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+# ON DISENTANGLED REPRESENTATIONSLEARNED FROM CORRELATED DATA
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Despite impressive progress in the last decade, it still remains an open challenge to build models that generalize well across multiple tasks and datasets. One path to achieve this is to learn meaningful and compact representations, in which different semantic aspects of data are structurally disentangled. The focus of disentanglement approaches has been on separating independent factors of variation despite the fact that real-world observations are often not structured into meaningful independent causal variables. In this work, we bridge the gap to real-world scenarios by analyzing the behavior of most prominent methods and disentanglement scores on correlated data in a large scale empirical study (including 4260 models). We show that systematically induced correlations in the dataset are being learned and reflected in the latent representations, while widely used disentanglement scores fall short of capturing these latent correlations. Finally, we demonstrate how to disentangle these latent correlations using weak supervision, even if we constrain this supervision to be causally plausible. Our results thus support the argument to learn independent mechanisms rather than independent factors of variations.
+
+# 1 INTRODUCTION
+
+Due to the induced structure, disentangled representations promise generalization to unseen scenarios (Higgins et al., 2017b), increased interpretability (Adel et al., 2018; Higgins et al., 2018) and faster learning on downstream tasks (van Steenkiste et al., 2019; Locatello et al., 2019a). While the advantages of disentangled representations have been well established, they generally assume the existence of natural factors that vary independently within the given dataset, which is rarely the case in real-world settings. As an example, consider a scene with a table and some chairs (see Fig. 1). The higher-level factors of this representation are in fact correlated and what we actually want to infer are independent (causal) mechanisms (Peters et al., 2017; Parascandolo et al., 2018; Suter et al., 2019; Goyal et al., 2019).
+
+A complex generative model can be thought of as the composition of independent mechanisms or “causal” modules, which generate highdimensional observations (such as images or videos). In the causality community, this is often considered a prerequisite to achieve representations which are robust to interventions upon variables determined by such models (Peters et al., 2017). One particular instantiation of this idea in the machine learning community is the notion of disentangled representations (Bengio et al., 2013). The goal of disentanglement learning is to find a representation of the data which captures all the ground-truth factors of variation (FoV) independently.
+
+![](images/7cd9c32eb4e5dfbc24deba208b2c328f8872a001b1f886a15b06da7c96872ccb.jpg)  
+Figure 1: While in principle we consider the presence of the objects (coffee cup, table, chair) to be independent mechanisms, they tend to appear together in observed data.
+
+Despite the recent growth of the field, the performance of state-of-the-art disentanglement learners remains unknown for more realistic settings where FoV are correlated during training. Given the potential societal impact in the medical domain (Chartsias et al., 2018) or fair decision making (Locatello et al., $2 0 1 9 \mathrm { a }$ ; Madras et al., 2018; Creager et al., 2019), the evaluation of the usefulness of disentangled representations trained on correlated data is of high importance.
+
+To go beyond the highly idealized settings considered thus far, we conducted a large scale empirical study to systematically assess the effect of induced correlations between pairs of factors of variation in training data on the learned representations. To provide a qualitative and quantitative evaluation, we investigate multiple datasets with access to ground-truth labels. Moreover, we study the generalization abilities of the representations learned on correlated data as well as their performance in particular for the downstream task of fair decision making.
+
+Contributions. Our main contributions can be summarized as follows:
+
+• We present the first large-scale empirical study (4260 models)1 that examines how modern disentanglement learners perform when ground truth factors of the observational data are correlated.   
+• We find that factorization-based inductive biases are insufficient to learn disentangled representations from observational data. Existing methods fail to disentangle correlated factors of variation, resulting in correlated latent space dimensions. Moreover, standard disentanglement metrics do not reveal these persisting correlations.   
+• We investigate the usefulness of semi-supervised and weakly-supervised approaches to resolve latent entanglement. For the latter setting, we focus on multiple observational and interventional distributions.
+
+# 2 BACKGROUND AND RELATED WORK
+
+Disentanglement. Current state-of-the-art disentanglement approaches use the framework of variational auto-encoders (VAEs) (Kingma & Welling, 2014; Rezende et al., 2014). The (high-dimensional) observations $_ { \textbf { \em x } }$ are modelled as being generated from some latent features $_ { z }$ with chosen prior $p ( z )$ according to the probabilistic model $p _ { \theta } ( \pmb { x } | \pmb { z } ) p ( \pmb { z } )$ . The generative model $p _ { \boldsymbol { \theta } } ( \boldsymbol { x } | \boldsymbol { z } )$ as well as the proxy posterior $q _ { \phi } ( \pmb { z } | \pmb { x } )$ can be parameterized by neural networks, which are optimized by maximizing the variational lower bound (ELBO) of $\log p ( \pmb { x } _ { 1 } , \ldots , \pmb { x } _ { N } )$ .
