test the corresponding classiﬁer and thus obtain a more accurate performance. The main difference is that for the stochastic bandit problem, the goal is to maximize the sum of obtained rewards, whereas ours is to maximize the performance of the ﬁnal classiﬁer. However, the strategies are similar: since the distributions are initially unknown, one should ﬁrst explore all the actions (exploration phase) but then rapidly select the best one (exploitation phase). This is called the exploration-exploitation trade-off. 3.2 The UCB-classif algorithm The task presentation strategy is a close variant of the Upper Conﬁdence Bound (UCB) algorithm of [10], which builds high probability Upper Conﬁdence Bounds (UCB) on the mean reward value of each action, and selects at each time step the action with highest bound. We adapt the idea of this UCB algorithm to our adaptive classiﬁcation problem and call this algorithm UCB-classif (see the pseudo-code in Table 1). The algorithm builds a sequence of values Bk,t deﬁned as Bk,t = ˆrk,t + a log N Tk,t−1 , (2) where ˆrk,t represents an estimation of the classiﬁcation rate built from a q-fold cross-validation technique and the a corresponds to Equation 1 (see Supplementary Material for the precise theoretical value). The cross-validation uses a linear SVM classiﬁer based on the Tk,t data samples obtained (at time t) from task k. Writing r∗ k the classiﬁcation rate for the optimal linear SVM classiﬁer (which would be obtained by using a inﬁnite number of samples), we have the property that Bk,t is a high probability upper bound on r∗ k : P(Bk,t < r∗ k) decreases to zero polynomially fast (with N). The intuition behind the algorithm is that it selects at time t an action kt either because it has a good classiﬁcation rate ˆrk,t (thus it is interesting to obtain more samples from it, to perform exploitation) or because its classiﬁcation rate is highly uncertain since it has not been sampled many times, i.e., Tk,t−1 is small and then a log N Tk,t−1 is large (thus it is important to explore it more). With this strategy, the action that has the highest classiﬁcation rate is presented more often. It is indeed important to 2The tasks correspond to the imaginary movements of moving the feet, tongue, right hand, and left hand, plus 4 additional degraded tasks (so a total of K = 8 actions). 5 The UCB-Classif Algorithm Parameters: a, N, q Present each image q = 3 times (thus set Tk,qK = q). for t = qK + 1, . . . , N do Evaluate the performance ˆrk,t of each action (by a 8-split Cross Validation or leave-one-out if Tk,t < 8). Compute the UCB: Bk,t = ˆrk,t + q a log N Tk,t−1 for each action 1 ≤k ≤K. Select the image to present: kt = arg maxk∈{1,...,K} Bk,t. Update T: Tkt,t = Tkt,t−1 + 1 and ∀k = kt, Tk,t = Tk,t−1 end for Table 1: Pseudo-code of the UCB-classif algorithm. gather as much data as possible from the best action in order to build the best possible classiﬁer. The UCB-classif algorithm guarantees that the non-optimal tasks are chosen only a negligible fraction of times (O(log N) times out of a total budget N). The best action is thus sampled N −O(log N) times (this is formally proven in the Supplementary Material)3. It is a huge gain when compared to actual unadaptive procedures for building training sets. Indeed, the unadaptive optimal strategy is to sample each action N/K times, and thus the best task is only sampled N/K times (and not N −O(log N)). More precisely, we prove the following Theorem. Theorem 1 For any N ≥2qK, with probability at least 1 −1 N , if Equation 1 is satisﬁed (e.g. if the data are i.i.d.) and if a ≥5(d + 1) we have that the number of times that the image of the best imaginary movement is displayed by algorithm UCB-classif is such that (where r∗= maxk r∗ k) T ∗ N ≥N − k 8a log(8NK) (r∗−r∗ k)2 . The proof of this Theorem is in the provided Supplementary Material, Appendix A. 3.3 Discussion on variants of this algorithm We stated that our objective, given a ﬁxed budget N, is to ﬁnd as fast as possible the image with highest classiﬁcation rate, and to train the classiﬁer with as many samples as possible. Depending on the objectives of the practitioner, other possible aims can however be pursued. We brieﬂy describe two other settings, and explain how ideas from the bandit setting can be used to adapt to these different scenarios. Best stopping time: A close, yet different, goal, is to ﬁnd the best time for stopping the training phase. In this setting, the practitioner’s aim is to stop the training phase as soon as the algorithm has built an almost optimal classiﬁer for the user. With ideas very similar to those developed in [16] (and extended for bandit problems in e.g. [17]), we can think of an adaptation of algorithm UCB-classif to this new formulation of the problem. Assume that the objective is to ﬁnd an −optimal classiﬁer with probability 1 −δ, and to stop the training phase as soon as this classiﬁer is built. Then using ideas similar to those presented in [17], an efﬁcient algorithm will at time t select the action that maximizes B k,t = ˆrk,t + a log(NK/δ) Tk,t−1 and will stop at the ﬁrst time ˆT when there is an action ˆk∗such that ∀k = ˆk∗, B ˆ k∗, ˆT −B k, ˆT > + 2 a log(NK/δ) Tk, ˆ T −1 . We thus shorten the training phase almost optimally on the class of adaptive algorithms (see [17] for more details). Choice of the best action with a limited budget: Another question that could be of interest for the practitioner is to ﬁnd the best action with a ﬁxed budget (and not train the classiﬁer at the same time). We can use ideas from paper [18] to modify UCB-classif. By selecting at each time t the action that maximizes B k,t = ˆrk,t + a(N−K) Tk,t−1 , we attain this objective in the sense that we guarantee that the probability of choosing a non-optimal action decreases exponentially fast with N. 4 Results We present some numerical experiments illustrating the efﬁciency of Bandit algorithms for this problem. Although the objective is to implement UCB-classif on the BCI device, in this paper we test the algorithm on real databases that we bootstrap (this is explained in details later). This kind of 3The ideas of the proof are very similar to the ideas in [10], with the difference that the upper bounds have to be computed using inequalities based on VC-dimension. 6 procedure is common for testing the performances of adaptive algorithms (see e.g. [19]). Acquiring data for BCI experiments is time-consuming because it requires a human subject to sit through the experiment. The advantage of bootstrapping is that several experiments can be performed with a single database, making it possible to provide conﬁdence bands for the results. In this Section, we present the experiments we performed, i.e. describe the kind of data we collect, and illustrate the performance of our algorithm on these data. 4.1 Performances of the different tasks The images that were displayed to the subjects correspond to movements of both feet, of the tongue, of the right hand, and of the left hand (4 actions in total). Six right-handed subjects went through the experiment with real movements and three of them went through an additional shorter experiment with imaginary movements. For four of the six subjects, the best performance for the real movement was achieved with the right hand, whereas the two other subjects’ best tasks corresponded to the left hand and the feet. We collected data for these four tasks. It is not a large number of tasks but we needed a large amount of data for each of them in order to do a signiﬁcant comparison. In order to have a larger number of tasks and place ourselves in a more realistic situation, we created some articicial tasks (see below). Results on only four tasks are presented in a companion article [20]. Surprisingly, two of the subjects who went through the imaginary experiment obtained better results while imagining moving their left hand than their right hand, which was the best task during the real movements experiment. For the third subject who did the imaginary experiment, the best task was the feet, as for the real movement experiment. As explained in section 2.2, for this study we chose to use a very small set of ﬁxed features (12 features, extracted from 3 electrodes, 2 frequency bands and 2 time-windows), calibrated on only one of the six subjects during a preliminary experiment. In this work, the features were not subject-speciﬁc. It would certainly improve the classiﬁcation results to tune the features. Using the bandit algorithm to tune the features and to select the tasks at the same time presents a risk overﬁtting, especially for an initially very small amount of data, and also a risk of biasing the task selection to those that have been the most sampled, and for which the features will thus be the best tuned. Although for all the subjects, the best task achieved a classiﬁcation accuracy above 85%, this accuracy could further be improved by using a larger set of subject-speciﬁc features [21] and more advanced techniques (like the CSP [22] or feature selection [23]). 4.2 Performances of the bandit algorithm We compare the performance of the UCB-classif sampling strategy to a uniform strategy, i.e. the standard way of selecting a task, consisting of N/K presentations of each image. Movement Number of presentations Off-line classiﬁcation rate Right hand 28.6 ± 12.8 88.1% Left hand 9.0 ± 7.5 80.5% Feet 11.6 ± 9.5 82.6% Tongue 4.5 ± 1.5 63.3% Feet 80% 5.1 ± 2.6 71.4% Feet 60% 4.0 ± 1.5 68.6% Feet 40% 3.5 ± 1.0 59.2% Feet 20% 3.5 ± 0.9 54.0% Total presentations 70 Table 2: Actions presented by the UCB-classif algorithm for subject 5 across 500 simulated online BCI experiments. Feet X% is a mixture of the features measured during feet movement and during the resting condition, with a X/100-X proportion. (The off-line classiﬁcation rate of each action gives an idea of the performance of each action). To obtain a realistic evaluation of the performance of our algorithm we use a bootstrap technique. More precisely, for each chosen budget N, for the UCB-classif strategy and the uniform strategy, we simulated 500 online BCI experiments by randomly sampling from the acquired data of each action. Table 2 shows, for one subject and for a ﬁxed budget of N = 70, the average number of presentations of each task Tk, and its standard deviation, across the 500 simulated experiments. It also contains the off-line classiﬁcation rate of each task to give an idea of the performances of the different tasks for this subject. We can see that very little budget is allocated to the tongue movement and to the most degraded feet 20% tasks, which are the less discriminative actions, and that most of the budget is devoted to the right hand, thus enabling a more efﬁcient training. 7 Figure 2 and Table 3 show, for different budgets (N), the performance of the UCB-classif algorithm versus the uniform technique. The training of the classiﬁer is done on the actions presented during the simulated BCI experiment, and the testing on the remaining data. For a budget N > 70 the UCB-classif could not be used for all the subjects because there was not enough data for the best action (One subject only underwent a session of 5 blocks and so only 50 samples of each motor task were recorded. If we try to simulate an on-line experiment using the UCB-classif with a budget higher than N = 70 it is likely to ask for a 51th presentation of the best task, which has not been recorded). The classiﬁcation results depend on which data is used to simulate the BCI experiment. To give an idea of this variability, the ﬁrst and last quartiles are plotted as error bars on the graphics. Budget (N) Length of the experiment Uniform strategy UCB-classif Beneﬁt 30 3min45 47.7% 64.4% +16.7% 40 5min 58.5% 77.2% +18.7% 50 6min15 63.4% 82.0% +18.5% 60 7min30 67.0% 84.0% +17.1% 70 8min45 70.1% 85.7% +15.6% 100 12min30 77.6% * 150 18min45 83.2% * 180 22min30 85.2% * Table 3: Comparison of the performances of the UCB-classif vs. the uniform strategy for different budgets, averaged over all subjects, for real movements. (The increases are signiﬁcant with p > 95%.) For each budget, we give an indication of the length of the experiment (without counting pauses between blocks) required to obtain this amount of data. The UCB-classif strategy signiﬁcantly outperforms the uniform strategy, even for relatively small N. On average on all the users it even gives better classiﬁcation rates when using only half of the available samples, compared to the uniform strategy. Indeed, Table 3 shows that, to achieve a classiﬁcation rate of 85% the UCB-classif only requires a budget of N = 70 whereas the uniform strategy needs N = 180. We believe that such gain in performance motivates the implementation of such a training algorithm in BCI devices, specially since the algorithm itself is quite simple and fast. 30 40 50 60 70 80 90 100 110 120 40 50 60 70 80 90 Budget N Classification rate of the chosen movement Sujet 1 real movement Adaptative Algorithm Uniform Strategy 30 40 50 60 40 50 60 70 80 90 Budget N Classification rate of the chosen movement Sujet 2 imaginary movement Adaptative Algorithm Uniform Strategy 30 40 50 60 70 40 50 60 70 80 90 Budget N Classification rate of the chosen movement Sujet 3 imaginary movement Adaptative Algorithm Uniform Strategy Figure 2: UCB-classif algorithm (full line, red) versus uniform strategy (dashed line, black). 5 Conclusion The method presented in this paper falls in the category of adaptive BCI based on Bandit Theory. To the best of our knowledge, this is the ﬁrst such method for dealing with automatic task selection. UCB-classif is a new adaptive algorithm that allows to automatically select a motor task in view of a brain-controlled button. By rapidly eliminating non-efﬁcient motor tasks and focusing on the most promising ones, it enables a better task selection procedure than a uniform strategy. Moreover, by more frequently presenting the best task it allows a good training of the classiﬁer. This algorithm enables to shorten the training period, or equivalently, to allow for a larger set of possible movements among which to select the best. In a paper due to appear [20], we implement this algorithm online. A future research direction is to learn several discriminant tasks in order to activate several buttons. Acknowledgements This work was partially supported by the French ANR grant Co-Adapt ANR-09-EMER-002, Nord-Pas-de-Calais Regional Council, French ANR grant EXPLO-RA (ANR08-COSI-004), the EC Seventh Framework Programme (FP7/2007-2013) under grant agreement 270327 (CompLACS project), and by Pascal-2. 8 References [1] D. J. McFarland, W. A. Sarnacki, and J. R Wolpaw. Electroencephalographic (EEG) control of threedimensional movement. Journal of Neural Engineering, 7(3):036007, 2010. [2] T. Solis-Escalante, G. Mller-Putz, C. Brunner, V. Kaiser, and G. Pfurtscheller. Analysis of sensorimotor rhythms for the implementation of a brain switch for healthy subjects. Biomedical Signal Processing and Control, 5(1):15 – 20, 2010. [3] B. Blankertz, G. Dornhege, M. Krauledat, K.-R. Mller, and G. Curio. The non-invasive berlin braincomputer interface: Fast acquisition of effective performance in untrained subjects. NeuroImage, 37(2):539 – 550, 2007. [4] J. Fruitet, M. Clerc, and T. Papadopoulo. 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4,492	Localizing 3D cuboids in single-view images Jianxiong Xiao Bryan C. Russell∗ Antonio Torralba Massachusetts Institute of Technology ∗University of Washington Abstract In this paper we seek to detect rectangular cuboids and localize their corners in uncalibrated single-view images depicting everyday scenes. In contrast to recent approaches that rely on detecting vanishing points of the scene and grouping line segments to form cuboids, we build a discriminative parts-based detector that models the appearance of the cuboid corners and internal edges while enforcing consistency to a 3D cuboid model. Our model copes with different 3D viewpoints and aspect ratios and is able to detect cuboids across many different object categories. We introduce a database of images with cuboid annotations that spans a variety of indoor and outdoor scenes and show qualitative and quantitative results on our collected database. Our model out-performs baseline detectors that use 2D constraints alone on the task of localizing cuboid corners. 1 Introduction Extracting a 3D representation from a single-view image depicting a 3D object has been a longstanding goal of computer vision [20]. Traditional approaches have sought to recover 3D properties, such as creases, folds, and occlusions of surfaces, from a line representation extracted from the image [18]. Among these are works that have characterized and detected geometric primitives, such as quadrics (or “geons”) and surfaces of revolution, which have been thought to form the components for many different object types [1]. While these approaches have achieved notable early successes, they could not be scaled-up due to their dependence on reliable contour extraction from natural images. In this work we focus on the task of detecting rectangular cuboids, which are a basic geometric primitive type and occur often in 3D scenes (e.g. indoor and outdoor man-made scenes [22, 23, 24]). Moreover, we wish to recover the shape parameters of the detected cuboids. The detection and recovery of shape parameters yield at least a partial geometric description of the depicted scene, which allows a system to reason about the affordances of a scene in an object-agnostic fashion [9, 15]. This is especially important when the category of the object is ambiguous or unknown. There have been several recent efforts that revisit this problem [9, 11, 12, 17, 19, 21, 26, 28, 29]. Although there are many technical differences amongst these works, the main pipeline of these approaches is similar. Most of them estimate the camera parameters using three orthogonal vanishing points with a Manhattan world assumption of a man-made scene. They detect line segments via Canny edges and recover surface orientations [13] to form 3D cuboid hypotheses using bottomup grouping of line and region segments. Then, they score these hypotheses based on the image evidence for lines and surface orientations [13]. In this paper we look to take a different approach for this problem. As shown in Figure 1, we aim to build a 3D cuboid detector to detect individual boxy volumetric structures. We build a discriminative parts-based detector that models the appearance of the corners and internal edges of cuboids while enforcing spatial consistency of the corners and edges to a 3D cuboid model. Our model is trained in a similar fashion to recent work that detects articulated human body joints [27]. 1 Input Image Output Detection Result 3D Cuboid Detector detect Synthesized New Views Figure 1: Problem summary. Given a single-view input image, our goal is to detect the 2D corner locations of the cuboids depicted in the image. With the output part locations we can subsequently recover information about the camera and 3D shape via camera resectioning. Our cuboid detector is trained across different 3D viewpoints and aspect ratios. This is in contrast to view-based approaches for object detection that train separate models for different viewpoints, e.g. [7]. Moreover, instead of relying on edge detection and grouping to form an initial hypothesis of a cuboid [9, 17, 26, 29], we use a 2D sliding window approach to exhaustively evaluate all possible detection windows. Also, our model does not rely on any preprocessing step, such as computing surface orientations [13]. Instead, we learn the parameters for our model using a structural SVM framework. This allows the detector to adapt to the training data to identify the relative importance of corners, edges and 3D shape constraints by learning the weights for these terms. We introduce an annotated database of images with geometric primitives labeled and validate our model by showing qualitative and quantitative results on our collected database. We also compare to baseline detectors that use 2D constraints alone on the tasks of geometric primitive detection and part localization. We show improved performance on the part localization task. 2 Model for 3D cuboid localization We represent the appearance of cuboids by a set of parts located at the corners of the cuboid and a set of internal edges. We enforce spatial consistency among the corners and edges by explicitly reasoning about its 3D shape. Let I be the image and pi = (xi, yi) be the 2D image location of the ith corner on the cuboid. We deﬁne an undirected loopy graph G = (V, E) over the corners of the cuboid, with vertices V and edges E connecting the corners of the cuboid. We illustrate our loopy graph layout in Figure 2(a). We deﬁne a scoring function associated with the corner locations in the image: S(I, p) = i∈V wH i · HOG(I, pi) + ij∈E wD ij · Displacement2D(pi, pj) + ij∈E wE ij · Edge(I, pi, pj) + wS · Shape3D(p) (1) where HOG(I, pi) is a HOG descriptor [4] computed at image location pi and Displacement2D(pi, pj) = −[(xi −xj)2, xi −xj, (yi −yj)2, yi −yj] is a 2D corner displacement term that is used in other pictorial parts-based models [7, 27]. By reasoning about the 3D shape, our model handles different 3D viewpoints and aspect ratios, as illustrated in Figure 2. Notice that our model is linear in the weights w. Moreover, the model is ﬂexible as it adapts to the training data by automatically learning weights that measure the relative importance of the appearance and spatial terms. We deﬁne the Edge and Shape3D terms as follows. Edge(I, pi, pj): The internal edge information on cuboids is informative and provides a salient feature for the locations of the corners. For this, we include a term that models the appearance of the internal edges, which is illustrated in Figure 3. For adjacent corners on the cuboid, we identify the edge between the two corners and calculate the image evidence to support the existence of such an edge. Given the corner locations pi and pj, we use Chamfer matching to align the straight line between the two corners to edges extracted from the image. We ﬁnd image edges using Canny edge detection [3] and efﬁciently compute the distance of each pixel along the line segment to the nearest edge via the truncated distance transform. We use Bresenham’s line algorithm [2] to efﬁciently ﬁnd the 2D image locations on the line between the two points. The ﬁnal edge term is the negative mean value of the Chamfer matching score for all pixels on the line. As there are usually 9 visible edges for a cuboid, we have 9 dimensions for the edge term. 2 (a) Our Full Model. (b) 2D Tree Model. (c) Example Part Detections. Figure 2: Model visualization. Corresponding model parts are colored consistently in the ﬁgure. In (a) and (b) the displayed corner locations are the average 2D locations across all viewpoints and aspect ratios in our database. In (a) the edge thickness corresponds to the learned weight for the edge term. We can see that the bottom edge is signiﬁcantly thicker, which indicates that it is informative for detection, possibly due to shadows and contact with a supporting plane. Shape3D(p): The 3D shape of a cuboid constrains the layout of the parts and edges in the image. We propose to deﬁne a shape term that measures how well the conﬁguration of corner locations respect the 3D shape. In other words, given the 2D locations p of the corners, we deﬁne a term that tells us how likely this conﬁguration of corner locations p can be interpreted as the reprojection of a valid cuboid in 3D. When combined with the weights wS, we get an overall evaluation of the goodness of the 2D locations with respect to the 3D shape. Let X (written in homogeneous coordinates) be the 3D points on the unit cube centered at the world origin. Then, X transforms as x = PLX, where L is a matrix that transforms the shape of the unit cube and P is a 3 × 4 camera matrix. For each corner, we use the other six 2D corner locations to estimate the product PL using camera resectioning [10]. The estimated matrix is used to predict the corner location. We use the negative L2 distance to the predicted corner location as a feature for the corner in our model. As we model 7 corners on the cuboid, there are 7 dimensions in the feature vector. When combined with the learned weights wS through dot-product, this represents a weighted reprojection error score. 2.1 Inference Our goal is to ﬁnd the 2D corner locations p over the HOG grid of I that maximizes the score given in Equation (1). Note that exact inference is hard due to the global shape term. Therefore, we propose a spanning tree approximation to the graph to obtain multiple initial solutions for possible corner locations. Then we adjust the corner locations using randomized simple hill climbing. For the initialization, it is important for the computation to be efﬁcient since we need to evaluate all possible detection windows in the image. Therefore, for simplicity and speed, we use a spanning tree T to approximate the full graph G, as shown in Figure 2(b). In addition to the HOG feature as a unary term, we use a popular pairwise spring term along the edges of the tree to establish weak spatial constraints on the corners. We use the following scoring function for the initialization: ST (I, p) = i∈V wH i · HOG(I, pi) + ij∈T wD ij · Displacement2D(pi, pj) (2) Note that the model used for obtaining initial solutions is similar to [7, 27], which is only able to handle a ﬁxed viewpoint and 2D aspect ratio. Nonetheless, we use it since it meets our speed requirement via dynamic programming and the distance transform [8]. With the tree approximation, we pick the top 1000 possible conﬁgurations of corner locations from each image and optimize our scoring function by adjusting the corner locations using randomized simple hill climbing. Given the initial corner locations for a single conﬁguration, we iteratively choose a random corner i with the goal of ﬁnding a new pixel location ˆpi that increases the scoring function given in Equation (1) while holding the other corner locations ﬁxed. We compute the scores at neighboring pixel locations to the current setting pi. We also consider the pixel location that the 3D rigid model predicts when estimated from the other corner locations. We randomly choose one of the locations and update pi if it yields a higher score. Otherwise, we choose another random corner and location. The algorithm terminates when no corner can reach a location that improves the score, which indicates that we have reached a local maxima. During detection, since the edge and 3D shape terms are non-positive and the weights are constrained to be positive, this allows us to upper-bound the scoring function and quickly reject candidate loca3 −100 −90 −80 −70 −60 −50 −40 −30 −20 −10 Image Distance Transformed Edge Map Pixels Covered by Line Segment Dot-product is the Edge Term Figure 3: Illustration of the edge term in our model. Given line endpoints, we compute a Chamfer matching score for pixels that lie on the line using the response from a Canny edge detector. tions without evaluating the entire function. Also, since only one corner can change locations at each iteration, we can reuse the computed scoring function from previous iterations during hill climbing. Finally, we perform non-maximal suppression among the parts and then perform non-maximal suppression over the entire object to get the ﬁnal detection result. 2.2 Learning For learning, we ﬁrst note that our scoring function in Equation (1) is linear in the weights w. This allows us to use existing structured prediction procedures for learning. To learn the weights, we adapt the structural SVM framework of [16]. Given positive training images with the 2D corner locations labeled {In, pn} and negative training images {In}, we wish to learn weights and bias term β = (wH, wD, wE, wS, b) that minimizes the following structured prediction objective function: min β,ξ≥0 1 2β · β + C X n ξn (3) ∀n ∈pos β · Φ (In, pn) ≥1 −ξn ∀n ∈neg, ∀p ∈P β · Φ (In, p) ≤−1 + ξn where all appearance and spatial feature vectors are concatenated into the vector Φ(In, p) and P is the set of all possible part locations. During training we constrain the weights wD, wE, wS ≥ 0.0001. We tried mining negatives from the wrong corner locations in the positive examples but found that it did not improve the performance. We also tried latent positive mining and empirically observed that it slightly helps. Since the latent positive mining helped, we also tried an offset compensation as post-processing to obtain the offset of corner locations introduced during latent positive mining. For this, we ran the trained detector on the training set to obtain the offsets and used the mean to compensate for the location changes. However, we observed empirically that it did not help performance. 