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+# AUTOMATICALLY DISCOVERING AND LEARNING NEW VISUAL CATEGORIES WITH RANKING STATISTICS
+
+Kai Han∗ Sylvestre-Alvise Rebuffi∗ Sebastien Ehrhardt∗ Andrea Vedaldi Andrew Zisserman Visual Geometry Group, Department of Engineering Science, University of Oxford {khan,srebuffi,hyenal,vedaldi,az}@robots.ox.ac.uk
+
+# ABSTRACT
+
+We tackle the problem of discovering novel classes in an image collection given labelled examples of other classes. This setting is similar to semi-supervised learning, but significantly harder because there are no labelled examples for the new classes. The challenge, then, is to leverage the information contained in the labelled images in order to learn a general-purpose clustering model and use the latter to identify the new classes in the unlabelled data. In this work we address this problem by combining three ideas: (1) we suggest that the common approach of bootstrapping an image representation using the labeled data only introduces an unwanted bias, and that this can be avoided by using self-supervised learning to train the representation from scratch on the union of labelled and unlabelled data; (2) we use rank statistics to transfer the model’s knowledge of the labelled classes to the problem of clustering the unlabelled images; and, (3) we train the data representation by optimizing a joint objective function on the labelled and unlabelled subsets of the data, improving both the supervised classification of the labelled data, and the clustering of the unlabelled data. We evaluate our approach on standard classification benchmarks and outperform current methods for novel category discovery by a significant margin.
+
+# 1 INTRODUCTION
+
+Modern machine learning systems can match or surpass human-level performance in tasks such as image classification (Deng et al., 2009), but at the cost of collecting large quantities of annotated training data. Semi-supervised learning (SSL) (Oliver et al., 2018) can alleviate this issue by mixing labelled with unlabelled data, which is usually much cheaper to obtain. However, these methods still require some annotations for each of the classes that one wishes to learn. We argue this is not always possible in real applications. For instance, consider the task of recognizing products in supermarkets. Thousands of new products are introduced in stores every week, and it would be very expensive to annotate them all. However, new products do not differ drastically from the existing ones, so it should be possible to discover them automatically as they arise in the data. Unfortunately, machines are still unable to effectively learn new classes without manual annotations.
+
+In this paper, we thus consider the problem of discovering new visual classes automatically, assuming that a certain number of classes are already known by the model (Hsu et al., 2018; 2019; Han et al., 2019). This knowledge comes in the form of a labelled dataset of images for a certain set of classes. Given that this data is labelled, off-the-shelf supervised learning techniques can be used to train a very effective classifier for the known classes, particularly if Convolutional Neural Networks (CNNs) are employed. However, this does not mean that the learned features are useful as a representation of the new classes. Furthermore, even if the representation transfers well, one still has the problem of identifying the new classes in an unlabelled dataset, which is a clustering problem.
+
+We tackle these problems by introducing a novel approach that combines three key ideas (section 2 and fig. 1). The first idea is to pre-train the image representation (a CNN) using all available images, both labelled and unlabelled, using a self-supervised learning objective. Crucially, this objective does not leverage the known labels, resulting in features that are much less biased towards the labelled classes. Labels are used only after pre-training to learn a classifier specific to the labelled data as well as to fine-tune the deepest layers of the CNN, for which self-supervision is not as effective.
+
+![](images/9c76455edd386d2f416ea32e19ed3799838374222b6c521e38a5561651f23df2.jpg)  
+Figure 1: Overview of our three step learning pipeline. The first step of the training consists in learning an unbiased image representation via self-supervision using both labelled and unlabelled data, which learns well the early layers of the representation; in the second step, we fine-tune only the last few layers of the model using supervision on the labelled set; finally, the fine-tuned representation is used, via rank statistics, to induce clusters in the unlabelled data, while maintaining a good representation on the labelled set.
+
+The second idea is a new approach to transfer the information contained in the labelled images to the problem of clustering the unlabelled ones. Information is transferred by sharing the same representation between labelled and unlabelled images, motivated by the fact that the new classes are often similar to the known ones. In more detail, pairs of unlabelled images are compared via their representation vectors. The comparison is done using robust rank statistics, by testing if two images share the same subset of $k$ maximally activated representation components. This test is used to decide if two unlabelled images belong to the same (new) class or not, generating a set of noisy pairwise pseudo-labels. The pseudo-labels are then used to learn a similarity function for the unlabelled images.
+
+The third and final idea is, after bootstrapping the representation, to optimise the model by minimizing a joint objective function, containing terms for both the labelled and unlabelled subsets, using respectively the given labels and the generated pseudo-labels, thus avoiding the forgetting issue that may arise with a sequential approach. A further boost is obtained by incorporating incremental learning of the discovered classes in the classification task, which allows information to flow between the labelled and unlabelled images.
+
+We evaluate our method on several public benchmarks (section 3), outperforming by a large margin all existing techniques (section 4) that can be applied to this problem, demonstrating the effectiveness of our approach. We conclude the paper by summarizing our findings (section 5). Our code can be found at http://www.robots.ox.ac.uk/˜vgg/research/auto_novel.
+
+# 2 METHOD
+
+Given an unlabelled dataset $D ^ { u } = \{ x _ { i } ^ { u } , i = 1 , \ldots , M \}$ of images $x _ { i } ^ { u } \in \mathbb { R } ^ { 3 \times H \times W }$ , our goal is to automatically cluster the images into a number of classes $C ^ { u }$ , which we assume to be known $a$ priori. We also assume to have a second labelled image dataset $\dot { D ^ { l } } = \{ ( x _ { i } ^ { l } , y _ { i } ^ { l } ) , i = 1 , \dots , N \}$ where $y _ { i } ^ { l } \in \{ 1 , \ldots , C ^ { l } \}$ is the class label for image $x _ { i } ^ { l }$ . We also assume that the set of $C ^ { l }$ labelled classes is disjoint from the set of $C ^ { u }$ unlabelled ones. While the statistics of $D ^ { l }$ and $D ^ { u }$ thus differ, we hypothesize that a general notion of what constitutes a “good class” can be extracted from $D ^ { l }$ and that the latter can be used to better cluster $D ^ { u }$ .
+
+We approach the problem by learning an image representation $\Phi : x \mapsto \Phi ( x ) \in \mathbb { R } ^ { d }$ in the form of a CNN. The goal of the representation is to help to recognize the known classes and to discover the new ones. In order to learn this representation, we combine three ideas, detailed in the next three sections.
+
+# 2.1 SELF-SUPERVISED LEARNING
+
+Given that we have a certain number of labelled images $D ^ { l }$ at our disposal, the obvious idea is to use these labels to bootstrap the representation $\Phi$ by minimizing a standard supervised objective such as the cross-entropy loss. However, experiments show that this causes the representation to overly-specialize for the classes in $D ^ { l }$ , providing a poor representation of the new classes in $D ^ { u }$ .
+
+Thus we resist the temptation of using the labels right away and use instead a self-supervised learning method to bootstrap the representation $\Phi$ . Self-supervised learning has been shown (Kolesnikov et al., 2019; Gidaris et al., 2018) to produce robust low-level features, especially for the first few layers of typical CNNs. It has the benefit that no data annotations are needed, and thus it can be applied to both labelled and unlabelled images during training. In this way, we achieve the key benefit of ensuring that the representation is initialized without being biased towards the labelled data.
+
+In detail, we first pre-train our model $\Phi$ with self-supervision on the union of $D ^ { l }$ and $D ^ { u }$ (ignoring all labels). We use the RotNet (Gidaris et al., 2018) approach1 due to its simplicity and efficacy, but any self-supervised method could be used instead. We then extend the pre-trained network $\Phi$ with a classification head $\eta ^ { l } : \mathbb { R } ^ { d }  \mathbb { R } ^ { C ^ { l } }$ implemented as a single linear layer followed by a softmax layer. The function $\eta ^ { l } \circ \boldsymbol \Phi$ is fine-tuned on the labelled dataset $D ^ { l }$ in order to learn a classifier for the $C ^ { l }$ known classes, this time using the labels $y _ { i }$ and optimizing the standard cross-entropy (CE) loss:
+
+$$
+L _ { \mathrm { C E } } = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \eta _ { y _ { i } } ^ { l } ( z _ { i } ^ { l } )
+$$
+
+where $z _ { i } ^ { l } = \Phi ( x _ { i } ^ { l } ) \in \mathbb { R } ^ { d }$ is the representation of image $x _ { i } ^ { l }$ . Only $\eta ^ { l }$ and the last macro-block of $\Phi$ (section 3) are updated in order to avoid overfitting the representation to the labelled data.
+
+# 2.2 TRANSFER LEARNING VIA RANK STATISTICS
+
+Once the representation $\Phi$ and the classifier $\eta ^ { l }$ have been trained, we are ready to look for the new classes in $D ^ { u }$ . Since the classes in $D ^ { u }$ are unknown, we represent them by defining a relation among pairs of unlabelled images $( x _ { i } ^ { u } , x _ { j } ^ { u } )$ . The idea is that similar images should belong to the same (new) class, which we denote by the symbol $s _ { i j } = 1$ , while dissimilar ones should not, which we denote by $s _ { i j } = 0$ . The problem is then to obtain the labels $s _ { i j }$ .
+
+Our assumption is that the new classes will have some degree of visual similarity with the known ones. Hence, the learned representation should be applicable to old and new classes equally well. As a consequence, we expect the descriptors $z _ { i } ^ { u } = \Phi ( x _ { i } ^ { u } )$ and $z _ { j } ^ { u } = \Phi ( x _ { j } ^ { u } )$ of two images $x _ { i } ^ { u }$ , $x _ { j } ^ { u }$ from the new classes to be close if they are from the same (new) class, and to be distinct otherwise.
+
+Rather than comparing vectors $z _ { i } ^ { u } , z _ { j } ^ { u }$ directly (e.g., by a scalar product), however, we use a more robust rank statistics. Specifically, we rank the values in vector $z _ { i } ^ { u }$ by magnitude. Then, if the rankings obtained for two unlabelled images $\boldsymbol { x } _ { i } ^ { u }$ and $x _ { j } ^ { u }$ are the same, they are very likely to belong to the same (new) class, so we set $s _ { i j } = 1$ . Otherwise, we set $s _ { i j } = 0$ . In practice, it is too strict to require the two rankings to be identical if the dimension of $z _ { i } ^ { u }$ is high (otherwise we may end up with $s _ { i j } = 0$ for all pairs $( i , j ) , i \neq j )$ . Therefore, we relax this requirement by only testing if the sets of the top- $k$ ranked dimensions are the same (we use $k = 5$ in our experiments), i.e.:
+
+$$
+s _ { i j } = \mathbb { 1 } \left\{ \mathrm { t o p } _ { k } ( \Phi ( \boldsymbol { x } _ { i } ^ { u } ) ) = \mathrm { t o p } _ { k } ( \Phi ( \boldsymbol { x } _ { j } ^ { u } ) ) \right\} ,
+$$
+
+where $\mathrm { t o p } _ { k } : \mathbb { R } ^ { d } \to \mathcal { P } ( \{ 1 , . . . , d \} )$ associates to a vector $z$ the subset of indices $\{ 1 , \ldots , d \}$ of its top- $k$ elements.
+
+Once the labels $s _ { i j }$ have been obtained, we use them as pseudo-labels to train a comparison function for the unlabelled data. In order to do this, we apply a new head $\eta ^ { u } : \mathbb { R } ^ { d }  \mathbb { R } ^ { C ^ { u } }$ to the image representation $z _ { i } ^ { u } = \Phi ( x _ { i } ^ { u } )$ to extract a new descriptor vector $\eta ^ { u } ( z _ { i } ^ { u } )$ optimized for the unlabelled data. As in section 2.1, the head is composed of a linear layer followed by a softmax. Then, the inner product $\eta ^ { u } ( z _ { i } ^ { u } ) ^ { \top } \eta ^ { u } ( z _ { j } ^ { u } )$ is used as a score for whether images $\boldsymbol { x } _ { i } ^ { u }$ and $x _ { j } ^ { u }$ belong to the same class or not. Note that $\eta ^ { u } ( z _ { i } ^ { u } )$ is a normalized vector due to the softmax layer in $\eta ^ { u }$ . This descriptor is trained by optimizing the binary cross-entropy (BCE) loss:
+
+$$
+L _ { \mathrm { B C E } } = - \frac { 1 } { M ^ { 2 } } \sum _ { i = 1 } ^ { M } \sum _ { j = 1 } ^ { M } [ s _ { i j } \log \eta ^ { u } ( z _ { i } ^ { u } ) ^ { \top } \eta ^ { u } ( z _ { j } ^ { u } ) + ( 1 - s _ { i j } ) \log ( 1 - \eta ^ { u } ( z _ { i } ^ { u } ) ^ { \top } \eta ^ { u } ( z _ { j } ^ { u } ) ) ] .
+$$
+
+Furthermore, we structure $\eta ^ { u }$ in a particular manner: We set its output dimension to be equal to the number of new classes $C ^ { u }$ . In this manner, we can use the index of the maximum element of each vector $\hat { y } _ { i } ^ { u } = \mathrm { a r g m a x } _ { y } [ \eta ^ { u } \circ \Phi ( x _ { i } ^ { u } ) ] _ { y }$ as prediction $\hat { y } _ { i } ^ { u }$ for the class of image $x _ { i } ^ { u }$ (as opposed to assigning labels via a clustering method such as $k$ -means).
+
+# 2.3 JOINT TRAINING ON LABELLED AND UNLABELLED DATA
+
+We now have two losses that involve the representation $\Phi$ : the CE loss $L _ { \mathrm { C E } }$ for the labelled data $D ^ { l }$ and the pairwise BCE loss $L _ { \mathrm { B C E } }$ for the unlabelled data $D ^ { u }$ . They both share the same image embedding $\Phi$ . This embedding can be trained sequentially, first on the labelled data, and then on the unlabelled data using the pseudo-labels obtained above. However, in this way the model will very likely forget the knowledge learned from the labelled data, which is known as catastrophic forgetting in incremental learning (Rebuffi et al., 2017; Lopez-Paz & Ranzato, 2017; Shmelkov et al., 2017; Aljundi et al., 2018).
+
+In contrast, we jointly fine-tune our model using both losses at the same time. Note that most of the model $\Phi$ is frozen; we only fine-tune the last macro-block of $\Phi$ together with the two heads $\eta ^ { u }$ and $\eta ^ { l }$ . Importantly, as we fine-tune the model, the labels $s _ { i j }$ are changing at every epoch as the embedding $\eta ^ { l }$ is updated. This in turn affects the rank statistics used to determine the labels $s _ { i j }$ as explained in section 2.2. This leads to a “moving target” phenomenon that can introduce some instability in learning the model. This potential issue is addressed in the next section.
+
+# 2.4 ENFORCING PREDICTIONS TO BE CONSISTENT
+
+In addition to the CE and BCE losses, we also introduce a consistency regularization term, which is used for both labelled and unlabelled data. In semi-supervised learning (Oliver et al., 2018; Tarvainen & Valpola, 2017; Laine & Aila, 2017), the idea of consistency is that the class predictions on an image $x$ and on a randomly-transformed counterpart $t x$ (for example an image rotation) should be the same. In our case, as will be shown in the experiments, consistency is very important to obtain good performance. One reason is that, as noted above, the pairwise pseudo-labels for the unlabelled data are subject to change on the fly during training. Indeed, for an image $x _ { i } ^ { u }$ and a randomly-transformed counterpart $t x _ { i } ^ { u }$ , if we do not enforce consistency, we can have $\mathrm { t o p } _ { k } ( \Phi ( x _ { i } ^ { u } ) ) \neq \mathrm { t o p } _ { k } ( \Phi ( t x _ { i } ^ { u } ) )$ . According to eq. (2) defining $s _ { i j }$ , it could result in different $s _ { i j }$ for $\boldsymbol { x } _ { i } ^ { u }$ depending on the data augmentation applied to the image. This variability of the ranking labels for a given pair could then confuse the training of the embedding.
+
+Following the common practice in semi-supervised learning, we use the Mean Squared Error (MSE) as the consistency cost. This is given by:
+
+$$
+L _ { \mathrm { M S E } } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } ( \eta ^ { l } ( z _ { i } ^ { l } ) - \eta ^ { l } ( \hat { z } _ { i } ^ { l } ) ) ^ { 2 } + \frac { 1 } { M } \sum _ { i = 1 } ^ { M } ( \eta ^ { u } ( z _ { i } ^ { u } ) - \eta ^ { u } ( \hat { z } _ { i } ^ { u } ) ) ^ { 2 } ,
+$$
+
+where $\hat { z }$ is the representation of $t x$
+
+The overall loss of our model can then be written as
+
+$$
+\begin{array} { r } { L = L _ { \mathrm { C E } } + L _ { \mathrm { B C E } } + \omega ( t ) L _ { \mathrm { M S E } } , } \end{array}
+$$
+
+where the coefficient $\omega ( t )$ is a ramp-up function. This is widely used in semi-supervised learning (Laine & Aila, 2017; Tarvainen & Valpola, 2017). Following (Laine & Aila, 2017; Tarvainen & Valpola, 2017), we use the sigmoid-shaped function $\omega ( t ) = \lambda e ^ { - 5 ( 1 - \frac { t } { T } ) ^ { 2 } }$ , where $t$ is current time step and $T$ is the ramp-up length and $\lambda \in \mathbb { R } _ { + }$ .
+
+# 2.5 INCREMENTAL LEARNING SCHEME
+
+We also explore a setting analogous to incremental learning. In this approach, after tuning on the labelled set (end of section 2.1), we extend the head $\eta ^ { l }$ to $C ^ { u }$ new classes, so that $\eta ^ { l } : \mathbb { R } ^ { d }  \mathbb { R } ^ { C ^ { l } + C ^ { u } }$ . The head parameters for the new classes are initialized randomly. The model is then trained using the same loss eq. (5), but the cross-entropy part of the loss is evaluated on both labelled and unlabelled data $D ^ { l }$ and $D ^ { u }$ . Since the cross-entropy requires labels, for the unlabelled data we use the pseudo-labels $\hat { y } _ { i } ^ { u }$ , which are generated on-the-fly from the head $\eta ^ { u }$ at each forward pass.
+
+The advantage is that this approach increments $\eta ^ { l }$ to discriminate both old and new classes, which is often desirable in applications. It also creates a feedback loop that causes the features $z _ { i } ^ { u }$ to be refined, which in turn generates better pseudo-labels $\hat { y } _ { i } ^ { u }$ for $D ^ { u }$ from the head $\eta ^ { u }$ . In this manner, further improvements can be obtained by this cycle of positive interactions between the two heads during training.
+
+Table 1: Ablation study. “w/o Con.” means without consistency constraints; “w/o CE” means without the cross entropy loss for training on labeled data. “w/o BCE” means without binary cross entropy loss for training on unlabeled data; “w/o S.S.” means without selfsupervision.   
+
+<table><tr><td></td><td>CIFAR-10</td><td>CIFAR-100</td><td>SVHN</td></tr><tr><td>Ours w/o Con</td><td>82.6±12.0%</td><td>61.8±3.6%</td><td>61.3±1.9%</td></tr><tr><td>Ours w/o CE</td><td>84.7±4.4%</td><td>58.4±2.7%</td><td>59.7±6.6%</td></tr><tr><td>Ours w/o BCE</td><td>26.2±2.0%</td><td>6.6±0.7%</td><td>24.5±0.5%</td></tr><tr><td>Ours w/o S.S.</td><td>89.4±1.4%</td><td>67.4±2.0%</td><td>72.9±5.0%</td></tr><tr><td>Ours full</td><td>90.4±0.5%</td><td>73.2±2.1%</td><td>95.0±0.2%</td></tr><tr><td>Ours w/ I.L.</td><td>91.7±0.9%</td><td>75.2±4.2%</td><td>95.2±0.3%</td></tr></table>
+
+# 3 EXPERIMENTS
+
+# 3.1 DATA AND EXPERIMENTAL DETAILS
+
+We evaluate our models on a variety of standard benchmark datasets: CIFAR-10 (Krizhevsky & Hinton, 2009), CIFAR-100 (Krizhevsky & Hinton, 2009), SVHN (Netzer et al., 2011), OmniGlot (Lake et al., 2015), and ImageNet (Deng et al., 2009). Following Han et al. (2019), we split these to have 5/20/5/654/30 classes respectively in the unlabelled set. In addition, for OmniGlot and ImageNet we use 20 and 3 different splits respectively, as in Han et al. (2019), and report average clustering accuracy. More details on the splits can be found in appendix A.
+
+Evaluation metrics. We adopt clustering accuracy (ACC) to evaluate the clustering performance of our approach. The ACC is defined as follow:
+
+$$
+\operatorname* { m a x } _ { g \in \mathrm { { S y m } } ( L ) } \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \mathbb { 1 } \left\{ \overline { { y } } _ { i } = g \left( y _ { i } \right) \right\} ,
+$$
+
+where ${ \overline { { y } } } _ { i }$ and $y _ { i }$ denote the ground-truth label and clustering assignment for each data point $x _ { i } ^ { u } \in D ^ { u }$ respectively, and $\mathrm { S y m } ( L )$ is the group permutations of $L$ elements (this discounts the fact that the cluster indices may not be in the same order as the ground-truth labels). Permutations are optimized using the Hungarian algorithm (Kuhn, 1955).
+
+Implementation details. We use the ResNet-18 (He et al., 2016) architecture, except for OmniGlot for which we use a VGG-like network (Simonyan & Zisserman, 2015) with six layers to make our setting directly comparable to prior work. We use SGD with momentum (Sutskever et al., 2013) as optimizer for all but the OmniGlot dataset, for which we use Adam (Kingma & Ba, 2014). For all experiments we use a batch size of 128 and $k = 5$ which we found worked consistently well across datasets (see appendix D). More details about the hyper-parameters can be found in appendix B.
+
+# 3.2 ABLATION STUDY
+
+We validate the effectiveness of the components of our method by ablating them and measuring the resulting ACC on the unlabelled data. Note that, since the evaluation is restricted to the unlabelled data, we are solving a clustering problem. The same unlabelled data points are used for both training and testing, except that data augmentation (i.e. image transformations) is not applied when computing the cluster assignments. As can be seen in table 1, all components have a significant effect as removing any of them causes the performance to drop substantially. Among them, the BCE loss is by far the most important one, since removing it results in a dramatic drop of $40 \mathrm { - } 6 0 \%$ absolute ACC points. For example, the full method has ACC $9 0 . 4 \%$ on CIFAR-10, while removing BCE causes the ACC to drop to $2 6 . 2 \%$ . This shows that that our rank-based embedding comparison can indeed generate reliable pairwise pseudo labels for the BCE loss. Without consistency, cross entropy, or self-supervision, the performance drops by a more modest but still significant $7 . 8 \%$ , $5 . 7 \%$ and $1 . 0 \%$ absolute ACC points, respectively, for CIFAR-10. It means that the consistency term plays a role as important as the cross-entropy term by preventing the “moving target” phenomenon described in section 2.4. Finally, by incorporating the discovered classes in the classification task, we get a further boost of $1 . 3 \%$ , $2 . 0 \%$ and $0 . 2 \%$ points on CIFAR-10, CIFAR-100 and SVHN respectively.
+
+Table 2: Novel category discovery results on CIFAR-10, CIFAR-100, and SVHN. ACC on the unlabelled set. “w/ S.S.” means with self-supervised learning.   
+
+<table><tr><td>No</td><td></td><td>CIFAR-10</td><td>CIFAR-100</td><td>SVHN</td></tr><tr><td>(1)</td><td>k-means (MacQueen,1967)</td><td>65.5±0.0 %</td><td>56.6±1.6%</td><td>42.6%±0.0</td></tr><tr><td>(2)</td><td>KCL (Hsu et al., 2018)</td><td>66.5±3.9%</td><td>14.3±1.3%</td><td>21.4%±0.6</td></tr><tr><td>(3)</td><td>MCL (Hsu et al., 2019)</td><td>64.2±0.1%</td><td>21.3±3.4%</td><td>38.6%±10.8</td></tr><tr><td>(4)</td><td>DTC (Han et al., 2019)</td><td>87.5±0.3%</td><td>56.7±1.2%</td><td>60.9%±1.6</td></tr><tr><td>(5)</td><td>k-means (MacQueen,1967) w/ S.S.</td><td>72.5±0.0%</td><td>56.3±1.7%</td><td>46.7±0.0%</td></tr><tr><td>(6)</td><td>KCL (Hsu et al., 2018) w/ S.S.</td><td>72.3±0.2%</td><td>42.1±1.8%</td><td>65.6±4.9%</td></tr><tr><td>(7)</td><td>MCL (Hsu et al.,2019) w/ S.S.</td><td>70.9±0.1%</td><td>21.5±2.3%</td><td>53.1±0.3%</td></tr><tr><td>(8)</td><td>DTC (Han et al., 2019) w/ S.S.</td><td>88.7±0.3%</td><td>67.3±1.2%</td><td>75.7±0.4%</td></tr><tr><td>(9)</td><td>Ours</td><td>90.4±0.5%</td><td>73.2±2.1%</td><td>95.0±0.2%</td></tr><tr><td>(10)</td><td>Ours w/ I.L.</td><td>91.7±0.9%</td><td>75.2±4.2%</td><td>95.2±0.2%</td></tr></table>
+
+![](images/1f699b05e8a4e57588ae83f600b928bef851c7fef98bdb359b1475d68b65a71c.jpg)  
+Figure 2: Evolution of the t-SNE during the training of CIFAR-10. Performed on unlabelled data (i.e., instances of dog, frog, horse, ship, truck). Colors of data points denote their ground-truth labels.
+
+# 3.3 NOVEL CATEGORY DISCOVERY
+
+We compare our method to baselines and state-of-the-art methods for new class discovery, starting from CIFAR-10, CIFAR-100, and SVHN in table 2. The first baseline (row 5 in table 2) amounts to applying $k$ -means (MacQueen, 1967) to the features extracted by the fine-tuned model (the second step in section 2.1), for which we use the $k$ -means $^ { + + }$ (Arthur & Vassilvitskii, 2007) initialization. The second baseline (row 1 in table 2) is similar, but uses as feature extractor a model trained from scratch using only the labelled images, which corresponds to a standard transfer learning setting. By comparing rows 1, 5 and 9 in table 2, we can see that our method substantially outperforms $k$ -means. Next, we compare with the KCL (Hsu et al., 2018), MCL (Hsu et al., 2019) and DTC (Han et al., 2019) methods. By comparing rows 2–4 to 9, we see that our method outperforms these by a large margin. We also try to improve KCL, MCL and DTC by using the same self-supervised initialization we adopt (section 2.1), which indeed results in an improvement (rows 2–4 vs 6–8). However, their overall performance still lags behind ours by a large margin. For example, our method of section 2.4 achieves $9 5 . 0 \%$ ACC on SVHN, while “KCL w/ S.S.”, “MCL w/ S.S.” and “DTC w/ S.S.” achieve only $6 5 . 6 \%$ , $5 3 . 1 \%$ and $7 5 . 7 \%$ ACC, respectively. Similar trends hold for CIFAR-10 and CIFAR-100. Finally, the incremental learning scheme of section 2.5 results in further improvements, as can be seen by comparing rows 9 and 10 of table 2.
+
+In fig. 2, we show the evolution of the learned representation on the unlabelled data on CIFAR-10 using t-SNE (van der Maaten & Hinton, 2008). As can be seen, while the clusters overlap in the beginning, they become more and more separated as the training progresses, showing that our model can effectively discover novel visual categories without labels and learn meaningful embeddings for them.
+
+We further compare our method to others on two more challenging datasets, OmniGlot and ImageNet, in table 3. For OmniGlot, results are averaged over the 20 alphabets in the evaluation set (see appendix A); for ImageNet, results are averaged over the three 30-class unlabelled sets used in (Hsu et al., 2018; 2019). Since we have a relatively larger number of labelled classes in these two datasets, we follow (Han et al., 2019) and use metric learning on the labelled classes to pre-train the feature extractor, instead of the self-supervised learning. We empirically found that self-supervision does not provide obvious gains for these two datasets. This is reasonable since the data in the labelled sets of these two datasets are rather diverse and abundant, so metric learning can provide good feature initialization as there is less class-specific bias due to the large number of pre-training classes. However, by comparing rows 1 and 5 in table 3, it is clear that metric learning alone is not sufficient for the task of novel category discovery. Our method substantially outperforms the $k$ -means results obtained using the features from metric learning — by $1 1 . 9 \%$ and $\mathrm { { \bar { 1 } 0 . 6 \% } }$ on OmniGlot and ImageNet respectively. Our method also substantially outperforms the current state-of-the-art, achieving ${ \bar { 8 9 } } . 1 \%$ and $8 2 . 5 \%$ ACC on OmniGlot and ImageNet respectively, compared with $8 9 . 0 \%$ and $7 8 . 8 \%$ of (Han et al., 2019), thus setting the new state-of-the-art.
+
+<table><tr><td>No</td><td></td><td>OmniGlot</td><td>ImageNet</td></tr><tr><td>(1)</td><td>k-means (MacQueen,1967)</td><td>77.2%</td><td>71.9%</td></tr><tr><td>(2)</td><td>KCL (Hsu et al., 2018)</td><td>82.4%</td><td>73.8%</td></tr><tr><td>(3)</td><td>MCL (Hsu et al., 2019)</td><td>83.3%</td><td>74.4%</td></tr><tr><td>(4)</td><td>DTC (Han et al., 2019)</td><td>89.0%</td><td>78.3%</td></tr><tr><td>(5)</td><td>Ours</td><td>89.1%</td><td>82.5%</td></tr></table>
+
+![](images/1d6429af68e17efa89c35d0bc22c398bc47d02a56a02d662b000d4ae57cbc16f.jpg)  
+Table 3: Novel category discovery results on OmniGlot and ImageNet. ACC on the unlabelled set.   
+Figure 3: t-SNE on CIFAR-10: impact of incremental Learning. (a) representation on the labelled and (b) unlabelled CIFAR classes. Colors of data points denote their ground-truth labels. We observe a bigger overlap in (a) between the “old” class 3 and the “new” class 5 when not incorporating Incremental Learning.
+
+# 3.4 INCREMENTAL LEARNING
+
+Table 4: Incremental Learning with the novel categories. “old” refers to the ACC on the labelled classes while “new” refers to the unlabelled classes in the testing set. “all” indicates the whole testing set. It should be noted that the predictions are not restricted to their respective subset. Standard deviation can be found in appendix C.   
+
+<table><tr><td></td><td colspan="3">CIFAR-10</td><td colspan="3">CIFAR-100</td><td colspan="3">SVHN</td></tr><tr><td>Classes</td><td>old</td><td>new</td><td>all</td><td>old</td><td>new</td><td>all</td><td>old</td><td>new</td><td>all</td></tr><tr><td>KCL w/ S.S.</td><td>79.4%</td><td>60.1%</td><td>69.8%</td><td>23.4%</td><td>29.4%</td><td>24.6%</td><td>90.3%</td><td>65.0%</td><td>81.0%</td></tr><tr><td>MCL w/ S.S.</td><td>81.4%</td><td>64.8%</td><td>73.1%</td><td>18.2%</td><td>18.0%</td><td>18.2%</td><td>94.0%</td><td>48.6%</td><td>77.2%</td></tr><tr><td>DTC w/ S.S.</td><td>58.7%</td><td>78.6%</td><td>68.7%</td><td>47.6%</td><td>49.1%</td><td>47.9%</td><td>90.5%</td><td>72.8%</td><td>84.0%</td></tr><tr><td>Ours w/ I.L.</td><td>90.6%</td><td>88.8%</td><td>89.7%</td><td>71.2%</td><td>56.8%</td><td>68.3%</td><td>96.3%</td><td>96.1%</td><td>96.2%</td></tr></table>
+
+Here, we further evaluate our incremental scheme for novel category discovery as described in section 2.5. Methods for novel category discovery such as (Han et al., 2019; Hsu et al., 2019; 2018) focus on obtaining the highest clustering accuracy for the new unlabelled classes, but may forget the existing labelled classes in the process. In practice, forgetting is not desirable as the model should be able to recognize both old and new classes. Thus, we argue that the classification accuracy on the labelled classes should be assessed as well, as for any incremental learning setting. Note however that our setup differs substantially from standard incremental learning (Rebuffi et al., 2017; Lopez-Paz & Ranzato, 2017; Shmelkov et al., 2017; Aljundi et al., 2018) where every class is labelled and the focus is on using limited memory. In our case, we can store and access the original data without memory constraints, but the new classes are unlabelled, which is often encountered in applications.
+
+By construction (section 2.5), our method learns the new classes on top of the old ones incrementally, out of the box. In order to compare to methods such as KCL, MCL and DTC that do not have this property, we proceed as follows. First, the method runs as usual to cluster the unlabelled portion of the data, thus obtaining pseudo-labels for it, and learning a feature extractor as a byproduct. Then, the feature extractor is used to compute features for both the labelled and unlabelled training data, and a linear classifier is trained using labels and pseudo-labels, jointly on all the classes, old and new.
+
+We report in table 4 the performance of the resulting joint classifier networks on the testing set of each dataset (this is now entirely disjoint from the training set). Our method has similar performances on the old and new classes for CIFAR-10 and SVHN, as might be expected as the split between old and new classes is balanced. In comparison, the feature extractor learned by KCL and MCL works much better for the old classes (e.g., the accuracy discrepancy between old and new classes is $2 5 . 3 \%$ for KCL on SVHN). Conversely, DTC learns features that work better for the new classes, as shown by the poor performance for the old classes on CIFAR-10. Thus, KCL, MCL and DTC learn representations that are biased to either the old or new classes, resulting overall in suboptimal performance. In contrast, our method works well on both old and new classes; furthermore, it drastically outperforms existing methods on both.
+
+# 4 RELATED WORK
+
+Our work draws inspiration from semi-supervised learning, transfer learning, clustering, and zero-shot learning. We review below the most relevant contributions.
+
+In semi-supervised learning (SSL) (Chapelle et al., 2006), a partially labelled training dataset is given and the objective is to learn a model that can propagate the labels from the labelled data to unlabelled data. Most SSL methods focus on the classification task where, usually both labelled and unlabelled points belong to the same set of classes. On the contrary, our goal is to handle the case where the unlabelled data classes differ from the labelled data. Oliver et al. (2018) summarizes the state-of-the-art SSL methods. Among them, the consistency-based methods appeared to be the most effective. Rasmus et al. (2015) propose a ladder network which is trained on both labelled and unlabelled data using a reconstruction loss. Laine & Aila (2017) simplifies this ladder network by enforcing prediction consistency between a data point and its augmented counterpart. As an alternative to data augmentation, they also consider a regularization method based on the exponential moving average (EMA) of the predictions. This idea is further improved by Tarvainen & Valpola (2017): instead of using the EMA of predictions, they propose to maintain the EMA of model parameters. The consistency is then measured between the predictions of the current model (student) and the predictions of the EMA model (teacher). More recently (and closer to our work) practitioners have also combined SSL with self-supervision(Rebuffi et al., 2019; Zhai et al., 2019) to leverage dataset with very few annotations.
+
+Transfer learning (Pan & Yang, 2010; Weiss et al., 2016; Tan et al., 2018) is an effective way to reduce the amount of data annotations required to train a model by pre-training the model on an existing dataset. In image classification, for example, it is customary to start from a model pre-trained on the ImageNet (Deng et al., 2009) dataset. In most transfer learning settings, however, both the source data and the target data are fully annotated. In contrast, our goal is to transfer information from a labelled dataset to an unlabelled one.
+
+Many classic (e.g., Aggarwal & Reddy (2013); MacQueen (1967); Comaniciu & Meer (1979); $\mathrm { N g }$ et al. (2001)) and deep learning (e.g., Xie et al. (2016); Chang et al. (2017); Dizaji et al. (2017); Yang et al. (2017; 2016); Hsu et al. (2018; 2019)) clustering methods have been proposed to automatically partition an unlabelled data collection into different classes. However, this task is usually ill-posed as there are multiple, equally valid criteria to partition most datasets. We address this challenge by learning the appropriate criterion by using a labelled dataset, narrowing down what constitutes a proper class. We call this setting “transfer clustering”.
+
+To the best of our knowledge, the work most related to ours are (Hsu et al., 2018; 2019; Han et al., 2019). Han et al. (2019) also consider discovering new classes as a transfer clustering problem. They first learn a data embedding by using metric learning on the labelled data, and then fine-tune the embedding and learn the cluster assignments on the unlabelled data. In (Hsu et al., 2018; 2019), the authors introduce KCL and MCL clustering methods. In both, a similarity prediction network (SPN), also used in (Hsu et al., 2016), is first trained on a labelled dataset. Afterwards, the pre-trained SPN is used to provide binary pseudo labels for training the main model on an unlabelled dataset. The overall pipelines of the two methods are similar, but the losses differ: KCL uses a Kullback-Leibler divergence based contrastive loss equivalent to the BCE used in this paper (eq. (3)), and MCL uses the Meta Classification Likelihood loss. Zero-shot learning (ZSL) (Xian et al., 2018; Fu et al., 2018) can also be used to recognize new classes. However, differently from our work, ZSL also requires additional side information (e.g., class attributes) in addition to the raw images.
+
+Finally, other works (Dean et al., 2013; Yagnik et al., 2011) discuss the application of rank statistics to measuring the similarity of vectors; however, to the best of our knowledge, we are the first to apply rank statistics to the task of novel category discovery using deep neural networks.
+
+# 5 CONCLUSIONS
+
+In this paper, we have looked at the problem of discovering new classes in an image collection, leveraging labels available for other, known classes. We have shown that this task can be addressed very successfully by a few new ideas. First, the use of self-supervised learning for bootstrapping the image representation trades off the representation quality with its generality, and for our problem this leads to a better solution overall. Second, we have shown that rank statistics are an effective method to compare noisy image descriptors, resulting in robust data clustering. Third, we have shown that jointly optimizing both labelled recognition and unlabelled clustering in an incremental learning setup can reinforce the two tasks while avoiding forgetting. On standard benchmarks, the combination of these ideas results in much better performance than existing methods that solve the same task. Finally, for larger datasets with more classes and diverse data (e.g., ImageNet) we note that self-supervision can be bypassed as the pretraining on labelled data already provides a powerful enough representation. In such cases, we still show that the rank statistics for clustering gives drastic improvement over existing methods.
+
+# 6 ACKNOWLEDGMENTS
+
+This work is supported by the EPSRC Programme Grant Seebibyte EP/M013774/1, Mathworks/DTA DFR02620, and ERC IDIU-638009.
+
+# REFERENCES
+
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+
+# A DATASET SPLITS
+
+For CIFAR-10 and SVHN we keep the labels of the five first categories (namely airplane, automobile, bird, cat, deer for CIFAR-10, 0–4 for SVHN) and keep the rest of the data as the unlabelled set. For CIFAR-100 we use the first 80 categories as labelled data while the rest are used for the unlabelled set. Following Hsu et al. (2018; 2019), for OmniGlot, each of the 20 alphabets in evaluation set (with 20–47 categories, 659 characters/class) is used as unlabelled data, and all the 30 alphabets in background set are used as labelled set (964 characters/class). For ImageNet, we follow Hsu et al. (2018; 2019) to use the 882/118 classes split proposed in Vinyals et al. (2016), and use the three 30-class subsets sampled from the 118 classes as unlabelled sets.
+
+# B IMPLEMENTATION DETAILS
+
+In the first self-supervised training step, otherwise mentioned, we trained our model with the pretext task of rotation predictions (i.e., a four-class classification: $0 ^ { \circ }$ , $9 0 °$ , $1 8 0 ^ { \circ }$ , and $2 7 0 ^ { \circ }$ ) for 200 epochs and a step-wise decaying learning rate starting from 0.1 and divided by 5 at epochs 60, 120, and 160.
+
+In the second step of our framework (i.e., supervised training using labelled data), we fine-tuned our model on the labelled set for 100 epochs and a step-wise decaying learning rate starting from 0.1 and halved every 10 epochs. From this step onward we fix the first three convolutional blocks of the model, and fine-tuned the last convolutional block together with the linear classifier.
+
+Finally, in the last joint training step, we fine-tuned our model for 200/100/90 epochs for {CIFAR-10, CIFAR-100, SVHN}/OmniGlot/ImageNet, which was randomly sampled from the merged set of both labelled and unlabelled data. The initial learning rate was set to 0.1 for all datasets, and was decayed with a factor of 10 at the $1 7 0 \mathrm { t h } / \{ 3 0 \mathrm { t h } , 6 0 \mathrm { t h } \}$ epoch for {CIFAR-10, CIFAR-100, SVHN}/ImageNet. The learning rate of 0.01 was kept fixed for OmniGlot. For the consistency regularization term, we used the ramp-up function as described in section 2.4 with $\lambda = \{ 5 . 0 , 5 0 . { \dot { 0 } } , 5 { \bar { 0 } } . 0 , 1 0 0 . 0 , 1 0 . 0 \}$ , and $T = \{ 5 0 , 1 5 \bar { 0 } , 8 \bar { 0 } , 1 , 5 0 \}$ for CIFAR-10, CIFAR-100, SVHN, OmniGlot, and ImageNet respectively.
+
+In the incremental learning setting, all previous hyper parameters remain the same for our method. We only add a ramp-up on the cross entropy loss on unlabelled data. The ramp-up length is the same as the one used for eq. (4) and we use for all experiments a coefficient of 0.05. For all other methods we trained the classifier for 150 epochs with SGD with momentum and learning rate of 0.1 divided by 10 at epoch 50.
+
+We implemented our method using PyTorch 1.1.0 and ran experiments on NVIDIA Tesla M40 GPUs. Following (Han et al., 2019), our results were averaged over 10 runs for all datasets, except for ImageNet which was averaged over the three 30-class subsets. In general, we found the results were stable. Our code is publicly available at http://www.robots.ox.ac.uk/˜vgg/ research/auto_novel.
+
+# C STANDARD DEVIATION OF INCREMENTAL LEARNING EXPERIMENT IN TABLE 4
+
+Table 5: Incremental Learning with the novel categories. “old” refers to the standard deviation ACC on the labelled classes while “new” refers to the unlabelled classes in the testing set. “all” indicates the whole testing set. It should be noted that the predictions are not restricted to their respective subset.   
+
+<table><tr><td></td><td colspan="3">CIFAR-10</td><td colspan="3">CIFAR-100</td><td colspan="3">SVHN</td></tr><tr><td>Classes</td><td>old</td><td>new</td><td>all</td><td>old</td><td>new</td><td>all</td><td>old</td><td>new</td><td>all</td></tr><tr><td>KCL w/ S.S.</td><td>0.6%</td><td>0.6%</td><td>0.1%</td><td>0.3%</td><td>0.3%</td><td>0.2%</td><td>0.3%</td><td>0.5%</td><td>0.1%</td></tr><tr><td>MCL w/ S.S.</td><td>0.4%</td><td>0.4%</td><td>0.1%</td><td>0.3%</td><td>0.1%</td><td>0.2%</td><td>0.2%</td><td>0.3%</td><td>0.1%</td></tr><tr><td>DTC w/ S.S.</td><td>0.6%</td><td>0.2%</td><td>0.3%</td><td>0.2%</td><td>0.2%</td><td>0.2%</td><td>0.3%</td><td>0.2%</td><td>0.1%</td></tr><tr><td>Ours w/ I.L.</td><td>0.2%</td><td>0.2%</td><td>0.1%</td><td>0.1%</td><td>0.3%</td><td>0.1%</td><td>0.1%</td><td>0.0%</td><td>0.1%</td></tr></table>
+
+![](images/711bdf42af75f07613e546052e4dc2926f4fe7c7f19bbb369d5b9b25a50a12e8.jpg)  
+Performance of our method w.r.t $k$   
+Figure 4: Performance evolution with respect to $k$ . We report results for $k \_ =$ $\{ 1 , 2 , 3 , 5 , 7 , 1 0 , 1 5 , 2 0 , 5 0 \}$ .
+
+# D IMPACT OF $k$ OVER RESULTS
+
+We provide an additional study of the evolution of performances of our method with respect to $k$ . We results on SVHN/CIFAR10/CIFAR100 in fig. 4. We found that $k = \{ 5 , 7 \}$ gave the best results overall. We also found that for all values of $k$ except 1 results were in general stable.
+
+# E RESULTS WITH AN UNKNOWN NUMBER OF CLASSES
+
+While in our work we assume the number of new classes $C ^ { u }$ to be known a priori, this hypothesis can be restrictive in practice. Instead, one can estimate the number of classes in $D ^ { u }$ using recent methods such as DTC (Han et al., 2019). In table 6 we compare ACC of KCL (Hsu et al., 2018), KCL (Hsu et al., 2019), DTC(Han et al., 2019) and our method on unlabelled splits of OmniGlot and ImageNet datasets with $C ^ { u }$ computed from DTC. We note that our method again reaches the state-of-the-art on ImageNet and is on par with the state-of-the-art on OmniGlot.
+
+Table 6: Novel category discovery results with unknown $C ^ { u }$ .   