+
+$$
+\mathcal { L } _ { V A E } = \sum _ { i = 1 } ^ { N } \mathbb { E } _ { q _ { \phi } ( z | \mathbf { x } ^ { ( i ) } ) } [ \log p _ { \theta } ( \mathbf { x } ^ { ( i ) } | z ) ] - D _ { K L } \big ( q _ { \phi } ( z | \mathbf { x } ^ { ( i ) } ) \| p ( z ) \big )
+$$
+
+The above objective does not enforce any structure on the latent space, except for similarity (in KL-divergence) to the prior $p ( z )$ (typically chosen as an isotropic Gaussian). However, the structure and semantic meaning of latent representations can be relevant to study generation properties. Consequently, various proposals for structure-imposing regularization and commonly used evaluation metrics measuring different notions of disentanglement of the learned representations have been made (Higgins et al., 2017a; Kim & Mnih, 2018; Burgess et al., 2018; Kumar et al., 2018; Chen et al., 2018; Eastwood & Williams, 2018; Mathieu et al., 2018). Recently, it has been shown that unsupervised disentangling by optimising marginal likelihood in a generative model is impossible without further inductive bias (Locatello et al., 2019b). To address this theoretical limitation, methods have been proposed that do not require explicitly labelled data but only some weak labeling information (Locatello et al., 2020; Shu et al., 2019). Ideas related to disentangling the factors of variations date back to the non-linear ICA literature (Bach & Jordan, 2002; Comon, 1994; Jutten & Karhunen, 2003; Hyvärinen & Pajunen, 1999; Hyvarinen et al., 2019; Hyvarinen & Morioka, 2016; Gresele et al., 2019). Recent work combines non-linear ICA with disentanglement (Khemakhem et al., 2020; Sorrenson et al., 2020; Klindt et al., 2020)
+
+Correlations. A set of random variables $X _ { i = 1 , \ldots , n }$ is not independent, if and only if their joint distribution does not factorize
+
+$$
+P ( X _ { 1 } , X _ { 2 } , \ldots , X _ { n } ) \neq \prod _ { i = 1 } ^ { n } P ( X _ { i } ) .
+$$
+
+In this case, we speak of dependence between the random variables, also commonly referred to as correlation.2 Correlation between two variables can either stem from a direct causal relationship (one variable affects the other), but can also be due to different unobserved circumstances (confounders) affecting both. Real-world datasets display many of these (“spurious” and often a priori unknown) correlations (Geirhos et al., 2020). However, most work on learning disentangled representations assumes that there is an underlying set of independent ground truth variables that govern the generative process of observable data. These methods are hence predominantly evaluated on data that obey independence in the true factors of variation, which we then consider to be the correct factorization. In the real world, the observation generating process is likely not always as clearly “disentangled” and we do expect correlations in the collected datasets. It is thus an open question to what degree existing inductive biases from the encoder/decoder architecture, but more importantly the dataset biases, affect the learned representation. In our experiments, we introduce dataset correlations in a controlled manner to understand to what degree state-of-the-art approaches can cope with such correlations. We believe these correlations to reflect a major feature of more realistic environments.
+
+![](images/8434d800ad72d18ee910812aa691bbcbaf1629e87d4199312f4297e9514c68d5.jpg)  
+Figure 2: Disentanglement metrics show no clear trend along different correlation strengths (each violin represents 180 models, higher $\sigma$ indicates less correlation).
+
+Other Related Work. Most popular datasets in the disentanglement literature exhibit perfect independence in their FoV. At some level this is sensible as it reflects the underlying assumption in the inductive bias being studied. However, this assumption is unlikely to hold in practice as shown by Li et al. (2019), who propose methods based on a pairwise independence assumption instead. The literature so far has not thoroughly measured how popular inductive biases such as factorized priors behave when learning from correlated datasets, although several smaller experiments along these lines can be acknowledged. Chen et al. (2018) studied correlated 3DFaces (Paysan et al., 2009) by fixing all except three factors in which the authors conclude that the $\beta$ -TC-VAE regularizer can help to disentangle imposed correlations. Brekelmans et al. (2019) show that Echo noise results in superior disentanglement compared to standard betaVAE in a small experiment on a downsampled dSprites variant where randomly selected factor pairs are excluded. However, the latent structure was not studied in detail; our findings suggest that global disentanglement metrics are insufficient to diagnose issues when models learn from correlated data. Creager et al. (2019) based some of the evaluations of a proposed new autoencoder architecture in the fairness context on a biased dSprites variant and Yang et al. (2020) study a linear SCM in a VAE architecure on datasets with dependent variables. However, their studies focused on representation learners that require strong supervision via FoV labels at train time.
+
+# 3 THE EFFECT OF CORRELATED DATA ON DISENTANGLEMENT LEARNERS
+
+In this section, we want to present the key findings from our empirical study of unsupervised disentanglement learning on a particular variant of correlated data. We start by outlining the experimental design of our study in Section 3.1. Based on this, we analyze the latent spaces in Section 3.2 and find that factorization-based inductive biases are insufficient to learn disentangled representations from observational data. Persisting pairwise correlations in the latent space are not sufficiently revealed by standard disentanglement metrics that might be particularly relevant and problematic for fairness applications. Finally, in Section 3.3, we show extrapolation and generalization capabilities of the learned representations towards unseen factor combinations due to the induced correlations.