2.3 Discussion Sliding window object detectors typically use a root ﬁlter that covers the entire object [4] or a combination of root ﬁlter and part ﬁlters [7]. The use of a root ﬁlter is sufﬁcient to capture the appearance for many object categories since they have canonical 3D viewpoints and aspect ratios. However, cuboids in general span a large number of object categories and do not have a consistent 3D viewpoint or aspect ratio. The diversity of 3D viewpoints and aspect ratios causes dramatic changes in the root ﬁlter response. However, we have observed that the responses for the part ﬁlters are less affected. Moreover, we argue that a purely view-based approach that trains separate models for the different viewpoints and aspect ratios may not capture well this diversity. For example, such a strategy would require dividing the training data to train each model. In contrast, we train our model for all 3D viewpoints and aspect ratios. We illustrate this in Figure 2, where detected parts are colored consistently in the ﬁgure. As our model handles different viewpoints and aspect ratios, we are able to make use of the entire database during training. Due to the diversity of cuboid appearance, our model is designed to capture the most salient features, namely the corners and edges. While the corners and edges may be occluded (e.g. by self-occlusion, 4 (a) (b) (c) 1° 9° 18° 26° 37° 43° 0 15 30 45 −45 0 45 90 Azimuth Elevation Figure 4: Illustration of the labeling tool and 3D viewpoint statistics. (a) A cuboid being labeled through the tool. A projection of the cuboid model is overlaid on the image and the user must select and drag anchor points to their corresponding location in the image. (b) Scatter plot of 3D azimuth and elevation angles for annotated cuboids with zenith angle close zero. We perform an image left/right swap to limit the rotation range. (c) Crops of cuboids at different azimuth angles for a ﬁxed elevation, with the shown examples marked as red points in the scatter plot of (b). other objects in front, or cropping), for now we do not handle these cases explicitly in our model. Furthermore, we do not make use of other appearance cues, such as the appearance within the cuboid faces, since they have a larger variation across the object categories (e.g. dice and ﬁre alarm trigger) and may not generalize as well. We also take into account the tractability of our model as adding additional appearance cues will increase the complexity of our model and the detector needs to be evaluated over a large number of possible sliding windows in an image. Compared with recent approaches that detect cuboids by reasoning about the shape of the entire scene [9, 11, 12, 17, 19, 29], one of the key differences is that we detect cuboids directly without consideration of the global scene geometry. These prior approaches rely heavily on the assumption that the camera is located inside a cuboid-like room and held at human height, with the parameters of the room cuboid inferred through vanishing points based on a Manhattan world assumption. Therefore, they cannot handle outdoor scenes or close-up snapshots of an object (e.g. the boxes on a shelf in row 1, column 3 of Figure 6). As our detector is agnostic to the scene geometry, we are able to detect cuboids even when these assumptions are violated. While previous approaches reason over rigid cuboids, our model is ﬂexible in that it can adapt to deformations of the 3D shape. We observe that not all cuboid-like objects are perfect cuboids in practice. Deformations of the shape may arise due to the design of the object (e.g. the printer in Figure 1), natural deformation or degradation of the object (e.g. a cardboard box), or a global transformation of the image (e.g. camera radial distortion). We argue that modeling the deformations is important in practice since a violation of the rigid constraints may make a 3D reconstructionbased approach numerically unstable. In our approach, we model the 3D deformation and allow the structural SVM to learn based on the training data how to weight the importance of the 3D shape term. Moreover, a rigid shape requires a perfect 3D reconstruction and it is usually done with nonlinear optimization [17], which is expensive to compute and becomes impractical in an exhaustive sliding-window search in order to maintain a high recall rate. With our approach, if a rigid cuboid is needed, we can recover the 3D shape parameters via camera resectioning, as shown in Figure 9. 3 Database of 3D cuboids To develop and evaluate any models for 3D cuboid detection in real-world environments, it is necessary to have a large database of images depicting everyday scenes with 3D cuboids labeled. In this work, we seek to build a database by manually labeling point correspondences between images and 3D cuboids. We have built a labeling tool that allows a user to select and drag key points on a projected 3D cuboid model to its corresponding location in the image. This is similar to existing tools, such as Google building maker [14], which has been used to build 3D models of buildings for maps. Figure 4(a) shows a screenshot of our tool. For the database, we have harvested images from four sources: (i) a subset of the SUN database [25], which contains images depicting a large variety of different scene categories, (ii) ImageNet synsets [5] with objects having one or more 3D cuboids depicted, (iii) images returned from an Internet search using keywords for objects that are wholly or partially described by 3D cuboids, and (iv) a set of images that we manually collected from our personal photographs. Given the corner correspondences, the parameters for the 3D cuboids and camera are estimated. The cuboid and camera parameters are estimated up to a similarity transformation via camera resectioning using Levenberg-Marquardt optimization [10]. 5 Figure 5: Single top 3D cuboid detection in each image. Yellow: ground truth, green: correct detection, red: false alarm. Bottom row - false positives. The false positives tend to occur when a part ﬁres on a “cuboid-like” corner region (e.g. row 3, column 5) or ﬁnds a smaller cuboid (e.g. the Rubik’s cube depicted in row 3, column 1). Figure 6: All 3D cuboid detections above a ﬁxed threshold in each image. Notice that our model is able to detect the presence of multiple cuboids in an image (e.g. row 1, columns 2-5) and handles partial occlusions (e.g. row 1, column 4), small objects, and a range of 3D viewpoints, aspect ratios, and object classes. Moreover, the depicted scenes have varying amount of clutter. Yellow - ground truth. Green - correct prediction. Red - false positive. Line thickness corresponds to detector conﬁdence. For our database, we have 785 images with 1269 cuboids annotated. We have also collected a negative set containing 2746 images that do contain any cuboid like objects. We perform an image left/right swap to limit the rotation range. As a result, the min/max azimuth, elevation, and zenith angles are 0/45, -90/90, -180/180 degrees respectively. In Figure 4(b) we show a scatter plot of the azimuth and elevation angles for all of the labeled cuboids with zenith angle close to zero. Notice that the cuboids cover a large range of azimuth angles for elevation angles between 0 (frontal view) and 45 degrees. We also show a number of cropped examples for a ﬁxed elevation angle in Figure 4(c), with their corresponding azimuth angles indicated by the red points in the scatter plot. Figure 8(c) shows the distribution of objects from the SUN database [25] that overlap with our cuboids (there are 326 objects total from 114 unique classes). Compared with [12], our database covers a larger set of object and scene categories, with images focusing on both objects and scenes (all images in [12] are indoor scene images). Moreover, we annotate objects closely resembling a 3D cuboid (in [12] there are many non-cuboids that are annotated with a bounding cuboid) and overall our cuboids are more accurately labeled. 4 Evaluation In this section we show qualitative results of our model on the 3D cuboids database and report quantitative results on two tasks: (i) 3D cuboid detection and (ii) corner localization accuracy. For training and testing, we randomly split equally the positive and negative images. As discussed in Section 3, there is rotational symmetry in the 3D cuboids. During training, we allow the image 6 (a) (b) (c) Figure 7: Corner localization comparison for detected geometric primitives. (a) Input image and ground truth annotation. (b) 2D tree-based initialization. (c) Our full model. Notice that our model is able to better localize cuboid corners over the baseline 2D tree-based model, which corresponds to 2D parts-based models used in object detection and articulated pose estimation [7, 27]. The last column shows a failure case where a part ﬁres on a “cuboid-like” corner region in the image. to mirror left-right and orient the 3D cuboid to minimize the variation in rotational angle. During testing, we run the detector on left-right mirrors of the image and select the output at each location with the highest detector response. For the parts we extract HOG features [4] in a window centered at each corner with scale of 10% of the object bounding box size. Figure 5 shows the single top cuboid detection in each image and Figure 6 shows all of the most conﬁdent detections in the image. Notice that our model is able to handle partial occlusions (e.g. row 1, column 4 of Figure 6), small objects, and a range of 3D viewpoints, aspect ratios, and object classes. Moreover, the depicted scenes have varying amount of clutter. We note that our model fails when a corner ﬁres on a “cuboid-like” corner region (e.g. row 3, column 5 of Figure 5). We compare the various components of our model against two baseline approaches. The ﬁrst baseline is a root HOG template [4] trained over the appearance within a bounding box covering the entire object. A single model using the root HOG template is trained for all viewpoints and aspect ratios. During detection, output corner locations corresponding to the average training corner locations relative to the bounding boxes are returned. The second baseline is the 2D tree-based approximation of Equation (2), which corresponds to existing 2D parts models used in object detection and articulated pose estimation [7, 27]. Figure 7 shows a qualitative comparison of our model against the 2D tree-based model. Notice that our model localizes well and often provides a tighter ﬁt to the image data than the baseline model. We evaluate geometric primitive detection accuracy using the bounding box overlap criteria in the Pascal VOC [6]. We report precision recall in Figure 8(a). We have observed that all of the cornerbased models achieve almost identical detection accuracy across all recall levels, and out-perform the root HOG template detector [4]. This is expected as we initialize our full model with the output of the 2D tree-based model and it generally does not drift too far from this initialization. This in effect does not allow us to detect additional cuboids but allows for better part localization. In addition to detection accuracy, we also measure corner localization accuracy for correctly detected examples for a given model. A corner is deemed correct if its predicted image location is within t pixels of the ground truth corner location. We set t to be 15% of the square root of the area of the ground truth bounding box for the object. The reported trends in the corner localization performance hold for nearby values of t. In Figure 8 we plot corner localization accuracy as a function of recall and compare our model against the two baselines. Moreover, we report performance when either the edge term or the 3D shape term is omitted from our model. Notice that our full model out-performs the other baselines. Also, the additional edge and 3D shape terms provide a gain in performance over using the appearance and 2D spatial terms alone. The edge term provides a slightly larger gain in performance over the 3D shape term, but when integrated together consistently provides the best performance on our database. 7 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 recall precision recall Root Filter [0.16] 2D Tree Approximation [0.23] Full Model−Edge [0.26] Full Model−Shape [0.24] Full Model [0.24] (a) Cuboid detection 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 recall reprojection accuracy (criteria=0.150) Root Filter [0.25] 2D Tree Approximation [0.30] Full Model−Edge [0.37] Full Model−Shape [0.37] Full Model [0.38] (b) Corner localization stove (5/13) refrigerator (5/8) night table occluded (5/12) kitchen island (5/6) cabinets (5/22) brick (5/5) stand (7/11) CPU (7/8) table (8/26) desk (8/22) box (9/18) chest of drawers (10/10) bed (15/22) cabinet (28/87) others 97 categories (168/883) night table (15/29) building (16/49) screen (5/16) (c) Object distribution Figure 8: Cuboid detection (precision vs. recall) and corner localization accuracy (accuracy vs. recall). The area under the curve is reported in the plot legends. Notice that all of the corner-based models achieve almost identical detection accuracy across all recall levels and out-perform the root HOG template detector [4]. For the task of corner localization, our full model out-performs the two baseline detectors or when either the Edge or Shape3D terms are omitted from our model. (c) Distribution of objects from the SUN database [25] that overlap with our cuboids. There are 326 objects total from 114 unique classes. The ﬁrst number within the parentheses indicates the number of instances in each object category that overlaps with a labeled cuboid, while the second number is the total number of labeled instances for the object category within our dataset. Figure 9: Detected cuboids and subsequent synthesized new views via camera resectioning. 5 Conclusion We have introduced a novel model that detects 3D cuboids and localizes their corners in single-view images. Our 3D cuboid detector makes use of both corner and edge information. Moreover, we have constructed a dataset with ground truth cuboid annotations. Our detector handles different 3D viewpoints and aspect ratios and, in contrast to recent approaches for 3D cuboid detection, does not make any assumptions about the scene geometry and allows for deformation of the 3D cuboid shape. As HOG is not invariant to viewpoint, we believe that part mixtures would allow the model to be invariant to viewpoint. We believe our approach extends to other shapes, such as cylinders and pyramids. Our work raises a number of (long-standing) issues that would be interesting to address. For instance, which objects can be described by one or more geometric primitives and how to best represent the compositionality of objects in general? By detecting geometric primitives, what applications and systems can be developed to exploit this? Our dataset and source code is publicly available at the project webpage: http://SUNprimitive.csail.mit.edu. Acknowledgments: Jianxiong Xiao is supported by Google U.S./Canada Ph.D. Fellowship in Computer Vision. Bryan Russell was funded by the Intel Science and Technology Center for Pervasive Computing (ISTC-PC). This work is funded by ONR MURI N000141010933 and NSF Career Award No. 0747120 to Antonio Torralba. 8 References [1] I. Biederman. Recognition by components: a theory of human image interpretation. Pyschological review, 94:115–147, 1987. [2] J. E. Bresenham. Algorithm for computer control of a digital plotter. IBM Systems Journal, 4(1):25–30, 1965. [3] J. F. Canny. A computational approach to edge detection. IEEE PAMI, 8(6):679–698, 1986. [4] N. Dalal and B. Triggs. Histograms of Oriented Gradients for Human Detection. In CVPR, 2005. [5] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. ImageNet: A large-scale hierarchical image database. In CVPR, 2009. [6] M. 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4,493	A Convex Formulation for Learning Scale-Free Networks via Submodular Relaxation Aaron J. Defazio NICTA/Australian National University Canberra, ACT, Australia aaron.defazio@anu.edu.au Tiberio S. Caetano NICTA/ANU/University of Sydney Canberra and Sydney, Australia tiberio.caetano@nicta.com.au Abstract A key problem in statistics and machine learning is the determination of network structure from data. We consider the case where the structure of the graph to be reconstructed is known to be scale-free. We show that in such cases it is natural to formulate structured sparsity inducing priors using submodular functions, and we use their Lov´asz extension to obtain a convex relaxation. For tractable classes such as Gaussian graphical models, this leads to a convex optimization problem that can be efﬁciently solved. We show that our method results in an improvement in the accuracy of reconstructed networks for synthetic data. We also show how our prior encourages scale-free reconstructions on a bioinfomatics dataset. Introduction Structure learning for graphical models is a problem that arises in many contexts. In applied statistics, undirected graphical models can be used as a tool for understanding the underlying conditional independence relations between variables in a dataset. For example, in bioinfomatics Gaussian graphical models are ﬁtted to data resulting from micro-array experiments, where the ﬁtted graph can be interpreted as a gene expression network [9]. In the context of Gaussian models, the structure learning problem is known as covariance selection [8]. The most common approach is the application of sparsity inducing regularization to the maximum likelihood objective. There is a signiﬁcant body of literature, more than 30 papers by our count, on various methods of optimizing the L1 regularized covariance selection objective alone (see the recent review by Scheinberg and Ma [17]). Recent research has seen the development of structured sparsity, where more complex prior knowledge about a sparsity pattern can be encoded. Examples include group sparsity [22], where parameters are linked so that they are regularized in groups. More complex sparsity patterns, such as region shape constraints in the case of pixels in an image [13], or hierarchical constraints [12] have also been explored. In this paper, we study the problem of recovering the structure of a Gaussian graphical model under the assumption that the graph recovered should be scale-free. Many real-world networks are known a priori to be scale-free and therefore enforcing that knowledge through a prior seems a natural idea. Recent work has offered an approach to deal with this problem which results in a non-convex formulation [14]. Here we present a convex formulation. We show that scale-free networks can be induced by enforcing submodular priors on the network’s degree distribution, and then using their convex envelope (the Lov´asz extension) as a convex relaxation [2]. The resulting relaxed prior has an interesting non-differentiable structure, which poses challenges to optimization. We outline a few options for solving the optimisation problem via proximal operators [3], in particular an efﬁcient dual decomposition method. Experiments on both synthetic data produced by scale-free network models and a real bioinformatics dataset suggest that the convex relaxation is not weak: we can infer scale-free networks with similar or superior accuracy than in [14]. 1 1 Combinatorial Objective Consider an undirected graph with edge set E and node set V , where n is the number of nodes. We denote the degree of node v as dE(v), and the complete graph with n nodes as Kn. We are concerned with placing priors on the degree distributions of graphs such as (V, E). By degree distribution, we mean the bag of degrees {dE(v)\|v ∈V }. A natural prior on degree distributions can be formed from the family of exponential random graphs [21]. Exponential random graph (ERG) models assign a probability to each n node graph using an exponential family model. The probability of each graph depends on a small set of sufﬁcient statistics, in our case we only consider the degree statistics. A ERG distribution with degree parametrization takes the form: p(G = (V, E); h) ≈ 1 Z(h) exp " − X v∈V h(dE(v)) # , (1) The degree weighting function h : Z+ →R encodes the preference for each particular degree. The function Z is chosen so that the distribution is correctly normalized over n node graphs. A number of choices for h are reasonable; A geometric series h(i) ∝1 −αi with α ∈(0, 1) has been proposed by Snijders et al. [20] and has been widely adopted. However for encouraging scale free graphs we require a more rapidly increasing sequence. It is instructive to observe that, under the strong assumption that each node’s degree is independent of the rest, h grows logarithmically. To see this, take a scale free model with scale α; the joint distribution takes the form: p(G = (V, E); ϵ, α) ≈ 1 Z(ϵ, α) Y v∈V (dE(v) + ϵ)−α, where ϵ > 0 is added to prevent inﬁnite weights. Putting this into ERG form gives the weight sequence h(i) = α log(i + ϵ). We will consider this and other functions h in Section 4. We intend to perform maximum a posteriori (MAP) estimation of a graph structure using such a distribution as a prior, so the object of our attention is the negative log-posterior, which we denote F: F(E) = X v∈V h(dE(v)) + const. (2) So far we have deﬁned a function on edge sets only, however in practice we want to optimize over a weighted graph, which is intractable when using discontinuous functions such as F. We now consider the properties of h that lead to a convex relaxation of F. 2 Submodularity A set function F : 2E →R on E is a non-decreasing submodular function if for all A ⊂B ⊂E and x ∈E\B the following conditions hold: F(A ∪{x}) −F(A) ≥F(B ∪{x}) −F(B) (submodularity) and F(A) ≤F(B). (non-decreasing) The ﬁrst condition can be interpreted as a diminishing returns condition; adding x to a set A increases F(A) by more than adding it to a larger set B, if B contains A. We now consider a set of conditions that can be placed on h so that F is submodular. Proposition 1. Denote h as tractable if h is non-decreasing, concave and h(0) = 0. For tractable h, F is a non-decreasing submodular function. Proof. First note that the degree function is a set cardinality function, and hence modular. A concave transformation of a modular function is submodular [1], and the sum of submodular functions is submodular. 2 The concavity restriction we impose on h is the key ingredient that allows us to use submodularity to enforce a prior for scale-free networks; any prior favouring long tailed degree distributions must place a lower weight on new edges joining highly connected nodes than on those joining other nodes. As far as we are aware, this is a novel way of mathematically modelling the ‘preferential attachment’ rule [4] that gives rise to scale-free networks: through non-decreasing submodular functions on the degree distribution. Let X denote a symmetric matrix of edge weights. A natural convex relaxation of F would be the convex envelope of F(Supp(X)) under some restricted domain. For tractable h, we have by construction that F satisﬁes the conditions of Proposition 1 in [2], so that the convex envelope of F(Supp(X)) on the L∞ball is precisely the Lov´asz extension evaluated on \|X\|. The Lov´asz extension for our function is easy to determine as it is a sum of “functions of cardinality” which are considered in [2]. Below is the result from [2] adapted to our problem. Proposition 2. Let Xi,(j) be the weight of the jth edge connected to i, under a decreasing ordering by absolute value (i.e \|Xi,(0)\| ≥\|Xi,(1)\| ≥... ≥\|Xi,(n−1)\|). The notation (i) maps from sorted order to the natural ordering, with the diagonal not included. Then the convex envelope of F for tractable h over the L∞norm unit ball is: Ω(X) = n X i=0 n−1 X k=0 (h(k + 1) −h(k)) \|Xi,(k)\|. This function is piece-wise linear and convex. The form of Ωis quite intuitive. It behaves like a L1 norm with an additional weight on each edge that depends on how the edge ranks with respect to the other edges of its neighbouring nodes. 3 Optimization We are interested in using Ωas a prior, for optimizations of the form minimizeX f(X) = g(X) + αΩ(X), for convex functions g and prior strength parameters α ∈R+, over symmetric X. We will focus on the simplest structure learning problem that occurs in graphical model training, that of Gaussian models. In which case we have g(X) = ⟨X, C⟩−log det X, where C is the observed covariance matrix of our data. The support of X will then be the set of edges in the undirected graphical model together with the node precisions. This function is a rescaling of the maximum likelihood objective. In order for the resulting X to deﬁne a normalizable distribution, X must be restricted to the cone of positive deﬁnite matrices. This is not a problem in practice as g(X) is inﬁnite on the boundary of the PSD cone, and hence the constraint can be handled by restricting optimization steps to the interior of the cone. In fact X can be shown to be in a strictly smaller cone, X∗⪰aI, for a derivable from C [15]. This restricted domain is useful as g(X) has Lipschitz continuous gradients over X ⪰aI but not over all positive deﬁnite matrices [18]. There are a number of possible algorithms that can be applied for optimizing a convex nondifferentiable objective such as f. Bach [2] suggests two approaches to optimizing functions involving submodular relaxation priors; a subgradient approach and a proximal approach. Subgradient methods are the simplest class of methods for optimizing non-smooth convex functions. They provide a good baseline for comparison with other methods. For our objective, a subgradient is simple to evaluate at any point, due to the piecewise continuous nature of Ω(X). Unfortunately (primal) subgradient methods for our problem will not return sparse solutions except in the limit of convergence. They will instead give intermediate values that oscillate around their limiting values. An alternative is the use of proximal methods [2]. Proximal methods exhibit superior convergence in comparison to subgradient methods, and produce sparse solutions. Proximal methods rely on solving a simpler optimization problem, known as the proximal operator at each iteration: arg min X αΩ(X) + 1 2 ∥X −Z∥2 2 , 3 where Z is a variable that varies at each iteration. For many problems of interest, the proximal operator can be evaluated using a closed form solution. For non-decreasing submodular relaxations, the proximal operator can be evaluated by solving a submodular minimization on a related (not necessarily non-decreasing) submodular function [2]. Bach [2] considers several example problems where the proximal operator can be evaluated using fast graph cut methods. For the class of functions we consider, graph-cut methods are not applicable. Generic submodular minimization algorithms could be as slow as O(n12) for a n-vertex graph, which is clearly impractical [11]. We will instead propose a dual decomposition method for solving this proximal operator problem in Section 3.2. For solving our optimisation problem, instead of using the standard proximal method (sometimes known as ISTA), which involves a gradient step followed by the proximal operator, we propose to use the alternating direction method of multipliers (ADMM), which has shown good results when applied to the standard L1 regularized covariance selection problem [18]. Next we show how to apply ADMM to our problem. 3.1 Alternating direction method of multipliers The alternating direction method of multipliers (ADMM, Boyd et al. [6]) is one approach to optimizing our objective that has a number of advantages over the basic proximal method. Let U be the matrix of dual variables for the decoupled problem: minimizeX g(X) + αΩ(Y ), s.t. X = Y. Following the presentation of the algorithm in Boyd et al. [6], given the values Y (l) and U (l) from iteration l, with U (0) = 0n and Y (0) = In the ADMM updates for iteration l + 1 are: X(l+1) = arg min X h ⟨X, C⟩−log det X + ρ 2\|\|X −Y (l) + U (l)\|\|2 2 i Y (l+1) = arg min Y h αΩ(Y ) + ρ 2\|\|X(l+1) −Y + U (l)\|\|2 2 i U (l+1) = U (l) + X(l+1) −Y (l+1), where ρ > 0 is a ﬁxed step-size parameter (we used ρ = 0.5). The advantage of this form is that both the X and Y updates are a proximal operation. It turns out that the proximal operator for g (i.e. the X(l+1) update) actually has a simple solution [18] that can be computed by taking an eigenvalue decomposition QT ΛQ = ρ(Y −U)−C, where Λ = diag(λ1, . . . , λn) and updating the eigenvalues using the formula λ′ i := λi + p λ2 i + 4ρ 2ρ to give X = QT Λ′Q. The stopping criterion we used was \|\|X(l+1) −Y (l+1)\|\| < ϵ and \|\|Y (l+1) −Y (l)\|\| < ϵ. In practice the ADMM method is one of the fastest methods for L1 regularized covariance selection. Scheinbert et al. [18] show that convergence is guaranteed if additional cone restrictions are placed on the minimization with respect to X, and small enough step sizes are used. For our degree prior regularizer, the difﬁcultly is in computing the proximal operator for Ω, as the rest of the algorithm is identical to that presented in Boyd et al. [6]. We now show how we solve the problem of computing the proximal operator for Ω. 3.2 Proximal operator using dual decomposition Here we describe the optimisation algorithm that we effectively use for computing the proximal operator. The regularizer Ωhas a quite complicated structure due to the interplay between the terms involving the two end points for each edge. We can decouple these terms using the dual decomposition technique, by writing the proximal operation for a given Z = Y −U as: minimizeX = α ρ n X i n−1 X k (h(k + 1) −h(k)) Xi,(k) + 1 2\|\|X −Z\|\|2 2 s.t. X = XT . 4 The only difference so far is that we have made the symmetry constraint explicit. Taking the dual gives a formulation where the upper and lower triangle are treated as separate variables. The dual variable matrix V corresponds to the Lagrange multipliers of the symmetry constraint, which for notational convenience we store in an anti-symmetric matrix. The dual decomposition method is given in Algorithm 1. Algorithm 1 Dual decomposition main input: matrix Z, constants α, ρ input: step-size 0 < η < 1 initialize: X = Z initialize: V = 0n repeat for l = 0 until n −1 do Xl∗= solveSubproblem(Zl∗, Vl∗) # Algorithm 2 end for V = V + η(X −XT ) until \|\|X −XT \|\| < 10−6 X = 1 2(X + XT ) # symmetrize round: any \|Xij\| < 10−15 to 0 return X We use the notation Xi∗to denote the ith row of X. Since this is a dual method, the primal variables X are not feasible (i.e. symmetric) until convergence. Essentially we have decomposed the original problem, so that now we only need to solve the proximal operation for each node in isolation, namely the subproblems: ∀i. X(l+1) i∗ = arg min x α ρ n−1 X k (h(k + 1) −h(k)) x(k) + \|\|x −Zi∗+ V (l) i∗\|\|2 2. (3) Note that the dual variable has been integrated into the quadratic term by completing the square. As the diagonal elements of X are not included in the sort ordering, they will be minimized by Xii = Zii, for all i. Each subproblem is strongly convex as they consist of convex terms plus a positive quadratic term. This implies that the dual problem is differentiable (as the subdifferential contains only one subgradient), hence the V update is actually gradient ascent. Since a ﬁxed step size is used, and the dual is Lipschitz continuous, for sufﬁciently small step-size convergence is guaranteed. In practice we used η = 0.9 for all our tests. This dual decomposition subproblem can also be interpreted as just a step within the ADMM framework. If applied in a standard way, only one dual variable update would be performed before another expensive eigenvalue decomposition step. Since each iteration of the dual decomposition is much faster than the eigenvalue decomposition, it makes more sense to treat it as a separate problem as we propose here. It also ensures that the eigenvalue decomposition is only performed on symmetric matrices. Each subproblem in our decomposition is still a non-trivial problem. They do have a closed form solution, involving a sort and several passes over the node’s edges, as described in Algorithm 2. Proposition 3. Algorithm 2 solves the subproblem in equation 3. Proof: See Appendix 1 in the supplementary material. The main subtlety is the grouping together of elements induced at the non-differentiable points. If multiple edges connected to the same node have the same absolute value, their subdifferential becomes the same, and they behave as a single point whose weight is the average. To handle this grouping, we use a disjoint-set data-structure, where each xj is either in a singleton set, or grouped in a set with other elements, whose absolute value is the same. 4 Alternative degree priors Under the restrictions on h detailed in Proposition 1, several other choices seem reasonable. The scale free prior can be smoothed somewhat, by the addition of a linear term, giving hϵ,β(i) = log(i + ϵ) + βi, 5 Algorithm 2 Dual decomposition subproblem (solveSubproblem) input: vectors z, v initialize: Disjoint-set datastructure with set membership function γ w = z −v # w gives the sort order u = 0n build: sorted-to-original position function µ under descending absolute value order of w, excluding the diagonal for k = 0 until n −1 do j = µ(k) uj = \|wj\| −α ρ (h(k + 1) −h(k)) γ(j).value = uj r = k while r > 1 and γ(µ(r)).value ≥γ(µ(r −1)).value do join: the sets containing µ(r) and µ(r −1) γ(µ(r)).value = 1 \|γ(µ(r))\| P i∈γ(µ(r)) ui set: r to the ﬁrst element of γ(µ(r)) by the sort ordering end while end for for i = 1 to N do xi = γ(i).value if xi < 0 then xj = 0 # negative values imply shrinkage to 0 end if if wi < 0 then xj = −xj # Correct orthant end if end for return x where β controls the strength of the smoothing. A slower diminishing choice would be a square-root function such as hβ(i) = (i + 1) 1 2 −1 + βi. This requires the linear term in order to correspond to a normalizable prior. Ideally we would choose h so that the expected degree distribution under the ERG model matches the particular form we wish to encourage. Finding such a h for a particular graph size and degree distribution amounts to maximum likelihood parameter learning, which for ERG models is a hard learning problem. The most common approach is to use sampling based inference. Approaches based on Markov chain Monte Carlo techniques have been applied widely to ERG models [19] and are therefore applicable to our model. 