+
+<table><tr><td>No</td><td>OmniGlot</td><td>ImageNet</td></tr><tr><td>(1)</td><td>KCL (Hsu et al., 2018) 80.3%</td><td>71.4%</td></tr><tr><td>(2) MCL (Hsu et al., 2019)</td><td>80.5%</td><td>72.9%</td></tr><tr><td>(3) DTC (Han et al., 2019)</td><td>87.0%</td><td>77.6%</td></tr><tr><td>(4) Ours</td><td>85.4%</td><td>80.5%</td></tr></table>

parse/train/BJl2_nVFPB/BJl2_nVFPB_content_list.json

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+        "type": "text",
+        "text": "AUTOMATICALLY DISCOVERING AND LEARNING NEW VISUAL CATEGORIES WITH RANKING STATISTICS ",
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+        "text": "Kai Han∗ Sylvestre-Alvise Rebuffi∗ Sebastien Ehrhardt∗ Andrea Vedaldi Andrew Zisserman Visual Geometry Group, Department of Engineering Science, University of Oxford {khan,srebuffi,hyenal,vedaldi,az}@robots.ox.ac.uk ",
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+        "text": "ABSTRACT ",
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+        "text": "We tackle the problem of discovering novel classes in an image collection given labelled examples of other classes. This setting is similar to semi-supervised learning, but significantly harder because there are no labelled examples for the new classes. The challenge, then, is to leverage the information contained in the labelled images in order to learn a general-purpose clustering model and use the latter to identify the new classes in the unlabelled data. In this work we address this problem by combining three ideas: (1) we suggest that the common approach of bootstrapping an image representation using the labeled data only introduces an unwanted bias, and that this can be avoided by using self-supervised learning to train the representation from scratch on the union of labelled and unlabelled data; (2) we use rank statistics to transfer the model’s knowledge of the labelled classes to the problem of clustering the unlabelled images; and, (3) we train the data representation by optimizing a joint objective function on the labelled and unlabelled subsets of the data, improving both the supervised classification of the labelled data, and the clustering of the unlabelled data. We evaluate our approach on standard classification benchmarks and outperform current methods for novel category discovery by a significant margin. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Modern machine learning systems can match or surpass human-level performance in tasks such as image classification (Deng et al., 2009), but at the cost of collecting large quantities of annotated training data. Semi-supervised learning (SSL) (Oliver et al., 2018) can alleviate this issue by mixing labelled with unlabelled data, which is usually much cheaper to obtain. However, these methods still require some annotations for each of the classes that one wishes to learn. We argue this is not always possible in real applications. For instance, consider the task of recognizing products in supermarkets. Thousands of new products are introduced in stores every week, and it would be very expensive to annotate them all. However, new products do not differ drastically from the existing ones, so it should be possible to discover them automatically as they arise in the data. Unfortunately, machines are still unable to effectively learn new classes without manual annotations. ",
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+        "type": "text",
+        "text": "In this paper, we thus consider the problem of discovering new visual classes automatically, assuming that a certain number of classes are already known by the model (Hsu et al., 2018; 2019; Han et al., 2019). This knowledge comes in the form of a labelled dataset of images for a certain set of classes. Given that this data is labelled, off-the-shelf supervised learning techniques can be used to train a very effective classifier for the known classes, particularly if Convolutional Neural Networks (CNNs) are employed. However, this does not mean that the learned features are useful as a representation of the new classes. Furthermore, even if the representation transfers well, one still has the problem of identifying the new classes in an unlabelled dataset, which is a clustering problem. ",
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+        "text": "We tackle these problems by introducing a novel approach that combines three key ideas (section 2 and fig. 1). The first idea is to pre-train the image representation (a CNN) using all available images, both labelled and unlabelled, using a self-supervised learning objective. Crucially, this objective does not leverage the known labels, resulting in features that are much less biased towards the labelled classes. Labels are used only after pre-training to learn a classifier specific to the labelled data as well as to fine-tune the deepest layers of the CNN, for which self-supervision is not as effective. ",
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+        "type": "image",
+        "img_path": "images/9c76455edd386d2f416ea32e19ed3799838374222b6c521e38a5561651f23df2.jpg",
+        "image_caption": [
+            "Figure 1: Overview of our three step learning pipeline. The first step of the training consists in learning an unbiased image representation via self-supervision using both labelled and unlabelled data, which learns well the early layers of the representation; in the second step, we fine-tune only the last few layers of the model using supervision on the labelled set; finally, the fine-tuned representation is used, via rank statistics, to induce clusters in the unlabelled data, while maintaining a good representation on the labelled set. "
+        ],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "The second idea is a new approach to transfer the information contained in the labelled images to the problem of clustering the unlabelled ones. Information is transferred by sharing the same representation between labelled and unlabelled images, motivated by the fact that the new classes are often similar to the known ones. In more detail, pairs of unlabelled images are compared via their representation vectors. The comparison is done using robust rank statistics, by testing if two images share the same subset of $k$ maximally activated representation components. This test is used to decide if two unlabelled images belong to the same (new) class or not, generating a set of noisy pairwise pseudo-labels. The pseudo-labels are then used to learn a similarity function for the unlabelled images. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "The third and final idea is, after bootstrapping the representation, to optimise the model by minimizing a joint objective function, containing terms for both the labelled and unlabelled subsets, using respectively the given labels and the generated pseudo-labels, thus avoiding the forgetting issue that may arise with a sequential approach. A further boost is obtained by incorporating incremental learning of the discovered classes in the classification task, which allows information to flow between the labelled and unlabelled images. ",
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+        "type": "text",
+        "text": "We evaluate our method on several public benchmarks (section 3), outperforming by a large margin all existing techniques (section 4) that can be applied to this problem, demonstrating the effectiveness of our approach. We conclude the paper by summarizing our findings (section 5). Our code can be found at http://www.robots.ox.ac.uk/˜vgg/research/auto_novel. ",
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+    {
+        "type": "text",
+        "text": "2 METHOD ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Given an unlabelled dataset $D ^ { u } = \\{ x _ { i } ^ { u } , i = 1 , \\ldots , M \\}$ of images $x _ { i } ^ { u } \\in \\mathbb { R } ^ { 3 \\times H \\times W }$ , our goal is to automatically cluster the images into a number of classes $C ^ { u }$ , which we assume to be known $a$ priori. We also assume to have a second labelled image dataset $\\dot { D ^ { l } } = \\{ ( x _ { i } ^ { l } , y _ { i } ^ { l } ) , i = 1 , \\dots , N \\}$ where $y _ { i } ^ { l } \\in \\{ 1 , \\ldots , C ^ { l } \\}$ is the class label for image $x _ { i } ^ { l }$ . We also assume that the set of $C ^ { l }$ labelled classes is disjoint from the set of $C ^ { u }$ unlabelled ones. While the statistics of $D ^ { l }$ and $D ^ { u }$ thus differ, we hypothesize that a general notion of what constitutes a “good class” can be extracted from $D ^ { l }$ and that the latter can be used to better cluster $D ^ { u }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We approach the problem by learning an image representation $\\Phi : x \\mapsto \\Phi ( x ) \\in \\mathbb { R } ^ { d }$ in the form of a CNN. The goal of the representation is to help to recognize the known classes and to discover the new ones. In order to learn this representation, we combine three ideas, detailed in the next three sections. ",
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+    {
+        "type": "text",
+        "text": "2.1 SELF-SUPERVISED LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Given that we have a certain number of labelled images $D ^ { l }$ at our disposal, the obvious idea is to use these labels to bootstrap the representation $\\Phi$ by minimizing a standard supervised objective such as the cross-entropy loss. However, experiments show that this causes the representation to overly-specialize for the classes in $D ^ { l }$ , providing a poor representation of the new classes in $D ^ { u }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Thus we resist the temptation of using the labels right away and use instead a self-supervised learning method to bootstrap the representation $\\Phi$ . Self-supervised learning has been shown (Kolesnikov et al., 2019; Gidaris et al., 2018) to produce robust low-level features, especially for the first few layers of typical CNNs. It has the benefit that no data annotations are needed, and thus it can be applied to both labelled and unlabelled images during training. In this way, we achieve the key benefit of ensuring that the representation is initialized without being biased towards the labelled data. ",
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+        "type": "text",
+        "text": "In detail, we first pre-train our model $\\Phi$ with self-supervision on the union of $D ^ { l }$ and $D ^ { u }$ (ignoring all labels). We use the RotNet (Gidaris et al., 2018) approach1 due to its simplicity and efficacy, but any self-supervised method could be used instead. We then extend the pre-trained network $\\Phi$ with a classification head $\\eta ^ { l } : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { C ^ { l } }$ implemented as a single linear layer followed by a softmax layer. The function $\\eta ^ { l } \\circ \\boldsymbol \\Phi$ is fine-tuned on the labelled dataset $D ^ { l }$ in order to learn a classifier for the $C ^ { l }$ known classes, this time using the labels $y _ { i }$ and optimizing the standard cross-entropy (CE) loss: ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e6c7c223bc0d95a0c20e98d056b094d1729057e2314e242558447c96ce825221.jpg",
+        "text": "$$\nL _ { \\mathrm { C E } } = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log \\eta _ { y _ { i } } ^ { l } ( z _ { i } ^ { l } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $z _ { i } ^ { l } = \\Phi ( x _ { i } ^ { l } ) \\in \\mathbb { R } ^ { d }$ is the representation of image $x _ { i } ^ { l }$ . Only $\\eta ^ { l }$ and the last macro-block of $\\Phi$ (section 3) are updated in order to avoid overfitting the representation to the labelled data. ",
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+    {
+        "type": "text",
+        "text": "2.2 TRANSFER LEARNING VIA RANK STATISTICS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Once the representation $\\Phi$ and the classifier $\\eta ^ { l }$ have been trained, we are ready to look for the new classes in $D ^ { u }$ . Since the classes in $D ^ { u }$ are unknown, we represent them by defining a relation among pairs of unlabelled images $( x _ { i } ^ { u } , x _ { j } ^ { u } )$ . The idea is that similar images should belong to the same (new) class, which we denote by the symbol $s _ { i j } = 1$ , while dissimilar ones should not, which we denote by $s _ { i j } = 0$ . The problem is then to obtain the labels $s _ { i j }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our assumption is that the new classes will have some degree of visual similarity with the known ones. Hence, the learned representation should be applicable to old and new classes equally well. As a consequence, we expect the descriptors $z _ { i } ^ { u } = \\Phi ( x _ { i } ^ { u } )$ and $z _ { j } ^ { u } = \\Phi ( x _ { j } ^ { u } )$ of two images $x _ { i } ^ { u }$ , $x _ { j } ^ { u }$ from the new classes to be close if they are from the same (new) class, and to be distinct otherwise. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Rather than comparing vectors $z _ { i } ^ { u } , z _ { j } ^ { u }$ directly (e.g., by a scalar product), however, we use a more robust rank statistics. Specifically, we rank the values in vector $z _ { i } ^ { u }$ by magnitude. Then, if the rankings obtained for two unlabelled images $\\boldsymbol { x } _ { i } ^ { u }$ and $x _ { j } ^ { u }$ are the same, they are very likely to belong to the same (new) class, so we set $s _ { i j } = 1$ . Otherwise, we set $s _ { i j } = 0$ . In practice, it is too strict to require the two rankings to be identical if the dimension of $z _ { i } ^ { u }$ is high (otherwise we may end up with $s _ { i j } = 0$ for all pairs $( i , j ) , i \\neq j )$ . Therefore, we relax this requirement by only testing if the sets of the top- $k$ ranked dimensions are the same (we use $k = 5$ in our experiments), i.e.: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4b05c857fe331d4f34bdda6ae934513b5d17f86e94efea6b77117f0cb5c8dbe9.jpg",
+        "text": "$$\ns _ { i j } = \\mathbb { 1 } \\left\\{ \\mathrm { t o p } _ { k } ( \\Phi ( \\boldsymbol { x } _ { i } ^ { u } ) ) = \\mathrm { t o p } _ { k } ( \\Phi ( \\boldsymbol { x } _ { j } ^ { u } ) ) \\right\\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "where $\\mathrm { t o p } _ { k } : \\mathbb { R } ^ { d } \\to \\mathcal { P } ( \\{ 1 , . . . , d \\} )$ associates to a vector $z$ the subset of indices $\\{ 1 , \\ldots , d \\}$ of its top- $k$ elements. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Once the labels $s _ { i j }$ have been obtained, we use them as pseudo-labels to train a comparison function for the unlabelled data. In order to do this, we apply a new head $\\eta ^ { u } : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { C ^ { u } }$ to the image representation $z _ { i } ^ { u } = \\Phi ( x _ { i } ^ { u } )$ to extract a new descriptor vector $\\eta ^ { u } ( z _ { i } ^ { u } )$ optimized for the unlabelled data. As in section 2.1, the head is composed of a linear layer followed by a softmax. Then, the inner product $\\eta ^ { u } ( z _ { i } ^ { u } ) ^ { \\top } \\eta ^ { u } ( z _ { j } ^ { u } )$ is used as a score for whether images $\\boldsymbol { x } _ { i } ^ { u }$ and $x _ { j } ^ { u }$ belong to the same class or not. Note that $\\eta ^ { u } ( z _ { i } ^ { u } )$ is a normalized vector due to the softmax layer in $\\eta ^ { u }$ . This descriptor is trained by optimizing the binary cross-entropy (BCE) loss: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a0d145cd140ba369dd17d8a9a7cf464c3ba7da14d0463c157298d0dd0e411da8.jpg",
+        "text": "$$\nL _ { \\mathrm { B C E } } = - \\frac { 1 } { M ^ { 2 } } \\sum _ { i = 1 } ^ { M } \\sum _ { j = 1 } ^ { M } [ s _ { i j } \\log \\eta ^ { u } ( z _ { i } ^ { u } ) ^ { \\top } \\eta ^ { u } ( z _ { j } ^ { u } ) + ( 1 - s _ { i j } ) \\log ( 1 - \\eta ^ { u } ( z _ { i } ^ { u } ) ^ { \\top } \\eta ^ { u } ( z _ { j } ^ { u } ) ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we structure $\\eta ^ { u }$ in a particular manner: We set its output dimension to be equal to the number of new classes $C ^ { u }$ . In this manner, we can use the index of the maximum element of each vector $\\hat { y } _ { i } ^ { u } = \\mathrm { a r g m a x } _ { y } [ \\eta ^ { u } \\circ \\Phi ( x _ { i } ^ { u } ) ] _ { y }$ as prediction $\\hat { y } _ { i } ^ { u }$ for the class of image $x _ { i } ^ { u }$ (as opposed to assigning labels via a clustering method such as $k$ -means). ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 JOINT TRAINING ON LABELLED AND UNLABELLED DATA ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We now have two losses that involve the representation $\\Phi$ : the CE loss $L _ { \\mathrm { C E } }$ for the labelled data $D ^ { l }$ and the pairwise BCE loss $L _ { \\mathrm { B C E } }$ for the unlabelled data $D ^ { u }$ . They both share the same image embedding $\\Phi$ . This embedding can be trained sequentially, first on the labelled data, and then on the unlabelled data using the pseudo-labels obtained above. However, in this way the model will very likely forget the knowledge learned from the labelled data, which is known as catastrophic forgetting in incremental learning (Rebuffi et al., 2017; Lopez-Paz & Ranzato, 2017; Shmelkov et al., 2017; Aljundi et al., 2018). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In contrast, we jointly fine-tune our model using both losses at the same time. Note that most of the model $\\Phi$ is frozen; we only fine-tune the last macro-block of $\\Phi$ together with the two heads $\\eta ^ { u }$ and $\\eta ^ { l }$ . Importantly, as we fine-tune the model, the labels $s _ { i j }$ are changing at every epoch as the embedding $\\eta ^ { l }$ is updated. This in turn affects the rank statistics used to determine the labels $s _ { i j }$ as explained in section 2.2. This leads to a “moving target” phenomenon that can introduce some instability in learning the model. This potential issue is addressed in the next section. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.4 ENFORCING PREDICTIONS TO BE CONSISTENT ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In addition to the CE and BCE losses, we also introduce a consistency regularization term, which is used for both labelled and unlabelled data. In semi-supervised learning (Oliver et al., 2018; Tarvainen & Valpola, 2017; Laine & Aila, 2017), the idea of consistency is that the class predictions on an image $x$ and on a randomly-transformed counterpart $t x$ (for example an image rotation) should be the same. In our case, as will be shown in the experiments, consistency is very important to obtain good performance. One reason is that, as noted above, the pairwise pseudo-labels for the unlabelled data are subject to change on the fly during training. Indeed, for an image $x _ { i } ^ { u }$ and a randomly-transformed counterpart $t x _ { i } ^ { u }$ , if we do not enforce consistency, we can have $\\mathrm { t o p } _ { k } ( \\Phi ( x _ { i } ^ { u } ) ) \\neq \\mathrm { t o p } _ { k } ( \\Phi ( t x _ { i } ^ { u } ) )$ . According to eq. (2) defining $s _ { i j }$ , it could result in different $s _ { i j }$ for $\\boldsymbol { x } _ { i } ^ { u }$ depending on the data augmentation applied to the image. This variability of the ranking labels for a given pair could then confuse the training of the embedding. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Following the common practice in semi-supervised learning, we use the Mean Squared Error (MSE) as the consistency cost. This is given by: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7f0bd37360cbf1f75946effba4a324bb0b908dc07af5f585668e17ff9def67f1.jpg",
+        "text": "$$\nL _ { \\mathrm { M S E } } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } ( \\eta ^ { l } ( z _ { i } ^ { l } ) - \\eta ^ { l } ( \\hat { z } _ { i } ^ { l } ) ) ^ { 2 } + \\frac { 1 } { M } \\sum _ { i = 1 } ^ { M } ( \\eta ^ { u } ( z _ { i } ^ { u } ) - \\eta ^ { u } ( \\hat { z } _ { i } ^ { u } ) ) ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { z }$ is the representation of $t x$ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The overall loss of our model can then be written as ",
+        "bbox": [
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+            672,
+            514,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/163dc34377640117bb8cda76e5761978dcb836ab118b01909f3673b48b888036.jpg",
+        "text": "$$\n\\begin{array} { r } { L = L _ { \\mathrm { C E } } + L _ { \\mathrm { B C E } } + \\omega ( t ) L _ { \\mathrm { M S E } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the coefficient $\\omega ( t )$ is a ramp-up function. This is widely used in semi-supervised learning (Laine & Aila, 2017; Tarvainen & Valpola, 2017). Following (Laine & Aila, 2017; Tarvainen & Valpola, 2017), we use the sigmoid-shaped function $\\omega ( t ) = \\lambda e ^ { - 5 ( 1 - \\frac { t } { T } ) ^ { 2 } }$ , where $t$ is current time step and $T$ is the ramp-up length and $\\lambda \\in \\mathbb { R } _ { + }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.5 INCREMENTAL LEARNING SCHEME ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We also explore a setting analogous to incremental learning. In this approach, after tuning on the labelled set (end of section 2.1), we extend the head $\\eta ^ { l }$ to $C ^ { u }$ new classes, so that $\\eta ^ { l } : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { C ^ { l } + C ^ { u } }$ . The head parameters for the new classes are initialized randomly. The model is then trained using the same loss eq. (5), but the cross-entropy part of the loss is evaluated on both labelled and unlabelled data $D ^ { l }$ and $D ^ { u }$ . Since the cross-entropy requires labels, for the unlabelled data we use the pseudo-labels $\\hat { y } _ { i } ^ { u }$ , which are generated on-the-fly from the head $\\eta ^ { u }$ at each forward pass. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The advantage is that this approach increments $\\eta ^ { l }$ to discriminate both old and new classes, which is often desirable in applications. It also creates a feedback loop that causes the features $z _ { i } ^ { u }$ to be refined, which in turn generates better pseudo-labels $\\hat { y } _ { i } ^ { u }$ for $D ^ { u }$ from the head $\\eta ^ { u }$ . In this manner, further improvements can be obtained by this cycle of positive interactions between the two heads during training. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/bb8142e393fd367556bc79ff7f4ccfa2b6216eaabc57f12afdb260dbdc798e6a.jpg",
+        "table_caption": [
+            "Table 1: Ablation study. “w/o Con.” means without consistency constraints; “w/o CE” means without the cross entropy loss for training on labeled data. “w/o BCE” means without binary cross entropy loss for training on unlabeled data; “w/o S.S.” means without selfsupervision. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>CIFAR-10</td><td>CIFAR-100</td><td>SVHN</td></tr><tr><td>Ours w/o Con</td><td>82.6±12.0%</td><td>61.8±3.6%</td><td>61.3±1.9%</td></tr><tr><td>Ours w/o CE</td><td>84.7±4.4%</td><td>58.4±2.7%</td><td>59.7±6.6%</td></tr><tr><td>Ours w/o BCE</td><td>26.2±2.0%</td><td>6.6±0.7%</td><td>24.5±0.5%</td></tr><tr><td>Ours w/o S.S.</td><td>89.4±1.4%</td><td>67.4±2.0%</td><td>72.9±5.0%</td></tr><tr><td>Ours full</td><td>90.4±0.5%</td><td>73.2±2.1%</td><td>95.0±0.2%</td></tr><tr><td>Ours w/ I.L.</td><td>91.7±0.9%</td><td>75.2±4.2%</td><td>95.2±0.3%</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "3.1 DATA AND EXPERIMENTAL DETAILS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate our models on a variety of standard benchmark datasets: CIFAR-10 (Krizhevsky & Hinton, 2009), CIFAR-100 (Krizhevsky & Hinton, 2009), SVHN (Netzer et al., 2011), OmniGlot (Lake et al., 2015), and ImageNet (Deng et al., 2009). Following Han et al. (2019), we split these to have 5/20/5/654/30 classes respectively in the unlabelled set. In addition, for OmniGlot and ImageNet we use 20 and 3 different splits respectively, as in Han et al. (2019), and report average clustering accuracy. More details on the splits can be found in appendix A. ",
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+    {
+        "type": "text",
+        "text": "Evaluation metrics. We adopt clustering accuracy (ACC) to evaluate the clustering performance of our approach. The ACC is defined as follow: ",
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+    {
+        "type": "equation",
+        "img_path": "images/0cc3b6a64e73aafe17eb0cf8eacfad23a1ff3d82ec63702766024ded05996e2d.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { g \\in \\mathrm { { S y m } } ( L ) } \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\mathbb { 1 } \\left\\{ \\overline { { y } } _ { i } = g \\left( y _ { i } \\right) \\right\\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\overline { { y } } } _ { i }$ and $y _ { i }$ denote the ground-truth label and clustering assignment for each data point $x _ { i } ^ { u } \\in D ^ { u }$ respectively, and $\\mathrm { S y m } ( L )$ is the group permutations of $L$ elements (this discounts the fact that the cluster indices may not be in the same order as the ground-truth labels). Permutations are optimized using the Hungarian algorithm (Kuhn, 1955). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Implementation details. We use the ResNet-18 (He et al., 2016) architecture, except for OmniGlot for which we use a VGG-like network (Simonyan & Zisserman, 2015) with six layers to make our setting directly comparable to prior work. We use SGD with momentum (Sutskever et al., 2013) as optimizer for all but the OmniGlot dataset, for which we use Adam (Kingma & Ba, 2014). For all experiments we use a batch size of 128 and $k = 5$ which we found worked consistently well across datasets (see appendix D). More details about the hyper-parameters can be found in appendix B. ",
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+    {
+        "type": "text",
+        "text": "3.2 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We validate the effectiveness of the components of our method by ablating them and measuring the resulting ACC on the unlabelled data. Note that, since the evaluation is restricted to the unlabelled data, we are solving a clustering problem. The same unlabelled data points are used for both training and testing, except that data augmentation (i.e. image transformations) is not applied when computing the cluster assignments. As can be seen in table 1, all components have a significant effect as removing any of them causes the performance to drop substantially. Among them, the BCE loss is by far the most important one, since removing it results in a dramatic drop of $40 \\mathrm { - } 6 0 \\%$ absolute ACC points. For example, the full method has ACC $9 0 . 4 \\%$ on CIFAR-10, while removing BCE causes the ACC to drop to $2 6 . 2 \\%$ . This shows that that our rank-based embedding comparison can indeed generate reliable pairwise pseudo labels for the BCE loss. Without consistency, cross entropy, or self-supervision, the performance drops by a more modest but still significant $7 . 8 \\%$ , $5 . 7 \\%$ and $1 . 0 \\%$ absolute ACC points, respectively, for CIFAR-10. It means that the consistency term plays a role as important as the cross-entropy term by preventing the “moving target” phenomenon described in section 2.4. Finally, by incorporating the discovered classes in the classification task, we get a further boost of $1 . 3 \\%$ , $2 . 0 \\%$ and $0 . 2 \\%$ points on CIFAR-10, CIFAR-100 and SVHN respectively. ",
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+    {
+        "type": "table",
+        "img_path": "images/dc38cd44861071cf66e0165f39d253f65bd0f17d84734066ffb6145a13f462e2.jpg",
+        "table_caption": [
+            "Table 2: Novel category discovery results on CIFAR-10, CIFAR-100, and SVHN. ACC on the unlabelled set. “w/ S.S.” means with self-supervised learning. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>No</td><td></td><td>CIFAR-10</td><td>CIFAR-100</td><td>SVHN</td></tr><tr><td>(1)</td><td>k-means (MacQueen,1967)</td><td>65.5±0.0 %</td><td>56.6±1.6%</td><td>42.6%±0.0</td></tr><tr><td>(2)</td><td>KCL (Hsu et al., 2018)</td><td>66.5±3.9%</td><td>14.3±1.3%</td><td>21.4%±0.6</td></tr><tr><td>(3)</td><td>MCL (Hsu et al., 2019)</td><td>64.2±0.1%</td><td>21.3±3.4%</td><td>38.6%±10.8</td></tr><tr><td>(4)</td><td>DTC (Han et al., 2019)</td><td>87.5±0.3%</td><td>56.7±1.2%</td><td>60.9%±1.6</td></tr><tr><td>(5)</td><td>k-means (MacQueen,1967) w/ S.S.</td><td>72.5±0.0%</td><td>56.3±1.7%</td><td>46.7±0.0%</td></tr><tr><td>(6)</td><td>KCL (Hsu et al., 2018) w/ S.S.</td><td>72.3±0.2%</td><td>42.1±1.8%</td><td>65.6±4.9%</td></tr><tr><td>(7)</td><td>MCL (Hsu et al.,2019) w/ S.S.</td><td>70.9±0.1%</td><td>21.5±2.3%</td><td>53.1±0.3%</td></tr><tr><td>(8)</td><td>DTC (Han et al., 2019) w/ S.S.</td><td>88.7±0.3%</td><td>67.3±1.2%</td><td>75.7±0.4%</td></tr><tr><td>(9)</td><td>Ours</td><td>90.4±0.5%</td><td>73.2±2.1%</td><td>95.0±0.2%</td></tr><tr><td>(10)</td><td>Ours w/ I.L.</td><td>91.7±0.9%</td><td>75.2±4.2%</td><td>95.2±0.2%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/1f699b05e8a4e57588ae83f600b928bef851c7fef98bdb359b1475d68b65a71c.jpg",
+        "image_caption": [
+            "Figure 2: Evolution of the t-SNE during the training of CIFAR-10. Performed on unlabelled data (i.e., instances of dog, frog, horse, ship, truck). Colors of data points denote their ground-truth labels. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.3 NOVEL CATEGORY DISCOVERY ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We compare our method to baselines and state-of-the-art methods for new class discovery, starting from CIFAR-10, CIFAR-100, and SVHN in table 2. The first baseline (row 5 in table 2) amounts to applying $k$ -means (MacQueen, 1967) to the features extracted by the fine-tuned model (the second step in section 2.1), for which we use the $k$ -means $^ { + + }$ (Arthur & Vassilvitskii, 2007) initialization. The second baseline (row 1 in table 2) is similar, but uses as feature extractor a model trained from scratch using only the labelled images, which corresponds to a standard transfer learning setting. By comparing rows 1, 5 and 9 in table 2, we can see that our method substantially outperforms $k$ -means. Next, we compare with the KCL (Hsu et al., 2018), MCL (Hsu et al., 2019) and DTC (Han et al., 2019) methods. By comparing rows 2–4 to 9, we see that our method outperforms these by a large margin. We also try to improve KCL, MCL and DTC by using the same self-supervised initialization we adopt (section 2.1), which indeed results in an improvement (rows 2–4 vs 6–8). However, their overall performance still lags behind ours by a large margin. For example, our method of section 2.4 achieves $9 5 . 0 \\%$ ACC on SVHN, while “KCL w/ S.S.”, “MCL w/ S.S.” and “DTC w/ S.S.” achieve only $6 5 . 6 \\%$ , $5 3 . 1 \\%$ and $7 5 . 7 \\%$ ACC, respectively. Similar trends hold for CIFAR-10 and CIFAR-100. Finally, the incremental learning scheme of section 2.5 results in further improvements, as can be seen by comparing rows 9 and 10 of table 2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In fig. 2, we show the evolution of the learned representation on the unlabelled data on CIFAR-10 using t-SNE (van der Maaten & Hinton, 2008). As can be seen, while the clusters overlap in the beginning, they become more and more separated as the training progresses, showing that our model can effectively discover novel visual categories without labels and learn meaningful embeddings for them. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We further compare our method to others on two more challenging datasets, OmniGlot and ImageNet, in table 3. For OmniGlot, results are averaged over the 20 alphabets in the evaluation set (see appendix A); for ImageNet, results are averaged over the three 30-class unlabelled sets used in (Hsu et al., 2018; 2019). Since we have a relatively larger number of labelled classes in these two datasets, we follow (Han et al., 2019) and use metric learning on the labelled classes to pre-train the feature extractor, instead of the self-supervised learning. We empirically found that self-supervision does not provide obvious gains for these two datasets. This is reasonable since the data in the labelled sets of these two datasets are rather diverse and abundant, so metric learning can provide good feature initialization as there is less class-specific bias due to the large number of pre-training classes. However, by comparing rows 1 and 5 in table 3, it is clear that metric learning alone is not sufficient for the task of novel category discovery. Our method substantially outperforms the $k$ -means results obtained using the features from metric learning — by $1 1 . 9 \\%$ and $\\mathrm { { \\bar { 1 } 0 . 6 \\% } }$ on OmniGlot and ImageNet respectively. Our method also substantially outperforms the current state-of-the-art, achieving ${ \\bar { 8 9 } } . 1 \\%$ and $8 2 . 5 \\%$ ACC on OmniGlot and ImageNet respectively, compared with $8 9 . 0 \\%$ and $7 8 . 8 \\%$ of (Han et al., 2019), thus setting the new state-of-the-art. ",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>No</td><td></td><td>OmniGlot</td><td>ImageNet</td></tr><tr><td>(1)</td><td>k-means (MacQueen,1967)</td><td>77.2%</td><td>71.9%</td></tr><tr><td>(2)</td><td>KCL (Hsu et al., 2018)</td><td>82.4%</td><td>73.8%</td></tr><tr><td>(3)</td><td>MCL (Hsu et al., 2019)</td><td>83.3%</td><td>74.4%</td></tr><tr><td>(4)</td><td>DTC (Han et al., 2019)</td><td>89.0%</td><td>78.3%</td></tr><tr><td>(5)</td><td>Ours</td><td>89.1%</td><td>82.5%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/1d6429af68e17efa89c35d0bc22c398bc47d02a56a02d662b000d4ae57cbc16f.jpg",
+        "image_caption": [
+            "Table 3: Novel category discovery results on OmniGlot and ImageNet. ACC on the unlabelled set. ",
+            "Figure 3: t-SNE on CIFAR-10: impact of incremental Learning. (a) representation on the labelled and (b) unlabelled CIFAR classes. Colors of data points denote their ground-truth labels. We observe a bigger overlap in (a) between the “old” class 3 and the “new” class 5 when not incorporating Incremental Learning. "
+        ],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.4 INCREMENTAL LEARNING ",
+        "text_level": 1,
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+    },
+    {
+        "type": "table",
+        "img_path": "images/bfc5fe586c300b413162e81d9e3c6c036246abf46be740ca2103e94cebc4488d.jpg",
+        "table_caption": [
+            "Table 4: Incremental Learning with the novel categories. “old” refers to the ACC on the labelled classes while “new” refers to the unlabelled classes in the testing set. “all” indicates the whole testing set. It should be noted that the predictions are not restricted to their respective subset. Standard deviation can be found in appendix C. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">CIFAR-10</td><td colspan=\"3\">CIFAR-100</td><td colspan=\"3\">SVHN</td></tr><tr><td>Classes</td><td>old</td><td>new</td><td>all</td><td>old</td><td>new</td><td>all</td><td>old</td><td>new</td><td>all</td></tr><tr><td>KCL w/ S.S.</td><td>79.4%</td><td>60.1%</td><td>69.8%</td><td>23.4%</td><td>29.4%</td><td>24.6%</td><td>90.3%</td><td>65.0%</td><td>81.0%</td></tr><tr><td>MCL w/ S.S.</td><td>81.4%</td><td>64.8%</td><td>73.1%</td><td>18.2%</td><td>18.0%</td><td>18.2%</td><td>94.0%</td><td>48.6%</td><td>77.2%</td></tr><tr><td>DTC w/ S.S.</td><td>58.7%</td><td>78.6%</td><td>68.7%</td><td>47.6%</td><td>49.1%</td><td>47.9%</td><td>90.5%</td><td>72.8%</td><td>84.0%</td></tr><tr><td>Ours w/ I.L.</td><td>90.6%</td><td>88.8%</td><td>89.7%</td><td>71.2%</td><td>56.8%</td><td>68.3%</td><td>96.3%</td><td>96.1%</td><td>96.2%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Here, we further evaluate our incremental scheme for novel category discovery as described in section 2.5. Methods for novel category discovery such as (Han et al., 2019; Hsu et al., 2019; 2018) focus on obtaining the highest clustering accuracy for the new unlabelled classes, but may forget the existing labelled classes in the process. In practice, forgetting is not desirable as the model should be able to recognize both old and new classes. Thus, we argue that the classification accuracy on the labelled classes should be assessed as well, as for any incremental learning setting. Note however that our setup differs substantially from standard incremental learning (Rebuffi et al., 2017; Lopez-Paz & Ranzato, 2017; Shmelkov et al., 2017; Aljundi et al., 2018) where every class is labelled and the focus is on using limited memory. In our case, we can store and access the original data without memory constraints, but the new classes are unlabelled, which is often encountered in applications. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "By construction (section 2.5), our method learns the new classes on top of the old ones incrementally, out of the box. In order to compare to methods such as KCL, MCL and DTC that do not have this property, we proceed as follows. First, the method runs as usual to cluster the unlabelled portion of the data, thus obtaining pseudo-labels for it, and learning a feature extractor as a byproduct. Then, the feature extractor is used to compute features for both the labelled and unlabelled training data, and a linear classifier is trained using labels and pseudo-labels, jointly on all the classes, old and new. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We report in table 4 the performance of the resulting joint classifier networks on the testing set of each dataset (this is now entirely disjoint from the training set). Our method has similar performances on the old and new classes for CIFAR-10 and SVHN, as might be expected as the split between old and new classes is balanced. In comparison, the feature extractor learned by KCL and MCL works much better for the old classes (e.g., the accuracy discrepancy between old and new classes is $2 5 . 3 \\%$ for KCL on SVHN). Conversely, DTC learns features that work better for the new classes, as shown by the poor performance for the old classes on CIFAR-10. Thus, KCL, MCL and DTC learn representations that are biased to either the old or new classes, resulting overall in suboptimal performance. In contrast, our method works well on both old and new classes; furthermore, it drastically outperforms existing methods on both. ",
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+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our work draws inspiration from semi-supervised learning, transfer learning, clustering, and zero-shot learning. We review below the most relevant contributions. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In semi-supervised learning (SSL) (Chapelle et al., 2006), a partially labelled training dataset is given and the objective is to learn a model that can propagate the labels from the labelled data to unlabelled data. Most SSL methods focus on the classification task where, usually both labelled and unlabelled points belong to the same set of classes. On the contrary, our goal is to handle the case where the unlabelled data classes differ from the labelled data. Oliver et al. (2018) summarizes the state-of-the-art SSL methods. Among them, the consistency-based methods appeared to be the most effective. Rasmus et al. (2015) propose a ladder network which is trained on both labelled and unlabelled data using a reconstruction loss. Laine & Aila (2017) simplifies this ladder network by enforcing prediction consistency between a data point and its augmented counterpart. As an alternative to data augmentation, they also consider a regularization method based on the exponential moving average (EMA) of the predictions. This idea is further improved by Tarvainen & Valpola (2017): instead of using the EMA of predictions, they propose to maintain the EMA of model parameters. The consistency is then measured between the predictions of the current model (student) and the predictions of the EMA model (teacher). More recently (and closer to our work) practitioners have also combined SSL with self-supervision(Rebuffi et al., 2019; Zhai et al., 2019) to leverage dataset with very few annotations. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Transfer learning (Pan & Yang, 2010; Weiss et al., 2016; Tan et al., 2018) is an effective way to reduce the amount of data annotations required to train a model by pre-training the model on an existing dataset. In image classification, for example, it is customary to start from a model pre-trained on the ImageNet (Deng et al., 2009) dataset. In most transfer learning settings, however, both the source data and the target data are fully annotated. In contrast, our goal is to transfer information from a labelled dataset to an unlabelled one. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Many classic (e.g., Aggarwal & Reddy (2013); MacQueen (1967); Comaniciu & Meer (1979); $\\mathrm { N g }$ et al. (2001)) and deep learning (e.g., Xie et al. (2016); Chang et al. (2017); Dizaji et al. (2017); Yang et al. (2017; 2016); Hsu et al. (2018; 2019)) clustering methods have been proposed to automatically partition an unlabelled data collection into different classes. However, this task is usually ill-posed as there are multiple, equally valid criteria to partition most datasets. We address this challenge by learning the appropriate criterion by using a labelled dataset, narrowing down what constitutes a proper class. We call this setting “transfer clustering”. ",
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+    },
+    {
+        "type": "text",
+        "text": "To the best of our knowledge, the work most related to ours are (Hsu et al., 2018; 2019; Han et al., 2019). Han et al. (2019) also consider discovering new classes as a transfer clustering problem. They first learn a data embedding by using metric learning on the labelled data, and then fine-tune the embedding and learn the cluster assignments on the unlabelled data. In (Hsu et al., 2018; 2019), the authors introduce KCL and MCL clustering methods. In both, a similarity prediction network (SPN), also used in (Hsu et al., 2016), is first trained on a labelled dataset. Afterwards, the pre-trained SPN is used to provide binary pseudo labels for training the main model on an unlabelled dataset. The overall pipelines of the two methods are similar, but the losses differ: KCL uses a Kullback-Leibler divergence based contrastive loss equivalent to the BCE used in this paper (eq. (3)), and MCL uses the Meta Classification Likelihood loss. Zero-shot learning (ZSL) (Xian et al., 2018; Fu et al., 2018) can also be used to recognize new classes. However, differently from our work, ZSL also requires additional side information (e.g., class attributes) in addition to the raw images. ",
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+    {
+        "type": "text",
+        "text": "Finally, other works (Dean et al., 2013; Yagnik et al., 2011) discuss the application of rank statistics to measuring the similarity of vectors; however, to the best of our knowledge, we are the first to apply rank statistics to the task of novel category discovery using deep neural networks. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we have looked at the problem of discovering new classes in an image collection, leveraging labels available for other, known classes. We have shown that this task can be addressed very successfully by a few new ideas. First, the use of self-supervised learning for bootstrapping the image representation trades off the representation quality with its generality, and for our problem this leads to a better solution overall. Second, we have shown that rank statistics are an effective method to compare noisy image descriptors, resulting in robust data clustering. Third, we have shown that jointly optimizing both labelled recognition and unlabelled clustering in an incremental learning setup can reinforce the two tasks while avoiding forgetting. On standard benchmarks, the combination of these ideas results in much better performance than existing methods that solve the same task. Finally, for larger datasets with more classes and diverse data (e.g., ImageNet) we note that self-supervision can be bypassed as the pretraining on labelled data already provides a powerful enough representation. In such cases, we still show that the rank statistics for clustering gives drastic improvement over existing methods. ",
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+    {
+        "type": "text",
+        "text": "",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "6 ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This work is supported by the EPSRC Programme Grant Seebibyte EP/M013774/1, Mathworks/DTA DFR02620, and ERC IDIU-638009. ",
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+    {
+        "type": "text",
+        "text": "A DATASET SPLITS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For CIFAR-10 and SVHN we keep the labels of the five first categories (namely airplane, automobile, bird, cat, deer for CIFAR-10, 0–4 for SVHN) and keep the rest of the data as the unlabelled set. For CIFAR-100 we use the first 80 categories as labelled data while the rest are used for the unlabelled set. Following Hsu et al. (2018; 2019), for OmniGlot, each of the 20 alphabets in evaluation set (with 20–47 categories, 659 characters/class) is used as unlabelled data, and all the 30 alphabets in background set are used as labelled set (964 characters/class). For ImageNet, we follow Hsu et al. (2018; 2019) to use the 882/118 classes split proposed in Vinyals et al. (2016), and use the three 30-class subsets sampled from the 118 classes as unlabelled sets. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "B IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+            265,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In the first self-supervised training step, otherwise mentioned, we trained our model with the pretext task of rotation predictions (i.e., a four-class classification: $0 ^ { \\circ }$ , $9 0 °$ , $1 8 0 ^ { \\circ }$ , and $2 7 0 ^ { \\circ }$ ) for 200 epochs and a step-wise decaying learning rate starting from 0.1 and divided by 5 at epochs 60, 120, and 160. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In the second step of our framework (i.e., supervised training using labelled data), we fine-tuned our model on the labelled set for 100 epochs and a step-wise decaying learning rate starting from 0.1 and halved every 10 epochs. From this step onward we fix the first three convolutional blocks of the model, and fine-tuned the last convolutional block together with the linear classifier. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Finally, in the last joint training step, we fine-tuned our model for 200/100/90 epochs for {CIFAR-10, CIFAR-100, SVHN}/OmniGlot/ImageNet, which was randomly sampled from the merged set of both labelled and unlabelled data. The initial learning rate was set to 0.1 for all datasets, and was decayed with a factor of 10 at the $1 7 0 \\mathrm { t h } / \\{ 3 0 \\mathrm { t h } , 6 0 \\mathrm { t h } \\}$ epoch for {CIFAR-10, CIFAR-100, SVHN}/ImageNet. The learning rate of 0.01 was kept fixed for OmniGlot. For the consistency regularization term, we used the ramp-up function as described in section 2.4 with $\\lambda = \\{ 5 . 0 , 5 0 . { \\dot { 0 } } , 5 { \\bar { 0 } } . 0 , 1 0 0 . 0 , 1 0 . 0 \\}$ , and $T = \\{ 5 0 , 1 5 \\bar { 0 } , 8 \\bar { 0 } , 1 , 5 0 \\}$ for CIFAR-10, CIFAR-100, SVHN, OmniGlot, and ImageNet respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In the incremental learning setting, all previous hyper parameters remain the same for our method. We only add a ramp-up on the cross entropy loss on unlabelled data. The ramp-up length is the same as the one used for eq. (4) and we use for all experiments a coefficient of 0.05. For all other methods we trained the classifier for 150 epochs with SGD with momentum and learning rate of 0.1 divided by 10 at epoch 50. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We implemented our method using PyTorch 1.1.0 and ran experiments on NVIDIA Tesla M40 GPUs. Following (Han et al., 2019), our results were averaged over 10 runs for all datasets, except for ImageNet which was averaged over the three 30-class subsets. In general, we found the results were stable. Our code is publicly available at http://www.robots.ox.ac.uk/˜vgg/ research/auto_novel. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C STANDARD DEVIATION OF INCREMENTAL LEARNING EXPERIMENT IN TABLE 4 ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/f600cdbdc66eed6df162d53340906c1f18a0ebc7deb3dacd006003208cb3495c.jpg",
+        "table_caption": [
+            "Table 5: Incremental Learning with the novel categories. “old” refers to the standard deviation ACC on the labelled classes while “new” refers to the unlabelled classes in the testing set. “all” indicates the whole testing set. It should be noted that the predictions are not restricted to their respective subset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">CIFAR-10</td><td colspan=\"3\">CIFAR-100</td><td colspan=\"3\">SVHN</td></tr><tr><td>Classes</td><td>old</td><td>new</td><td>all</td><td>old</td><td>new</td><td>all</td><td>old</td><td>new</td><td>all</td></tr><tr><td>KCL w/ S.S.</td><td>0.6%</td><td>0.6%</td><td>0.1%</td><td>0.3%</td><td>0.3%</td><td>0.2%</td><td>0.3%</td><td>0.5%</td><td>0.1%</td></tr><tr><td>MCL w/ S.S.</td><td>0.4%</td><td>0.4%</td><td>0.1%</td><td>0.3%</td><td>0.1%</td><td>0.2%</td><td>0.2%</td><td>0.3%</td><td>0.1%</td></tr><tr><td>DTC w/ S.S.</td><td>0.6%</td><td>0.2%</td><td>0.3%</td><td>0.2%</td><td>0.2%</td><td>0.2%</td><td>0.3%</td><td>0.2%</td><td>0.1%</td></tr><tr><td>Ours w/ I.L.</td><td>0.2%</td><td>0.2%</td><td>0.1%</td><td>0.1%</td><td>0.3%</td><td>0.1%</td><td>0.1%</td><td>0.0%</td><td>0.1%</td></tr></table>",
+        "bbox": [
+            222,
+            796,
+            774,
+            901
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/711bdf42af75f07613e546052e4dc2926f4fe7c7f19bbb369d5b9b25a50a12e8.jpg",
+        "image_caption": [
+            "Performance of our method w.r.t $k$ ",
+            "Figure 4: Performance evolution with respect to $k$ . We report results for $k \\_ =$ $\\{ 1 , 2 , 3 , 5 , 7 , 1 0 , 1 5 , 2 0 , 5 0 \\}$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            333,
+            112,
+            660,
+            296
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D IMPACT OF $k$ OVER RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            337,
+            444,
+            353
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We provide an additional study of the evolution of performances of our method with respect to $k$ . We results on SVHN/CIFAR10/CIFAR100 in fig. 4. We found that $k = \\{ 5 , 7 \\}$ gave the best results overall. We also found that for all values of $k$ except 1 results were in general stable. ",
+        "bbox": [
+            174,
+            362,
+            826,
+            404
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "E RESULTS WITH AN UNKNOWN NUMBER OF CLASSES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            419,
+            640,
+            434
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "While in our work we assume the number of new classes $C ^ { u }$ to be known a priori, this hypothesis can be restrictive in practice. Instead, one can estimate the number of classes in $D ^ { u }$ using recent methods such as DTC (Han et al., 2019). In table 6 we compare ACC of KCL (Hsu et al., 2018), KCL (Hsu et al., 2019), DTC(Han et al., 2019) and our method on unlabelled splits of OmniGlot and ImageNet datasets with $C ^ { u }$ computed from DTC. We note that our method again reaches the state-of-the-art on ImageNet and is on par with the state-of-the-art on OmniGlot. ",
+        "bbox": [
+            173,
+            443,
+            826,
+            526
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/9d6065aebd7a8260c4b0103a8ded71b84eb6921d673616fcee27aeba396a767a.jpg",
+        "table_caption": [
+            "Table 6: Novel category discovery results with unknown $C ^ { u }$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>No</td><td>OmniGlot</td><td>ImageNet</td></tr><tr><td>(1)</td><td>KCL (Hsu et al., 2018) 80.3%</td><td>71.4%</td></tr><tr><td>(2) MCL (Hsu et al., 2019)</td><td>80.5%</td><td>72.9%</td></tr><tr><td>(3) DTC (Han et al., 2019)</td><td>87.0%</td><td>77.6%</td></tr><tr><td>(4) Ours</td><td>85.4%</td><td>80.5%</td></tr></table>",
+        "bbox": [
+            320,
+            550,
+            676,
+            633
+        ],
+        "page_idx": 12
+    }
+]

parse/train/G8A_Nl0yim6/G8A_Nl0yim6.md

@@ -0,0 +1,421 @@
+# Bridging Offline Reinforcement Learning and Imitation Learning: A Tale of Pessimism
+
+Paria Rashidinejad Department of EECS UC Berkeley Berkeley, CA, 94709 paria.rashidinejad@berkeley.edu
+
+Banghua Zhu Department of EECS UC Berkeley Berkeley, CA, 94709 banghua@berkeley.edu
+
+# Cong Ma
+
+# Stuart Russell
+
+Jiantao Jiao   
+Department of EECS   
+UC Berkeley   
+Berkeley, CA, 94709   
+jiantao@berkeley.edu
+
+Department of Statistics University of Chicago Chicago, IL, 60637 congm@uchicago.edu
+
+Department of EECS   
+UC Berkeley   
+Berkeley, CA, 94709   
+russell@berkeley.edu
+
+# Abstract
+
+Offline (or batch) reinforcement learning (RL) algorithms seek to learn an optimal policy from a fixed dataset without active data collection. Based on the composition of the offline dataset, two main methods are used: imitation learning which is suitable for expert datasets, and vanilla offline RL which often requires uniform coverage datasets. From a practical standpoint, datasets often deviate from these two extremes and the exact data composition is usually unknown. To bridge this gap, we present a new offline RL framework that smoothly interpolates between the two extremes of data composition, hence unifying imitation learning and vanilla offline RL. The new framework is centered around a weak version of the concentrability coefficient that measures the deviation of the behavior policy from the expert policy alone. Under this new framework, we ask: can one develop an algorithm that achieves a minimax optimal rate adaptive to unknown data composition? To address this question, we consider a lower confidence bound (LCB) algorithm developed based on pessimism in the face of uncertainty in offline RL. We study finite-sample properties of LCB as well as information-theoretic limits in multi-armed bandits, contextual bandits, and Markov decision processes (MDPs). Our analysis reveals surprising facts about optimality rates. In particular, in both contextual bandits and RL, LCB achieves a faster rate of $1 / N$ for nearlyexpert datasets compared to the usual rate of $1 / \sqrt { N }$ in offline RL, where $N$ is the batch dataset sample size. In contextual bandits, we prove that LCB is adaptively optimal for the entire data composition range, achieving a smooth transition from imitation learning to offline RL. We further show that LCB is almost adaptively optimal in tabular MDPs.
+
+# 1 Introduction
+
+Reinforcement learning (RL) algorithms have recently achieved tremendous empirical success including beating Go champions [45, 46] and surpassing professionals in Atari games [30, 31], to name a few. Most success stories, however, are in the realm of online RL in which active data collection is necessary. This online paradigm falls short of leveraging previously-collected datasets and dealing with scenarios where online exploration is not possible [10]. To tackle these issues, offline (or batch) reinforcement learning [24, 26] arises in which the agent aims at achieving competence by exploiting a batch dataset without access to online exploration. This paradigm is useful in a diverse array of application domains such as healthcare [51, 15, 35], autonomous driving [57, 4, 36], and recommendation systems [47, 12, 49].
+
+![](images/0e84d6afbae4309843aaee872e27706043bb985418e4fa772cde827bd44ca0ee.jpg)  
+Figure 1: Dataset composition range for offline RL problems.
+
+The key component of offline RL is a pre-collected dataset from an unknown stochastic environment. Broadly speaking, there exist two types of data composition for which offline RL algorithms have shown promising empirical and theoretical success; see Figure 1 for an illustration.
+
+Expert data. One end of the spectrum includes datasets collected by following an expert policy. For such datasets, imitation learning algorithms (e.g., behavior cloning [40]) are shown to be effective in achieving a small sub-optimality w.r.t. the expert policy. Recently, Rajaraman et al. [39] showed that the behavior cloning algorithm achieves the minimal sub-optimality of $1 / N$ in episodic Markov decision processes (MDPs), where $N$ is the sample size in the expert dataset.
+
+Uniform coverage data. On the other end of the spectrum lies the datasets with uniform coverage, which aim to cover all states and actions, even the states never visited or actions never taken by satisfactory policies. Most vanilla offline RL algorithms are only suited in this region and are shown to diverge—both empirically [11, 22] and theoretically [2, 7]—for narrower datasets [10, 20], such as those collected via human demonstrations or hand-crafted policies. In this regime, a widely-adopted requirement is the bounded uniform concentrability coefficient which assumes that the ratio of the state-action occupancy density of any policy and the data distribution is bounded uniformly over all states and actions [32, 9, 6, 52]. Another common assumption is uniformly lower bounded data distribution on all states and actions [43, 1], which ensures all states and actions are visited with√ sufficient probabilities. Algorithms developed for this regime are demonstrated to achieve a $1 / \sqrt { N }$ sub-optimality competing with the optimal policy [53, 16, 50].
+
+# 1.1 Motivating questions
+
+Both of these two ends impose strong assumptions on the dataset: at one extreme, we hope for a solely expert-driven dataset; at the other extreme, we require the dataset to cover every, even sub-optimal, actions. In practice, there are numerous scenarios where the dataset deviates from these two extremes, which has motivated new offline RL benchmark datasets with different data compositions [10, 20]. With this need in mind, the first and foremost question is regarding offline RL formulations:
+
+Question 1 (Formulation) Can we propose an offline RL framework that accommodates the entire data composition range?
+
+We answer this question affirmatively by proposing a new formulation for offline RL that smoothly interpolates between two regimes: expert data and data with uniform coverage. More specifically, we characterize the data composition in terms of the ratio between the state-action occupancy density of an optimal policy $d ^ { \star } ( s , a ) ^ { 1 }$ and that of the data distribution $\mu ( s , a )$ , i.e., we define $C ^ { \star }$ to be the smallest constant that satisfies $d ^ { \star } ( s , a ) / \mu ( s , a ) \leq C ^ { \star }$ for all $s \in S$ and $a \in { \mathcal { A } }$ ; see Definition 1 for a precise characterization.
+
+In words, $C ^ { \star }$ can be viewed as a measure of the deviation between the data distribution and the distribution induced by the optimal policy. $C ^ { \star } = 1$ describes the expert datasets as by definition, the behavior policy is identical to the optimal policy. In contrast, when $C ^ { \star } > 1$ , the dataset is no longer purely expert-driven: it could contain “spurious” samples—states and actions that are not visited by the optimal policy. As another example, when the data distribution is lower bounded by $\mu _ { \mathrm { { m i n } } }$ over all states and actions, C? is upper bounded by µ−1min.
+
+Assuming a finite $C ^ { \star }$ is the weakest concentrability requirement [42, 13, 52] that is currently enjoyed only by some online algorithms such as CPI [18]. $C ^ { \star }$ imposes a much weaker assumption in contrast to other concentrability requirements which involve taking a maximum over all policies; see [42] for a hierarchy of different concentrability definitions. We would like to immediately point out that existing works on offline RL either do not specify the dependency of sub-optimality on data coverage [17, 55], or do not have a batch data coverage assumption that accommodates the entire data spectrum [54, 19]. For instance, Yin et al. [54] requires a uniformly lower bounded data distribution that traces an optimal policy, which implies that optimal actions should be included in states not visited by the optimal policy. Furthermore, this characterization of data coverage does not recover imitation learning: even if the behavior policy is exactly equal to the optimal policy, data distribution lower bound can be arbitrarily small. Further discussion of related work is presented in Appendix A.
+
+With this formulation in mind, a natural next step is designing offline RL algorithms that can handle various data compositions, i.e., for all $C ^ { \star } \geq 1$ . Recently, efforts have been made toward reducing the offline dataset requirements based on a shared intuition: the agent should act conservatively and avoid states and actions less covered in the offline dataset. Based on this intuition, a variety of model-based [55, 19, 56] and model-free [22, 33, 11, 34, 25, 37, 44, 14, 28, 23, 3] offline RL algorithms are proposed that achieve promising empirical results. However, it is observed empirically that existing model-free methods perform better when the dataset is nearly expert-driven whereas existing model-based methods perform better when the dataset is randomly-collected [55, 5, 56].
+
+It remains unclear whether a single algorithm exists that performs well regardless of data composition— an important challenge from a practical perspective [21, 10, 20]. More importantly, the knowledge of the dataset composition may not be available a priori to assist in selecting the right algorithm. In practice, imitation learning often succeeds with very few samples in contrast to offline RL [41]. Unifying offline RL and imitation learning via a single algorithm is thus beneficial, as it can result in tremendous sample savings, in case the dataset has good coverage on an expert policy. This motivates the second question on the algorithm design:
+
+Question 2 (Adaptive algorithm design) Can we design algorithms that can achieve minimal suboptimality when facing different dataset compositions (i.e., different $C ^ { \star }$ )? Furthermore, can this be achieved in an adaptive manner, i.e., without knowing $C ^ { \star }$ beforehand?
+
+To answer the second question, we analyze a pessimistic variant of a value-based method in which we first form a lower confidence bound (LCB) for the value function of a policy using the batch data and then seek to find a policy that maximizes the LCB. The idea of pessimism has appeared in the literature of risk minimization [48, 8]. A similar algorithm design has recently appeared in [17]. It turns out that such a simple algorithm—fully agnostic to the data composition—achieves almost optimal performance in multi-armed bandits and MDPs, and optimally solves the offline learning problem in contextual bandits.
+
+Results summary. Figure 2 summarizes our theoretical findings. For multi-armed bandits, we prove that LCB achieves a $\sqrt { C ^ { \star } / N }$ sub-optimality for any $C ^ { \star } \geq 1$ . Yet, we prove lower bounds showing that LCB cannot be adaptively optimal for any $C ^ { \star } \geq 1$ if the knowledge of $C ^ { \star }$ is not available. For contextual bandits with at least two contexts, we prove that LCB enjoys a rate of $\sqrt { ( C ^ { \star } - 1 ) / N } + 1 / N$ , which translates to a fast rate of $1 / N$ when $C ^ { \star } \approx 1$ akin to the performance of behavioral cloning and smoothly transitions from $1 / N$ to $1 / \sqrt { N }$ as $C ^ { \star }$ increases. This rate matches the information theoretic limit, showing adaptive optimality of LCB in contextual bandits. Establishing the $C ^ { \star } - 1$ dependency requires a novel analysis based on a careful policy sub-optimality decomposition and directly analyzing the probability of taking wrong actions. For MDPs, we similarly show that LCB achieves a fast rate of $1 / N$ for $C ^ { \star } \approx 1$ and a rate of $\sqrt { C ^ { \star } / N }$ for larger values of $C ^ { \star }$ . We conjecture that LCB upper bound also has the form $\sqrt { ( C ^ { \star } - 1 ) / N }$ . We verify this conjecture in a simple example and show that establishing the $C ^ { \star } - 1$ dependency in MDPs requires a delicate analysis that accounts for the value gap between optimal and sub-optimal actions.
+
+# 2 Background and problem formulation
+
+Notation. The probability simplex over set $\mathcal { X }$ is denoted by $\Delta ( \mathcal { X } )$ . We write $x \lesssim y$ when there exists $c > 0$ such that $x \leq c y$ and write $x \asymp y$ if $c _ { 1 } , c _ { 2 } > 0$ exist such that $c _ { 1 } | x | \leq | y | \leq c _ { 2 } | x |$ . We write $x \vee y$ to denote the supremum of $x$ and $y$ . We write $f ( x ) = O ( g ( x ) )$ if $M > 0 , x _ { 0 }$ exist such
+
+![](images/be61fed8429c390344e44d284f508a228af194dd325bb9b48ce9e9fcd7a6d4e1.jpg)  
+Figure 2: The sub-optimality upper bounds and information-theoretic lower bounds for the LCB-based algorithms. In all setting, $C ^ { \star }$ is unknown to the algorithm.
+
+that $| f ( x ) | \leq M g ( x )$ for all $x \geq x _ { 0 }$ . We use ${ \widetilde { O } } ( \cdot )$ to be the big- $O$ notation ignoring logarithmic factors. We write $f ( x ) = \Omega ( g ( x ) )$ if $M > 0 , x _ { 0 }$ eexist such that $| { \bar { f } } ( x ) | \geq M g ( { \bar { x } } )$ for all $x \geq x _ { 0 }$ .
+
+# 2.1 Markov decision processes
+
+An infinite-horizon discounted MDP is described by a tuple $M = ( \mathcal { S } , \mathcal { A } , P , R , \rho , \gamma )$ , where $s$ is a finite state space with $S = | S |$ , $\mathcal { A }$ is a finite action space, $P : \mathcal { S } \times \mathcal { A } \mapsto \Delta ( \mathcal { S } )$ is a transition matrix, $R : \mathcal { S } \times \mathcal { A } \stackrel { - } { \mapsto } \Delta ( [ 0 , 1 ] )$ encodes a family of reward distributions with $r : \mathcal { S \times A } \mapsto [ 0 , 1 ]$ as the expected reward function, $\rho : { \mathcal { S } } \mapsto \Delta ( { \mathcal { S } } )$ is the initial distribution, and $\gamma \in [ 0 , 1 )$ is a discount factor.
+
+A stationary deterministic policy $\pi : { \mathcal { S } } \mapsto A$ is a function that maps a state to an action. Correspondingly, the (normalized) state-action discounted occupancy measure $\begin{array} { r } { d ^ { \bar { \boldsymbol { \pi } } } ( \boldsymbol { s } , \boldsymbol { a } ) : = ( 1 - \gamma ) \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \mathbb { P } _ { t } ( s _ { t } = \boldsymbol { s } , \boldsymbol { a } _ { t } = \boldsymbol { a } ; \boldsymbol { \pi } ) } \end{array}$ for all $s \in S , a \in A$ $d ^ { \pi } : S \times \mathcal { A } \mapsto [ 0 , 1 ]$ , where $\mathbb { P } _ { t } ( s _ { t } = s , a _ { t } = a ; \pi )$ is defined as denotes the probability of $s _ { t } = s , a _ { t } = a$ after executing policy $\pi$ and starting from $s _ { 0 } \sim \rho ( \cdot )$ .
+
+The value function $V ^ { \pi } : \mathcal { S } \mapsto \mathbb { R }$ of the policy $\pi$ is defined as $\begin{array} { r } { V ^ { \pi } ( s ) : = \mathbb { E } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r _ { t } \mid s _ { 0 } = s , a _ { t } = } \end{array}$ $\pi ( s _ { t } )$ for all $t \geq 0 ]$ for $s \in S$ . The quality function (or Q-function) $Q ^ { \pi } : S \times \mathcal { A }  \mathbb { R }$ of policy $\pi$ is defined analogously $\begin{array} { r } { Q ^ { \pi } ( s , a ) : = \bar { \mathbb { E } } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r _ { t } \ | \ s _ { 0 } = s , a _ { 0 } = a , a _ { t } = \pi ( s _ { t } ) } \end{array}$ for all $t \geq 1 ]$ ]. It is convenient to define a scalar summary of the performance of a policy $\pi$ as $J ( \pi ) : = \mathbb { E } _ { s \sim \rho } [ V ^ { \pi } { \bar { ( s ) } } ]$ . It is well known [38] that a stationary deterministic policy $\pi ^ { \star }$ exists that simultaneously maximizes $V ^ { \pi } ( s )$ for all $s \in S$ , and hence maximizing the expected value $J ( \pi )$ . We use shorthands $V ^ { \star } : = V ^ { \pi ^ { \star } }$ and $Q ^ { \star } : = Q ^ { \pi ^ { \star } }$ to denote the optimal value function and Q-function.
+
+# 2.2 Offline data and offline RL
+
+The current paper focuses on offline RL, where the agent cannot interact with the MDP and instead is given a batch dataset $\mathcal { D }$ consisting of tuples $( s , a , r , s ^ { \prime } )$ , where $r \sim R ( \cdot \mid s , a )$ and $s ^ { \prime } \sim P ( \cdot \mid s , a )$ . For simplicity, we assume $( s , a )$ pairs are generated i.i.d. according to a data distribution $\mu$ over $\boldsymbol { S } \times \boldsymbol { A }$ , which is unknown to the agent.2 We denote by $N ( s , a ) \geq 0$ the number of times $( s , a )$ is observed in $\mathcal { D }$ and by $N = | \mathcal { D } |$ the number of samples. The goal of offline RL is to find a policy $\hat { \pi }$ —based on $\mathcal { D } -$ so as to minimize the expected sub-optimality with respect to the optimal policy $\pi ^ { \star }$ , i.e., $\mathbb { E } _ { \mathcal { D } } \left[ J ( \pi ^ { \star } ) - J ( \hat { \pi } ) \right]$ , where the expectation is taken with respect to the randomness in the dataset.
+
+# 2.3 Dataset coverage assumption
+
+Definition 1 (Single policy concentrability) Given a policy $\pi$ , define $C ^ { \pi }$ to be the smallest constant that satisfies $d ^ { \pi } ( s , a ) / \mu ( s , a ) \leq C ^ { \pi }$ for all $s \in S$ and $a \in { \mathcal { A } }$ .
+
+In words, $C ^ { \pi }$ characterizes the distribution shift between the occupancy measure induced by $\pi$ and data distribution $\mu$ . For a stationary deterministic optimal policy, $C ^ { \star } : = C ^ { \pi ^ { \star } }$ is the “best” concentrability coefficient definition [42, 13, 2, 52] which is often much smaller than the widely-used uniform concentrability coefficient $C : = \operatorname* { m a x } _ { \pi } C ^ { \pi }$ which takes the maximum over all policies. A small $C ^ { \pi }$ implies that data distribution covers $( s , a )$ pairs visited by policy $\pi$ , whereas a small $C$ requires the coverage of $( s , a )$ visited by all policies. A similar notion of concentratbility coefficient in $\ell _ { 2 }$ norm instead of $\ell _ { \infty }$ norm has appeared in the literature of off-policy evaluation [27, 29].
+
+# 3 A warm-up: LCB in multi-armed bandits
+
+We begin with the simplest example of an MDP, the multi-armed bandit (MAB) model, where $S = 1$ and $\gamma = 0$ . For MABs, the offline dataset simplifies to $\mathcal { D } = \{ ( a _ { i } , r _ { i } ) \} _ { i = 1 } ^ { N }$ , where $a _ { i } \sim \mu ( \cdot )$ and $r _ { i } \ \sim \ R ( a _ { i } )$ . Competing with an optimal arm $a ^ { \star }$ , the data coverage assumption becomes $1 / \mu ( a ^ { \star } ) \leq C ^ { \star }$ . The goal of offline learning in MAB is to select an arm $\hat { a }$ that minimizes the expected sub-optimality $\mathbb { E } _ { \mathcal { D } } [ J ( \pi ^ { \star } ) - J ( \hat { \pi } ) ] = \mathbb { E } _ { \mathcal { D } } [ r ( a ^ { \star } ) - r ( \hat { a } ) ]$ , where $r ( a )$ is the expected reward of arm $a$ .
+
+# 3.1 Why does the best empirical arm fail?
+
+A natural choice for solving the offline learning problem is to select the arm with the highest empirical average reward, i.e., $\hat { a } : = \arg \operatorname* { m a x } _ { a } \hat { r } ( a )$ . Though intuitive, the empirical best arm is quite sensitive to the arms with a small $N ( a )$ : a less-explored sub-optimal arm might have a high empirical mean just by chance (due to large variance) and overwhelm the true optimal arm. The following proposition formalizes this intuition; see Appendix B.1 for a proof.
+
+Proposition 1 (Failure of the best empirical arm) For any $\epsilon < 0 . 3$ , $N \geq 5 0 0 ,$ , there exists a bandit problem with two arms such that for $\hat { a } = \arg \operatorname* { m a x } _ { a } \hat { r } ( a )$ , one has $\mathbb { E } _ { \mathcal { D } } [ r ( a ^ { \star } ) - r ( \hat { a } ) ] \geq \epsilon .$ .
+
+# 3.2 LCB: The benefit of pessimism
+
+Given the failure of best empirical arm, one soon realizes that it is not sensible to put every arm on an equal footing: one should be pessimistic about the true mean reward of the arms pulled less often. Strategically, pessimism can be deployed by first constructing a penalty function $\bar { b } ( a )$ that shrinks as $N ( a )$ increases and then returning ${ \hat { a } } \in \arg \operatorname* { m a x } _ { a } { \hat { r } } ( a ) - b ( a )$ . When $b ( a )$ captures a confidence level about the empirical reward, ${ \hat { r } } ( a ) - b ( a )$ can be viewed as a lower confidence bound (LCB) on the true mean reward $r ( a )$ . Algorithm 1 shows one instance of the LCB, in which the penalty function originates from Hoeffding’s inequality. The following theorem captures the performance of this algorithm in the MAB setting. The proof can be found in Appendix B.2.
+
+Theorem 1 (LCB sub-optimality, MAB) For a MAB, assume that $1 / \mu ( a ^ { \star } ) \leq C ^ { \star }$ for some $C ^ { \star } \geq 1$ Provided that $N \geq 8 C ^ { \star } \log N$ , arm $\hat { a }$ returned by Algorithm $I$ with $\delta = 1 / N$ obeys
+
+$$
+\mathbb { E } _ { \mathcal { D } } [ r ( a ^ { \star } ) - r ( \hat { a } ) ] \lesssim \operatorname* { m i n } \left( 1 , \sqrt { \frac { C ^ { \star } \log ( 2 N | \cal A | ) } { N } } \right) .
+$$
+
+Applying the above guarantee to the failure instance given in Proposition 1, one sees that LCB secures√ a sub-optimality of $\bar { \tilde { O } } ( 1 / \sqrt { N } )$ , which beats the best empirical arm. This is because the LCB approach applies larger penalties to the arms with a small number of samples, which helps to rule them out.
+
+# 3.3 Is LCB optimal for solving offline multi-armed bandits?
+
+Given the performance bound (1), it is natural to ask whether LCB is optimal for solving offline MAB problems. To address this question, we resort to the usual minimax criterion. Define the following MAB family: $\mathsf { M A B } ( C ^ { \star } ) = \{ ( \mu , R ) | 1 / \mu ( a ^ { \star } ) \leq C ^ { \star } \}$ , which includes all possible pairs of $\mu$ and $R$ such that the data coverage assumption $1 / \mu ( a ^ { \star } ) \leq C ^ { \star }$ holds. We define the worst-case risk of any estimator $\hat { a }$ to be $\begin{array} { r } { \operatorname* { s u p } _ { ( \mu , R ) \in \mathsf { M A B } ( C ^ { \star } ) } \mathbb { E } _ { \mathcal { D } } [ r ( a ^ { \star } ) - r ( \hat { a } ) ] } \end{array}$ . An estimator $\hat { a }$ is a measureable function of the dataset $\mathcal { D }$ collected under the MAB instance $\mu$ and $R$ . The following theorem shows that LCB is optimal up to a logarithmic factor when $C ^ { \star } \geq 2$ ; see Appendix B.3 for the proof.
+
+Theorem 2 (Information-theoretic limit, MAB) For $C ^ { \star } \geq 2$ , one has
+
+$$
+\operatorname* { i n f } _ { \hat { a } } \operatorname* { s u p } _ { ( \mu , R ) \in \mathsf { M A B } ( C ^ { \star } ) } \mathbb { E } _ { \mathcal { D } } [ r ( a ^ { \star } ) - r ( \hat { a } ) ] \gtrsim \operatorname* { m i n } \left( 1 , \sqrt { \frac { C ^ { \star } } { N } } \right) .
+$$
+
+For $C ^ { \star } \in ( 1 , 2 )$ , one has
+
+$$
+\operatorname* { i n f } _ { \hat { a } } \operatorname* { s u p } _ { ( \mu , R ) \in \mathrm { M A B } ( C ^ { \star } ) } \mathbb { E } _ { \mathcal { D } } [ r ( a ^ { \star } ) - r ( \hat { a } ) ] \gtrsim \exp \left( - ( 2 - C ^ { \star } ) \log \left( \frac { 2 } { C ^ { \star } - 1 } \right) \cdot N \right) .
+$$
+
+# 3.4 Imitation learning in bandit: the most played arm achieves a better rate
+
+Theorem 2 reveals that when $C ^ { \star } \geq 2$ , the best possible expected sub-optimality is $\sqrt { C ^ { \star } / N }$ , which is achieved by LCB. On the other hand, in the case of $C ^ { \star } \in [ 1 , 2 )$ , which corresponds to $\dot { \mu } ( a ^ { \star } ) > 1 / 2$ , we can simply use imitation learning to improve the rate by picking the most frequently selected arm in the dataset, i.e., $\hat { a } = \mathrm { a r g } \operatorname* { m a x } _ { a } N ( a )$ . The performance guarantee of the most played arm is stated in the following proposition. The proof is deferred to Appendix B.4.