5 Related Work The covariance selection problem has recently been addressed by Liu and Ihler [14] using reweighted L1 regularization. They minimize the following objective: f(X) = ⟨X, C⟩−log det X + α X v∈V log (∥X¬v∥+ ϵ) + β X v \|Xvv\| . The regularizer is split into an off diagonal term which is designed to encourage sparsity in the edge parameters, and a more traditional diagonal term. Essentially they use ∥X¬v∥as the continuous counterpart of node v’s degree. The biggest difﬁculty with this objective is the log term, which makes f highly non-convex. This can be contrasted to our approach, where we start with essentially the same combinatorial prior, but we use an alternative, convex relaxation. The reweighted L1 [7] aspect refers to the method of optimization applied. A double loop method is used, in the same class as EM methods and difference of convex programming, where each L1 inner problem gives a monotonically improving lower bound on the true solution. 6 0.00 0.05 0.10 0.15 0.20 0.25 False Positives 0.70 0.75 0.80 0.85 0.90 0.95 1.00 True Positives L1 Reweighted L1 Submodular log Submodular root 0.00 0.05 0.10 0.15 0.20 0.25 False Positives 0.70 0.75 0.80 0.85 0.90 0.95 1.00 True Positives L1 Reweighted L1 Submodular log Submodular root Figure 1: ROC curves for BA model (left) and ﬁxed degree distribution model (right) Figure 2: Reconstruction of a gene association network using L1 (left), submodular relaxation (middle), and reweighted L1 (right) methods 6 Experiments Reconstruction of synthetic networks. We performed a comparison against the reweighted L1 method of Liu and Ihler [14], and a standard L1 regularized method, both implemented using ADMM for optimization. Although Liu and Ihler [14] use the glasso [10] method for the inner loop, ADMM will give identical results, and is usually faster [18]. Graphs with 60 nodes were generated using both the Barabasi-Albert model [4] and a predeﬁned degree distribution model sampled using the method from Bayati et al. [5] implemented in the NetworkX software package. Both methods generate scale-free graphs; the BA model exhibits a scale parameter of 3.0, whereas we ﬁxed the scale parameter at 2.0 for the other model. To deﬁne a valid Gaussian model, edge weights of Xij = −0.2 were assigned, and the node weights were set at Xii = 0.5 −P i̸=j Xij so as to make the resulting precision matrix diagonally dominant. The resulting Gaussian graphical model was sampled 500 times. The covariance matrix of these samples was formed, then normalized to have diagonal uniformly 1.0. We tested with the two h sequences described in section 4. The parameters for the degree weight sequences were chosen by grid search on random instances separate from those we tested on. The resulting ROC curves for the Hamming reconstruction loss are shown in Figure 1. Results were averaged over 30 randomly generated graphs for each each ﬁgure. We can see from the plots that our method with the square-root weighting presents results superior to those from Liu and Ihler [14] for these datasets. This is encouraging particularly since our formulation is convex while the one from Liu and Ihler [14] isn’t. Interestingly, the log based weights give very similar but not identical results to the reweighting scheme which also uses a log term. The only case where it gives inferior reconstructions is when it is forced to give a sparser reconstruction than the original graph. Reconstruction of a gene activation network. A common application of sparse covariance selection is the estimation of gene association networks from experimental data. A covariance matrix of gene co-activations from a number of independent micro-array experiments is typically formed, on which a number of methods, including sparse covariance selection, can be applied. Sparse estimation is key for a consistent reconstruction due to the small number of experiments performed. Many biological networks are conjectured to be scale-free, and additionally ERG modelling techniques are known to produce good results on biological networks [16]. So we consider micro-array datasets a natural test-bed for our method. We ran our method and the L1 reconstruction method on the ﬁrst 7 500 genes from the GDS1429 dataset (http://www.ncbi.nlm.nih.gov/gds/1429), which contains 69 samples for 8565 genes. The parameters for both methods were tuned to produce a network with near to 50 edges for visualization purposes. The major connected component for each is shown in Figure 2. While these networks are too small for valid statistical analysis of the degree distribution, the submodular relaxation method produces a network with structure that is commonly seen in scale free networks. The star subgraph centered around gene 60 is more clearly deﬁned in the submodular relaxation reconstruction, and the tight cluster of genes in the right is less clustered in the L1 reconstruction. The reweighted L1 method produced a quite different reconstruction, with greater clustering. 0 20 40 60 80 100 Iteration 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Distance from solution Dual decomp Subgradient MNP Figure 3: Comparison of proximal operators Runtime comparison: different proximal operator methods. We performed a comparison against two other methods for computing the proximal operator: subgradient descent and the minimum norm point (MNP) algorithm. The MNP algorithm is a submodular minimization method that can be adapted for computing the proximal operator [2]. We took the input parameters from the last invocation of the proximal operator in the BA test, at a prior strength of 0.7. We then plotted the convergence rate of each of the methods, shown in Figure 3. As the tests are on randomly generated graphs, we present only a representative example. It is clear from this and similar tests that we performed that the subgradient descent method converges too slowly to be of practical applicability for this problem. Subgradient methods can be a good choice when only a low accuracy solution is required; for convergence of ADMM the error in the proximal operator needs to be smaller than what can be obtained by the subgradient method. The MNP method also converges slowly for this problem, however it achieves a low but usable accuracy quickly enough that it could be used in practice. The dual decomposition method achieves a much better rate of convergence, converging quickly enough to be of use even for strong accuracy requirements. The time for individual iterations of each of the methods was 0.65ms for subgradient descent, 0.82ms for dual decomposition and 15ms for the MNP method. The speed difference is small between a subgradient iteration and a dual decomposition iteration as both are dominated by the cost of a sort operation. The cost of a MNP iteration is dominated by two least squares solves, whose running time in the worst case is proportional to the square of the current iteration number. Overall, it is clear that our dual decomposition method is signiﬁcantly more efﬁcient. Runtime comparison: submodular relaxation against other approaches. The running time of the three methods we tested is highly dependent on implementation details, so the following speed comparison should be taken as a rough guide. For a sparse reconstruction of a BA model graph with 100 vertices and 200 edges, the average running time per 10−4 error reconstruction over 10 random graphs was 16 seconds for the reweighted L1 method and 5.0 seconds for the submodular relaxation method. This accuracy level was chosen so that the active edge set for both methods had stabilized between iterations. For comparison, the standard L1 method was signiﬁcantly faster, taking only 0.72 seconds on average. Conclusion We have presented a new prior for graph reconstruction, which enforces the recovery of scale-free networks. This prior falls within the growing class of structured sparsity methods. Unlike previous approaches to regularizing the degree distribution, our proposed prior is convex, making training tractable and convergence predictable. Our method can be directly applied in contexts where sparse covariance selection is currently used, where it may improve the reconstruction quality. Acknowledgements NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program. 8 References [1] Francis Bach. Convex analysis and optimization with submodular functions: a tutorial. Technical report, INRIA, 2010. [2] Francis Bach. Structured sparsity-inducing norms through submodular functions. NIPS, 2010. [3] Francis Bach, Rodolphe Jenatton, Julien Mairal, and Guillaume Obozinski. Optimization with sparsityinducing penalties. Foundations and Trends in Machine Learning, 2012. [4] Albert-Laszlo Barabasi and Reka Albert. Emergence of scaling in random networks. Science, 286:509– 512, 1999. [5] Moshen Bayati, Jeong Han Kim, and Amin Saberi. A sequential algorithm for generating random graphs. Algorithmica, 58, 2009. [6] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning, 3, 2011. [7] Emmanuel J. Candes, Michael B. Wakin, and Stephen P. Boyd. Enhancing sparsity by reweighted l1 minimization. Journal of Fourier Analysis and Applications, 2008. [8] A. P. Dempster. Covariance selection. Biometrics, 28:157–175, 1972. [9] Adrian Dobra, Chris Hans, Beatrix Jones, Joseph R Nevins, and Mike West. Sparse graphical models for exploring gene expression data. Journal of Multivariate Analysis, 2004. [10] Jerome Friedman, Trevor Hastie, and Robert Tibshirani. Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 2007. [11] Saruto Fujishige. Submodular Functions and Optimization. Elsevier, 2005. [12] Rodolphe Jenatton, Julien Mairal, Guillaume Obozinski, and Francis Bach. Proximal methods for sparse hierarchical dictionary learning. ICML, 2010. [13] Rodolphe Jenatton, Guillaume Obozinski, and Francis Bach. Structured sparse principal component analysis. AISTATS, 2010. [14] Qiang Liu and Alexander Ihler. Learning scale free networks by reweighted l1 regularization. AISTATS, 2011. [15] Zhaosong Lu. Smooth optimization approach for sparse covariance selection. SIAM J. Optim., 2009. [16] Zachary M. Saul and Vladimir Filkov. Exploring biological network structure using exponential random graph models. Bioinformatics, 2007. [17] Katya Scheinberg and Shiqian Ma. Optimization for Machine Learning, chapter 17. optimization methods for sparse inverse covariance selection. MIT Press, 2011. [18] Katya Scheinbert, Shiqian Ma, and Donald Goldfarb. Sparse inverse covariance selection via alternating linearization methods. In NIPS, 2010. [19] T. Snijders. Markov chain monte carlo estimation of exponential random graph models. Journal of Social Structure, 2002. [20] Tom A.B. Snijders, Philippa E. Pattison, and Mark S. Handcock. New speciﬁcations for exponential random graph models. Technical report, University of Washington, 2004. [21] Alan Terry. Exponential random graphs. Master’s thesis, University of York, 2005. [22] M. Yuan and Y. Lin. Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society, 2007. 9	2012	137
4,494	Hamming Distance Metric Learning Mohammad Norouzi† David J. Fleet† Ruslan Salakhutdinov†,‡ Departments of Computer Science† and Statistics‡ University of Toronto [norouzi,fleet,rsalakhu]@cs.toronto.edu Abstract Motivated by large-scale multimedia applications we propose to learn mappings from high-dimensional data to binary codes that preserve semantic similarity. Binary codes are well suited to large-scale applications as they are storage efﬁcient and permit exact sub-linear kNN search. The framework is applicable to broad families of mappings, and uses a ﬂexible form of triplet ranking loss. We overcome discontinuous optimization of the discrete mappings by minimizing a piecewise-smooth upper bound on empirical loss, inspired by latent structural SVMs. We develop a new loss-augmented inference algorithm that is quadratic in the code length. We show strong retrieval performance on CIFAR-10 and MNIST, with promising classiﬁcation results using no more than kNN on the binary codes. 1 Introduction Many machine learning algorithms presuppose the existence of a pairwise similarity measure on the input space. Examples include semi-supervised clustering, nearest neighbor classiﬁcation, and kernel-based methods, When similarity measures are not given a priori, one could adopt a generic function such as Euclidean distance, but this often produces unsatisfactory results. The goal of metric learning techniques is to improve matters by incorporating side information, and optimizing parametric distance functions such as the Mahalanobis distance [7, 12, 30, 34, 36]. Motivated by large-scale multimedia applications, this paper advocates the use of discrete mappings, from input features to binary codes. Compact binary codes are remarkably storage efﬁcient, allowing one to store massive datasets in memory. The Hamming distance, a natural similarity measure on binary codes, can be computed with just a few machine instructions per comparison. Further, it has been shown that one can perform exact nearest neighbor search in Hamming space signiﬁcantly faster than linear search, with sublinear run-times [15, 25]. By contrast, retrieval based on Mahalanobis distance requires approximate nearest neighbor (ANN) search, for which state-of-the-art methods (e.g., see [18, 23]) do not always perform well, especially with massive, high-dimensional datasets when storage overheads and distance computations become prohibitive. Most approaches to discrete (binary) embeddings have focused on preserving the metric (e.g. Euclidean) structure of the input data, the canonical example being locality-sensitive hashing (LSH) [4, 17]. Based on random projections, LSH and its variants (e.g., [26]) provide guarantees that metric similarity is preserved for sufﬁciently long codes. To ﬁnd compact codes, recent research has turned to machine learning techniques that optimize mappings for speciﬁc datasets (e.g., [20, 28, 29, 32, 3]). However, most such methods aim to preserve Euclidean structure (e.g. [13, 20, 35]). In metric learning, by comparison, the goal is to preserve semantic structure based on labeled attributes or parameters associated with training exemplars. There are papers on learning binary hash functions that preserve semantic similarity [29, 28, 32, 24], but most have only considered ad hoc datasets and uninformative performance measures, for which it is difﬁcult to judge performance with anything but the qualitative appearance of retrieval results. The question of whether or not it is possible to learn hash functions capable of preserving complex semantic structure, with high ﬁdelity, has remained unanswered. 1 To address this issue, we introduce a framework for learning a broad class of binary hash functions based on a triplet ranking loss designed to preserve relative similarity (c.f. [11, 5]). While certainly useful for preserving metric structure, this loss function is very well suited to the preservation of semantic similarity. Notably, it can be viewed as a form of local ranking loss. It is more ﬂexible than the pairwise hinge loss of [24], and is shown below to produce superior hash functions. Our formulation is inspired by latent SVM [10] and latent structural SVM [37] models, and it generalizes the minimal loss hashing (MLH) algorithm of [24]. Accordingly, to optimize hash function parameters we formulate a continuous upper-bound on empirical loss, with a new form of lossaugmented inference designed for efﬁcient optimization with the proposed triplet loss on the Hamming space. To our knowledge, this is one of the most general frameworks for learning a broad class of hash functions. In particular, many previous loss-based techniques [20, 24] are not capable of optimizing mappings that involve non-linear projections, e.g., by neural nets. Our experiments indicate that the framework is capable of preserving semantic structure on challenging datasets, namely, MNIST [1] and CIFAR-10 [19]. We show that k-nearest neighbor (kNN) search on the resulting binary codes retrieves items that bear remarkable similarity to a given query item. To show that the binary representation is rich enough to capture salient semantic structure, as is common in metric learning, we also report classiﬁcation performance on the binary codes. Surprisingly, on these datasets, simple kNN classiﬁers in Hamming space are competitive with sophisticated discriminative classiﬁers, including SVMs and neural networks. An important appeal of our approach is the scalability of kNN search on binary codes to billions of data points, and of kNN classiﬁcation to millions of class labels. 2 Formulation The task is to learn a mapping b(x) that projects p-dimensional real-valued inputs x ∈Rp onto q-dimensional binary codes h ∈H ≡{−1, 1}q, while preserving some notion of similarity. This mapping, referred to as a hash function, is parameterized by a real-valued vector w as b(x; w) = sign (f(x; w)) , (1) where sign(.) denotes the element-wise sign function, and f(x; w) : Rp →Rq is a real-valued transformation. Different forms of f give rise to different families of hash functions: 1. A linear transform f(x) = Wx, where W ∈Rq×p and w ≡vec(W), is the simplest and most well-studied case [4, 13, 24, 33]. Under this mapping the kth bit is determined by a hyperplane in the input space whose normal is given by the kth row of W. 1 2. In [35], linear projections are followed by an element-wise cosine transform, i.e. f(x) = cos(Wx). For such mappings the bits correspond to stripes of 1 and −1 regions, oriented parallel to the corresponding hyperplanes, in the input space. 3. Kernelized hash functions [20, 21]. 4. More complex hash functions are obtained with multilayer neural networks [28, 32]. For example, a two-layer network with a p′-dimensional hidden layer and weight matrices W1 ∈ Rp′×p and W2 ∈Rq×p′ can be expressed as f(x) = tanh(W2 tanh(W1x)), where tanh(.) is the element-wise hyperbolic tangent function. Our Hamming distance metric learning framework applies to all of the above families of hash functions. The only restriction is that f must be differentiable with respect to its parameters, so that one is able to compute the Jacobian of f(x; w) with respect to w. 2.1 Loss functions The choice of loss function is crucial for learning good similarity measures. To this end, most existing supervised binary hashing techniques [13, 22, 24] formulate learning objectives in terms of pairwise similarity, where pairs of inputs are labelled as either similar or dissimilar. Similaritypreserving hashing aims to ensure that Hamming distances between binary codes for similar (dissimilar) items are small (large). For example, MLH [24] uses a pairwise hinge loss function. For 1For presentation clarity, in linear and nonlinear cases, we omit bias terms. They are incorporated by adding one dimension to the input vectors, and to the hidden layers of neural networks, with a ﬁxed value of one. 2 two binary codes h, g ∈H with Hamming distance2 ∥h−g∥H, and a similarity label s ∈{0, 1}, the pairwise hinge loss is deﬁned as: ℓpair(h, g, s) = [ ∥h−g∥H −ρ + 1 ]+ for s = 1 (similar) [ ρ −∥h−g∥H + 1 ]+ for s = 0 (dissimilar) , (2) where [α]+ ≡max(α, 0), and ρ is a Hamming distance threshold that separates similar from dissimilar codes. This loss incurs zero cost when a pair of similar inputs map to codes that differ by less than ρ bits. The loss is zero for dissimilar items whose Hamming distance is more than ρ bits. One problem with such loss functions is that ﬁnding a suitable threshold ρ with cross-validation is slow. Furthermore, for many problems one cares more about the relative magnitudes of pairwise distances than their precise numerical values. So, constraining pairwise Hamming distances over all pairs of codes with a single threshold is overly restrictive. More importantly, not all datasets are amenable to labeling input pairs as similar or dissimilar. One way to avoid some of these problems is to deﬁne loss in terms of relative similarity. Such loss functions have been used in metric learning [5, 11], and, as shown below, they are also naturally suited to Hamming distance metric learning. To deﬁne relative similarity, we assume that the training data includes triplets of items (x, x+, x−), such that the pair (x, x+) is more similar than the pair (x, x−). Our goal is to learn a hash function b such that b(x) is closer to b(x+) than to b(x−) in Hamming distance. Accordingly, we propose a ranking loss on the triplet of binary codes (h, h+, h−), obtained from b applied to (x, x+, x−): ℓtriplet(h, h+, h−) = ∥h−h+∥H −∥h−h−∥H + 1 + . (3) This loss is zero when the Hamming distance between the more-similar pair, ∥h −h+∥H, is at least one bit smaller than the Hamming distance between the less-similar pair, ∥h−h−∥H. This loss function is more ﬂexible than the pairwise loss function ℓpair, as it can be used to preserve rankings among similar items, for example based on Euclidean distance, or perhaps using path distance between category labels within a phylogenetic tree. 3 Optimization Given a training set of triplets, D = (xi, x+ i , x− i ) n i=1, our objective is the sum of the empirical loss and a simple regularizer on the vector of unknown parameters w: L(w) = X (x,x+,x−)∈D ℓtriplet b(x; w), b(x+; w), b(x−; w) + λ 2 ∥w∥2 2 . (4) This objective is discontinuous and non-convex. The hash function is a discrete mapping and empirical loss is piecewise constant. Hence optimization is very challenging. We cannot overcome the non-convexity, but the problems owing to the discontinuity can be mitigated through the construction of a continuous upper bound on the loss. The upper bound on loss that we adopt is inspired by previous work on latent structural SVMs [37]. The key observation that relates our Hamming distance metric learning framework to structured prediction is as follows, b(x; w) = sign (f(x; w)) = argmax h∈H hTf(x; w) , (5) where H ≡{−1, +1}q. The argmax on the RHS effectively means that for dimensions of f(x; w) with positive values, the optimal code should take on values +1, and when elements of f(x; w) are negative the corresponding bits of the code should be −1. This is identical to the sign function. 3.1 Upper bound on empirical loss The upper bound on loss that we exploit for learning hash functions takes the following form: ℓtriplet b(x; w), b(x+; w), b(x−; w) ≤ max g,g+,g− n ℓtriplet g, g+, g− + gTf(x; w) + g+Tf(x+; w) + g−Tf(x−; w) o −max h hTf(x; w) −max h+ n h+Tf(x+; w) o −max h− n h−Tf(x−; w) o , (6) 2The Hamming norm ∥v∥H is deﬁned as the number of non-zero entries of vector v. 3 where g, g+, g−, h, h+, and h−are constrained to be q-dimensional binary vectors. To prove the inequality in Eq. 6, note that if the ﬁrst term on the RHS were maximized3 by (g, g+, g−) = (b(x), b(x+), b(x−)), then using Eq. 5, it is straightforward to show that Eq. 6 would become an equality. In all other cases of (g, g+, g−) which maximize the ﬁrst term, the RHS can only be as large or larger than when (g, g+, g−) = (b(x), b(x+), b(x−)), hence the inequality holds. Summing the upper bound instead of the loss in Eq. 4 yields an upper bound on the regularized empirical loss in Eq. 4. Importantly, the resulting bound is easily shown to be continuous and piecewise smooth in w as long as f is continuous in w. The upper bound of Eq. 6 is a generalization of a bound introduced in [24] for the linear case, f(x) = Wx. In particular, when f is linear in w, the bound on regularized empirical loss becomes piecewise linear and convex-concave. While the bound in Eq. 6 is more challenging to optimize than the bound in [24], it allows us to learn hash functions based on non-linear functions f, e.g. neural networks. While the bound in [24] was deﬁned for ℓpair-type loss functions and pairwise similarity labels, the bound here applies to the more ﬂexible class of triplet loss functions. 3.2 Loss-augmented inference To use the upper bound in Eq. 6 for optimization, we must be able to ﬁnd the binary codes given by (ˆg, ˆg+, ˆg−) = argmax (g,g+,g−) n ℓtriplet g, g+, g− + gTf(x) + g+Tf(x+) + g−Tf(x−) o . (7) In the structured prediction literature this maximization is called loss-augmented inference. The challenge stems from the 23q possible binary codes over which one has to maximize the RHS. Fortunately, we can show that this loss-augmented inference problem can be solved efﬁciently for the class of triplet loss functions that depend only on the value of d(g, g+, g−) ≡∥g−g+∥H −∥g−g−∥H . Importantly, such loss functions do not depend on the speciﬁc binary codes, but rather just the differences. Further, note that d(g, g+, g−) can take on only 2q+1 possible values, since it is an integer between −q and +q. Clearly the triplet ranking loss only depends on d since ℓtriplet g, g+, g− = ℓ′d(g, g+, g−) , where ℓ′(α) = [ α −1 ]+ . (8) For this family of loss functions, given the values of f(.) in Eq. 7, loss-augmented inference can be performed in time O(q2). To prove this, ﬁrst consider the case d(g, g+, g−) = m, where m is an integer between −q and q. In this case we can replace the loss augmented inference problem with ℓ′(m) + max g,g+,g− n gTf(x) + g+Tf(x+) + g−Tf(x−) o s.t. d(g, g+, g−) = m . (9) One can solve Eq. 9 for each possible value of m. It is straightforward to see that the largest of those 2q + 1 maxima is the solution to Eq. 7. Then, what remains for us is to solve Eq. 9. To solve Eq. 9, consider the ith bit for each of the three codes, i.e. a = g[i], b = g+[i], and c = g−[i], where v[i] denotes the ith element of vector v. There are 8 ways to select a, b and c, but no matter what values they take on, they can only change the value of d(g, g+, g−) by −1, 0, or +1. Accordingly, let ei ∈{−1, 0, +1} denote the effect of the ith bits on d(g, g+, g−). For each value of ei, we can easily compute the maximal contribution of (a, b, c) to Eq. 9 by: cont(i, ei) = max a,b,c af(x)[i] + bf(x+)[i] + cf(x−)[i] (10) such that a, b, c ∈{−1, +1} and ∥a−b∥H −∥a−c∥H = ei. Therefore, to solve Eq. 9, we aim to select values for ei, for all i, such that Pq i=1 ei = m and Pq i=1 cont(i, ei) is maximized. This can be solved for any m using a dynamic programming algorithm, similar to knapsack, in O(q2). Finally, we choose m that maximizes Eq. 9 and set the bits to the conﬁgurations that maximized cont(i, ei). 3For presentation clarity we will sometimes drop the dependence of f and b on w, and write b(x) and f(x). 4 3.3 Perceptron-like learning Our learning algorithm is a form of stochastic gradient descent, where in the tth iteration we sample a triplet (x, x+, x−) from the dataset, and then take a step in the direction that decreases the upper bound on the triplet’s loss in Eq. 6. To this end, we randomly initialize w(0). Then, at each iteration t + 1, given w(t), we use the following procedure to update the parameters, w(t+1): 1. Select a random triplet (x, x+, x−) from dataset D. 2. Compute (ˆh, ˆh+, ˆh−) = (b(x; w(t)), b(x+; w(t)), b(x−; w(t))) using Eq. 5. 3. Compute (ˆg, ˆg+, ˆg−), the solution to the loss-augmented inference problem in Eq. 7 . 4. Update model parameters using w(t+1) =w(t) + η ∂f(x) ∂w ˆh−ˆg + ∂f(x+) ∂w ˆh+−ˆg+ + ∂f(x−) ∂w ˆh−−ˆg− −λw(t) , where η is the learning rate, and ∂f(x)/∂w ≡∂f(x; w)/∂w\|w=w(t) ∈R\|w\|×q is the transpose of the Jacobian matrix, where \|w\| is the number of parameters. This update rule can be seen as gradient descent in the upper bound of the regularized empirical loss. Although the upper bound in Eq. 6 is not differentiable at isolated points (owing to the max terms), in our experiments we ﬁnd that this update rule consistently decreases both the upper bound and the actual regularized empirical loss L(w). 4 Asymmetric Hamming distance When Hamming distance is used to score and retrieve the nearest neighbors to a given query, there is a high probability of a tie, where multiple items are equidistant from the query in Hamming space. To break ties and improve the similarity measure, previous work suggests the use of an asymmetric Hamming (AH) distance [9, 14]. With an AH distance, one stores dataset entries as binary codes (for storage efﬁciency) but the queries are not binarized. An asymmetric distance function is therefore deﬁned on a real-valued query vector, v ∈Rq, and a database binary code, h ∈H. Computing AH distance is slightly less efﬁcient than Hamming distance, and efﬁcient retrieval algorithms, such as [25], are not directly applicable. Nevertheless, the AH distance can also be used to re-rank items retrieved using Hamming distance, with a negligible increase in run-time. To improve efﬁciency further when there are many codes to be re-ranked, AH distance from the query to binary codes can be pre-computed for each 8 or 16 consecutive bits, and stored in a query-speciﬁc lookup table. In this work, we use the following asymmetric Hamming distance function AH(h, v; s) = 1 4 ∥h −tanh(Diag(s) v) ∥2 2 , (11) where s ∈Rq is a vector of scaling parameters that control the slope of hyperbolic tangent applied to different bits; Diag(s) is a diagonal matrix with the elements of s on its diagonal. As the scaling factors in s approach inﬁnity, AH and Hamming distances become identical. Here we use the AH distance between a database code b(x′) and the real-valued projection for the query f(x). Based on our validation sets, the AH distance of Eq. 11 is relatively insensitive to values in s. For the experiments we simply use s to scale the average absolute values of the elements of f(x) to be 0.25. 5 Implementation details In practice, the basic learning algorithm described in Sec. 3 is implemented with several modiﬁcations. First, instead of using a single training triplet to estimate the gradients, we use mini-batches comprising 100 triplets and average the gradient. Second, for each triplet (x, x+, x−), we replace x−with a “hard” example by selecting an item among all negative examples in the mini-batch that is closest in the current Hamming distance to b(x). By harvesting hard negative examples, we ensure that the Hamming constraints for the triplets are not too easily satisﬁed. Third, to ﬁnd good binary codes, we encourage each bit, averaged over the training data, to be mean-zero before quantization (motivated in [35]). This is accomplished by adding the following penalty to the objective function: 1 2∥mean x f(x; w) ∥2 2 , (12) 5 10 100 1000 10000 0.87 0.9 0.93 0.96 0.99 k Precision @k Two−layer net, triplet Two−layer net, pairwise Linear, triplet Linear, pairwise [24] 10 100 1000 10000 0.87 0.9 0.93 0.96 0.99 k Precision @k 128−bit, linear, triplet 64−bit, linear, triplet 32−bit, linear, triplet Euclidean distance Figure 1: MNIST precision@k: (left) four methods (with 32-bit codes); (right) three code lengths. where mean(f(x; w)) denotes the mean of f(x; w) across the training data. In our implementation, for efﬁciency, the stochastic gradient of Eq. 12 is computed per mini-batch using the Jacobian matrix in the update rule (see Sec. 3.3). Empirically, we observe that including this term in the objective improves the quality of binary codes, especially with the triplet ranking loss. We use a heuristic to adapt learning rates, known as bold driver [2]. For each mini-batch we evaluate the learning objective before the parameters are updated. As long as the objective is decreasing we slowly increase the learning rate η, but when the objective increases, η is halved. In particular, after every 25 epochs, if the objective, averaged over the last 25 epochs, decreased, we increase η by 5%, otherwise we decrease η by 50%. We also used a momentum term; i.e. the previous gradient update is scaled by 0.9 and then added to the current gradient. All experiments are run on a GPU for 2, 000 passes through the datasets. The training time for our current implementation is under 4 hours of GPU time for most of our experiments. The two exceptions involve CIFAR-10 with 6400-D inputs and relatively long code-lengths of 256 and 512 bits, for which the training times are approximated 8 and 16 hours respectively. 6 Experiments Our experiments evaluate Hamming distance metric learning using two families of hash functions, namely, linear transforms and multilayer neural networks (see Sec. 2). For each, we examine two loss functions, the pairwise hinge loss (Eq. 2) and the triplet ranking loss (Eq. 3). Experiments are conducted on two well-known image corpora, MNIST [1] and CIFAR-10 [19]. Ground-truth similarity labels are derived from class labels; items from the same class are deemed similar4. This deﬁnition of similarity ignores intra-class variations and the existence of subcategories, e.g. styles of handwritten fours, or types of airplanes. Nevertheless, we use these coarse similarity labels to evaluate our framework. To that end, using items from the test set as queries, we report precision@k, i.e. the fraction of k closest items in Hamming distance that are same-class neighbors. We also show kNN retrieval results for qualitative inspection. Finally, we report Hamming (H) and asymmetric Hamming (AH) kNN classiﬁcation rates on the test sets. Datasets. The MNIST [1] digit dataset contains 60, 000 training and 10, 000 test images (28×28 pixels) of ten handwritten digits (0 to 9). Of the 60, 000 training images, we set aside 5, 000 for validation. CIFAR-10 [19] comprises 50, 000 training and 10, 000 test color images (32×32 pixels). Each image belongs to one of 10 classes, namely airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck. The large variability in scale, viewpoint, illumination, and background clutter poses a signiﬁcant challenge for classiﬁcation. Instead of using raw pixel values, we borrow a bagof-words representation from Coates et al [6]. Its 6400-D feature vector comprises one 1600-bin histogram per image quadrant, the codewords of which are learned from 6×6 image patches. Such high-dimensional inputs are challenging for learning similarity-preserving hash functions. Of the 50, 000 training images, we set aside 5, 000 for validation. MNIST: We optimize binary hash functions, mapping raw MNIST images to 32, 64, and 128-bit codes. For each test code we ﬁnd the k closest training codes using Hamming distance, and report precision@k in Fig. 1. As one might expect, the non-linear mappings5 signiﬁcantly outperform linear mappings. We also ﬁnd that the triplet loss (Eq. 3) yields better performance than the pairwise 4Training triplets are created by taking two items from the same class, and one item from a different class. 5The two-layer neural nets for Fig. 1 and Table 1 had 1 hidden layer with 512 units. Weights were initialized randomly, and the Jacobian with respect to the parameters was computed with the backprop algorithm [27]. 6 Hash function, Loss Distance kNN 32 bits 64 bits 128 bits Linear, pairwise hinge [24] Hamming 2 NN 4.66 3.16 2.61 Linear, triplet ranking 2 NN 4.44 3.06 2.44 Two-layer Net, pairwise hinge 30 NN 1.50 1.45 1.44 Two-layer Net, triplet ranking 30 NN 1.45 1.38 1.27 Linear, pairwise hinge Asym. Hamming 3 NN 4.30 2.78 2.46 Linear, triplet ranking 3 NN 3.88 2.90 2.51 Two-layer Net, pairwise hinge 30 NN 1.50 1.36 1.35 Two-layer Net, triplet ranking 30 NN 1.45 1.29 1.20 Baseline Error Deep neural nets with pre-training [16] 1.2 Large margin nearest neighbor [34] 1.3 RBF-kernel SVM [8] 1.4 Neural network [31] 1.6 Euclidean 3NN 2.89 Table 1: Classiﬁcation error rates on MNIST test set. loss (Eq. 2). The sharp drop in precision at k = 6000 is a consequence of the fact that each digit in MNIST has approximately 6000 same-class neighbors. Fig. 1 (right) shows how precision improves as a function of the binary code length. Notably, kNN retrieval, for k > 10 and all code lengths, yields higher precision than Euclidean NN on the 784-D input space. Further, note that these Euclidian results effectively provide an upper bound on the performance one would expect with existing hashing methods that preserve Eucliean distances (e.g., [13, 17, 20, 35]). One can also evaluate the ﬁdelity of the Hamming space represenation in terms of classiﬁcation performance from the Hamming codes. To focus on the quality of the hash functions, and the speed of retrieval for large-scale multimedia datasets, we use a kNN classiﬁer; i.e. we just use the retrieved neighbors to predict class labels for each test code. Table 1 reports classiﬁcation error rates using kNN based on Hamming and asymmetric Hamming distance. Non-linear mappings, even with only 32-bit codes, signiﬁcantly outperform linear mappings (e.g.with 128 bits). The ranking hinge loss also improves upon the pairwise hinge loss, even though the former has no hyperparameters. Table 1 also indicates that AH distance provides a modest boost in performance. For each method the parameter k in the kNN classiﬁer is chosen based on the validation set. For baseline comparison, Table 1 reports state-of-the-art performance on MNIST with sophisticated discriminative classiﬁers (excluding those using examplar deformations and convolutional nets). Despite the simplicity of a kNN classiﬁer, our model achieves error rates of 1.29% and 1.20% using 64- and 128-bit codes. This is compared to 1.4% with RBF-SVM [8], and to 1.6%, the best published neural net result for this version of the task [31]. Our model also out performs the metric learning approach of [34], and is competitive with the best known Deep Belief Network [16]; although they used unsupervised pre-training while we do not. The above results show that our Hamming distance metric learning framework can preserve sufﬁcient semantic similarity, to the extent that Hamming kNN classiﬁcation becomes competitive with state-of-the-art discriminative methods. Nevertheless, our method is not solely a classiﬁer, and it can be used within many other machine learning algorithms. In comparison, another hashing technique called iterative quantization (ITQ) [13] achieves 8.5% error on MNIST and 78% accuracy on CIFAR-10. Our method compares favorably, especially on MNIST. However, ITQ [13] inherently binarizes the outcome of a supervised classiﬁer (Canonical Correlation Analysis with labels), and does not explicitly learn a similarity measure on the input features based on pairs or triplets. CIFAR-10: On CIFAR-10 we optimize hash functions for 64, 128, 256, and 512-bit codes. The supplementary material includes precision@k curves, showing superior quality of hash functions learned by the ranking loss compared to the pairwise loss. Here, in Fig. 2, we depict the quality of retrieval results for two queries, showing the 16 nearest neighbors using 256-bit codes, 64-bit codes (both learned with the triplet ranking loss), and Euclidean distance in the original 6400-D feature space. The number of class-based retrieval errors is much smaller in Hamming space, and the similarity in visual appearance is also superior. More such results, including failure modes, are shown in the supplementary material. 7 (Hamming on 256 bit codes) (Hamming on 64 bit codes) (Euclidean distance) Figure 2: Retrieval results for two CIFAR-10 test images using Hamming distance on 256-bit and 64-bit codes, and Euclidean distance on bag-of-words features. Red rectangles indicate mistakes. Hashing, Loss Distance kNN 64 bits 128 bits 256 bits 512 bits Linear, pairwise hinge [24] H 7 NN 72.2 72.8 73.8 74.6 Linear, pairwise hinge AH 8 NN 72.3 73.5 74.3 74.9 Linear, triplet ranking H 2 NN 75.1 75.9 77.1 77.9 Linear, triplet ranking AH 2 NN 75.7 76.8 77.5 78.0 Baseline Accuracy One-vs-all linear SVM [6] 77.9 Euclidean 3NN 59.3 Table 2: Recognition accuracy on the CIFAR-10 test set (H ≡Hamming, AH ≡Asym. Hamming). Table 2 reports classiﬁcation performance (showing accuracy instead of error rates for consistency with previous papers). Euclidean NN on the 6400-D input features yields under 60% accuracy, while kNN with the binary codes obtains 76−78%. As with MNIST data, this level of performance is comparable to one-vs-all SVMs applied to the same features [6]. Not surprisingly, training fully-connected neural nets on 6400-dimensional features with only 50, 000 training examples is challenging and susceptible to over-ﬁtting, hence the results of neural nets on CIFAR-10 were not competitive. Previous work [19] had some success training convolutional neural nets on this dataset. Note that our framework can easily incorporate convolutional neural nets, which are intuitively better suited to the intrinsic spatial structure of natural images. 7 Conclusion We present a framework for Hamming distance metric learning, which entails learning a discrete mapping from the input space onto binary codes. This framework accommodates different families of hash functions, including quantized linear transforms, and multilayer neural nets. By using a piecewise-smooth upper bound on a triplet ranking loss, we optimize hash functions that are shown to preserve semantic similarity on complex datasets. In particular, our experiments show that a simple kNN classiﬁer on the learned binary codes is competitive with sophisticated discriminative classiﬁers. While other hashing papers have used CIFAR or MNIST, none report kNN classiﬁcation performance, often because it has been thought that the bar established by state-of-the-art classiﬁers is too high. On the contrary our kNN classiﬁcation performance suggests that Hamming space can be used to represent complex semantic structures with high ﬁdelity. One appeal of this approach is the scalability of kNN search on binary codes to billions of data points, and of kNN classiﬁcation to millions of class labels. 8 References [1] http://yann.lecun.com/exdb/mnist/. [2] R. Battiti. Accelerated backpropagation learning: Two optimization methods. Complex Systems, 1989. [3] A. Bergamo, L. Torresani, and A. Fitzgibbon. Picodes: Learning a compact code for novel-category recognition. NIPS, 2011. [4] M. Charikar. 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4,495	Co-Regularized Hashing for Multimodal Data Yi Zhen and Dit-Yan Yeung Department of Computer Science and Engineering Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong {yzhen,dyyeung}@cse.ust.hk Abstract Hashing-based methods provide a very promising approach to large-scale similarity search. To obtain compact hash codes, a recent trend seeks to learn the hash functions from data automatically. In this paper, we study hash function learning in the context of multimodal data. We propose a novel multimodal hash function learning method, called Co-Regularized Hashing (CRH), based on a boosted coregularization framework. The hash functions for each bit of the hash codes are learned by solving DC (difference of convex functions) programs, while the learning for multiple bits proceeds via a boosting procedure so that the bias introduced by the hash functions can be sequentially minimized. We empirically compare CRH with two state-of-the-art multimodal hash function learning methods on two publicly available data sets. 1 Introduction Nearest neighbor search, a.k.a. similarity search, plays a fundamental role in many important applications, including document retrieval, object recognition, and near-duplicate detection. Among the methods proposed thus far for nearest neighbor search [1], hashing-based methods [2, 3] have attracted considerable interest in recent years. The major advantage of hashing-based methods is that they index data using binary hash codes which enjoy not only low storage requirements but also high computational efﬁciency. To preserve similarity in the data, a family of algorithms called locality sensitive hashing (LSH) [4, 5] has been developed over the past decade. The basic idea of LSH is to hash the data into bins so that the collision probability reﬂects data similarity. LSH is very appealing in that it has theoretical guarantee and is also simple to implement. However, in practice LSH algorithms often generate long hash codes in order to achieve acceptable performance because the theoretical guarantee only holds asymptotically. This shortcoming can be attributed largely to their data-independent nature which cannot capture the data characteristics very accurately in the hash codes. Besides, in many applications, neighbors cannot be deﬁned easily using some generic distance or similarity measures. As such, a new research trend has emerged over the past few years by learning the hash functions from data automatically. In the sequel, we refer to this new trend as hash function learning (HFL). Boosting, as one of the most popular machine learning approaches, was ﬁrst applied to learning hash functions for pose estimation [6]. Later, impressive performance for HFL using restricted Boltzmann machines was reported [7]. These two early HFL methods have been successfully applied to content-based image retrieval in which large-scale data sets are commonly encountered [8]. A number of algorithms have been proposed since then. Spectral hashing (SH) [9] treats HFL as a special case of manifold learning and uses an efﬁcient algorithm based on eigenfunctions. One shortcoming of spectral hashing is in its assumption, which requires that the data be uniformly distributed. To overcome this limitation, several methods have been proposed, including binary reconstructive embeddings [10], shift-invariant kernel hashing [11], distribution matching [12], optmized kernel hashing [13], and minimal loss hashing [14]. Recently, some semi-supervised hashing models have 1 been developed to combine both feature similarity and semantic similarity for HFL [15, 16, 17, 18]. To further improve the scalability of these methods, Liu et al. [19] presented a fast algorithm based on anchor graphs. Existing HFL algorithms have enjoyed wide success in challenging applications. Nevertheless, they can only be applied to a single type of data, called unimodal data, which refer to data from a single modality such as image, text, or audio. Nowadays, it is common to ﬁnd similarity search applications that involve multimodal data. For example, given an image of a tourist attraction as query, one would like to retrieve some textual documents that provide more detailed information about the place of interest. Because data from different modalities reside in different feature spaces, performing multimodal similarity search will be made much easier and faster if the multimodal data can be mapped into a common Hamming space. However, it is challenging to do so because data from different modalities generally have very different representations. As far as we know, there exist only two multimodal HFL methods. Bronstein et al. [20] made the ﬁrst attempt to learn linear hash functions using eigendecomposition and boosting, while Kumar et al. [21] extended spectral hashing to the multiview setting and proposed a cross-view hashing model. One major limitation of these two methods is that they both rely on eigendecomposition operations which are computationally very demanding when the data dimensionality is high. Moreover, they consider applications for shape retrieval, image alignment, and people search which are quite different from the multimodal retrieval applications of interest here. In this paper, we propose a novel multimodal HFL method, called Co-Regularized Hashing (CRH), based on a boosted co-regularization framework. For each bit of the hash codes, CRH learns a group of hash functions, one for each modality, by minimizing a novel loss function. Although the loss function is non-convex, it is in a special form which can be expressed as a difference of convex functions. As a consequence, the Concave-Convex Procedure (CCCP) [22] can be applied to solve the optimization problem iteratively. We use a stochastic sub-gradient method, which converges very fast, in each CCCP iteration to ﬁnd a local optimum. After learning the hash functions for one bit, CRH proceeds to learn more bits via a boosting procedure such that the bias introduced by the hash functions can be sequentially minimized. In the next section, we present the CRH method in detail. Extensive empirical study using two data sets is reported in Section 3. Finally, Section 4 concludes the paper. 2 Co-Regularized Hashing We use boldface lowercase letters and calligraphic letters to denote vectors and sets, respectively. For a vector x, xT denotes its transpose and ∥x∥its ℓ2 norm. 2.1 Objective Function Suppose that there are two sets of data points from two modalities,1 e.g., {xi ∈X}I i=1 for a set of I images from some feature space X and {yj ∈Y}J j=1 for a set of J textual documents from another feature space Y. We also have a set of N inter-modality point pairs Θ = {(xa1, yb1), (xa2, yb2), . . . , (xaN , ybN )}, where, for the nth pair, an and bn are indices of the points in X and Y, respectively. We further assume that each pair has a label sn = 1 if xan and ybn are similar and sn = 0 otherwise. The notion of inter-modality similarity varies from application to application. For example, if an image includes a tiger and a textual document is a research paper on tigers, they should be labeled as similar. On the other hand, it is highly unlikely to label the image as similar to a textual document on basketball. For each bit of the hash codes, we deﬁne two linear hash functions as follows: f(x) = sgn(wT x x) and g(y) = sgn(wT y y), where sgn(·) denotes the sign function, and wx and wy are projection vectors which, ideally, should map similar points to the same hash bin and dissimilar points to different bins. Our goal is to achieve HFL by learning wx and wy from the multimodal data. 1For simplicity of our presentation, we focus on the bimodal case here and leave the discussion on extension to more than two modalities to Section 2.4. 2 To achieve this goal, we propose to minimize the following objective function w.r.t. (with respect to) wx and wy: O = 1 I I X i=1 ℓx i + 1 J J X j=1 ℓy j + γ N X n=1 ωnℓ∗ n + λx 2 ∥wx∥2 + λy 2 ∥wy∥2, (1) where ℓx i and ℓy j are intra-modality loss terms for modalities X and Y, respectively. In this work, we deﬁne them as: ℓx i = 1 −f(xi)(wT x xi) + = 1 −\|wT x xi\| +, ℓy j = 1 −g(yj)(wT y yj) + = 1 −\|wT y yj\| +, where [a]+ is equal to a if a ≥0 and 0 otherwise. We note that the intra-modality loss terms are similar to the hinge loss in the (linear) support vector machine but have quite different meaning. Conceptually, we want the projected values to be far away from 0 and hence expect the hash functions learned to have good generalization ability [16]. For the inter-modality loss term ℓ∗ n, we associate with each point pair a weight ωn, with PN n=1 ωn = 1, to normalize the loss as well as compute the bias of the hash functions. In this paper, we deﬁne ℓ∗ n as ℓ∗ n = snd2 n + (1 −sn)τ(dn), where dn = wT x xan −wT y ybn and τ(d) is called the smoothly clipped inverted squared deviation (SCISD) function. The loss function such deﬁned requires that the similar inter-modality points, i.e., sn = 1, have small distance after projection, and the dissimilar ones, i.e., sn = 0, have large distance. With these two kinds of loss terms, we expect that the learned hash functions can enjoy the large-margin property while effectively preserving the inter-modality similarity. The SCISD function was ﬁrst proposed in [23]. It can be deﬁned as follows: τ(d) =      −1 2d2 + aλ2 2 if \|d\| ≤λ d2−2aλ\|d\|+a2λ2 2(a−1) if λ < \|d\| ≤aλ 0 if aλ < \|d\|, where a and λ are two user-speciﬁed parameters. The SCISD function penalizes projection vectors that result in small distance between dissimilar points after projection. A more important property is that it can be expressed as a difference of two convex functions. Speciﬁcally, we can express τ(d) = τ1(d) −τ2(d) where τ1(d) =      0 if \|d\| ≤λ ad2−2aλ\|d\|+aλ2 2(a−1) if λ < \|d\| ≤aλ 1 2d2 −aλ2 2 if aλ < \|d\| and τ2(d) = 1 2d2 −aλ2 2 . 2.2 Optimization Though the objective function (1) is nonconvex w.r.t. wx and wy, we can optimize it w.r.t. wx and wy in an alternating manner. Take wx for example, we remove the irrelevant terms and get the following objective: 1 I I X i=1 ℓx i + λx 2 ∥wx∥2 + γ N X n=1 ωnℓ∗ n, (2) where ℓx i =    0 if \|wT x xi\| ≥1 1 −wT x xi if 0 ≤wT x xi < 1 1 + wT x xi if −1 < wT x xi < 0. It is easy to realize that the objective function (2) can be expressed as a difference of two convex functions in different cases. As a consequence, we can use CCCP to solve the nonconvex optimization problem iteratively with each iteration minimizing a convex upper bound of the original objective function. 3 Brieﬂy speaking, given an objective function f0(x)−g0(x) where both f0 and g0 are convex, CCCP works iteratively as follows. The variable x is ﬁrst randomly initialized to x(0). At the tth iteration, CCCP minimizes the following convex upper bound of f0(x) −g0(x) at location x(t): f0(x) − g0(x(t)) + ∂xg0(x(t))(x −x(t)) , where ∂xg0(x(t)) is the ﬁrst derivative of g0(x) at x(t). This optimization problem can be solved using any convex optimization solver to obtain x(t+1). Given an initial value x(0), the solution sequence {x(t)} found by CCCP is guaranteed to reach a local minimum or a saddle point. For our problem, the optimization problem at the tth iteration minimizes the following upper bound of Equation (2) w.r.t. wx: Ox = λx∥wx∥2 2 + γ N X n=1 ωn snd2 n + (1 −sn)ζx n + 1 I I X i=1 ℓx i , (3) where ζx n = τ1(dn) −τ2(d(t) n ) −d(t) n xT an(wx −w(t) x ), d(t) n = (w(t) x )T xan −wT y ybn, and w(t) x is the value of wx at the tth iteration. To ﬁnd a locally optimal solution to problem (3), we can use any gradient-based method. In this work, we develop a stochastic sub-gradient solver based on Pegasos [24], which is known to be one of the fastest solvers for margin-based classiﬁers. Speciﬁcally, we randomly select k points from each modality and l point pairs to evaluate the sub-gradient at each iteration. The key step of our method is to evaluate the sub-gradient of objective function (3) w.r.t. wx, which can be computed as ∂Ox ∂wx = 2γ N X n=1 ωnsndnxan + γ N X n=1 ωnµx n + λxwx −1 I I X i=1 πx i , (4) where µx n = (1 −sn) ∂τ1 ∂dn −d(t) n xan, ∂τ1 ∂dn =    0 if \|dn\| ≤λ adn−2aλ sgn(dn) (a−1) if λ < \|dn\| ≤aλ dn if aλ < \|dn\| and πx i = 0 if \|wT x xi\| ≥1 sgn wT x xi xi if \|wT x xi\| < 1. Similarly, the objective function for the optimization problem w.r.t. wy at the tth CCCP iteration is: Oy = λy∥wy∥2 2 + γ N X n=1 ωn snd2 n + (1 −sn)ζy n + 1 J I X j=1 ℓy j , (5) where ζy n = τ1(dn) −τ2(d(t) n ) + d(t) n yT bn(wy −w(t) y ), d(t) n = wT x xan −(w(t) y )T ybn, w(t) y is the value of wy at the tth iteration and ℓy j =    0 if \|wT y yj\| ≥1 1 −wT y yj if 0 ≤wT y yj < 1 1 + wT y yj if −1 < wT y yj < 0. The corresponding sub-gradient is given by ∂Oy ∂wy = −2γ N X n=1 ωnsndnybn −γ N X n=1 ωnµy n + λywy −1 J I X j=1 πy j , (6) where µy n = (1 −sn) ∂τ1 ∂dn −d(t) n ybn and πy j = 0 if \|wT y yj\| ≥1 sgn wT y yj yj if \|wT y yj\| < 1. 4 2.3 Algorithm So far we have only discussed how to learn the hash functions for one bit of the hash codes. To learn the hash functions for multiple bits, one could repeat the same procedure and treat the learning for each bit independently. However, as reported in previous studies [15, 19], it is very important to take into consideration the relationships between different bits in HFL. In other words, to learn compact hash codes, we should coordinate the learning of hash functions for different bits. To this end, we take the standard AdaBoost [25] approach to learn multiple bits sequentially. Intuitively, this approach allows learning of the hash functions in later stages to be aware of the bias introduced by their antecedents. The overall algorithm of CRH is summarized in Algorithm 1. Algorithm 1 Co-Regularized Hashing Input: X, Y – multimodal data Θ – inter-modality point pairs K – code length λx, λy, γ – regularization parameters a, λ – parameters for SCISD function Output: w(k) x , k = 1, . . . , K – projection vectors for X w(k) y , k = 1, . . . , K – projection vectors for Y Procedure: Initialize ω(1) n = 1/N, ∀n ∈{1, 2, . . . , N}. for k = 1 to K do repeat Optimize Equation (3) to get w(k) x ; Optimize Equation (5) to get w(k) y ; until convergence. Compute error of current hash functions ϵk = XN n=1 ω(k) n I[sn̸=hn], where I[a] = 1 if a is true and I[a] = 0 otherwise, and hn = 1 if f(xan) = g(ybn) 0 if f(xan) ̸= g(ybn). Set βk = ϵk/(1 −ϵk). Update the weight for each point pair: ω(k+1) n = ω(k) n β 1−I[sn̸=hn] k . end for In the following, we brieﬂy analyze the time complexity of Algorithm 1 for one bit. The ﬁrst computationally expensive part of the algorithm is to evaluate the sub-gradients. The time complexity is O((k + l)d), where d is the data dimensionality, and k and l are the numbers of random points and random pairs, respectively, for the stochastic sub-gradient solver. In our experiments, we set k = 1 and l = 500. We notice that further increasing the two numbers brings no signiﬁcant performance improvement. We leave the theoretical study of the impact of k and l to our future work. Another major computational cost comes from updating the weights of the inter-modality point pairs. The time complexity is O(dN), where N is the number of inter-modality point pairs. To summarize, our algorithm scales linearly with the number of inter-modality point pairs and the data dimensionality. In practice, the number of inter-modality point pairs is usually small, making our algorithm very efﬁcient. 2.4 Extensions We brieﬂy discuss two possible extensions of CRH in this subsection. First, we note that it is easy to extend CRH to learn nonlinear hash functions via the kernel trick [26]. Speciﬁcally, according to the generalized representer theorem [27], we can represent the projection vectors wx and wy as wx = XI i=1 αiφx(xi) and wy = XJ j=1 βjφy(yj), where φx(·) and φy(·) are kernel-induced feature maps for modalities X and Y, respectively. Then the objective function (1) can be expressed in kernel form and kernel-based hash functions can be learned by minimizing a new but very similar objective function. Another possible extension is to make CRH support more than two modalities. Taking a new modality Z for example, we need to incorporate into Equation (1) the following terms: loss and regularization terms for Z, and all pairwise loss terms involving Z and other modalities, e.g., X and Y. 5 For both extensions, it is straightforward to adapt the algorithm presented above to solve the new optimization problems. 2.5 Discussions CRH is closely related to a recent multimodal metric learning method called Multiview Neighborhood Preserving Projections (Multi-NPP) [23], because CRH uses a loss function for inter-modality point pairs which is similar to Multi-NPP. However, CRH is a general framework and other loss functions for inter-modality point pairs can also be adopted. The two methods have at least three signiﬁcant differences. First, our focus is on HFL while Multi-NPP is on metric learning through embedding. Second, in addition to the inter-modality loss term, the objective function in CRH includes two intra-modality loss terms for large margin HFL while Multi-NPP only has a loss term for the inter-modality point pairs. Third, CRH uses boosting to sequentially learn the hash functions but Multi-NPP does not take this aspect into consideration. As discussed brieﬂy in [23], one may ﬁrst use Multi-NPP to map multimodal data into a common real space and then apply any unimodal HFL method for multimodal hashing. However, this naive two-stage approach has some limitations. First, both stages can introduce information loss which impairs the quality of the hash functions learned. Second, a two-stage approach generally needs more computational resources. These two limitations can be overcome by using a one-stage method such as CRH. 3 Experiments 3.1 Experimental Settings In our experiments, we compare CRH with two state-of-the-art multimodal hashing methods, namely, Cross-Modal Similarity Sensitive Hashing (CMSSH) [20]2 and Cross-View Hashing (CVH) [21],3 for two cross-modal retrieval tasks: (1) image query vs. text database; (2) text query vs. image database. The goal of each retrieval task is to ﬁnd from the text (image) database the nearest neighbors for the image (text) query. We use two benchmark data sets which are, to the best of our knowledge, the largest fully paired and labeled multimodal data sets. We further divide each data set into a database set and a query set. To train the models, we randomly select a group of documents from the database set to form the training set. Moreover, we randomly select 0.1% of the point pairs from the training set. For fair comparison, all models are trained on the same training set and the experiments are repeated with 5 independent training sets. The mean average precision (mAP) is used as the performance measure. To compute the mAP, we ﬁrst evaluate the average precision (AP) of a set of R retrieved documents by AP = 1 L PR r=1 P(r) δ(r), where L is the number of true neighbors in the retrieved set, P(r) denotes the precision of the top r retrieved documents, and δ(r) = 1 if the rth retrieved document is a true neighbor and δ(r) = 0 otherwise. The mAP is then computed by averaging the AP values over all the queries in the query set. The larger the mAP, the better the performance. In the experiments, we set R = 50. Besides, we also report the precision and recall within a ﬁxed Hamming radius. We use cross-validation to choose the parameters for CRH and ﬁnd that the model performance is only mildly sensitive to the parameters. As a result, in all experiments, we set λx = 0.01, λy = 0.01, γ = 1000, a = 3.7, and λ = 1/a. Besides, unless speciﬁed otherwise, we ﬁx the training set size to 2,000 and the code length K to 24. 