+
+Proposition 2 (Sub-optimality of the most played arm) Assume that $1 / \mu ( a ^ { \star } ) ~ \leq ~ C ^ { \star }$ for some $C ^ { \star } \in [ 1 , 2 )$ . For $\hat { a } = \mathrm { a r g } \operatorname* { m a x } _ { a } N ( a )$ , we have
+
+$$
+\begin{array} { r } { \mathbb { E } _ { \mathcal { D } } [ r ( a ^ { \star } ) - r ( \hat { a } ) ] \leq \exp \left( - N \cdot { \sf K L } \left( \mathrm { B e r n } \left( \frac { 1 } { 2 } \right) \bigg | \Big | \mathrm { B e r n } \left( \frac { 1 } { C ^ { \star } } \right) \right) \right) . } \end{array}
+$$
+
+When $C ^ { \star } \in [ 1 , 2 )$ , the most played arm achieves an exponential rate in √ $N$ , whereas the upper bound for LCB is only $1 / \sqrt { N }$ . On the other hand, the most played arm algorithm completely fails when $C ^ { \star } > 2$ , while LCB secures the rate $1 / \sqrt { N }$ . In terms of $C ^ { \star }$ dependence, the KL divergence above evaluates to $\log ( C ^ { \star } / 2 ) + \log ( 1 / ( C ^ { \star } - 1 ) ) / 2$ . As $C ^ { \star }  1$ , the rate increases to the order of $1 / ( C ^ { \star } - 1 ) ^ { N }$ , matching the lower bound in Theorem 2.
+
+# 3.5 Non-adaptivity of LCB
+
+One may wonder whether LCB can achieve optimal rate under both cases of $C ^ { \star } \in [ 1 , 2 )$ and $C ^ { \star } \geq 2$ . Unfortunately, we show in the following theorem that regardless of the parameter $\delta$ in Algorithm 1, LCB cannot be optimally adaptive in both regimes. The proof is deferred to Appendix B.5.
+
+Theorem 3 (Non-adaptivity of LCB, MAB) Let $C ^ { \star } = 1 . 5$ . There exists a two-armed bandit instance $( \mu _ { 0 } , R _ { 0 } ) \in \mathsf { M A B } ( C ^ { \star } )$ such that Algorithm $^ { l }$ with $L : = \sqrt { \log ( 2 | \mathcal { A } | / \delta ) / 2 }$ satisfies
+
+$$
+\begin{array} { r } { \mathbb { E } _ { \mathcal { D } } [ r ( a ^ { \star } ) - r ( \hat { a } ) ] \gtrsim \operatorname* { m i n } \left( \frac { \sqrt { L } } { N } , \frac { 1 } { \sqrt { N } } \right) \cdot \exp ( - 3 2 L ) . } \end{array}
+$$
+
+On the other hand, when $C ^ { \star } = 6$ , there exists $( \mu _ { 1 } , R _ { 1 } ) \in \mathsf { M A B } ( C ^ { \star } )$ such that
+
+$$
+\begin{array} { r } { \mathbb { E } _ { \mathcal { D } } [ r ( a ^ { \star } ) - r ( \hat { a } ) ] \gtrsim \operatorname* { m i n } \left( 1 , \sqrt { \frac { L } { N } } \right) . } \end{array}
+$$
+
+Intuitively, a larger $L$ means that we put greater weight on penalty instead of empirical average. As $L  \infty$ , the LCB algorithm recovers the most played arm algorithm; while as $L  0$ , the LCB algorithm recovers the best empirical arm algorithm. When $C ^ { \star } \in ( 1 , 2 )$ , to achieve an exponential rate similar to the most played arm (Theorem 2), we need to select $\delta$ such that $L \gtrsim N ^ { \alpha }$ for $\alpha > 0$ . However, under this choice of $L$ , the algorithm fails to achieve $1 / \sqrt { N }$ rate when $C ^ { \star } \geq 6$ , which can be achieved by setting $\delta = 1 / N$ (and thus $L = \log ( 2 | \mathcal { A } | N ) )$ based on Theorem 1. Hence, it is impossible for LCB to achieve optimal rate in both $C ^ { \star } \in ( 1 , 2 )$ and $C ^ { \star } \geq 2$ regimes simultaneously.
+
+# 4 LCB in contextual bandits
+
+We take the analysis one step further by studying offline learning in contextual bandits (CBs). CB is a special case of MDP described in Section 2.1 with $\gamma = 0$ . In CB setting, the batch dataset is $\mathcal { D } \overset { \cdot } { = } \{ ( s _ { i } , a _ { i } , r _ { i } ) \} _ { i = 1 } ^ { N }$ and the coverage assumption simplifies to $\smash { \operatorname* { m a x } _ { s } \rho ( s ) \bar { / } \mu ( s , \pi ^ { \star } ( s ) ) } \le C ^ { \star }$ . The offline learning objective in CB is to find a policy $\hat { \pi }$ based on $\mathcal { D }$ that minimizes the expected sub-optimality $\mathbb { E } _ { \mathcal { D } } [ \boldsymbol { J } ( \pi ^ { \star } ) - \boldsymbol { J } ( \hat { \pi } ) ] = \mathbb { E } _ { \mathcal { D } , \rho } [ r ( s , \bar { \pi } ^ { \star } ( s ) ) - r ( s , \hat { \pi } ( s ) ) ]$ .
+
+# 4.1 LCB algorithm and its performance guarantee
+
+The pessimism principle introduced for MAB can be naturally extended to CB by subtracting a penalty function $b ( s , a )$ from the empirical rewards ${ \hat { r } } ( s , a )$ and returning $\hat { \pi } ( s ) \in \arg \operatorname* { m a x } _ { a } \hat { r } ( s , a ) - \bar { b } ( s , \bar { a } )$ for every state $s$ . The following theorem establishes an upper bound on the expected sub-optimality of the policy returned by Algorithm 1; see Appendix C.1 for a complete proof.
+
+# Algorithm 1 LCB for bandits and contextual bandits
+
+1: Inputs: Batch dataset $\mathcal { D } = \{ ( s _ { i } , a _ { i } , r _ { i } ) \} _ { i = 1 } ^ { N }$ , and confidence level $\delta$   
+2: for $s \in \mathcal S , a \in \mathcal A$ do   
+3: if $N ( s , a ) = 0$ then Set ${ \hat { r } } ( s , a ) \gets 0$ .   
+4: else Set $\begin{array} { r } { \hat { r } ( s , a ) \longleftrightarrow \frac { 1 } { N ( s , a ) } \sum _ { i = 1 } ^ { N } r _ { i } \mathbb { 1 } \{ ( s _ { i } , a _ { i } ) = ( s , a ) \} . } \end{array}$ .   
+5: Compute the penalty $\begin{array} { r } { b ( s , a ) = \sqrt { \frac { 2 0 0 0 \log ( 2 S | \mathcal { A } | / \delta ) } { N ( s , a ) \vee 1 } } } \end{array}$ .   
+6: Return: $\hat { \pi } ( s ) \in \arg \operatorname* { m a x } _ { a } \hat { r } ( s , a ) - b ( s , a )$ for each $s \in S$ .
+
+Theorem 4 (LCB sub-optimality, CB) For a $C B$ with $S \geq 2$ , assume $\begin{array} { r } { \operatorname* { m a x } _ { s } \rho ( s ) / \mu ( s , \pi ^ { \star } ( s ) ) \ \leq } \end{array}$ $C ^ { \star }$ , for some $C ^ { \star } \geq 1$ . The policy $\hat { \pi }$ returned by Algorithm $I$ with $\delta = 1 / N$ obeys
+
+$$
+\begin{array} { r } { \mathbb { E } _ { \mathcal { D } } [ J ( \pi ^ { \star } ) - J ( \hat { \pi } ) ] \lesssim \operatorname* { m i n } \left( 1 , \widetilde { O } \left( \sqrt { \frac { S ( C ^ { \star } - 1 ) } { N } } + \frac { S } { N } \right) \right) . } \end{array}
+$$
+
+The sub-optimality in Theorem 4 consists of two terms. The first term has the usual statistical estimation rate of $1 / \sqrt { N }$ . The second term is due to missing mass, which captures the suboptimality incurred in states for which an optimal arm is never observed in the dataset. Importantly, the dependency of the first term on data composition is $C ^ { \star } - 1$ . When $C ^ { \star }$ is close to one, LCB enjoys a faster rate of $1 / N$ , reminiscent of the behavioral cloning rate. Furthermore, the convergence rate smoothly transitions from $1 / N$ to $1 / \sqrt { N }$ as $C ^ { \star }$ increases.
+
+# 4.2 Optimality of LCB for solving offline contextual bandits
+
+We now establish an information-theoretic lower bound for the contextual bandit setup described above. Define the following family of CB problems ${ \mathsf { C B } } ( C ^ { \star } ) \ : = \ \{ ( \rho , \mu , { \bar { R } } ) \} \ |$ $\smash  \operatorname* { m a x } _ { s } \rho ( s ) / \mu ( s , \pi ^ { \star } ( s ) ) \ \leq \ C ^ { \star } \}$ . Let ${ \hat { \pi } } : S \mapsto A$ be an arbitrary estimator of the best arm $\pi ( s )$ for any state $s$ , which is a measurable function of the data. For the worst-case risk of $\hat { \pi }$ defined as $\begin{array} { r l } & { \operatorname* { s u p } _ { ( \rho , \mu , R ) \in \mathsf { C B } ( C ^ { \star } ) } \mathbb { E } _ { \mathcal { D } } [ J ( \pi ^ { \star } ) - J ( \hat { \pi } ) ] } \end{array}$ , we have the following minimax lower bound:
+
+Theorem 5 (Information-theoretic limit, CB) Assume that $S \geq 2$ . For any $C ^ { \star } \geq 1$ , one has
+
+$$
+\operatorname* { i n f } _ { \hat { \pi } } \operatorname* { s u p } _ { ( \rho , \mu , R ) \in \mathsf { C B } ( C ^ { \star } ) } \mathbb { E } _ { \mathcal { D } } [ J ( \pi ^ { \star } ) - J ( \hat { \pi } ) ] \gtrsim \operatorname* { m i n } \left( 1 , \sqrt { \frac { S ( C ^ { \star } - 1 ) } { N } } + \frac { S } { N } \right) .
+$$
+
+The proof is provided in Appendix C.2. Comparing with Theorem 4, one sees that LCB enjoys a near-optimal rate in CB with $S \geq 2$ regardless of $C ^ { \star }$ . This is in stark contrast to the MAB case.
+
+On a closer inspection, in the $C ^ { \star } \in [ 1 , 2 )$ regime, there is a clear separation between the informationtheoretic difficulty of offline learning in MAB, which has an exponential rate in $N$ , and CB with at least 2 states, which has a $1 / N$ rate. The reason behind this separation is the missing mass rate when $S \geq 2$ . Informally, when there is only one state, the probability that an optimal action is never observed in the dataset decays exponentially. On the other hand, when there are more than one states, the probability that an optimal action is never observed in at least one state has a $1 / N$ rate.
+
+Assume hypothetically that we know $C ^ { \star } \in ( 1 , 2 )$ . Under this circumstance, one might wonder whether simply picking the most played arm in every state achieves a fast rate, analogous to MAB. Strikingly, the answer is negative as the following proposition shows that the most played arm fails to achieve a vanishing rate when $C ^ { \star } \in ( 1 , 2 )$ . The proof of this theorem is deferred to Appendix C.3.
+
+Proposition 3 (Failure of the most played arm, CB) For any $C ^ { \star } \in ( 1 , 2 )$ , there exists a contextual bandit problem $( \rho , \mu , R ) \in \mathsf { C B } ( C ^ { \star } )$ such that for the policy $\hat { \pi } ( s ) = \arg \operatorname* { m a x } _ { a } N ( s , a )$ ,
+
+$$
+\operatorname* { l i m } _ { N \to \infty } \mathbb { E } _ { \mathcal { D } } [ J ( \pi ^ { \star } ) - J ( \hat { \pi } ) ] \geq C ^ { \star } - 1 .
+$$
+
+# 5 LCB in Markov decision processes
+
+# 5.1 Offline value iteration with LCB
+
+Now we are ready to instantiate the LCB principle to the full-fledged MDP case. Our algorithm design builds upon the classic value iteration algorithm. As we do not have access to the true expected
+
+2: Set 3: Spl 1: Inputs: Batch dataset $\begin{array} { r } { T : = \frac { \log N } { 1 - \gamma } , L : = 2 0 0 0 \log ( 2 ( T + 1 ) S | A | / \delta ) , V _ { \operatorname* { m a x } } = ( 1 - \gamma ) ^ { - 1 } } \end{array}$ $\mathcal { D }$ , discount factor $\gamma$ , and confidence level $\delta$ . .   
+$\mathcal { D }$ $T + 1$ $\mathcal { D } _ { t } = \{ ( s _ { i } , a _ { i } , r _ { i } , s _ { i } ^ { \prime } ) \} _ { i = 1 } ^ { m }$ $t \in \{ 0 , 1 , \ldots , T \}$   
+4: Set $\begin{array} { r } { m _ { t } ( s , a ) : = \sum _ { i = 1 } ^ { m } \mathbb { 1 } \{ ( s _ { i } , \grave { a _ { i } } ) = ( s , a ) \} } \end{array}$ i i=1    based on dataset $\mathcal { D } _ { t }$ for $t \in \{ 0 , 1 , \ldots , T \}$ .   
+5: Initialize $Q _ { 0 } ( s , a ) = 0$ , $V _ { 0 } ( s ) = 0$ and set $\pi _ { 0 } ( s ) = \arg \operatorname* { m a x } _ { a } m _ { 0 } ( s , a )$ , for $a \in { \mathcal { A } }$ and $s \in S$ .   
+6: for $t = 1 , \dots , T$ do   
+7: for $( s , a ) \in ( S , { \mathcal { A } } )$ do   
+8: if $m _ { t } ( s , a ) = 0$ then Set $r _ { t } ( s , a ) = 0$ and $P ^ { t } ( \cdot \mid s , a )$ to be a random probability vector.   
+9: else Set $P ^ { t } ( \cdot \mid s , a )$ to be empirical transitions and $r _ { t } ( s , a )$ be empirical rewards.   
+10: Compute penalty bt(s, a) := Vmax · q $\begin{array} { r } { b _ { t } ( s , a ) : = V _ { \operatorname* { m a x } } \cdot \sqrt { \frac { L } { m _ { t } ( s , a ) \vee 1 } } } \end{array}$ .   
+11: Set $\begin{array} { r } { Q _ { t } ( s , a ) \longleftrightarrow r _ { t } ( s , a ) - b _ { t } ( s , a ) + \overset { \cdot } { \gamma } \sum _ { s ^ { \prime } } P ^ { t } ( s ^ { \prime } \mid s , a ) V _ { t - 1 } ( s ^ { \prime } ) . } \end{array}$   
+12: Compute $V _ { t } ^ { \mathrm { m i d } }  \operatorname* { m a x } _ { a } Q _ { t } ( s , a )$ and $\pi _ { t } ^ { \mathrm { m i d } } ( s ) \in \arg \operatorname* { m a x } _ { a } Q _ { t } ( s , a )$ .   
+13: for $s \in S$ do   
+14: if $V _ { t } ^ { \mathrm { m i d } } ( s ) \leq V _ { t - 1 } ( s )$ then $V _ { t } ( s )  V _ { t - 1 } ( s )$ and $\pi _ { t } ( s )  \pi _ { t - 1 } ( s )$ .   
+15: else $V _ { t } ( s )  V _ { t } ^ { \mathrm { m i d } } ( s )$ and $\pi _ { t } ( s )  \pi _ { t } ^ { \mathrm { m i d } } ( s )$ .   
+16: Return $\hat { \pi } : = \pi _ { T }$ .
+
+rewards $r$ and transitions $P$ , we replace them with the empirical counterparts $\hat { r }$ and $\hat { P }$ . Furthermore, mimicking the LCB algorithmic design for MABs and CBs, we subtract a penalty function $b ( s , a )$ from the $Q$ update as the finishing touch, which yields the value iteration algorithm with LCB:
+
+$$
+\begin{array} { l } { { Q ( s , a ) \longleftrightarrow \hat { r } ( s , a ) - b ( s , a ) + \gamma \displaystyle \sum _ { s ^ { \prime } } \hat { P } ( s ^ { \prime } \mid s , a ) V ( s ^ { \prime } ) , \qquad \mathrm { f o r ~ a l l ~ } ( s , a ) , } } \\ { { \mathrm { } } } \\ { { V ( s ) \longleftrightarrow \displaystyle \operatorname* { m a x } _ { a } Q ( s , a ) , \qquad \mathrm { f o r ~ a l l ~ } s . } } \end{array}
+$$
+
+Algorithm 2 uses the update rule (4) as its key component as well as a few other tricks:
+
+• Data splitting. Instead of using the full dataset $\mathcal { D }$ to form $\hat { r }$ and $\hat { P }$ , Algorithm 2 splits $\mathcal { D }$ and uses different samples in each update (4). This procedure is not needed in practice, however, it alleviates the dependency issues in the analysis, which removes an extra factor of $S$ in the sample complexity. • Monotonic update. Algorithm 2 updates value function $V$ and policy $\pi$ only when the corresponding value estimate is larger than that in the previous iteration. The key benefit of the monotonic update is to shave a $1 / ( 1 - \gamma )$ factor in the sample complexity; see [43] for further discussions.
+
+Now we turn to the performance guarantee of VI-LCB, whose proof is given in Appendix D.6.
+
+Theorem 6 (LCB sub-optimality, MDP) For a MDP, assume that $\begin{array} { r } { \operatorname* { m a x } _ { s , a } d ^ { \star } ( s , a ) / \mu ( s , a ) \leq C ^ { \star } } \end{array}$ . Then, for all $C ^ { \star } \geq 1$ , policy $\hat { \pi }$ returned by Algorithm 2 with $\delta = 1 / N$ achieves
+
+$$
+\begin{array} { r } { \mathbb { E } _ { \mathcal { D } } \left[ J ( \pi ^ { \star } ) - J ( \hat { \pi } ) \right] \lesssim \operatorname* { m i n } \left( \frac { 1 } { 1 - \gamma } , \widetilde { O } \left( \sqrt { \frac { S C ^ { \star } } { ( 1 - \gamma ) ^ { 5 } N } } \right) \right) . } \end{array}
+$$
+
+In addition, $\begin{array} { r } { i f 1 \le C ^ { \star } \le 1 + \frac { L \log ( N ) } { 2 0 0 ( 1 - \gamma ) N } } \end{array}$ , we have a tighter performance upper bound
+
+$$
+\begin{array} { r } { \mathbb { E } _ { \mathcal { D } } \left[ J ( \pi ^ { \star } ) - J ( \hat { \pi } ) \right] \lesssim \operatorname* { m i n } \left( \frac { 1 } { 1 - \gamma } , \widetilde { O } \left( \frac { S } { ( 1 - \gamma ) ^ { 4 } { N } } \right) \right) . } \end{array}
+$$
+
+The upper bound shows that for all $C ^ { \star } \geq 1$ , we can guarantee a rate of $\widetilde { O } ( { \sqrt { S C ^ { \star } / ( ( 1 - \gamma ) ^ { 5 } N ) } } )$ , which is similar to the rate of CB when the $C ^ { \star } = 1 + \Omega ( 1 )$ by taking $\gamma = 0$ . When $C = 1 + \widetilde { O } ( 1 / N )$ , we have a rate $S / ( ( 1 - \gamma ^ { 4 } ) N )$ , which also recovers the result in the CB case. However, in the regime of $C ^ { \star } \in [ 1 + \widetilde \Omega ( 1 / N ) , 1 + O ( 1 ) ]$ , while CB enjoys $\sqrt { S ( C ^ { \star } - 1 ) / N }$ rate, we fail to give the same dependence on $C ^ { \star }$ in MDP; see Section 6 for further discussion on sub-optimality in this regime.
+
+# 5.2 Information-theoretic lower bound for offline RL in MDPs
+
+To capture the statistical limits of offline learning in MDPs, as before we define the following family of instances MDP(C?) := {(ρ, µ, P, R) | maxs,a d?(s,a)µ(s,a) ≤ C?}. We have the following minimax lower bound for offline policy learning in MDPs, with the proof deferred to Appendix D.7.
+
+# Theorem 7 (Information-theoretic limit, MDP) For any $C ^ { \star } \geq 1 , \gamma \geq 0 . 5 ,$ , one has
+
+$$
+\operatorname* { i n f } _ { \hat { \pi } } \operatorname* { s u p } _ { ( \rho , \mu , P , R ) \in \mathsf { M D P } ( C ^ { \star } ) } \mathbb { E } _ { \mathcal { D } } [ J ( \pi ^ { \star } ) - J ( \hat { \pi } ) ] \gtrsim \operatorname* { m i n } \left( \frac { 1 } { 1 - \gamma } , \frac { S } { ( 1 - \gamma ) ^ { 2 } N } + \sqrt { \frac { S ( C ^ { \star } - 1 ) } { ( 1 - \gamma ) ^ { 3 } N } } \right) .
+$$
+
+Imitation learning and offline learning. Similar to the CB lower bound, the statistical limit in Theorem 7 involves two terms. The first term captures the imitation learning regime under which a fast√ rate $1 / N$ is expected, while the second term deals with the large $C ^ { \star }$ regime with a rate $1 / \sqrt { N }$ . More interestingly, the dependence on $C ^ { \star }$ appears to be $C ^ { \star } - 1$ , which is different from the performance upper bound of VI-LCB in Theorem 6. We will comment more on this in the coming section.
+
+Dependence on the effective horizon. Comparing the upper bound in Theorem 6 with the lower bound in Theorem 7, one sees that the sample complexity of VI-LCB is loose by an extra $1 / ( 1 - \gamma ) ^ { 2 }$ factor in sample complexity. We believe that this extra factor can be shaved by replacing the Hoeffding-based penalty to a Bernstein-based one and using variance reduction similar to [43].
+
+# 6 Proof techniques and conjecture
+
+To prove the crude rate of $\sqrt { C ^ { \star } / N }$ for any $C ^ { \star } \geq 1$ in all three MAB, CB, and MDP settings, it is sufficient to bound the sub-optimality by an expectation over penalty followed by the coverage√ √ assumption. Since the penalty is proportional to $1 / \sqrt { N }$ , this technique only yields a $1 / \sqrt { N }$ rate.
+
+The $\sqrt { ( C ^ { \star } - 1 ) / N } + S / N$ rate in CB. We carefully decompose the sub-optimality and characterize the probability of choosing sub-optimal arms to prove the tighter bound in CB, which closes the gap between upper and lower bounds. We achieve this goal by directly analyzing the policy sub-optimality via a gradual decomposition of the sub-optimality of $\hat { \pi }$ as illustrated in Figure 3.
+
+$$
+\begin{array} { r }  \mathbb { E } _ { \mathcal { D } } [ J ( \pi ^ { \star } ) - J ( \hat { \pi } ) ] \begin{array} { l }  \{ \begin{array} { l l } { N ( s , \pi ^ { \star } ( s ) ) = 0  T _ { 1 } } \\ & { } \\ { N ( s , \pi ^ { \star } ( s ) ) \geq 1 } \end{array} \} ^ { \mathrm { { I } } \{ \mathcal { E } ^ { c } \}  T _ { 2 } } \\ { \mathbb { E } _ { \mathcal { I } } \{ \mathcal { E } \} \end{array} \begin{array} { l l } { \{ \begin{array} { l l } { \rho ^ { ( s ) < \frac { 2 C ^ { \star } L } { N }  T _ { 3 } }  T _ { 3 } } \\ { 1 \{ \mathcal { E } \} } \end{array}  } \end{array} } \end{array}
+$$
+
+Figure 3: Decomposition of the sub-optimality of the policy $\hat { \pi }$ returned by Algorithm 1.
+
+In the first level of decomposition, we separate the error based on whether $N ( s , \pi ^ { \star } ( s ) )$ is zero for a certain state $s$ . When ${ \cal N } ( s , \pi ^ { \star } ( s ) ) = 0$ , there is absolutely no basis for the LCB approach to figure out the correct action $\pi ^ { \star } ( s )$ . Fortunately, this type of error, incurred by missing mass, can be bounded by $\begin{array} { r } { T _ { 1 } \lesssim \frac { C ^ { \star } S } { N } } \end{array}$ ·
+
+The second level of decomposition hinges on the following clean/good event: $\mathcal { E } : = \{ \forall s , a : \ | r ( s , a ) - $ ${ \hat { r } } ( s , a ) | \leq b ( s , a ) \}$ . In words, the event $\mathcal { E }$ captures the scenario in which the penalty function provides valid confidence bounds for every state-action pair. Standard concentration arguments tell us that $\mathcal { E }$ takes place with high probability, i.e., the term $T _ { 2 }$ in the figure is no larger than $\delta$ . By setting $\delta$ small, say $1 \bar { / } N$ , we are allowed to concentrate on the case when $\mathcal { E }$ holds.
+
+The third level of decomposition relies on the observation that states with small weights (i.e., $\rho ( s )$ is small) have negligible effects on the sub-optimality $J ( \pi ^ { \star } ) - J ( \hat { \pi } )$ . More specifically, the aggregated contribution focus on the $T _ { 3 }$ from the states with ates with large weig $\begin{array} { r } { \rho ( s ) \lesssim \frac { C ^ { \star } L } { N } } \end{array}$ is upper bounded by rd an immediate con $\begin{array} { r } { T _ { 3 } \lesssim \frac { C ^ { \star } S L } { N } } \end{array}$ . Thilarge ws us toand the $\rho ( s )$ data coverage assumption, that is $\bar { \mu } ( s , \pi ^ { \star } ( s ) ) \geq \rho ( s ) / C ^ { \star } \asymp L / N$ .
+
+Now comes the most important part of the error decomposition, which is not present in the MAB analysis. We decompose the error based on whether the optimal action has a higher data probability $\mu ( s , \pi ^ { \star } ( s ) )$ than the total probability of sub-optimal actions $\begin{array} { r } { \overline { { \mu } } ( s ) : = \sum _ { a \neq \pi ^ { \star } ( s ) } \bar { \mu } ( s , a ) } \end{array}$ . In particular, when $\mu ( s , \pi ^ { \star } ( s ) ) < 1 0 \overline { { \mu } } ( s )$ , we can repeat the analysis of MAB and show that $\begin{array} { r } { T _ { 4 } \lesssim \sqrt { \frac { S ( C ^ { \star } - 1 ) L } { N } } } \end{array}$ . Here, the appearance of $C ^ { \star } - 1$ , as opposed to $C ^ { \star }$ is due to the restriction $\mu ( s , \pi ^ { \star } ( s ) ) < 1 0 \overline { { \mu } } ( s )$ . One can verify that $\mu ( s , \pi ^ { \star } ( s ) ) < 1 0 \overline { { \mu } } ( s )$ together with the data coverage assumption ensures that $\begin{array} { r } { \sum _ { s : \rho ( s ) \geq 2 C ^ { \star } L / N , \mu ( s , \pi ^ { \star } ( s ) ) < 1 0 \overline { { \mu } } ( s ) } \rho ( s ) \lesssim C ^ { \star } - 1 . } \end{array}$ . On the other hand, when $\mu ( \mathfrak { s } , \overline { { \pi ^ { \star } ( \mathfrak { s } ) } } ) \geq 1 0 \overline { { \mu } } ( \mathfrak { s } )$ , i.e., when the optimal action is more likely to be seen in the dataset, the penalty function $b ( s , \pi ^ { \star } ( s ) )$ associated with the optimal action would be much smaller than those of the sub-optimal actions. Thanks to the LCB approach, the optimal action will be chosen with high probability, i.e., $T _ { 5 } \lesssim$ $1 / N ^ { 1 0 }$ .
+
+$C ^ { \star }$ dependency in MDPs. Ignoring the dependency on $1 / ( 1 - \gamma )$ , by comparing Theorems 6 and 7, one realizes that VI-LCB is optimal both when $C ^ { \star } \geq 1 + \Theta ( 1 )$ and $C ^ { \star } \le 1 + \Theta ( 1 / N )$ . However, in the middle region, the upper and lower bounds differ in their dependency on $C ^ { \star }$ . We conjecture that VI-LCB is optimal even this regime and the current gap is an artifact of our analysis.
+
+Conjecture 1 (Adaptive optimality of LCB) The LCB approach, together with value iteration is adaptively optimal for solving offline MDPs for all ranges of $C ^ { \star }$ .
+
+Technical hurdle. It turns out that closing the gap in MDPs is significantly more challenging due to error propagation. Naively applying the decomposition in the CB case fails to achieve the $C ^ { \breve { \star } } - 1$ dependence in the regime $C ^ { \check { \star } } \stackrel { } { \in } [ 1 ^ { \check { + } } \breve { \Omega } ( 1 / N ) , 1 \dot { + } O ( 1 ) ]$ . Major difficulties arise in controlling case (i), where $\begin{array} { r } { \mu ( s , \pi ^ { \star } ( s ) ) \gg \sum _ { a \neq \pi ^ { \star } ( s ) } \mu ( s , a ) } \end{array}$ . Recall that VI-LCB picks the right action if
+
+$$
+\mathrm { ~ } _ { \mathsf { t } } \left( s , \pi ^ { \star } ( s ) \right) - \sqrt { \frac { L } { m _ { t } ( s , \pi ^ { \star } ( s ) ) \vee 1 } } + \gamma P ^ { t } ( \cdot | s , \pi ^ { \star } ( s ) ) \cdot V _ { t - 1 } > r _ { t } ( s , a ) - \sqrt { \frac { L } { m _ { t } ( s , a ) \vee 1 } } + \gamma P ^ { t } ( \cdot | s , a ) \cdot V _ { t - 1 } ,
+$$
+
+for all $a \neq \pi ^ { \star } ( s )$ . The presence of the previous values $V _ { t - 1 }$ drastically changes the picture: even if we know that $m _ { t } ( s , \pi ^ { \star } ( s ) ) \gg m _ { t } ( s , a )$ , the current analysis does not guarantee the above inequality. It is likely that the value gap $g ( s ) \mathrel { \mathop : } = Q ^ { \star } ( s , \pi ^ { \star } ( s ) ) - Q ^ { \star } ( s , a )$ affects whether VI-LCB chooses the optimal action. How to study the interplay between the gap and the policy chosen by VI-LCB forms the main obstacle to obtaining tight performance guarantees when $\bar { C } ^ { \star } \in \dot { [ 1 + \Omega ( 1 / N ) , 1 + O ( 1 ) ] }$ .
+
+A confirmation from an episodic MDP. In Appendix E.6, we present an episodic example to demonstrate that (1) an episodic variant of VI-LCB achieves the optimal dependency on $C ^ { \star }$ and hence closes the gap between upper and lower bounds, and (2) a tight analysis of the sub-optimality is rather intricate and depends on a delicate decomposition based on the gap $Q ^ { \star } ( s , \pi ^ { \star } ( s ) ) - \bar { Q } ^ { \star } ( s , a )$ . As a preview, our example is an episodic MDP with $H = 3$ . To tackle case (i) above, we decompose the error based on whether $g ( s )$ is small. If $g ( s )$ is small for state $s$ , the contribution to the suboptimality is well controlled. Otherwise, we manage to show that VI-LCB selects the right action with high probability. What is more interesting and surprising is that the right threshold for value gap is given by $\sqrt { ( C ^ { \star } - 1 ) / N }$ . Ultimately, this allows us to achieve the optimal dependency on $C ^ { \star }$ .
+
+# 7 Discussion
+
+We propose a new batch RL framework based on the single policy concentrability coefficient $C ^ { \star }$ that smoothly interpolates the two extremes of data composition encountered in practice, namely the expert data and uniform coverage data. Under this new framework, we pursue the statistically optimal algorithms that can even be implemented without the knowledge of the data composition. More specifically, focusing on the lower confidence bound (LCB) approach inspired by the principle of pessimism, we find that LCB is adaptively minimax optimal for addressing the offline learning problems in most settings. Under the new framework, there exist numerous avenues for future study. One interesting direction is to provide a tighter bound for LCB in MDP for the regime where a significant fraction of the data comes from the optimal policy (Conjecture 1). Furthermore, it would be important to extend this work to function approximation setting. We expect to see our characterization of offline RL via single-policy concentrability to be extended to the function approximation setting and used in the development of new offline RL algorithms that only require partial coverage. Another interesting direction for future work is to analyze whether alternative conservative methods such as value regularization can achieve adaptivity and/or minimax optimality.
+
+# Acknowledgements
+
+The authors are grateful to Nan Jiang, Aviral Kumar, Yao Liu, and Zhaoran Wang for helpful discussions and suggestions. PR was partially supported by the Open Philanthropy Foundation and the Leverhulme Trust. BZ and JJ were partially supported by NSF Grants IIS-1901252, CCF-1909499, and DMS-2023505.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] In the abstract, we describe our contributions on proposing a new offline RL framework and analyzing the adaptive optimality of pessimism under this framework. In the introduction, we discuss these contributions in more detail, compare them with related work, and give a summary Figure 2 on our theoretical findings.   
+(b) Did you describe the limitations of your work? [Yes] We discuss our analysis limitation in proving adaptive optimality of LCB in MDP and provide extensive discussion on our conjecture in Section 6. In Section 5.2, we also point out that Algorithm 2 does not achieve optimal dependency in effective horizon and discuss possible methods to address this limitation. Furthermore, as explained in Section 7, this work is limited to the tabular setting and analyzes adaptive optimality of one conservative algorithm developed based on the pessimism principle.   
+(c) Did you discuss any potential negative societal impacts of your work? [No] The framework, algorithm, and theoretical analysis presented in this paper are general-purpose and can help RL algorithms utilize previously-collected datasets more effectively. We do not foresee any direct negative societal impacts.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] We have read the ethics review guidelines and our paper conforms to them.
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] Assumptions are layed out in theorem statements.   
+(b) Did you include complete proofs of all theoretical results? [Yes] All proofs are presented in the Appendix.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [N/A] We do not include any experiments.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [N/A]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A]
+
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [N/A]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [N/A] Our work does not use any assets.   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] We did not use crowdsourcing or conduct research with human subjects.   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/SJaP_-xAb/SJaP_-xAb.md

@@ -0,0 +1,360 @@
+# DEEP LEARNING WITH LOGGED BANDIT FEEDBACK
+
+Thorsten Joachims Cornell University tj@cs.cornell.edu
+
+Adith Swaminathan Microsoft Research adswamin@microsoft.com
+
+Maarten de Rijke University of Amsterdam derijke@uva.nl
+
+# ABSTRACT
+
+We propose a new output layer for deep neural networks that permits the use of logged contextual bandit feedback for training. Such contextual bandit feedback can be available in huge quantities (e.g., logs of search engines, recommender systems) at little cost, opening up a path for training deep networks on orders of magnitude more data. To this effect, we propose a counterfactual risk minimization approach for training deep networks using an equivariant empirical risk estimator with variance regularization, BanditNet, and show how the resulting objective can be decomposed in a way that allows stochastic gradient descent training. We empirically demonstrate the effectiveness of the method by showing how deep networks – ResNets in particular – can be trained for object recognition without conventionally labeled images.
+
+# 1 INTRODUCTION
+
+Log data can be recorded from online systems such as search engines, recommender systems, or online stores at little cost and in huge quantities. For concreteness, consider the interaction logs of an ad-placement system for banner ads. Such logs typically contain a record of the input to the system (e.g., features describing the user, banner ad, and page), the action that was taken by the system (e.g., a specific banner ad that was placed) and the feedback furnished by the user (e.g., clicks on the ad, or monetary payoff). This feedback, however, provides only partial information – “contextual-bandit feedback” – limited to the actions taken by the system. We do not get to see how the user would have responded, if the system had chosen a different action (e.g., other ads or banner types). Thus, the feedback for all other actions the system could have taken is typically not known. This makes learning from log data fundamentally different from traditional supervised learning, where “correct” predictions and a loss function provide feedback for all actions.
+
+In this paper, we propose a new output layer for deep neural networks that allows training on logged contextual bandit feedback. By circumventing the need for full-information feedback, our approach opens a new and intriguing pathway for acquiring knowledge at unprecedented scale, giving deep neural networks access to this abundant and ubiquitous type of data. Similarly, it enables the application of deep learning even in domains where manually labeling full-information feedback is not viable.
+
+In contrast to online learning with contextual bandit feedback (e.g., (Williams, 1992; Agarwal et al., 2014)), we perform batch learning from bandit feedback (BLBF) (Beygelzimer & Langford, 2009; Swaminathan & Joachims, 2015a;b;c) and the algorithm does not require the ability to make interactive interventions. At the core of the new output layer for BLBF training of deep neural networks lies a counterfactual training objective that replaces the conventional cross-entropy objective. Our approach – called BanditNet – follows the view of a deep neural network as a stochastic policy. We propose a counterfactual risk minimization (CRM) objective that is based on an equivariant estimator of the true error that only requires propensity-logged contextual bandit feedback. This makes our training objective fundamentally different from the conventional cross-entropy objective for supervised classification, which requires full-information feedback. Equivariance in our context means that the learning result is invariant to additive translations of the loss, and it is more formally defined in Section 3.2. To enable large-scale training, we show how this training objective can be decomposed to allow stochastic gradient descent (SGD) optimization.
+
+In addition to the theoretical derivation of BanditNet, we present an empirical evaluation that verifies the applicability of the theoretical argument. It demonstrates how a deep neural network architecture can be trained in the BLBF setting. In particular, we derive a BanditNet version of ResNet (He et al., 2016) for visual object classification. Despite using potentially much cheaper data, we find that Bandit-ResNet can achieve the same classification performance given sufficient amounts of contextual bandit feedback as ResNet trained with cross-entropy on conventionally (full-information) annotated images. To easily enable experimentation on other applications, we share an implementation of BanditNet.1
+
+# 2 RELATED WORK
+
+Several recent works have studied weak supervision approaches for deep learning. Weak supervision has been used to pre-train good image features (Joulin et al., 2016) and for information retrieval (Dehghani et al., 2017). Closely related works have studied label corruption on CIFAR10 recently (Zhang et al., 2016). However, all these approaches use weak supervision/corruption to construct noisy proxies for labels, and proceed with traditional supervised training (using crossentropy or mean-squared-error loss) with these proxies. In contrast, we work in the BLBF setting, which is an orthogonal data-source, and modify the loss functions optimized by deep nets to directly implement risk minimization.
+
+Virtually all previous methods that can learn from logged bandit feedback employ some form of risk minimization principle (Vapnik, 1998) over a model class. Most of the methods (Beygelzimer & Langford, 2009; Bottou et al., 2013; Swaminathan & Joachims, 2015a) employ an inverse propensity scoring (IPS) estimator (Rosenbaum & Rubin, 1983) as empirical risk and use stochastic gradient descent (SGD) to optimize the estimate over large datasets. Recently, the self-normalized estimator (Trotter & Tukey, 1956) has been shown to be a more suitable estimator for BLBF (Swaminathan & Joachims, 2015c). The self-normalized estimator, however, is not amenable to stochastic optimization and scales poorly with dataset size. In our work, we demonstrate how we can efficiently optimize a reformulation of the self-normalized estimator using SGD.
+
+Previous BLBF methods focus on simple model classes: log-linear and exponential models (Swaminathan & Joachims, 2015a) or tree-based reductions (Beygelzimer & Langford, 2009). In contrast, we demonstrate how current deep learning models can be trained effectively via batch learning from bandit feedback (BLBF), and compare these with existing approaches on a benchmark dataset (Krizhevsky & Hinton, 2009).
+
+Our work, together with independent concurrent work (Serban et al., 2017), demonstrates success with off-policy variants of the REINFORCE (Williams, 1992) algorithm. In particular, our algorithm employs a Lagrangian reformulation of the self-normalized estimator, and the objective and gradients of this reformulation are similar in spirit to the updates of the REINFORCE algorithm. This connection sheds new light on the role of the baseline hyper-parameters in REINFORCE: rather than simply reduce the variance of policy gradients, our work proposes a constructive algorithm for selecting the baseline in the off-policy setting and it suggests that the baseline is instrumental in creating an equivariant counterfactual learning objective.
+
+# 3 BANDITNET: COUNTERFACTUAL RISK MINIMIZATION FOR DEEP NETS
+
+To formalize the problem of batch learning from bandit feedback for deep neural networks, consider the contextual bandit setting where a policy $\pi$ takes as input $x \in \mathcal { X }$ and outputs an action $y \in \mathcal { V }$ . In response, we observe the loss (or payoff) $\delta ( x , y )$ of the selected action $y$ , where $\delta ( x , y )$ is an arbitrary (unknown) function that maps actions and contexts to a bounded real number. For example, in display advertising, the context $x$ could be a representation of the user and page, $y$ denotes the displayed ad, and $\delta ( x , y )$ could be the monetary payoff from placing the ad (zero if no click, or dollar amount if clicked). The contexts are drawn i.i.d. from a fixed but unknown distribution $\mathrm { P r } ( X )$ .
+
+In this paper, a (deep) neural network is viewed as implementing a stochastic policy $\pi$ . We can think of such a network policy as a conditional distribution $\pi _ { w } ( Y \mid x )$ over actions $y \in Y$ , where $w$ are the parameters of the network. The network makes a prediction by sampling an action $y \sim \pi _ { w } ( Y \mid$ $x$ ), where deterministic $\pi _ { w } ( Y \mid x )$ are a special case. As we will show as part of the empirical evaluation, many existing network architectures are compatible with this stochastic-policy view. For example, any network $f _ { w } ( x , y )$ with a softmax output layer
+
+$$
+\pi _ { w } ( y \mid x ) = \frac { \exp ( f _ { w } ( x , y ) ) } { \sum _ { y ^ { \prime } \in \mathcal { y } } \exp ( f _ { w } ( x , y ^ { \prime } ) ) }
+$$
+
+can be re-purposed as a conditional distribution from which one can sample actions, instead of interpreting it as a conditional likelihood like in full-information supervised learning.
+
+The goal of learning is to find a policy $\pi _ { w }$ that minimizes the risk (analogously: maximizes the payoff) defined as
+
+$$
+R ( \pi _ { w } ) = \mathbb { E } \mathbb { E } _ { x \sim \operatorname* { P r } ( X ) } \mathbb { E } _ { y \sim \pi _ { w } ( Y \mid x ) } [ \delta ( x , y ) ] .
+$$
+
+Any data collected from an interactive system depends on the policy $\pi _ { 0 }$ that was running on the system at the time, determining which actions $y$ and losses $\delta ( x , y )$ are observed. We call $\pi _ { 0 }$ the logging policy, and for simplicity assume that it is stationary. The logged data $D$ are $n$ tuples of observed context $x _ { i } \sim \operatorname* { P r } ( { \bar { X } } )$ , action $y _ { i } \sim \pi _ { 0 } ( Y \mid x _ { i } )$ taken by the logging policy, the probability of this action $p _ { i } \equiv \pi _ { 0 } ( y _ { i } \mid x _ { i } )$ , which we call the propensity, and the received loss $\delta _ { i } \equiv \delta ( x _ { i } , y _ { i } )$ :
+
+$$
+D = \left[ \left( x _ { 1 } , y _ { 1 } , p _ { 1 } , \delta _ { 1 } \right) , \ldots , \left( x _ { n } , y _ { n } , p _ { n } , \delta _ { n } \right) \right] .
+$$
+
+We will now discuss how we can use this logged contextual bandit feedback to train a neural network policy $\pi _ { w } ( Y \mid x )$ that has low risk $R ( \pi _ { w } )$ .
+
+# 3.1 COUNTERFACTUAL RISK MINIMIZATION
+
+While conditional maximum likelihood is a standard approach for training deep neural networks, it requires that the loss $\delta ( x _ { i } , y )$ is known for all $y \in \mathcal { V }$ . However, we only know $\delta ( x _ { i } , y _ { i } )$ for the particular $y _ { i }$ chosen by the logging policy $\pi _ { 0 }$ . We therefore take a different approach following (Langford et al., 2008; Swaminathan $\&$ Joachims, 2015b), where we directly minimize an empirical risk that can be estimated from the logged bandit data $D$ . This approach is called counterfactual risk minimization (CRM) (Swaminathan $\&$ Joachims, 2015b), since for any policy $\pi _ { w }$ it addresses the counterfactual question of how well that policy would have performed, if it had been used instead of $\pi _ { 0 }$ .
+
+While minimizing an empirical risk as an estimate of the true risk $R ( \pi _ { w } )$ is a common principle in machine learning (Vapnik, 1998), getting a reliable estimate based on the training data $D$ produced by $\pi _ { 0 }$ is not straightforward. The logged bandit data $D$ is not only incomplete (i.e., we lack knowledge of $\delta ( x _ { i } , y )$ for many $y \in \mathcal { V }$ that $\pi _ { w }$ would have chosen differently from $\pi _ { 0 }$ ), but it is also biased (i.e., the actions preferred by $\pi _ { 0 }$ are over-represented). This is why existing work on training deep neural networks either requires full knowledge of the loss function, or requires the ability to interactively draw new samples $y _ { i } \sim \pi _ { w } ( Y \mid x _ { i } )$ for any new policy $\pi _ { w }$ . In our setting we can do neither – we have a fixed dataset $D$ that is limited to samples from $\pi _ { 0 }$ .
+
+To nevertheless get a useful estimate of the empirical risk, we explicitly address both the bias and the variance of the risk estimate. To correct for sampling bias and handle missing data, we approach the risk estimation problem using importance sampling and thus remove the distribution mismatch between $\pi _ { 0 }$ and $\pi _ { w }$ (Langford et al., 2008; Owen, 2013; Swaminathan & Joachims, 2015b):
+
+$$
+R ( \pi _ { w } ) = \underset { x \sim \operatorname* { P r } ( X ) } { \mathbb { E } } \ \underset { y \sim \pi _ { w } ( Y \mid x ) } { \mathbb { E } } \left[ \delta ( x , y ) \right] = \underset { x \sim \operatorname* { P r } ( X ) } { \mathbb { E } } \ \underset { y \sim \pi _ { 0 } ( Y \mid x ) } { \mathbb { E } } \left[ \delta ( x , y ) \frac { \pi _ { w } ( y \mid x ) } { \pi _ { 0 } ( y \mid x ) } \right] .
+$$
+
+The latter expectation can be estimated on a sample $D$ of $n$ bandit-feedback examples using the following IPS estimator (Langford et al., 2008; Owen, 2013; Swaminathan $\&$ Joachims, 2015b):
+
+$$
+\hat { R } _ { I P S } ( \pi _ { w } ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \delta _ { i } \frac { \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } .
+$$
+
+This IPS estimator is unbiased and has bounded variance, if the logging policy has full support in the sense that $\forall x , y : \pi _ { 0 } ( y \mid x ) \ge \epsilon > 0$ . While at first glance it may seem natural to directly train the parameters $w$ of a network to optimize this IPS estimate as an empirical risk, there are at least three obstacles to overcome. First, we will argue in the following section that the naive IPS estimator’s lack of equivariance makes it sub-optimal for use as an empirical risk for high-capacity models. Second, we have to find an efficient algorithm for minimizing the empirical risk, especially making it accessible to stochastic gradient descent (SGD) optimization. And, finally, we are faced with an unusual type of bias-variance trade-off since “distance” from the exploration policy impacts the variance of the empirical risk estimate for different $w$ .
+
+# 3.2 EQUIVARIANT COUNTERFACTUAL RISK MINIMIZATION
+
+While Eq. (5) provides an unbiased empirical risk estimate, it exhibits the – possibly severe – problem of “propensity overfitting” when directly optimized within a learning algorithm (Swaminathan & Joachims, 2015c). It is a problem of overfitting to the choices $y _ { i }$ of the logging policy, and it occurs on top of the normal overfitting to the $\delta _ { i }$ . Propensity overfitting is linked to the lack of equivariance of the IPS estimator: while the minimizer of true risk $R ( \pi _ { w } )$ does not change when translating the loss by a constant (i.e., $\forall x , y : \delta ( x , y ) + c )$ by linearity of expectation,
+
+$$
+\begin{array} { r l r } { c + \underset { w } { \operatorname* { m i n } } \underset { x \sim \mathrm { P r } ( X ) } { \mathbb { E } } \underset { y \sim \pi _ { w } ( Y \mid x ) } { \mathbb { E } } [ \delta ( x , y ) ] } & { = } & { \underset { w } { \operatorname* { m i n } } \underset { x \sim \mathrm { P r } ( X ) } { \mathbb { E } } \underset { y \sim \pi _ { w } ( Y \mid x ) } { \mathbb { E } } [ \delta ( x , y ) + c ] } \end{array}
+$$
+
+the minimizer of the IPS-estimated empirical risk $\hat { R } _ { I P S } ( \pi _ { w } )$ can change dramatically for finite training samples, and
+
+$$
+c + \operatorname* { m i n } _ { w } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \delta _ { i } \frac { \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } \neq \operatorname* { m i n } _ { w } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( \delta _ { i } + c ) \frac { \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } .
+$$
+
+Intuitively, when $c$ shifts losses to be positive numbers, policies $\pi _ { w }$ that put as little probability mass as possible on the observed actions have low risk estimates. If $c$ shifts the losses to the negative range, the exact opposite is the case. For either choice of $c$ , the choice of the policy eventually selected by the learning algorithm can be dominated by where $\pi _ { 0 }$ happens to sample data, not by which actions have low loss.
+
+The following self-normalized IPS estimator (SNIPS) addresses the propensity overfitting problem (Swaminathan & Joachims, 2015c) and is equivariant:
+
+$$
+\begin{array} { r l r } { \hat { R } _ { S N I P S } ( \pi _ { w } ) } & { = } & { \frac { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \delta _ { i } \frac { \pi _ { w } ( y _ { i } | x _ { i } ) } { \pi _ { 0 } ( y _ { i } | x _ { i } ) } } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \pi _ { w } ( y _ { i } | x _ { i } ) } { \pi _ { 0 } ( y _ { i } | x _ { i } ) } } . } \end{array}
+$$
+
+In addition to being equivariant, this estimate can also have substantially lower variance than Eq. (5), since it exploits the knowledge that the denominator
+
+$$
+S \ : = \ \sum _ { n } ^ { 1 } \sum _ { i = 1 } ^ { n } { \frac { \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } }
+$$
+
+always has expectation 1:
+
+$$
+\mathbb { E } [ S ] = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \int \frac { \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } \pi _ { 0 } ( y _ { i } \mid x _ { i } ) \operatorname* { P r } ( x _ { i } ) d y _ { i } d x _ { i } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \int 1 \operatorname* { P r } ( x _ { i } ) d x _ { i } = 1 .
+$$
+
+The SNIPS estimator uses this knowledge as a multiplicative control variate (Swaminathan & Joachims, 2015c). While the SNIPS estimator has some bias, this bias asymptotically vanishes at a rate of $O ( { \textstyle { \frac { 1 } { n } } } )$ (Hesterberg, 1995). Using the SNIPS estimator as our empirical risk implies that we need to solve the following optimization problem for training:
+
+$$
+\hat { w } = \underset { w \in \Re ^ { N } } { \arg \operatorname* { m i n } } \hat { R } _ { S N I P S } ( \pi _ { w } ) .
+$$
+
+Thus, we now turn to designing efficient optimization methods for this training objective.
+
+# 3.3 TRAINING ALGORITHM
+
+Unfortunately, the training objective in Eq. (11) does not permit stochastic gradient descent (SGD) optimization in the given form (see Appendix C), which presents an obstacle to efficient and effective training of the network. To remedy this problem, we will now develop a reformulation that retains both the desirable properties of the SNIPS estimator, as well as the ability to reuse established SGD training algorithms. Instead of optimizing a ratio as in Eq. (11), we will reformulate the problem into a series of constrained optimization problems. Let $\hat { w }$ be a solution of Eq. (11), and at that solution let $S ^ { * }$ be the value of the control variate for $\pi _ { \hat { w } }$ as defined in Eq. (9). For simplicity, assume that the minimizer $\hat { w }$ is unique. If we knew $S ^ { * }$ , we could equivalently solve the following constrained optimization problem:
+
+$$
+\hat { w } = \underset { w \in \Re ^ { N } } { \arg \operatorname* { m i n } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \delta _ { i } \frac { \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } \mathrm { s u b j e c t } { \mathrm { t o } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } = S ^ { * } .
+$$
+
+Of course, we do not actually know $S ^ { * }$ . However, we can do a grid search in $\{ S _ { 1 } , \ldots , S _ { k } \}$ for $S ^ { * }$ and solve the above optimization problem for each value, giving us a set of solutions $\{ \hat { w } _ { 1 } , \hdots , \hat { w } _ { k } \}$ . Note that $S$ is just a one-dimensional quantity, and that the sensible range we need to search for $S ^ { * }$ concentrates around 1 as $n$ increases (see Appendix B). To find the overall (approximate) $\hat { w }$ that optimizes the SNIPS estimate, we then simply take the minimum:
+
+$$
+\begin{array} { r c l } { \hat { w } } & { = } & { \underset { ( \hat { w } _ { j } , S _ { j } ) } { \arg \operatorname* { m i n } } \frac { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \delta _ { i } \frac { \pi _ { \hat { w } _ { j } } ( y _ { i } | x _ { i } ) } { \pi _ { 0 } ( y _ { i } | x _ { i } ) } } { S _ { j } } . } \end{array}
+$$
+
+This still leaves the question of how to solve each equality constrained risk minimization problem using SGD. Fortunately, we can perform an equivalent search for $S ^ { * }$ without constrained optimization. To this effect, consider the Lagrangian of the constrained optimization problem in Eq. (12) with $S _ { j }$ in the constraint instead of $S ^ { * }$ :
+
+$$
+\therefore ( w , \lambda ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \delta _ { i } \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } - \lambda \Bigg [ \frac { 1 } { n } \left( \sum _ { i = 1 } ^ { n } \frac { \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } \right) - S _ { j } \Bigg ] = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { ( \delta _ { i } - \lambda ) \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } + \lambda S _ { j } .
+$$
+
+The variable $\lambda$ is an unconstrained Lagrange multiplier. To find the minimum of Eq. (12) for a particular $S _ { j }$ , we need to minimize $L ( w , \lambda )$ w.r.t. $w$ and maximize w.r.t. $\lambda$ .
+
+$$
+\hat { w } _ { j } = \underset { w \in \Re ^ { N } } { \arg \operatorname* { m i n } } \underset { \lambda } { \operatorname* { m a x } } L ( w , \lambda )
+$$
+
+However, we are not actually interested in the constrained solution of Eq. (12) for any specific $S _ { j }$ . We are merely interested in exploring a certain range $S \in [ S _ { 1 } , S _ { k } ]$ in our search for $S ^ { * }$ . So, we can reverse the roles of $\lambda$ and $S$ , where we keep $\lambda$ fixed and determine the corresponding $S$ in hindsight. In particular, for each $\left\{ \lambda _ { 1 } , \ldots , \lambda _ { k } \right\}$ we solve
+
+$$
+\begin{array} { c c l } { \hat { w } _ { j } } & { = } & { \underset { w \in \Re ^ { N } } { \arg \operatorname* { m i n } } L ( w , \lambda _ { j } ) . } \end{array}
+$$
+
+Note that the solution $\hat { w } _ { j }$ does not depend on $S _ { j }$ , so we can compute $S _ { j }$ after we have found the minimum $\hat { w } _ { j }$ . In particular, we can determine the $S _ { j }$ that corresponds to the given $\lambda _ { j }$ using the necessary optimality conditions,
+
+$$
+\frac { \partial L } { \partial w } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \partial \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \partial w } \frac { ( \delta _ { i } - \lambda _ { j } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } = 0 ~ \mathrm { a n d } ~ \frac { \partial L } { \partial \lambda _ { j } } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } - S _ { j } = 0 ,
+$$
+
+by solving the second equality of Eq. (16). In this way, the sequence of $\lambda _ { j }$ produces solutions $\hat { w } _ { j }$ corresponding to a sequence of $\{ S _ { 1 } , \ldots , S _ { k } \}$ .
+
+To identify the sensible range of $S$ to explore, we can make use of the fact that Eq. (9) concentrates around its expectation of 1 for each $\pi _ { w }$ as $n$ increases. Theorem 2 in Appendix B provides a characterization of how large the range needs to be. Furthermore, we can steer the exploration of $S$ via $\lambda$ , since the resulting $S$ changes monotonically with $\lambda$ :
+
+$\left( \lambda _ { a } < \lambda _ { b } \right)$ and $\hat { w } _ { a } \neq \hat { w } _ { b }$ are not equivalent optima in Eq. $( 1 5 ) ) \Rightarrow ( S _ { a } < S _ { b } ) .$
+
+A more formal statement and proof are given as Theorem 1 in Appendix A. In the simplest form one could therefore perform a grid search on $\lambda$ , but more sophisticated search methods are possible too.
+
+After this reformulation, the key computational problem is finding the solution of Eq. (15) for each $\lambda _ { j }$ . Note that in this unconstrained optimization problem, the Lagrange multiplier effectively translates the loss values in the conventional IPS estimate:
+
+$$
+\begin{array} { r c l } { { \hat { w } _ { j } } } & { { = } } & { { \displaystyle \arg \operatorname* { m i n } _ { w } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( \delta _ { i } - \lambda _ { j } \right) \frac { \pi _ { w } \left( y _ { i } \mid x _ { i } \right) } { \pi _ { 0 } \left( y _ { i } \mid x _ { i } \right) } \ ~ = ~ \displaystyle \arg \operatorname* { m i n } _ { w } \hat { R } _ { I P S } ^ { \lambda _ { j } } ( \pi _ { w } ) . } } \end{array}
+$$
+
+![](images/cd8a0b92b22cb4698570f765a38cf13020c79b4e44fb48bd878528ee1fc73bba.jpg)  
+Figure 1: Learning curve of BanditNet. The $\mathbf { X }$ -axis is the amount of bandit feedback, the y-axis is the test error. Given enough bandit feedback, Bandit-ResNet converges to the skyline performance.
+
+We denote this $\lambda$ -translated IPS estimate with $\hat { R } _ { I P S } ^ { \lambda } ( \pi _ { w } )$ . Note that each such optimization problem is now in the form required for SGD, where we merely weight the derivative of the stochastic policy network $\pi _ { w } ( y \mid x )$ by a factor $( \delta _ { i } - \lambda _ { j } ) / \pi _ { 0 } ( y _ { i } \mid x _ { i } )$ . This opens the door for re-purposing existing fast methods for training deep neural networks, and we demonstrate experimentally that SGD with momentum is able to optimize our objective scalably.
+
+Similar loss translations have previously been used in on-policy reinforcement learning (Williams, 1992), where they are motivated as minimizing the variance of the gradient estimate (Weaver & Tao, 2001; Greensmith et al., 2004). However, the situation is different in the off-policy setting we consider. First, we cannot sample new roll-outs from the current policy under consideration, which means we cannot use the standard variance-optimal estimator used in REINFORCE. Second, we tried using the (estimated) expected loss of the learned policy as the baseline as is commonly done in REINFORCE, but will see in the experiment section that this value for $\lambda$ is far from optimal. Finally, it is unclear whether gradient variance, as opposed to variance of the ERM objective, is really the key issue in batch learning from bandit feedback. In this sense, our approach provides a rigorous justification and a constructive way of picking the value of $\lambda$ in the off-policy setting – namely the value for which the corresponding $S _ { j }$ minimizes Eq. (13). In addition, one can further add variance regularization (Swaminathan & Joachims, 2015b) to improve the robustness of the risk estimate in Eq. (18) (see Appendix $\mathrm { D }$ for details).
+
+# 4 EMPIRICAL EVALUATION
+
+The empirical evaluation is designed to address three key questions. First, it verifies that deep models can indeed be trained effectively using our approach. Second, we will compare how the same deep neural network architecture performs under different types of data and training objectives – in particular, conventional cross-entropy training using full-information data. In order to be able to do this comparison, we focus on synthetic contextual bandit feedback data for training BanditNet that is sampled from the full-information labels. Third, we explore the effectiveness and fidelity of the approximate SNIPS objective.