3.2 Results on Wiki The Wiki data set, generated from Wikipedia featured articles, consists of 2,866 image-text pairs.4 In each pair, the text is an article describing some events or people and the image is closely related to 2We used the implementation generously provided by the authors. 3We implemented the method ourselves because the code is not publicly available. 4http://www.svcl.ucsd.edu/projects/crossmodal/ 6 the content of the article. The images are represented by 128-dimensional SIFT [28] feature vectors, while the text articles are represented by the probability distributions over 10 topics learned by a latent Dirichlet allocation (LDA) model [29]. Each pair is labeled with one of 10 semantic classes. We simply use these class labels to identify the neighbors. Moreover, we use 80% of the data as the database set and the remaining 20% to form the query set. The mAP values of the three methods are reported in Table 1. We can see that CRH outperforms CVH and CMSSH under all settings and CVH performs slightly better than CMSSH. We note that CMSSH ignores the intra-modality relational information and CVH simply treats each bit independently. Hence the performance difference is expected. Table 1: mAP comparison on Wiki Task Method Code Length K = 24 K = 48 K = 64 Image Query vs. Text Database CRH 0.2537 ± 0.0206 0.2399 ± 0.0185 0.2392 ± 0.0131 CVH 0.2043 ± 0.0150 0.1788 ± 0.0149 0.1732 ± 0.0072 CMSSH 0.1965 ± 0.0123 0.1780 ± 0.0080 0.1624 ± 0.0073 Text Query vs. Image Database CRH 0.2896 ± 0.0214 0.2882 ± 0.0261 0.2989 ± 0.0293 CVH 0.2714 ± 0.0164 0.2304 ± 0.0104 0.2156 ± 0.0202 CMSSH 0.2179 ± 0.0161 0.2094 ± 0.0072 0.2040 ± 0.0135 We further compare the three methods on several aspects in Figure 1. We ﬁrst vary the size of the training set in subﬁgures 1(a) and 1(d). Although CVH performs the best when the training set is small, its performance is gradually surpassed by CRH as the size increases. We then plot the precision-recall curves and recall curves for all three methods in the remaining subﬁgures. It is clear that CRH outperforms its two counterparts by a large margin. 0 500 1000 1500 2000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Size of Training Set Precision within Hamming Radius 2 Image Query vs. Text Database CRH CVH CMSSH (a) Varying training set size 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 Recall Precision Image Query vs. Text Database CRH CVH CMSSH (b) Precision-recall curve 0 5 10 15 x 10 5 0 0.2 0.4 0.6 0.8 1 No. of Retrieved Points Recall Image Query vs. Text Database CRH CVH CMSSH (c) Recall curve 0 500 1000 1500 2000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Size of Training Set Precision within Hamming Radius 2 Text Query vs. Image Database CRH CVH CMSSH (d) Varying training set size 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 Recall Precision Text Query vs. Image Database CRH CVH CMSSH (e) Precision-recall curve 0 5 10 15 x 10 5 0 0.2 0.4 0.6 0.8 1 No. of Retrieved Points Recall Text Query vs. Image Database CRH CVH CMSSH (f) Recall curve Figure 1: Results on Wiki 3.3 Results on Flickr The Flickr data set consists of 186,577 image-tag pairs pruned from the NUS data set5 [30] by keeping the pairs that belong to one of the 10 largest classes. The images are represented by 500dimensional SIFT vectors. To obtain more compact representations of the tags, we perform PCA on the original tag occurrence features and obtain 1000-dimensional feature vectors. Each pair is annotated by at least one of 10 semantic labels, and two points are deﬁned as neighbors if they share at least one label. We use 99% of the data as the database set and the remaining 1% to form the query set. 5http://lms.comp.nus.edu.sg/research/NUS-WIDE.htm 7 The mAP values of the three methods are reported in Table 2. In the task of image query vs. text database, CRH performs comparably to CMSSH, which is better than CVH. However, in the other task, CRH achieves the best performance. Table 2: mAP comparison on Flickr Task Method Code Length K = 24 K = 48 K = 64 Image Query vs. Text Database CRH 0.5259 ± 0.0094 0.4990 ± 0.0075 0.4929 ± 0.0064 CVH 0.4717 ± 0.0035 0.4515 ± 0.0041 0.4471 ± 0.0023 CMSSH 0.5287 ± 0.0123 0.5098 ± 0.0141 0.4911 ± 0.0220 Text Query vs. Image Database CRH 0.5364 ± 0.0021 0.5185 ± 0.0050 0.5064 ± 0.0055 CVH 0.4598 ± 0.0020 0.4519 ± 0.0029 0.4477 ± 0.0058 CMSSH 0.5029 ± 0.0321 0.4815 ± 0.0101 0.4660 ± 0.0298 Similar to the previous subsection, we have conducted a group of experiments to compare the three methods on several aspects and report the results in Figure 2. The results for varying the size of the training set are plotted in subﬁgures 2(a) and 2(d). As more training data are used, CRH always performs better but the performance of CVH and CMSSH has high variance. The precision-recall curves and recall curves are shown in the remaining subﬁgures. Similar to the results on Wiki, CRH performs the best. However, the performance gap is smaller here. 0 500 1000 1500 2000 0 0.1 0.2 0.3 0.4 0.5 Size of Training Set Precision within Hamming Radius 2 Image Query vs. Text Database CRH CVH CMSSH (a) Varying training set size 0 0.2 0.4 0.6 0.8 1 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Recall Precision Image Query vs. Text Database CRH CVH CMSSH (b) Precision-recall curve 0 1 2 3 4 x 10 8 0 0.2 0.4 0.6 0.8 1 No. of Retrieved Points Recall Image Query vs. Text Database CRH CVH CMSSH (c) Recall curve 0 500 1000 1500 2000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Size of Training Set Precision within Hamming Radius 2 Text Query vs. Image Database CRH CVH CMSSH (d) Varying training set size 0 0.2 0.4 0.6 0.8 1 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Recall Precision Text Query vs. Image Database CRH CVH CMSSH (e) Precision-recall curve 0 1 2 3 4 x 10 8 0 0.2 0.4 0.6 0.8 1 No. of Retrieved Points Recall Text Query vs. Image Database CRH CVH CMSSH (f) Recall curve Figure 2: Results on Flickr 4 Conclusions In this paper, we have presented a novel method for multimodal hash function learning based on a boosted co-regularization framework. Because the objective function of the optimization problem is in the form of a difference of convex functions, we can devise an efﬁcient learning algorithm based on CCCP and a stochastic sub-gradient method. Comparative studies based on two benchmark data sets show that CRH outperforms two state-of-the-art multimodal hashing methods. To take this work further, we would like to conduct theoretical analysis of CRH and apply it to some other tasks such as multimodal medical image alignment. Another possible research issue is to develop more efﬁcient optimization algorithms to further improve the scalability of CRH. Acknowledgement This research has been supported by General Research Fund 621310 from the Research Grants Council of Hong Kong. 8 References [1] Gregory Shakhnarovich, Trevor Darrell, and Piotr Indyk, editors. Nearest-Neighbor Methods in Learning and Vision: Theory and Practice. MIT Press, March 2006. [2] Piotr Indyk and Rajeev Motwani. 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4,496	Truncation-free Stochastic Variational Inference for Bayesian Nonparametric Models Chong Wang∗ Machine Learning Department Carnegie Mellon University chongw@cs.cmu.edu David M. Blei Computer Science Department Princeton Univeristy blei@cs.princeton.edu Abstract We present a truncation-free stochastic variational inference algorithm for Bayesian nonparametric models. While traditional variational inference algorithms require truncations for the model or the variational distribution, our method adapts model complexity on the ﬂy. We studied our method with Dirichlet process mixture models and hierarchical Dirichlet process topic models on two large data sets. Our method performs better than previous stochastic variational inference algorithms. 1 Introduction Bayesian nonparametric (BNP) models [1] have emerged as an important tool for building probability models with ﬂexible latent structure and complexity. BNP models use posterior inference to adapt the model complexity to the data. For example, as more data are observed, Dirichlet process (DP) mixture models [2] can create new mixture components and hierarchical Dirichlet process (HDP) topic models [3] can create new topics. In general, posterior inference in BNP models is intractable and we must approximate the posterior. The most widely-used approaches are Markov chain Monte Carlo (MCMC) [4] and variational inference [5]. For BNP models, the advantage of MCMC is that it directly operates in the unbounded latent space; whether to increase model complexity (such as adding a new mixture component) naturally folds in to the sampling steps [6, 3]. However MCMC does not easily scale—it requires storing many conﬁgurations of hidden variables, each one on the order of the number of data points. For scalable MCMC one typically needs parallel hardware, and even then the computational complexity scales linearly with the data, which is not fast enough for massive data [7, 8, 9]. The alternative is variational inference, which ﬁnds the member of a simpliﬁed family of distributions to approximate the true posterior [5, 10]. This is generally faster than MCMC, and recent innovations let us use stochastic optimization to approximate posteriors with massive and streaming data [11, 12, 13]. Unlike MCMC, however, variational inference algorithms for BNP models do not operate in an unbounded latent space. Rather, they truncate the model or the variational distribution to a maximum model complexity [13, 14, 15, 16, 17, 18].1 This is particularly limiting in the stochastic approach, where we might hope for a Bayesian nonparametric posterior seamlessly adapting its model complexity to an endless stream of data. In this paper, we develop a truncation-free stochastic variational inference algorithm for BNP models. This lets us more easily apply Bayesian nonparametric data analysis to massive and streaming data. ∗Work was done when the author was with Princeton University. 1In [17, 18], split-merge techniques were used to grow/shrink truncations. However, split-merge operations are model-speciﬁc and difﬁcult to design. It is also unknown how to apply these to the stochastic variational inference setting we consider. 1 In particular, we present a new general inference algorithm, locally collapsed variational inference. When applied to BNP models, it does not require truncations and gives a principled mechanism for increasing model complexity on the ﬂy. We demonstrate our algorithm on DP mixture models and HDP topic models with two large data sets, showing improved performance over truncated algorithms. 2 Truncation-free stochastic variational inference for BNP models Although our goal is to develop an efﬁcient stochastic variational inference algorithm for BNP models, it is more succinct to describe our algorithm for a wider class of hierarchical Bayesian models [19]. We will show how we apply our algorithm for BNP models in §2.3. We consider the general class of hierarchical Bayesian models shown in Figure 1. Let the global hidden variables be β with prior p(β \| η) (η is the hyperparameter) and local variables for each data sample be zi (hidden) and xi (observed) for i = 1, . . . , n. The joint distribution of all variables (hidden and observed) factorizes as, p(β, z1:n, x1:n \| η) = p(β \| η) Qn i=1 p(xi, zi \| β) = p(β \| η) Qn i=1 p(xi \| zi, β)p(zi \| β). (1) The idea behind the nomenclature is that the local variables are conditionally independent of each other given the global variables. For convenience, we assume global variables β are continuous and local variables zi are discrete. (This assumption is not necessary.) A large range of models can be represented using this form, e.g., mixture models [20, 21], mixed-membership models [3, 22], latent factor models [23, 24] and tree-based hierarchical models [25]. As an example, consider a DP mixture model for document clustering. Each document is modeled as a bag of words drawn from a distribution over the vocabulary. The mixture components are the distributions over the vocabulary θ and the mixture proportions π are represented with a stick-breaking process [26]. The global variables β ≜(π, θ) contain the proportions and components, and the local variables zi are the mixture assignments for each document xi. The generative process is: 1. Draw mixture component θk and sticks πk for k = 1, 2, · · · , θk ∼Dirichlet(η), πk = ¯πk Qk−1 ℓ=1 (1 −¯πℓ), ¯πk ∼Beta(1, a). 2. For each document xi, (a) Draw mixture assignment zi ∼Mult(π). (b) For each word xij, draw the word xij ∼Mult(θzi). We now return to the general model in Eq. 1. In inference, we are interested in the posterior of the hidden variables β and z1:n given the observed data x1:n, i.e., p(β, z1:n \| x1:n, η). For many models, this posterior is intractable. We will approximate it using mean-ﬁeld variational inference. 2.1 Variational inference In variational inference we try to ﬁnd a distribution in a simple family that is close to the true posterior. We describe the mean-ﬁeld approach, the simplest variational inference algorithm [5]. It assumes the fully factorized family of distributions over the hidden variables, q(β, z1:n) = q(β) Qn i=1 q(zi). (2) We call q(β) the global variational distribution and q(zi) the local variational distribution. We want to minimize the KL-divergence between this variational distribution and the true posterior. Under the standard variational theory [5], this is equivalent to maximizing a lower bound of the log marginal likelihood of the observed data x1:n. We obtain this bound with Jensen’s inequality, log p(x1:n \| η) = log R P z1:n p(x1:n, z1:n, β \| η)dβ ≥Eq [log p(β) −log q(β) + Pn i=1 log p(xi, zi\|β) −log q(zi)] ≜L(q). (3) 2 zi xi b n h Figure 1: Graphical model for hierarchical Bayesian models with global hidden variables β, local hidden and observed variables zi and xi, i = 1, . . . , n. Hyperparameter η is ﬁxed, not a random variable. A: q(theta1)=Dirichlet(0.1,1,...,1) B: q(theta1)=Dirichlet(1,1,...,1) -14 -12 -10 -8 -6 -4 -2 0 10 20 30 1 2 3 4 5 1 2 3 4 5 w1: frequency of word 1 in the new doc log odds method mean-field our method Figure 2: Results on assigning document d = {w1, 0, . . . , 0} to q(θ1) (case A and B, shown in the ﬁgure above) or q(θ2) = Dirichlet(0.1, 0.1, . . . , 0.1). The y axis is the log-odds of q(z = 1) to q(z = 2)—if it is larger than 0, it is more likely to be assigned to component 1. The mean-ﬁeld approach underestimates the uncertainty around θ2, assigning d incorrectly for case B. The locally collapsed approach does it correctly in both cases. Algorithm 1 Mean-ﬁeld variational inference. 1: Initialize q(β). 2: for iter = 1 to M do 3: for i = 1 to n do 4: Set local variational distribution q(zi) ∝ exp Eq(β) [log p(xi, zi \| β)] . 5: end for 6: Set global variational distribution q(β) ∝ exp Eq(z1:n) [log p(x1:n, z1:n, β)] . 7: end for 8: return q(β). Algorithm 2 Locally collapsed variational inference. 1: Initialize q(β). 2: for iter = 1 to M do 3: for i = 1 to n do 4: Set local distribution q(zi) ∝Eq(β) [p(xi, zi \| β)]. 5: Sample from q(zi) to obtain its empirical ˆq(zi). 6: end for 7: Set global variational distribution q(β) ∝ exp Eˆq(z1:n) [log p(x1:n, z1:n, β)] . 8: end for 9: return q(β). Maximizing L(q) w.r.t. q(β, z1:n) deﬁned in Eq. 2 (with the optimal conditions given in [27]) gives q(β) ∝exp Eq(z1:n) [log p(x1:n, z1:n, β \| η)] (4) q(zi) ∝exp Eq(β) [log p(xi, zi \| β)] . (5) Typically these equations are used in a coordinate ascent algorithm, iteratively optimizing each factor while holding the others ﬁxed (see Algorithm 1). The factorization into global and local variables ensures that the local updates only depend on the global factors, which facilitates speed-ups like parallel [28] and stochastic variational inference [11, 12, 13, 29]. In BNP models, however, the value of zi is potentially unbounded (e.g., the mixture assignment in a DP mixture). Thus we need to truncate the variational distribution [13, 14]. Truncation is necessary in variational inference because of the mathematical structure of BNP models. Moreover, it is difﬁcult to grow the truncation in mean-ﬁeld variational inference even in an ad-hoc way because it tends to underestimate posterior variance [30, 31]. In contrast, its mathematical structure and that it gets the variance right in the conditional distribution allow Gibbs sampling for BNP models to effectively explore the unbounded latent space [6]. 2.2 Locally collapsed variational inference We now describe locally collapsed variational inference, which mitigates the problem of underestimating posterior variance in mean-ﬁeld variational inference. Further, when applied to BNP models, it is truncation-free—it gives a good mechanism to increase truncation on the ﬂy. Algorithm 2 outlines the approach. The difference between traditional mean-ﬁeld variational inference and our algorithm lies in the update of the local distribution q(zi). In our algorithm, it is q(zi) ∝Eq(β) [p(xi, zi \| β)] , (6) as opposed to the mean-ﬁeld update in Eq. 5. Because we collapse out the global variational distribution q(β) locally, we call this method locally collapsed variational inference. Note the two algorithms are similar when q(β) has low variance. However, when the uncertainty modeled in q(β) is high, these two approaches lead to different approximations of the posterior. 3 In our implementation, we use a collapsed Gibbs sampler to sample from Equation 6. This is a local Gibbs sampling step and thus is very fast. Further, this is where our algorithm does not require truncation because Gibbs samplers for BNP models can operate in an unbounded space [6, 3]. Now we update q(β). Suppose we have a set of samples from q(zi) to construct its empirical distribution ˆq(zi). Plugging this into Eq. 3 gives the solution to q(β), q(β) ∝exp Eˆq(z1:n) [log p(x1:n, z1:n, β \| η)] , (7) which has the same form as in Eq. 4 for the mean-ﬁeld approach. This ﬁnishes Algorithm 2. To give an intuitive comparison of locally collapsed (Algorithm 2) and mean-ﬁeld (Algorithm 1) variational inference, we consider a toy document clustering problem with vocabulary size V = 10. We use a two-component Bayesian mixture model with ﬁxed and equal prior proportions π1 = π2 = 1/2. Suppose at some stage, component 1 has some documents assignments while component 2 has not yet and we have obtained the (approximate) posterior for the two component parameters θ1 and θ2 as q(θ1) and q(θ2). For θ1, we consider two cases, A) q(θ1) = Dirichlet(0.1, 1, . . . , 1); B) q(θ1) = Dirichlet(1, 1, . . . , 1). For θ2, we only consider q(θ2) = Dirichlet(0.1, 0.1, . . . , 0.1). In both cases, q(θ1) has relatively low variance while q(θ2) has high variance. The difference is that the q(θ1) in case A has a lower probability on word 1 than that in case B. Now we have a new document d = {w1, 0, . . . , 0}, where word 1 is the only word and its frequency is w1. In both cases, document d is more likely be assigned to component 2 when w1 becomes larger. Figure 2 shows the difference between mean-ﬁeld and locally collapsed variational inference. In case A, the mean-ﬁeld approach does it correctly, since word 1 already has a very low probability in θ1. But in case B, it ignores the uncertainty around θ2, resulting in incorrect clustering. Our approach does it correctly in both cases. What justiﬁes this approach? Alas, as for some other adaptations of variational inference, we do not yet have an airtight justiﬁcation [32, 33, 34]. We are not optimizing q(zi) and so the corresponding lower bound must be looser than the optimized lower bound from the mean-ﬁeld approach if the issue of local modes is excluded. However, our experiments show that we ﬁnd a better predictive distribution than mean-ﬁeld inference. One possible explanation is outlined in S.1 (section 1 of the supplement), where we show that our algorithm can be understood as an approximate Expectation Propagation (EP) algorithm [35]. Related algorithms. Our algorithm is closely related to collapsed variational inference (CVI) [15, 16, 36, 32, 33]. CVI applies variational inference to the marginalized model, integrating out the global hidden variable β. This gives better estimates of posterior variance. In CVI, however, the optimization for each local variable zi depends on all other local variables, and this makes it difﬁcult to apply it at large scale. Our algorithm is akin to applying CVI for the intermediate model that treats q(β) as a prior and considers a single data point xi with its hidden structure zi. This lets us develop stochastic algorithms that can be ﬁt to massive data sets (as we show below). Our algorithm is also related to the recently proposed a hybrid approach of using Gibbs sampling inside stochastic variational inference to take advantage of the sparsity in text documents in topic modeling [37]. Their approach still uses the mean-ﬁeld update as in Eq. 5, where all local hidden topic variables (for a document) are grouped together and the optimal q(zi) is approximated by a Gibbs sampler. With some adaptations, their fast sparse update idea can be used inside our algorithm. Stochastic locally collapsed variational inference. We now extend our algorithm to stochastic variational inference, allowing us to ﬁt approximate posteriors to massive data sets. To do this, we assume the model in Figure 1 is in the exponential family and satisﬁes the conditional conjugacy [11, 13, 29]—the global distribution p(β \| η) is the conjugate prior for the local distribution p(xi, zi \| β), p(β \| η) = h(β) exp η⊤t(β) −a(η) , (8) p(xi, zi \| β) = h(xi, zi) exp β⊤t(xi, zi) −a(β) , (9) where we overload the notation for base measures h(·), sufﬁcient statistics t(·), and log normalizers a(·). (These will often be different for the two families.) Due to the conjugacy, the term t(β) has the form t(β) = [β; −a(β)]. Also assume the global variational distribution q(β \| λ) is in the same family as the prior q(β \| η). Given these conditions, the batch update for q(β \| λ) in Eq. 7 is λ = η + Pn i=1 Eˆq(zi) [¯t(xi, zi)] . (10) 4 The term ¯t(xi, zi) is deﬁned as ¯t(xi, zi) ≜[t(xi, zi); 1]. Analysis in [12, 13, 29] shows that given the conditional conjugacy assumption, the batch update of parameter λ in Eq. 10 can be easily turned into a stochastic update using natural gradient [38]. Suppose our parameter is λt at step t. Given a random observation xt, we sample from q(zt \| xt, λt) to obtain the empirical distribution ˆq(zt). With an appropriate learning rate ρt, we have λt+1 ←λt + ρt −λt + η + nEˆq(zi) [¯t(xt, zt)] . (11) This corresponds to an stochastic update using the noisy natural gradient to optimize the lower bound in Eq. 3 [39]. (We note that the natural gradient is an approximation since our q(zi) in Eq. 6 is suboptimal for the lower bound Eq. 3.) Mini-batch. A common strategy used in stochastic variational inference [12, 13] is to use a small batch of samples at each update. Suppose we have a batch size S, and the set of samples xt, t ∈S. Using our formulation, the q(zt, t ∈S) becomes q(zt,t∈S) ∝Eq(β \| λt) Q t∈S p(xt, zt\|β) . We choose not to factorize zt,t∈S, since factorization will potentially lead to the label-switching problem when new components are instantiated for BNP models [7]. 2.3 Truncation-free stochastic variational inference for BNP models We have described locally collapsed variational inference in a general setting. Our main interests in this paper are BNP models, and we now show how this approach leads to truncation-free variational algorithms. We describe the approach for a DP mixture model [21], whose full description was presented in the beginning of §2.1. See S.2 for the details on the HDP topic models [3]. The global variational distribution. The variational distribution for the global hidden variables, mixture components β and stick proportions ¯π is q(θ, ¯π \| λ, u, v) = Q k q(θ \| λk)q(¯πk \| uk, vk), where λk is the Dirichlet parameter and (uk, vk) is the Beta parameter. The sufﬁcient statistic term t(xi, zi) deﬁned in Eq. 9 can be summarized as t(xi, zi)λkw = 1[zi=k] P j 1[xij=w]; t(xi, zi)uk = 1[zi=k], t(xi, zi)vk = P j=k+1 1[zi=j], where 1[·] is the indicator function. Suppose at time t, we have obtained the empirical distribution ˆq(zi) for observation xi, we use Eq. 11 to update Dirichlet parameter λ and Beta parameter (u, v), λkw ←λkw + ρt(−λkw + η + nˆq(zi = k) P j 1[xij=w]) uk ←uk + ρt(−uk + 1 + nˆq(zi = k)) vk ←vk + ρt(−vk + a + n P ℓ=k+1 ˆq(zi = ℓ)). Although we have a unbounded number of mixture components, we do not need to represent them explicitly. Suppose we have T components that are associated with some data. These updates indicate q(θk \| λk) = Dirichlet(η) and q(¯πk) = Beta(1, a), i.e., their prior distributions, when k > T. Similar to a Gibbs sampler [6], the model is “truncated” automatically. (We re-ordered the sticks according to their sizes [15].) The local empirical distribution ˆq(zi). Since the mixture assignment zi is the only hidden variable, we obtain its analytical form using Eq. 6, q(zi = k) ∝ R p(xi \| θk)p(zi = k \| π)q(θk \| λk)q(¯π)dθkd¯π = Γ(P w λkw) Q w Γ(λkw) Q w Γ(λkw+P j 1[xij=w]) Γ(P w λkw+\|xi\|) uk uk+vk Qk−1 ℓ=1 vℓ uℓ+vℓ, where \|xi\| is the document length and Γ(·) is the Gamma function. (For mini batches, we do not have an analytical form, but we can sample from it.) The probability of creating a new component is q(zi > T) ∝Γ(ηV ) ΓV (η) Q w Γ(η+P j 1[xij=w]) Γ(ηV +\|xi\|) QT k=1 vk uk+vk . We sample from q(zi) to obtain its empirical distribution ˆq(zi). If zi > T, we create a new component. 5 Discussion. Why is “locally collapsed” enough? This is analogous to the collapsed Gibbs sampling algorithm in DP mixture models [6]— whether or not exploring a new mixture component is initialized by one single sample. The locally collapsed variational inference is powerful enough to trigger this. In the toy example above, the role of distribution q(θ2) = Dirichlet(0.1, . . . , 0.1) is similar to that of the potential new component we want to maintain in Gibbs sampling for DP mixture models. Note the difference between this approach and those found in [17, 18], which use mean-ﬁeld methods that can grow or shrink the truncation using split-merge moves. These approaches are model-speciﬁc and difﬁcult to design. Further, they do not transfer to the stochastic setting. In contrast, the approach presented here grows the truncation as a natural consequence of the inference algorithm and is easily adapted to stochastic inference. 3 Experiments We evaluate our methods on DP mixtures and HDP topic models, comparing them to truncation-based stochastic mean-ﬁeld variational inference. We focus on stochastic methods and large data sets. Datasets. We analyzed two large document collections. The Nature data contains about 350K documents from the journal Nature from years 1869 to 2008, with about 58M tokens and a vocabulary size of 4,253. The New York Times dataset contains about 1.8M documents from the years 1987 to 2007, with about 461M tokens and a vocabulary size of 8,000. Standard stop words and those words that appear less than 10 times or in more than 20 percent of the documents are removed, and the ﬁnal vocabulary is chosen by TF-IDF. We set aside a test set of 10K documents from each corpus on which to evaluate its predictive power; these test sets were not given for training. Evaluation Metric. We evaluate the different algorithms using held-out per-word likelihood, likelihoodpw ≜log p(Dtest \| Dtrain)/P xi∈Dtest \|xi\|, Higher likelihood is better. Since exact computing the held-out likelihood is intractable, we use approximations. See S.3 for details of approximating the likelihood. There is some question as to the meaningfulness of held-out likelihood as a metric for comparing different models [40]. Held-out likelihood metrics are nonetheless suited to measuring how well an inference algorithm accomplishes the speciﬁc optimization task deﬁned by a model. Experimental Settings. For DP mixtures, we set component Dirichlet parameter η = 0.5 and the concentration parameter of DP a = 1. For HDP topic models, we set topic Dirichlet parameter η = 0.01, and the ﬁrst-level and second-level concentration parameters of DP a = b = 1 as in [13]. (See S.2 for the full description of HDP topic models.) For stochastic mean-ﬁeld variational inference, we set the truncation level at 300 for both DP and HDP. We run all algorithms for 10 hours and took the model at the ﬁnal stage as the output, without assessing the convergence. We vary the mini-batch size S = {1, 2, 5, 10, 50, 100, 500}. (We do not intend to compare DP and HDP; we want to show our algorithm works on both methods.) For stochastic mean-ﬁeld approach, we set the learning rate according to [13] with ρt = (τ0 + t)−κ with κ = 0.6 and τ0 = 1. We start our new method with 0 components without seeing any data. We cannot use the learning rate schedule as in [13], since it gives very large weights to the ﬁrst several components, effectively leaving no room for creating new components on the ﬂy. We set the learning rate ρt = S/nt, for nt < n, where nt is the size of corpus that the algorithm has seen at time t. After we see all the documents, nt = n. For both stochastic mean-ﬁeld and our algorithm, we set the lower bound of learning rate as S/n. We found this works well in practice. This mimics the usual trick of running Gibbs sampler—one uses sequential prediction for initialization and after all data points have been initialized, one runs the full Gibbs sampler [41]. We remove components with fewer than 1 document for DP and topics with fewer than 1 word for HDP topic models each time when we process 20K documents. 