+
+For the following BanditNet experiments, we adapted the ResNet20 architecture (He et al., 2016) by replacing the conventional cross-entropy objective with our counterfactual risk minimization objective. We evaluate the performance of this Bandit-ResNet on the CIFAR-10 (Krizhevsky & Hinton, 2009) dataset, where we can compare training on full-information data with training on bandit feedback, and where there is a full-information test set for estimating prediction error.
+
+To simulate logged bandit feedback, we perform the standard supervised to bandit conversion (Beygelzimer & Langford, 2009). We use a hand-coded logging policy that achieves about $49 \%$ error rate on the training data, which is substantially worse than what we hope to achieve after learning. This emulates a real world scenario where one would bootstrap an operational system with a mediocre policy (e.g., derived from a small hand-labeled dataset) and then deploys it to log bandit feedback. This logged bandit feedback data is then used to train the Bandit-ResNet.
+
+We evaluate the trained model using error rate on the held out (full-information) test set. We compare this model against the skyline of training a conventional ResNet using the full-information feedback from the 50,000 training examples. Both the conventional full-information ResNet as well as the Bandit-ResNet use the same network architecture, the same hyperparameters, the same data augmentation scheme, and the same optimization method that were set in the CNTK implementation of ResNet20. Since CIFAR10 does not come with a validation set for tuning the variance-regularization constant $\gamma$ , we do not use variance regularization for Bandit-ResNet. The Lagrange multiplier $\lambda \in \{ 0 . 6 5 , 0 . 7 , 0 . 7 5 , 0 . 8 , 0 . 8 5 , 0 . 9 , 0 . 9 5 , 1 . 0 , 1 . 0 5 \}$ is selected on the training set via Eq. (13). The only parameter we adjusted for Bandit-ResNet is lowering the learning rate to 0.1 and slowing down the learning rate schedule. The latter was done to avoid confounding the Bandit-ResNet results with potential effects from early stopping, and we report test performance after 1000 training epochs, which is well beyond the point of convergence in all runs.
+
+![](images/e4862e220661d6ee257b7d8cad93054cf108ff766a686f231636136e2f318cee.jpg)  
+Figure 2: The $\mathbf { X }$ -axis shows the value of the Lagrange multiplier $\lambda$ used for training. Left plot shows the test error. Right plot shows the value of the SNIPS objective and the normalizer $S$ . The size of the training set is $5 0 \mathrm { k }$ bandit-feedback examples.
+
+Learning curve. Figure 1 shows the prediction error of the Bandit-ResNet as more and more bandit feedback is provided for training. First, even though the logging policy that generated the bandit feedback has an error rate of $49 \%$ , the prediction error of the policy learned by the Bandit-ResNet is substantially better. It is between $13 \%$ and $8 . 2 \%$ , depending on the amount of training data. Second, the horizontal line is the performance of a conventional ResNet trained on the full-information training set. It serves as a skyline of how good Bandit-ResNet could possibly get given that it is sampling bandit feedback from the same full-information training set. The learning curve in Figure 1 shows that Bandit-ResNet converges to the skyline performance given enough bandit feedback training data, providing strong evidence that our training objective and method can effectively extract the available information provided in the bandit feedback.
+
+Effect of the choice of Lagrange multiplier. The left-hand plot in Figure 2 shows the test error of solutions $\hat { w } _ { j }$ depending on the value of the Lagrange multiplier $\lambda _ { j }$ used during training. It shows that $\lambda$ in the range 0.8 to 1.0 results in good prediction performance, but that performance degrades outside this area. The SNIPS estimates in the right-hand plot of Figure 2 roughly reflects this optimal range, given empirical support for both the SNIPS estimator and the use of Eq. (13).
+
+We also explored two other methods for selecting $\lambda$ . First, we used the straightforward IPS estimator as the objective (i.e., $\lambda = 0$ ), which leads to prediction performance worse than that of the logging policy (not shown). Second, we tried using the (estimated) expected loss of the learned policy as the baseline as is commonly done in REINFORCE. As Figure 1 shows, it is between 0.130 and 0.083 for the best policies we found. Figure 2 (left) shows that these baseline values are well outside of the optimum range.
+
+Also shown in the right-hand plot of Figure 2 is the value of the control variate in the denominator of the SNIPS estimate. As expected, it increases from below 1 to above 1 as $\lambda$ is increased. Note that large deviations of the control variate from 1 are a sign of propensity overfitting (Swaminathan & Joachims, 2015c). In particular, for all solutions $\hat { w } _ { j }$ the estimated standard error of the control variate $S _ { j }$ was less than 0.013, meaning that the normal $9 5 \%$ confidence interval for each $S _ { j }$ is contained in [0.974, 1.026]. If we see a $\hat { w } _ { j }$ with control variate $S _ { j }$ outside this range, we should be suspicious of propensity overfitting to the choices of the logging policy and discard this solution.
+
+# 5 CONCLUSIONS AND FUTURE WORK
+
+We proposed a new output layer for deep neural networks that enables the use of logged contextual bandit feedback for training. This type of feedback is abundant and ubiquitous in the form of interaction logs from autonomous systems, opening up the possibility of training deep neural networks on unprecedented amounts of data. In principle, this new output layer can replace the conventional cross-entropy layer for any network architecture. We provide a rigorous derivation of the training objective, linking it to an equivariant counterfactual risk estimator that enables counterfactual risk minimization. Most importantly, we show how the resulting training objective can be decomposed and reformulated to make it feasible for SGD training. We find that the BanditNet approach applied to the ResNet architecture achieves predictive accuracy comparable to conventional full-information training for visual object recognition.
+
+The paper opens up several directions for future work. First, it enables many new applications where contextual bandit feedback is readily available. Second, in settings where it is infeasible to log propensity-scored data, it would be interesting to combine BanditNet with propensity estimation techniques. Third, there may be improvements to BanditNet, like smarter search techniques for $S$ , more efficient counterfactual estimators beyond SNIPS, and the ability to handle continuous outputs.
+
+# ACKNOWLEDGMENTS
+
+This research was supported in part by NSF Award IIS-1615706, a gift from Bloomberg, the Criteo Faculty Research Award program, and the Netherlands Organisation for Scientific Research (NWO) under project nr. 612.001.116. All content represents the opinion of the authors, which is not necessarily shared or endorsed by their respective employers and/or sponsors.
+
+# REFERENCES
+
+A. Agarwal, D. Hsu, S. Kale, J. Langford, Lihong Li, and R. Schapire. Taming the monster: A fast and simple algorithm for contextual bandits. In ICML, 2014.   
+A. Beygelzimer and J. Langford. The offset tree for learning with partial labels. In KDD, pp. 129–138, 2009.   
+L. Bottou, J. Peters, J. Quinonero-Candela, D. Charles, M. Chickering, E. Portugaly, D. Ray, P. Simard, and E. Snelson. Counterfactual reasoning and learning systems: The example of computational advertising. JMLR, 14:3207–3260, 2013.   
+M. Dehghani, H. Zamani, A. Severyn, J. Kamps, and W. B. Croft. Neural ranking models with weak supervision. In SIGIR, pp. 65–74, 2017.   
+E. Greensmith, P.L. Bartlett, and J. Baxter. Variance reduction techniques for gradient estimates in reinforcement learning. JMLR, 5:1471–1530, 2004.   
+K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In CVPR, 2016.   
+T. Hesterberg. Weighted average importance sampling and defensive mixture distributions. Technometrics, 37:185–194, 1995.   
+A. Joulin, L. van der Maaten, A. Jabri, and N. Vasilache. Learning visual features from large weakly supervised data. In ECCV, pp. 67–84, 2016.   
+A. Krizhevsky and G. Hinton. Learning multiple layers of features from tiny images. Technical report, Computer Science Department, University of Toronto, 2009.   
+J. Langford, A. Strehl, and J. Wortman. Exploration scavenging. In ICML, pp. 528–535, 2008.   
+A.B. Owen. Monte Carlo theory, methods and examples. 2013.   
+P. Rosenbaum and D. Rubin. The central role of propensity score in observational studies for causal effects. Biometrica, 70:41–55, 1983.   
+I. Serban, C. Sankar, M. Germain, S. Zhang, Z. Lin, S. Subramanian, T. Kim, M. Pieper, S. Chandar, N. R. Ke, S. Mudumba, A. de Brebisson, J. M. R. Sotelo, D. Suhubdy, V. Michalski, A. Nguyen, J. Pineau, and Y. Bengio. A deep reinforcement learning chatbot. ArXiv e-prints, September 2017.   
+A. Swaminathan and T. Joachims. Counterfactual risk minimization: Learning from logged bandit feedback. In ICML, pp. 814–823, 2015a.   
+A. Swaminathan and T. Joachims. Batch learning from logged bandit feedback through counterfactual risk minimization. JMLR, 16:1731–1755, 2015b.   
+A. Swaminathan and T. Joachims. The self-normalized estimator for counterfactual learning. In NIPS, 2015c.   
+H. F. Trotter and J. W. Tukey. Conditional monte carlo for normal samples. In Symposium on Monte Carlo Methods, pp. 64–79, 1956.   
+V. Vapnik. Statistical Learning Theory. Wiley, Chichester, GB, 1998.   
+L. Weaver and N. Tao. The optimal reward baseline for gradient-based reinforcement learning. In UAI, pp. 538–545, 2001.   
+R. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8(3-4), May 1992.   
+C. Zhang, S. Bengio, M. Hardt, B. Recht, and O. Vinyals. Understanding deep learning requires rethinking generalization. CoRR, abs/1611.03530, 2016.
+
+# A APPENDIX: STEERING THE EXPLORATION OF $S$ THROUGH $\lambda$ .
+
+Theorem 1. Let $\lambda _ { a } < \lambda _ { b }$ and let
+
+$$
+\begin{array} { r c l } { { \hat { w } _ { a } } } & { { = } } & { { \underset { w } { \arg \operatorname* { m i n } } \hat { R } _ { I P S } ^ { \lambda _ { a } } ( \pi _ { w } ) } } \\ { { } } & { { } } & { { } } \\ { { \hat { w } _ { b } } } & { { = } } & { { \underset { w } { \arg \operatorname* { m i n } } \hat { R } _ { I P S } ^ { \lambda _ { b } } ( \pi _ { w } ) . } } \end{array}
+$$
+
+If the optima $\hat { w } _ { a }$ and $\hat { w } _ { b }$ are not equivalent in the sense that $\hat { R } _ { I P S } ^ { \lambda _ { a } } ( \pi _ { \hat { w } _ { a } } ) \neq \hat { R } _ { I P S } ^ { \lambda _ { a } } ( \pi _ { \hat { w } _ { b } } )$ and $\hat { R } _ { I P S } ^ { \lambda _ { b } } ( \pi _ { \hat { w } _ { a } } ) \neq \hat { R } _ { I P S } ^ { \lambda _ { b } } ( \pi _ { \hat { w } _ { b } } )$ , then
+
+$$
+S _ { a } < S _ { b } .
+$$
+
+Proof. Abbreviate $\begin{array} { r } { f ( w ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \delta _ { i } \frac { \pi _ { w } ( y _ { i } | x _ { i } ) } { \pi _ { 0 } ( y _ { i } | x _ { i } ) } } \end{array}$ )) and g(w) = 1n Pni=1 πw(yi|xi)π0(yi|xi) .
+
+$$
+\hat { R } _ { I P S } ^ { \lambda } ( \pi _ { w } ) = f ( w ) - \lambda g ( w ) ,
+$$
+
+where $g ( w )$ corresponds to the value of the control variate $S$ . Since $\hat { w } _ { a }$ and $\hat { w } _ { b }$ are not equivalent optima, we know that
+
+$$
+\begin{array} { r l r } { f ( \hat { w } _ { a } ) - \lambda _ { a } g ( \hat { w } _ { a } ) } & { < } & { f ( \hat { w } _ { b } ) - \lambda _ { a } g ( \hat { w } _ { b } ) } \\ { f ( \hat { w } _ { b } ) - \lambda _ { b } g ( \hat { w } _ { b } ) } & { < } & { f ( \hat { w } _ { a } ) - \lambda _ { b } g ( \hat { w } _ { a } ) } \end{array}
+$$
+
+Adding the two inequalities and solving implies that
+
+$$
+\begin{array} { r l } & { f ( \hat { w } _ { a } ) - \lambda _ { a } g \big ( \hat { w } _ { a } \big ) + f \big ( \hat { w } _ { b } \big ) - \lambda _ { b } g \big ( \hat { w } _ { b } \big ) < f \big ( \hat { w } _ { b } \big ) - \lambda _ { a } g \big ( \hat { w } _ { b } \big ) + f \big ( \hat { w } _ { a } \big ) - \lambda _ { b } g \big ( \hat { w } _ { a } \big ) } \\ & { \lambda _ { a } g \big ( \hat { w } _ { a } \big ) + \lambda _ { b } g \big ( \hat { w } _ { b } \big ) > \lambda _ { a } g \big ( \hat { w } _ { b } \big ) + \lambda _ { b } g \big ( \hat { w } _ { a } \big ) } \\ & { \big ( \lambda _ { b } - \lambda _ { a } \big ) g \big ( \hat { w } _ { b } \big ) > \big ( \lambda _ { b } - \lambda _ { a } \big ) g \big ( \hat { w } _ { a } \big ) } \\ & { g \big ( \hat { w } _ { b } \big ) > g \big ( \hat { w } _ { a } \big ) } \\ & { S _ { b } > S _ { a } \quad \boxed { 1 } } \end{array}
+$$
+
+# B APPENDIX: CHARACTERIZING THE RANGE OF $S$ TO EXPLORE.
+
+Theorem 2. Let $p \ \leq \ \pi _ { 0 } ( y \mid x )$ be a lower bound on the propensity for the logging policy, then constraining the solution of Eq. (11) to the $w$ with control variate $S \in [ 1 - \epsilon , 1 + \epsilon ]$ for a training set of size n will not exclude the minimizer of the true risk $\begin{array} { r } { w ^ { * } = \arg \operatorname* { m i n } _ { w \in W } R ( \pi _ { w } ) } \end{array}$ in the policy space $W$ with probability at least
+
+$$
+\begin{array} { r } { 1 - 2 \exp \left( - 2 n \epsilon ^ { 2 } p ^ { 2 } \right) . } \end{array}
+$$
+
+Proof. For the optimal $w ^ { * }$ , let
+
+$$
+\begin{array} { r c l } { S } & { = } & { \displaystyle \sum _ { i = 1 } ^ { n } { \frac { \pi _ { w ^ { * } } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } } } \end{array}
+$$
+
+be the control variate in the denominator of the SNIPS estimator. $S$ is a random variable that is a sum of bounded random variables between 0 and
+
+$$
+\operatorname* { m a x } _ { x , y } { \frac { \pi _ { w ^ { * } } ( y \mid x ) } { \pi _ { 0 } ( y \mid x ) } } \leq { \frac { 1 } { p } } .
+$$
+
+We can bound the probability that the control variate $S$ of the optimum $w ^ { * }$ lies outside of $[ 1 - \epsilon , 1 + \epsilon ]$ via Hoeffding’s inequality:
+
+$$
+\begin{array} { r c l } { P ( | S - 1 | \ge \epsilon ) } & { \le } & { 2 \exp \left( \displaystyle \frac { - 2 n ^ { 2 } \epsilon ^ { 2 } } { n ( 1 / p ) ^ { 2 } } \right) } \\ & { = } & { 2 \exp \left( - 2 n \epsilon ^ { 2 } p ^ { 2 } \right) . \quad \bigsqcup } \end{array}
+$$
+
+The same argument applies to any individual policy $\pi _ { w }$ , not just $w ^ { * }$ . Note, however, that it can still be highly likely that at least one policy $\pi _ { w }$ with $w \in W$ shows a large deviation in the control variate for high-capacity $W$ , which can lead to propensity overfitting when using the naive IPS estimator.
+
+# C APPENDIX: WHY DIRECT STOCHASTIC OPTIMIZATION OF RATIO ESTIMATORS IS NOT POSSIBLE.
+
+Suppose we have a dataset of $n$ BLBF samples $D = \{ ( x _ { 1 } , y _ { 1 } , \delta _ { 1 } , p _ { 1 } ) \dots ( x _ { n } , y _ { n } , \delta _ { n } , p _ { n } ) \}$ where each instance is an i.i.d. sample from the data generating distribution. In the sequel we will be considering two datasets of $n + 1$ samples, $D ^ { \prime } = \bar { D } \cup \{ ( x ^ { \prime } , \bar { y ^ { \prime } } , \delta ^ { \prime } , p ^ { \prime } ) \}$ and $D ^ { \prime \prime } { = } D \bar { \cup } \{ ( x ^ { \prime \prime } , y ^ { \prime \prime } , \delta ^ { \prime \prime } , p ^ { \prime \prime } ) \}$ where $\bar { ( } x ^ { \prime } , y ^ { \prime } , \delta ^ { \prime } , p ^ { \prime } ) \neq ( x ^ { \prime \prime } , y ^ { \prime \prime } , \delta ^ { \prime \prime } , p ^ { \prime \prime } )$ and $( x ^ { \prime } , y ^ { \prime } , \delta ^ { \prime } , p ^ { \prime } ) , ( x ^ { \prime \prime } , y ^ { \prime \prime } , \delta ^ { \prime \prime } , p ^ { \prime \prime } ) \notin D$ .
+
+For notational convenience, let $\begin{array} { r } { f _ { i } : = \delta _ { i } \frac { \pi _ { w } \left( y _ { i } | x _ { i } \right) } { \pi _ { 0 } \left( y _ { i } | x _ { i } \right) } } \end{array}$ , and ˙fi := ∇wfi; gi := πw(yi|xi)π (y |x ) , and $\dot { g } _ { i } : = \nabla _ { w } g _ { i }$ .   
+First consider the vanilla IPS risk estimate of Eq. (5).
+
+$$
+\hat { R } _ { I P S } ( \pi _ { w } ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \delta _ { i } \frac { \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } f _ { i } .
+$$
+
+To maximize this estimate using stochastic optimization, we must construct an unbiased gradient estimate. That is, we randomly select one sample from $D$ and compute a gradient $\alpha ( ( x _ { i } , y _ { i } , \delta _ { i } , p _ { i } ) )$ and we require that
+
+$$
+\nabla _ { \boldsymbol { w } } \hat { R } _ { I P S } ( \pi _ { \boldsymbol { w } } ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \dot { f } _ { i } = \mathbb { E } _ { i \sim D } \left[ \alpha ( ( x _ { i } , y _ { i } , \delta _ { i } , p _ { i } ) ) \right] .
+$$
+
+Here the expectation is over our random choice of 1 out of $n$ samples. Observe that $\alpha ( ( x _ { i } , y _ { i } , \delta _ { i } , \bar { p } _ { i } ) ) = \dot { f } _ { i }$ suffices (and indeed, this corresponds to vanilla SGD):
+
+$$
+\mathbb { E } _ { i \sim D } \left[ \alpha \big ( ( x _ { i } , y _ { i } , \delta _ { i } , p _ { i } ) \big ) \right] = \sum _ { i = 1 } ^ { n } \frac { 1 } { n } \alpha \big ( ( x _ { i } , y _ { i } , \delta _ { i } , p _ { i } ) \big ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \dot { f } _ { i } = \nabla _ { w } \hat { R } _ { I P S } \big ( \pi _ { w } \big ) .
+$$
+
+Other choices of $\alpha ( \cdot )$ can also produce unbiased gradient estimates, and this leads to the study of stochastic variance-reduced gradient optimization.
+
+Now let us attempt to construct an unbiased gradient estimate for Eq. (8):
+
+$$
+\hat { R } _ { S N I P S } ( \pi _ { w } ) = \frac { \sum _ { i = 1 } ^ { n } f _ { i } } { \sum _ { i = 1 } ^ { n } g _ { i } } .
+$$
+
+Suppose such a gradient estimate exists, $\beta ( ( x _ { i } , y _ { i } , \delta _ { i } , p _ { i } ) )$ . Then,
+
+$$
+\nabla _ { w } \hat { R } _ { S N I P S } ( \pi _ { w } ) = \nabla _ { w } \frac { \sum _ { i = 1 } ^ { n } f _ { i } } { \sum _ { i = 1 } ^ { n } g _ { i } } = \mathbb { E } _ { i \sim D } \left[ \beta ( ( x _ { i } , y _ { i } , \delta _ { i } , p _ { i } ) ) \right] = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \beta ( ( x _ { i } , y _ { i } , \delta _ { i } , p _ { i } ) ) .
+$$
+
+This identity is true for any sample of BLBF instances – in particular, for $D ^ { \prime }$ and $D ^ { \prime \prime }$ :
+
+$$
+\begin{array} { l l } { \displaystyle \nabla _ { w } \frac { \sum _ { i = 1 } ^ { n } f _ { i } + f ^ { \prime } } { \sum _ { i = 1 } ^ { n } g _ { i } + g ^ { \prime } } = \sum _ { i = 1 } ^ { n } \frac { 1 } { n + 1 } \beta ( ( x _ { i } , y _ { i } , \delta _ { i } , p _ { i } ) ) + \frac { \beta ( ( x ^ { \prime } , y ^ { \prime } , \delta ^ { \prime } , p ^ { \prime } ) ) } { n + 1 } , } \\ { \displaystyle \nabla _ { w } \frac { \sum _ { i = 1 } ^ { n } f _ { i } + f ^ { \prime \prime } } { \sum _ { i = 1 } ^ { n } g _ { i } + g ^ { \prime \prime } } = \sum _ { i = 1 } ^ { n } \frac { 1 } { n + 1 } \beta ( ( x _ { i } , y _ { i } , \delta _ { i } , p _ { i } ) ) + \frac { \beta ( ( x ^ { \prime \prime } , y ^ { \prime \prime } , \delta ^ { \prime \prime } , p ^ { \prime \prime } ) ) } { n + 1 } . } \end{array}
+$$
+
+Subtracting these two equations,
+
+$$
+\nabla _ { w } \left( \frac { \sum _ { i = 1 } ^ { n } f _ { i } + f ^ { \prime } } { \sum _ { i = 1 } ^ { n } g _ { i } + g ^ { \prime } } - \frac { \sum _ { i = 1 } ^ { n } f _ { i } + f ^ { \prime \prime } } { \sum _ { i = 1 } ^ { n } g _ { i } + g ^ { \prime \prime } } \right) = \frac { \beta ( ( x ^ { \prime } , y ^ { \prime } , \delta ^ { \prime } , p ^ { \prime } ) ) - \beta ( ( x ^ { \prime \prime } , y ^ { \prime \prime } , \delta ^ { \prime \prime } , p ^ { \prime \prime } ) ) } { n + 1 } .
+$$
+
+The LHS clearly depends on $\{ ( x _ { i } , y _ { i } , \delta _ { i } , p _ { i } ) \} _ { i = 1 } ^ { n }$ in general, while the RHS does not! This contradiction indicates that no construction of $\beta$ that only looks at a sub-sample of the data can yield an unbiased gradient estimate of $\hat { R } _ { S N I P S } ( \pi _ { w } )$ .
+
+# D APPENDIX: VARIANCE REGULARIZATION
+
+Unlike in conventional supervised learning, a counterfactual empirical risk estimator like $\hat { R } _ { I P S } ( \pi _ { w } )$ can have vastly different variances $\mathrm { V a r } ( \hat { R } _ { I P S } ( \pi _ { w } ) )$ for different $\pi _ { w }$ in the hypothesis space (and $\hat { R } _ { S N I P S } ( \pi _ { w } )$ as well) (Swaminathan $\&$ Joachims, 2015b). Intuitively, the “closer” the particular $\pi _ { w }$ is to the exploration policy $\pi _ { 0 }$ , the larger the effective sample size (Owen, 2013) will be and the smaller the variance of the empirical risk estimate. For the optimization problems we solve in Eq. (18), this means that we should trust the $\lambda$ -translated risk estimate $\hat { R } _ { I P S } ^ { \lambda _ { j } } ( \pi _ { w } )$ more for some $w$ than for others, as we use $\hat { R } _ { I P S } ^ { \lambda _ { j } } ( \pi _ { w } )$ only as a proxy for finding the policy that minimizes its expected value (i.e., the true loss). To this effect, generalization error bounds that account for this variance difference (Swaminathan & Joachims, 2015b) motivate a new type of overfitting control. This leads to the following training objective (Swaminathan $\&$ Joachims, 2015b), which can be thought of as a more reliable version of Eq. (18):
+
+$$
+\hat { w } _ { j } = \underset { w } { \arg \operatorname* { m i n } } \left[ \hat { R } _ { I P S } ^ { \lambda _ { j } } ( \pi _ { w } ) + \gamma \sqrt { \frac { \widehat { \mathrm { V a r } } ( \hat { R } _ { I P S } ^ { \lambda _ { j } } ( \pi _ { w } ) ) } { n } } \right] .
+$$
+
+Here, $\widehat { \mathrm { V a r } } ( \hat { R } _ { I P S } ^ { \lambda _ { j } } ( \pi _ { w } ) )$ is the estimated variance of $\hat { R } _ { I P S } ^ { \lambda _ { j } } ( \pi _ { w } )$ on the training data, and $\gamma$ is a regwe optimize the upper confidence interval, which depends on the variance of the risk estimate for each $\pi _ { w }$ . While this objective again does not permit SGD optimization in its given form, it has been shown that a Taylor-majorization can be used to successively upper bound the objective in Eq. (35), and that typically a small number of iterations suffices to converge to a local optimum (Swaminathan & Joachims, 2015b). Each such Taylor-majorization is again of a form
+
+$$
+{ \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } \left[ A \left( { \frac { \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } } \right) + B \left( { \frac { \pi _ { w } ( y _ { i } \mid x _ { i } ) } { \pi _ { 0 } ( y _ { i } \mid x _ { i } ) } } \right) ^ { 2 } \right]
+$$
+
+for easily computable constants $A$ and $B$ (Swaminathan & Joachims, 2015b), which allows for SGD optimization.

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+    {
+        "type": "text",
+        "text": "DEEP LEARNING WITH LOGGED BANDIT FEEDBACK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Thorsten Joachims Cornell University tj@cs.cornell.edu ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Adith Swaminathan Microsoft Research adswamin@microsoft.com ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Maarten de Rijke University of Amsterdam derijke@uva.nl ",
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+        "type": "text",
+        "text": "ABSTRACT ",
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+        "text": "We propose a new output layer for deep neural networks that permits the use of logged contextual bandit feedback for training. Such contextual bandit feedback can be available in huge quantities (e.g., logs of search engines, recommender systems) at little cost, opening up a path for training deep networks on orders of magnitude more data. To this effect, we propose a counterfactual risk minimization approach for training deep networks using an equivariant empirical risk estimator with variance regularization, BanditNet, and show how the resulting objective can be decomposed in a way that allows stochastic gradient descent training. We empirically demonstrate the effectiveness of the method by showing how deep networks – ResNets in particular – can be trained for object recognition without conventionally labeled images. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Log data can be recorded from online systems such as search engines, recommender systems, or online stores at little cost and in huge quantities. For concreteness, consider the interaction logs of an ad-placement system for banner ads. Such logs typically contain a record of the input to the system (e.g., features describing the user, banner ad, and page), the action that was taken by the system (e.g., a specific banner ad that was placed) and the feedback furnished by the user (e.g., clicks on the ad, or monetary payoff). This feedback, however, provides only partial information – “contextual-bandit feedback” – limited to the actions taken by the system. We do not get to see how the user would have responded, if the system had chosen a different action (e.g., other ads or banner types). Thus, the feedback for all other actions the system could have taken is typically not known. This makes learning from log data fundamentally different from traditional supervised learning, where “correct” predictions and a loss function provide feedback for all actions. ",
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+    {
+        "type": "text",
+        "text": "In this paper, we propose a new output layer for deep neural networks that allows training on logged contextual bandit feedback. By circumventing the need for full-information feedback, our approach opens a new and intriguing pathway for acquiring knowledge at unprecedented scale, giving deep neural networks access to this abundant and ubiquitous type of data. Similarly, it enables the application of deep learning even in domains where manually labeling full-information feedback is not viable. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In contrast to online learning with contextual bandit feedback (e.g., (Williams, 1992; Agarwal et al., 2014)), we perform batch learning from bandit feedback (BLBF) (Beygelzimer & Langford, 2009; Swaminathan & Joachims, 2015a;b;c) and the algorithm does not require the ability to make interactive interventions. At the core of the new output layer for BLBF training of deep neural networks lies a counterfactual training objective that replaces the conventional cross-entropy objective. Our approach – called BanditNet – follows the view of a deep neural network as a stochastic policy. We propose a counterfactual risk minimization (CRM) objective that is based on an equivariant estimator of the true error that only requires propensity-logged contextual bandit feedback. This makes our training objective fundamentally different from the conventional cross-entropy objective for supervised classification, which requires full-information feedback. Equivariance in our context means that the learning result is invariant to additive translations of the loss, and it is more formally defined in Section 3.2. To enable large-scale training, we show how this training objective can be decomposed to allow stochastic gradient descent (SGD) optimization. ",
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+    {
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+        "text": "In addition to the theoretical derivation of BanditNet, we present an empirical evaluation that verifies the applicability of the theoretical argument. It demonstrates how a deep neural network architecture can be trained in the BLBF setting. In particular, we derive a BanditNet version of ResNet (He et al., 2016) for visual object classification. Despite using potentially much cheaper data, we find that Bandit-ResNet can achieve the same classification performance given sufficient amounts of contextual bandit feedback as ResNet trained with cross-entropy on conventionally (full-information) annotated images. To easily enable experimentation on other applications, we share an implementation of BanditNet.1 ",
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+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Several recent works have studied weak supervision approaches for deep learning. Weak supervision has been used to pre-train good image features (Joulin et al., 2016) and for information retrieval (Dehghani et al., 2017). Closely related works have studied label corruption on CIFAR10 recently (Zhang et al., 2016). However, all these approaches use weak supervision/corruption to construct noisy proxies for labels, and proceed with traditional supervised training (using crossentropy or mean-squared-error loss) with these proxies. In contrast, we work in the BLBF setting, which is an orthogonal data-source, and modify the loss functions optimized by deep nets to directly implement risk minimization. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Virtually all previous methods that can learn from logged bandit feedback employ some form of risk minimization principle (Vapnik, 1998) over a model class. Most of the methods (Beygelzimer & Langford, 2009; Bottou et al., 2013; Swaminathan & Joachims, 2015a) employ an inverse propensity scoring (IPS) estimator (Rosenbaum & Rubin, 1983) as empirical risk and use stochastic gradient descent (SGD) to optimize the estimate over large datasets. Recently, the self-normalized estimator (Trotter & Tukey, 1956) has been shown to be a more suitable estimator for BLBF (Swaminathan & Joachims, 2015c). The self-normalized estimator, however, is not amenable to stochastic optimization and scales poorly with dataset size. In our work, we demonstrate how we can efficiently optimize a reformulation of the self-normalized estimator using SGD. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Previous BLBF methods focus on simple model classes: log-linear and exponential models (Swaminathan & Joachims, 2015a) or tree-based reductions (Beygelzimer & Langford, 2009). In contrast, we demonstrate how current deep learning models can be trained effectively via batch learning from bandit feedback (BLBF), and compare these with existing approaches on a benchmark dataset (Krizhevsky & Hinton, 2009). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our work, together with independent concurrent work (Serban et al., 2017), demonstrates success with off-policy variants of the REINFORCE (Williams, 1992) algorithm. In particular, our algorithm employs a Lagrangian reformulation of the self-normalized estimator, and the objective and gradients of this reformulation are similar in spirit to the updates of the REINFORCE algorithm. This connection sheds new light on the role of the baseline hyper-parameters in REINFORCE: rather than simply reduce the variance of policy gradients, our work proposes a constructive algorithm for selecting the baseline in the off-policy setting and it suggests that the baseline is instrumental in creating an equivariant counterfactual learning objective. ",
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+    {
+        "type": "text",
+        "text": "3 BANDITNET: COUNTERFACTUAL RISK MINIMIZATION FOR DEEP NETS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To formalize the problem of batch learning from bandit feedback for deep neural networks, consider the contextual bandit setting where a policy $\\pi$ takes as input $x \\in \\mathcal { X }$ and outputs an action $y \\in \\mathcal { V }$ . In response, we observe the loss (or payoff) $\\delta ( x , y )$ of the selected action $y$ , where $\\delta ( x , y )$ is an arbitrary (unknown) function that maps actions and contexts to a bounded real number. For example, in display advertising, the context $x$ could be a representation of the user and page, $y$ denotes the displayed ad, and $\\delta ( x , y )$ could be the monetary payoff from placing the ad (zero if no click, or dollar amount if clicked). The contexts are drawn i.i.d. from a fixed but unknown distribution $\\mathrm { P r } ( X )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, a (deep) neural network is viewed as implementing a stochastic policy $\\pi$ . We can think of such a network policy as a conditional distribution $\\pi _ { w } ( Y \\mid x )$ over actions $y \\in Y$ , where $w$ are the parameters of the network. The network makes a prediction by sampling an action $y \\sim \\pi _ { w } ( Y \\mid$ $x$ ), where deterministic $\\pi _ { w } ( Y \\mid x )$ are a special case. As we will show as part of the empirical evaluation, many existing network architectures are compatible with this stochastic-policy view. For example, any network $f _ { w } ( x , y )$ with a softmax output layer ",
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+        "text": "",
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+    {
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+        "img_path": "images/8fe8207f603de3f94bea1e5b219d27d3548635a4442ea3fed39c693dd693add9.jpg",
+        "text": "$$\n\\pi _ { w } ( y \\mid x ) = \\frac { \\exp ( f _ { w } ( x , y ) ) } { \\sum _ { y ^ { \\prime } \\in \\mathcal { y } } \\exp ( f _ { w } ( x , y ^ { \\prime } ) ) }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "can be re-purposed as a conditional distribution from which one can sample actions, instead of interpreting it as a conditional likelihood like in full-information supervised learning. ",
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+    {
+        "type": "text",
+        "text": "The goal of learning is to find a policy $\\pi _ { w }$ that minimizes the risk (analogously: maximizes the payoff) defined as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/e2cb828b4ed7dec3bccd3bd571907b46ddace97484424689aced8c4896c5e5b5.jpg",
+        "text": "$$\nR ( \\pi _ { w } ) = \\mathbb { E } \\mathbb { E } _ { x \\sim \\operatorname* { P r } ( X ) } \\mathbb { E } _ { y \\sim \\pi _ { w } ( Y \\mid x ) } [ \\delta ( x , y ) ] .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Any data collected from an interactive system depends on the policy $\\pi _ { 0 }$ that was running on the system at the time, determining which actions $y$ and losses $\\delta ( x , y )$ are observed. We call $\\pi _ { 0 }$ the logging policy, and for simplicity assume that it is stationary. The logged data $D$ are $n$ tuples of observed context $x _ { i } \\sim \\operatorname* { P r } ( { \\bar { X } } )$ , action $y _ { i } \\sim \\pi _ { 0 } ( Y \\mid x _ { i } )$ taken by the logging policy, the probability of this action $p _ { i } \\equiv \\pi _ { 0 } ( y _ { i } \\mid x _ { i } )$ , which we call the propensity, and the received loss $\\delta _ { i } \\equiv \\delta ( x _ { i } , y _ { i } )$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/680051497bb94a806cbad00eca901e733ee082626bf1b12859b90626332b2f5b.jpg",
+        "text": "$$\nD = \\left[ \\left( x _ { 1 } , y _ { 1 } , p _ { 1 } , \\delta _ { 1 } \\right) , \\ldots , \\left( x _ { n } , y _ { n } , p _ { n } , \\delta _ { n } \\right) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We will now discuss how we can use this logged contextual bandit feedback to train a neural network policy $\\pi _ { w } ( Y \\mid x )$ that has low risk $R ( \\pi _ { w } )$ . ",
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+    {
+        "type": "text",
+        "text": "3.1 COUNTERFACTUAL RISK MINIMIZATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "While conditional maximum likelihood is a standard approach for training deep neural networks, it requires that the loss $\\delta ( x _ { i } , y )$ is known for all $y \\in \\mathcal { V }$ . However, we only know $\\delta ( x _ { i } , y _ { i } )$ for the particular $y _ { i }$ chosen by the logging policy $\\pi _ { 0 }$ . We therefore take a different approach following (Langford et al., 2008; Swaminathan $\\&$ Joachims, 2015b), where we directly minimize an empirical risk that can be estimated from the logged bandit data $D$ . This approach is called counterfactual risk minimization (CRM) (Swaminathan $\\&$ Joachims, 2015b), since for any policy $\\pi _ { w }$ it addresses the counterfactual question of how well that policy would have performed, if it had been used instead of $\\pi _ { 0 }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While minimizing an empirical risk as an estimate of the true risk $R ( \\pi _ { w } )$ is a common principle in machine learning (Vapnik, 1998), getting a reliable estimate based on the training data $D$ produced by $\\pi _ { 0 }$ is not straightforward. The logged bandit data $D$ is not only incomplete (i.e., we lack knowledge of $\\delta ( x _ { i } , y )$ for many $y \\in \\mathcal { V }$ that $\\pi _ { w }$ would have chosen differently from $\\pi _ { 0 }$ ), but it is also biased (i.e., the actions preferred by $\\pi _ { 0 }$ are over-represented). This is why existing work on training deep neural networks either requires full knowledge of the loss function, or requires the ability to interactively draw new samples $y _ { i } \\sim \\pi _ { w } ( Y \\mid x _ { i } )$ for any new policy $\\pi _ { w }$ . In our setting we can do neither – we have a fixed dataset $D$ that is limited to samples from $\\pi _ { 0 }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To nevertheless get a useful estimate of the empirical risk, we explicitly address both the bias and the variance of the risk estimate. To correct for sampling bias and handle missing data, we approach the risk estimation problem using importance sampling and thus remove the distribution mismatch between $\\pi _ { 0 }$ and $\\pi _ { w }$ (Langford et al., 2008; Owen, 2013; Swaminathan & Joachims, 2015b): ",
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+        "type": "equation",
+        "img_path": "images/439d9c8b47193f65c9b49a018a3c1e76d63effcdcc1fa247ad0a15fcc947ca9d.jpg",
+        "text": "$$\nR ( \\pi _ { w } ) = \\underset { x \\sim \\operatorname* { P r } ( X ) } { \\mathbb { E } } \\ \\underset { y \\sim \\pi _ { w } ( Y \\mid x ) } { \\mathbb { E } } \\left[ \\delta ( x , y ) \\right] = \\underset { x \\sim \\operatorname* { P r } ( X ) } { \\mathbb { E } } \\ \\underset { y \\sim \\pi _ { 0 } ( Y \\mid x ) } { \\mathbb { E } } \\left[ \\delta ( x , y ) \\frac { \\pi _ { w } ( y \\mid x ) } { \\pi _ { 0 } ( y \\mid x ) } \\right] .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The latter expectation can be estimated on a sample $D$ of $n$ bandit-feedback examples using the following IPS estimator (Langford et al., 2008; Owen, 2013; Swaminathan $\\&$ Joachims, 2015b): ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/c8a6a9ed5da2488b7304db204fa7beee515d0e16b5774a29f149ba52a32ba2a3.jpg",
+        "text": "$$\n\\hat { R } _ { I P S } ( \\pi _ { w } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\delta _ { i } \\frac { \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "This IPS estimator is unbiased and has bounded variance, if the logging policy has full support in the sense that $\\forall x , y : \\pi _ { 0 } ( y \\mid x ) \\ge \\epsilon > 0$ . While at first glance it may seem natural to directly train the parameters $w$ of a network to optimize this IPS estimate as an empirical risk, there are at least three obstacles to overcome. First, we will argue in the following section that the naive IPS estimator’s lack of equivariance makes it sub-optimal for use as an empirical risk for high-capacity models. Second, we have to find an efficient algorithm for minimizing the empirical risk, especially making it accessible to stochastic gradient descent (SGD) optimization. And, finally, we are faced with an unusual type of bias-variance trade-off since “distance” from the exploration policy impacts the variance of the empirical risk estimate for different $w$ . ",
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+    {
+        "type": "text",
+        "text": "3.2 EQUIVARIANT COUNTERFACTUAL RISK MINIMIZATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While Eq. (5) provides an unbiased empirical risk estimate, it exhibits the – possibly severe – problem of “propensity overfitting” when directly optimized within a learning algorithm (Swaminathan & Joachims, 2015c). It is a problem of overfitting to the choices $y _ { i }$ of the logging policy, and it occurs on top of the normal overfitting to the $\\delta _ { i }$ . Propensity overfitting is linked to the lack of equivariance of the IPS estimator: while the minimizer of true risk $R ( \\pi _ { w } )$ does not change when translating the loss by a constant (i.e., $\\forall x , y : \\delta ( x , y ) + c )$ by linearity of expectation, ",
+        "bbox": [
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+        "img_path": "images/438aa4b0c6df3ec6406f6c97f25843eeb6d92da69ff69cec36206c06ca9b2cf7.jpg",
+        "text": "$$\n\\begin{array} { r l r } { c + \\underset { w } { \\operatorname* { m i n } } \\underset { x \\sim \\mathrm { P r } ( X ) } { \\mathbb { E } } \\underset { y \\sim \\pi _ { w } ( Y \\mid x ) } { \\mathbb { E } } [ \\delta ( x , y ) ] } & { = } & { \\underset { w } { \\operatorname* { m i n } } \\underset { x \\sim \\mathrm { P r } ( X ) } { \\mathbb { E } } \\underset { y \\sim \\pi _ { w } ( Y \\mid x ) } { \\mathbb { E } } [ \\delta ( x , y ) + c ] } \\end{array}\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "the minimizer of the IPS-estimated empirical risk $\\hat { R } _ { I P S } ( \\pi _ { w } )$ can change dramatically for finite training samples, and ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/315a78ca03d0be9dbf8cece8e16bcd91c72bbbeef883072f1a63a3c3062fe36a.jpg",
+        "text": "$$\nc + \\operatorname* { m i n } _ { w } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\delta _ { i } \\frac { \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } \\neq \\operatorname* { m i n } _ { w } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } ( \\delta _ { i } + c ) \\frac { \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Intuitively, when $c$ shifts losses to be positive numbers, policies $\\pi _ { w }$ that put as little probability mass as possible on the observed actions have low risk estimates. If $c$ shifts the losses to the negative range, the exact opposite is the case. For either choice of $c$ , the choice of the policy eventually selected by the learning algorithm can be dominated by where $\\pi _ { 0 }$ happens to sample data, not by which actions have low loss. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The following self-normalized IPS estimator (SNIPS) addresses the propensity overfitting problem (Swaminathan & Joachims, 2015c) and is equivariant: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/1d1a80b3d979398b67ff32531670d6a59df1b5bf709b882208e2c7539881dfb5.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\hat { R } _ { S N I P S } ( \\pi _ { w } ) } & { = } & { \\frac { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\delta _ { i } \\frac { \\pi _ { w } ( y _ { i } | x _ { i } ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi _ { w } ( y _ { i } | x _ { i } ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In addition to being equivariant, this estimate can also have substantially lower variance than Eq. (5), since it exploits the knowledge that the denominator ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0684c5ce8d7591e27608955a9bcd3901995655f51823c6093e75420c26ad594c.jpg",
+        "text": "$$\nS \\ : = \\ \\sum _ { n } ^ { 1 } \\sum _ { i = 1 } ^ { n } { \\frac { \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "always has expectation 1: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8a3a2ecfd66a9af3a3b790984c373e6115a001938a0c58700730d01d7a18edf2.jpg",
+        "text": "$$\n\\mathbb { E } [ S ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\int \\frac { \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) \\operatorname* { P r } ( x _ { i } ) d y _ { i } d x _ { i } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\int 1 \\operatorname* { P r } ( x _ { i } ) d x _ { i } = 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The SNIPS estimator uses this knowledge as a multiplicative control variate (Swaminathan & Joachims, 2015c). While the SNIPS estimator has some bias, this bias asymptotically vanishes at a rate of $O ( { \\textstyle { \\frac { 1 } { n } } } )$ (Hesterberg, 1995). Using the SNIPS estimator as our empirical risk implies that we need to solve the following optimization problem for training: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e41f4d925bdd2c219ca17e91e194eabc0779798997e6c4a55580ee5e1cdb3dad.jpg",
+        "text": "$$\n\\hat { w } = \\underset { w \\in \\Re ^ { N } } { \\arg \\operatorname* { m i n } } \\hat { R } _ { S N I P S } ( \\pi _ { w } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Thus, we now turn to designing efficient optimization methods for this training objective. ",
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+    {
+        "type": "text",
+        "text": "3.3 TRAINING ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Unfortunately, the training objective in Eq. (11) does not permit stochastic gradient descent (SGD) optimization in the given form (see Appendix C), which presents an obstacle to efficient and effective training of the network. To remedy this problem, we will now develop a reformulation that retains both the desirable properties of the SNIPS estimator, as well as the ability to reuse established SGD training algorithms. Instead of optimizing a ratio as in Eq. (11), we will reformulate the problem into a series of constrained optimization problems. Let $\\hat { w }$ be a solution of Eq. (11), and at that solution let $S ^ { * }$ be the value of the control variate for $\\pi _ { \\hat { w } }$ as defined in Eq. (9). For simplicity, assume that the minimizer $\\hat { w }$ is unique. If we knew $S ^ { * }$ , we could equivalently solve the following constrained optimization problem: ",
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+        "type": "equation",
+        "img_path": "images/cf41f3191e570aef299202e6d4f7022283101e576e8cab07bfe3979d828eed7f.jpg",
+        "text": "$$\n\\hat { w } = \\underset { w \\in \\Re ^ { N } } { \\arg \\operatorname* { m i n } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\delta _ { i } \\frac { \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } \\mathrm { s u b j e c t } { \\mathrm { t o } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } = S ^ { * } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Of course, we do not actually know $S ^ { * }$ . However, we can do a grid search in $\\{ S _ { 1 } , \\ldots , S _ { k } \\}$ for $S ^ { * }$ and solve the above optimization problem for each value, giving us a set of solutions $\\{ \\hat { w } _ { 1 } , \\hdots , \\hat { w } _ { k } \\}$ . Note that $S$ is just a one-dimensional quantity, and that the sensible range we need to search for $S ^ { * }$ concentrates around 1 as $n$ increases (see Appendix B). To find the overall (approximate) $\\hat { w }$ that optimizes the SNIPS estimate, we then simply take the minimum: ",
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+        "text": "$$\n\\begin{array} { r c l } { \\hat { w } } & { = } & { \\underset { ( \\hat { w } _ { j } , S _ { j } ) } { \\arg \\operatorname* { m i n } } \\frac { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\delta _ { i } \\frac { \\pi _ { \\hat { w } _ { j } } ( y _ { i } | x _ { i } ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } } { S _ { j } } . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "This still leaves the question of how to solve each equality constrained risk minimization problem using SGD. Fortunately, we can perform an equivalent search for $S ^ { * }$ without constrained optimization. To this effect, consider the Lagrangian of the constrained optimization problem in Eq. (12) with $S _ { j }$ in the constraint instead of $S ^ { * }$ : ",
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+        "text": "$$\n\\therefore ( w , \\lambda ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\delta _ { i } \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } - \\lambda \\Bigg [ \\frac { 1 } { n } \\left( \\sum _ { i = 1 } ^ { n } \\frac { \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } \\right) - S _ { j } \\Bigg ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { ( \\delta _ { i } - \\lambda ) \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } + \\lambda S _ { j } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The variable $\\lambda$ is an unconstrained Lagrange multiplier. To find the minimum of Eq. (12) for a particular $S _ { j }$ , we need to minimize $L ( w , \\lambda )$ w.r.t. $w$ and maximize w.r.t. $\\lambda$ . ",
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+        "text": "$$\n\\hat { w } _ { j } = \\underset { w \\in \\Re ^ { N } } { \\arg \\operatorname* { m i n } } \\underset { \\lambda } { \\operatorname* { m a x } } L ( w , \\lambda )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "However, we are not actually interested in the constrained solution of Eq. (12) for any specific $S _ { j }$ . We are merely interested in exploring a certain range $S \\in [ S _ { 1 } , S _ { k } ]$ in our search for $S ^ { * }$ . So, we can reverse the roles of $\\lambda$ and $S$ , where we keep $\\lambda$ fixed and determine the corresponding $S$ in hindsight. In particular, for each $\\left\\{ \\lambda _ { 1 } , \\ldots , \\lambda _ { k } \\right\\}$ we solve ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/a66a20455f025ee61e21eed9f46204b842ee180adec15b8271e6db0011806bd7.jpg",
+        "text": "$$\n\\begin{array} { c c l } { \\hat { w } _ { j } } & { = } & { \\underset { w \\in \\Re ^ { N } } { \\arg \\operatorname* { m i n } } L ( w , \\lambda _ { j } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Note that the solution $\\hat { w } _ { j }$ does not depend on $S _ { j }$ , so we can compute $S _ { j }$ after we have found the minimum $\\hat { w } _ { j }$ . In particular, we can determine the $S _ { j }$ that corresponds to the given $\\lambda _ { j }$ using the necessary optimality conditions, ",
+        "bbox": [
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+        "img_path": "images/fe2f6a0688332c923108e0f6346cc9cf65ee85485915f6bfd502b8b32b46baf7.jpg",
+        "text": "$$\n\\frac { \\partial L } { \\partial w } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\partial \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\partial w } \\frac { ( \\delta _ { i } - \\lambda _ { j } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } = 0 ~ \\mathrm { a n d } ~ \\frac { \\partial L } { \\partial \\lambda _ { j } } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } - S _ { j } = 0 ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "by solving the second equality of Eq. (16). In this way, the sequence of $\\lambda _ { j }$ produces solutions $\\hat { w } _ { j }$ corresponding to a sequence of $\\{ S _ { 1 } , \\ldots , S _ { k } \\}$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To identify the sensible range of $S$ to explore, we can make use of the fact that Eq. (9) concentrates around its expectation of 1 for each $\\pi _ { w }$ as $n$ increases. Theorem 2 in Appendix B provides a characterization of how large the range needs to be. Furthermore, we can steer the exploration of $S$ via $\\lambda$ , since the resulting $S$ changes monotonically with $\\lambda$ : ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "$\\left( \\lambda _ { a } < \\lambda _ { b } \\right)$ and $\\hat { w } _ { a } \\neq \\hat { w } _ { b }$ are not equivalent optima in Eq. $( 1 5 ) ) \\Rightarrow ( S _ { a } < S _ { b } ) .$ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A more formal statement and proof are given as Theorem 1 in Appendix A. In the simplest form one could therefore perform a grid search on $\\lambda$ , but more sophisticated search methods are possible too. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "After this reformulation, the key computational problem is finding the solution of Eq. (15) for each $\\lambda _ { j }$ . Note that in this unconstrained optimization problem, the Lagrange multiplier effectively translates the loss values in the conventional IPS estimate: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/45573bcad24f886ad739ba7b07852e252cb4c14b7ffdead2d1052380187df6fd.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { \\hat { w } _ { j } } } & { { = } } & { { \\displaystyle \\arg \\operatorname* { m i n } _ { w } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\left( \\delta _ { i } - \\lambda _ { j } \\right) \\frac { \\pi _ { w } \\left( y _ { i } \\mid x _ { i } \\right) } { \\pi _ { 0 } \\left( y _ { i } \\mid x _ { i } \\right) } \\ ~ = ~ \\displaystyle \\arg \\operatorname* { m i n } _ { w } \\hat { R } _ { I P S } ^ { \\lambda _ { j } } ( \\pi _ { w } ) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/cd8a0b92b22cb4698570f765a38cf13020c79b4e44fb48bd878528ee1fc73bba.jpg",
+        "image_caption": [
+            "Figure 1: Learning curve of BanditNet. The $\\mathbf { X }$ -axis is the amount of bandit feedback, the y-axis is the test error. Given enough bandit feedback, Bandit-ResNet converges to the skyline performance. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We denote this $\\lambda$ -translated IPS estimate with $\\hat { R } _ { I P S } ^ { \\lambda } ( \\pi _ { w } )$ . Note that each such optimization problem is now in the form required for SGD, where we merely weight the derivative of the stochastic policy network $\\pi _ { w } ( y \\mid x )$ by a factor $( \\delta _ { i } - \\lambda _ { j } ) / \\pi _ { 0 } ( y _ { i } \\mid x _ { i } )$ . This opens the door for re-purposing existing fast methods for training deep neural networks, and we demonstrate experimentally that SGD with momentum is able to optimize our objective scalably. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Similar loss translations have previously been used in on-policy reinforcement learning (Williams, 1992), where they are motivated as minimizing the variance of the gradient estimate (Weaver & Tao, 2001; Greensmith et al., 2004). However, the situation is different in the off-policy setting we consider. First, we cannot sample new roll-outs from the current policy under consideration, which means we cannot use the standard variance-optimal estimator used in REINFORCE. Second, we tried using the (estimated) expected loss of the learned policy as the baseline as is commonly done in REINFORCE, but will see in the experiment section that this value for $\\lambda$ is far from optimal. Finally, it is unclear whether gradient variance, as opposed to variance of the ERM objective, is really the key issue in batch learning from bandit feedback. In this sense, our approach provides a rigorous justification and a constructive way of picking the value of $\\lambda$ in the off-policy setting – namely the value for which the corresponding $S _ { j }$ minimizes Eq. (13). In addition, one can further add variance regularization (Swaminathan & Joachims, 2015b) to improve the robustness of the risk estimate in Eq. (18) (see Appendix $\\mathrm { D }$ for details). ",
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+    {
+        "type": "text",
+        "text": "4 EMPIRICAL EVALUATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The empirical evaluation is designed to address three key questions. First, it verifies that deep models can indeed be trained effectively using our approach. Second, we will compare how the same deep neural network architecture performs under different types of data and training objectives – in particular, conventional cross-entropy training using full-information data. In order to be able to do this comparison, we focus on synthetic contextual bandit feedback data for training BanditNet that is sampled from the full-information labels. Third, we explore the effectiveness and fidelity of the approximate SNIPS objective. ",
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+    {
+        "type": "text",
+        "text": "For the following BanditNet experiments, we adapted the ResNet20 architecture (He et al., 2016) by replacing the conventional cross-entropy objective with our counterfactual risk minimization objective. We evaluate the performance of this Bandit-ResNet on the CIFAR-10 (Krizhevsky & Hinton, 2009) dataset, where we can compare training on full-information data with training on bandit feedback, and where there is a full-information test set for estimating prediction error. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To simulate logged bandit feedback, we perform the standard supervised to bandit conversion (Beygelzimer & Langford, 2009). We use a hand-coded logging policy that achieves about $49 \\%$ error rate on the training data, which is substantially worse than what we hope to achieve after learning. This emulates a real world scenario where one would bootstrap an operational system with a mediocre policy (e.g., derived from a small hand-labeled dataset) and then deploys it to log bandit feedback. This logged bandit feedback data is then used to train the Bandit-ResNet. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate the trained model using error rate on the held out (full-information) test set. We compare this model against the skyline of training a conventional ResNet using the full-information feedback from the 50,000 training examples. Both the conventional full-information ResNet as well as the Bandit-ResNet use the same network architecture, the same hyperparameters, the same data augmentation scheme, and the same optimization method that were set in the CNTK implementation of ResNet20. Since CIFAR10 does not come with a validation set for tuning the variance-regularization constant $\\gamma$ , we do not use variance regularization for Bandit-ResNet. The Lagrange multiplier $\\lambda \\in \\{ 0 . 6 5 , 0 . 7 , 0 . 7 5 , 0 . 8 , 0 . 8 5 , 0 . 9 , 0 . 9 5 , 1 . 0 , 1 . 0 5 \\}$ is selected on the training set via Eq. (13). The only parameter we adjusted for Bandit-ResNet is lowering the learning rate to 0.1 and slowing down the learning rate schedule. The latter was done to avoid confounding the Bandit-ResNet results with potential effects from early stopping, and we report test performance after 1000 training epochs, which is well beyond the point of convergence in all runs. ",
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+        "img_path": "images/e4862e220661d6ee257b7d8cad93054cf108ff766a686f231636136e2f318cee.jpg",
+        "image_caption": [
+            "Figure 2: The $\\mathbf { X }$ -axis shows the value of the Lagrange multiplier $\\lambda$ used for training. Left plot shows the test error. Right plot shows the value of the SNIPS objective and the normalizer $S$ . The size of the training set is $5 0 \\mathrm { k }$ bandit-feedback examples. "