6 Nature New York Times -7.5 -7.4 -7.3 -7.2 -7.1 -8.0 -7.9 -7.8 -7.7 -7.6 100 200 300 400 500 100 200 300 400 500 batchsize likelihood method mean-field our method (a) on both corpora (b) on New York Times 50 100 150 100 200 300 400 500 batchsize number of mixtures At the end 50 100 150 200 250 300 0 2 4 6 8 10 time (hours) Batchsize=100 method mean-field our method DP mixtures Nature New York Times -7.8 -7.6 -7.4 -7.2 -8.4 -8.2 -8.0 -7.8 100 200 300 400 500 100 200 300 400 500 batchsize heldout likelihood method mean-field our method 0 50 100 100 200 300 400 500 batchsize number of topics At the end 50 100 150 200 250 300 0 2 4 6 8 time (hours) Batchsize=100 method mean-field our method (a) on both corpora (b) on Nature Nature New York Times -7.5 -7.4 -7.3 -7.2 -7.1 -8.0 -7.9 -7.8 -7.7 -7.6 100 200 300 400 500 100 200 300 400 500 batchsize likelihood method mean-field our method 50 100 150 100 200 300 400 500 batchsize number of topics At the end 50 100 150 200 250 300 0 2 4 6 8 10 time (hours) Batchsize=100 method mean-field our method (a) on both corpora (b) on New York Times HDP topic models Nature New York Times -7.5 -7.4 -7.3 -7.2 -7.1 -8.0 -7.9 -7.8 -7.7 -7.6 100 200 300 400 500 100 200 300 400 500 batchsize likelihood method mean-field our method 50 100 150 100 200 300 400 500 batchsize number of topics At the end 50 100 150 200 250 300 0 2 4 6 8 time (hours) Batchsize=100 method mean-field our method (a) on both corpora (b) on New York Times stochastic Nature New York Times -7.5 -7.4 -7.3 -7.2 -7.1 -8.0 -7.9 -7.8 -7.7 -7.6 100 200 300 400 500 100 200 300 400 500 batchsize likelihood method mean-field our method 50 100 150 100 200 300 400 500 batchsize number of topics At the end 50 100 150 200 250 300 0 2 4 6 8 10 time (hours) Batchsize=100 method mean-field our method (a) on both corpora (b) on New York Times Figure 3: Results on DP mixtures. (a) Held-out likelihood comparison on both corpora. Our approach is more robust to batch sizes and gives better predictive performance. (b) The inferred number of mixtures on New York Times. (Nature is similar.) The left of ﬁgure (b) shows the number of mixture components inferred after 10 hours; our method tends to give more mixtures. Small batch sizes for the stochastic mean-ﬁeld approach do not really work, resulting in very small number of mixtures. The right of ﬁgure (b) shows how different methods infer the number of mixtures. The stochastic mean ﬁeld approach shrinks it while our approach grows it. Nature New York Times -7.8 -7.6 -7.4 -7.2 -8.4 -8.2 -8.0 -7.8 100 200 300 400 500 100 200 300 400 500 batchsize heldout likelihood method mean-field our method 0 50 100 100 200 300 400 500 batchsize number of topics At the end 50 100 150 200 250 300 0 2 4 6 8 time (hours) Batchsize=100 method mean-field our method (a) on both corpora (b) on Nature HDP topic models Nature New York Times -7.5 -7.4 -7.3 -7.2 -7.1 -8.0 -7.9 -7.8 -7.7 -7.6 100 200 300 400 500 100 200 300 400 500 batchsize likelihood method mean-field our method 50 100 150 100 200 300 400 500 batchsize number of topics At the end 50 100 150 200 250 300 0 2 4 6 time (hours) Batchsize=10 method mean-field our method (a) on both corpora (b) on New York Times stochastic Nature New York Times -7.5 -7.4 -7.3 -7.2 -7.1 -8.0 -7.9 -7.8 -7.7 -7.6 100 200 300 400 500 100 200 300 400 500 batchsize likelihood method mean-field our method 50 100 150 100 200 300 400 500 batchsize number of topics At the end 50 100 150 200 250 300 0 2 4 6 8 10 time (hours) Batchsize=100 method mean-field our method (a) on both corpora (b) on New York Times Figure 4: Results on HDP topic models. (a) Held-out likelihood comparison on both corpora. Our approach is more robust to batch sizes and gives better predictive performance most of time. (b) The inferred number of topics on Nature. (New York Times is similar.) The left of ﬁgure (b) show the number of topics inferred after 10 hours; our method tends to give more topics. Small batch sizes for the stochastic mean-ﬁeld approach do not really work, resulting in very small number of topics. The right of ﬁgure (b) shows how different methods infer the number of topics. Similar to DP, the stochastic mean ﬁeld approach shrinks it while our approach grows it. Results. Figure 3 shows the results for DP mixture models. Figure 3(a) shows the held-out likelihood comparison on both datasets. Our approach is more robust to batch sizes and usually gives better predictive performance. Small batch sizes of the stochastic mean-ﬁeld approach do not work well. Figure 3(b) shows the inferred number of mixtures on New York Times. (Nature is similar.) Our method tends to give more mixtures than the stochastic mean-ﬁeld approach. The stochastic mean-ﬁeld approach shrinks the preset truncation; our approach does not need a truncation and grows the number of mixtures when data requires. Figure 4 shows the results for HDP topic models. Figure 4(a) shows the held-out likelihood comparison on both datasets. Similar to DP mixtures, our approach is more robust to batch sizes and gives better predictive performance most of time. And small batch sizes of the stochastic mean-ﬁeld approach do not work well. Figure 4(b) shows the inferred number of topics on Nature. (New York Times is similar.) This is also similar to DP. Our method tends to give more topics than the stochastic mean-ﬁeld approach. The stochastic mean-ﬁeld approach shrinks the preset truncation while our approach grows the number of topics when data requires. One possible explanation that our method gives better results than the truncation-based stochastic mean-ﬁeld approach is as follows. For truncation-based approach, the algorithm relies more on the random initialization placed on the parameters within the preset truncations. If the random initialization is not used well, performance degrades. This also explains that smaller batch sizes in stochastic mean-ﬁelds tend to work much worse—the ﬁrst fewer samples might dominate the effect from the random initialization, leaving no room for later samples. Our approach mitigates this problem by allowing new components/topics to be created as data requires. If we compare DP and HDP, the best result of DP is better than that of HDP. But this comparison is not meaningful. Besides the different settings of hyperparameters, computing the held-out likelihood for DP is tractable, but intractable for HDP. We used importance sampling to approximate. (See S.3 7 for details.) [42] shows that importance sampling usually gives the correct ranking of different topic models but signiﬁcantly underestimates the probability. 4 Conclusion and future work In this paper, we have developed truncation-free stochastic variational inference algorithms for Bayesian nonparametric models (BNP models) and applied them to two large datasets. Extensions to other BNP models, such as Pitman-Yor process [43], Indian buffet process (IBP) [23, 24] and the nested Chinese restaurant process [18, 25] are straightforward by using their stick-breaking constructions. Exploring how this algorithm behaves in the true streaming setting where the program never stops—a “never-ending learning machine” [44]—is an interesting future direction. Acknowledgements. Chong Wang was supported by Google PhD and Siebel Scholar Fellowships. David M. Blei is supported by ONR N00014-11-1-0651, NSF CAREER 0745520, AFOSR FA955009-1-0668, the Alfred P. Sloan foundation, and a grant from Google. 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4,497	The Coloured Noise Expansion and Parameter Estimation of Diffusion Processes Simon M.J. Lyons School of Informatics University of Edinburgh 10 Crichton Street, Edinburgh, EH8 9AB S.Lyons-4@sms.ed.ac.uk Simo S¨arkk¨a Aalto University Department of Biomedical Engineering and Computational Science Rakentajanaukio 2, 02150 Espoo simo.sarkka@aalto.fi Amos J. Storkey School of Informatics University of Edinburgh 10 Crichton Street, Edinburgh, EH8 9AB a.storkey@ed.ac.uk Abstract Stochastic differential equations (SDE) are a natural tool for modelling systems that are inherently noisy or contain uncertainties that can be modelled as stochastic processes. Crucial to the process of using SDE to build mathematical models is the ability to estimate parameters of those models from observed data. Over the past few decades, signiﬁcant progress has been made on this problem, but we are still far from having a deﬁnitive solution. We describe a novel method of approximating a diffusion process that we show to be useful in Markov chain Monte-Carlo (MCMC) inference algorithms. We take the ‘white’ noise that drives a diffusion process and decompose it into two terms. The ﬁrst is a ‘coloured noise’ term that can be deterministically controlled by a set of auxilliary variables. The second term is small and enables us to form a linear Gaussian ‘small noise’ approximation. The decomposition allows us to take a diffusion process of interest and cast it in a form that is amenable to sampling by MCMC methods. We explain why many state-of-the-art inference methods fail on highly nonlinear inference problems, and we demonstrate experimentally that our method performs well in such situations. Our results show that this method is a promising new tool for use in inference and parameter estimation problems. 1 Introduction Diffusion processes are a ﬂexible and useful tool in stochastic modelling. Many important real world systems are currently modelled and best understood in terms of stochastic differential equations in general and diffusions in particular. Diffusions have been used to model prices of ﬁnancial instruments [1], chemical reactions [2], ﬁring patterns of individual neurons [3], weather patterns [4] and fMRI data [5, 6, 7] among many other phenomena. The analysis of diffusions dates back to Feller and Kolmogorov, who studied them as the scaling limits of certain Markov processes (see [8]). The theory of diffusion processes was revolutionised by Itˆo, who interpreted a diffusion process as the solution to a stochastic differential equation [9, 10]. This viewpoint allows one to see a diffusion process as the randomised counterpart of an ordinary differential equation. One can argue that stochastic differential equations are the natural 1 tool for modelling continuously evolving systems of real valued quantities that are subject to noise or stochastic inﬂuences. The classical approach to mathematical modelling starts with a set of equations that describe the evolution of a system of interest. These equations are goverened by a set of input parameters (for example particle masses, reaction rates, or more general constants of proportionality) that determine the behaviour of the system. For practical purposes, it is of considerable interest to solve the inverse problem. Given the output of some system, what can be said about the parameters that govern it? In the present setting, we observe data which we hypothesize are generated by a diffusion. We would like to know what the nature of this diffusion is. For example, we may begin with a parametric model of a physical system, with a prior distribution over the parameters. In principle, one can apply Bayes’ theorem to deduce the posterior distribution. In practice, this is computationally prohibitive: it is necessary to solve a partial differential equation known as the Fokker-Planck equation (see [11]) in order to ﬁnd the transition density of the diffusion of interest. This solution is rarely available in closed form, and must be computed numerically. In this paper, we propose a novel approximation for a nonlinear diffusion process X. One heuristic way of thinking about a diffusion is as an ordinary differential equation that is perturbed by white noise. We demonstrate that one can replace the white noise by a ‘coloured’ approximation without inducing much error. The nature of the coloured noise expansion method enables us to control the behaviour of the diffusion over various length-scales. This allows us to produce samples from the diffusion process that are consistent with observed data. We use these samples in a Markov chain Monte-Carlo (MCMC) inference algorithm. The main contributions of this paper are: • Novel development of a method for sampling from the time-t marginal distribution of a diffusion process based on a ‘coloured’ approximation of white noise. • Demonstration that this approximation is a powerful and scalable tool for making parameter estimation feasible for general diffusions at minimal cost. The paper is structured as follows: in Section 2, we describe the structure of our problem. In Section 3 we conduct a brief survey of existing approaches to the problem. In Section 4, we discuss the coloured noise expansion and its use in controlling the behaviour of a diffusion process. Our inference algorithm is described in Section 5. We describe some numerical experiments in Section 6, and future work is discussed in Section 7. 2 Parametric Diffusion Processes In this section we develop the basic notation and formalism for the diffusion processes used in this work. First, we assume our data are generated by observing a k-dimensional diffusion processes with dynamics dXt = aθ(Xt)dt + BθdWt, X0 ∼p(x0), (1) where the initial condition is drawn from some known distribution. Observations are assumed to occur at times t1, . . . , tn, with ti−ti−1 := Ti. We require that aθ : IRk →IRk is sufﬁciently regular to guarantee the existence of a unique strong solution to (1), and we assume Bθ ∈IRk×d. Both terms depend on a set of potentially unknown parameters θ ∈IRdθ. We impose a prior distribution p(θ) on the parameters. The driving noise W is a d-dimensional Brownian motion, and the equation is interpreted in the Itˆo sense. Observations are subject to independent Gaussian perturbations centered at the true value of X. That is, Yti = Xti + ϵti, ϵti ∼N(0, Σi) (2) We use the notation X to refer to the entire sample path of the diffusion, and Xt to denote the value of the process at time t. We will also employ the shorthand Y1:n = {Yt1, . . . , Ytn}. Many systems can be modelled using the form (1). Such systems are particularly relevant in physics and natural sciences. In situations where this is not explicitly the case, one can often hope to reduce a diffusion to this form via the Lamperti transform. One can almost always accomplish this in the univariate case, but the multivariate setting is somewhat more involved. A¨ıt-Sahalia [12] characterises the set of multivariate diffusions to which this transform can be applied. 2 3 Background Work Most approaches to parameter estimation of diffusion processes rely on the Monte-Carlo approximation. Beskos et al. [13] [14] employ a method based on rejection sampling to estimate parameters without introducing any discretisation error. Golightly and Wilkinson [15] extend the work of Chib et al. [16] and Durham and Gallant [17] to construct a Gibbs sampler that can be applied to the parameter estimation problem. Roughly speaking, Gibbs samplers that exist in the literature alternate between drawing samples from some representation of the diffusion process X conditional on parameters θ, and samples from θ conditional on the current sample path of X. Note that draws from X must be consistent with the observations Y1:n. The usual approach to the consistency issue is to make a proposal by conditioning a linear diffusion to hit some neighbourhood of the observation Yk, then to make a correction via a rejection sampling [18] or a Metropolis-Hastings [16] step. However, as the inter-observation time grows, the qualitative difference between linear and nonlinear diffusions gets progressively more pronounced, and the rate of rejection grows accordingly. Figure 1 shows the disparity between a sample from a nonlinear process and a sample from the linear proposal. One can see that the target sample path is constrained to stay near the mode γ = 2.5, whereas the proposal can move more freely. One should expect to make many proposals before ﬁnding one that ‘behaves’ like a typical draw from the true process. 0 1 2 3 −3 −2 −1 0 1 t X(t) Nonlinear sample path and proposal 0 5 10 15 20 −3 −2 −1 0 1 2 t X(t) Sample path with noisy observations (a) (b) Figure 1: (a) Sample path of a double well process (see equation (18)) with α = 2, γ = 2.5, B = 2 (blue line). Current Gibbs samplers use linear proposals (dashed red line) with a rejection step to draw conditioned nonlinear paths. In this case, the behaviour of the proposal is very different to that of the target, and the rate of rejection is high. (b) Sample path of a double well process (solid blue line) with noisy observations (red dots). We use this as an initial dataset on which to test our algorithm. Parameters are α = 2, γ = 1, B = 1. Observation errors have variance Σ = .25. For low-dimensional inference problems, algorithms that employ sequential Monte-Carlo (SMC) methods [19] [20] typically yield good results. However, unlike the Gibbs samplers mentioned above, SMC-based methods often do not scale well with dimension. The number of particles that one needs to maintain a given accuracy is known to scale exponentially with the dimension of the problem [21]. A¨ıt-Sahalia [12, 22] uses a deterministic technique based on Edgeworth expansions to approximate the transition density. Other approaches include variational methods [23, 24] that can compute continuous time Gaussian process approximations to more general stochastic differential systems, as well as various non-linear Kalman ﬁltering and smoothing based approximations [25, 26, 27] . 4 Coloured Noise Expansions and Brownian Motion We now introduce a method of approximating a nonlinear diffusion that allows us to gain a considerable amount of control over the behaviour of the process. Similar methods have been used 3 for stratiﬁed sampling of diffusion processes [28] and the solution of stochastic partial differential equations [29] . One of the major challenges of using MCMC methods for parameter estimation in the present context is that it is typically very difﬁcult to draw samples from a diffusion process conditional on observed data. If one only knows the initial condition of a diffusion, then it is straightforward to simulate a sample path of the process. However, simulating a sample path conditional on both initial and ﬁnal conditions is a challenging problem. Our approximation separates the diffusion process X into the sum of a linear and nonlinear component. The linear component of the sum allows us to condition the approximation to ﬁt observed data more easily than in conventional methods. On the other hand, the nonlinear component captures the ‘gross’ variation of a typical sample path. In this section, we ﬁx a generic time interval [0, T], though one can apply the same derivation for any given interval Ti = ti −ti−1. Heuristically, one can think of the random process that drives the process deﬁned in equation (1) as white noise. In our approximation, we project this white noise into an N-dimensional subspace of L2[0, T], the Hilbert space of square-integrable functions deﬁned on the interval [0, T]. This gives a ‘coloured noise’ process that approaches white noise asymptotically as N →∞. The coloured noise process is then used to drive an approximation of (1). We can choose the space into which to project the white noise in such a way that we will gain some control over its behaviour. This is analagous to the way that Fourier analysis allows us to manipulate properties of signals Recall that a standard Brownian motion on the interval [0, T] is a one-dimentional Gaussian process with zero mean and covariance function k(s, t) = min{s, t}. By deﬁnition of the Itˆo integral, we can write Wt = Z t 0 dWs = Z T 0 I[0,t](s)dWs. (3) Suppose {φi}i≥1 is an orthonormal basis of L2[0, T]. We can interpret the indicator function in (3) as an element of L2[0, T] and expand it in terms of the basis functions as follows: I[0,t](s) = ∞ X i=1 ⟨I[0,t](·), φi(·)⟩φi(s) = ∞ X i=1 Z t 0 φi(u)du φi(s). (4) Substituting (4) into (3), we see that Wt = ∞ X i=1 Z T 0 φi(s)dWs ! Z t 0 φi(u)du. (5) We will employ the shorthand Zi = R T 0 φi(s)dWs. Since the functions {φi} are deterministic and orthonormal, we know from standard results of Itˆo calculus that the random variables {Zi} are i.i.d standard normal. The inﬁnite series in equation (5) can be truncated after N terms to derive an approximation, ˆWt of Brownian motion. Taking the derivative with respect to time, the result is a ‘coloured’ approximation of white noise, taking the form d ˆWt dt = N X i=1 Ziφi(t). (6) The multivariate approximation is similar. We seperate a d-dimensional Brownian motion into onedimensional components and decompose the individual components as in (6). In principle, one can choose a different value of N for each component of the Brownian motion, but for ease of exposition we do not do so here. We can substitute this approximation into equation (1), which gives dXNL t dt = aθ(XNL t ) + Bθ N X i=1 ΦiZi, XNL 0 ∼p(x0), (7) where Φi is the diagonal d × d matrix with entries (φi1, . . . , φid), and Zi = (Zi1, . . . , Zid)⊺. This derivation is useful because equation (7) gives us an alternative to the Euler-Maruyama discretisation for sampling approximately from the time-t marginal distribution of a diffusion process. We 4 draw coefﬁcients Zij from a standard normal distribution, and solve the appropriate vector-valued ordinary differential equation. While the Euler discretisation is the de facto standard method for numerical approximation of SDE, other methods do exist. Kloeden and Platen [30] discuss higher order methods such as the stochastic Runge-Kutta scheme [31]. In the Euler-Maruyama approximation, one discretises the driving Brownian motion into increments Wti −Wti−1 = √TiZi. One must typically employ a ﬁne discretisation to get a good approximation to the true diffusion process. Empirically, we ﬁnd that one needs far fewer Gaussian inputs Zi for an accurate representation of XT using the coloured noise approximation. This more parsimonious representation has advantages. For example, Corlay and Pages [28] employ related ideas to conduct stratiﬁed sampling of a diffusion process. The coefﬁcients Zi are also more amenable to interpretation than the Gaussian increments in the Euler-Maruyama expansion. Suppose we have a one-dimensional process in which we use the Fourier cosine basis φk(t) = p 2/T cos((2k −1)πt/2T). (8) If we change Z1 while holding the other coefﬁcients ﬁxed, we will typically see a change in the large-scale behaviour of the path. On the other hand, a change in ZN will typically result in a change to the small-scale oscillations in the path. The seperation of behaviours across coefﬁcients gives us a means to obtain ﬁne-grained control over the behaviour of a diffusion process within a Metropolis-Hastings algorithm. We can improve our approximation by attempting to correct for the fact that we truncated the sum in equation (6). Instead of simply discarding the terms ZiΦi for i > N, we attempt to account for their effect as follows. We assume the existence of some ‘correction’ process XC such that X = XNL + XC. We know that the dynamics of X satisfy dXt = aθ XNL t + XC t dt + BθdWt. (9) Taylor expanding the drift term around XNL, we see that to ﬁrst order, dXt ≈ aθ XNL t + Ja(XNL t )XC t dt + BθdWt = aθ XNL t + Bθd ˆ Wt dt + Ja(XNL t )XC t dt + Bθ dWt −d ˆ Wt . (10) Here, Ja(x) is the Jacobian matrix of the function a evaluated at x. This motivates the use of a linear time-dependent approximation to the correction process. We will refer to this linear approximation as XL. The dynamics of XL satisfy dXL t = Ja(XNL t )XL t dt + BθdRt, XL 0 = 0, (11) where the driving noise is the ‘residual’ term R = W −ˆ W. Conditional on XNL, XL is a linear Gaussian process, and equation (11) can be solved in semi-closed form. First, we compute a numerical approximation to the solution of the homogenous matrix-valued equation d dtΨ(t) = Ja(XNL t )Ψ(t), Ψ(0) = In. (12) One can compute Ψ−1(t) in a similar fashion via the relationship dΨ−1/dt = −Ψ−1(dΨ/dt)Ψ−1. We then have XL t = Ψ(t) Z t 0 Ψ(u)−1BdRu = Ψ(t) Z t 0 Ψ(u)−1BdWu − N X i=1 Ψ(t) Z t 0 Ψ(u)−1BΦi(u)du Zi. (13) 5 It follows that XL has mean 0 and covariance k(s, t) = Ψ(s) Z s∧t 0 Ψ(u)−1BB⊺Ψ⊺(u)−1du Ψ⊺(t) − N X i=1 Ψ(s) Z s 0 Ψ(u)−1BΦi(u)du Z t 0 Ψ(u)−1BΦi(u)du ⊺ Ψ⊺(t). (14) The process XNL is designed to capture the most signiﬁcant nonlinear features of the original diffusion X, while the linear process XL corrects for the truncation of the sum (6), and can be understood using tools from the theory of Gaussian processes. One can think of the linear term as the result of a ‘small-noise’ expansion about the nonlinear trajectory. Small-noise techniques have been applied to diffusions in the past [11], but the method described above has the advantage of being inherently nonlinear. In the supplement to this paper, we show that ˆX = XNL +XL converges to X in L2[0, T] as N →∞under the assumption that a is Lipschitz continuous. If the drift function is linear, then ˆX = X regardless of the choice of N. 5 Parameter Estimation In this section, we describe a novel modiﬁcation of the Gibbs sampler that does not suffer the drawbacks of the linear proposal strategy. In Section 6, we demonstrate that for highly nonlinear problems it will perform signiﬁcantly better than standard methods because of the nonlinear component of our approximation. Suppose for now that we make a single noiseless observation at time t1 = T (for ease of notation, we will assume that observations are uniformly spaced through time with ti+1 −ti = T, though this is not necessary). Our aim is to sample from the posterior distribution p θ, Z1:N\| XNL 1 + XL 1 = Y1 ∝N(Y1 \| XNL 1 , k1(T, T))N(Z1:N)p(θ). (15) We adopt the convention that N(·\| µ, Σ) represents the normal distribution with mean µ and covariance Σ, whereas N(·) represents the standard normal distribution. Note that we have left dependence of k1 on Z and θ implicit. The right-hand side of this expression allows us to evaluate the posterior up to proportionality; hence it can be targeted with a Metropolis-Hastings sampler. With multiple observations, the situation is similar. However, we now have a set of Gaussian inputs Z(i) for each transition ˆXi\| ˆXi−1. If we attempt to update θ and {Z(i)}i≤n all at once, the rate of rejection will be unacceptably high. For this reason, we update each Z(i) in turn, holding θ and the other Gaussian inputs ﬁxed. We draw Z(i)∗from the proposal distribution, and compute XNL∗ i with initial condition Yi−1. We also compute the covariance k∗ i (T, T) of the linear correction. The acceptance probability for this update is α = 1 ∧N(Yi \| XNL∗ i , k∗ i (T, T))N(Z(i)∗ 1:N)p(Z(i)∗ 1:N →Z(i) 1:N) N(Yi \| XNL i , ki(T, T))N(Z(i) 1:N)p(Z(i) 1:N →Z(i)∗ 1:N) (16) After updating the Gaussian inputs, we make a global update for the θ parameter. The acceptance probability for this move is α = 1 ∧ n Y i=1 N(Yi \| XNL∗ i , k∗ i (T, T))p(θ∗)p(θ∗→θ) N(Yi \| XNL i , ki(T, T))p(θ)p(θ →θ∗) , (17) where XNL∗ i and k∗ i (T, T) are computed using the proposed value of θ∗. We noted earlier that when j is large, Zj governs the small-time oscillations of the diffusion process. One should not expect to gain much information about the value of Zj when we have large interobservation times. We ﬁnd this to be the case in our experiments - the posterior distribution of Zj:N approaches a spherical Gaussian distribution when j > 3. For this reason, we employ a Gaussian random walk proposal in Z1 with stepsize σRW = .45, and proposals for Z2:N are drawn independently from the standard normal distribution. 6 In the presence of observation noise, we proceed roughly as before. Recall that we make observations Yi = Xi + ϵi. We draw proposals Z(i)∗ 1:N and ϵ∗ i . The initial condition for XNL i is now Yi−1 −ϵi−1. However, one must make an important modiﬁcation to the algorithm. Suppose we propose an update of ˆXi and it is accepted. If we subsequently propose an update for ˆXi+1 and it is rejected, then the initial condition for ˆXi+1 will be inconsistent with the current state of the chain (it will be Yi −ϵi instead of Yi −ϵ∗ i ). For this reason, we must propose joint updates for ( ˆXi, ϵi, ˆXi+1). If the variance of the observation noise is high, it may be more efﬁcient to target the joint posterior distribution p θ, {Zi 1:N, XL i } \| Y1:n . 6 Numerical Experiments The double-well diffusion is a widely-used benchmark for nonlinear inference problems [24, 32, 33, 34]. It has been used to model systems that exhibit switching behaviour or bistability [11, 35]. It possesses nonlinear features that are sufﬁcient to demonstrate the shortcomings of some existing inference methods, and how our approach overcomes these issues. The dynamics of the process are given by dXt = αXt γ2 −X2 t dt + BdWt. (18) The process X has a bimodal stationary distribution, with modes at x = ±γ. The parameter α governs the rate at which sample trajectories are ’pushed’ toward either mode. If B is small in comparison to α, mode-switching occurs relatively rarely. Figure 1(b) shows a trajectory of a double-well diffusion over 20 units of time, with observations at times {1, 2, . . . , 20} . We used the parameters α = 2, γ = 1, B = 1. The variance of the observation noise was set to Σ = .25. As we mentioned earlier, particle MCMC performs well in low-dimensional inference problems. For this reason, the results of a particle MCMC inference algorithm (with N = 1, 000) particles are used as ’ground truth’. Our algorithm used N = 3 Gaussian inputs with a linear correction. We used the Fourier cosine series (8) as an orthonormal basis. We compare our Gibbs sampler to that of Golightly and Wilkinson [15], for which we use an Euler discretisation with stepsize ∆t = .05. Each algorithm drew 70, 000 samples from the posterior distribution, moving through the parameter space in a Gaussian random walk. We placed an exponential(4) prior on γ and an exponential(1) prior on α and B. For this particular choice of parameters, both Gibbs samplers give a good approximation to the true posterior. Figure 2 shows histograms of the marginal posterior distributions of (α, γ, B) for each algorithm. 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 α p(α) (a) p(α\|Y1:20) 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 γ p(γ) (b) p(γ\|Y1:20) 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 B p(B) (c) p(B\|Y1:20) Figure 2: Marginal posterior distributions for (α, γ, B) conditional on observed data. The solid black line is the output of a particle MCMC method, taken as ground truth. The broken red line is the output of the linear proposal method, and the broken and dotted blue line is the density estimate from the coloured noise expansion method. We see that both methods give a good approximation to the ground truth. Gibbs samplers that have been used in the past rely on making proposals by conditioning a linear diffusion to hit a target, and subsequently accepting or rejecting those proposals. Over short timescales, or for problems that are not highly nonlinear, this can be an effective strategy. However, as the timescale increases, the proposal and target become quite dissimilar (see Figure 1(a)). 