+        ],
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+    {
+        "type": "text",
+        "text": "Learning curve. Figure 1 shows the prediction error of the Bandit-ResNet as more and more bandit feedback is provided for training. First, even though the logging policy that generated the bandit feedback has an error rate of $49 \\%$ , the prediction error of the policy learned by the Bandit-ResNet is substantially better. It is between $13 \\%$ and $8 . 2 \\%$ , depending on the amount of training data. Second, the horizontal line is the performance of a conventional ResNet trained on the full-information training set. It serves as a skyline of how good Bandit-ResNet could possibly get given that it is sampling bandit feedback from the same full-information training set. The learning curve in Figure 1 shows that Bandit-ResNet converges to the skyline performance given enough bandit feedback training data, providing strong evidence that our training objective and method can effectively extract the available information provided in the bandit feedback. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Effect of the choice of Lagrange multiplier. The left-hand plot in Figure 2 shows the test error of solutions $\\hat { w } _ { j }$ depending on the value of the Lagrange multiplier $\\lambda _ { j }$ used during training. It shows that $\\lambda$ in the range 0.8 to 1.0 results in good prediction performance, but that performance degrades outside this area. The SNIPS estimates in the right-hand plot of Figure 2 roughly reflects this optimal range, given empirical support for both the SNIPS estimator and the use of Eq. (13). ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "We also explored two other methods for selecting $\\lambda$ . First, we used the straightforward IPS estimator as the objective (i.e., $\\lambda = 0$ ), which leads to prediction performance worse than that of the logging policy (not shown). Second, we tried using the (estimated) expected loss of the learned policy as the baseline as is commonly done in REINFORCE. As Figure 1 shows, it is between 0.130 and 0.083 for the best policies we found. Figure 2 (left) shows that these baseline values are well outside of the optimum range. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Also shown in the right-hand plot of Figure 2 is the value of the control variate in the denominator of the SNIPS estimate. As expected, it increases from below 1 to above 1 as $\\lambda$ is increased. Note that large deviations of the control variate from 1 are a sign of propensity overfitting (Swaminathan & Joachims, 2015c). In particular, for all solutions $\\hat { w } _ { j }$ the estimated standard error of the control variate $S _ { j }$ was less than 0.013, meaning that the normal $9 5 \\%$ confidence interval for each $S _ { j }$ is contained in [0.974, 1.026]. If we see a $\\hat { w } _ { j }$ with control variate $S _ { j }$ outside this range, we should be suspicious of propensity overfitting to the choices of the logging policy and discard this solution. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS AND FUTURE WORK ",
+        "text_level": 1,
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "We proposed a new output layer for deep neural networks that enables the use of logged contextual bandit feedback for training. This type of feedback is abundant and ubiquitous in the form of interaction logs from autonomous systems, opening up the possibility of training deep neural networks on unprecedented amounts of data. In principle, this new output layer can replace the conventional cross-entropy layer for any network architecture. We provide a rigorous derivation of the training objective, linking it to an equivariant counterfactual risk estimator that enables counterfactual risk minimization. Most importantly, we show how the resulting training objective can be decomposed and reformulated to make it feasible for SGD training. We find that the BanditNet approach applied to the ResNet architecture achieves predictive accuracy comparable to conventional full-information training for visual object recognition. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "The paper opens up several directions for future work. First, it enables many new applications where contextual bandit feedback is readily available. Second, in settings where it is infeasible to log propensity-scored data, it would be interesting to combine BanditNet with propensity estimation techniques. Third, there may be improvements to BanditNet, like smarter search techniques for $S$ , more efficient counterfactual estimators beyond SNIPS, and the ability to handle continuous outputs. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "This research was supported in part by NSF Award IIS-1615706, a gift from Bloomberg, the Criteo Faculty Research Award program, and the Netherlands Organisation for Scientific Research (NWO) under project nr. 612.001.116. All content represents the opinion of the authors, which is not necessarily shared or endorsed by their respective employers and/or sponsors. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "A. Agarwal, D. Hsu, S. Kale, J. Langford, Lihong Li, and R. Schapire. Taming the monster: A fast and simple algorithm for contextual bandits. In ICML, 2014.   \nA. Beygelzimer and J. Langford. The offset tree for learning with partial labels. In KDD, pp. 129–138, 2009.   \nL. Bottou, J. Peters, J. Quinonero-Candela, D. Charles, M. Chickering, E. Portugaly, D. Ray, P. Simard, and E. Snelson. Counterfactual reasoning and learning systems: The example of computational advertising. JMLR, 14:3207–3260, 2013.   \nM. Dehghani, H. Zamani, A. Severyn, J. Kamps, and W. B. Croft. Neural ranking models with weak supervision. In SIGIR, pp. 65–74, 2017.   \nE. Greensmith, P.L. Bartlett, and J. Baxter. Variance reduction techniques for gradient estimates in reinforcement learning. JMLR, 5:1471–1530, 2004.   \nK. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In CVPR, 2016.   \nT. Hesterberg. Weighted average importance sampling and defensive mixture distributions. Technometrics, 37:185–194, 1995.   \nA. Joulin, L. van der Maaten, A. Jabri, and N. Vasilache. Learning visual features from large weakly supervised data. In ECCV, pp. 67–84, 2016.   \nA. Krizhevsky and G. Hinton. Learning multiple layers of features from tiny images. Technical report, Computer Science Department, University of Toronto, 2009.   \nJ. Langford, A. Strehl, and J. Wortman. Exploration scavenging. In ICML, pp. 528–535, 2008.   \nA.B. Owen. Monte Carlo theory, methods and examples. 2013.   \nP. Rosenbaum and D. Rubin. The central role of propensity score in observational studies for causal effects. Biometrica, 70:41–55, 1983.   \nI. Serban, C. Sankar, M. Germain, S. Zhang, Z. Lin, S. Subramanian, T. Kim, M. Pieper, S. Chandar, N. R. Ke, S. Mudumba, A. de Brebisson, J. M. R. Sotelo, D. Suhubdy, V. Michalski, A. Nguyen, J. Pineau, and Y. Bengio. A deep reinforcement learning chatbot. ArXiv e-prints, September 2017.   \nA. Swaminathan and T. Joachims. Counterfactual risk minimization: Learning from logged bandit feedback. In ICML, pp. 814–823, 2015a.   \nA. Swaminathan and T. Joachims. Batch learning from logged bandit feedback through counterfactual risk minimization. JMLR, 16:1731–1755, 2015b.   \nA. Swaminathan and T. Joachims. The self-normalized estimator for counterfactual learning. In NIPS, 2015c.   \nH. F. Trotter and J. W. Tukey. Conditional monte carlo for normal samples. In Symposium on Monte Carlo Methods, pp. 64–79, 1956.   \nV. Vapnik. Statistical Learning Theory. Wiley, Chichester, GB, 1998.   \nL. Weaver and N. Tao. The optimal reward baseline for gradient-based reinforcement learning. In UAI, pp. 538–545, 2001.   \nR. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8(3-4), May 1992.   \nC. Zhang, S. Bengio, M. Hardt, B. Recht, and O. Vinyals. Understanding deep learning requires rethinking generalization. CoRR, abs/1611.03530, 2016. ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
+            103,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX: STEERING THE EXPLORATION OF $S$ THROUGH $\\lambda$ . ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            101,
+            714,
+            119
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Let $\\lambda _ { a } < \\lambda _ { b }$ and let ",
+        "bbox": [
+            174,
+            132,
+            395,
+            147
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fe2fede9245725bb951ca68d9fc04feac769fe6a0b33d256c5ec76377878bb7b.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { \\hat { w } _ { a } } } & { { = } } & { { \\underset { w } { \\arg \\operatorname* { m i n } } \\hat { R } _ { I P S } ^ { \\lambda _ { a } } ( \\pi _ { w } ) } } \\\\ { { } } & { { } } & { { } } \\\\ { { \\hat { w } _ { b } } } & { { = } } & { { \\underset { w } { \\arg \\operatorname* { m i n } } \\hat { R } _ { I P S } ^ { \\lambda _ { b } } ( \\pi _ { w } ) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            150,
+            598,
+            207
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "If the optima $\\hat { w } _ { a }$ and $\\hat { w } _ { b }$ are not equivalent in the sense that $\\hat { R } _ { I P S } ^ { \\lambda _ { a } } ( \\pi _ { \\hat { w } _ { a } } ) \\neq \\hat { R } _ { I P S } ^ { \\lambda _ { a } } ( \\pi _ { \\hat { w } _ { b } } )$ and $\\hat { R } _ { I P S } ^ { \\lambda _ { b } } ( \\pi _ { \\hat { w } _ { a } } ) \\neq \\hat { R } _ { I P S } ^ { \\lambda _ { b } } ( \\pi _ { \\hat { w } _ { b } } )$ , then ",
+        "bbox": [
+            173,
+            210,
+            828,
+            244
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/993812cb7a249e0cfa231bab6d2fe86b9a62ba67764a07322f25824184751f69.jpg",
+        "text": "$$\nS _ { a } < S _ { b } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            449,
+            250,
+            516,
+            265
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Proof. Abbreviate $\\begin{array} { r } { f ( w ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\delta _ { i } \\frac { \\pi _ { w } ( y _ { i } | x _ { i } ) } { \\pi _ { 0 } ( y _ { i } | x _ { i } ) } } \\end{array}$ )) and g(w) = 1n Pni=1 πw(yi|xi)π0(yi|xi) . ",
+        "bbox": [
+            176,
+            279,
+            732,
+            301
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/817331a7ea04712e92efcb5347f1321e50253b3c8afc29df93748f832d18908e.jpg",
+        "text": "$$\n\\hat { R } _ { I P S } ^ { \\lambda } ( \\pi _ { w } ) = f ( w ) - \\lambda g ( w ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            306,
+            580,
+            325
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where $g ( w )$ corresponds to the value of the control variate $S$ . Since $\\hat { w } _ { a }$ and $\\hat { w } _ { b }$ are not equivalent optima, we know that ",
+        "bbox": [
+            169,
+            329,
+            823,
+            358
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4688449dab8260a53ece273af8df03c0b757320de7fd5770526a367de3bed56e.jpg",
+        "text": "$$\n\\begin{array} { r l r } { f ( \\hat { w } _ { a } ) - \\lambda _ { a } g ( \\hat { w } _ { a } ) } & { < } & { f ( \\hat { w } _ { b } ) - \\lambda _ { a } g ( \\hat { w } _ { b } ) } \\\\ { f ( \\hat { w } _ { b } ) - \\lambda _ { b } g ( \\hat { w } _ { b } ) } & { < } & { f ( \\hat { w } _ { a } ) - \\lambda _ { b } g ( \\hat { w } _ { a } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            361,
+            647,
+            400
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Adding the two inequalities and solving implies that ",
+        "bbox": [
+            173,
+            401,
+            519,
+            416
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6391a2c18e9244450678042b2b3b572e25c5278c6d98d5c2570808de9806dc69.jpg",
+        "text": "$$\n\\begin{array} { r l } & { f ( \\hat { w } _ { a } ) - \\lambda _ { a } g \\big ( \\hat { w } _ { a } \\big ) + f \\big ( \\hat { w } _ { b } \\big ) - \\lambda _ { b } g \\big ( \\hat { w } _ { b } \\big ) < f \\big ( \\hat { w } _ { b } \\big ) - \\lambda _ { a } g \\big ( \\hat { w } _ { b } \\big ) + f \\big ( \\hat { w } _ { a } \\big ) - \\lambda _ { b } g \\big ( \\hat { w } _ { a } \\big ) } \\\\ & { \\lambda _ { a } g \\big ( \\hat { w } _ { a } \\big ) + \\lambda _ { b } g \\big ( \\hat { w } _ { b } \\big ) > \\lambda _ { a } g \\big ( \\hat { w } _ { b } \\big ) + \\lambda _ { b } g \\big ( \\hat { w } _ { a } \\big ) } \\\\ & { \\big ( \\lambda _ { b } - \\lambda _ { a } \\big ) g \\big ( \\hat { w } _ { b } \\big ) > \\big ( \\lambda _ { b } - \\lambda _ { a } \\big ) g \\big ( \\hat { w } _ { a } \\big ) } \\\\ & { g \\big ( \\hat { w } _ { b } \\big ) > g \\big ( \\hat { w } _ { a } \\big ) } \\\\ & { S _ { b } > S _ { a } \\quad \\boxed { 1 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            419,
+            784,
+            510
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "B APPENDIX: CHARACTERIZING THE RANGE OF $S$ TO EXPLORE.",
+        "text_level": 1,
+        "bbox": [
+            174,
+            526,
+            728,
+            544
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. Let $p \\ \\leq \\ \\pi _ { 0 } ( y \\mid x )$ be a lower bound on the propensity for the logging policy, then constraining the solution of Eq. (11) to the $w$ with control variate $S \\in [ 1 - \\epsilon , 1 + \\epsilon ]$ for a training set of size n will not exclude the minimizer of the true risk $\\begin{array} { r } { w ^ { * } = \\arg \\operatorname* { m i n } _ { w \\in W } R ( \\pi _ { w } ) } \\end{array}$ in the policy space $W$ with probability at least ",
+        "bbox": [
+            173,
+            556,
+            825,
+            614
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/de77d36fbbf49ad97e0daf936fbb3723fc582a327b63f254b1c9225c8949fba1.jpg",
+        "text": "$$\n\\begin{array} { r } { 1 - 2 \\exp \\left( - 2 n \\epsilon ^ { 2 } p ^ { 2 } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            617,
+            560,
+            637
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Proof. For the optimal $w ^ { * }$ , let ",
+        "bbox": [
+            174,
+            650,
+            375,
+            665
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/63ebf73786b8e5c43cf49ac56401462622ca8ed19a1dcde869f3548c6f85b3e5.jpg",
+        "text": "$$\n\\begin{array} { r c l } { S } & { = } & { \\displaystyle \\sum _ { i = 1 } ^ { n } { \\frac { \\pi _ { w ^ { * } } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            669,
+            584,
+            710
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "be the control variate in the denominator of the SNIPS estimator. $S$ is a random variable that is a sum of bounded random variables between 0 and ",
+        "bbox": [
+            173,
+            713,
+            825,
+            742
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/713360765d851009c4e06b7dabe047589fbc2614d0c007f96fc09530388f09f8.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { x , y } { \\frac { \\pi _ { w ^ { * } } ( y \\mid x ) } { \\pi _ { 0 } ( y \\mid x ) } } \\leq { \\frac { 1 } { p } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            743,
+            558,
+            779
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We can bound the probability that the control variate $S$ of the optimum $w ^ { * }$ lies outside of $[ 1 - \\epsilon , 1 + \\epsilon ]$ via Hoeffding’s inequality: ",
+        "bbox": [
+            174,
+            782,
+            825,
+            811
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8523c59571ebb04094a5e68dbfc605f382dbbc21a0374ce08bc1e31d2b3afae3.jpg",
+        "text": "$$\n\\begin{array} { r c l } { P ( | S - 1 | \\ge \\epsilon ) } & { \\le } & { 2 \\exp \\left( \\displaystyle \\frac { - 2 n ^ { 2 } \\epsilon ^ { 2 } } { n ( 1 / p ) ^ { 2 } } \\right) } \\\\ & { = } & { 2 \\exp \\left( - 2 n \\epsilon ^ { 2 } p ^ { 2 } \\right) . \\quad \\bigsqcup } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            815,
+            651,
+            873
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The same argument applies to any individual policy $\\pi _ { w }$ , not just $w ^ { * }$ . Note, however, that it can still be highly likely that at least one policy $\\pi _ { w }$ with $w \\in W$ shows a large deviation in the control variate for high-capacity $W$ , which can lead to propensity overfitting when using the naive IPS estimator. ",
+        "bbox": [
+            173,
+            881,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "C APPENDIX: WHY DIRECT STOCHASTIC OPTIMIZATION OF RATIO ESTIMATORS IS NOT POSSIBLE. ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            101,
+            753,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Suppose we have a dataset of $n$ BLBF samples $D = \\{ ( x _ { 1 } , y _ { 1 } , \\delta _ { 1 } , p _ { 1 } ) \\dots ( x _ { n } , y _ { n } , \\delta _ { n } , p _ { n } ) \\}$ where each instance is an i.i.d. sample from the data generating distribution. In the sequel we will be considering two datasets of $n + 1$ samples, $D ^ { \\prime } = \\bar { D } \\cup \\{ ( x ^ { \\prime } , \\bar { y ^ { \\prime } } , \\delta ^ { \\prime } , p ^ { \\prime } ) \\}$ and $D ^ { \\prime \\prime } { = } D \\bar { \\cup } \\{ ( x ^ { \\prime \\prime } , y ^ { \\prime \\prime } , \\delta ^ { \\prime \\prime } , p ^ { \\prime \\prime } ) \\}$ where $\\bar { ( } x ^ { \\prime } , y ^ { \\prime } , \\delta ^ { \\prime } , p ^ { \\prime } ) \\neq ( x ^ { \\prime \\prime } , y ^ { \\prime \\prime } , \\delta ^ { \\prime \\prime } , p ^ { \\prime \\prime } )$ and $( x ^ { \\prime } , y ^ { \\prime } , \\delta ^ { \\prime } , p ^ { \\prime } ) , ( x ^ { \\prime \\prime } , y ^ { \\prime \\prime } , \\delta ^ { \\prime \\prime } , p ^ { \\prime \\prime } ) \\notin D$ . ",
+        "bbox": [
+            173,
+            151,
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+            210
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For notational convenience, let $\\begin{array} { r } { f _ { i } : = \\delta _ { i } \\frac { \\pi _ { w } \\left( y _ { i } | x _ { i } \\right) } { \\pi _ { 0 } \\left( y _ { i } | x _ { i } \\right) } } \\end{array}$ , and ˙fi := ∇wfi; gi := πw(yi|xi)π (y |x ) , and $\\dot { g } _ { i } : = \\nabla _ { w } g _ { i }$ .   \nFirst consider the vanilla IPS risk estimate of Eq. (5). ",
+        "bbox": [
+            173,
+            215,
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+            258
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1588150c3eec7d7df5aca22c72b2ab5a04dbb1394bb4248520f5d8e0955f8e61.jpg",
+        "text": "$$\n\\hat { R } _ { I P S } ( \\pi _ { w } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\delta _ { i } \\frac { \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } f _ { i } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            267,
+            653,
+            308
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To maximize this estimate using stochastic optimization, we must construct an unbiased gradient estimate. That is, we randomly select one sample from $D$ and compute a gradient $\\alpha ( ( x _ { i } , y _ { i } , \\delta _ { i } , p _ { i } ) )$ and we require that ",
+        "bbox": [
+            173,
+            316,
+            825,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1ab720c7769afb3ebdb8914216805263124d87b3de99a71efc748aa2ac21f818.jpg",
+        "text": "$$\n\\nabla _ { \\boldsymbol { w } } \\hat { R } _ { I P S } ( \\pi _ { \\boldsymbol { w } } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\dot { f } _ { i } = \\mathbb { E } _ { i \\sim D } \\left[ \\alpha ( ( x _ { i } , y _ { i } , \\delta _ { i } , p _ { i } ) ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            315,
+            368,
+            681,
+            410
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Here the expectation is over our random choice of 1 out of $n$ samples. Observe that $\\alpha ( ( x _ { i } , y _ { i } , \\delta _ { i } , \\bar { p } _ { i } ) ) = \\dot { f } _ { i }$ suffices (and indeed, this corresponds to vanilla SGD): ",
+        "bbox": [
+            173,
+            417,
+            825,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d33be3e59c7e86f74a41738c8be88ce8982b4cd0c087e512206944d2c878155e.jpg",
+        "text": "$$\n\\mathbb { E } _ { i \\sim D } \\left[ \\alpha \\big ( ( x _ { i } , y _ { i } , \\delta _ { i } , p _ { i } ) \\big ) \\right] = \\sum _ { i = 1 } ^ { n } \\frac { 1 } { n } \\alpha \\big ( ( x _ { i } , y _ { i } , \\delta _ { i } , p _ { i } ) \\big ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\dot { f } _ { i } = \\nabla _ { w } \\hat { R } _ { I P S } \\big ( \\pi _ { w } \\big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            230,
+            457,
+            766,
+            500
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Other choices of $\\alpha ( \\cdot )$ can also produce unbiased gradient estimates, and this leads to the study of stochastic variance-reduced gradient optimization. ",
+        "bbox": [
+            174,
+            508,
+            823,
+            537
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Now let us attempt to construct an unbiased gradient estimate for Eq. (8): ",
+        "bbox": [
+            173,
+            542,
+            655,
+            559
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2065fde062f2448390b9a36989f12b7892500ea33a7035f2eca2d9548d6f4dfd.jpg",
+        "text": "$$\n\\hat { R } _ { S N I P S } ( \\pi _ { w } ) = \\frac { \\sum _ { i = 1 } ^ { n } f _ { i } } { \\sum _ { i = 1 } ^ { n } g _ { i } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            568,
+            586,
+            604
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Suppose such a gradient estimate exists, $\\beta ( ( x _ { i } , y _ { i } , \\delta _ { i } , p _ { i } ) )$ . Then, ",
+        "bbox": [
+            173,
+            612,
+            599,
+            628
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e94667e9e4c21d682dd7f8783a5f0066fa3b5857652f22b14f0b6f3e13ccb6e4.jpg",
+        "text": "$$\n\\nabla _ { w } \\hat { R } _ { S N I P S } ( \\pi _ { w } ) = \\nabla _ { w } \\frac { \\sum _ { i = 1 } ^ { n } f _ { i } } { \\sum _ { i = 1 } ^ { n } g _ { i } } = \\mathbb { E } _ { i \\sim D } \\left[ \\beta ( ( x _ { i } , y _ { i } , \\delta _ { i } , p _ { i } ) ) \\right] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\beta ( ( x _ { i } , y _ { i } , \\delta _ { i } , p _ { i } ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            207,
+            637,
+            789,
+            678
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This identity is true for any sample of BLBF instances – in particular, for $D ^ { \\prime }$ and $D ^ { \\prime \\prime }$ : ",
+        "bbox": [
+            174,
+            694,
+            733,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/52240fb1035be530901b71bccb035367b37afa1e10f9108d3b2ee59ebe26d51d.jpg",
+        "text": "$$\n\\begin{array} { l l } { \\displaystyle \\nabla _ { w } \\frac { \\sum _ { i = 1 } ^ { n } f _ { i } + f ^ { \\prime } } { \\sum _ { i = 1 } ^ { n } g _ { i } + g ^ { \\prime } } = \\sum _ { i = 1 } ^ { n } \\frac { 1 } { n + 1 } \\beta ( ( x _ { i } , y _ { i } , \\delta _ { i } , p _ { i } ) ) + \\frac { \\beta ( ( x ^ { \\prime } , y ^ { \\prime } , \\delta ^ { \\prime } , p ^ { \\prime } ) ) } { n + 1 } , } \\\\ { \\displaystyle \\nabla _ { w } \\frac { \\sum _ { i = 1 } ^ { n } f _ { i } + f ^ { \\prime \\prime } } { \\sum _ { i = 1 } ^ { n } g _ { i } + g ^ { \\prime \\prime } } = \\sum _ { i = 1 } ^ { n } \\frac { 1 } { n + 1 } \\beta ( ( x _ { i } , y _ { i } , \\delta _ { i } , p _ { i } ) ) + \\frac { \\beta ( ( x ^ { \\prime \\prime } , y ^ { \\prime \\prime } , \\delta ^ { \\prime \\prime } , p ^ { \\prime \\prime } ) ) } { n + 1 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            258,
+            718,
+            740,
+            804
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Subtracting these two equations, ",
+        "bbox": [
+            173,
+            810,
+            388,
+            825
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c9120367f8e18ee530059b49b02efe8c0c839c8b75628460a0aeafecbb4b6527.jpg",
+        "text": "$$\n\\nabla _ { w } \\left( \\frac { \\sum _ { i = 1 } ^ { n } f _ { i } + f ^ { \\prime } } { \\sum _ { i = 1 } ^ { n } g _ { i } + g ^ { \\prime } } - \\frac { \\sum _ { i = 1 } ^ { n } f _ { i } + f ^ { \\prime \\prime } } { \\sum _ { i = 1 } ^ { n } g _ { i } + g ^ { \\prime \\prime } } \\right) = \\frac { \\beta ( ( x ^ { \\prime } , y ^ { \\prime } , \\delta ^ { \\prime } , p ^ { \\prime } ) ) - \\beta ( ( x ^ { \\prime \\prime } , y ^ { \\prime \\prime } , \\delta ^ { \\prime \\prime } , p ^ { \\prime \\prime } ) ) } { n + 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            223,
+            834,
+            774,
+            871
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The LHS clearly depends on $\\{ ( x _ { i } , y _ { i } , \\delta _ { i } , p _ { i } ) \\} _ { i = 1 } ^ { n }$ in general, while the RHS does not! This contradiction indicates that no construction of $\\beta$ that only looks at a sub-sample of the data can yield an unbiased gradient estimate of $\\hat { R } _ { S N I P S } ( \\pi _ { w } )$ . ",
+        "bbox": [
+            174,
+            878,
+            823,
+            925
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "D APPENDIX: VARIANCE REGULARIZATION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            558,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Unlike in conventional supervised learning, a counterfactual empirical risk estimator like $\\hat { R } _ { I P S } ( \\pi _ { w } )$ can have vastly different variances $\\mathrm { V a r } ( \\hat { R } _ { I P S } ( \\pi _ { w } ) )$ for different $\\pi _ { w }$ in the hypothesis space (and $\\hat { R } _ { S N I P S } ( \\pi _ { w } )$ as well) (Swaminathan $\\&$ Joachims, 2015b). Intuitively, the “closer” the particular $\\pi _ { w }$ is to the exploration policy $\\pi _ { 0 }$ , the larger the effective sample size (Owen, 2013) will be and the smaller the variance of the empirical risk estimate. For the optimization problems we solve in Eq. (18), this means that we should trust the $\\lambda$ -translated risk estimate $\\hat { R } _ { I P S } ^ { \\lambda _ { j } } ( \\pi _ { w } )$ more for some $w$ than for others, as we use $\\hat { R } _ { I P S } ^ { \\lambda _ { j } } ( \\pi _ { w } )$ only as a proxy for finding the policy that minimizes its expected value (i.e., the true loss). To this effect, generalization error bounds that account for this variance difference (Swaminathan & Joachims, 2015b) motivate a new type of overfitting control. This leads to the following training objective (Swaminathan $\\&$ Joachims, 2015b), which can be thought of as a more reliable version of Eq. (18): ",
+        "bbox": [
+            173,
+            132,
+            825,
+            299
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f51cd62017e9b17e83ef1ea7d024f710266bbc133855afa671b5a2fec855aad7.jpg",
+        "text": "$$\n\\hat { w } _ { j } = \\underset { w } { \\arg \\operatorname* { m i n } } \\left[ \\hat { R } _ { I P S } ^ { \\lambda _ { j } } ( \\pi _ { w } ) + \\gamma \\sqrt { \\frac { \\widehat { \\mathrm { V a r } } ( \\hat { R } _ { I P S } ^ { \\lambda _ { j } } ( \\pi _ { w } ) ) } { n } } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            305,
+            673,
+            356
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\widehat { \\mathrm { V a r } } ( \\hat { R } _ { I P S } ^ { \\lambda _ { j } } ( \\pi _ { w } ) )$ is the estimated variance of $\\hat { R } _ { I P S } ^ { \\lambda _ { j } } ( \\pi _ { w } )$ on the training data, and $\\gamma$ is a regwe optimize the upper confidence interval, which depends on the variance of the risk estimate for each $\\pi _ { w }$ . While this objective again does not permit SGD optimization in its given form, it has been shown that a Taylor-majorization can be used to successively upper bound the objective in Eq. (35), and that typically a small number of iterations suffices to converge to a local optimum (Swaminathan & Joachims, 2015b). Each such Taylor-majorization is again of a form ",
+        "bbox": [
+            173,
+            363,
+            825,
+            464
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/702cb1e2bdf44f79bc5398baea730acdeaf3fcf9b298632eb8a8e89cbf650196.jpg",
+        "text": "$$\n{ \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } \\left[ A \\left( { \\frac { \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } } \\right) + B \\left( { \\frac { \\pi _ { w } ( y _ { i } \\mid x _ { i } ) } { \\pi _ { 0 } ( y _ { i } \\mid x _ { i } ) } } \\right) ^ { 2 } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            470,
+            645,
+            513
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "for easily computable constants $A$ and $B$ (Swaminathan & Joachims, 2015b), which allows for SGD optimization. ",
+        "bbox": [
+            174,
+            518,
+            823,
+            547
+        ],
+        "page_idx": 11
+    }
+]

parse/train/ZyugLlWzdO/ZyugLlWzdO.md

@@ -0,0 +1,474 @@
+# Is Heterophily A Real Nightmare For Graph Neural Networks To Do Node Classification?
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Graph Neural Networks (GNNs) extend basic Neural Networks (NNs) by using   
+2 the graph structures based on the relational inductive bias (homophily assumption).   
+3 Though GNNs are believed to outperform NNs in real-world tasks, performance   
+4 advantages of GNNs over graph-agnostic NNs seem not generally satisfactory.   
+5 Heterophily has been considered as a main cause and numerous works have been   
+6 put forward to address it. In this paper, we first show that not all cases of heterophily   
+7 are harmful for GNNs with aggregation operation. Then, we propose new metrics   
+8 based on a similarity matrix which considers the influence of graph structure and   
+9 input features on GNNs. The metrics demonstrate advantages over the commonly   
+10 used homophily metrics by tests on synthetic graphs. From the metrics and the   
+11 observations, we find some cases of harmful heterophily can be addressed by   
+12 diversification operation. With this fact and knowledge of filterbanks, we propose   
+13 the Adaptive Channel Mixing (ACM) framework to adaptively exploit aggregation,   
+14 diversification and identity operations in each GNN layer to address harmful   
+15 heterophily. We validate the ACM-augmented baselines with 11 real-world node   
+16 classification tasks. They consistently achieve significant performance gain and   
+17 exceed the state-of-the-art GNNs on most of the tasks without incurring significant   
+18 computational burden.
+
+# 19 1 Introduction
+
+20 Deep Neural Networks (NNs) $[ \overline { { 1 8 } } ]$ have revolutionized many machine learning areas, including   
+21 image recognition $ { \mathbb { I } } ^ { [ 1 2 ] }$ , speech recognition $\mathbb { m }$ and natural language processing [2], etc.One major   
+22 strength is their capacity and effectiveness of learning latent representation from Euclidean data.   
+23 Recently, the focus has been put on its applications on non-Euclidean data [4], e.g., relational data   
+24 or graphs. Combining graph signal processing and convolutional neural networks $\mathbb { \lVert \rVert }$ , numerous   
+25 Graph Neural Networks (GNNs) [29, 7, 12, 30, 15, 24] have been proposed which empirically out  
+26 perform traditional neural networks on graph-based machine learning tasks, e.g., node classification,   
+27 graph classification, link prediction and graph generation, etc.GNNs are built on the homophily   
+28 assumption[26], i.e., connected nodes tend to share similar attributes with each other $\mathbb { \ m }$ , which   
+29 offers additional information besides node features. Such relational inductive bias $\mathbb { \left[ 3 \right] }$ is believed to   
+30 be a key factor leading to GNNs’ superior performance over NNs’ in many tasks.   
+31 Nevertheless, growing evidence shows that GNNs do not always gain advantages over traditional NNs   
+32 when dealing with relational data. In some cases, even simple Multi-Layer Perceptrons (MLPs) can   
+33 outperform GNNs by a large margin [35, 22, 5]. An important reason for the performance degradation   
+34 is believed to be the heterophily problem, i.e., connected nodes tend to have different labels which   
+35 makes the homophily assumption fail. Heterophily challenge has received attention recently and   
+36 there are increasing number of models being put forward to address this problem [35, 22, 5, 34, 33].   
+37 Contributions In this paper, we first demonstrate that not all heterophilous graphs are harmful for   
+38 aggregation-based GNNs and the existing metrics of homophily are insufficient to decide whether the   
+39 aggregation operation will make nodes less distinguishable or not. By constructing a similarity matrix   
+40 from backpropagation analysis, we derive new metrics to depict how much GNNs are influenced by   
+41 the graph structure and node features. We show the advantage of our metrics over the existing metrics   
+42 by comparing the ability of characterizing the performance of two baseline GNNs on synthetic graphs   
+43 of different levels of homophily. From the similarity matrix, we find that diversification operation   
+44 is able to address some harmful heterophily cases, and then based on which we propose Adaptive   
+45 Channel Mixing (ACM) GNN framework. The experiments on the synthetic datasets, real-world   
+46 datasets and the ablation studies consistently show that baseline GNN augmented by ACM framework   
+47 is able to obtain significant performance boost on node classification tasks on heterophilous graphs.   
+48 The rest of this paper is mainly organized as follows: In section 2, we introduce the notation and the   
+49 background knowledge. In section $\boxed { 3 }$ we conduct node-wise heterophily analysis, derive new metrics   
+50 based on a similarity matrix and conduct experiments to show their advantage. In section $\boxed { 4 . 3 }$ we   
+51 propose the ACM-GNN framework to adaptively utilize the information from different filterbank   
+52 channels to address heterophily problem. In section $5 ,$ we discuss the related works and clarify   
+53 the differences to our method. In section $\textcircled{6}$ we provide empirical evaluations on ACM framework,   
+54 including ablation study and tests on 11 real-world node classification tasks.
+
+# 55 2 Preliminaries
+
+56 We introduce the related notation and background knowledge. We use bold fonts for vectors (e.g.,   
+57 $v )$ . Suppose we have an undirected connected graph $\mathcal { G } = ( \nu , \mathcal { E } , A )$ , where $\nu$ is the node set with   
+58 $| \nu | = N ;$ ; $\mathcal { E }$ is the edge set without self-loop; $\bar { A ( \in \mathbb { R } ^ { N \times N } }$ is the symmetric adjacency matrix with   
+59 $A _ { i , j } = 1$ iff $e _ { i j } \in \mathcal { E }$ , otherwise $A _ { i , j } = 0$ . We use $D$ to denote the diagonal degree matrix of $\mathcal { G }$ , i.e.,   
+60 $\begin{array} { r } { D _ { i , i } = d _ { i } = \check { \sum } _ { j } A _ { i , j } } \end{array}$ and use ${ \mathcal { N } } _ { i }$ to denote the neighborhood set of node $i$ , i.e., ${ \mathcal { N } } _ { i } = \{ j : e _ { i j } \in { \mathcal { E } } \}$ .   
+61 A graph signal is a vector $\pmb { x } \in \mathbb { R } ^ { N }$ defined on $\nu$ , where $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ is defined on the node $i$ . We also have a   
+62 feature matrix $X \in \mathbb { R } ^ { N \times F }$ , whose columns are graph signals and whose $i$ -th row $X _ { i , \astrosun }$ : is a feature   
+63 vector of node $i$ . We use $Z \in \mathbb { R } ^ { N \times C }$ to denote the label encoding matrix, whose $i$ -th row $Z _ { i , \cdot }$ : is the   
+64 one-hot encoding of the label of node $i$ . The $i$ -th column of the identity matrix $I$ is denoted by $e _ { i }$ .
+
+# 65 2.1 Graph Laplacian, Affinity Matrix and Their Variants
+
+66 The (combinatorial) graph Laplacian is defined as $L = D - A$ , which is Symmetric Positive   
+67 Semi-Definite (SPSD) $\dot { \left[ 6 \right] }$ . Its eigendecomposition gives ${ \cal L } = U \Lambda U ^ { T }$ , where the columns $\mathbf { \Delta } \mathbf { u } _ { i }$ of   
+68 $U \in \mathbb { R } ^ { N \times N }$ are orthonormal eigenvectors, namely the graph Fourier basis, $\boldsymbol { \Lambda } = \operatorname { d i a g } ( \lambda _ { 1 } , \ldots , \lambda _ { N } )$   
+69 with $\lambda _ { 1 } \leq \cdots \leq \lambda _ { N }$ , and these eigenvalues are also called frequencies. The graph Fourier transform   
+70 of the graph signal $_ { \textbf { \em x } }$ is defined as $\pmb { x } _ { \mathcal { F } } = { \cal U } ^ { - 1 } \pmb { x } = { \cal U } ^ { T } \pmb { x } = [ \pmb { u } _ { 1 } ^ { T } \pmb { x } , \dots , \pmb { u } _ { N } ^ { T } \pmb { x } ] ^ { \hat { T } }$ , where $\pmb { u } _ { i } ^ { T } \pmb { x }$ is the   
+71 component of $_ { \textbf { \em x } }$ in the direction of $\mathbf { \Delta } \mathbf { u } _ { i }$ .   
+72 In additional to $L$ , some variants are also commonly used, e.g., the symmetric normalized Laplacian   
+73 $L _ { \mathrm { s v m } } = D ^ { - 1 / 2 } L D ^ { - 1 / 2 } = I - D ^ { - 1 / 2 } A D ^ { - 1 / 2 }$ and the random walk normalized Laplacian $L _ { \mathrm { r w } } =$   
+74 $D ^ { \dot { - } 1 } L = I - D ^ { - 1 } A$ . The affinity (transition) matrices can be derived from the Laplacians, e.g.,   
+75 $A _ { \mathrm { r w } } = I - L _ { \mathrm { r w } } = D ^ { - 1 } A$ , $A _ { \mathrm { s y m } } = \bar { I } - L _ { \mathrm { s y m } } = D ^ { - 1 / 2 } A D ^ { - 1 / 2 }$ and are considered to be low-pass   
+76 filters $\lVert 2 5 \rVert$ . Their eigenvalues satisfy $\lambda _ { i } ( A _ { \mathrm { r w } } ) = \lambda _ { i } ( A _ { \mathrm { s y m } } ) = 1 - \lambda _ { i } ( L _ { \mathrm { s y m } } ) = 1 - \lambda _ { i } ( L _ { \mathrm { r w } } ) \in ( - \bar { 1 } , 1 ]$   
+77 Applying the renormalization trick $\mathbb { I } \breve { \Sigma } \mathbb { I }$ to affinity and Laplacian matrices respectively leads to   
+78 $\hat { A } _ { \mathrm { s y m } } ^ { - 1 } = \bar { \tilde { D } } ^ { - 1 / 2 } \tilde { A } \tilde { D } ^ { - 1 / 2 }$ and $\hat { L } _ { \mathrm { s y m } } = I - \hat { A } _ { \mathrm { s y m } }$ , where ${ \tilde { A } } \equiv A + I$ and $\tilde { D } \equiv D + I$ . The renormalized   
+79 affinity matrix essentially adds a self-loop to each node in the graph, and is widely used in Graph   
+80 Convolutional Network (GCN) $\mathbb { \lVert 1 5 \rVert }$ as follows,
+
+$$
+Y = \mathrm { s o f t m a x } ( { \hat { A } } _ { \mathrm { s y m } } \mathrm { R e L U } ( { \hat { A } } _ { \mathrm { s y m } } X W _ { 0 } ) W _ { 1 } )
+$$
+
+where 81 $W _ { 0 } \in \mathbb { R } ^ { F \times F _ { 1 } }$ and $W _ { 1 } \in \mathbb { R } ^ { F _ { 1 } \times O }$ are learnable parameter matrices. GCN can be trained by 82 minimizing the following cross entropy loss
+
+$$
+\mathcal { L } = - \mathrm { t r a c e } ( Z ^ { T } \log Y )
+$$
+
+83 where $\log ( \cdot )$ is a component-wise logarithm operation. The random walk renormalized matrix   
+84 $\hat { A } _ { \mathrm { r w } } = \tilde { D } ^ { - 1 } \tilde { A }$ , which shares the same eigenvalues as $\hat { A } _ { \mathrm { s y m } }$ , can also be applied in GCN. The
+
+corresponding Laplacian is defined as 85 $\hat { L } _ { \mathrm { r w } } = I - \hat { A } _ { \mathrm { r w } }$ . $\hat { A } _ { \mathrm { r w } }$ is essentially a random walk matrix and 86 behaves as a mean aggregator that is applied in spatial-based GNNs [12, 11]. To bridge the spectral and spatial methods, we use 87 $\hat { A } _ { r w }$ in the paper.
+
+# 2.2 Metrics of Homophily
+
+89 The metrics of homophily are defined by considering different relations between node labels and   
+90 graph structures defined by adjacency matrix. There are three commonly used homophilies: edge   
+91 homophily [1, 35], node homophily $\pm \overbrace { \mathbb { L } 8 } \Vert$ , and class homophily [21] 1 defined as follows:
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { I } _ { \mathrm { e d g e } } ( \mathcal { G } ) = \frac { \big | \{ e _ { u v } \mid e _ { u v } \in \mathcal { E } , Z _ { u , : } = Z _ { v , : } \} \big | } { | \mathcal { E } | } , ~ H _ { \mathrm { n o t e } } ( \mathcal { G } ) = \frac { 1 } { | \mathcal { V } | } \displaystyle \sum _ { v \in \mathcal { V } } \frac { \big | \{ u \mid u \in \mathcal { N } _ { v } , Z _ { u , : } = Z _ { v , : } \} \big | } { d _ { v } } , } \\ { \displaystyle \mathcal { I } _ { \mathrm { c l a s } } ( \mathcal { G } ) = \frac { 1 } { C - 1 } \displaystyle \sum _ { k = 1 } ^ { C } \bigg [ h _ { k } - \frac { \big | \{ v \mid Z _ { v , k } = 1 \} \big | } { N } \bigg ] _ { + } , ~ h _ { k } = \frac { \sum _ { v \in \mathcal { V } } \big | \{ u \mid Z _ { v , k } = 1 , u \in \mathcal { N } _ { v } , Z _ { u , : } = Z _ { v , : } \} \big | } { \sum _ { v \in \{ v | Z _ { v , k } = 1 \} } d _ { v } } , ~ } \end{array}
+$$
+
+92 where $[ a ] _ { + } = \operatorname* { m a x } ( a , 0 )$ ; $h _ { k }$ is the class-wise homophily metric $[ [ 2 1 ] ]$ . They are all in the range   
+93 of $[ 0 , 1 ]$ and a value close to 1 corresponds to strong homophily while a value close to 0 indicates   
+94 strong heterophily. $H _ { \mathrm { e d g e } } ( { \mathcal { G } } )$ measures the proportion of edges that connect two nodes in the same   
+95 class; $H _ { \mathrm { n o d e } } ( \mathcal { G } )$ evaluates the average proportion of edge-label consistency of all nodes; $H _ { \mathrm { c l a s s } } ( \mathcal G )$   
+96 tries to avoid the sensitivity to imbalanced class, which can cause $H _ { \mathrm { e d g e } }$ misleadingly large. The   
+97 above definitions are all based on the graph-label consistency and imply that the inconsistency will   
+98 cause harmful effect to GNNs. With this in mind, we will show a counter example to illustrate the   
+99 insufficiency of the above metrics and propose new metrics.
+
+# 3 Analysis of Heterophily
+
+# 3.1 Motivation and Aggregation Homophily
+
+02 Heterophily is believed to be harmful for   
+103 message-passing based GNNs [35, 28, 5] be  
+104 cause intuitively features of nodes in different   
+105 classes will be falsely mixed and this will lead   
+106 nodes indistinguishable [35]. Nevertheless, it   
+107 is not always the case, e.g., the bipartite graph   
+108 shown in Figure $\boxed { 1 }$ is highly heterophilous ac  
+109 cording to the homophily metrics in $\textcircled{3}$ , but   
+10 after mean aggregation, the nodes in classes 1   
+111 and 2 only exchange colors and are still dis  
+112 tinguishable. Authors in $\pmb { \Vert 5 \Vert }$ also point out the   
+113 insufficiency of $H _ { \mathrm { n o d e } }$ by examples to show that   
+14 different graph typologies with the same $H _ { \mathrm { n o d e } }$   
+15 can carry different label information.   
+116 To analyze to what extent the graph structure can affect the output of a GNN, we first simplify the   
+117 GCN by removing its non-linearity as $\textcircled { \scriptsize { 1 3 1 } }$ . Let $\hat { A } \in \mathbb { R } ^ { N \times N }$ denote a general aggregation operator.   
+118 Then, equation $\bar { \textcircled { 1 } }$ can be simplified as,
+
+![](images/5b5783d6c3caa3dda73f278dd5cdfb7972b9a93b715f044df70e8cb1572c854a.jpg)  
+Figure 1: Example of harmless heterophily
+
+$$
+Y = \mathrm { s o f t m a x } ( \hat { A } X W ) = \mathrm { s o f t m a x } ( Y ^ { \prime } )
+$$
+
+After each gradient decent step 119 $\begin{array} { r } { \Delta W = \gamma \frac { \partial \mathcal { L } } { \partial W } } \end{array}$ , where $\gamma$ is the learning rate, and the update of $Y ^ { \prime }$ will 120 be (see Appendix $\boxed { \mathbf { B } }$ for derivation),
+
+$$
+\Delta Y ^ { \prime } = { \hat { A } } X \Delta W = \gamma { \hat { A } } X { \frac { \partial { \mathcal { L } } } { \partial W } } \propto { \hat { A } } X { \frac { \partial { \mathcal { L } } } { \partial W } } = { \hat { A } } X X ^ { T } { \hat { A } } ^ { T } ( Z - Y ) = S ( { \hat { A } } , X ) ( Z - Y )
+$$
+
+121 where $S ( \hat { A } , X ) \equiv \hat { A } X ( \hat { A } X ) ^ { T }$ is a node similarity matrix after aggregation, $Z - Y$ is the prediction   
+122 error matrix. The update direction of node $i$ is essentially a weighted sum of the prediction error, $i . e .$ .,   
+123 $\begin{array} { r } { \Delta ( Y ^ { \prime } ) _ { i , : } = \sum _ { j \in \mathcal { V } } \Big [ S ( \hat { A } , X ) \Big ] _ { i , j } \big ( Z - Y \big ) _ { j , : } . } \end{array}$
+
+1 The authors in [21] did not name this homophily metric. We name it class homophily based on its definition.
+
+125 Definition 1. Aggregation similarity score
+
+$$
+\begin{array}{c} \operatorname { S } _ { a g g } \Big ( S ( \hat { A } , X ) \Big ) = \frac { \Big | \Big \{ v \big | \operatorname { M e a n } _ { u } \big ( \{ S ( \hat { A } , X ) _ { v , u } | Z _ { u ; \cdot } = Z _ { v , \cdot } \} \big ) \geq \operatorname { M e a n } _ { u } \big ( \{ S ( \hat { A } , X ) _ { v , u } | Z _ { u ; \cdot } \neq Z _ { v , \cdot } \} \big ) \Big . } \\ { \Big . | \mathcal { V } | } \end{array}
+$$
+
+26 where ${ \mathrm { M e a n } } _ { u } \left( \{ \cdot \} \right)$ takes the average over u of a given multiset of values or variables.
+
+127 $S _ { \mathrm { a g g } } ( S ( \hat { A } , X ) )$ is the proportion of nodes $v \in \mathcal V$ that will put relatively larger similarity weights on   
+128 nodes in the same class than in other classes after aggregation. It is easy to see that $S _ { \mathrm { a g g } } ( S ( \hat { A } , X ) ) \in$   
+129 [0, 1]. But in practice, we observe that in most datasets, we will have $S _ { \mathrm { a g g } } ( S ( \hat { A } , X ) ) \geq 0 . 5$ . Based on   
+130 this observation, we rescale $\textcircled{6}$ to the following modified aggregation similarity for practical usage,
+
+$$
+S _ { \mathrm { a g g } } ^ { M } \left( S ( \hat { A } , X ) \right) = \Bigl [ 2 S _ { \mathrm { a g g } } \left( S ( \hat { A } , X ) \right) - 1 \Bigr ] _ { + }
+$$
+
+131 In order to measure the consistency between labels and graph structures without considering node   
+132 features and make a fair comparison with the existing homophily metrics in $\textcircled{3}$ , we define the graph   
+133 $( { \mathcal { G } } )$ aggregation $( \hat { A } )$ homophily and its modified version as
+
+$$
+H _ { \mathrm { a g g } } ( \mathcal { G } ) = S _ { \mathrm { a g g } } \left( S ( \hat { A } , Z ) \right) , H _ { \mathrm { a g g } } ^ { M } ( \mathcal { G } ) = S _ { \mathrm { a g g } } ^ { M } \left( S ( \hat { A } , Z ) \right)
+$$
+
+134 In practice, we will only check $H _ { \mathrm { a g g } } ( { \mathcal { G } } )$ when $H _ { \mathrm { a g g } } ^ { M } ( { \mathcal G } ) = 0$ . As Figure $^ 1$ shows, when $\hat { A } = \hat { A } _ { \mathrm { r w } }$ ,   
+135 $H _ { \mathrm { a g g } } ( \mathcal G ) = H _ { \mathrm { a g g } } ^ { M } ( \mathcal G ) = 1$ . Thus, this new metric reflects the fact that nodes in classes 1 and 2 are still   
+136 highly distinguishable after aggregation, while other metrics mentioned before fail to capture the   
+137 information and misleadingly give value 0. This shows the advantage of $H _ { \mathrm { a g g } } ( \mathcal { G } )$ and $H _ { \mathrm { a g g } } ^ { \dot { M } } ( { \mathcal G } )$ by   
+138 additionally considering information from aggregation operator $\hat { A }$ and the similarity matrix.
+
+To comprehensively compare 139 $H _ { \mathrm { a g g } } ^ { M } ( { \mathcal { G } } )$ and the metrics in $\textcircled{3}$ in terms of how they reveal the influence 140 of graph structure on the GNN performance, we generate synthetic graphs and evaluate SGC [31] 141 and GCN $[ \overline { { | 1 5 | } }$ on them in the next subsection.
+
+# 3.2 Evaluation and Comparison on Synthetic Graphs
+
+143 Data Generation & Experimental Setup For one dataset, we generate 95 graphs in total with 19   
+144 edge homophily levels varied from 0.05 to 0.95, each corresponding to 5 graphs. For every generated   
+145 graph, we have 5 classes with 400 nodes in each class. In each class, there are randomly generated   
+146 800 intra-class edges and $( 8 0 0 - 8 0 0 H _ { \mathrm { e d g e } } ( \mathcal G ) ) / H _ { \mathrm { e d g e } } ( \mathcal G ) ] _ {  } ^ {  } ]$ inter-class edges. The features of nodes   
+147 in each class are sampled from node features in the corresponding class of the base dataset. Nodes   
+148 are randomly split into $6 0 \% / 2 0 \% / 2 0 \%$ for train/validation/test. We train 1-hop SGC (sgc-1 [31] and   
+149 $\mathrm { G C N } \left\| \Sigma \right\|$ on synthetic data (see appendix $\boxed { \mathrm { A . 1 } }$ for hyperparameter searching range). For each value   
+150 of $H _ { \mathrm { e d g e } } ( { \mathcal { G } } )$ , we take the average test accuracy and standard deviation of runs over 5 generated graphs.   
+151 For each generated graph, we also calculate its $H _ { \mathrm { n o d e } } ( \mathcal { G } )$ , $H _ { \mathrm { c l a s s } } ( \mathcal { G } )$ and $H _ { \mathrm { a g g } } ^ { M } ( { \mathcal { G } } )$ . Model performance   
+with respect to different homophily values are shown in Figure 2.   
+153 Comparison of Homophily Metrics The performance of SGC-1 and GCN are expected to be   
+154 monotonically increasing with a proper and informative homophily metric. However, Figure $2 ( \mathrm { a } ) ( \mathrm { b } ) ( \mathrm { c } )$   
+155 show that the performance curves under $H _ { \mathrm { e d g e } } ( \mathcal { G } ) , H _ { \mathrm { n o d e } } ( \mathcal { \bar { G } } )$ and $H _ { \mathrm { c l a s s } } ( \mathcal G )$ are $U$ -shaped $\mathbb { B }$ while   
+156 Figure $\bigstar \bigstar \bigstar$ reveals a nearly monotonic curve with a little perturbation around 1. This indicates that   
+157 $H _ { \mathrm { a g g } } ^ { \overline { { M } } } ( \mathcal { G } )$ can describe how the graph structure affects the performance of SGC-1 and GCN.   
+58 In addition, we notice that in Figure $2 ( \mathrm { a } )$ , both SGC-1 and GCN get the worst performance on all   
+9 datasets when $H _ { \mathrm { e d g e } } ( { \mathcal { G } } )$ is around somewhere between 0.1 and 0.2. This interesting phenomenon can   
+0 be explained by the following theorem.   
+161 Theorem 1. (See Appendix $\bigtriangledown$ for proof). Suppose there are $C$ classes in the graph $\mathcal { G }$ , edges for each   
+162 node are i.i.d.generated such that each edge of any node has probability $h$ of connecting with nodes in   
+163 the same class and probability $1 - h$ of connecting with nodes in different classes, and $\mathbb { E } ( d _ { v } ) = d$ for   
+164 all nodes. Let the aggregation operator $\hat { A } = \hat { A } _ { \mathrm { r w } }$ . Then, for nodes $v , u _ { 1 }$ and $u _ { 2 }$ , where $Z _ { u _ { 1 } , : } = Z _ { v , }$ ,:   
+165 and $Z _ { u _ { 2 } , : } \neq Z _ { v , : }$ , we have
+
+![](images/84bd4e741367d6258d606b257499e6bd2fde9facc78f8372bb4352314bd1309e.jpg)  
+Figure 2: Comparison of baseline performance under different homophily metrics.
+
+$$
+g ( h ) \equiv {  { \mathbb E } } \left( S ( {  { \hat { A } } } , Z ) _ { v , u _ { 1 } } \right) - {  { \mathbb E } } \left( S ( {  { \hat { A } } } , Z ) _ { v , u _ { 2 } } \right) = \left( \frac { ( C - 1 ) ( h d + 1 ) - ( 1 - h ) d } { ( C - 1 ) ( d + 1 ) } \right) ^ { 2 }
+$$
+
+and the minimum of $g ( h )$ is reached at
+
+$$
+h = \frac { d + 1 - C } { C d } = \frac { d _ { \mathrm { i n t r a } } / h + 1 - C } { C ( d _ { \mathrm { i n t r a } } / h ) } \Rightarrow h = \frac { d _ { \mathrm { i n t r a } } } { C d _ { \mathrm { i n t r a } } + C - 1 }
+$$
+
+166 where $d _ { \mathrm { i n t r a } } = d h$ , which is the expectation of the number of neighbors of a node that have the same   
+167 label as the node.   
+168 The value of $g ( h )$ in $\textcircled{9}$ is the expected differences of the similarity values between nodes in the   
+169 same class as $v$ and nodes in other classes. $g ( h )$ is strongly related to the definition of aggregation   
+170 homophily and its minimum potentially implies the worst value of $H _ { \mathrm { a g g } } ( { \mathcal { G } } )$ . In the synthetic experi  
+171 ments, we have $d _ { \mathrm { i n t r a } } = 2 , C = 5$ and the minimum of $g ( h )$ is reached at $h = 1 / 7 \approx 0 . 1 4$ , which   
+172 corresponds to the lowest point in the performance curve in Figure $\bigstar \bigstar \bigstar$ . In other words, the $h$ where   
+173 SGC-1 and GCN perform worst is where $g ( h )$ gets the smallest value, instead of the point with the   
+174 smallest edge homophily value. This again shows the advantage of $H _ { \mathrm { a g g } } ( { \mathcal { G } } )$ over $H _ { \mathrm { e d g e } } ( { \mathcal { G } } )$ by taking   
+175 use of the similarity matrix.
+
+![](images/fe1663ea7f22a0ed0bfdf07bb223ad73527ea58208148ab9c322c548e07bcd51.jpg)  
+Figure 3: Example of how HP filter addresses harmful heterophily
+
+# 176 4 Adaptive Channel Mixing (ACM) Framework
+
+# 177 4.1 How Diversification Operation Helps with Harmful Heterophily
+
+178 We first consider the example shown in Figure $\boxed { 3 }$ . From $S ( { \hat { A } } , X )$ , nodes 1,3 assign relatively large   
+179 positive weights to nodes in class 2, which will negatively affect information aggregation. Despite   
+180 the fact, we can still distinguish between nodes 1,3 and 4,5,6,7 by considering their neighborhood   
+181 difference: nodes 1,3 are distinguishable from their neighbors while nodes 4,5,6,7 are homogeneous to   
+182 their neighbors. This indicates, in some cases, although some nodes become similar after aggregation,   
+183 they are still distinguishable via surrounding dissimilarities. This suggests the possibility of using   
+184 diversification operation to address harmful heterophily i.e., high-pass (HP) filter $I - { \hat { A } }$ [8] (will be   
+185 introduced in next subsection). As $S ( I - { \hat { A } } , Z )$ in Figure $3$ shows, nodes 1,3 assign negative weights   
+186 to nodes 4,5,6,7, i.e., nodes 1,3 treat nodes 4,5,6,7 as negative samples and will move away from   
+187 them. Base on this example, we propose diversification distinguishability as follows,
+
+Definition 2. Diversification Distinguishability $( D D )$ based on $S ( I - { \hat { A } } , X )$ .
+
+Given $S ( I - { \hat { A } } , X )$ , a node $v$ is diversification distinguishable if the following two conditions are satisfied at the same time,
+
+$$
+\begin{array} { r } { I . \mathrm { \mathrm { ~ M e a n } } _ { u } \left( \{ S ( I - \hat { A } , X ) _ { v , u } | u \in \mathcal { V } \wedge Z _ { u , : } = Z _ { v , : } \} \right) \geq 0 ; } \\ { 2 . \mathrm { \mathrm { ~ M e a n } } _ { u } \left( \{ S ( I - \hat { A } , X ) _ { v , u } | u \in \mathcal { V } \wedge Z _ { u , : } \neq Z _ { v , : } \} \right) \leq 0 } \end{array}
+$$
+
+191 Then, graph diversification distinguishability value is defined as
+
+$$
+\mathrm { D D } _ { \hat { A } , X } ( { \mathcal G } ) = \frac { 1 } { | \mathcal V | } \Big | \{ v | v i s d i v e r s i f i c a t i o n d i s t i n g u i s h a b l e \} \Big |
+$$
+
+92 $\mathrm { D D } _ { \hat { A } , X } ( { \mathcal G } ) \in [ 0 , 1 ]$ measures the proportion of nodes that HP filter is helpful for. Its effectiveness   
+93 can be proved for binary classification problems under certain conditions, leading us to:
+
+Theorem 2. (See Appendix94 $\boxed { \mathrm { E } }$ for proof). Suppose $X = Z , \hat { A } = \hat { A } _ { \mathrm { r w } } .$ . Then, for a binary classifica5 tion problem, i.e., $C = 2$ , all nodes are diversification distinguishable and $\mathrm { D D } _ { \hat { A } , Z } ( { \mathcal G } ) = \mathrm { 1 }$ .
+
+Conducting both aggregation and diversification operations to distinctively extract the low- and highfrequency information from graph signals is the same as using filterbanks in graph signal processing. We introduce filterbanks in next subsection.
+
+# 4.2 Filterbank in Spectral and Spatial Forms
+
+Filterbank For the graph signal $_ { \textbf { \em x } }$ defined on $\mathcal { G }$ , a 2-channel linear (analysis) filterbank [8] 4 includes a pair of filters $H _ { \mathrm { L P } } , H _ { \mathrm { H P } }$ , where $H _ { \mathrm { L P } }$ and $H _ { \mathrm { H P } }$ retain the low-frequency and high-frequency content of $_ { \textbf { \em x } }$ , respectively.
+
+Most existing GNNs are under uni-channel filtering architecture $[ \overline { { 1 5 } } , \overline { { 3 0 } } , \overline { { 1 2 } } ]$ with either $H _ { \mathrm { L P } }$ or $H _ { \mathrm { H P } }$ channel that only partially preserves the input information. Unlike the uni-channel architecture, filterbanks with $H _ { \mathrm { L P } } + H _ { \mathrm { H P } } = I$ will not lose any information of the input signal, i.e., perfect reconstruction property $\mathbb { B } \mathbb { B } \mathbb { Z } \mathbb { Z }$ .
+
+Generally, the Laplacian matrices207 $( L _ { \mathrm { s y m } } , L _ { \mathrm { r w } } , \hat { L } _ { \mathrm { s y m } } , \hat { L } _ { \mathrm { r w } } )$ can be regarded as HP filters $\pmb { \bigtriangledown }$ and affinity matrices 208 $( A _ { \mathrm { s y m } } , A _ { \mathrm { r w } } , \hat { A } _ { \mathrm { s y m } } , \hat { A } _ { \mathrm { r w } } )$ can be treated as LP filters $[ [ 2 5 , \boxed { 1 1 } ] ]$ . Moreover, MLPs can be 209 considered as owing a special identity filterbank with matrix $I$ that satisfies $H _ { \mathrm { L P } } + H _ { \mathrm { H P } } = I + 0 = I$ .
+
+210 Filterbank in Spatial Form Filterbank methods can also be extended to spatial GNNs. Formally,   
+211 on the node level, left multiplying $H _ { \mathrm { L P } }$ and $H _ { \mathrm { H P } }$ on $_ { \textbf { \em x } }$ performs as aggregation and diversification   
+212 operations, respectively. For example, suppose $H _ { \mathrm { L P } } = { \hat { A } }$ and $H _ { \mathrm { H P } } = I - { \hat { A } }$ , then for node $i$ we have
+
+$$
+( H _ { \mathrm { L P } } \pmb { x } ) _ { i } = \sum _ { j \in \{ \mathcal { N } _ { i } \cup i \} } \hat { A } _ { i , j } \pmb { x } _ { j } , ( H _ { \mathrm { H P } } \pmb { x } ) _ { i } = \pmb { x } _ { i } - \sum _ { j \in \{ \mathcal { N } _ { i } \cup i \} } \hat { A } _ { i , j } \pmb { x } _ { j }
+$$
+
+213 where $\hat { A } _ { i , j }$ is the connection weight between two nodes. To leverage HP and identity channels in   
+214 GNNs, we propose the Adaptive Channel Mixing (ACM) architecture in the following subsection.   
+216 ACM framework can be applied in lots of baseline GNNs and in this subsection, we use GCN as an   
+217 example and introduce ACM framework in matrix form. We use $H _ { \mathrm { L P } }$ and $H _ { \mathrm { H P } }$ to represent general   
+218 LP and HP filters. The ACM framework includes 3 steps as follows,
+
+# Step 1. Feature Extraction for Each Channel:
+
+$\begin{array} { r } { H _ { L } ^ { l } = H _ { \mathrm { L P } } \mathrm { R e L U } \left( H ^ { l - 1 } W _ { L } ^ { l - 1 } \right) , H _ { H } ^ { l } = H _ { \mathrm { H P } } \mathrm { R e L U } \left( H ^ { l - 1 } W _ { H } ^ { l - 1 } \right) , H _ { I } ^ { l } = I \mathrm { R e L U } \left( H ^ { l - 1 } W _ { I } ^ { l - 1 } \right) , } \end{array}$ $W _ { L } ^ { l - 1 } , W _ { H } ^ { l - 1 } , W _ { I } ^ { l - 1 } \in \mathbb { R } ^ { F _ { l - 1 } \times F _ { l } }$ ;
+
+# Step 2. Feature-based Weight Learning with Row Normalization (RN):
+
+$$
+\begin{array} { r l } & { \tilde { H } _ { I } ^ { l } = \mathrm { R N } \left( H _ { I } ^ { l } \right) , \tilde { H } _ { L } ^ { l } = \mathrm { R N } \left( H _ { L } ^ { l } \right) , \tilde { H } _ { H } ^ { l } = \mathrm { R N } \left( H _ { H } ^ { l } \right) ; } \\ & { \alpha _ { L } ^ { l } = \sigma \left( \mathrm { E L U } \left( \tilde { H } _ { L } ^ { l } \tilde { W } _ { L } ^ { l } \right) \right) , \alpha _ { H } ^ { l } = \sigma \left( \mathrm { E L U } \left( \tilde { H } _ { H } ^ { l } \tilde { W } _ { H } ^ { l } \right) \right) , \alpha _ { I } ^ { l } = \sigma \left( \mathrm { E L U } \left( \tilde { H } _ { I } ^ { l } \tilde { W } _ { I } ^ { l } \right) \right) , } \\ & { \tilde { W } _ { L } ^ { l - 1 } , \tilde { W } _ { H } ^ { l - 1 } , \tilde { W } _ { I } ^ { l - 1 } \in \mathbb { R } ^ { F _ { l } \times 1 } ; } \end{array}
+$$
+
+Step 3. Channel Mixing:
+
+$$
+H ^ { l } = \left( { \mathrm { d i a g } } ( \alpha _ { L } ^ { l } ) H _ { L } ^ { l } + { \mathrm { d i a g } } ( \alpha _ { H } ^ { l } ) H _ { H } ^ { l } + { \mathrm { d i a g } } ( \alpha _ { I } ^ { l } ) H _ { I } ^ { l } \right) . \nonumber
+$$
+
+219 ACM-GCN first implements distinct non-linear feature extractions for 3 channels, respectively. After   
+220 processed by a set of filterbanks, 3 filtered components $H _ { L } ^ { l } , H _ { H } ^ { l } , H _ { I } ^ { l }$ are obtained. Different nodes   
+221 may have different needs for the information in the 3 channels, e.g., in Figure ${ \underline { { 3 } } } ,$ nodes 1,3 demand   
+222 high-frequency information while node 2 only needs low-frequency information. To adaptively exploit   
+223 information from different channels, ACM-GCN learns rowwise (nodewise) feature-conditioned   
+224 (un-normalized) weights to combine the 3 channels. ACM can be easily plugged into spatial GNNs by   
+225 replacing $H _ { \mathrm { L P } }$ and $H _ { \mathrm { H P } }$ by aggregation and diversification operations as shown in $( 1 2 )$ . See Appendix   
+226 F for a detailed discussion of model comparison on synthetic datasets.   
+27 Complexity Number of learnable parameters in layer $l$ of ACM-GCN is $3 F _ { l - 1 } ( F _ { l } + 1 )$ , while it is   
+28 $F _ { l - 1 } F _ { l }$ in GCN. The computation of step 1-3 takes $N F _ { l } ( 2 0 + F _ { l - 1 } ) + 2 F _ { l } ( \mathrm { n n z } ( H _ { \mathrm { L P } } ) + \mathrm { n n z } ( H _ { \mathrm { H P } } ) )$   
+29 flops, while GCN layer takes $2 N F _ { l - 1 } F _ { l } + 2 F _ { l } ( \mathrm { n n z } ( H _ { \mathrm { L P } } ) )$ flops, where $\mathrm { n n z } ( \cdot )$ is the number of   
+30 non-zero elements. A detailed experiments on running time is conducted in section 6.1.
+
+Limitations Diversification operation does not work well in all harmful heterophily cases. For example, consider an imbalanced dataset where several small clusters with distinctive labels are densely connected to a large cluster. In this case, the surrounding differences of nodes in small clusters are similar, i.e., the neighborhood differences are mainly from their connection to the same large cluster, and this possibly makes diversification operation fail to discriminate them. See a more detailed demonstration and discussion in Appendix G.
+
+# 5 Prior Work
+
+GNNs on Addressing Heterophily We discuss relevant work of GNNs on addressing heterophily challenge in this part. [1] acknowledges the difficulty of learning on graphs with weak homophily and propose MixHop to extract features from multi-hop neighborhood to get more information. Geom-GCN $\pmb { \left. 2 8 \right. }$ precomputes unsupervised node embeddings and uses graph structure defined by geometric relationships in the embedding space to define the bi-level aggregation process. [13] proposes measurements based on feature smoothness and label smoothness that are potentially helpful to guide GNNs on dealing with heterophilous graphs. $\mathrm { H _ { 2 } G C N }$ [35] combines 3 key designs to address heterophily: (1) ego- and neighbor-embedding separation; (2) higher-order neighborhoods; (3) combination of intermediate representations. CPGNN $[ \textcircled { 3 4 } ]$ models label correlations by the compatibility matrix, which is beneficial for heterophily settings, and propagates a prior belief estimation into GNNs by the compatibility matrix. GPRGNN $\pmb { \Vert 5 \Vert }$ uses learnable weights that can be both positive and negative for feature propagation, it allows GRPGNN to adapt heterophily structure of graph and is able to handle both high and low frequency parts of the graph signals.
+
+251 GNNs with Filterbanks Previously, there are geometric scattering networks $\mathbb { P } , \mathbb { Z } \mathbb { Z }$ that apply   
+252 filterbanks to address over-smoothing $\pmb { \left. 2 0 \right. }$ problem. The scattering construction captures different   
+253 channels of variation from node features or labels. In geometric learning and graph signal processing,   
+254 the band-pass filtering operations extract geometric information beyond smooth signals, thus it is   
+255 believed that filterbanks can alleviate over-smoothing in GNNs. In ACM framework, we aim to   
+56 design a framework with the help of filterbanks to adaptively utilize different channels to address the   
+7 challenge of learning on heterophilous graph. We deal with different problem as in $\mathbb { B } \mathbb { Z } \mathbb { Z }$ .
+
+# 6 Experiments on Real-World Datasets
+
+In this section, we evaluate ACM framework on real-world datasets. We first conduct ablation studies in subsection $6 . 1$ to validate different components. Then, we compare with the state-of-the-arts models in subsection $6 . 2 .$
+
+# 262 6.1 Ablation Study & Efficiency
+
+Table 1: Ablation study on 9 real-world datasets $\pmb {  \pmb {   2 8  }  }$ . Cell with Xmeans the component is applied to the baseline model. The best test results are highlighted.   
+
+<table><tr><td>Ablation Study on Different Components in ACM-SGC and ACM-GCN(%)</td><td colspan="10"></td></tr><tr><td>Baseline</td><td colspan="2">Model Components</td><td></td><td>Cornell</td><td>Wisconsin</td><td>Texas</td><td>Film</td><td>Chameleon</td><td>Squirrel</td><td>Cora CiteSeer</td><td>PubMed</td></tr><tr><td>Models</td><td>|LP HP Identity Mixing|</td><td></td><td></td><td>Acc ± Std</td><td>Acc ± Std Acc ± Std</td><td>Acc ± Std</td><td>Acc ± Std</td><td>Acc ± Std</td><td>Acc ± Std</td><td>Acc ± Std</td><td>Acc ± Std</td></tr><tr><td></td><td>√</td><td></td><td></td><td></td><td>74.43 ± 6.01 69.75 ± 5.02 84.1 ± 2.32 25.34 ± 2.41 64.55 ± 1.38</td><td></td><td></td><td>42.8 ± 1.1</td><td>85.24±1.85 79.85±1.04 84.44±0.38</td><td></td><td></td></tr><tr><td rowspan="5">SGC-1 w/</td><td>√ √ √</td><td></td><td>√</td><td></td><td></td><td></td><td>84.92± 4.59 91.75 ± 4.05 89.34 ± 3.67 36.94 ± 1.07 63.11 ± 1.64 44.8 ± 1.35</td><td></td><td></td><td></td><td>85.6±1.33 80.33±1.25 84.5±0.42</td></tr><tr><td></td><td>√</td><td>√</td><td></td><td></td><td></td><td>92.3± 3.893±2.11 91.64±3.65 38.25±1.657±1.93</td><td></td><td></td><td></td><td>40.2 ±2.18 85.98±0.84 80.2±2.01 84.37 ±0.44</td></tr><tr><td>√ √</td><td>√</td><td></td><td></td><td>88.2±3.88 90.75±2.3792.3±3.88 36.58±1.36 61.64±2.52 41.59±2.2984.98±1.2 79.81±1.287.13±0.58</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>√ √</td><td>√</td><td>√</td><td></td><td>92.46±2.1093.38±2.6891.97±3.2338.71±1.22 62.39±2.4545.65±1.4486.52±1.5580.79±1.6587.69±0.6</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="5">GCN w/</td><td>√ √</td><td></td><td></td><td></td><td>81.31±3.13 70.25±4.782.13±4.05 34.45±0.83 64.86±1.56 45.11±1.3987.47±0.8281.3±0.9587.85±0.44</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>√</td><td>√</td><td>√</td><td></td><td>82.95±5.178.63±2.5188.03±2.6740.16±1.0668.12±1.7352.08±1.478.44±1.62 81.45±0.990.09 ±0.29</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>√</td><td>√</td><td>√</td><td></td><td></td><td>92.13±2.6594.37±3.2793.11±2.4840.3±1.636.67±2.1649.45±0.838.46±1.3181.42±1.1391.21±1.17</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>√</td><td>√</td><td>√</td><td></td><td></td><td></td><td>88.52±4.5195±2.2592.3±2.2140.25±1.78 65.97±2.2451.02±1.64.7±1.6880.93±1.53 90.66 ±0.32</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>92.62 ±3.04 95.37±2.1 94.75±1.7741.48±0.7867.79±1.7952.86±1.96 89.11±0.87 82.16±0.8490.72 ±0.7</td></tr><tr><td colspan="9">Average Running Time Per Epoch/Average Total Running Time Comparison</td><td rowspan="5"></td><td rowspan="10"></td><td rowspan="10"></td><td rowspan="10"></td><td rowspan="10"></td></tr><tr><td rowspan="5">SGC-1 w/</td><td>√</td><td></td><td></td><td></td><td>2.70ms/0.59s2.53ms/0.51s2.63ms/0.55s3.62ms/1.13s4.96ms/3.99s4.09ms/0.87s5.34ms/8.22s4.79ms/4.s5.58ms/7.70s</td><td></td><td></td></tr><tr><td>√ √</td><td>√</td><td>√</td><td></td><td>4.93ms/1.04s5.03ms/1.04s.67ms/1.58s6.68ms/1.37s6.42ms/1.96s7.41ms/1.93s6.68ms/2.43s6.69ms/1.96s7.0ms/2.8s</td><td></td><td></td><td></td></tr><tr><td>√</td><td>√</td><td>√</td><td></td><td>4.73ms/0.98s4.99ms/1.09s4.79ms/102s5.53ms/1.28s5.89ms/1.50s6.48ms/1.50s6.50ms/2.09s6.23ms/1.76s6.73ms/2.4s</td><td></td><td></td><td></td></tr><tr><td>√ √</td><td>√</td><td>√</td><td></td><td>4.30ms/0.88s4.51ms/0.91s4.58ms/0.95s5.86ms/1.19s5.99ms/1.43s6.84ms/1.63s5.44ms/1.37s5.72ms/1.44s6.36ms/2.4s 5.15ms/1.08s5.82ms/1.28s.55ms/1.18s6.28ms/1.50s6.60ms/1.96s7.27ms/1.52s7.05ms/2.40s6.99ms/1.94s7.28ms/2.7s</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td colspan="6"></td></tr><tr><td rowspan="5">GCN w/</td><td>√ √</td><td></td><td>√</td><td></td><td rowspan="5">3.78ms/0.78s3.91ms/0.79s3.80ms/0.78s4.42ms/0.89s4.44ms/0.89s6.85ms/1.48s4.19ms/0.87s5.22ms/1.13s4.81ms/0.99s 7.63ms/1.54s7.99ms/1.92s7.26ms/1.48s8.42ms/1.73s9.74ms/2.76s11.19ms/2.38s7.74ms/1.61s9.98ms/3.56s 9.10ms/1.s</td><td></td><td></td><td></td><td rowspan="5"></td><td rowspan="5"></td><td rowspan="5"></td></tr><tr><td>√</td><td>√ √</td><td></td><td></td><td>6.75ms/1.36s6.3ms/1.41s99ms/1.46s7.62ms/1.4s7.80ms/1.67s9.76ms/2.02s7.59ms/1.54s7.43ms/1.54s.28ms/1.0s</td><td></td></tr><tr><td>√</td><td>√ √</td><td></td><td>√</td><td>7.3ms/1.49s6.80ms/1.38s6.99ms/1.41s8.76ms/2.19s7.81ms/1.59s11.26ms/2.9s7.77ms/1.59s7.66ms/1.56s8.36ms/.0s</td><td></td></tr><tr><td>√</td><td></td><td>√</td><td>√</td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>8.04ms/1.63s8.98ms/1.83s8.17ms/1.65s 9.29ms/2.00s9.33ms/1.96s12.15ms/2.53s 9.16ms/1.85s9.48ms/1.95s9.54ms/1.92s</td><td></td></tr></table>
+
+263 We investigate the effectiveness and efficiency of adding HP, identity channels and the adaptive mixing   
+264 mechanism in ACM framework by ablation study. Specifically, we apply the above components to   
+265 SGC-1 and GCN separately, run 10 times on each dataset used in $\bar { \| 2 8 \| }$ with $6 0 \% / 2 0 \% / 2 0 \%$ random   
+266 splits for train/validation/test and report the average test accuracy as well as the standard deviation.   
+267 We also record the average running time per epoch(in milliseconds)/average total running time(in   
+268 seconds) to compare the efficiency. (See Appendix A for hyperparameter searching space.)   
+269 From the results we can see that on most datasets, the additional HP and identity channels are helpful,   
+270 even on strong homophily datasets, such as Cora, CiteSeer and PubMed. The adaptive mixing   
+271 mechanism also shows its advantage over the method that directly adds the three channels together.   
+272 This illustrates the necessity of learning to customize the channel usage adaptively for different nodes.   
+273 As for efficiency, we can see that the running time is approximately doubled in ACM framework than   
+274 the original model.
+
+# 275 6.2 Comparison with State-of-the-art Models
+
+Datasets & Experimental Setup In this section, we implement SGC $\textcircled { \scriptsize { 1 3 1 } }$ with 1 hop and 2 hop (SGC-1, SGC-2), GCN [15] and GraphSAGE and apply them [12] in ACM framework: we use $\hat { A } _ { \mathrm { r w } }$ and mean aggregator as LP filter and the corresponding HP filter can be derived from $\textcircled { 1 2 }$ We compare them with several baselines and state-of-the-art models: MLP with 2 layers (MLP-2), GAT $\bar { \mathbb { B } } \bar { 0 } \mathbb { I }$ , APPNP [16], GPRGNN [5], $\mathrm { H _ { 2 } G C N }$ [35], MixHop [1], GCN+JK [15, 32, 21], $\mathrm { G A T + J K }$ [30, 32, 21] and Geom-GCN $[ [ 2 8 ] ]$ . Besides the 9 benchmark datasets used in $\left\| \widehat { 2 8 } \right\|$ , we further tests the above models on 2 new benchmark datasets, Deezer-Europe and YelpChi, that are proposed in
+
+<table><tr><td></td><td>Cornell</td><td>Wisconsin</td><td>Texas</td><td>Film</td><td>Chameleon</td><td>Squirrel</td><td>Deezer-Europe</td><td>YelpChi</td><td>Cora</td><td>CiteSeer</td><td>PubMed</td><td></td></tr><tr><td>#nodes</td><td>183</td><td>251</td><td>183</td><td>7,600</td><td>2.277</td><td>5,201</td><td>28.281</td><td>45,954</td><td>2,708</td><td>3,327</td><td>19,717</td><td></td></tr><tr><td>#edges</td><td>295</td><td>499</td><td>309</td><td>33,544</td><td>36,101</td><td>217,073</td><td>92,752</td><td>3,846.979</td><td>5,429</td><td>4,732</td><td>44,338</td><td></td></tr><tr><td>#features</td><td>1,703</td><td>1,703</td><td>1,703</td><td>931</td><td>2.325</td><td>2.089</td><td>31,241</td><td>32</td><td>1,433</td><td>3,703</td><td>500</td><td></td></tr><tr><td>#classes</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>2</td><td>2</td><td>7</td><td>6</td><td>3</td><td></td></tr><tr><td>Hedge(G)</td><td>0.5669</td><td>0.4480</td><td>0.4106</td><td>0.3750</td><td>0.2795</td><td>0.2416</td><td>0.5251</td><td>0.7730</td><td>0.8100</td><td>0.7362</td><td>0.8024</td><td></td></tr><tr><td>Hnode(9)</td><td>0.3855</td><td>0.1498</td><td>0.0968</td><td>0.2210</td><td>0.2470</td><td>0.2156</td><td>0.5299</td><td>0.7698</td><td>0.8252</td><td>0.7175</td><td>0.7924</td><td></td></tr><tr><td>Hclass(9)</td><td>0.0468</td><td>0.0941</td><td>0.0013</td><td>0.0110</td><td>0.0620</td><td>0.0254</td><td>0.0304</td><td>0.0520</td><td>0.7657</td><td>0.6270</td><td>0.6641</td><td></td></tr><tr><td>H（）</td><td>0.8032</td><td>0.7768</td><td>0.694</td><td>0.6822</td><td>0.61</td><td>0.3566</td><td>0.5790</td><td>0.7206</td><td>0.9904</td><td>0.9826</td><td>0.9432</td><td></td></tr><tr><td>Data Splits(%)</td><td>60/20/20</td><td>60/20/20</td><td>60/20/20</td><td>60/20/20</td><td>60/20/20</td><td>60/20/20</td><td>50/25/25</td><td>50/25/25</td><td>60/20/20</td><td>60/20/20</td><td>60/20/20</td><td></td></tr><tr><td></td><td colspan="10">Test Accuracy(%)of State-of-the-art Models, Baseline GNN Models and ACM-GNN models</td><td colspan="3">|Rank</td></tr><tr><td>MLP-2*</td><td colspan="10"></td></tr><tr><td></td><td colspan="10">910±0.70.908505406.0.5.4±0020</td></tr><tr><td>GAT* APPNP*</td><td colspan="10">76.00±1.014660.5.980.0.46.00720.4067</td></tr><tr><td>GPRGNN*</td><td colspan="10">91.80±0.639.00±3.59918±07038.602451.91±0.63.770.3467.1±0.5675.6±0.4879.0.3868.59±030850</td></tr><tr><td>H2GCN</td><td colspan="10">91.36±0.7093.5±27.9±639077.0.04.9±0.56.90.5059±0.389.51±0.367.±80</td></tr><tr><td>MixHop</td><td colspan="10">86.23±4.717.5±177.5170489±122207.09.76 62.52</td></tr><tr><td>GCN+JK GAT+JK</td><td colspan="10">.56±13.82650±15.758066±.92.7±62646±.855.196.99±0.14650866.9±15173.77±59</td></tr><tr><td>Geom-GCNt</td><td colspan="10">74.43±10.2469.50±3.1275.41±7.1835.41±0.9768.14±1.1852.28±3.6159.66±0.9290.04±0.6189.52±0.43 74.49 ± 2.76 89.15 ± 0.87</td></tr><tr><td></td><td colspan="10">60.81 64.12 67.57 31.63 60.9 38.14 NA NA 85.27 77.99 90.05</td></tr><tr><td>SGC-1 SGC-2</td><td colspan="10">74.43±6.016.55.24.±2.42.34±46.4±1.922.81.159.70.58.60.858560.8279.930380970</td></tr><tr><td>GCN</td><td colspan="10">77.7±4.47 72.75 ±3.91 81.48 ±3.88 29.39 ±0.20 63.02±0.4337.41±1 61.56±0.51 57.18 ±0.75 86.58±0.26 76.23±0.2981.14±0.71 81.3± 0.95 87.85 ± 0.44</td></tr><tr><td>GraphSAGE</td><td colspan="10">81.31±3.1370.25±4.782.13±4.0534.45±0.8364.86±1.5645.11±1.39 62.23±0.5363.62±1.0087.47±0.82 71.41±1.2464.85±5.1479.03±1.2036.37±0.21 62.15±0.42 41.26±0.26 62.55±0.48 62.57±1.12 86.58±0.26</td></tr><tr><td>ACM-SGC-1</td><td colspan="10"></td></tr><tr><td>ACM-SGC-2</td><td colspan="10">91.31±2.947654.665455±5</td></tr><tr><td>ACM-GCN</td><td colspan="10">90.66±3.369.3±5.069066±284377±17458.51±2.4239.7±1.416.98±0.85.4±1177.44±0.80.3±1260. 82.16 ± 0.84 90.72 ± 0.7</td></tr><tr><td>ACM-SAGE</td><td colspan="10">92.62±3.0495.37±2.195.08±1.841.48±0.7867.79±1.7952.86±1.96 66.85±0.9589.91±1.0289.11±0.87 80.87±1.36 88.51±0.9</td></tr><tr><td></td><td colspan="10">91.31±2.9490.13±2.6791.97±3.1536.68±2.46 61.84±2.7144.63±3.02 66.21±0.898.73±1.4586.24±1.25</td></tr><tr><td></td><td colspan="10"></td></tr></table>
+
+Table 2: Experimental results: average test accuracy $\pm$ standard deviation on 11 real-world benchmark datasets. The best results are highlighted. The "†" results are from $\left[ \left[ 2 8 \right] \right]$ and NA means the reported results are not available. Results "\*" are from [5, 21].
+
+[21] 5 . We test these models 10 times on Cornell, Wisconsin, Texas, Film, Chameleon, Squirrel, Cora, Citeseer and Pubmed following the same early stopping strategy, the same data splitting and Adam [14] optimizer used in GPRGNN [5]. For Deezer-Europe and YelpChi, we test the above models 5 times with the same early stopping strategy, the same splits and AdamW [23] used in $\mathbb { \left| \left[ 2 \right] \right| }$ . The details of hyperparameter search are reported in appendix A.
+
+The main results of this set of experiments with statistics of datasets are summarized in Table 2, where we report the mean accuracy and standard deviation. We can see that after applied in ACM framework, the performance of baseline models are boosted on almost all tasks. Especially, ACM-GCN performs the best in terms of average rank (1.73) across all datasets and achieves SOTA performance on 6 out of 11 datasets. Overall, It suggests that ACM framework can help GNNs to generalize better on node classification tasks on heterophilous graphs.
+
+# 94 7 Future Work
+
+The similarity matrix and the new metrics defined in this paper mainly capture the linear relations of the aggregated nodes. But this might be insufficient sometimes when nonlinearity information in feature vectors are important for classification. In the future, similarity matrix that is able to capture nonlinear relations between nodes can be proposed to define new homophily metrics.
+
+From experimental results, the standard deviation of ACM-GNNs are relatively higher than GNNs on some tasks and this is suspiciously caused by the feature-based weight learning mechanism. In the future, a stabilizer or a more robust weight learning method can be proposed to reduce the variance.
+
+# 8 Social Impact
+
+303 We do not find any direct path of this work to any negative social impact.
+
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+381 networks with heterophily. arXiv preprint arXiv:2009.13566, 2020.   
+382 [35] J. Zhu, Y. Yan, L. Zhao, M. Heimann, L. Akoglu, and D. Koutra. Beyond homophily in graph   
+383 neural networks: Current limitations and effective designs. Advances in Neural Information   
+384 Processing Systems, 33, 2020.
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] In the Appendix G, we discussion cases that high-pass filter cannot tackle.   
+(c) Did you discuss any potential negative societal impacts of your work? [No] It is in Section 8, we have not come up with significant social negative impact.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] In Section 3&4, we mainly define a new homophily metric and it is followed by two theorems.   
+(b) Did you include complete proofs of all theoretical results? [Yes] In Appendix B&C& E, we justify the new metric and two theorems.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The settings are provided in details and the source code is submitted in the supplemental material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] In Section 6, we specify model details.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We include average test accuracy of times of running with standard deviation.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We include hardware details in Appendix, which is not computationally expensive.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] In Section 6, we specify the datasets with their data split sources in footnotes.   
+(b) Did you mention the license of the assets? [No]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] None included.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [No] None included.   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [No] None included.   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [No] None included.

parse/train/ZyugLlWzdO/ZyugLlWzdO_content_list.json

@@ -0,0 +1,1348 @@
+[
+    {
+        "type": "text",
+        "text": "Is Heterophily A Real Nightmare For Graph Neural Networks To Do Node Classification? ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+            313,
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+            329
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Graph Neural Networks (GNNs) extend basic Neural Networks (NNs) by using   \n2 the graph structures based on the relational inductive bias (homophily assumption).   \n3 Though GNNs are believed to outperform NNs in real-world tasks, performance   \n4 advantages of GNNs over graph-agnostic NNs seem not generally satisfactory.   \n5 Heterophily has been considered as a main cause and numerous works have been   \n6 put forward to address it. In this paper, we first show that not all cases of heterophily   \n7 are harmful for GNNs with aggregation operation. Then, we propose new metrics   \n8 based on a similarity matrix which considers the influence of graph structure and   \n9 input features on GNNs. The metrics demonstrate advantages over the commonly   \n10 used homophily metrics by tests on synthetic graphs. From the metrics and the   \n11 observations, we find some cases of harmful heterophily can be addressed by   \n12 diversification operation. With this fact and knowledge of filterbanks, we propose   \n13 the Adaptive Channel Mixing (ACM) framework to adaptively exploit aggregation,   \n14 diversification and identity operations in each GNN layer to address harmful   \n15 heterophily. We validate the ACM-augmented baselines with 11 real-world node   \n16 classification tasks. They consistently achieve significant performance gain and   \n17 exceed the state-of-the-art GNNs on most of the tasks without incurring significant   \n18 computational burden. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "19 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "20 Deep Neural Networks (NNs) $[ \\overline { { 1 8 } } ]$ have revolutionized many machine learning areas, including   \n21 image recognition $ { \\mathbb { I } } ^ { [ 1 2 ] }$ , speech recognition $\\mathbb { m }$ and natural language processing [2], etc.One major   \n22 strength is their capacity and effectiveness of learning latent representation from Euclidean data.   \n23 Recently, the focus has been put on its applications on non-Euclidean data [4], e.g., relational data   \n24 or graphs. Combining graph signal processing and convolutional neural networks $\\mathbb { \\lVert \\rVert }$ , numerous   \n25 Graph Neural Networks (GNNs) [29, 7, 12, 30, 15, 24] have been proposed which empirically out  \n26 perform traditional neural networks on graph-based machine learning tasks, e.g., node classification,   \n27 graph classification, link prediction and graph generation, etc.GNNs are built on the homophily   \n28 assumption[26], i.e., connected nodes tend to share similar attributes with each other $\\mathbb { \\ m }$ , which   \n29 offers additional information besides node features. Such relational inductive bias $\\mathbb { \\left[ 3 \\right] }$ is believed to   \n30 be a key factor leading to GNNs’ superior performance over NNs’ in many tasks.   \n31 Nevertheless, growing evidence shows that GNNs do not always gain advantages over traditional NNs   \n32 when dealing with relational data. In some cases, even simple Multi-Layer Perceptrons (MLPs) can   \n33 outperform GNNs by a large margin [35, 22, 5]. An important reason for the performance degradation   \n34 is believed to be the heterophily problem, i.e., connected nodes tend to have different labels which   \n35 makes the homophily assumption fail. Heterophily challenge has received attention recently and   \n36 there are increasing number of models being put forward to address this problem [35, 22, 5, 34, 33].   \n37 Contributions In this paper, we first demonstrate that not all heterophilous graphs are harmful for   \n38 aggregation-based GNNs and the existing metrics of homophily are insufficient to decide whether the   \n39 aggregation operation will make nodes less distinguishable or not. By constructing a similarity matrix   \n40 from backpropagation analysis, we derive new metrics to depict how much GNNs are influenced by   \n41 the graph structure and node features. We show the advantage of our metrics over the existing metrics   \n42 by comparing the ability of characterizing the performance of two baseline GNNs on synthetic graphs   \n43 of different levels of homophily. From the similarity matrix, we find that diversification operation   \n44 is able to address some harmful heterophily cases, and then based on which we propose Adaptive   \n45 Channel Mixing (ACM) GNN framework. The experiments on the synthetic datasets, real-world   \n46 datasets and the ablation studies consistently show that baseline GNN augmented by ACM framework   \n47 is able to obtain significant performance boost on node classification tasks on heterophilous graphs.   \n48 The rest of this paper is mainly organized as follows: In section 2, we introduce the notation and the   \n49 background knowledge. In section $\\boxed { 3 }$ we conduct node-wise heterophily analysis, derive new metrics   \n50 based on a similarity matrix and conduct experiments to show their advantage. In section $\\boxed { 4 . 3 }$ we   \n51 propose the ACM-GNN framework to adaptively utilize the information from different filterbank   \n52 channels to address heterophily problem. In section $5 ,$ we discuss the related works and clarify   \n53 the differences to our method. In section $\\textcircled{6}$ we provide empirical evaluations on ACM framework,   \n54 including ablation study and tests on 11 real-world node classification tasks. ",
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+    {
+        "type": "text",
+        "text": "55 2 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
+            148,
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+        ],
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+        "text": "56 We introduce the related notation and background knowledge. We use bold fonts for vectors (e.g.,   \n57 $v )$ . Suppose we have an undirected connected graph $\\mathcal { G } = ( \\nu , \\mathcal { E } , A )$ , where $\\nu$ is the node set with   \n58 $| \\nu | = N ;$ ; $\\mathcal { E }$ is the edge set without self-loop; $\\bar { A ( \\in \\mathbb { R } ^ { N \\times N } }$ is the symmetric adjacency matrix with   \n59 $A _ { i , j } = 1$ iff $e _ { i j } \\in \\mathcal { E }$ , otherwise $A _ { i , j } = 0$ . We use $D$ to denote the diagonal degree matrix of $\\mathcal { G }$ , i.e.,   \n60 $\\begin{array} { r } { D _ { i , i } = d _ { i } = \\check { \\sum } _ { j } A _ { i , j } } \\end{array}$ and use ${ \\mathcal { N } } _ { i }$ to denote the neighborhood set of node $i$ , i.e., ${ \\mathcal { N } } _ { i } = \\{ j : e _ { i j } \\in { \\mathcal { E } } \\}$ .   \n61 A graph signal is a vector $\\pmb { x } \\in \\mathbb { R } ^ { N }$ defined on $\\nu$ , where $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ is defined on the node $i$ . We also have a   \n62 feature matrix $X \\in \\mathbb { R } ^ { N \\times F }$ , whose columns are graph signals and whose $i$ -th row $X _ { i , \\astrosun }$ : is a feature   \n63 vector of node $i$ . We use $Z \\in \\mathbb { R } ^ { N \\times C }$ to denote the label encoding matrix, whose $i$ -th row $Z _ { i , \\cdot }$ : is the   \n64 one-hot encoding of the label of node $i$ . The $i$ -th column of the identity matrix $I$ is denoted by $e _ { i }$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "65 2.1 Graph Laplacian, Affinity Matrix and Their Variants ",
+        "text_level": 1,
+        "bbox": [
+            151,
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+        "text": "66 The (combinatorial) graph Laplacian is defined as $L = D - A$ , which is Symmetric Positive   \n67 Semi-Definite (SPSD) $\\dot { \\left[ 6 \\right] }$ . Its eigendecomposition gives ${ \\cal L } = U \\Lambda U ^ { T }$ , where the columns $\\mathbf { \\Delta } \\mathbf { u } _ { i }$ of   \n68 $U \\in \\mathbb { R } ^ { N \\times N }$ are orthonormal eigenvectors, namely the graph Fourier basis, $\\boldsymbol { \\Lambda } = \\operatorname { d i a g } ( \\lambda _ { 1 } , \\ldots , \\lambda _ { N } )$   \n69 with $\\lambda _ { 1 } \\leq \\cdots \\leq \\lambda _ { N }$ , and these eigenvalues are also called frequencies. The graph Fourier transform   \n70 of the graph signal $_ { \\textbf { \\em x } }$ is defined as $\\pmb { x } _ { \\mathcal { F } } = { \\cal U } ^ { - 1 } \\pmb { x } = { \\cal U } ^ { T } \\pmb { x } = [ \\pmb { u } _ { 1 } ^ { T } \\pmb { x } , \\dots , \\pmb { u } _ { N } ^ { T } \\pmb { x } ] ^ { \\hat { T } }$ , where $\\pmb { u } _ { i } ^ { T } \\pmb { x }$ is the   \n71 component of $_ { \\textbf { \\em x } }$ in the direction of $\\mathbf { \\Delta } \\mathbf { u } _ { i }$ .   \n72 In additional to $L$ , some variants are also commonly used, e.g., the symmetric normalized Laplacian   \n73 $L _ { \\mathrm { s v m } } = D ^ { - 1 / 2 } L D ^ { - 1 / 2 } = I - D ^ { - 1 / 2 } A D ^ { - 1 / 2 }$ and the random walk normalized Laplacian $L _ { \\mathrm { r w } } =$   \n74 $D ^ { \\dot { - } 1 } L = I - D ^ { - 1 } A$ . The affinity (transition) matrices can be derived from the Laplacians, e.g.,   \n75 $A _ { \\mathrm { r w } } = I - L _ { \\mathrm { r w } } = D ^ { - 1 } A$ , $A _ { \\mathrm { s y m } } = \\bar { I } - L _ { \\mathrm { s y m } } = D ^ { - 1 / 2 } A D ^ { - 1 / 2 }$ and are considered to be low-pass   \n76 filters $\\lVert 2 5 \\rVert$ . Their eigenvalues satisfy $\\lambda _ { i } ( A _ { \\mathrm { r w } } ) = \\lambda _ { i } ( A _ { \\mathrm { s y m } } ) = 1 - \\lambda _ { i } ( L _ { \\mathrm { s y m } } ) = 1 - \\lambda _ { i } ( L _ { \\mathrm { r w } } ) \\in ( - \\bar { 1 } , 1 ]$   \n77 Applying the renormalization trick $\\mathbb { I } \\breve { \\Sigma } \\mathbb { I }$ to affinity and Laplacian matrices respectively leads to   \n78 $\\hat { A } _ { \\mathrm { s y m } } ^ { - 1 } = \\bar { \\tilde { D } } ^ { - 1 / 2 } \\tilde { A } \\tilde { D } ^ { - 1 / 2 }$ and $\\hat { L } _ { \\mathrm { s y m } } = I - \\hat { A } _ { \\mathrm { s y m } }$ , where ${ \\tilde { A } } \\equiv A + I$ and $\\tilde { D } \\equiv D + I$ . The renormalized   \n79 affinity matrix essentially adds a self-loop to each node in the graph, and is widely used in Graph   \n80 Convolutional Network (GCN) $\\mathbb { \\lVert 1 5 \\rVert }$ as follows, ",
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+        "bbox": [
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+    {
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+        "img_path": "images/c6cb1d5120a39b31c21b2f716b37bf3b3bbb34dded5048f8b77cb0fb4c8ae971.jpg",
+        "text": "$$\nY = \\mathrm { s o f t m a x } ( { \\hat { A } } _ { \\mathrm { s y m } } \\mathrm { R e L U } ( { \\hat { A } } _ { \\mathrm { s y m } } X W _ { 0 } ) W _ { 1 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where 81 $W _ { 0 } \\in \\mathbb { R } ^ { F \\times F _ { 1 } }$ and $W _ { 1 } \\in \\mathbb { R } ^ { F _ { 1 } \\times O }$ are learnable parameter matrices. GCN can be trained by 82 minimizing the following cross entropy loss ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e00880a9736dd68f50076c137d4521ee5bc0f554c7f63e0bd0f967278d4417de.jpg",
+        "text": "$$\n\\mathcal { L } = - \\mathrm { t r a c e } ( Z ^ { T } \\log Y )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "83 where $\\log ( \\cdot )$ is a component-wise logarithm operation. The random walk renormalized matrix   \n84 $\\hat { A } _ { \\mathrm { r w } } = \\tilde { D } ^ { - 1 } \\tilde { A }$ , which shares the same eigenvalues as $\\hat { A } _ { \\mathrm { s y m } }$ , can also be applied in GCN. The ",
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+    },
+    {
+        "type": "text",
+        "text": "corresponding Laplacian is defined as 85 $\\hat { L } _ { \\mathrm { r w } } = I - \\hat { A } _ { \\mathrm { r w } }$ . $\\hat { A } _ { \\mathrm { r w } }$ is essentially a random walk matrix and 86 behaves as a mean aggregator that is applied in spatial-based GNNs [12, 11]. To bridge the spectral and spatial methods, we use 87 $\\hat { A } _ { r w }$ in the paper. ",
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+    {
+        "type": "text",
+        "text": "2.2 Metrics of Homophily ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "89 The metrics of homophily are defined by considering different relations between node labels and   \n90 graph structures defined by adjacency matrix. There are three commonly used homophilies: edge   \n91 homophily [1, 35], node homophily $\\pm \\overbrace { \\mathbb { L } 8 } \\Vert$ , and class homophily [21] 1 defined as follows: ",
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+        "img_path": "images/e6fb37c92277f6177c7d8a41347269085adb7b2f296d5104d81e6a4de0143a4b.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { I } _ { \\mathrm { e d g e } } ( \\mathcal { G } ) = \\frac { \\big | \\{ e _ { u v } \\mid e _ { u v } \\in \\mathcal { E } , Z _ { u , : } = Z _ { v , : } \\} \\big | } { | \\mathcal { E } | } , ~ H _ { \\mathrm { n o t e } } ( \\mathcal { G } ) = \\frac { 1 } { | \\mathcal { V } | } \\displaystyle \\sum _ { v \\in \\mathcal { V } } \\frac { \\big | \\{ u \\mid u \\in \\mathcal { N } _ { v } , Z _ { u , : } = Z _ { v , : } \\} \\big | } { d _ { v } } , } \\\\ { \\displaystyle \\mathcal { I } _ { \\mathrm { c l a s } } ( \\mathcal { G } ) = \\frac { 1 } { C - 1 } \\displaystyle \\sum _ { k = 1 } ^ { C } \\bigg [ h _ { k } - \\frac { \\big | \\{ v \\mid Z _ { v , k } = 1 \\} \\big | } { N } \\bigg ] _ { + } , ~ h _ { k } = \\frac { \\sum _ { v \\in \\mathcal { V } } \\big | \\{ u \\mid Z _ { v , k } = 1 , u \\in \\mathcal { N } _ { v } , Z _ { u , : } = Z _ { v , : } \\} \\big | } { \\sum _ { v \\in \\{ v | Z _ { v , k } = 1 \\} } d _ { v } } , ~ } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "92 where $[ a ] _ { + } = \\operatorname* { m a x } ( a , 0 )$ ; $h _ { k }$ is the class-wise homophily metric $[ [ 2 1 ] ]$ . They are all in the range   \n93 of $[ 0 , 1 ]$ and a value close to 1 corresponds to strong homophily while a value close to 0 indicates   \n94 strong heterophily. $H _ { \\mathrm { e d g e } } ( { \\mathcal { G } } )$ measures the proportion of edges that connect two nodes in the same   \n95 class; $H _ { \\mathrm { n o d e } } ( \\mathcal { G } )$ evaluates the average proportion of edge-label consistency of all nodes; $H _ { \\mathrm { c l a s s } } ( \\mathcal G )$   \n96 tries to avoid the sensitivity to imbalanced class, which can cause $H _ { \\mathrm { e d g e } }$ misleadingly large. The   \n97 above definitions are all based on the graph-label consistency and imply that the inconsistency will   \n98 cause harmful effect to GNNs. With this in mind, we will show a counter example to illustrate the   \n99 insufficiency of the above metrics and propose new metrics. ",
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+    {
+        "type": "text",
+        "text": "3 Analysis of Heterophily ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 Motivation and Aggregation Homophily ",
+        "text_level": 1,
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+        "text": "02 Heterophily is believed to be harmful for   \n103 message-passing based GNNs [35, 28, 5] be  \n104 cause intuitively features of nodes in different   \n105 classes will be falsely mixed and this will lead   \n106 nodes indistinguishable [35]. Nevertheless, it   \n107 is not always the case, e.g., the bipartite graph   \n108 shown in Figure $\\boxed { 1 }$ is highly heterophilous ac  \n109 cording to the homophily metrics in $\\textcircled{3}$ , but   \n10 after mean aggregation, the nodes in classes 1   \n111 and 2 only exchange colors and are still dis  \n112 tinguishable. Authors in $\\pmb { \\Vert 5 \\Vert }$ also point out the   \n113 insufficiency of $H _ { \\mathrm { n o d e } }$ by examples to show that   \n14 different graph typologies with the same $H _ { \\mathrm { n o d e } }$   \n15 can carry different label information.   \n116 To analyze to what extent the graph structure can affect the output of a GNN, we first simplify the   \n117 GCN by removing its non-linearity as $\\textcircled { \\scriptsize { 1 3 1 } }$ . Let $\\hat { A } \\in \\mathbb { R } ^ { N \\times N }$ denote a general aggregation operator.   \n118 Then, equation $\\bar { \\textcircled { 1 } }$ can be simplified as, ",
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+        "image_caption": [
+            "Figure 1: Example of harmless heterophily "
+        ],
+        "image_footnote": [],
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+        "img_path": "images/fdf29c6bf3a111c00e30d02b4114a50cf6854ce7e45642036553c8b0855e65fe.jpg",
+        "text": "$$\nY = \\mathrm { s o f t m a x } ( \\hat { A } X W ) = \\mathrm { s o f t m a x } ( Y ^ { \\prime } )\n$$",
+        "text_format": "latex",
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+        "text": "After each gradient decent step 119 $\\begin{array} { r } { \\Delta W = \\gamma \\frac { \\partial \\mathcal { L } } { \\partial W } } \\end{array}$ , where $\\gamma$ is the learning rate, and the update of $Y ^ { \\prime }$ will 120 be (see Appendix $\\boxed { \\mathbf { B } }$ for derivation), ",
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+        "text": "$$\n\\Delta Y ^ { \\prime } = { \\hat { A } } X \\Delta W = \\gamma { \\hat { A } } X { \\frac { \\partial { \\mathcal { L } } } { \\partial W } } \\propto { \\hat { A } } X { \\frac { \\partial { \\mathcal { L } } } { \\partial W } } = { \\hat { A } } X X ^ { T } { \\hat { A } } ^ { T } ( Z - Y ) = S ( { \\hat { A } } , X ) ( Z - Y )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
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+        "type": "text",
+        "text": "121 where $S ( \\hat { A } , X ) \\equiv \\hat { A } X ( \\hat { A } X ) ^ { T }$ is a node similarity matrix after aggregation, $Z - Y$ is the prediction   \n122 error matrix. The update direction of node $i$ is essentially a weighted sum of the prediction error, $i . e .$ .,   \n123 $\\begin{array} { r } { \\Delta ( Y ^ { \\prime } ) _ { i , : } = \\sum _ { j \\in \\mathcal { V } } \\Big [ S ( \\hat { A } , X ) \\Big ] _ { i , j } \\big ( Z - Y \\big ) _ { j , : } . } \\end{array}$ ",
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+    {
+        "type": "text",
+        "text": "1 The authors in [21] did not name this homophily metric. We name it class homophily based on its definition. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "125 Definition 1. Aggregation similarity score ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/154a21d17ef9feb2de1e262c19ecd61737c15c08061e3961bad3256666907b87.jpg",
+        "text": "$$\n\\begin{array}{c} \\operatorname { S } _ { a g g } \\Big ( S ( \\hat { A } , X ) \\Big ) = \\frac { \\Big | \\Big \\{ v \\big | \\operatorname { M e a n } _ { u } \\big ( \\{ S ( \\hat { A } , X ) _ { v , u } | Z _ { u ; \\cdot } = Z _ { v , \\cdot } \\} \\big ) \\geq \\operatorname { M e a n } _ { u } \\big ( \\{ S ( \\hat { A } , X ) _ { v , u } | Z _ { u ; \\cdot } \\neq Z _ { v , \\cdot } \\} \\big ) \\Big . } \\\\ { \\Big . | \\mathcal { V } | } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "26 where ${ \\mathrm { M e a n } } _ { u } \\left( \\{ \\cdot \\} \\right)$ takes the average over u of a given multiset of values or variables. ",
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+        "type": "text",
+        "text": "127 $S _ { \\mathrm { a g g } } ( S ( \\hat { A } , X ) )$ is the proportion of nodes $v \\in \\mathcal V$ that will put relatively larger similarity weights on   \n128 nodes in the same class than in other classes after aggregation. It is easy to see that $S _ { \\mathrm { a g g } } ( S ( \\hat { A } , X ) ) \\in$   \n129 [0, 1]. But in practice, we observe that in most datasets, we will have $S _ { \\mathrm { a g g } } ( S ( \\hat { A } , X ) ) \\geq 0 . 5$ . Based on   \n130 this observation, we rescale $\\textcircled{6}$ to the following modified aggregation similarity for practical usage, ",
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+        "type": "equation",
+        "img_path": "images/6d31da42ad19cfec77a9c86d59899a5bd58f71c2333fc77f948b5a4faa94b997.jpg",
+        "text": "$$\nS _ { \\mathrm { a g g } } ^ { M } \\left( S ( \\hat { A } , X ) \\right) = \\Bigl [ 2 S _ { \\mathrm { a g g } } \\left( S ( \\hat { A } , X ) \\right) - 1 \\Bigr ] _ { + }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "131 In order to measure the consistency between labels and graph structures without considering node   \n132 features and make a fair comparison with the existing homophily metrics in $\\textcircled{3}$ , we define the graph   \n133 $( { \\mathcal { G } } )$ aggregation $( \\hat { A } )$ homophily and its modified version as ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/7ccd34461e326b77ce0c3e8fc6a7b44653d686f1dae23d28e4c3d90c31deed93.jpg",
+        "text": "$$\nH _ { \\mathrm { a g g } } ( \\mathcal { G } ) = S _ { \\mathrm { a g g } } \\left( S ( \\hat { A } , Z ) \\right) , H _ { \\mathrm { a g g } } ^ { M } ( \\mathcal { G } ) = S _ { \\mathrm { a g g } } ^ { M } \\left( S ( \\hat { A } , Z ) \\right)\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "134 In practice, we will only check $H _ { \\mathrm { a g g } } ( { \\mathcal { G } } )$ when $H _ { \\mathrm { a g g } } ^ { M } ( { \\mathcal G } ) = 0$ . As Figure $^ 1$ shows, when $\\hat { A } = \\hat { A } _ { \\mathrm { r w } }$ ,   \n135 $H _ { \\mathrm { a g g } } ( \\mathcal G ) = H _ { \\mathrm { a g g } } ^ { M } ( \\mathcal G ) = 1$ . Thus, this new metric reflects the fact that nodes in classes 1 and 2 are still   \n136 highly distinguishable after aggregation, while other metrics mentioned before fail to capture the   \n137 information and misleadingly give value 0. This shows the advantage of $H _ { \\mathrm { a g g } } ( \\mathcal { G } )$ and $H _ { \\mathrm { a g g } } ^ { \\dot { M } } ( { \\mathcal G } )$ by   \n138 additionally considering information from aggregation operator $\\hat { A }$ and the similarity matrix. ",
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+    },
+    {
+        "type": "text",
+        "text": "To comprehensively compare 139 $H _ { \\mathrm { a g g } } ^ { M } ( { \\mathcal { G } } )$ and the metrics in $\\textcircled{3}$ in terms of how they reveal the influence 140 of graph structure on the GNN performance, we generate synthetic graphs and evaluate SGC [31] 141 and GCN $[ \\overline { { | 1 5 | } }$ on them in the next subsection. ",
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+    {
+        "type": "text",
+        "text": "3.2 Evaluation and Comparison on Synthetic Graphs ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "143 Data Generation & Experimental Setup For one dataset, we generate 95 graphs in total with 19   \n144 edge homophily levels varied from 0.05 to 0.95, each corresponding to 5 graphs. For every generated   \n145 graph, we have 5 classes with 400 nodes in each class. In each class, there are randomly generated   \n146 800 intra-class edges and $( 8 0 0 - 8 0 0 H _ { \\mathrm { e d g e } } ( \\mathcal G ) ) / H _ { \\mathrm { e d g e } } ( \\mathcal G ) ] _ {  } ^ {  } ]$ inter-class edges. The features of nodes   \n147 in each class are sampled from node features in the corresponding class of the base dataset. Nodes   \n148 are randomly split into $6 0 \\% / 2 0 \\% / 2 0 \\%$ for train/validation/test. We train 1-hop SGC (sgc-1 [31] and   \n149 $\\mathrm { G C N } \\left\\| \\Sigma \\right\\|$ on synthetic data (see appendix $\\boxed { \\mathrm { A . 1 } }$ for hyperparameter searching range). For each value   \n150 of $H _ { \\mathrm { e d g e } } ( { \\mathcal { G } } )$ , we take the average test accuracy and standard deviation of runs over 5 generated graphs.   \n151 For each generated graph, we also calculate its $H _ { \\mathrm { n o d e } } ( \\mathcal { G } )$ , $H _ { \\mathrm { c l a s s } } ( \\mathcal { G } )$ and $H _ { \\mathrm { a g g } } ^ { M } ( { \\mathcal { G } } )$ . Model performance   \nwith respect to different homophily values are shown in Figure 2.   \n153 Comparison of Homophily Metrics The performance of SGC-1 and GCN are expected to be   \n154 monotonically increasing with a proper and informative homophily metric. However, Figure $2 ( \\mathrm { a } ) ( \\mathrm { b } ) ( \\mathrm { c } )$   \n155 show that the performance curves under $H _ { \\mathrm { e d g e } } ( \\mathcal { G } ) , H _ { \\mathrm { n o d e } } ( \\mathcal { \\bar { G } } )$ and $H _ { \\mathrm { c l a s s } } ( \\mathcal G )$ are $U$ -shaped $\\mathbb { B }$ while   \n156 Figure $\\bigstar \\bigstar \\bigstar$ reveals a nearly monotonic curve with a little perturbation around 1. This indicates that   \n157 $H _ { \\mathrm { a g g } } ^ { \\overline { { M } } } ( \\mathcal { G } )$ can describe how the graph structure affects the performance of SGC-1 and GCN.   \n58 In addition, we notice that in Figure $2 ( \\mathrm { a } )$ , both SGC-1 and GCN get the worst performance on all   \n9 datasets when $H _ { \\mathrm { e d g e } } ( { \\mathcal { G } } )$ is around somewhere between 0.1 and 0.2. This interesting phenomenon can   \n0 be explained by the following theorem.   \n161 Theorem 1. (See Appendix $\\bigtriangledown$ for proof). Suppose there are $C$ classes in the graph $\\mathcal { G }$ , edges for each   \n162 node are i.i.d.generated such that each edge of any node has probability $h$ of connecting with nodes in   \n163 the same class and probability $1 - h$ of connecting with nodes in different classes, and $\\mathbb { E } ( d _ { v } ) = d$ for   \n164 all nodes. Let the aggregation operator $\\hat { A } = \\hat { A } _ { \\mathrm { r w } }$ . Then, for nodes $v , u _ { 1 }$ and $u _ { 2 }$ , where $Z _ { u _ { 1 } , : } = Z _ { v , }$ ,:   \n165 and $Z _ { u _ { 2 } , : } \\neq Z _ { v , : }$ , we have ",
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+        "type": "image",
+        "img_path": "images/84bd4e741367d6258d606b257499e6bd2fde9facc78f8372bb4352314bd1309e.jpg",
+        "image_caption": [
+            "Figure 2: Comparison of baseline performance under different homophily metrics. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "equation",
+        "img_path": "images/2a22f83182a233db2e55f4635d303464b3ca3e65355c118c3f2f039fdbf511fa.jpg",
+        "text": "$$\ng ( h ) \\equiv {  { \\mathbb E } } \\left( S ( {  { \\hat { A } } } , Z ) _ { v , u _ { 1 } } \\right) - {  { \\mathbb E } } \\left( S ( {  { \\hat { A } } } , Z ) _ { v , u _ { 2 } } \\right) = \\left( \\frac { ( C - 1 ) ( h d + 1 ) - ( 1 - h ) d } { ( C - 1 ) ( d + 1 ) } \\right) ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            338
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "and the minimum of $g ( h )$ is reached at ",
+        "bbox": [
+            173,
+            344,
+            429,
+            359
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5875d391837e4f1fe0105c35da30ba3d9038a45f215a44a689db272dac7de306.jpg",
+        "text": "$$\nh = \\frac { d + 1 - C } { C d } = \\frac { d _ { \\mathrm { i n t r a } } / h + 1 - C } { C ( d _ { \\mathrm { i n t r a } } / h ) } \\Rightarrow h = \\frac { d _ { \\mathrm { i n t r a } } } { C d _ { \\mathrm { i n t r a } } + C - 1 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "166 where $d _ { \\mathrm { i n t r a } } = d h$ , which is the expectation of the number of neighbors of a node that have the same   \n167 label as the node.   \n168 The value of $g ( h )$ in $\\textcircled{9}$ is the expected differences of the similarity values between nodes in the   \n169 same class as $v$ and nodes in other classes. $g ( h )$ is strongly related to the definition of aggregation   \n170 homophily and its minimum potentially implies the worst value of $H _ { \\mathrm { a g g } } ( { \\mathcal { G } } )$ . In the synthetic experi  \n171 ments, we have $d _ { \\mathrm { i n t r a } } = 2 , C = 5$ and the minimum of $g ( h )$ is reached at $h = 1 / 7 \\approx 0 . 1 4$ , which   \n172 corresponds to the lowest point in the performance curve in Figure $\\bigstar \\bigstar \\bigstar$ . In other words, the $h$ where   \n173 SGC-1 and GCN perform worst is where $g ( h )$ gets the smallest value, instead of the point with the   \n174 smallest edge homophily value. This again shows the advantage of $H _ { \\mathrm { a g g } } ( { \\mathcal { G } } )$ over $H _ { \\mathrm { e d g e } } ( { \\mathcal { G } } )$ by taking   \n175 use of the similarity matrix. ",
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+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/fe1663ea7f22a0ed0bfdf07bb223ad73527ea58208148ab9c322c548e07bcd51.jpg",
+        "image_caption": [
+            "Figure 3: Example of how HP filter addresses harmful heterophily "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "176 4 Adaptive Channel Mixing (ACM) Framework ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "177 4.1 How Diversification Operation Helps with Harmful Heterophily ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "178 We first consider the example shown in Figure $\\boxed { 3 }$ . From $S ( { \\hat { A } } , X )$ , nodes 1,3 assign relatively large   \n179 positive weights to nodes in class 2, which will negatively affect information aggregation. Despite   \n180 the fact, we can still distinguish between nodes 1,3 and 4,5,6,7 by considering their neighborhood   \n181 difference: nodes 1,3 are distinguishable from their neighbors while nodes 4,5,6,7 are homogeneous to   \n182 their neighbors. This indicates, in some cases, although some nodes become similar after aggregation,   \n183 they are still distinguishable via surrounding dissimilarities. This suggests the possibility of using   \n184 diversification operation to address harmful heterophily i.e., high-pass (HP) filter $I - { \\hat { A } }$ [8] (will be   \n185 introduced in next subsection). As $S ( I - { \\hat { A } } , Z )$ in Figure $3$ shows, nodes 1,3 assign negative weights   \n186 to nodes 4,5,6,7, i.e., nodes 1,3 treat nodes 4,5,6,7 as negative samples and will move away from   \n187 them. Base on this example, we propose diversification distinguishability as follows, ",
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+    {
+        "type": "text",
+        "text": "Definition 2. Diversification Distinguishability $( D D )$ based on $S ( I - { \\hat { A } } , X )$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Given $S ( I - { \\hat { A } } , X )$ , a node $v$ is diversification distinguishable if the following two conditions are satisfied at the same time, ",
+        "bbox": [
+            158,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3322b5544d10c9c4bafff4962ddf6c89b23ee1c86484effe629d9e6c5c6f9996.jpg",
+        "text": "$$\n\\begin{array} { r } { I . \\mathrm { \\mathrm { ~ M e a n } } _ { u } \\left( \\{ S ( I - \\hat { A } , X ) _ { v , u } | u \\in \\mathcal { V } \\wedge Z _ { u , : } = Z _ { v , : } \\} \\right) \\geq 0 ; } \\\\ { 2 . \\mathrm { \\mathrm { ~ M e a n } } _ { u } \\left( \\{ S ( I - \\hat { A } , X ) _ { v , u } | u \\in \\mathcal { V } \\wedge Z _ { u , : } \\neq Z _ { v , : } \\} \\right) \\leq 0 } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "191 Then, graph diversification distinguishability value is defined as ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e35e91c710c34a32d2dcfa2c14ef3a03325c84331e14318aa44eddf59ec52fc9.jpg",
+        "text": "$$\n\\mathrm { D D } _ { \\hat { A } , X } ( { \\mathcal G } ) = \\frac { 1 } { | \\mathcal V | } \\Big | \\{ v | v i s d i v e r s i f i c a t i o n d i s t i n g u i s h a b l e \\} \\Big |\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "92 $\\mathrm { D D } _ { \\hat { A } , X } ( { \\mathcal G } ) \\in [ 0 , 1 ]$ measures the proportion of nodes that HP filter is helpful for. Its effectiveness   \n93 can be proved for binary classification problems under certain conditions, leading us to: ",
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+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. (See Appendix94 $\\boxed { \\mathrm { E } }$ for proof). Suppose $X = Z , \\hat { A } = \\hat { A } _ { \\mathrm { r w } } .$ . Then, for a binary classifica5 tion problem, i.e., $C = 2$ , all nodes are diversification distinguishable and $\\mathrm { D D } _ { \\hat { A } , Z } ( { \\mathcal G } ) = \\mathrm { 1 }$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Conducting both aggregation and diversification operations to distinctively extract the low- and highfrequency information from graph signals is the same as using filterbanks in graph signal processing. We introduce filterbanks in next subsection. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 Filterbank in Spectral and Spatial Forms ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Filterbank For the graph signal $_ { \\textbf { \\em x } }$ defined on $\\mathcal { G }$ , a 2-channel linear (analysis) filterbank [8] 4 includes a pair of filters $H _ { \\mathrm { L P } } , H _ { \\mathrm { H P } }$ , where $H _ { \\mathrm { L P } }$ and $H _ { \\mathrm { H P } }$ retain the low-frequency and high-frequency content of $_ { \\textbf { \\em x } }$ , respectively. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Most existing GNNs are under uni-channel filtering architecture $[ \\overline { { 1 5 } } , \\overline { { 3 0 } } , \\overline { { 1 2 } } ]$ with either $H _ { \\mathrm { L P } }$ or $H _ { \\mathrm { H P } }$ channel that only partially preserves the input information. Unlike the uni-channel architecture, filterbanks with $H _ { \\mathrm { L P } } + H _ { \\mathrm { H P } } = I$ will not lose any information of the input signal, i.e., perfect reconstruction property $\\mathbb { B } \\mathbb { B } \\mathbb { Z } \\mathbb { Z }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Generally, the Laplacian matrices207 $( L _ { \\mathrm { s y m } } , L _ { \\mathrm { r w } } , \\hat { L } _ { \\mathrm { s y m } } , \\hat { L } _ { \\mathrm { r w } } )$ can be regarded as HP filters $\\pmb { \\bigtriangledown }$ and affinity matrices 208 $( A _ { \\mathrm { s y m } } , A _ { \\mathrm { r w } } , \\hat { A } _ { \\mathrm { s y m } } , \\hat { A } _ { \\mathrm { r w } } )$ can be treated as LP filters $[ [ 2 5 , \\boxed { 1 1 } ] ]$ . Moreover, MLPs can be 209 considered as owing a special identity filterbank with matrix $I$ that satisfies $H _ { \\mathrm { L P } } + H _ { \\mathrm { H P } } = I + 0 = I$ . ",