7 For our second experiment, we simulate a double well process with (α, γ, B) = (2, 2.5, 2). We make noisy observations with ti −ti−1 = 3 and Σ = .1. The algorithms target the posterior distribution over γ, with α and B ﬁxed at their true values. From our previous discussion, one might expect the linear proposal strategy to perform poorly in this more nonlinear setting. This is indeed the case. As in the previous experiment, we used a linear proposal Gibbs sampler with Euler stepsize dt = 0.05. In the ‘path update’ stage, fewer than .01% of proposals were accepted. On the other hand, the coloured noise expansion method used N = 7 Gaussian inputs with a linear correction and was able to approximate the posterior accurately. Figure 3 shows histograms of the results. Note the different scaling of the rightmost plot. 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 γ p(γ) (a) Particle MCMC 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 γ p(γ) (b) Coloured noise expansion method 1 1.5 2 2.5 3 0 2 4 6 8 γ p(γ) (c) Linear proposal method Figure 3: p(γ\|Y1:10, B, α) after ten observations with a relatively large inter-observation time. We drew data from a double well process with (α, γ, B) = (2, 2.5, 2). The coloured noise expansion method matches the ground truth, whereas the linear proposal method is inconsistent with the data. 7 Discussion and Future Work We have seen that the standard linear proposal/correction strategy can fail for highly nonlinear problems. Our inference method avoids the linear correction step, instead targeting the posterior over input variables directly. With regard to computational efﬁciency, it is difﬁcult to give an authoritative analysis because both our method and the linear proposal method are complex, with several parameters to tune. In our experiments, the algorithms terminated in a roughly similar length of time (though no serious attempt was made to optimise the runtime of either method). With regard to our method, several questions remain open. The accuracy of our algorithm depends on the choice of basis functions {φi}. At present, it is not clear how to make this choice optimally in the general setting. In the linear case, it is possible to show that one can achieve the accuracy of the Karhunen-Loeve decomposition, which is theoretically optimal. One can also set the error at a single time t to zero with a judicious choice of a single basis function. We aim to present these results in a paper that is currently under preparation. We used a Taylor expansion to compute the covariance of the correction term. However, it may be fruitful to use more sophisticated ideas, collectively known as statistical linearisation methods. In this paper, we restricted our attention to processes with a state-independent diffusion coefﬁcient so that the covariance of the correction term could be computed. We may be able to extend this methodology to process with state-dependent noise - certainly one could achieve this by taking a 0-th order Taylor expansion about XNL. Whether it is possible to improve upon this idea is a matter for further investigation. Acknowledgments Simon Lyons was supported by Microsoft Research, Cambridge. References [1] R.C. Merton. Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4:141–183, 1973. [2] D.T. Gillespie. The chemical Langevin equation. 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4,498	How Prior Probability Inﬂuences Decision Making: A Unifying Probabilistic Model Yanping Huang University of Washington huangyp@cs.washington.edu Abram L. Friesen University of Washington afriesen@cs.washington.edu Timothy D. Hanks Princeton University thanks@princeton.edu Michael N. Shadlen Columbia University Howard Hughes Medical Institute ms4497@columbia.edu Rajesh P. N. Rao University of Washington rao@cs.washington.edu Abstract How does the brain combine prior knowledge with sensory evidence when making decisions under uncertainty? Two competing descriptive models have been proposed based on experimental data. The ﬁrst posits an additive offset to a decision variable, implying a static effect of the prior. However, this model is inconsistent with recent data from a motion discrimination task involving temporal integration of uncertain sensory evidence. To explain this data, a second model has been proposed which assumes a time-varying inﬂuence of the prior. Here we present a normative model of decision making that incorporates prior knowledge in a principled way. We show that the additive offset model and the time-varying prior model emerge naturally when decision making is viewed within the framework of partially observable Markov decision processes (POMDPs). Decision making in the model reduces to (1) computing beliefs given observations and prior information in a Bayesian manner, and (2) selecting actions based on these beliefs to maximize the expected sum of future rewards. We show that the model can explain both data previously explained using the additive offset model as well as more recent data on the time-varying inﬂuence of prior knowledge on decision making. 1 Introduction A fundamental challenge faced by the brain is to combine noisy sensory information with prior knowledge in order to perceive and act in the natural world. It has been suggested (e.g., [1, 2, 3, 4]) that the brain may solve this problem by implementing an approximate form of Bayesian inference. These models however leave open the question of how actions are chosen given probabilistic representations of hidden state obtained through Bayesian inference. Daw and Dayan [5, 6] were among the ﬁrst to study decision theoretic and reinforcement learning models with the goal of interpreting results from various neurobiological experiments. Bogacz and colleagues proposed a model that combines a traditional decision making model with reinforcement learning [7] (see also [8, 9]). In the decision making literature, two apparently contradictory models have been suggested to explain how the brain utilizes prior knowledge in decision making: (1) a model that adds an offset to a 1 decision variable, implying a static effect of changes to the prior probability [10, 11, 12], and (2) a model that adds a time varying weight to the decision variable, representing the changing inﬂuence of prior probability over time [13]. The LATER model (Linear Approach to Threshold with Ergodic Rate), an instance of the additive offset model, incorporates prior probability as the starting point of a linearly rising decision variable and successfully predicts changes to saccade latency when discriminating between two low contrast stimuli [10]. However, the LATER model fails to explain data from the random dots motion discrimination task [14] in which the agent is presented with noisy, time-varying stimuli and must continually process this data in order to make a correct choice and receive reward. The drift diffusion model (DDM), which uses a random walk accumulation, instead of a linear rise to a boundary, has been successful in explaining behavioral and neurophysiological data in various perceptual discrimination tasks [14, 15, 16]. However, in order to explain behavioral data from recent variants of random dots tasks in which the prior probability of motion direction is manipulated [13], DDMs require the additional assumption of dynamic reweighting of the inﬂuence of the prior over time. Here, we present a normative framework for decision making that incorporates prior knowledge and noisy observations under a reward maximization hypothesis. Our work is inspired by models which cast human and animal decision making in a rational, or optimal, framework. Frazier & Yu [17] used dynamic programming to derive an optimal strategy for two-alternative forced choice tasks under a stochastic deadline. Rao [18] proposed a neural model for decision making based on the framework of partially observable Markov decision processes (POMDPs) [19]; the model focuses on network implementation and learning but assumes a ﬁxed deadline to explain the collapsing decision threshold seen in many decision making tasks. Drugowitsch et al. [9] sought to explain the collapsing decision threshold by combining a traditional drift diffusion model with reward rate maximization; their model also requires knowledge of decision time in hindsight. In this paper, we derive a novel POMDP model from which we compute the optimal behavior for sequential decision making tasks. We demonstrate our model’s explanatory power on two such tasks: the random dots motion discrimination task [13] and Carpenter and Williams’ saccadic eye movement task [10]. We show that the urgency signal, hypothesized in previous models, emerges naturally as a collapsing decision boundary with no assumption of a decision deadline. Furthermore, our POMDP formulation enables incorporation of partial or incomplete prior knowledge about the environment. By ﬁtting model parameters to the psychometric function in the neutral prior condition (equal prior probability of either direction), our model accurately predicts both the psychometric function and the reaction times for the biased (unequal prior probability) case, without introducing additional free parameters. Finally, the same model also accurately predicts the effect of prior probability changes on the distribution of reaction times in the Carpenter and Williams task, data that was previously interpreted in terms of the additive offset model. 2 Decision Making in a POMDP framework 2.1 Model Setup We model a decision making task using a POMDP, which assumes that at any particular time step, t, the environment is in a particular hidden state, x ∈X, that is not directly observable by the animal. The animal can make sensory measurements in order to observe noisy samples of this hidden state. At each time step, the animal receives an observation (stimulus), st, from the environment as determined by an emission distribution, Pr(st\|x). The animal must maintain a belief over the set of possible true world states, given the observations it has made so far: bt(x) = Pr(x\|s1:t), where s1:t represents the sequence of stimuli that the animal has received so far, and b0(x) represents the animal’s prior knowledge about the environment. At each time step, the animal chooses an action, a ∈A and receives an observation and a reward, R(x, a), from the environment, depending on the current state and the action taken. The animal uses Bayes rule to update its belief about the environment after each observation. Through these interactions, the animal learns a policy, π(b) ∈A for all b, which dictates the action to take for each belief state. The goal is to ﬁnd an optimal policy, π∗(b), that maximizes the animal’s total expected future reward in the task. For example, in the random dots motion discrimination task, the hidden state, x, is composed of both the coherence of the random dots c ∈[0, 1] and the direction d ∈{−1, 1} (corresponding to leftward and rightward motion, respectively), neither of which are known to the animal. The 2 animal is shown a movie of randomly moving dots, a fraction of which are moving in the same direction (this fraction is the coherence). The movie is modeled as a sequence of time varying stimuli s1:t. Each frame, st, is a snapshot of the changes in dot positions, sampled from the emission distribution st ∼Pr(st\|kc, d), where k > 0 is a free parameter that determines the scale of st. In order to discriminate the direction given the stimuli, the animal uses Bayes rule to compute the posterior probability of the static joint hidden state, Pr(x = kdc\|s1:t)1. At each time step, the animal chooses one of three actions, a ∈{AR, AL, AS}, denoting rightward eye movement, leftward eye movement, and sampling (i.e., waiting for one more observation), respectively. When the animal makes a correct choice (i.e., a rightward eye movement a = AR when x > 0 or a leftward eye movement a = AL when x < 0), the animal receives a positive reward RP > 0. The animal receives a negative reward (penalty) or no reward when an incorrect action is chosen, RN ≤0. We assume that the animal is motivated by hunger or thirst to make a decision as quickly as possible and model this with a unit penalty RS = −1, representing the cost the agent needs to pay when choosing the sampling action AS. 2.2 Bayesian Inference of Hidden State from Prior Information and Noisy Observations In a POMDP, decisions are made based on the belief state bt(x) = Pr(x\|s1:t), which is the posterior probability distribution over x given a sequence of observations s1:t. The initial belief b0(x) represents the animal’s prior knowledge about x. In both the Carpenter and William’s task [10] and the random dots motion discrimination task [13], prior information about the probability of a speciﬁc direction (we use rightward direction here, dR, without loss of generality) is learned by the subjects, Pr(dR) = Pr(d = 1) = Pr(x > 0) = 1 −Pr(dL). Consider the random dots motion discrimination task. Unlike the traditional case where a full prior distribution is given, this direction-only prior information provides only partial knowledge about the hidden state which also includes coherence. In the least informative case, only Pr(dR) is known and we model the distribution over the remaining components of x as a uniform distribution. Combining this with the direction prior, which is Bernoulli distributed, gives a piecewise uniform distribution for the prior, b0(x). In the general case, we can express the distribution over coherence as a normal distribution parameterized by µ0 and σ0, resulting in a piecewise normal prior over x, b0(x) = Z−1 0 N(x \| µ0, σ0) × Pr(dR) x ≥0 Pr(dL) x < 0, (1) where Zt = Pr(dR)(1 −Φ (0 \| µt, σt)) + Pr(dL)Φ (0 \| µt, σt) is the normalization factor and Φ(x \| µ, σ) = R x −∞N(x \| µ, σ)dx is the cumulative distribution function (CDF) of the normal distribution. The piecewise uniform prior is then just a special case with µ0 = 0 and σ0 = ∞. We assume the emission distribution is also normally-distributed, Pr(st\|x) = N(st\|x, σ2 e), which, from Bayes’ rule, results in a piecewise normal posterior distribution bt(x) = Z−1 t N(x \| µt, σt) × Pr(dR) x ≥0 Pr(dL) x < 0 (2) where µt = µ0 σ2 0 + t¯st σ2e / 1 σ2 0 + t σ2e , (3) σ2 t = 1 σ2 0 + t σ2e −1 , (4) and the running average ¯st = Pt t′=1 st′/t. Consequently, the posterior distribution depends only on ¯s and t, the two sufﬁcient statistics of the sequence s1:t. For the case of a piecewise uniform prior, the variance σ2 t = σ2 e t , which decreases inversely in proportion to elapsed time. Unless otherwise mentioned, we ﬁx σe = 1, σ0 = ∞and µ0 = 0 for the rest of this paper because we can rescale the POMDP time step t′ = t σe to compensate. 1In the decision making tasks that we model in this paper, the hidden state is ﬁxed within a trial and thus there is no transition distribution to include in the belief update equation. However, the POMDP framework is entirely valid for time-varying states. 3 2.3 Finding the optimal policy by reward maximization Within the POMDP framework, the animal’s goal is to ﬁnd an optimal policy π∗(bt) that maximizes its expected reward, starting at bt. This is encapsulated in the value function vπ(bt) = E " ∞ X k=1 r(bt+k, π(bt+k)) \| bt, π # (5) where the expectation is taken with respect to all future belief states (bt+1, . . . , bt+k, . . .) given that the animal is using π to make decisions, and r(b, a) is the reward function over belief states or, equivalently, the expected reward over hidden states, r(b, a) = R x R(x, a)b(x)dx. Given the value function, the optimal policy is simply π∗(b) = arg maxπ vπ(b). In this model, the belief b is parameterized by ¯st and t, so the animal only needs to keep track of these instead of encoding the entire posterior distribution bt(x) explicitly. In our model, the expected reward r(b, a) = R x R(x, a)b(x)dx is r(b, a) =      RS, when a = AS Z−1 t [ RP Pr(dR) (1 −Φ(0 \| µt, σt)) + RNPr(dL)Φ(0 \| µt, σt) ], when a = AR Z−1 t [ RNPr(dR) (1 −Φ(0 \| µt, σt)) + RP Pr(dL)Φ(0 \| µt, σt) ], when a = AL (6) where µt and σt are given by (3) and (4), respectively. The above equations can be interpreted as follows. With probability Pr(dL) · Φ(0 \| µt, σt), the hidden state x is less than 0, making AR an incorrect decision and resulting in a penalty RN if chosen. Similarly, action AR is correct with probability Pr(dR)·[1 −Φ(0 \| µt, σt)] and earns a reward of RP . The inverse is true for AL. When AS is selected, the animal simply receives an observation at a cost of RS. Computing the value function deﬁned in (5) involves an expectation with respect to future belief. Therefore, we need to compute the transition probabilities over belief states, T(bt+1\|bt, a), for each action. When the animal chooses to sample, at = AS, the animal’s belief distribution at the next time step is computed by marginalizing over all possible observations [19] T(bt+1\|bt, AS) = Z s Pr(bt+1\|s, bt, AS)Pr(s\|bt, AS)ds (7) where Pr(bt+1 \| s, bt, AS) = 1 if bt+1(x) = Pr(s\|x)bt(x)/Pr(s\|bt, AS), ∀x 0 otherwise; (8) and Pr(s \| bt, AS) = Z x Pr(s\|x)Pr(x\|b, a)dx = Ex∼b[Pr(s\|x)] (9) When choosing AS, the agent does not affect the world state, so, given the current state bt and an observation s, the updated belief bt+1 is deterministic and thus Pr(bt+1 \| s, bt, AS) is a delta function, following Bayes’ rule. The probability Pr(s \| bt, AS) can be treated as a normalization factor and is independent of hidden state2. Thus, the transition probability function, T(bt+1 \| bt, AS), is solely a function of the belief bt and is a stationary distribution over the belief space. When the selected action is AL or AR, the animal stops sampling and makes an eye movement to the left or the right, respectively. To account for these cases, we include a terminal state, Γ, with zeroreward (i.e., R(Γ, a) = 0, ∀a), and absorbing behavior, T(Γ\|Γ, a) = 1, ∀a. Moreover, whenever the animal chooses AL or AR, the POMDP immediately transitions into Γ: T(Γ\|b, a ∈{AL, AR}) = 1, ∀b, indicating the end of a trial. Given the transition probability between belief states T(bt+1\|bt, a) and the reward function, we can convert our POMDP model into a Markov Decision Process (MDP) over the belief state. Standard dynamic programming techniques (e.g., value iteration [20]) can then be applied to compute the value function in (5). A neurally plausible method for learning the optimal policy by trial and error using temporal difference (TD) learning was suggested in [18]. Here, we derive the optimal policy from ﬁrst principles and focus on comparisons between our model’s predictions and behavioral data. 2Explicitly, Pr(s\|bt, AS) = Z−1 t N(s\|µt, σ2 e +σ2 t )[Pr(dR)+(1−2Pr(dR))Φ(0\| µt σ2 t + s σ2e 1 σ2 t + 1 σ2e , ( 1 σ2 t + 1 σ2e )−1]). 4 3 Model Predictions 3.1 Optimal Policy (a) (b) Figure 1: Optimal policy for Pr(dR) = 0.5, and 0.9. (a–b) Optimal policy as a joint function of ¯s and t. Every point in these ﬁgures represents a belief state determined by equations (2), (3) and (4). The color of each point represents the corresponding optimal action. The boundaries ψR(t) and ψL(t) divide the belief space into three areas ΠS (center), ΠR (upper) and ΠL (lower), respectively. Model parameters: RN−RP RS = 1, 000. Figure 1(a) shows the optimal policy π∗as a joint function of ¯s and t for the unbiased case where the prior probability Pr(dR) = Pr(dL) = 0.5. π∗partitions the belief space into three regions: ΠR, ΠL, and ΠS, representing the set of belief states preferring actions AR, AL and AS, respectively. We deﬁne the boundary between AR and AS, and the boundary between AL and AS as ψR(t) and ψL(t), respectively. Early in a trial, the model selects the sampling action AS regardless of the value of the observed evidence. This is because the variance of the running average ¯s is high for small t. Later in the trial, the model will choose AR or AL when ¯s is only slightly above 0 because this variance decreases as the model receives more observations. For this reason, the width of ΠS diminishes over time. This gradual decrease in the threshold for choosing one of the non-sampling actions AR or AL has been called a “collapsing bound” in the decision making literature [21, 17, 22]. For this unbiased prior case, the expected reward function is symmetric, r(bt(x), AR) = r(Pr(x\|¯st, t), AR) = r(Pr(x\| −¯st, t), AL), and thus the decision boundaries are also symmetric around 0: ψR(t) = −ψL(t). The optimal policy π∗is entirely determined by the reward parameters {RP , RN, RS} and the prior probability (the standard deviation of the emission distribution σe only determines the temporal resolution of the POMDP). It applies to both Carpenter and Williams’ task and the random dots task (these two tasks differ only in the interpretation of the hidden state x). The optimal action at a speciﬁc belief state is determined by the relative, not the absolute, value of the expected future reward. From (6), we have r(b, AL) −r(b, AR) ∝RN −RP . (10) Moreover, if the unit of reward is speciﬁed by the sampling penalty, the optimal policy π∗is entirely determined by the ratio RN−RP RS and the prior. As the prior probability becomes biased, the optimal policy becomes asymmetric. When the prior probability, Pr(dR), increases, the decision boundary for the more likely direction (ψR(t)) shifts towards the center (the dashed line at ¯s = 0 in ﬁgure 1), while the decision boundary for the opposite direction (ψL(t)) shifts away from the center, as illustrated in Figure 1(b) for prior Pr(dR = 0.9). Early in a trial, ΠS has approximately the same width as in the neutral prior case, but it is shifted downwards to favor more sampling for dL (¯s < 0). Later in a trial, even for some belief states with ¯s < 0, the optimal action is still AR, because the effect of the prior is stronger than that of the observed data. 3.2 Psychometric function and reaction times in the random dots task We now construct a decision model from the learned policy for the reaction time version of the motion discrimination task [14], and compare the model’s predictions to the psychometric and 5 (a) Human SK (b) Human LH (c) Monkey Pr(dR) = .8 (d) Monkey Pr(dR) = .7 Figure 2: Comparison of Psychometric (upper panels) and Chronometric (lower panels) functions between the Model and Experiments. The dots with error bars represent experimental data from human subject SK, and LH, and the combined results from four monkeys. Blue solid curves are model predictions in the neutral case while green dotted curves are model predictions from the biased case. The R2 ﬁts are shown in the plots. Model parameters: (a) RN−RP RS = 1, 000, k = 1.45. (b) RN−RP RS = 1, 000, µ = 1.45. (c) Pr(dR) = 0.8, RN−RP RS = 1, 000, k = 1.4. (d) Pr(dR) = 0.7, RN−RP RS = 1, 000, k = 1.4. chronometric functions of a monkey performing the same task [13, 14]. Recall that the belief b is parametrized by ¯st and t, so the animal only needs to know the elapsed time and compute a running average ¯st of the observations in order to maintain the posterior belief bt(x). Given its current belief, the animal selects an action from the optimal policy π∗(bt). When bt ∈ΠS, the animal chooses the sampling action and gets a new observation st+1. Otherwise the animal terminates the trial by making an eye movement to the right or to the left, for ¯st > ψR(t) or ¯st < ψL(t), respectively. The performance on the task using the optimal policy can be measured in terms of both the accuracy of direction discrimination (the so-called psychometric function), and the reaction time required to reach a decision (the chronometric function). The hidden variable x = kdc encapsulates the unknown direction and coherence, as well as the free parameter k that determines the scale of stimulus st. Without loss of generality, we ﬁx d = 1 (rightward direction), and set the hidden direction dR as the biased direction. Given an optimal policy, we compute both the psychometric and chronometric function by simulating a large number of trials (10000 trials per data point) and collecting the reaction time and chosen direction from each trial. The upper panels of ﬁgure 2(a) and 2(b) (blue curves) show the performance accuracy as a function of coherence for both the model (blue solid curve) and the human subjects (blue dots) for neutral prior Pr(dR) = 0.5. We ﬁt our simulation results to the experimental data by adjusting the only two free parameters in our model: RN−RP RS and k. The lower panels of ﬁgure 2(a) and 2(b) (blue solid curves) shows the predicted mean reaction time for correct choices as a function of coherence c for our model (blue solid curve, with same model parameters) and the data (blue dots). Note that our model’s predicted reaction times represent the expected number of POMDP time steps before making a rightward eye movement AR, which we can directly compare to the monkey’s experimental data in units of real time. A linear regression is used to determine the duration τ of a single time step and the onset of decision time tnd. This offset, tnd, can be naturally interpreted as the non-decision residual time. We applied the experimental mean reaction time reported in [13] with motion coherence c = 0.032, 0.064, 0.128, 0.256 and 0.512 to compute the slope and offset, τ and tnd. Linear regression gives the unit duration per POMDP step as τ = 5.74ms , and the offset tnd = 314.6ms, for human SK. For human LH, similar results are obtained with τ = 5.20ms and tnd = 250.0ms. Our predicted offsets compare well with the 300ms non-decision time on average reported in the literature [23, 24]. 6 When the human subject is verbally told that the prior probability is Pr(dR) = Pr(d = 1) = 0.8, the experimental data is inconsistent with the predictions of the classic drift diffusion model [14] unless an additional assumption of a dynamic bias signal is introduced. In the POMDP model we propose, we predict both the accuracy and reaction times in the biased setting (green curves in ﬁgure 2) with the parameters learned in the neutral case, and achieve a good ﬁt (with the coefﬁcients of determination shown in ﬁg. 2) to the experimental data reported by Hanks et al. [13]. Our model predictions for the biased cases are a direct result of the reward maximization component of our framework and require no additional parameter ﬁtting. Combined behavioral data from four monkeys is shown by the dotted curves in ﬁgure 2(c). We ﬁt our model parameters to the psychometric function in the neutral case, with τ = 8.20ms and tnd = 312.50ms, and predict both the psychometric function and the reaction times in the biased case. However, our results do not match the monkey data as well as the human data when Pr(dR) = 0.8. This may be due to the fact that the monkeys cannot receive verbal instructions from the experimenters and must learn an estimate of the prior during training. As a result, the monkeys’ estimate of the prior probability might be inaccurate. To test this hypothesis, we simulated our model with Pr(dR) = 0.7 (see ﬁgure 2(d)) and these results ﬁt the experimental data much more accurately (even though the actual probability was 0.8). 3.3 Reaction times in the Carpenter and Williams’ task (a) (b) Figure 3: Model predictions of saccadic eye movement in Carpenter & Williams’ experiments [10]. (a) Saccadic latency distributions from model simulations plotted in the form of probitscale cumulative mass function, as a function of reciprocal latency. For different values of Pr(dR), the simulated data are well ﬁt by straight lines, indicating that the reciprocal of latency follows a normal distribution. The solid lines are linear functions ﬁt to the data with the constraint that all lines must pass through the same intercept for inﬁnite time (see [10]). (b) Median latency plotted as a function of log prior probability. Black dots are from experimental data and blue dots are model predictions. The two (overlapping) straight lines are the linear least squares ﬁts to the experimental data and model data. These lines do not differ noticeably in either slope or offset. Model parameters: RN−RP RS = 1, 000, k = 0.3, σe = 0.46. In Carpenter and Williams’ task, the animal needs to decide on which side d ∈{−1, 1} (denoting left or right side) a target light appeared at a ﬁxed distance from a central ﬁxation light. After the sudden appearance of the target light, a constant stimulus st = s is observed by the animal, where s can be regarded as the perceived location of the target. Due to noise and uncertainty in the nervous system, we assume that s varies from trial to trial, centered at the location of the target light and with standard deviation σe (i.e., s ∼N(s \| k, σ2 e)), where k is the distance between the target and the ﬁxation light. Inference over the direction d thus involves joint inference over (d, k) where the emission probability follows Pr(s\|d, k). Then the joint state (k, d) can be one-on-one-mapped to kd = x, where x represents the actual location of the target light. Under the POMDP framework, Carpenter and Williams’ task and the random dots task differ in the interpretation of hidden state x and stimulus s, but they follow the same optimal policy given the same reward parameters. Without loss of generality, we set the hidden variable x > 0 and say that the animal makes a correct choice at a hitting time tH when the animal’s belief state reaches the right boundary. The 7 saccadic latency can be computed by inverting the boundary function ψ−1 R (s) = tH. Since, for small t, ψR(t) behaves like a simple reciprocal function of t, the reciprocal of the reaction time is approximately proportional to a normal distribution with 1 tH ∼N(1/tH \| k, σ2 e). In ﬁgure 3(a), we plot the distribution of reciprocal reaction time with different values of Pr(dR) on a probit scale (similar to [10]). Note that we label the y-axis using the CDF of the corresponding probit value and the x-axis in ﬁgure 3(a) has been reversed. If the reciprocal of reaction time (with the same prior Pr(dR)) follows a normal distribution, each point on the graph will fall on a straight line with y-intercept k √ 2 σe that is independent of Pr(dR). We ﬁt straight lines to the points on the graph, with the constraint that all lines should pass through the same intercept for inﬁnite time (see [10]). We obtain an intercept of 6.19, consistent with the intercept 6.20 obtained from experimental data in [10]. Figure 3(b) demonstrates that the median of our model’s reaction times is a linear function of the log of the prior probability. Increasing the prior probability lowers the decision boundary ψR(t), effectively decreasing the latency. The slope and intercept of the best ﬁt line are consistent with experimental data (see ﬁg. 3(b)). 