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+        "type": "text",
+        "text": "210 Filterbank in Spatial Form Filterbank methods can also be extended to spatial GNNs. Formally,   \n211 on the node level, left multiplying $H _ { \\mathrm { L P } }$ and $H _ { \\mathrm { H P } }$ on $_ { \\textbf { \\em x } }$ performs as aggregation and diversification   \n212 operations, respectively. For example, suppose $H _ { \\mathrm { L P } } = { \\hat { A } }$ and $H _ { \\mathrm { H P } } = I - { \\hat { A } }$ , then for node $i$ we have ",
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+    {
+        "type": "equation",
+        "img_path": "images/92f3d4c6e486fc873de5d28956daa7c4f43e73eaa899eb6638c3b0c729db0408.jpg",
+        "text": "$$\n( H _ { \\mathrm { L P } } \\pmb { x } ) _ { i } = \\sum _ { j \\in \\{ \\mathcal { N } _ { i } \\cup i \\} } \\hat { A } _ { i , j } \\pmb { x } _ { j } , ( H _ { \\mathrm { H P } } \\pmb { x } ) _ { i } = \\pmb { x } _ { i } - \\sum _ { j \\in \\{ \\mathcal { N } _ { i } \\cup i \\} } \\hat { A } _ { i , j } \\pmb { x } _ { j }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "213 where $\\hat { A } _ { i , j }$ is the connection weight between two nodes. To leverage HP and identity channels in   \n214 GNNs, we propose the Adaptive Channel Mixing (ACM) architecture in the following subsection.   \n216 ACM framework can be applied in lots of baseline GNNs and in this subsection, we use GCN as an   \n217 example and introduce ACM framework in matrix form. We use $H _ { \\mathrm { L P } }$ and $H _ { \\mathrm { H P } }$ to represent general   \n218 LP and HP filters. The ACM framework includes 3 steps as follows, ",
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+        "type": "text",
+        "text": "Step 1. Feature Extraction for Each Channel: ",
+        "text_level": 1,
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+        "text": "$\\begin{array} { r } { H _ { L } ^ { l } = H _ { \\mathrm { L P } } \\mathrm { R e L U } \\left( H ^ { l - 1 } W _ { L } ^ { l - 1 } \\right) , H _ { H } ^ { l } = H _ { \\mathrm { H P } } \\mathrm { R e L U } \\left( H ^ { l - 1 } W _ { H } ^ { l - 1 } \\right) , H _ { I } ^ { l } = I \\mathrm { R e L U } \\left( H ^ { l - 1 } W _ { I } ^ { l - 1 } \\right) , } \\end{array}$ $W _ { L } ^ { l - 1 } , W _ { H } ^ { l - 1 } , W _ { I } ^ { l - 1 } \\in \\mathbb { R } ^ { F _ { l - 1 } \\times F _ { l } }$ ; ",
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+    {
+        "type": "text",
+        "text": "Step 2. Feature-based Weight Learning with Row Normalization (RN): ",
+        "text_level": 1,
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+        "type": "equation",
+        "img_path": "images/c35dc210976604583cb7de6612ad151e7caa7c9a8b664a67a78558e629517e58.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\tilde { H } _ { I } ^ { l } = \\mathrm { R N } \\left( H _ { I } ^ { l } \\right) , \\tilde { H } _ { L } ^ { l } = \\mathrm { R N } \\left( H _ { L } ^ { l } \\right) , \\tilde { H } _ { H } ^ { l } = \\mathrm { R N } \\left( H _ { H } ^ { l } \\right) ; } \\\\ & { \\alpha _ { L } ^ { l } = \\sigma \\left( \\mathrm { E L U } \\left( \\tilde { H } _ { L } ^ { l } \\tilde { W } _ { L } ^ { l } \\right) \\right) , \\alpha _ { H } ^ { l } = \\sigma \\left( \\mathrm { E L U } \\left( \\tilde { H } _ { H } ^ { l } \\tilde { W } _ { H } ^ { l } \\right) \\right) , \\alpha _ { I } ^ { l } = \\sigma \\left( \\mathrm { E L U } \\left( \\tilde { H } _ { I } ^ { l } \\tilde { W } _ { I } ^ { l } \\right) \\right) , } \\\\ & { \\tilde { W } _ { L } ^ { l - 1 } , \\tilde { W } _ { H } ^ { l - 1 } , \\tilde { W } _ { I } ^ { l - 1 } \\in \\mathbb { R } ^ { F _ { l } \\times 1 } ; } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Step 3. Channel Mixing: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/cf261b2fe3b983a3c7177b703df3d122b2af89fcf9d5be2afabdaab933e91077.jpg",
+        "text": "$$\nH ^ { l } = \\left( { \\mathrm { d i a g } } ( \\alpha _ { L } ^ { l } ) H _ { L } ^ { l } + { \\mathrm { d i a g } } ( \\alpha _ { H } ^ { l } ) H _ { H } ^ { l } + { \\mathrm { d i a g } } ( \\alpha _ { I } ^ { l } ) H _ { I } ^ { l } \\right) . \\nonumber\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "219 ACM-GCN first implements distinct non-linear feature extractions for 3 channels, respectively. After   \n220 processed by a set of filterbanks, 3 filtered components $H _ { L } ^ { l } , H _ { H } ^ { l } , H _ { I } ^ { l }$ are obtained. Different nodes   \n221 may have different needs for the information in the 3 channels, e.g., in Figure ${ \\underline { { 3 } } } ,$ nodes 1,3 demand   \n222 high-frequency information while node 2 only needs low-frequency information. To adaptively exploit   \n223 information from different channels, ACM-GCN learns rowwise (nodewise) feature-conditioned   \n224 (un-normalized) weights to combine the 3 channels. ACM can be easily plugged into spatial GNNs by   \n225 replacing $H _ { \\mathrm { L P } }$ and $H _ { \\mathrm { H P } }$ by aggregation and diversification operations as shown in $( 1 2 )$ . See Appendix   \n226 F for a detailed discussion of model comparison on synthetic datasets.   \n27 Complexity Number of learnable parameters in layer $l$ of ACM-GCN is $3 F _ { l - 1 } ( F _ { l } + 1 )$ , while it is   \n28 $F _ { l - 1 } F _ { l }$ in GCN. The computation of step 1-3 takes $N F _ { l } ( 2 0 + F _ { l - 1 } ) + 2 F _ { l } ( \\mathrm { n n z } ( H _ { \\mathrm { L P } } ) + \\mathrm { n n z } ( H _ { \\mathrm { H P } } ) )$   \n29 flops, while GCN layer takes $2 N F _ { l - 1 } F _ { l } + 2 F _ { l } ( \\mathrm { n n z } ( H _ { \\mathrm { L P } } ) )$ flops, where $\\mathrm { n n z } ( \\cdot )$ is the number of   \n30 non-zero elements. A detailed experiments on running time is conducted in section 6.1. ",
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+        "text": "Limitations Diversification operation does not work well in all harmful heterophily cases. For example, consider an imbalanced dataset where several small clusters with distinctive labels are densely connected to a large cluster. In this case, the surrounding differences of nodes in small clusters are similar, i.e., the neighborhood differences are mainly from their connection to the same large cluster, and this possibly makes diversification operation fail to discriminate them. See a more detailed demonstration and discussion in Appendix G. ",
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+    {
+        "type": "text",
+        "text": "5 Prior Work ",
+        "text_level": 1,
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+        "text": "GNNs on Addressing Heterophily We discuss relevant work of GNNs on addressing heterophily challenge in this part. [1] acknowledges the difficulty of learning on graphs with weak homophily and propose MixHop to extract features from multi-hop neighborhood to get more information. Geom-GCN $\\pmb { \\left. 2 8 \\right. }$ precomputes unsupervised node embeddings and uses graph structure defined by geometric relationships in the embedding space to define the bi-level aggregation process. [13] proposes measurements based on feature smoothness and label smoothness that are potentially helpful to guide GNNs on dealing with heterophilous graphs. $\\mathrm { H _ { 2 } G C N }$ [35] combines 3 key designs to address heterophily: (1) ego- and neighbor-embedding separation; (2) higher-order neighborhoods; (3) combination of intermediate representations. CPGNN $[ \\textcircled { 3 4 } ]$ models label correlations by the compatibility matrix, which is beneficial for heterophily settings, and propagates a prior belief estimation into GNNs by the compatibility matrix. GPRGNN $\\pmb { \\Vert 5 \\Vert }$ uses learnable weights that can be both positive and negative for feature propagation, it allows GRPGNN to adapt heterophily structure of graph and is able to handle both high and low frequency parts of the graph signals. ",
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+        "type": "text",
+        "text": "251 GNNs with Filterbanks Previously, there are geometric scattering networks $\\mathbb { P } , \\mathbb { Z } \\mathbb { Z }$ that apply   \n252 filterbanks to address over-smoothing $\\pmb { \\left. 2 0 \\right. }$ problem. The scattering construction captures different   \n253 channels of variation from node features or labels. In geometric learning and graph signal processing,   \n254 the band-pass filtering operations extract geometric information beyond smooth signals, thus it is   \n255 believed that filterbanks can alleviate over-smoothing in GNNs. In ACM framework, we aim to   \n56 design a framework with the help of filterbanks to adaptively utilize different channels to address the   \n7 challenge of learning on heterophilous graph. We deal with different problem as in $\\mathbb { B } \\mathbb { Z } \\mathbb { Z }$ . ",
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+        "type": "text",
+        "text": "6 Experiments on Real-World Datasets ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we evaluate ACM framework on real-world datasets. We first conduct ablation studies in subsection $6 . 1$ to validate different components. Then, we compare with the state-of-the-arts models in subsection $6 . 2 .$ ",
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+    {
+        "type": "text",
+        "text": "262 6.1 Ablation Study & Efficiency ",
+        "text_level": 1,
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+    {
+        "type": "table",
+        "img_path": "images/1067515f6fcc7273e642ac14bde608dfe8fa2abf49bcf37f4313af5c1d1a5ca1.jpg",
+        "table_caption": [
+            "Table 1: Ablation study on 9 real-world datasets $\\pmb {  \\pmb {   2 8  }  }$ . Cell with Xmeans the component is applied to the baseline model. The best test results are highlighted. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Ablation Study on Different Components in ACM-SGC and ACM-GCN(%)</td><td colspan=\"10\"></td></tr><tr><td>Baseline</td><td colspan=\"2\">Model Components</td><td></td><td>Cornell</td><td>Wisconsin</td><td>Texas</td><td>Film</td><td>Chameleon</td><td>Squirrel</td><td>Cora CiteSeer</td><td>PubMed</td></tr><tr><td>Models</td><td>|LP HP Identity Mixing|</td><td></td><td></td><td>Acc ± Std</td><td>Acc ± Std Acc ± Std</td><td>Acc ± Std</td><td>Acc ± Std</td><td>Acc ± Std</td><td>Acc ± Std</td><td>Acc ± Std</td><td>Acc ± Std</td></tr><tr><td></td><td>√</td><td></td><td></td><td></td><td>74.43 ± 6.01 69.75 ± 5.02 84.1 ± 2.32 25.34 ± 2.41 64.55 ± 1.38</td><td></td><td></td><td>42.8 ± 1.1</td><td>85.24±1.85 79.85±1.04 84.44±0.38</td><td></td><td></td></tr><tr><td rowspan=\"5\">SGC-1 w/</td><td>√ √ √</td><td></td><td>√</td><td></td><td></td><td></td><td>84.92± 4.59 91.75 ± 4.05 89.34 ± 3.67 36.94 ± 1.07 63.11 ± 1.64 44.8 ± 1.35</td><td></td><td></td><td></td><td>85.6±1.33 80.33±1.25 84.5±0.42</td></tr><tr><td></td><td>√</td><td>√</td><td></td><td></td><td></td><td>92.3± 3.893±2.11 91.64±3.65 38.25±1.657±1.93</td><td></td><td></td><td></td><td>40.2 ±2.18 85.98±0.84 80.2±2.01 84.37 ±0.44</td></tr><tr><td>√ √</td><td>√</td><td></td><td></td><td>88.2±3.88 90.75±2.3792.3±3.88 36.58±1.36 61.64±2.52 41.59±2.2984.98±1.2 79.81±1.287.13±0.58</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>√ √</td><td>√</td><td>√</td><td></td><td>92.46±2.1093.38±2.6891.97±3.2338.71±1.22 62.39±2.4545.65±1.4486.52±1.5580.79±1.6587.69±0.6</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"5\">GCN w/</td><td>√ √</td><td></td><td></td><td></td><td>81.31±3.13 70.25±4.782.13±4.05 34.45±0.83 64.86±1.56 45.11±1.3987.47±0.8281.3±0.9587.85±0.44</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>√</td><td>√</td><td>√</td><td></td><td>82.95±5.178.63±2.5188.03±2.6740.16±1.0668.12±1.7352.08±1.478.44±1.62 81.45±0.990.09 ±0.29</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>√</td><td>√</td><td>√</td><td></td><td></td><td>92.13±2.6594.37±3.2793.11±2.4840.3±1.636.67±2.1649.45±0.838.46±1.3181.42±1.1391.21±1.17</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>√</td><td>√</td><td>√</td><td></td><td></td><td></td><td>88.52±4.5195±2.2592.3±2.2140.25±1.78 65.97±2.2451.02±1.64.7±1.6880.93±1.53 90.66 ±0.32</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>92.62 ±3.04 95.37±2.1 94.75±1.7741.48±0.7867.79±1.7952.86±1.96 89.11±0.87 82.16±0.8490.72 ±0.7</td></tr><tr><td colspan=\"9\">Average Running Time Per Epoch/Average Total Running Time Comparison</td><td rowspan=\"5\"></td><td rowspan=\"10\"></td><td rowspan=\"10\"></td><td rowspan=\"10\"></td><td rowspan=\"10\"></td></tr><tr><td rowspan=\"5\">SGC-1 w/</td><td>√</td><td></td><td></td><td></td><td>2.70ms/0.59s2.53ms/0.51s2.63ms/0.55s3.62ms/1.13s4.96ms/3.99s4.09ms/0.87s5.34ms/8.22s4.79ms/4.s5.58ms/7.70s</td><td></td><td></td></tr><tr><td>√ √</td><td>√</td><td>√</td><td></td><td>4.93ms/1.04s5.03ms/1.04s.67ms/1.58s6.68ms/1.37s6.42ms/1.96s7.41ms/1.93s6.68ms/2.43s6.69ms/1.96s7.0ms/2.8s</td><td></td><td></td><td></td></tr><tr><td>√</td><td>√</td><td>√</td><td></td><td>4.73ms/0.98s4.99ms/1.09s4.79ms/102s5.53ms/1.28s5.89ms/1.50s6.48ms/1.50s6.50ms/2.09s6.23ms/1.76s6.73ms/2.4s</td><td></td><td></td><td></td></tr><tr><td>√ √</td><td>√</td><td>√</td><td></td><td>4.30ms/0.88s4.51ms/0.91s4.58ms/0.95s5.86ms/1.19s5.99ms/1.43s6.84ms/1.63s5.44ms/1.37s5.72ms/1.44s6.36ms/2.4s 5.15ms/1.08s5.82ms/1.28s.55ms/1.18s6.28ms/1.50s6.60ms/1.96s7.27ms/1.52s7.05ms/2.40s6.99ms/1.94s7.28ms/2.7s</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td colspan=\"6\"></td></tr><tr><td rowspan=\"5\">GCN w/</td><td>√ √</td><td></td><td>√</td><td></td><td rowspan=\"5\">3.78ms/0.78s3.91ms/0.79s3.80ms/0.78s4.42ms/0.89s4.44ms/0.89s6.85ms/1.48s4.19ms/0.87s5.22ms/1.13s4.81ms/0.99s 7.63ms/1.54s7.99ms/1.92s7.26ms/1.48s8.42ms/1.73s9.74ms/2.76s11.19ms/2.38s7.74ms/1.61s9.98ms/3.56s 9.10ms/1.s</td><td></td><td></td><td></td><td rowspan=\"5\"></td><td rowspan=\"5\"></td><td rowspan=\"5\"></td></tr><tr><td>√</td><td>√ √</td><td></td><td></td><td>6.75ms/1.36s6.3ms/1.41s99ms/1.46s7.62ms/1.4s7.80ms/1.67s9.76ms/2.02s7.59ms/1.54s7.43ms/1.54s.28ms/1.0s</td><td></td></tr><tr><td>√</td><td>√ √</td><td></td><td>√</td><td>7.3ms/1.49s6.80ms/1.38s6.99ms/1.41s8.76ms/2.19s7.81ms/1.59s11.26ms/2.9s7.77ms/1.59s7.66ms/1.56s8.36ms/.0s</td><td></td></tr><tr><td>√</td><td></td><td>√</td><td>√</td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>8.04ms/1.63s8.98ms/1.83s8.17ms/1.65s 9.29ms/2.00s9.33ms/1.96s12.15ms/2.53s 9.16ms/1.85s9.48ms/1.95s9.54ms/1.92s</td><td></td></tr></table>",
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+        "type": "text",
+        "text": "263 We investigate the effectiveness and efficiency of adding HP, identity channels and the adaptive mixing   \n264 mechanism in ACM framework by ablation study. Specifically, we apply the above components to   \n265 SGC-1 and GCN separately, run 10 times on each dataset used in $\\bar { \\| 2 8 \\| }$ with $6 0 \\% / 2 0 \\% / 2 0 \\%$ random   \n266 splits for train/validation/test and report the average test accuracy as well as the standard deviation.   \n267 We also record the average running time per epoch(in milliseconds)/average total running time(in   \n268 seconds) to compare the efficiency. (See Appendix A for hyperparameter searching space.)   \n269 From the results we can see that on most datasets, the additional HP and identity channels are helpful,   \n270 even on strong homophily datasets, such as Cora, CiteSeer and PubMed. The adaptive mixing   \n271 mechanism also shows its advantage over the method that directly adds the three channels together.   \n272 This illustrates the necessity of learning to customize the channel usage adaptively for different nodes.   \n273 As for efficiency, we can see that the running time is approximately doubled in ACM framework than   \n274 the original model. ",
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+    {
+        "type": "text",
+        "text": "275 6.2 Comparison with State-of-the-art Models ",
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+        "type": "text",
+        "text": "Datasets & Experimental Setup In this section, we implement SGC $\\textcircled { \\scriptsize { 1 3 1 } }$ with 1 hop and 2 hop (SGC-1, SGC-2), GCN [15] and GraphSAGE and apply them [12] in ACM framework: we use $\\hat { A } _ { \\mathrm { r w } }$ and mean aggregator as LP filter and the corresponding HP filter can be derived from $\\textcircled { 1 2 }$ We compare them with several baselines and state-of-the-art models: MLP with 2 layers (MLP-2), GAT $\\bar { \\mathbb { B } } \\bar { 0 } \\mathbb { I }$ , APPNP [16], GPRGNN [5], $\\mathrm { H _ { 2 } G C N }$ [35], MixHop [1], GCN+JK [15, 32, 21], $\\mathrm { G A T + J K }$ [30, 32, 21] and Geom-GCN $[ [ 2 8 ] ]$ . Besides the 9 benchmark datasets used in $\\left\\| \\widehat { 2 8 } \\right\\|$ , we further tests the above models on 2 new benchmark datasets, Deezer-Europe and YelpChi, that are proposed in ",
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+        "table_body": "<table><tr><td></td><td>Cornell</td><td>Wisconsin</td><td>Texas</td><td>Film</td><td>Chameleon</td><td>Squirrel</td><td>Deezer-Europe</td><td>YelpChi</td><td>Cora</td><td>CiteSeer</td><td>PubMed</td><td></td></tr><tr><td>#nodes</td><td>183</td><td>251</td><td>183</td><td>7,600</td><td>2.277</td><td>5,201</td><td>28.281</td><td>45,954</td><td>2,708</td><td>3,327</td><td>19,717</td><td></td></tr><tr><td>#edges</td><td>295</td><td>499</td><td>309</td><td>33,544</td><td>36,101</td><td>217,073</td><td>92,752</td><td>3,846.979</td><td>5,429</td><td>4,732</td><td>44,338</td><td></td></tr><tr><td>#features</td><td>1,703</td><td>1,703</td><td>1,703</td><td>931</td><td>2.325</td><td>2.089</td><td>31,241</td><td>32</td><td>1,433</td><td>3,703</td><td>500</td><td></td></tr><tr><td>#classes</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>2</td><td>2</td><td>7</td><td>6</td><td>3</td><td></td></tr><tr><td>Hedge(G)</td><td>0.5669</td><td>0.4480</td><td>0.4106</td><td>0.3750</td><td>0.2795</td><td>0.2416</td><td>0.5251</td><td>0.7730</td><td>0.8100</td><td>0.7362</td><td>0.8024</td><td></td></tr><tr><td>Hnode(9)</td><td>0.3855</td><td>0.1498</td><td>0.0968</td><td>0.2210</td><td>0.2470</td><td>0.2156</td><td>0.5299</td><td>0.7698</td><td>0.8252</td><td>0.7175</td><td>0.7924</td><td></td></tr><tr><td>Hclass(9)</td><td>0.0468</td><td>0.0941</td><td>0.0013</td><td>0.0110</td><td>0.0620</td><td>0.0254</td><td>0.0304</td><td>0.0520</td><td>0.7657</td><td>0.6270</td><td>0.6641</td><td></td></tr><tr><td>H（）</td><td>0.8032</td><td>0.7768</td><td>0.694</td><td>0.6822</td><td>0.61</td><td>0.3566</td><td>0.5790</td><td>0.7206</td><td>0.9904</td><td>0.9826</td><td>0.9432</td><td></td></tr><tr><td>Data Splits(%)</td><td>60/20/20</td><td>60/20/20</td><td>60/20/20</td><td>60/20/20</td><td>60/20/20</td><td>60/20/20</td><td>50/25/25</td><td>50/25/25</td><td>60/20/20</td><td>60/20/20</td><td>60/20/20</td><td></td></tr><tr><td></td><td colspan=\"10\">Test Accuracy(%)of State-of-the-art Models, Baseline GNN Models and ACM-GNN models</td><td colspan=\"3\">|Rank</td></tr><tr><td>MLP-2*</td><td colspan=\"10\"></td></tr><tr><td></td><td colspan=\"10\">910±0.70.908505406.0.5.4±0020</td></tr><tr><td>GAT* APPNP*</td><td colspan=\"10\">76.00±1.014660.5.980.0.46.00720.4067</td></tr><tr><td>GPRGNN*</td><td colspan=\"10\">91.80±0.639.00±3.59918±07038.602451.91±0.63.770.3467.1±0.5675.6±0.4879.0.3868.59±030850</td></tr><tr><td>H2GCN</td><td colspan=\"10\">91.36±0.7093.5±27.9±639077.0.04.9±0.56.90.5059±0.389.51±0.367.±80</td></tr><tr><td>MixHop</td><td colspan=\"10\">86.23±4.717.5±177.5170489±122207.09.76 62.52</td></tr><tr><td>GCN+JK GAT+JK</td><td colspan=\"10\">.56±13.82650±15.758066±.92.7±62646±.855.196.99±0.14650866.9±15173.77±59</td></tr><tr><td>Geom-GCNt</td><td colspan=\"10\">74.43±10.2469.50±3.1275.41±7.1835.41±0.9768.14±1.1852.28±3.6159.66±0.9290.04±0.6189.52±0.43 74.49 ± 2.76 89.15 ± 0.87</td></tr><tr><td></td><td colspan=\"10\">60.81 64.12 67.57 31.63 60.9 38.14 NA NA 85.27 77.99 90.05</td></tr><tr><td>SGC-1 SGC-2</td><td colspan=\"10\">74.43±6.016.55.24.±2.42.34±46.4±1.922.81.159.70.58.60.858560.8279.930380970</td></tr><tr><td>GCN</td><td colspan=\"10\">77.7±4.47 72.75 ±3.91 81.48 ±3.88 29.39 ±0.20 63.02±0.4337.41±1 61.56±0.51 57.18 ±0.75 86.58±0.26 76.23±0.2981.14±0.71 81.3± 0.95 87.85 ± 0.44</td></tr><tr><td>GraphSAGE</td><td colspan=\"10\">81.31±3.1370.25±4.782.13±4.0534.45±0.8364.86±1.5645.11±1.39 62.23±0.5363.62±1.0087.47±0.82 71.41±1.2464.85±5.1479.03±1.2036.37±0.21 62.15±0.42 41.26±0.26 62.55±0.48 62.57±1.12 86.58±0.26</td></tr><tr><td>ACM-SGC-1</td><td colspan=\"10\"></td></tr><tr><td>ACM-SGC-2</td><td colspan=\"10\">91.31±2.947654.665455±5</td></tr><tr><td>ACM-GCN</td><td colspan=\"10\">90.66±3.369.3±5.069066±284377±17458.51±2.4239.7±1.416.98±0.85.4±1177.44±0.80.3±1260. 82.16 ± 0.84 90.72 ± 0.7</td></tr><tr><td>ACM-SAGE</td><td colspan=\"10\">92.62±3.0495.37±2.195.08±1.841.48±0.7867.79±1.7952.86±1.96 66.85±0.9589.91±1.0289.11±0.87 80.87±1.36 88.51±0.9</td></tr><tr><td></td><td colspan=\"10\">91.31±2.9490.13±2.6791.97±3.1536.68±2.46 61.84±2.7144.63±3.02 66.21±0.898.73±1.4586.24±1.25</td></tr><tr><td></td><td colspan=\"10\"></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 2: Experimental results: average test accuracy $\\pm$ standard deviation on 11 real-world benchmark datasets. The best results are highlighted. The \"†\" results are from $\\left[ \\left[ 2 8 \\right] \\right]$ and NA means the reported results are not available. Results \"\\*\" are from [5, 21]. ",
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+    {
+        "type": "text",
+        "text": "[21] 5 . We test these models 10 times on Cornell, Wisconsin, Texas, Film, Chameleon, Squirrel, Cora, Citeseer and Pubmed following the same early stopping strategy, the same data splitting and Adam [14] optimizer used in GPRGNN [5]. For Deezer-Europe and YelpChi, we test the above models 5 times with the same early stopping strategy, the same splits and AdamW [23] used in $\\mathbb { \\left| \\left[ 2 \\right] \\right| }$ . The details of hyperparameter search are reported in appendix A. ",
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+    {
+        "type": "text",
+        "text": "The main results of this set of experiments with statistics of datasets are summarized in Table 2, where we report the mean accuracy and standard deviation. We can see that after applied in ACM framework, the performance of baseline models are boosted on almost all tasks. Especially, ACM-GCN performs the best in terms of average rank (1.73) across all datasets and achieves SOTA performance on 6 out of 11 datasets. Overall, It suggests that ACM framework can help GNNs to generalize better on node classification tasks on heterophilous graphs. ",
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+    {
+        "type": "text",
+        "text": "94 7 Future Work ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "The similarity matrix and the new metrics defined in this paper mainly capture the linear relations of the aggregated nodes. But this might be insufficient sometimes when nonlinearity information in feature vectors are important for classification. In the future, similarity matrix that is able to capture nonlinear relations between nodes can be proposed to define new homophily metrics. ",
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+    },
+    {
+        "type": "text",
+        "text": "From experimental results, the standard deviation of ACM-GNNs are relatively higher than GNNs on some tasks and this is suspiciously caused by the feature-based weight learning mechanism. In the future, a stabilizer or a more robust weight learning method can be proposed to reduce the variance. ",
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+    {
+        "type": "text",
+        "text": "8 Social Impact ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "303 We do not find any direct path of this work to any negative social impact. ",
+        "bbox": [
+            147,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "References [1] S. Abu-El-Haija, B. Perozzi, A. Kapoor, N. Alipourfard, K. Lerman, H. Harutyunyan, G. Ver Steeg, and A. Galstyan. Mixhop: Higher-order graph convolutional architectures via sparsified neighborhood mixing. In international conference on machine learning, pages 21–29. PMLR, 2019. [2] D. Bahdanau, K. Cho, and Y. Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014. [3] P. W. Battaglia, J. B. Hamrick, V. Bapst, A. Sanchez-Gonzalez, V. Zambaldi, M. Malinowski, A. Tacchetti, D. Raposo, A. Santoro, R. Faulkner, et al. Relational inductive biases, deep learning, and graph networks. arXiv preprint arXiv:1806.01261, 2018. [4] M. M. Bronstein, J. Bruna, Y. LeCun, A. Szlam, and P. Vandergheynst. Geometric deep learning: going beyond euclidean data. arXiv, abs/1611.08097, 2016. [5] E. Chien, J. Peng, P. Li, and O. Milenkovic. Adaptive universal generalized pagerank graph neural network. In International Conference on Learning Representations. https://openreview. net/forum, 2021. [6] F. R. Chung and F. C. Graham. Spectral graph theory. Number 92. American Mathematical Soc., 1997. [7] M. Defferrard, X. Bresson, and P. Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. arXiv, abs/1606.09375, 2016. [8] V. N. Ekambaram. Graph structured data viewed through a fourier lens. University of California, Berkeley, 2014. [9] F. Gao, G. Wolf, and M. Hirn. Geometric scattering for graph data analysis. In International Conference on Machine Learning, pages 2122–2131. PMLR, 2019. [10] A. Graves, A.-r. Mohamed, and G. Hinton. Speech recognition with deep recurrent neural networks. In 2013 IEEE international conference on acoustics, speech and signal processing, pages 6645–6649. Ieee, 2013. [11] W. L. Hamilton. Graph representation learning. Synthesis Lectures on Artifical Intelligence and Machine Learning, 14(3):1–159, 2020. [12] W. L. Hamilton, R. Ying, and J. Leskovec. Inductive representation learning on large graphs. arXiv, abs/1706.02216, 2017. [13] Y. Hou, J. Zhang, J. Cheng, K. Ma, R. T. Ma, H. Chen, and M.-C. Yang. Measuring and improving the use of graph information in graph neural networks. In International Conference on Learning Representations, 2019. [14] D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. [15] T. N. Kipf and M. Welling. Semi-supervised classification with graph convolutional networks. arXiv, abs/1609.02907, 2016. [16] J. Klicpera, A. Bojchevski, and S. Günnemann. Predict then propagate: Graph neural networks meet personalized pagerank. arXiv preprint arXiv:1810.05997, 2018. [17] A. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pages 1097–1105, 2012. [18] Y. LeCun, Y. Bengio, and G. Hinton. Deep learning. nature, 521(7553):436, 2015. 347 [19] Y. LeCun, L. Bottou, Y. Bengio, P. Haffner, et al. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998. ",
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+        "text": "1. For all authors... ",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] In the Appendix G, we discussion cases that high-pass filter cannot tackle.   \n(c) Did you discuss any potential negative societal impacts of your work? [No] It is in Section 8, we have not come up with significant social negative impact.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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parse/train/rJl3yM-Ab/rJl3yM-Ab.md

@@ -0,0 +1,308 @@
+# EVIDENCE AGGREGATION FOR ANSWER RE-RANKING IN OPEN-DOMAIN QUESTION ANSWERING
+
+Shuohang Wang1∗, Mo $\mathbf { Y } \mathbf { u } ^ { 2 }$ ∗, Jing Jiang1,Wei Zhang2, Xiaoxiao $\mathbf { G u o ^ { 2 } }$ , Shiyu Chang2, Zhiguo Wang2   
+Tim Klinger2, Gerald Tesauro2 and Murray Campbell2
+
+1School of Information System, Singapore Management University 2AI Foundations - Learning, IBM Research AI shwang.2014@smu.edu.sg, yum@us.ibm.com, jingjiang@smu.edu.sg
+
+# ABSTRACT
+
+A popular recent approach to answering open-domain questions is to first search for question-related passages and then apply reading comprehension models to extract answers. Existing methods usually extract answers from single passages independently. But some questions require a combination of evidence from across different sources to answer correctly. In this paper, we propose two models which make use of multiple passages to generate their answers. Both use an answerreranking approach which reorders the answer candidates generated by an existing state-of-the-art QA model. We propose two methods, namely, strengthbased re-ranking and coverage-based re-ranking, to make use of the aggregated evidence from different passages to better determine the answer. Our models have achieved state-of-the-art results on three public open-domain QA datasets: Quasar-T, SearchQA and the open-domain version of TriviaQA, with about 8 percentage points of improvement over the former two datasets.
+
+# 1 INTRODUCTION
+
+Open-domain question answering (QA) aims to answer questions from a broad range of domains by effectively marshalling evidence from large open-domain knowledge sources. Such resources can be Wikipedia (Chen et al., 2017), the whole web (Ferrucci et al., 2010), structured knowledge bases (Berant et al., 2013; Yu et al., 2017) or combinations of the above (Baudis &ˇ Sediv ˇ y, 2015). \` Recent work on open-domain QA has focused on using unstructured text retrieved from the web to build machine comprehension models (Chen et al., 2017; Dhingra et al., 2017b; Wang et al., 2017). These studies adopt a two-step process: an information retrieval (IR) model to coarsely select passages relevant to a question, followed by a reading comprehension (RC) model (Wang & Jiang, 2017; Seo et al., 2017; Chen et al., 2017) to infer an answer from the passages. These studies have made progress in bringing together evidence from large data sources, but they predict an answer to the question with only a single retrieved passage at a time. However, answer accuracy can often be improved by using multiple passages. In some cases, the answer can only be determined by combining multiple passages.
+
+In this paper, we propose a method to improve open-domain QA by explicitly aggregating evidence from across multiple passages. Our method is inspired by two notable observations from previous open-domain QA results analysis:
+
+• First, compared with incorrect answers, the correct answer is often suggested by more passages repeatedly. For example, in Figure 1(a), the correct answer “danny boy” has more passages providing evidence relevant to the question compared to the incorrect one. This observation can be seen as multiple passages collaboratively enhancing the evidence for the correct answer. • Second, sometimes the question covers multiple answer aspects, which spreads over multiple passages. In order to infer the correct answer, one has to find ways to aggregate those multiple passages in an effective yet sensible way to try to cover all aspects. In Figure 1(b), for example,
+
+![](images/03d079f47a4c2e959d64277cac4da049041cbaac593f3bf98e7d000cda2cc72d.jpg)
+
+Figure 1: Two examples of questions and candidate answers. (a) A question benefiting from the repetition of evidence. Correct answer A2 has multiple passages that could support A2 as answer. The wrong answer A1 has only a single supporting passage. (b) A question benefiting from the union of multiple pieces of evidence to support the answer. The correct answer A2 has evidence passages that can match both the first half and the second half of the question. The wrong answer A1 has evidence passages covering only the first half.
+
+the correct answer “Galileo Galilei” at the bottom has passages P1, “Galileo was a physicist ...” and P2, “Galileo discovered the first 4 moons of Jupiter”, mentioning two pieces of evidence to match the question. In this case, the aggregation of these two pieces of evidence can help entail the ground-truth answer “Galileo Galilei”. In comparison, the incorrect answer “Isaac Newton” has passages providing partial evidence on only “physicist, mathematician and astronomer”. This observation illustrates the way in which multiple passages may provide complementary evidence to better infer the correct answer to a question.
+
+To provide more accurate answers for open-domain QA, we hope to make better use of multiple passages for the same question by aggregating both the strengthened and the complementary evidence from all the passages. We formulate the above evidence aggregation as an answer re-ranking problem. Re-ranking has been commonly used in NLP problems, such as in parsing and translation, in order to make use of high-order or global features that are too expensive for decoding algorithms (Collins & Koo, 2005; Shen et al., 2004; Huang, 2008; Dyer et al., 2016). Here we apply the idea of re-ranking; for each answer candidate, we efficiently incorporate global information from multiple pieces of textual evidence without significantly increasing the complexity of the prediction of the RC model. Specifically, we first collect the top- $K$ candidate answers based on their probabilities computed by a standard RC/QA system, and then we use two proposed re-rankers to re-score the answer candidates by aggregating each candidate’s evidence in different ways. The re-rankers are:
+
+• A strength-based re-ranker, which ranks the answer candidates according to how often their evidence occurs in different passages. The re-ranker is based on the first observation if an answer candidate has multiple pieces of evidence, and each passage containing some evidence tends to predict the answer with a relatively high score (although it may not be the top score), then the candidate is more likely to be correct. The passage count of each candidate, and the aggregated probabilities for the candidate, reflect how strong its evidence is, and thus in turn suggest how likely the candidate is the corrected answer.
+
+• A coverage-based re-ranker, which aims to rank an answer candidate higher if the union of all its contexts in different passages could cover more aspects included in the question. To achieve this, for each answer we concatenate all the passages that contain the answer together. The result is a new context that aggregates all the evidence necessary to entail the answer for the question. We then treat the new context as one sequence to represent the answer, and build an attention-based match-LSTM model (Wang & Jiang, 2017) between the sequence and the question to measure how well the new aggregated context could entail the question.
+
+![](images/788a51c44091f6bbd4eaaf1f7716bb592da9c8f9716be2d27f9008e68aae5b7c.jpg)  
+Figure 2: An overview of the full re-ranker. It consists of strength-based and coverage-based reranking.
+
+Overall, our contributions are as follows: 1) We propose a re-ranking-based framework to make use of the evidence from multiple passages in open-domain QA, and two re-rankers, namely, a strengthbased re-ranker and a coverage-based re-ranker, to perform evidence aggregation in existing opendomain QA datasets. We find the second re-ranker performs better than the first one on two of the three public datasets. 2) Our proposed approach leads to the state-of-the-art results on three different datasets (Quasar-T (Dhingra et al., 2017b), SearchQA (Dunn et al., 2017) and TriviaQA (Joshi et al., 2017)) and outperforms previous state of the art by large margins. In particular, we achieved up to $8 \%$ improvement on F1 on both Quasar-T and SearchQA compared to the previous best results.
+
+# 2 METHOD
+
+Given a question q, we are trying to find the correct answer $\mathbf { a } ^ { g }$ to $\mathbf { q }$ using information retrieved from the web. Our method proceeds in two phases. First, we run an IR model (with the help of a search engine such as google or bing) to find the top- $N$ web passages $\mathbf { p } _ { 1 } , \mathbf { p } _ { 2 } , . . . , \mathbf { p } _ { N }$ most related to the question. Then a reading comprehension (RC) model is used to extract the answer from these passages. This setting is different from standard reading comprehension tasks (e.g. (Rajpurkar et al., 2016)), where a single fixed passage is given, from which the answer is to be extracted. When developing a reading comprehension system, we can use the specific positions of the answer sequence in the given passage for training. By contrast, in the open-domain setting, the RC models are usually trained under distant supervision (Chen et al., 2017; Dhingra et al., 2017b; Joshi et al., 2017). Specifically, since the training data does not have labels indicating the positions of the answer spans in the passages, during the training stage, the RC model will match all passages that contain the ground-truth answer with the question one by one. In this paper we apply an existing RC model called ${ \tt R } ^ { 3 }$ (Wang et al., 2017) to extract these candidate answers.
+
+After the candidate answers are extracted, we aggregate evidence from multiple passages by reranking the answer candidates. Given a question $\mathbf { q }$ , suppose we have a baseline open-domain QA system that can generate the top- $K$ answer candidates $\mathbf { a } _ { 1 } , \ldots , \mathbf { a } _ { K }$ , each being a text span in some passage $\mathbf { p } _ { i }$ . The goal of the re-ranker is to rank this list of candidates so that the top-ranked candidates are more likely to be the correct answer $\mathbf { a } ^ { g }$ . With access to these additional features, the re-ranking step has the potential to prioritize answers not easily discoverable by the base system alone. We investigate two re-ranking strategies based on evidence strength and evidence coverage. An overview of our method is shown in Figure 2.
+
+# 2.1 EVIDENCE AGGREGATION FOR STRENGTH-BASED RE-RANKER
+
+In open-domain QA, unlike the standard RC setting, we have more passages retrieved by the IR model and the ground-truth answer may appear in different passages, which means different answer spans may correspond to the same answer. To exploit this property, we provide two features to further re-rank the top- $K$ answers generated by the RC model.
+
+Measuring Strength by Count This method is based on the hypothesis that the more passages that entail a particular answer, the stronger the evidence for that answer and the higher it should be ranked. To implement this we count the number of occurrences of each answer in the top- $K$ answer spans generated by the baseline QA model and return the answer with the highest count.
+
+Measuring Strength by Probability Since we can get the probability of each answer span in a passages based on the RC model, we can also sum up the probabilities of the answer spans that are referring to the same answer. In this method, the answer with the highest probability is the final prediction 1. In the re-ranking scenario, it is not necessary to exhaustively consider all the probabilities of all the spans in the passages, as there may be a large number of different answer spans and most of them are irrelevant to the ground-truth answer.
+
+Remark: Note that neither of the above methods require any training. Both just take the candidate predictions from the baseline QA system and perform counting or probability calculations. At test time, the time complexity of strength-based re-ranking is negligible.
+
+# 2.2 EVIDENCE AGGREGATION FOR COVERAGE-BASED RE-RANKER
+
+Consider Figure 1 where the two answer candidates both have evidence matching the first half of the question. Note that only the correct answer has evidence that could also match the second half. In this case, the strength-based re-ranker will treat both answer candidates the same due to the equal amount of supporting evidence, while the second answer has complementary evidence satisfying all aspects of the question. To handle this case, we propose a coverage-based re-ranker that ranks the answer candidates according to how well the union of their evidence from different passages covers the question.
+
+In order to take the union of evidence into consideration, we first concatenate the passages containing the answer into a single “pseudo passage” then measure how well this passage entails the answer for the question. As in the examples shown in Figure 1(b), we hope the textual entailment model will reflect (i) how each aspect of the question is matched by the union of multiple passages; and (ii) whether all the aspects of the question can be matched by the union of multiple passages. In our implementation an “aspect” of the question is a hidden state of a bi-directional LSTM (Hochreiter & Schmidhuber, 1997). The match-LSTM (Wang & Jiang, 2016) model is one way to achieve the above effect in entailment. Therefore we build our coverage-based re-ranker on top of the concatenated pseudo passages using the match-LSTM. The detailed method is described below.
+
+Passage Aggregation We consider the top- $K$ answers, $\mathbf { a } _ { 1 } , \ldots , \mathbf { a } _ { K }$ , provided by the baseline QA system. For each answer $\mathbf { a } _ { k } , k \in [ 1 , K ]$ , we concatenate all the passages that contain ${ \bf a } _ { k }$ , $\{ \mathbf { p } _ { n } | \mathbf { a } _ { k } \in$ $\mathbf { p } _ { n } , n \in [ 1 , N ] \}$ , to form the union passage $\hat { \bf p } _ { k }$ . Our further model is to identify which union passage, e.g., $\hat { \bf p } _ { k }$ , could better entail its answer, e.g., ${ \bf a } _ { k }$ , for the question.
+
+Measuring Aspect(Word)-Level Matching As discussed earlier, the first mission of the coverage-based re-ranker is to measure how each aspect of the question is matched by the union of multiple passages. We achieve this with word-by-word attention followed by a comparison module.
+
+First, we write the answer candidate a, question q and the union passage $\hat { \bf p }$ of a as matrices $\mathbf { A } , \mathbf { Q } , { \hat { \mathbf { P } } }$ , with each column being the embedding of a word in the sequence. We then feed them to the bidirectional LSTM as follows:
+
+$$
+\mathbf { H } ^ { \mathrm { a } } = \mathrm { B i L S T M } ( \mathbf { A } ) , \quad \mathbf { H } ^ { \mathrm { q } } = \mathrm { B i L S T M } ( \mathbf { Q } ) , \quad \mathbf { H } ^ { \mathrm { p } } = \mathrm { B i L S T M } ( \hat { \mathbf { P } } ) ,
+$$
+
+where $\mathbf { H } ^ { \mathrm { a } } \in \mathbb { R } ^ { l \times A }$ , $\mathbf { H } ^ { \mathrm { q } } \in \mathbb { R } ^ { l \times Q }$ and $\mathbf { H } ^ { \mathrm { p } } \in \mathbb { R } ^ { l \times P }$ are the hidden states for the answer candidate, question and passage respectively; $l$ is the dimension of the hidden states, and $A$ , $Q$ and $P$ are the length of the three sequences, respectively.
+
+Next, we enhance the question representation ${ \bf H } ^ { \mathrm { q } }$ with ${ \bf H } ^ { \mathrm { a } }$ :
+
+$$
+\begin{array} { r l r } { { \bf H } ^ { \mathrm { a q } } } & { { } = } & { [ { \bf H } ^ { \mathrm { a } } ; { \bf H } ^ { \mathrm { q } } ] , } \end{array}
+$$
+
+where $[ \cdot ; \cdot ]$ is the concatenation of two matrices in row and $\mathbf { H } ^ { \mathrm { a q } } \in \mathbb { R } ^ { l \times ( A + Q ) }$ . As most of the answer candidates do not appear in the question, this is for better matching with the passage and finding more answer-related information from the passage.2 Now we can view each aspect of the question as a column vector (i.e. a hidden state at each word position in the answer-question concatenation) in the enhanced question representation $\mathbf { H } ^ { \mathrm { a q } }$ . Then the task becomes to measure how well each column vector can be matched by the union passage; and we achieve this by computing the attention vector Parikh et al. (2016) for each hidden state of sequences a and $\mathbf { q }$ as follows:
+
+$$
+\alpha = \mathrm { S o f t M a x } \left( ( \mathbf { H } ^ { \mathrm { p } } ) ^ { \mathrm { T } } \mathbf { H } ^ { \mathrm { a q } } \right) , \quad \overline { { \mathbf { H } } } ^ { \mathrm { a q } } = \mathbf { H } ^ { \mathrm { p } } \alpha ,
+$$
+
+where $\alpha \in \mathbb { R } ^ { P \times ( A + Q ) }$ is the attention weight matrix which is normalized in column through softmax. $\overline { { \mathbf { H } } } ^ { \mathrm { a q } } \in \mathbb { R } ^ { l \times ( A + Q ) }$ are the attention vectors for each word of the answer and the question by weighted summing all the hidden states of the passage $\hat { \bf p }$ . Now in order to see whether the aspects in the question can be matched by the union passage, we use the following matching function:
+
+$$
+\begin{array} { r l r } { \mathbf { M } } & { = } & { \mathrm { R e L U } \left( \mathbf { W } ^ { \mathrm { m } } \left[ \mathbf { \overline { { H } } } ^ { \mathrm { a q } } \mathbf { \bigodot } \mathbf { \overline { { H } } } ^ { \mathrm { a q } } \right] + \mathbf { b } ^ { \mathrm { m } } \otimes \mathbf { e } _ { ( A + Q ) } \right) , } \\ & { } & { \mathbf { M } } \end{array}
+$$
+
+where $\cdot \otimes \mathbf { e } _ { ( A + Q ) }$ is to repeat the vector (or scalar) on the left $A + Q$ times; $( \cdot \odot \cdot )$ and $( \cdot - \cdot )$ are the element-wise operations for checking whether the word in the answer and question can be matched by the evidence in the passage. We also concatenate these matching representations with the hidden state representations $\mathbf { H } ^ { \mathrm { a q } }$ and $\overline { { \mathbf { H } } } ^ { \mathrm { a q } }$ , so that the lexical matching representations are also integrated into the the final aspect-level matching representations3 $\mathbf { M } \in \hat { \mathbb { R } } ^ { 2 l \times ( A + Q ) }$ , which is computed through the non-linear transformation on four different representations with parameters $\mathbf { W ^ { m } } \in \mathbb { R } ^ { 2 l \times 4 l }$ and $b ^ { \mathrm { m } } \in \mathbb { R } ^ { l }$ .
+
+Measuring the Entire Question Matching Next, in order to measure how the entire question is matched by the union passage $\hat { \bf p }$ by taking into consideration of the matching result at each aspect, we add another bi-directional LSTM on top of it to aggregate the aspect-level matching information 4:
+
+$$
+\mathrm { \bf H ^ { m } = B i L S T M ( M ) , \quad h ^ { s } = M a x P o o l i n g ( \bf H ^ { m } ) , }
+$$
+
+where $\mathbf { H } ^ { \mathrm { m } } \in \mathbb { R } ^ { l \times ( A + Q ) }$ is to denote all the hidden states and $\mathbf { h } ^ { \mathrm { s } } \in \mathbb { R } ^ { l }$ , the max of pooling of each dimension of $\mathbf { H } ^ { \mathrm { m } }$ , is the entire matching representation which reflects how well the evidences in questions could be matched by the union passage.
+
+Re-ranking Objective Function Our re-ranking is based on the entire matching representation. For each candidate answer $\mathbf { a } _ { k } , k \in [ 1 , K ]$ , we can get a matching representation $\mathbf { h } _ { k } ^ { \mathrm { s } }$ between the answer ${ \bf a } _ { k }$ , question $\mathbf { q }$ and the union passage $\hat { { \bf p } } _ { k }$ through Eqn. (1-5). Then we transform all representations into scalar values followed by a normalization process for ranking:
+
+$$
+{ \mathbf { R } } = \operatorname { T a n h } \big ( { \mathbf { W } } ^ { \mathsf { r } } [ { \mathbf { h } } _ { 1 } ^ { \mathsf { s } } ; { \mathbf { h } } _ { 2 } ^ { \mathsf { s } } ; . . . ; { \mathbf { h } } _ { K } ^ { \mathsf { s } } ] + { \mathbf { b } } ^ { \mathsf { r } } \otimes { \mathbf { e } } _ { K } \big ) , \quad { \mathbf { o } } = \operatorname { S o f t m a x } \big ( { \mathbf { w } } ^ { \circ } { \mathbf { R } } + b ^ { \sigma } \otimes { \mathbf { e } } _ { K } \big ) ,
+$$
+
+where we concatenate the match representations for each answer in row through $[ \cdot ; \cdot ]$ , and do a non-linear transformation by parameters $\mathbf { W } ^ { \mathrm { r } } \in \mathbb { R } ^ { l \times l }$ and $\mathbf { b } ^ { \mathrm { r } } \in \mathbb { R } ^ { l }$ to get hidden representation $\mathbf { R } \in \mathbb { R } ^ { l \times K }$ . Finally, we map the transformed matching representations into scalar values through parameters $\mathbf { w } ^ { o } \in \mathbb { R } ^ { l }$ and $\mathbf { w } ^ { o } \in \mathbb { R }$ . $\mathbf { o } \in \mathbb { R } ^ { K }$ is the normalized probability for the candidate answers to be ground-truth. Due to the aliases of the ground-truth answer, there may be multiple answers in the candidates are ground-truth, we use KL distance as our objective function:
+
+$$
+\sum _ { k = 1 } ^ { K } y _ { k } \left( \log ( y _ { k } ) - \log ( o _ { k } ) \right) ,
+$$
+
+ere is t $y _ { k }$ indicates whether ranking output of o ${ \bf a } _ { k }$ the groumodel for truth answer or not and is normalized by . $\sum _ { k = 1 } ^ { K } y _ { k }$ $O _ { k }$ ${ \bf a } _ { k }$
+
+# 2.3 COMBINATION OF DIFFERENT TYPES OF AGGREGATIONS
+
+Although the coverage-based re-ranker tries to deal with more difficult cases compared to the strength-based re-ranker, the strength-based re-ranker works on more common cases according to the distributions of most open-domain QA datasets. We can try to get the best of both worlds by combining the two approaches. The full re-ranker is a weighted combination of the outputs of the above different re-rankers without further training. Specifically, we first use softmax to re-normalize the top-5 answer scores provided by the two strength-based rankers and the one coverage-based reranker; we then weighted sum up the scores for the same answer and select the answer with the largest score as the final prediction.
+
+# 3 EXPERIMENTAL SETTINGS
+
+We conduct experiments on three publicly available open-domain QA datasets, namely, QuasarT (Dhingra et al., 2017b), SearchQA (Dunn et al., 2017) and TriviaQA (Joshi et al., 2017). These datasets contain passages retrieved for all questions using a search engine such as Google or Bing. We do not retrieve more passages but use the provided passages only.
+
+# 3.1 DATASETS
+
+The statistics of the three datasets are shown in Table 1.
+
+Quasar-T 5 (Dhingra et al., 2017b) is based on a trivia question set. The data set makes use of the “Lucene index” on the ClueWeb09 corpus. For each question, 100 unique sentence-level passages were collected. The human performance is evaluated in an open-book setting, i.e., the human subjects had access to the same passages retrieved by the IR model and tried to find the answers from the passages.
+
+SearchQA 6 (Dunn et al., 2017) is based on Jeopardy! questions and uses Google to collect about 50 web page snippets as passages for each question. The human performance is evaluated in a similar way to the Quasar-T dataset.
+
+TriviaQA (Open-Domain Setting) 7 (Joshi et al., 2017) collected trivia questions coming from 14 trivia and quiz-league websites, and makes use of the Bing Web search API to collect the top 50
+
+<table><tr><td></td><td></td><td></td><td></td><td></td><td></td><td>#q(train） #q(dev) #q(test）#p #p(truth) #p(aggregated)</td></tr><tr><td>Quasar-T</td><td>28,496</td><td>3,000</td><td>3.000</td><td>100</td><td>14.8</td><td>5.2</td></tr><tr><td>SearchQA</td><td>99,811</td><td>13,893</td><td>27,247</td><td>50</td><td>16.5</td><td>5.4</td></tr><tr><td>TriviaQA</td><td>66,828</td><td>11,313</td><td>10,832</td><td>100</td><td>16.0</td><td>5.6</td></tr></table>
+
+Table 1: Statistics of the datasets. #q represents the number of questions for training (not counting the questions that don’t have ground-truth answer in the corresponding passages for training set), development, and testing datasets. #p is the number of passages for each question. For TriviaQA, we split the raw documents into sentence level passages and select the top 100 passages based on the its overlaps with the corresponding question. #p(golden) means the number of passages that contain the ground-truth answer in average. #p(aggregated) is the number of passages we aggregated in average for top 10 candidate answers provided by RC model.
+
+webpages most related to the questions. We focus on the open domain setting (the unfiltered passage set) of the dataset 8 and our model uses all the information retrieved by the IR model.
+
+# 3.2 BASELINES
+
+Our baseline models 9 include the following: GA (Dhingra et al., 2017a;b), a reading comprehension model with gated-attention; BiDAF (Seo et al., 2017), a RC model with bidirectional attention flow; AQA (Buck et al., 2017), a reinforced system learning to aggregate the answers generated by the re-written questions; $\mathbb { R } ^ { 3 }$ (Wang et al., 2017), a reinforced model making use of a ranker for selecting passages to train the RC model. As $\mathbb { R } ^ { 3 }$ is the first step of our system for generating candidate answers, the improvement of our re-ranking methods can be directly compared to this baseline.
+
+TriviaQA does not provide the leaderboard under the open-domain setting. As a result, there is no public baselines in this setting and we only compare with the R3 baseline.10
+
+# 3.3 IMPLEMENTATION DETAILS
+
+We first use a pre-trained $\mathbb { R } ^ { 3 }$ model (Wang et al., 2017), which gets the state-of-the-art performance on the three public datasets we consider, to generate the top 50 candidate spans for the training, development and test datasets, and we use them for further ranking. During training, if the groundtruth answer does not appear in the answer candidates, we will manually add it into the answer candidate list.
+
+For the coverage-based re-ranker, we use Adam (Kingma & Ba, 2015) to optimize the model. Word embeddings are initialized by GloVe (Pennington et al., 2014) and are not updated during training. We set all the words beyond Glove as zero vectors. We set $l$ to 300, batch size to 30, learning rate to 0.002. We tune the dropout probability from 0 to 0.5 and the number of candidate answers for re-ranking $( K )$ in [3, 5, 10] 11.
+
+# 4 RESULTS AND ANALYSIS
+
+In this section, we present results and analysis of our different re-ranking methods on the three different public datasets.
+
+Table 2: Experiment results on three open-domain QA test datasets: Quasar-T, SearchQA and TriviaQA (open-domain setting). EM: Exact Match. Full Re-ranker is the combination of three different re-rankers.   