4 Summary and Conclusion Our results suggest that decision making in the primate brain may be governed by the dual principles of Bayesian inference and reward maximization as implemented within the framework of partially observable Markov decision processes (POMDPs). The model provides a uniﬁed explanation for experimental data previously explained by two competing models, namely, the additive offset model and the dynamic weighting model for incorporating prior knowledge. In particular, the model predicts psychometric and chronometric data for the random dots motion discrimination task [13] as well as Carpenter and Williams’ saccadic eye movement task [10]. Previous models of decision making, such as the LATER model [10] and the drift diffusion model [25, 15], have provided descriptive accounts of reaction time and accuracy data but often require assumptions such as a collapsing bound, urgency signal, or dynamic weighting to fully explain the data [26, 21, 22, 13]. Our model provides a normative account of the data, illustrating how the subject’s choices can be interpreted as being optimal under the framework of POMDPs. Our model relies on the principle of reward maximization to explain how an animal’s decisions are inﬂuenced by changes in prior probability. The same principle also allows us to predict how an animal’s choice is inﬂuenced by changes in the reward function. Speciﬁcally, the model predicts that the optimal policy π∗is determined by the ratio RN−RP RS and the prior probability Pr(dR). Thus, a testable prediction of the model is that the speed-accuracy trade-off in tasks such as the random dots task is governed by the ratio RN−RP RS : smaller penalties for sampling (RS) will increase accuracy and reaction time, as will larger rewards for correct choices (RP ) or greater penalties for errors (RN). Since the reward parameters in our model represent internal reward, our model also provides a bridge to study the relationship between physical reward and subjective reward. In our model of the random dots discrimination task, belief is expressed in terms of a piecewise normal distribution with the domain of the hidden variable x ∈(−∞, ∞). A piecewise beta distribution with domain x ∈[−1, 1] ﬁts the experimental data equally well. However, the beta distribution’s conjugate prior is the multinomial, which can limit the application of this model. For example, the observations in the Carpenter and Williams’ model cannot easily be described by a discrete value. The belief in our model can be expressed by any distribution, even a non-parametric one, as long as the observation model provides a faithful representation of the stimuli and captures the essential relationship between the stimuli and the hidden world state. The POMDP model provides a unifying framework for a variety of perceptual decision making tasks. Our state variable x and action variable a work with arbitrary state and action spaces, ranging from multiple alternative choices to high dimensional real value choices. The state variables can also be dynamic, with xt following a Markov chain. Currently, we have assumed that the stimuli are independent from one time step to the next, but most real world stimuli are temporally correlated. Our model is suitable for decision tasks with time-varying state and observations that are time dependent within a trial (as long as they are conditional independent given the time-varying hidden state sequence). 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4,499	Structured Learning of Gaussian Graphical Models Karthik Mohan∗, Michael Jae-Yoon Chung†, Seungyeop Han†, Daniela Witten‡, Su-In Lee§, Maryam Fazel∗ Abstract We consider estimation of multiple high-dimensional Gaussian graphical models corresponding to a single set of nodes under several distinct conditions. We assume that most aspects of the networks are shared, but that there are some structured differences between them. Speciﬁcally, the network differences are generated from node perturbations: a few nodes are perturbed across networks, and most or all edges stemming from such nodes differ between networks. This corresponds to a simple model for the mechanism underlying many cancers, in which the gene regulatory network is disrupted due to the aberrant activity of a few speciﬁc genes. We propose to solve this problem using the perturbed-node joint graphical lasso, a convex optimization problem that is based upon the use of a row-column overlap norm penalty. We then solve the convex problem using an alternating directions method of multipliers algorithm. Our proposal is illustrated on synthetic data and on an application to brain cancer gene expression data. 1 Introduction Probabilistic graphical models are widely used in a variety of applications, from computer vision to natural language processing to computational biology. As this modeling framework is used in increasingly complex domains, the problem of selecting from among the exponentially large space of possible network structures is of paramount importance. This problem is especially acute in the high-dimensional setting, in which the number of variables or nodes in the graphical model is much larger than the number of observations that are available to estimate it. As a motivating example, suppose that we have access to gene expression measurements for n1 lung cancer patients and n2 brain cancer patients, and that we would like to estimate the gene regulatory networks underlying these two types of cancer. We can consider estimating a single network on the basis of all n1+n2 patients. However, this approach is unlikely to be successful, due to fundamental differences between the true lung cancer and brain cancer gene regulatory networks that stem from tissue speciﬁcity of gene expression as well as differing etiology of the two diseases. As an alternative, we could simply estimate a gene regulatory network using the n1 lung cancer patients and a separate gene regulatory network using the n2 brain cancer patients. However, this approach fails to exploit the fact that the two underlying gene regulatory networks likely have substantial commonality, such as tumor-speciﬁc pathways. In order to effectively make use of the available data, we need a principled approach for jointly estimating the lung cancer and brain cancer networks in such a way that the two network estimates are encouraged to be quite similar to each other, while allowing for certain structured differences. In fact, these differences themselves may be of scientiﬁc interest. In this paper, we propose a general framework for jointly learning the structure of K networks, under the assumption that the networks are similar overall, but may have certain structured differences. ∗Electrical Engineering, Univ. of Washington. {karna,mfazel}@uw.edu †Computer Science and Engineering, Univ. of Washington. {mjyc,syhan}@cs.washington.edu ‡Biostatistics, Univ. of Washington. dwitten@uw.edu §Computer Science and Engineering, and Genome Sciences, Univ. of Washington. suinlee@uw.edu 1 Speciﬁcally, we assume that the network differences result from node perturbation – that is, certain nodes are perturbed across the conditions, and so all or most of the edges associated with those nodes differ across the K networks. We detect such differences through the use of a row-column overlap norm penalty. Figure 1 illustrates a toy example in which a pair of networks are identical to each other, except for a single perturbed node (X2) that will be detected using our proposal. The problem of estimating multiple networks that differ due to node perturbations arises in a number of applications. For instance, the gene regulatory networks in cancer patients and in normal individuals are likely to be similar to each other, with speciﬁc node perturbations that arise from a small set of genes with somatic (cancer-speciﬁc) mutations. Another example arises in the analysis of the conditional independence relationships among p stocks at two distinct points in time. We might be interested in detecting stocks that have differential connectivity with all other edges across the two time points, as these likely correspond to companies that have undergone signiﬁcant changes. Still another example can be found in the ﬁeld of neuroscience, where we are interested in learning how the connectivity of neurons in the human brain changes over time. Figure 1: An example of two networks that differ due to node perturbation of X2. (a) Network 1 and its adjacency matrix. (b) Network 2 and its adjacency matrix. (c) Left: Edges that differ between the two networks. Right: Shaded cells indicate edges that differ between Networks 1 and 2. Our proposal for estimating multiple networks in the presence of node perturbation can be formulated as a convex optimization problem, which we solve using an efﬁcient alternating directions method of multipliers (ADMM) algorithm that signiﬁcantly outperforms general-purpose optimization tools. We test our method on synthetic data generated from known graphical models, and on one real-world task that involves inferring gene regulatory networks from experimental data. The rest of this paper is organized as follows. In Section 2, we present recent work in the estimation of Gaussian graphical models (GGMs). In Section 3, we present our proposal for structured learning of multiple GGMs using the row-column overlap norm penalty. In Section 4, we present an ADMM algorithm that solves the proposed convex optimization problem. Applications to synthetic and real data are in Section 5, and the discussion is in Section 6. 2 Background 2.1 The graphical lasso Suppose that we wish to estimate a GGM on the basis of n observations, X1, . . . , Xn ∈Rp, which are independent and identically distributed N(0, Σ). It is well known that this amounts to learning the sparsity structure of Σ−1 [1, 2]. When n > p, one can estimate Σ−1 by maximum likelihood, but when p > n this is not possible because the empirical covariance matrix is singular. Consequently, a number of authors [3, 4, 5, 6, 7, 8, 9] have considered maximizing the penalized log likelihood maximize Θ∈Sp ++ {log det Θ −trace(SΘ) −λ∥Θ∥1} , (1) where S is the empirical covariance matrix based on the n observations, λ is a positive tuning parameter, Sp ++ denotes the set of positive deﬁnite matrices of size p, and ∥Θ∥1 is the entrywise ℓ1 norm. The ˆΘ that solves (1) serves as an estimate of Σ−1. This estimate will be positive deﬁnite for any λ > 0, and sparse when λ is sufﬁciently large, due to the ℓ1 penalty [10] in (1). We refer to (1) as the graphical lasso formulation. This formulation is convex, and efﬁcient algorithms for solving it are available [6, 4, 5, 7, 11]. 2 2.2 The fused graphical lasso In recent literature, convex formulations have been proposed for extending the graphical lasso (1) to the setting in which one has access to a number of observations from K distinct conditions. The goal of the formulations is to estimate a graphical model for each condition under the assumption that the K networks share certain characteristics [12, 13]. Suppose that Xk 1 , . . . , Xk nk ∈Rp are independent and identically distributed from a N(0, Σk) distribution, for k = 1, . . . , K. Letting Sk denote the empirical covariance matrix for the kth class, one can maximize the penalized log likelihood maximize Θ1∈Sp ++,...,ΘK∈Sp ++   L(Θ1, . . . , ΘK) −λ1 K ∑ k=1 ∥Θk∥1 −λ2 ∑ i̸=j P(Θ1 ij, . . . , ΘK ij )   , (2) where L(Θ1, . . . , ΘK) = ∑K k=1 nk ( log det Θk −trace(SkΘk) ) , λ1 and λ2 are nonnegative tuning parameters, and P(Θ1 ij, . . . , ΘK ij ) is a penalty applied to each off-diagonal element of Θ1, . . . , ΘK in order to encourage similarity among them. Then the ˆΘ1, . . . , ˆΘK that solve (2) serve as estimates for (Σ1)−1, . . . , (ΣK)−1. In particular, [13] considered the use of P(Θ1 ij, . . . , ΘK ij ) = ∑ k<k′ \|Θk ij −Θk′ ij\|, (3) a fused lasso penalty [14] on the differences between pairs of network edges. When λ1 is large, the network estimates will be sparse, and when λ2 is large, pairs of network estimates will have identical edges. We refer to (2) with penalty (3) as the fused graphical lasso formulation (FGL). Solving the FGL formulation allows for much more accurate network inference than simply learning each of the K networks separately, because FGL borrows strength across all available observations in estimating each network. But in doing so, it implicitly assumes that differences among the K networks arise from edge perturbations. Therefore, this approach does not take full advantage of the structure of the learning problem, which is that differences between the K networks are driven by nodes that differ across networks, rather than differences in individual edges. 3 The perturbed-node joint graphical lasso 3.1 Why is detecting node perturbation challenging? At ﬁrst glance, the problem of detecting node perturbation seems simple: in the case K = 2, we could simply modify (2) as follows, maximize Θ1∈Sp ++,Θ2∈Sp ++   L(Θ1, Θ2) −λ1∥Θ1∥1 −λ1∥Θ2∥1 −λ2 p ∑ j=1 ∥Θ1 j −Θ2 j∥2   , (4) where Θk j is the jth column of the matrix Θk. This amounts to applying a group lasso [15] penalty to the columns of Θ1 −Θ2. Since a group lasso penalty simultaneously shrinks all elements to which it is applied to zero, it appears that this will give the desired node perturbation structure. We will refer to this as the naive group lasso approach. Unfortunately, a problem arises due to the fact that the optimization problem (4) must be performed subject to a symmetry constraint on Θ1 and Θ2. This symmetry constraint effectively imposes overlap among the elements in the p group lasso penalties in (4), since the (i, j)th element of Θ1 − Θ2 is in both the ith (row) and jth (column) groups. In the presence of overlapping groups, the group lasso penalty yields estimates whose support is the complement of the union of groups [16, 17]. Figure 2 shows a simple example of (Σ1)−1−(Σ2)−1 in the case of node perturbation, as well as the estimate obtained using (4). The ﬁgure reveals that (4) cannot be used to detect node perturbation, since this task requires a penalty that yields estimates whose support is the union of groups. 3.2 Proposed approach A node-perturbation in a GGM can be equivalently represented through a perturbation of the entries of a row and column of the corresponding precision matrix (Figure 1). In other words, we can 3 Figure 2: A toy example with p = 6 variables, of which two are perturbed (in red). Each panel shows an estimate of (Σ1)−1 −(Σ2)−1, displayed as a network and as an adjacency matrix. Shaded elements of the adjacency matrix indicate non-zero elements of ˆΘ1−ˆΘ2, as do edges in the network. Results are shown for (a): PNJGL with q = 2, which gives the correct sparsity pattern; (b)-(c): the naive group lasso. The naive group lasso is unable to detect the pattern of node perturbation. detect a single node perturbation by looking for a row and a corresponding column of Θ1 −Θ2 that has nonzero elements. We deﬁne a row-column group as a group that consists of a row and the corresponding column in a matrix. Note that in a p × p matrix, there exist p such groups, which overlap. If several nodes of a GGM are perturbed, then this will correspond to the union of the corresponding row-column groups in Θ1 −Θ2. Therefore, in order to detect node perturbations in a GGM (Figure 1), we must construct a regularizer that can promote estimates whose support is the union of row-column groups. For this task, we propose the row-column overlap norm as a penalty. Deﬁnition 3.1. The row-column overlap norm (RCON) induced by a matrix norm f is deﬁned as Ωf(A) = min V:A=V+VT f(V). (5) RCON satisﬁes the following properties that are easy to check: (1) Ωf is indeed a norm. Consequently, it is convex. (2) When f is symmetric in its argument, i.e., f(V) = f(VT ), then Ωf(A) = f(A)/2. In this paper, we are interested in the particular class of RCON penalty where f is given by f(V) = p ∑ j=1 ∥Vj∥q, (6) where 1 ≤q ≤∞. The norm in (6) is known as the ℓ1/ℓq norm since it can be interpreted as the ℓ1 norm of the ℓq norms of the columns of a matrix. With a little abuse of notation, we will let Ωq denote Ωf with an ℓ1/ℓq norm of the form (6). We note that Ωq is closely related to the overlap group lasso penalty [17, 16], and in fact can be derived from it (for the case of q = 2). However, our deﬁnition naturally and elegantly handles the grouping structure induced by the overlap of rows and columns, and can accommodate any ℓq norm with q ≥1, and more generally any norm f. As discussed in [17], when applied to Θ1 −Θ2, the penalty Ωq (with q = 2) will encourage the support of the matrix ˆΘ1 −ˆΘ2 to be the union of a set of rows and columns. Now, consider the task of jointly estimating two precision matrices by solving maximize Θ1∈Sp ++,Θ2∈Sp ++ { L(Θ1, Θ2) −λ1∥Θ1∥1 −λ1∥Θ2∥1 −λ2Ωq(Θ1 −Θ2) } . (7) We refer to the convex optimization problem (7) as the perturbed-node joint graphical lasso (PNJGL) formulation. In (7), λ1 and λ2 are nonnegative tuning parameters, and q ≥1. Note that f(V) = ∥V∥1 satisﬁes property 2 of the RCON penalty. Thus we have the following observation. Remark 3.1. The FGL formulation (2) is a special case of the PNJGL formulation (7) with q = 1. Let ˆΘ1, ˆΘ2 be the optimal solution to (7). Note that the FGL formulation is an edge-based approach that promotes many entries (or edges) in ˆΘ1−ˆΘ2 to be set to zero. However, setting q = 2 or q = ∞ in (7) gives us a node-based approach, where the support of ˆΘ1 −ˆΘ2 is encouraged to be a union of a few rows and the corresponding columns [17, 16]. Thus the nodes that have been perturbed can be clearly detected using PNJGL with q = 2, ∞. An example of the sparsity structure detected by PNJGL with q = 2 is shown in the left-hand panel of Figure 2. We note that the above formulation can be easily extended to the estimation of K > 2 GGMs by including K(K−1) 2 RCON penalty terms in (7), one for each pair of models. However we restrict ourselves to the case of K = 2 in this paper. 4 4 An ADMM algorithm for the PNJGL formulation The PNJGL optimization problem (7) is convex, and so can be directly solved in the modeling environment cvx [18], which calls conic interior-point solvers such as SeDuMi or SDPT3. However, such a general approach does not fully exploit the structure of the problem and will not scale well to large-scale instances. Other algorithms proposed for overlapping group lasso penalties [19, 20, 21] do not apply to our setting since the PNJGL formulation has a combination of Gaussian log-likelihood loss (instead of squared error loss) and the RCON penalty along with a positivedeﬁnite constraint. We also note that other ﬁrst-order methods are not easily applied to solve the PNJGL formulation because the subgradient of the RCON is not easy to compute and in addition the proximal operator to RCON is non-trivial to compute. In this section we present a fast and scalable alternating directions method of multipliers (ADMM) algorithm [22] to solve the problem (7). We ﬁrst reformulate (7) by introducing new variables, so as to decouple some of the terms in the objective function that are difﬁcult to optimize jointly. This will result in a simple algorithm with closed-form updates. The reformulation is as follows: minimize Θ1∈Sp ++,Θ2∈Sp ++,Z1,Z2,V,W   −L(Θ1, Θ2) + λ1∥Z1∥1 + λ1∥Z2∥1 + λ2 p ∑ j=1 ∥Vj∥q    subject to Θ1 −Θ2 = V + W, V = WT , Θ1 = Z1, Θ2 = Z2. (8) An ADMM algorithm can now be obtained in a standard fashion from the augmented Lagrangian to (8). We defer the details to a longer version of this paper. The complete algorithm for (8) is given in Algorithm 1, in which the operator Expand is given by Expand(A, ρ, nk) = argmin Θ∈Sp ++ { −nk log det(Θ) + ρ∥Θ −A∥2 F } = 1 2U ( D + √ D2 + 2nk ρ I ) UT , where UDUT is the eigenvalue decomposition of A, and as mentioned earlier, nk is the number of observations in the kth class. The operator Tq is given by Tq(A, λ) = argmin X    1 2∥X −A∥2 F + λ p ∑ j=1 ∥Xj∥q   , and is also known as the proximal operator corresponding to the ℓ1/ℓq norm. For q = 1, 2, ∞, Tq takes a simple form, which we omit here due to space constraints. A description of these operators can also be found in Section 5 of [25]. Algorithm 1 can be interpreted as an approximate dual gradient ascent method. The approximation is due to the fact that the gradient of the dual to the augmented Lagrangian in each iteration is computed inexactly, through a coordinate descent cycling through the primal variables. Typically ADMM algorithms iterate over only two groups of primal variables. For such algorithms, the convergence properties are well-known (see e.g. [22]). However, in our case we cycle through more than two such groups. Although investigation of the convergence properties of ADMM algorithms for an arbitrary number of groups is an ongoing research area in the optimization literature [23, 24] and speciﬁc convergence results for our algorithm are not known, we empirically observe very good convergence behavior. Further study of this issue is a direction for future work. We initialize the primal variables to the identity matrix, and the dual variables to the matrix of zeros. We set µ = 5, and tmax = 1000. In our implementation, the stopping criterion is that the difference between consecutive iterates becomes smaller than a tolerance ϵ. The ADMM algorithm is orders of magnitude faster than an interior point method and also comparable in accuracy. Note that the per-iteration complexity of the ADMM algorithm is O(p3) (complexity of computing SVD). On the other hand, the complexity of an interior point method is O(p6). When p = 30, the interior point method (using cvx, which calls Sedumi) takes 7 minutes to run while ADMM takes only 10 seconds. When p = 50, the times are 3.5 hours and 2 minutes, respectively. Also, we observe that the average error between the cvx and ADMM solution when averaged over many random generations of the data is of O(10−4). 5 Algorithm 1: ADMM algorithm for the PNJGL optimization problem (7) input: ρ > 0, µ > 1, tmax > 0, ϵ > 0; for t = 1:tmax do ρ ←µρ ; while Not converged do Θ1 ←Expand ( 1 2(Θ2 + V + W + Z1) −1 2ρ(Q1 + n1S1 + F), ρ, n1 ) ; Θ2 ←Expand ( 1 2(Θ1 −(V + W) + Z2) −1 2ρ(Q2 + n2S2 −F), ρ, n2 ) ; Zi ←T1 ( Θi + Qi ρ , λ1 ρ ) for i = 1, 2 ; V ←Tq ( 1 2(WT −W + (Θ1 −Θ2)) + 1 2ρ(F −G), λ2 2ρ ) ; W ←1 2(VT −V + (Θ1 −Θ2)) + 1 2ρ(F + GT ) ; F ←F + ρ(Θ1 −Θ2 −(V + W)) ; G ←G + ρ(V −WT ); Qi ←Qi + ρ(Θi −Zi) for i = 1, 2 5 Experiments We describe experiments and report results on both synthetically generated data and real data. 5.1 Synthetic experiments Synthetic data generation. We generated two networks as follows. The networks share individual edges as well as hub nodes, or nodes that are highly-connected to many other nodes. There are also perturbed nodes that differ between the networks. We ﬁrst create a p × p symmetric matrix A, with diagonal elements equal to one. For i < j, we set Aij ∼i.i.d. {0 with probability 0.98 Unif([−0.6, −0.3] ∪[0.3, 0.6]) otherwise , and then we set Aji to equal Aij. Next, we randomly selected seven hub nodes, and set the elements of the corresponding rows and columns to be i.i.d. from a Unif([−0.6, −0.3]∪[0.3, 0.6]) distribution. These steps resulted in a background pattern of structure common to both networks. Next, we copied A into two matrices, A1 and A2. We randomly selected m perturbed nodes that differ between A1 and A2, and set the elements of the corresponding row and column of either A1 or A2 (chosen at random) to be i.i.d. draws from a Unif([−1.0, −0.5] ∪[0.5, 1.0]) distribution. Finally, we computed c = min(λmin(A1), λmin(A2)), the smallest eigenvalue of A1 and A2. We then set (Σ1)−1 equal to A1 + (0.1 −c)I and set (Σ2)−1 equal to A2 + (0.1 −c)I. This last step is performed in order to ensure positive deﬁniteness. We generated n independent observations each from a N(0, Σ1) and a N(0, Σ2) distribution, and used these to compute the empirical covariance matrices S1 and S2. We compared the performances of graphical lasso, FGL, and PNJGL with q = 2 with p = 100, m = 2, and n = {10, 25, 50, 200}. Results. Results (averaged over 100 iterations) are shown in Figure 3. Increasing n yields more accurate results for PNJGL with q = 2, FGL, and graphical lasso. Furthermore, PNJGL with q = 2 identiﬁes non-zero edges and differing edges much more accurately than does FGL, which is in turn more accurate than graphical lasso. PNJGL also leads to the most accurate estimates of Θ1 and Θ2. The extent to which PNJGL with q = 2 outperforms others is more apparent when n is small. 5.2 Inferring biological networks We applied the PNJGL method to a recently-published cancer gene expression data set [26], with mRNA expression measurements for 11,861 genes in 220 patients with glioblastoma multiforme (GBM), a brain cancer. Each patient has one of four distinct clinical subtypes: Proneural, Neural, Classical, and Mesenchymal. We selected two subtypes – Proneural (53 patients) and Mesenchymal 6 Figure 3: Simulation study results for PNJGL with q = 2, FGL, and the graphical lasso (GL), for (a) n = 10, (b) n = 25, (c) n = 50, (d) n = 200, when p = 100. Within each panel, each line corresponds to a ﬁxed value of λ2 (for PNJGL with q = 2 and for FGL). Each plot’s x-axis denotes the number of edges estimated to be non-zero. The y-axes are as follows. Left: Number of edges correctly estimated to be non-zero. Center: Number of edges correctly estimated to differ across networks, divided by the number of edges estimated to differ across networks. Right: The Frobenius norm of the error in the estimated precision matrices, i.e. (∑ i̸=j(θ1 ij −ˆθ1 ij)2) 1/2 + (∑ i̸=j(θ2 ij −ˆθ2 ij)2) 1/2. (56 patients) – for our analysis. In this experiment, we aim to reconstruct the gene regulatory networks of the two subtypes, as well as to identify genes whose interactions with other genes vary signiﬁcantly between the subtypes. Such genes are likely to have many somatic (cancer-speciﬁc) mutations. Understanding the molecular basis of these subtypes will lead to better understanding of brain cancer, and eventually, improved patient treatment. We selected the 250 genes with the highest within-subtype variance, as well as 10 genes known to be frequently mutated across the four GBM subtypes [26]: TP53, PTEN, NF1, EGFR, IDH1, PIK3R1, RB1, ERBB2, PIK3CA, PDGFRA. Two of these genes (EGFR, PDGFRA) were in the initial list of 250 genes selected based on the withinsubtype variance. This led to a total of 258 genes. We then applied PNJGL with q = 2 and FGL to the resulting 53 × 258 and 56 × 258 gene expression datasets, after standardizing each gene to have variance one. Tuning parameters were selected so that each approach results in a per-network estimate of approximately 6,000 non-zero edges, as well as approximately 4,000 edges that differ 7 across the two network estimates. However, the results that follow persisted across a wide range of tuning parameter values. Figure 4: PNJGL with q = 2 and FGL were performed on the brain cancer data set corresponding to 258 genes in patients with Proneural and Mesenchymal subtypes. (a)-(b): NPj is plotted for each gene, based on (a) the FGL estimates and (b) the PNJGL estimates. (c)-(d): A heatmap of ˆΘ1 −ˆΘ2 is shown for (c) FGL and (d) PNJGL; zero values are in white, and non-zero values are in black. We quantify the extent of node perturbation (NP) in the network estimates as follows: NPj = ∑ i \|Vij\|; for FGL we get V from the PNJGL formulation as 1 2( ˆΘ1−ˆΘ2). If NPj = 0 (using a zerothreshold of 10−6), then the jth gene has the same edge weights in the two conditions. In Figure 4(a)(b), we plotted the resulting values for each of the 258 genes in FGL and PNJGL. Although the network estimates resulting from PNJGL and FGL have approximately the same number of edges that differ across cancer subtypes, PNJGL results in estimates in which only 37 genes appear to have node perturbation. FGL results in estimates in which all 258 genes appear to have node perturbation. In Figure 4(c)-(d), the non-zero elements of ˆΘ1−ˆΘ2 for FGL and for PNJGL are displayed. Clearly, the pattern of network differences resulting from PNJGL is far more structured. The genes known to be frequently mutated across GBM subtypes are somewhat enriched out of those that appear to be perturbed according to the PNJGL estimates (3 out of 10 mutated genes were detected by PNJGL; 37 out of 258 total genes were detected by PNJGL; hypergeometric p-value = 0.1594). In contrast, FGL detects every gene as having node perturbation (Figure 4(a)). The gene with the highest NPj value (according to both FGL and PNJGL with q = 2) is CXCL13, a small cytokine that belongs to the CXC chemokine family. Together with its receptor CXCR5, it controls the organization of B-cells within follicles of lymphoid tissues. This gene was not identiﬁed as a frequently mutated gene in GBM [26]. However, there is recent evidence that CXCL13 plays a critical role in driving cancerous pathways in breast, prostate, and ovarian tissue [27, 28]. Our results suggest the possibility of a previously unknown role of CXCL13 in brain cancer. 6 Discussion and future work We have proposed the perturbed-node joint graphical lasso, a new approach for jointly learning Gaussian graphical models under the assumption that network differences result from node perturbations. We impose this structure using a novel RCON penalty, which encourages the differences between the estimated networks to be the union of just a few rows and columns. We solve the resulting convex optimization problem using ADMM, which is more efﬁcient and scalable than standard interior point methods. Our proposed approach leads to far better performance on synthetic data than two alternative approaches: learning Gaussian graphical models assuming edge perturbation [13], or simply learning each model separately. Future work will involve other forms of structured sparsity beyond simply node perturbation. For instance, if certain subnetworks are known a priori to be related to the conditions under study, then the RCON penalty can be modiﬁed in order to encourage some subnetworks to be perturbed across the conditions. In addition, the ADMM algorithm described in this paper requires computation of the eigen decomposition of a p × p matrix at each iteration; we plan to develop computational improvements that mirror recent results on related problems in order to reduce the computations involved in solving the FGL optimization problem [6, 13]. Acknowledgments D.W. was supported by NIH Grant DP5OD009145, M.F. was supported in part by NSF grant ECCS-0847077. 8 References [1] K.V. Mardia, J. Kent, and J.M. Bibby. Multivariate Analysis. Academic Press, 1979. [2] S.L. Lauritzen. Graphical Models. Oxford Science Publications, 1996. [3] M. Yuan and Y. Lin. Model selection and estimation in the Gaussian graphical model. Biometrika, 94(10):19–35, 2007. [4] J. Friedman, T. Hastie, and R. Tibshirani. 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