+
+<table><tr><td></td><td colspan="2">Quasar-T</td><td colspan="2">SearchQA</td><td colspan="2">TriviaQA (open)</td></tr><tr><td></td><td>EM</td><td>F1</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td></tr><tr><td>GA (Dhingra et al., 2017a)</td><td>26.4</td><td>26.4</td><td>-</td><td>1</td><td>-</td><td>1</td></tr><tr><td>BiDAF (Seo et al., 2017)</td><td>25.9</td><td>28.5</td><td>28.6</td><td>34.6</td><td>1</td><td>-</td></tr><tr><td>AQA (Buck et al., 2017)</td><td>-</td><td>1</td><td>40.5</td><td>47.4</td><td>-</td><td>1</td></tr><tr><td>R (Wang et al., 2017)</td><td>35.3</td><td>41.7</td><td>49.0</td><td>55.3</td><td>47.3</td><td>53.7</td></tr><tr><td>Baseline Re-Ranker (BM25)</td><td>33.6</td><td>45.2</td><td>51.9</td><td>60.7</td><td>44.6</td><td>55.7</td></tr><tr><td>Our Full Re-Ranker</td><td>42.3</td><td>49.6</td><td>57.0</td><td>63.2</td><td>50.6</td><td>57.3</td></tr><tr><td>Strength-Based Re-Ranker (Probability)</td><td>36.1</td><td>42.4</td><td>50.4</td><td>56.5</td><td>49.2</td><td>55.1</td></tr><tr><td>Strength-Based Re-Ranker (Counting)</td><td>37.1</td><td>46.7</td><td>54.2</td><td>61.6</td><td>46.1</td><td>55.8</td></tr><tr><td>Coverage-Based Re-Ranker</td><td>40.6</td><td>49.1</td><td>54.1</td><td>61.4</td><td>50.0</td><td>57.0</td></tr><tr><td>Human Performance</td><td>51.5</td><td>60.6</td><td>43.9</td><td>1</td><td>-</td><td>1</td></tr></table>
+
+# 4.1 OVERALL RESULTS
+
+The performance of our models is shown in Table 2. We use F1 score and Exact Match (EM) as our evaluation metrics 12. From the results, we can clearly see that the full re-ranker, the combination of different re-rankers, significantly outperforms the previous best performance by a large margin, especially on Quasar-T and SearchQA. Moreover, our model is much better than the human performance on the SearchQA dataset. In addition, we see that our coverage-based re-ranker achieves consistently good performance on the three datasets, even though its performance is marginally lower than the strength-based re-ranker on the SearchQA dataset.
+
+# 4.2 ANALYSIS
+
+In this subsection, we analyze the benefits of our re-ranking models.
+
+BM25 as an alternative coverage-based re-ranker We use the classical BM25 retrieval model (Robertson et al., 2009) to re-rank the aggregated passages the same way as the coveragebased re-ranker, where the IDF values are first computed from the raw passages before aggregation. From the results in Table 2, we see that the BM25-based re-ranker improves the F1 scores compared with the $R ^ { 3 }$ model, but it is still lower than our coverage-based re-ranker with neural network models. Moreover, with respect to EM scores, the BM25-based re-ranker sometimes gives lower performance. We hypothesize that there are two reasons behind the relatively poor performance of BM25. First, because BM25 relies on a bag-of-words representation, context information is not taken into consideration and it cannot model the phrase similarities. Second, shorter answers tend to be preferred by BM25. For example, in our method of constructing pseudo-passages, when an answer sequence $A$ is a subsequence of another answer sequence $B$ , the pseudo passage of $A$ is always a superset of the pseudo passage of $B$ that could better cover the question. Therefore the F1 score could be improved but the EM score sometimes becomes worse.
+
+Re-ranking performance versus answer lengths and question types Figure 3 decomposes the performance according to the length of the ground truth answers and the types of questions on TriviaQA and Quasar-T. We do not include the analysis on SearchQA because, for the Jeopardy! style questions, it is more difficult to distinguish the questions types, and the range of answer lengths is narrower.
+
+Our results show that the coverage-based re-ranker outperforms the baseline in different lengths of answers and different types of questions. The strength-based re-ranker (counting) also gives improvement but is less stable across different datasets, while the strength-based re-ranker (probability)
+
+![](images/bde245f862d68fe219b0348f7b257fc6761ed375ac734688422ae1527c09b7b5.jpg)  
+Figure 3: Performance decomposition according to the length of answers and the question types.
+
+<table><tr><td></td><td colspan="2">Quasar-T</td><td colspan="2">SearchQA</td><td colspan="2">TriviaQA (open)</td></tr><tr><td>Top-K</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td></tr><tr><td>1</td><td>35.1</td><td>41.6</td><td>51.2</td><td>57.3</td><td>47.6</td><td>53.5</td></tr><tr><td>3</td><td>46.2</td><td>53.5</td><td>63.9</td><td>68.9</td><td>54.1</td><td>60.4</td></tr><tr><td>5</td><td>51.0</td><td>58.9</td><td>69.1</td><td>73.9</td><td>58.0</td><td>64.5</td></tr><tr><td>10</td><td>56.1</td><td>64.8</td><td>75.5</td><td>79.6</td><td>62.1</td><td>69.0</td></tr></table>
+
+Table 3: The upper bound (recall) of the Top-K answer candidates generated by the baseline ${ \tt R } ^ { 3 }$ system (on dev set), which indicates the potential of the coverage-based re-ranker.
+
+tends to have results and trends that are close to the baseline curves, which is probably because the method is dominated by the probabilities predicted by the baseline.
+
+The coverage-based re-ranker and the strength-based re-ranker (counting) have similar trends on most of the question types. The only exception is that the strength-based re-ranker performs significantly worse compared to the coverage-based re-ranker on the “why” questions. This is possibly because those questions usually have non-factoid answers, which are less likely to have exactly the same text spans predicted on different passages by the baseline.
+
+Potential improvement of re-rankers Table 3 shows the percentage of times the correct answer is included in the top- $K$ answer predictions of the baseline ${ \tt R } ^ { 3 }$ method. More concretely, the scores are computed by selecting the answer from the top- $K$ predictions with the best EM/F1 score. Therefore the final top- $K$ EM and F1 can be viewed as the recall or an upper bound of the top- $K$ predictions. From the results, we can see that although the top-1 prediction of ${ \tt R } ^ { 3 }$ is not very accurate, there is high probability that a top- $K$ list with small $K$ could cover the correct answer. This explains why our re-ranking approach achieves large improvement. Also by comparing the upper bound performance of top-5 and our re-ranking performance in Table 2, we can see there is still a clear gap of about $10 \%$ on both datasets and on both F1 and EM, showing the great potential improvement for the re-ranking model in future work.
+
+Effect of the selection of $K$ for the coverage-based re-ranker As shown in Table 3, as $K$ ranges from 1 to 10, the recall of top- $K$ predictions from the baseline ${ \tt R } ^ { 3 }$ system increases significantly.
+
+Table 4: Results of running coverage-based re-ranker on different number of the top- $K$ answer candidates on Quasar-T (dev set).   
+
+<table><tr><td>Candidate Set</td><td>Re-Ranker Results EM</td><td>F1</td><td>Upper Bound EM</td><td>F1</td></tr><tr><td>top-3</td><td>40.5</td><td>47.8</td><td>46.2</td><td>53.5</td></tr><tr><td>top-5</td><td>41.8</td><td>50.1</td><td>51.0</td><td>58.9</td></tr><tr><td>top-10</td><td>41.3</td><td>50.8</td><td>56.1</td><td>64.8</td></tr></table>
+
+Table 5: Results of running strength-based re-ranker (counting) on different number of top- $K$ answer candidates on Quasar-T (dev set).   
+
+<table><tr><td>Candidate Set</td><td>Re-Ranker Results EM</td><td>EM</td><td>Upper Bound F1</td></tr><tr><td></td><td>37.9</td><td>F1 46.1</td><td>64.8</td></tr><tr><td>top-10 top-50</td><td>37.8 47.8</td><td>56.1 64.1</td><td>74.1</td></tr><tr><td>top-100</td><td>36.4 47.3</td><td>66.5</td><td>77.1</td></tr><tr><td>top-200</td><td>33.7 45.8</td><td>68.7</td><td>79.5</td></tr></table>
+
+Ideally, if we use a larger $K$ , then the candidate lists will be more likely to contain good answers. At the same time, the lists to be ranked are longer thus the re-ranking problem is harder. Therefore, there is a trade-off between the coverage of rank lists and the difficulty of re-ranking; and selecting an appropriate $K$ becomes important. Table 4 shows the effects of $K$ on the performance of coveragebased re-ranker. We train and test the coverage-based re-ranker on the top- $K$ predictions from the baseline, where $K \in \{ 3 , 5 , 1 0 \}$ . The upper bound results are the same ones from Table 3. The results show that when $K$ is small, like $K { = } 3$ , the performance is not very good due to the low coverage (thus low upper bound) of the candidate list. With the increase of $K$ , the performance becomes better, but the top-5 and top-10 results are on par with each other. This is because the higher upper bound of top-10 results counteracts the harder problem of re-ranking longer lists. Since there is no significant advantage of the usage of $K { = } 1 0$ while the computation cost is higher, we report all testing results with $K { = } 5$ .
+
+Effect of the selection of $K$ for the strength-based re-ranker Similar to Table 4, we conduct experiments to show the effects of $K$ on the performance of the strength-based re-ranker. We run the strength-based re-ranker (counting) on the top- $K$ predictions from the baseline, where $K \in \{ 1 0 , 5 0 , 1 0 0 , 2 0 0 \}$ . We also evaluate the upper bound results for these $K \mathrm { s }$ . Note that the strength-based re-ranker is very fast and the different values of $K$ do not affect the computation speed significantly compared to the other QA components.
+
+The results are shown in Table 5, where we achieve the best results when $K { = } 5 0$ . The performance drops significantly when $K$ increases to 200. This is because the ratio of incorrect answers increases notably, making incorrect answers also likely to have high counts. When $K$ is smaller, such incorrect answers appear less because statistically they have lower prediction scores. We report all testing results with $K { = } 5 0$ .
+
+Examples Table 6 shows an example from Quasar-T where the re-ranker successfully corrected the wrong answer predicted by the baseline. This is a case where the coverage-based re-ranker helped: the correct answer “Sesame Street” has evidence from different passages that covers the aspects “Emmy Award” and “children ’s television shows”. Although it still does not fully match all the facts in the question, it still helps to rank the correct answer higher than the top-1 prediction “Great Dane” from the ${ \tt R } ^ { 3 }$ baseline, which only has evidence covering “TV” and $" l9 6 9 "$ in the question.
+
+Table 6: An example from Quasar-T dataset. The ground-truth answer is ”Sesame Street”. Q: question, A: answer, P: passages containing corresponding answer.   
+
+<table><tr><td>Q: 122 Emmy Awards in all categories ?</td><td colspan="3">Which children &#x27;s TV programme,which first appeared in November 1969,has won a record</td></tr><tr><td>A1:</td><td>GreatDane</td><td>A2:</td><td>Sesame Street In its long history，Sesame Street has re-</td></tr><tr><td>P1</td><td>The world ’s most famous Great Dane frst appeared on television screens on Sept. 13 , 1969 .</td><td>P1:</td><td>ceived more Emmy Awards than any other program，.</td></tr><tr><td>P2</td><td>premiered on broadcast television（CBS） Saturday morning,Sept.13,1969 ，. yet beloved great Dane .</td><td>P2:</td><td>Sesame Street ... is recognized as a pioneer of the contemporary standard which com- bines education and entertainment in chil- dren &#x27;s television shows.</td></tr></table>
+
+# 5 RELATED WORK
+
+Open Domain Question Answering The task of open domain question answering dates back to as early as (Green Jr et al., 1961) and was popularized by TREC-8 (Voorhees, 1999). The task is to produce the answer to a question by exploiting resources such as documents (Voorhees, 1999), webpages (Kwok et al., 2001) or structured knowledge bases (Berant et al., 2013; Bordes et al., 2015; Yu et al., 2017).
+
+Recent efforts (Chen et al., 2017; Dunn et al., 2017; Dhingra et al., 2017b; Wang et al., 2017) benefit from the advances of machine reading comprehension (RC) and follow the search-and-read QA direction. These deep learning based methods usually rely on a document retrieval module to retrieve a list of passages for RC models to extract answers. As there is no passage-level annotation about which passages entail the answer, the model has to find proper ways to handle the noise introduced in the IR step. Chen et al. (2017) uses bi-gram passage index to improve the retrieval step; Dunn et al. (2017); Dhingra et al. (2017b) propose to reduce the length of the retrieved passages. Wang et al. (2017) focus more on noise reduction in the passage ranking step, in which a ranker module is jointly trained with the RC model with reinforcement learning.
+
+To the best of our knowledge, our work is the first to improve neural open-domain QA systems by using multiple passages for evidence aggregation. Moreover, we focus on the novel problem of “text evidence aggregation”, where the problem is essentially modeling the relationship between the question and multiple passages (i.e. text evidence). In contrast, previous answer re-ranking research did not address the above problem: (1) traditional QA systems like (Ferrucci et al., 2010) have similar passage retrieval process with answer candidates added to the queries. The retrieved passages were used for extracting answer scoring features, but the features were all extracted from single-passages thus did not utilize the information of union/co-occurrence of multiple passages. (2) KB-QA systems (Bast & Haussmann, 2015; Yih et al., 2015; Xu et al., 2016) sometimes use text evidence to enhance answer re-ranking, where the features are also extracted on the pair of question and a single-passage but ignored the union information among multiple passages.
+
+Multi-Step Approaches for Reading Comprehension We are the first to introduce re-ranking methods to neural open-domain QA and multi-passage RC. Meanwhile, our two-step approach shares some similarities to the previous multi-step approaches proposed for standard single-passage RC, in terms of the purposes of either using additional information or re-fining answer predictions that are not easily handled by the standard answer extraction models for RC.
+
+On cloze-test tasks (Hermann et al., 2015), Epireader Trischler et al. (2016) relates to our work in the sense that it is a two-step extractor-reasoner model, which first extracts $K$ most probable singletoken answer candidates and then constructs a hypothesis by combining each answer candidate to the question and compares the hypothesis with all the sentences in the passage. Their model differs from ours in several aspects: (i) Epireader matches a hypothesis to every single sentence, including all the “noisy” ones that does not contain the answer, that makes the model inappropriate for open-domain QA setting; (ii) The sentence matching is based on the sentence embedding vectors computed by a convolutional neural network, which makes it hard to distinguish redundant and complementary evidence in aggregation; (iii) Epireader passes the probabilities predicted by the extractor to the reasoner directly to sustain differentiability, which cannot be easily adapted to our problem to handle phrases as answers or to use part of passages.
+
+Similarly, (Cui et al., 2017) also combined answer candidates to the question to form hypotheses, and then explicitly use language models trained on documents to re-rank the hypotheses. This method benefits from the consistency between the documents and gold hypotheses (which are titles of the documents) in cloze-test datasets, but does not handle multiple evidence aggregation like our work.
+
+S-Net (Tan et al., 2017) proposes a two-step approach for generative QA. The model first extracts an text span as the answer clue and then generates the answer according to the question, passage and the text span. Besides the different goal on answer generation instead of re-ranking like this work, their approach also differs from ours on that it extracts only one text span from a single selected passage.
+
+# 6 CONCLUSIONS
+
+We have observed that open-domain QA can be improved by explicitly combining evidence from multiple retrieved passages. We experimented with two types of re-rankers, one for the case where evidence is consistent and another when evidence is complementary. Both re-rankers helped to significantly improve our results individually, and even more together. Our results considerably advance the state-of-the-art on three open-domain QA datasets.
+
+Although our proposed methods achieved some successes in modeling the union or co-occurrence of multiple passages, there are still much harder problems in open-domain QA that require reasoning and commonsense inference abilities. In future work, we will explore the above directions, and we believe that our proposed approach could be potentially generalized to these more difficult multipassage reasoning scenarios.
+
+# 7 ACKNOWLEDGMENTS
+
+This work was partially supported by DSO grant DSOCL15223.
+
+We thank Mandar Joshi for testing our model on the unfiltered TriviaQA hidden test dataset.
+
+# REFERENCES
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+        "text": "EVIDENCE AGGREGATION FOR ANSWER RE-RANKING IN OPEN-DOMAIN QUESTION ANSWERING ",
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+        "text": "Shuohang Wang1∗, Mo $\\mathbf { Y } \\mathbf { u } ^ { 2 }$ ∗, Jing Jiang1,Wei Zhang2, Xiaoxiao $\\mathbf { G u o ^ { 2 } }$ , Shiyu Chang2, Zhiguo Wang2   \nTim Klinger2, Gerald Tesauro2 and Murray Campbell2 ",
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+        "text": "1School of Information System, Singapore Management University 2AI Foundations - Learning, IBM Research AI shwang.2014@smu.edu.sg, yum@us.ibm.com, jingjiang@smu.edu.sg ",
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+        "text": "ABSTRACT ",
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+        "text": "A popular recent approach to answering open-domain questions is to first search for question-related passages and then apply reading comprehension models to extract answers. Existing methods usually extract answers from single passages independently. But some questions require a combination of evidence from across different sources to answer correctly. In this paper, we propose two models which make use of multiple passages to generate their answers. Both use an answerreranking approach which reorders the answer candidates generated by an existing state-of-the-art QA model. We propose two methods, namely, strengthbased re-ranking and coverage-based re-ranking, to make use of the aggregated evidence from different passages to better determine the answer. Our models have achieved state-of-the-art results on three public open-domain QA datasets: Quasar-T, SearchQA and the open-domain version of TriviaQA, with about 8 percentage points of improvement over the former two datasets. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Open-domain question answering (QA) aims to answer questions from a broad range of domains by effectively marshalling evidence from large open-domain knowledge sources. Such resources can be Wikipedia (Chen et al., 2017), the whole web (Ferrucci et al., 2010), structured knowledge bases (Berant et al., 2013; Yu et al., 2017) or combinations of the above (Baudis &ˇ Sediv ˇ y, 2015). \\` Recent work on open-domain QA has focused on using unstructured text retrieved from the web to build machine comprehension models (Chen et al., 2017; Dhingra et al., 2017b; Wang et al., 2017). These studies adopt a two-step process: an information retrieval (IR) model to coarsely select passages relevant to a question, followed by a reading comprehension (RC) model (Wang & Jiang, 2017; Seo et al., 2017; Chen et al., 2017) to infer an answer from the passages. These studies have made progress in bringing together evidence from large data sources, but they predict an answer to the question with only a single retrieved passage at a time. However, answer accuracy can often be improved by using multiple passages. In some cases, the answer can only be determined by combining multiple passages. ",
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+        "text": "In this paper, we propose a method to improve open-domain QA by explicitly aggregating evidence from across multiple passages. Our method is inspired by two notable observations from previous open-domain QA results analysis: ",
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+        "text": "• First, compared with incorrect answers, the correct answer is often suggested by more passages repeatedly. For example, in Figure 1(a), the correct answer “danny boy” has more passages providing evidence relevant to the question compared to the incorrect one. This observation can be seen as multiple passages collaboratively enhancing the evidence for the correct answer. • Second, sometimes the question covers multiple answer aspects, which spreads over multiple passages. In order to infer the correct answer, one has to find ways to aggregate those multiple passages in an effective yet sensible way to try to cover all aspects. In Figure 1(b), for example, ",
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+        "text": "Figure 1: Two examples of questions and candidate answers. (a) A question benefiting from the repetition of evidence. Correct answer A2 has multiple passages that could support A2 as answer. The wrong answer A1 has only a single supporting passage. (b) A question benefiting from the union of multiple pieces of evidence to support the answer. The correct answer A2 has evidence passages that can match both the first half and the second half of the question. The wrong answer A1 has evidence passages covering only the first half. ",
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+        "text": "the correct answer “Galileo Galilei” at the bottom has passages P1, “Galileo was a physicist ...” and P2, “Galileo discovered the first 4 moons of Jupiter”, mentioning two pieces of evidence to match the question. In this case, the aggregation of these two pieces of evidence can help entail the ground-truth answer “Galileo Galilei”. In comparison, the incorrect answer “Isaac Newton” has passages providing partial evidence on only “physicist, mathematician and astronomer”. This observation illustrates the way in which multiple passages may provide complementary evidence to better infer the correct answer to a question. ",
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+        "text": "To provide more accurate answers for open-domain QA, we hope to make better use of multiple passages for the same question by aggregating both the strengthened and the complementary evidence from all the passages. We formulate the above evidence aggregation as an answer re-ranking problem. Re-ranking has been commonly used in NLP problems, such as in parsing and translation, in order to make use of high-order or global features that are too expensive for decoding algorithms (Collins & Koo, 2005; Shen et al., 2004; Huang, 2008; Dyer et al., 2016). Here we apply the idea of re-ranking; for each answer candidate, we efficiently incorporate global information from multiple pieces of textual evidence without significantly increasing the complexity of the prediction of the RC model. Specifically, we first collect the top- $K$ candidate answers based on their probabilities computed by a standard RC/QA system, and then we use two proposed re-rankers to re-score the answer candidates by aggregating each candidate’s evidence in different ways. The re-rankers are: ",
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+        "text": "• A strength-based re-ranker, which ranks the answer candidates according to how often their evidence occurs in different passages. The re-ranker is based on the first observation if an answer candidate has multiple pieces of evidence, and each passage containing some evidence tends to predict the answer with a relatively high score (although it may not be the top score), then the candidate is more likely to be correct. The passage count of each candidate, and the aggregated probabilities for the candidate, reflect how strong its evidence is, and thus in turn suggest how likely the candidate is the corrected answer. ",
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+        "text": "• A coverage-based re-ranker, which aims to rank an answer candidate higher if the union of all its contexts in different passages could cover more aspects included in the question. To achieve this, for each answer we concatenate all the passages that contain the answer together. The result is a new context that aggregates all the evidence necessary to entail the answer for the question. We then treat the new context as one sequence to represent the answer, and build an attention-based match-LSTM model (Wang & Jiang, 2017) between the sequence and the question to measure how well the new aggregated context could entail the question. ",
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+        "img_path": "images/788a51c44091f6bbd4eaaf1f7716bb592da9c8f9716be2d27f9008e68aae5b7c.jpg",
+        "image_caption": [
+            "Figure 2: An overview of the full re-ranker. It consists of strength-based and coverage-based reranking. "
+        ],
+        "image_footnote": [],
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+        "text": "Overall, our contributions are as follows: 1) We propose a re-ranking-based framework to make use of the evidence from multiple passages in open-domain QA, and two re-rankers, namely, a strengthbased re-ranker and a coverage-based re-ranker, to perform evidence aggregation in existing opendomain QA datasets. We find the second re-ranker performs better than the first one on two of the three public datasets. 2) Our proposed approach leads to the state-of-the-art results on three different datasets (Quasar-T (Dhingra et al., 2017b), SearchQA (Dunn et al., 2017) and TriviaQA (Joshi et al., 2017)) and outperforms previous state of the art by large margins. In particular, we achieved up to $8 \\%$ improvement on F1 on both Quasar-T and SearchQA compared to the previous best results. ",
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+        "type": "text",
+        "text": "2 METHOD ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Given a question q, we are trying to find the correct answer $\\mathbf { a } ^ { g }$ to $\\mathbf { q }$ using information retrieved from the web. Our method proceeds in two phases. First, we run an IR model (with the help of a search engine such as google or bing) to find the top- $N$ web passages $\\mathbf { p } _ { 1 } , \\mathbf { p } _ { 2 } , . . . , \\mathbf { p } _ { N }$ most related to the question. Then a reading comprehension (RC) model is used to extract the answer from these passages. This setting is different from standard reading comprehension tasks (e.g. (Rajpurkar et al., 2016)), where a single fixed passage is given, from which the answer is to be extracted. When developing a reading comprehension system, we can use the specific positions of the answer sequence in the given passage for training. By contrast, in the open-domain setting, the RC models are usually trained under distant supervision (Chen et al., 2017; Dhingra et al., 2017b; Joshi et al., 2017). Specifically, since the training data does not have labels indicating the positions of the answer spans in the passages, during the training stage, the RC model will match all passages that contain the ground-truth answer with the question one by one. In this paper we apply an existing RC model called ${ \\tt R } ^ { 3 }$ (Wang et al., 2017) to extract these candidate answers. ",
+        "bbox": [
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+        "type": "text",
+        "text": "After the candidate answers are extracted, we aggregate evidence from multiple passages by reranking the answer candidates. Given a question $\\mathbf { q }$ , suppose we have a baseline open-domain QA system that can generate the top- $K$ answer candidates $\\mathbf { a } _ { 1 } , \\ldots , \\mathbf { a } _ { K }$ , each being a text span in some passage $\\mathbf { p } _ { i }$ . The goal of the re-ranker is to rank this list of candidates so that the top-ranked candidates are more likely to be the correct answer $\\mathbf { a } ^ { g }$ . With access to these additional features, the re-ranking step has the potential to prioritize answers not easily discoverable by the base system alone. We investigate two re-ranking strategies based on evidence strength and evidence coverage. An overview of our method is shown in Figure 2. ",
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+    {
+        "type": "text",
+        "text": "2.1 EVIDENCE AGGREGATION FOR STRENGTH-BASED RE-RANKER ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In open-domain QA, unlike the standard RC setting, we have more passages retrieved by the IR model and the ground-truth answer may appear in different passages, which means different answer spans may correspond to the same answer. To exploit this property, we provide two features to further re-rank the top- $K$ answers generated by the RC model. ",
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+    {
+        "type": "text",
+        "text": "Measuring Strength by Count This method is based on the hypothesis that the more passages that entail a particular answer, the stronger the evidence for that answer and the higher it should be ranked. To implement this we count the number of occurrences of each answer in the top- $K$ answer spans generated by the baseline QA model and return the answer with the highest count. ",
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+    {
+        "type": "text",
+        "text": "Measuring Strength by Probability Since we can get the probability of each answer span in a passages based on the RC model, we can also sum up the probabilities of the answer spans that are referring to the same answer. In this method, the answer with the highest probability is the final prediction 1. In the re-ranking scenario, it is not necessary to exhaustively consider all the probabilities of all the spans in the passages, as there may be a large number of different answer spans and most of them are irrelevant to the ground-truth answer. ",
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+        "type": "text",
+        "text": "Remark: Note that neither of the above methods require any training. Both just take the candidate predictions from the baseline QA system and perform counting or probability calculations. At test time, the time complexity of strength-based re-ranking is negligible. ",
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+    {
+        "type": "text",
+        "text": "2.2 EVIDENCE AGGREGATION FOR COVERAGE-BASED RE-RANKER ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Consider Figure 1 where the two answer candidates both have evidence matching the first half of the question. Note that only the correct answer has evidence that could also match the second half. In this case, the strength-based re-ranker will treat both answer candidates the same due to the equal amount of supporting evidence, while the second answer has complementary evidence satisfying all aspects of the question. To handle this case, we propose a coverage-based re-ranker that ranks the answer candidates according to how well the union of their evidence from different passages covers the question. ",
+        "bbox": [
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+        "type": "text",
+        "text": "In order to take the union of evidence into consideration, we first concatenate the passages containing the answer into a single “pseudo passage” then measure how well this passage entails the answer for the question. As in the examples shown in Figure 1(b), we hope the textual entailment model will reflect (i) how each aspect of the question is matched by the union of multiple passages; and (ii) whether all the aspects of the question can be matched by the union of multiple passages. In our implementation an “aspect” of the question is a hidden state of a bi-directional LSTM (Hochreiter & Schmidhuber, 1997). The match-LSTM (Wang & Jiang, 2016) model is one way to achieve the above effect in entailment. Therefore we build our coverage-based re-ranker on top of the concatenated pseudo passages using the match-LSTM. The detailed method is described below. ",
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+    {
+        "type": "text",
+        "text": "Passage Aggregation We consider the top- $K$ answers, $\\mathbf { a } _ { 1 } , \\ldots , \\mathbf { a } _ { K }$ , provided by the baseline QA system. For each answer $\\mathbf { a } _ { k } , k \\in [ 1 , K ]$ , we concatenate all the passages that contain ${ \\bf a } _ { k }$ , $\\{ \\mathbf { p } _ { n } | \\mathbf { a } _ { k } \\in$ $\\mathbf { p } _ { n } , n \\in [ 1 , N ] \\}$ , to form the union passage $\\hat { \\bf p } _ { k }$ . Our further model is to identify which union passage, e.g., $\\hat { \\bf p } _ { k }$ , could better entail its answer, e.g., ${ \\bf a } _ { k }$ , for the question. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Measuring Aspect(Word)-Level Matching As discussed earlier, the first mission of the coverage-based re-ranker is to measure how each aspect of the question is matched by the union of multiple passages. We achieve this with word-by-word attention followed by a comparison module. ",
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+        "type": "text",
+        "text": "First, we write the answer candidate a, question q and the union passage $\\hat { \\bf p }$ of a as matrices $\\mathbf { A } , \\mathbf { Q } , { \\hat { \\mathbf { P } } }$ , with each column being the embedding of a word in the sequence. We then feed them to the bidirectional LSTM as follows: ",
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+    {
+        "type": "equation",
+        "img_path": "images/e9f211d4e38020175f89a73edd3023d99d7f6d3f9a6b4482020aea3914f757d6.jpg",
+        "text": "$$\n\\mathbf { H } ^ { \\mathrm { a } } = \\mathrm { B i L S T M } ( \\mathbf { A } ) , \\quad \\mathbf { H } ^ { \\mathrm { q } } = \\mathrm { B i L S T M } ( \\mathbf { Q } ) , \\quad \\mathbf { H } ^ { \\mathrm { p } } = \\mathrm { B i L S T M } ( \\hat { \\mathbf { P } } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\mathbf { H } ^ { \\mathrm { a } } \\in \\mathbb { R } ^ { l \\times A }$ , $\\mathbf { H } ^ { \\mathrm { q } } \\in \\mathbb { R } ^ { l \\times Q }$ and $\\mathbf { H } ^ { \\mathrm { p } } \\in \\mathbb { R } ^ { l \\times P }$ are the hidden states for the answer candidate, question and passage respectively; $l$ is the dimension of the hidden states, and $A$ , $Q$ and $P$ are the length of the three sequences, respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Next, we enhance the question representation ${ \\bf H } ^ { \\mathrm { q } }$ with ${ \\bf H } ^ { \\mathrm { a } }$ : ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/85570e5e4b4e87f612fb91d8d8ec0f643124d4193a18958fee7d4db0958e4e74.jpg",
+        "text": "$$\n\\begin{array} { r l r } { { \\bf H } ^ { \\mathrm { a q } } } & { { } = } & { [ { \\bf H } ^ { \\mathrm { a } } ; { \\bf H } ^ { \\mathrm { q } } ] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $[ \\cdot ; \\cdot ]$ is the concatenation of two matrices in row and $\\mathbf { H } ^ { \\mathrm { a q } } \\in \\mathbb { R } ^ { l \\times ( A + Q ) }$ . As most of the answer candidates do not appear in the question, this is for better matching with the passage and finding more answer-related information from the passage.2 Now we can view each aspect of the question as a column vector (i.e. a hidden state at each word position in the answer-question concatenation) in the enhanced question representation $\\mathbf { H } ^ { \\mathrm { a q } }$ . Then the task becomes to measure how well each column vector can be matched by the union passage; and we achieve this by computing the attention vector Parikh et al. (2016) for each hidden state of sequences a and $\\mathbf { q }$ as follows: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/3ce8d3fd5e45b0b97c0a74205d3588a7900dbb83ca67698ad0cbae5befbcdf3e.jpg",
+        "text": "$$\n\\alpha = \\mathrm { S o f t M a x } \\left( ( \\mathbf { H } ^ { \\mathrm { p } } ) ^ { \\mathrm { T } } \\mathbf { H } ^ { \\mathrm { a q } } \\right) , \\quad \\overline { { \\mathbf { H } } } ^ { \\mathrm { a q } } = \\mathbf { H } ^ { \\mathrm { p } } \\alpha ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            309,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha \\in \\mathbb { R } ^ { P \\times ( A + Q ) }$ is the attention weight matrix which is normalized in column through softmax. $\\overline { { \\mathbf { H } } } ^ { \\mathrm { a q } } \\in \\mathbb { R } ^ { l \\times ( A + Q ) }$ are the attention vectors for each word of the answer and the question by weighted summing all the hidden states of the passage $\\hat { \\bf p }$ . Now in order to see whether the aspects in the question can be matched by the union passage, we use the following matching function: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/9a6895a3c97816d6f718a3d3df29abd37417fcb4a74ae24bdda5e8eeb199cfba.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\mathbf { M } } & { = } & { \\mathrm { R e L U } \\left( \\mathbf { W } ^ { \\mathrm { m } } \\left[ \\mathbf { \\overline { { H } } } ^ { \\mathrm { a q } } \\mathbf { \\bigodot } \\mathbf { \\overline { { H } } } ^ { \\mathrm { a q } } \\right] + \\mathbf { b } ^ { \\mathrm { m } } \\otimes \\mathbf { e } _ { ( A + Q ) } \\right) , } \\\\ & { } & { \\mathbf { M } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\cdot \\otimes \\mathbf { e } _ { ( A + Q ) }$ is to repeat the vector (or scalar) on the left $A + Q$ times; $( \\cdot \\odot \\cdot )$ and $( \\cdot - \\cdot )$ are the element-wise operations for checking whether the word in the answer and question can be matched by the evidence in the passage. We also concatenate these matching representations with the hidden state representations $\\mathbf { H } ^ { \\mathrm { a q } }$ and $\\overline { { \\mathbf { H } } } ^ { \\mathrm { a q } }$ , so that the lexical matching representations are also integrated into the the final aspect-level matching representations3 $\\mathbf { M } \\in \\hat { \\mathbb { R } } ^ { 2 l \\times ( A + Q ) }$ , which is computed through the non-linear transformation on four different representations with parameters $\\mathbf { W ^ { m } } \\in \\mathbb { R } ^ { 2 l \\times 4 l }$ and $b ^ { \\mathrm { m } } \\in \\mathbb { R } ^ { l }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Measuring the Entire Question Matching Next, in order to measure how the entire question is matched by the union passage $\\hat { \\bf p }$ by taking into consideration of the matching result at each aspect, we add another bi-directional LSTM on top of it to aggregate the aspect-level matching information 4: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/190b4566e9fa6d80787c88bb40e827766aaae2e3f1afb0fcaa7148b4317a4495.jpg",
+        "text": "$$\n\\mathrm { \\bf H ^ { m } = B i L S T M ( M ) , \\quad h ^ { s } = M a x P o o l i n g ( \\bf H ^ { m } ) , }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { H } ^ { \\mathrm { m } } \\in \\mathbb { R } ^ { l \\times ( A + Q ) }$ is to denote all the hidden states and $\\mathbf { h } ^ { \\mathrm { s } } \\in \\mathbb { R } ^ { l }$ , the max of pooling of each dimension of $\\mathbf { H } ^ { \\mathrm { m } }$ , is the entire matching representation which reflects how well the evidences in questions could be matched by the union passage. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Re-ranking Objective Function Our re-ranking is based on the entire matching representation. For each candidate answer $\\mathbf { a } _ { k } , k \\in [ 1 , K ]$ , we can get a matching representation $\\mathbf { h } _ { k } ^ { \\mathrm { s } }$ between the answer ${ \\bf a } _ { k }$ , question $\\mathbf { q }$ and the union passage $\\hat { { \\bf p } } _ { k }$ through Eqn. (1-5). Then we transform all representations into scalar values followed by a normalization process for ranking: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f28f42058676da878d455106cd3f0ec521313a7d7097dd4ee7b91317865af705.jpg",
+        "text": "$$\n{ \\mathbf { R } } = \\operatorname { T a n h } \\big ( { \\mathbf { W } } ^ { \\mathsf { r } } [ { \\mathbf { h } } _ { 1 } ^ { \\mathsf { s } } ; { \\mathbf { h } } _ { 2 } ^ { \\mathsf { s } } ; . . . ; { \\mathbf { h } } _ { K } ^ { \\mathsf { s } } ] + { \\mathbf { b } } ^ { \\mathsf { r } } \\otimes { \\mathbf { e } } _ { K } \\big ) , \\quad { \\mathbf { o } } = \\operatorname { S o f t m a x } \\big ( { \\mathbf { w } } ^ { \\circ } { \\mathbf { R } } + b ^ { \\sigma } \\otimes { \\mathbf { e } } _ { K } \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where we concatenate the match representations for each answer in row through $[ \\cdot ; \\cdot ]$ , and do a non-linear transformation by parameters $\\mathbf { W } ^ { \\mathrm { r } } \\in \\mathbb { R } ^ { l \\times l }$ and $\\mathbf { b } ^ { \\mathrm { r } } \\in \\mathbb { R } ^ { l }$ to get hidden representation $\\mathbf { R } \\in \\mathbb { R } ^ { l \\times K }$ . Finally, we map the transformed matching representations into scalar values through parameters $\\mathbf { w } ^ { o } \\in \\mathbb { R } ^ { l }$ and $\\mathbf { w } ^ { o } \\in \\mathbb { R }$ . $\\mathbf { o } \\in \\mathbb { R } ^ { K }$ is the normalized probability for the candidate answers to be ground-truth. Due to the aliases of the ground-truth answer, there may be multiple answers in the candidates are ground-truth, we use KL distance as our objective function: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3d1c01ddf49b20b5444fd21dc214217776d42be1de4ecaa11c3cbb373b0a0b36.jpg",
+        "text": "$$\n\\sum _ { k = 1 } ^ { K } y _ { k } \\left( \\log ( y _ { k } ) - \\log ( o _ { k } ) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ere is t $y _ { k }$ indicates whether ranking output of o ${ \\bf a } _ { k }$ the groumodel for truth answer or not and is normalized by . $\\sum _ { k = 1 } ^ { K } y _ { k }$ $O _ { k }$ ${ \\bf a } _ { k }$ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "2.3 COMBINATION OF DIFFERENT TYPES OF AGGREGATIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Although the coverage-based re-ranker tries to deal with more difficult cases compared to the strength-based re-ranker, the strength-based re-ranker works on more common cases according to the distributions of most open-domain QA datasets. We can try to get the best of both worlds by combining the two approaches. The full re-ranker is a weighted combination of the outputs of the above different re-rankers without further training. Specifically, we first use softmax to re-normalize the top-5 answer scores provided by the two strength-based rankers and the one coverage-based reranker; we then weighted sum up the scores for the same answer and select the answer with the largest score as the final prediction. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3 EXPERIMENTAL SETTINGS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We conduct experiments on three publicly available open-domain QA datasets, namely, QuasarT (Dhingra et al., 2017b), SearchQA (Dunn et al., 2017) and TriviaQA (Joshi et al., 2017). These datasets contain passages retrieved for all questions using a search engine such as Google or Bing. We do not retrieve more passages but use the provided passages only. ",
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+    {
+        "type": "text",
+        "text": "3.1 DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The statistics of the three datasets are shown in Table 1. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Quasar-T 5 (Dhingra et al., 2017b) is based on a trivia question set. The data set makes use of the “Lucene index” on the ClueWeb09 corpus. For each question, 100 unique sentence-level passages were collected. The human performance is evaluated in an open-book setting, i.e., the human subjects had access to the same passages retrieved by the IR model and tried to find the answers from the passages. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "SearchQA 6 (Dunn et al., 2017) is based on Jeopardy! questions and uses Google to collect about 50 web page snippets as passages for each question. The human performance is evaluated in a similar way to the Quasar-T dataset. ",
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+    {
+        "type": "text",
+        "text": "TriviaQA (Open-Domain Setting) 7 (Joshi et al., 2017) collected trivia questions coming from 14 trivia and quiz-league websites, and makes use of the Bing Web search API to collect the top 50 ",
+        "bbox": [
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td></td><td></td><td></td><td></td><td>#q(train） #q(dev) #q(test）#p #p(truth) #p(aggregated)</td></tr><tr><td>Quasar-T</td><td>28,496</td><td>3,000</td><td>3.000</td><td>100</td><td>14.8</td><td>5.2</td></tr><tr><td>SearchQA</td><td>99,811</td><td>13,893</td><td>27,247</td><td>50</td><td>16.5</td><td>5.4</td></tr><tr><td>TriviaQA</td><td>66,828</td><td>11,313</td><td>10,832</td><td>100</td><td>16.0</td><td>5.6</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 1: Statistics of the datasets. #q represents the number of questions for training (not counting the questions that don’t have ground-truth answer in the corresponding passages for training set), development, and testing datasets. #p is the number of passages for each question. For TriviaQA, we split the raw documents into sentence level passages and select the top 100 passages based on the its overlaps with the corresponding question. #p(golden) means the number of passages that contain the ground-truth answer in average. #p(aggregated) is the number of passages we aggregated in average for top 10 candidate answers provided by RC model. ",
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+    },
+    {
+        "type": "text",
+        "text": "webpages most related to the questions. We focus on the open domain setting (the unfiltered passage set) of the dataset 8 and our model uses all the information retrieved by the IR model. ",
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+    {
+        "type": "text",
+        "text": "3.2 BASELINES",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our baseline models 9 include the following: GA (Dhingra et al., 2017a;b), a reading comprehension model with gated-attention; BiDAF (Seo et al., 2017), a RC model with bidirectional attention flow; AQA (Buck et al., 2017), a reinforced system learning to aggregate the answers generated by the re-written questions; $\\mathbb { R } ^ { 3 }$ (Wang et al., 2017), a reinforced model making use of a ranker for selecting passages to train the RC model. As $\\mathbb { R } ^ { 3 }$ is the first step of our system for generating candidate answers, the improvement of our re-ranking methods can be directly compared to this baseline. ",
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+    {
+        "type": "text",
+        "text": "TriviaQA does not provide the leaderboard under the open-domain setting. As a result, there is no public baselines in this setting and we only compare with the R3 baseline.10 ",
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+    {
+        "type": "text",
+        "text": "3.3 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We first use a pre-trained $\\mathbb { R } ^ { 3 }$ model (Wang et al., 2017), which gets the state-of-the-art performance on the three public datasets we consider, to generate the top 50 candidate spans for the training, development and test datasets, and we use them for further ranking. During training, if the groundtruth answer does not appear in the answer candidates, we will manually add it into the answer candidate list. ",
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+    {
+        "type": "text",
+        "text": "For the coverage-based re-ranker, we use Adam (Kingma & Ba, 2015) to optimize the model. Word embeddings are initialized by GloVe (Pennington et al., 2014) and are not updated during training. We set all the words beyond Glove as zero vectors. We set $l$ to 300, batch size to 30, learning rate to 0.002. We tune the dropout probability from 0 to 0.5 and the number of candidate answers for re-ranking $( K )$ in [3, 5, 10] 11. ",
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+    {
+        "type": "text",
+        "text": "4 RESULTS AND ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we present results and analysis of our different re-ranking methods on the three different public datasets. ",
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+        "type": "table",
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+        "table_caption": [
+            "Table 2: Experiment results on three open-domain QA test datasets: Quasar-T, SearchQA and TriviaQA (open-domain setting). EM: Exact Match. Full Re-ranker is the combination of three different re-rankers. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">Quasar-T</td><td colspan=\"2\">SearchQA</td><td colspan=\"2\">TriviaQA (open)</td></tr><tr><td></td><td>EM</td><td>F1</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td></tr><tr><td>GA (Dhingra et al., 2017a)</td><td>26.4</td><td>26.4</td><td>-</td><td>1</td><td>-</td><td>1</td></tr><tr><td>BiDAF (Seo et al., 2017)</td><td>25.9</td><td>28.5</td><td>28.6</td><td>34.6</td><td>1</td><td>-</td></tr><tr><td>AQA (Buck et al., 2017)</td><td>-</td><td>1</td><td>40.5</td><td>47.4</td><td>-</td><td>1</td></tr><tr><td>R (Wang et al., 2017)</td><td>35.3</td><td>41.7</td><td>49.0</td><td>55.3</td><td>47.3</td><td>53.7</td></tr><tr><td>Baseline Re-Ranker (BM25)</td><td>33.6</td><td>45.2</td><td>51.9</td><td>60.7</td><td>44.6</td><td>55.7</td></tr><tr><td>Our Full Re-Ranker</td><td>42.3</td><td>49.6</td><td>57.0</td><td>63.2</td><td>50.6</td><td>57.3</td></tr><tr><td>Strength-Based Re-Ranker (Probability)</td><td>36.1</td><td>42.4</td><td>50.4</td><td>56.5</td><td>49.2</td><td>55.1</td></tr><tr><td>Strength-Based Re-Ranker (Counting)</td><td>37.1</td><td>46.7</td><td>54.2</td><td>61.6</td><td>46.1</td><td>55.8</td></tr><tr><td>Coverage-Based Re-Ranker</td><td>40.6</td><td>49.1</td><td>54.1</td><td>61.4</td><td>50.0</td><td>57.0</td></tr><tr><td>Human Performance</td><td>51.5</td><td>60.6</td><td>43.9</td><td>1</td><td>-</td><td>1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.1 OVERALL RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The performance of our models is shown in Table 2. We use F1 score and Exact Match (EM) as our evaluation metrics 12. From the results, we can clearly see that the full re-ranker, the combination of different re-rankers, significantly outperforms the previous best performance by a large margin, especially on Quasar-T and SearchQA. Moreover, our model is much better than the human performance on the SearchQA dataset. In addition, we see that our coverage-based re-ranker achieves consistently good performance on the three datasets, even though its performance is marginally lower than the strength-based re-ranker on the SearchQA dataset. ",
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+    {
+        "type": "text",
+        "text": "4.2 ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this subsection, we analyze the benefits of our re-ranking models. ",
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+        "type": "text",
+        "text": "BM25 as an alternative coverage-based re-ranker We use the classical BM25 retrieval model (Robertson et al., 2009) to re-rank the aggregated passages the same way as the coveragebased re-ranker, where the IDF values are first computed from the raw passages before aggregation. From the results in Table 2, we see that the BM25-based re-ranker improves the F1 scores compared with the $R ^ { 3 }$ model, but it is still lower than our coverage-based re-ranker with neural network models. Moreover, with respect to EM scores, the BM25-based re-ranker sometimes gives lower performance. We hypothesize that there are two reasons behind the relatively poor performance of BM25. First, because BM25 relies on a bag-of-words representation, context information is not taken into consideration and it cannot model the phrase similarities. Second, shorter answers tend to be preferred by BM25. For example, in our method of constructing pseudo-passages, when an answer sequence $A$ is a subsequence of another answer sequence $B$ , the pseudo passage of $A$ is always a superset of the pseudo passage of $B$ that could better cover the question. Therefore the F1 score could be improved but the EM score sometimes becomes worse. ",
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+        "type": "text",
+        "text": "Re-ranking performance versus answer lengths and question types Figure 3 decomposes the performance according to the length of the ground truth answers and the types of questions on TriviaQA and Quasar-T. We do not include the analysis on SearchQA because, for the Jeopardy! style questions, it is more difficult to distinguish the questions types, and the range of answer lengths is narrower. ",
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+        "type": "text",
+        "text": "Our results show that the coverage-based re-ranker outperforms the baseline in different lengths of answers and different types of questions. The strength-based re-ranker (counting) also gives improvement but is less stable across different datasets, while the strength-based re-ranker (probability) ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/bde245f862d68fe219b0348f7b257fc6761ed375ac734688422ae1527c09b7b5.jpg",
+        "image_caption": [
+            "Figure 3: Performance decomposition according to the length of answers and the question types. "
+        ],
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+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/0774b0cb0fcb9bec35e5f3309707e45b035e8c686a5fb1a7ad4af9ea6d3774b7.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">Quasar-T</td><td colspan=\"2\">SearchQA</td><td colspan=\"2\">TriviaQA (open)</td></tr><tr><td>Top-K</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td></tr><tr><td>1</td><td>35.1</td><td>41.6</td><td>51.2</td><td>57.3</td><td>47.6</td><td>53.5</td></tr><tr><td>3</td><td>46.2</td><td>53.5</td><td>63.9</td><td>68.9</td><td>54.1</td><td>60.4</td></tr><tr><td>5</td><td>51.0</td><td>58.9</td><td>69.1</td><td>73.9</td><td>58.0</td><td>64.5</td></tr><tr><td>10</td><td>56.1</td><td>64.8</td><td>75.5</td><td>79.6</td><td>62.1</td><td>69.0</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 3: The upper bound (recall) of the Top-K answer candidates generated by the baseline ${ \\tt R } ^ { 3 }$ system (on dev set), which indicates the potential of the coverage-based re-ranker. ",
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+    {
+        "type": "text",
+        "text": "tends to have results and trends that are close to the baseline curves, which is probably because the method is dominated by the probabilities predicted by the baseline. ",
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+    {
+        "type": "text",
+        "text": "The coverage-based re-ranker and the strength-based re-ranker (counting) have similar trends on most of the question types. The only exception is that the strength-based re-ranker performs significantly worse compared to the coverage-based re-ranker on the “why” questions. This is possibly because those questions usually have non-factoid answers, which are less likely to have exactly the same text spans predicted on different passages by the baseline. ",
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+    {
+        "type": "text",
+        "text": "Potential improvement of re-rankers Table 3 shows the percentage of times the correct answer is included in the top- $K$ answer predictions of the baseline ${ \\tt R } ^ { 3 }$ method. More concretely, the scores are computed by selecting the answer from the top- $K$ predictions with the best EM/F1 score. Therefore the final top- $K$ EM and F1 can be viewed as the recall or an upper bound of the top- $K$ predictions. From the results, we can see that although the top-1 prediction of ${ \\tt R } ^ { 3 }$ is not very accurate, there is high probability that a top- $K$ list with small $K$ could cover the correct answer. This explains why our re-ranking approach achieves large improvement. Also by comparing the upper bound performance of top-5 and our re-ranking performance in Table 2, we can see there is still a clear gap of about $10 \\%$ on both datasets and on both F1 and EM, showing the great potential improvement for the re-ranking model in future work. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Effect of the selection of $K$ for the coverage-based re-ranker As shown in Table 3, as $K$ ranges from 1 to 10, the recall of top- $K$ predictions from the baseline ${ \\tt R } ^ { 3 }$ system increases significantly. ",
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+    {
+        "type": "table",
+        "img_path": "images/7663713b9e571d3618f881d4276a629c28208e5f656cea6abf9ccce02611dad0.jpg",
+        "table_caption": [
+            "Table 4: Results of running coverage-based re-ranker on different number of the top- $K$ answer candidates on Quasar-T (dev set). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Candidate Set</td><td>Re-Ranker Results EM</td><td>F1</td><td>Upper Bound EM</td><td>F1</td></tr><tr><td>top-3</td><td>40.5</td><td>47.8</td><td>46.2</td><td>53.5</td></tr><tr><td>top-5</td><td>41.8</td><td>50.1</td><td>51.0</td><td>58.9</td></tr><tr><td>top-10</td><td>41.3</td><td>50.8</td><td>56.1</td><td>64.8</td></tr></table>",
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+    },
+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 5: Results of running strength-based re-ranker (counting) on different number of top- $K$ answer candidates on Quasar-T (dev set). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Candidate Set</td><td>Re-Ranker Results EM</td><td>EM</td><td>Upper Bound F1</td></tr><tr><td></td><td>37.9</td><td>F1 46.1</td><td>64.8</td></tr><tr><td>top-10 top-50</td><td>37.8 47.8</td><td>56.1 64.1</td><td>74.1</td></tr><tr><td>top-100</td><td>36.4 47.3</td><td>66.5</td><td>77.1</td></tr><tr><td>top-200</td><td>33.7 45.8</td><td>68.7</td><td>79.5</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Ideally, if we use a larger $K$ , then the candidate lists will be more likely to contain good answers. At the same time, the lists to be ranked are longer thus the re-ranking problem is harder. Therefore, there is a trade-off between the coverage of rank lists and the difficulty of re-ranking; and selecting an appropriate $K$ becomes important. Table 4 shows the effects of $K$ on the performance of coveragebased re-ranker. We train and test the coverage-based re-ranker on the top- $K$ predictions from the baseline, where $K \\in \\{ 3 , 5 , 1 0 \\}$ . The upper bound results are the same ones from Table 3. The results show that when $K$ is small, like $K { = } 3$ , the performance is not very good due to the low coverage (thus low upper bound) of the candidate list. With the increase of $K$ , the performance becomes better, but the top-5 and top-10 results are on par with each other. This is because the higher upper bound of top-10 results counteracts the harder problem of re-ranking longer lists. Since there is no significant advantage of the usage of $K { = } 1 0$ while the computation cost is higher, we report all testing results with $K { = } 5$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Effect of the selection of $K$ for the strength-based re-ranker Similar to Table 4, we conduct experiments to show the effects of $K$ on the performance of the strength-based re-ranker. We run the strength-based re-ranker (counting) on the top- $K$ predictions from the baseline, where $K \\in \\{ 1 0 , 5 0 , 1 0 0 , 2 0 0 \\}$ . We also evaluate the upper bound results for these $K \\mathrm { s }$ . Note that the strength-based re-ranker is very fast and the different values of $K$ do not affect the computation speed significantly compared to the other QA components. ",
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+    {
+        "type": "text",
+        "text": "The results are shown in Table 5, where we achieve the best results when $K { = } 5 0$ . The performance drops significantly when $K$ increases to 200. This is because the ratio of incorrect answers increases notably, making incorrect answers also likely to have high counts. When $K$ is smaller, such incorrect answers appear less because statistically they have lower prediction scores. We report all testing results with $K { = } 5 0$ . ",
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+    {
+        "type": "text",
+        "text": "Examples Table 6 shows an example from Quasar-T where the re-ranker successfully corrected the wrong answer predicted by the baseline. This is a case where the coverage-based re-ranker helped: the correct answer “Sesame Street” has evidence from different passages that covers the aspects “Emmy Award” and “children ’s television shows”. Although it still does not fully match all the facts in the question, it still helps to rank the correct answer higher than the top-1 prediction “Great Dane” from the ${ \\tt R } ^ { 3 }$ baseline, which only has evidence covering “TV” and $\" l9 6 9 \"$ in the question. ",
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+    {
+        "type": "table",
+        "img_path": "images/1f922e9eca1fa21c5f35d5d0fc1b1810cdf60521e8c582e49c833ee657d7a781.jpg",
+        "table_caption": [
+            "Table 6: An example from Quasar-T dataset. The ground-truth answer is ”Sesame Street”. Q: question, A: answer, P: passages containing corresponding answer. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Q: 122 Emmy Awards in all categories ?</td><td colspan=\"3\">Which children &#x27;s TV programme,which first appeared in November 1969,has won a record</td></tr><tr><td>A1:</td><td>GreatDane</td><td>A2:</td><td>Sesame Street In its long history，Sesame Street has re-</td></tr><tr><td>P1</td><td>The world ’s most famous Great Dane frst appeared on television screens on Sept. 13 , 1969 .</td><td>P1:</td><td>ceived more Emmy Awards than any other program，.</td></tr><tr><td>P2</td><td>premiered on broadcast television（CBS） Saturday morning,Sept.13,1969 ，. yet beloved great Dane .</td><td>P2:</td><td>Sesame Street ... is recognized as a pioneer of the contemporary standard which com- bines education and entertainment in chil- dren &#x27;s television shows.</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Open Domain Question Answering The task of open domain question answering dates back to as early as (Green Jr et al., 1961) and was popularized by TREC-8 (Voorhees, 1999). The task is to produce the answer to a question by exploiting resources such as documents (Voorhees, 1999), webpages (Kwok et al., 2001) or structured knowledge bases (Berant et al., 2013; Bordes et al., 2015; Yu et al., 2017). ",
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+    {
+        "type": "text",
+        "text": "Recent efforts (Chen et al., 2017; Dunn et al., 2017; Dhingra et al., 2017b; Wang et al., 2017) benefit from the advances of machine reading comprehension (RC) and follow the search-and-read QA direction. These deep learning based methods usually rely on a document retrieval module to retrieve a list of passages for RC models to extract answers. As there is no passage-level annotation about which passages entail the answer, the model has to find proper ways to handle the noise introduced in the IR step. Chen et al. (2017) uses bi-gram passage index to improve the retrieval step; Dunn et al. (2017); Dhingra et al. (2017b) propose to reduce the length of the retrieved passages. Wang et al. (2017) focus more on noise reduction in the passage ranking step, in which a ranker module is jointly trained with the RC model with reinforcement learning. ",
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+    {
+        "type": "text",
+        "text": "To the best of our knowledge, our work is the first to improve neural open-domain QA systems by using multiple passages for evidence aggregation. Moreover, we focus on the novel problem of “text evidence aggregation”, where the problem is essentially modeling the relationship between the question and multiple passages (i.e. text evidence). In contrast, previous answer re-ranking research did not address the above problem: (1) traditional QA systems like (Ferrucci et al., 2010) have similar passage retrieval process with answer candidates added to the queries. The retrieved passages were used for extracting answer scoring features, but the features were all extracted from single-passages thus did not utilize the information of union/co-occurrence of multiple passages. (2) KB-QA systems (Bast & Haussmann, 2015; Yih et al., 2015; Xu et al., 2016) sometimes use text evidence to enhance answer re-ranking, where the features are also extracted on the pair of question and a single-passage but ignored the union information among multiple passages. ",
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+    {
+        "type": "text",
+        "text": "Multi-Step Approaches for Reading Comprehension We are the first to introduce re-ranking methods to neural open-domain QA and multi-passage RC. Meanwhile, our two-step approach shares some similarities to the previous multi-step approaches proposed for standard single-passage RC, in terms of the purposes of either using additional information or re-fining answer predictions that are not easily handled by the standard answer extraction models for RC. ",
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+    {
+        "type": "text",
+        "text": "On cloze-test tasks (Hermann et al., 2015), Epireader Trischler et al. (2016) relates to our work in the sense that it is a two-step extractor-reasoner model, which first extracts $K$ most probable singletoken answer candidates and then constructs a hypothesis by combining each answer candidate to the question and compares the hypothesis with all the sentences in the passage. Their model differs from ours in several aspects: (i) Epireader matches a hypothesis to every single sentence, including all the “noisy” ones that does not contain the answer, that makes the model inappropriate for open-domain QA setting; (ii) The sentence matching is based on the sentence embedding vectors computed by a convolutional neural network, which makes it hard to distinguish redundant and complementary evidence in aggregation; (iii) Epireader passes the probabilities predicted by the extractor to the reasoner directly to sustain differentiability, which cannot be easily adapted to our problem to handle phrases as answers or to use part of passages. ",
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "Similarly, (Cui et al., 2017) also combined answer candidates to the question to form hypotheses, and then explicitly use language models trained on documents to re-rank the hypotheses. This method benefits from the consistency between the documents and gold hypotheses (which are titles of the documents) in cloze-test datasets, but does not handle multiple evidence aggregation like our work. ",
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+    },
+    {
+        "type": "text",
+        "text": "S-Net (Tan et al., 2017) proposes a two-step approach for generative QA. The model first extracts an text span as the answer clue and then generates the answer according to the question, passage and the text span. Besides the different goal on answer generation instead of re-ranking like this work, their approach also differs from ours on that it extracts only one text span from a single selected passage. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We have observed that open-domain QA can be improved by explicitly combining evidence from multiple retrieved passages. We experimented with two types of re-rankers, one for the case where evidence is consistent and another when evidence is complementary. Both re-rankers helped to significantly improve our results individually, and even more together. Our results considerably advance the state-of-the-art on three open-domain QA datasets. ",
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+    {
+        "type": "text",
+        "text": "Although our proposed methods achieved some successes in modeling the union or co-occurrence of multiple passages, there are still much harder problems in open-domain QA that require reasoning and commonsense inference abilities. In future work, we will explore the above directions, and we believe that our proposed approach could be potentially generalized to these more difficult multipassage reasoning scenarios. ",
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+    {
+        "type": "text",
+        "text": "7 ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This work was partially supported by DSO grant DSOCL15223. ",
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+    {
+        "type": "text",
+        "text": "We thank Mandar Joshi for testing our model on the unfiltered TriviaQA hidden test dataset. ",
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