+
+# 3.1 EXPERIMENTAL DESIGN
+
+For our first experiments we introduce correlations between single pairs of factors of variation on the following three datasets: Shapes3D with object size and azimuth (denoted “A”), dSprites with orientation and X-position (“B”) and finally the real-world observations dataset MPI-3D with first and second degree of freedom $\left( \ \stackrel { 6 6 } { \ C } \right)$ . We focus on linear correlations with Gaussian noise between the two variables, which we denote by $c 1 , c 2$ . The ground truth factors for $c 1 , c 2$ take values $z _ { c 1 } \in \{ 0 , \dots , z _ { c 1 } ^ { \operatorname* { m a x } } \}$ and $z _ { c 2 } \in \{ 0 , \dots , z _ { c 2 } ^ { \operatorname* { m a x } } \}$ respectively. We then parameterize correlations by sampling the training dataset from the joint $P ( z _ { c 1 } , z _ { c 2 } ) \propto \mathcal { N } ( z _ { c 2 } - \alpha z _ { c 1 } , \sigma )$ where $\alpha = z _ { c 2 } ^ { m a x } / z _ { c 1 } ^ { m a \bar { x } }$ The strength of the correlations can be tuned by $\sigma$ , for which we choose 0.2, 0.4, 0.7 in normalized units with respect to the range of values in $z _ { c 1 , c 2 }$ . Lower $\sigma$ indicates stronger correlation. See Fig. 5 for an example of $P ( z _ { c 1 } , z _ { c 2 } )$ for correlating azimuth and object size in Shapes3D with $\sigma = 0 . 2$ . Additionally, we study the uncorrelated limit $\sigma = \infty$ ), which amounts to the case typically studied in the literature. We train the same six VAE methods as discussed in Locatello et al. (2019b), including $\beta$ -VAE, FactorVAE, AnnealedVAE, DIP-VAE-I, DIP-VAE-II and $\beta$ -TC-VAE, each with six hyperparameter settings. Each method has been trained using five different random seeds. All remaining factors of variation are sampled uniformly at random. This first study sums up to a total of 2160 trained models, or 180 models per dataset and correlation strength3. Appendix A describes additional implementation details.
+
+![](images/c8e59e5dfb217a4b7e563ded39aa232ed4236c8bcac9947b1dbceacc1ab965b7.jpg)  
+Figure 3: We show latent traversals (left) of the best DCI score model among all 180 trained models with strongest correlation $\sigma = 0 . 2$ ) on Shapes3D (A). The traversals in latent code dimensions 4 and 8 (highlighted in black), suggest that these dimensions encode a mixture of azimuth and object size, reflecting one major axis along the correlation line of the joint distribution and one smaller, locally orthogonal axis. This is supported by a heat map of the GBT feature importance matrix of this model (right) indicating an entanglement of azimuth and object size encoded into both latent codes.
+
+# 3.2 CAN UNSUPERVISED METHODS ACHIEVE DISENTANGLEMENT OF CORRELATED DATA?
+
+Shortcomings of existing metrics. Following recent studies, we evaluate the trained models with the help of a broad range of disentanglement metrics that aim at quantifying overall success by a single scalar measure. Perhaps surprisingly, Fig. 2 shows no clear trend among all implemented disentanglement scores w.r.t. correlation strength (see Fig. 9 in the Appendix for the full result across all datasets and metrics). The metrics have been evaluated by sampling from the correlated data distribution although they do not differ substantially when evaluated on the uncorrelated distribution. However, as we will demonstrate along the following analysis, the latent spaces in this setting show some characteristic differences when trained on a strongly correlated pair of FoVs. We thus argue that common disentanglement metrics are limited when correlations are introduced into the training data. To conduct a more careful analysis of the inductive data bias applied on the learned representations we will instead evaluate pairwise metrics. Note that regarding BetaVAE and FactorVAE this observed trend is to some degree expected as they would yield perfect disentanglement scores even if we would take the correlated ground truth factors or a linear transformation in the case of BetaVAE as the representation.
+
+Latent structure and pairwise entanglement. We start by analysing latent traversals of some trained models on Shapes3D (A). For strong correlations $\sigma = 0 . 2$ and $\sigma = 0 . 4$ ), we typically observe trained models with two latent codes encoding the two correlated variables simultaneously. In these cases, one of the latent codes corresponds to data along the major axis of the correlation line whereas the other latent code dimension manifests in an orthogonal change of the two variables along the minor axis. Still, a full traversal of the code corresponding to the minor axis often seems to cover only observations within the correlation line. Fig. 3 (left) shows this effect for the latent space of a model trained on Shapes3D (A) with strongest correlation $\sigma = 0 . 2$ ).
+
+![](images/f824ae85040c71c66286db291a42526822f07e8f2335325a9be194b7b8a49246.jpg)
+
+Figure 4: Left: Pairwise entanglement scores help to uncover still existent correlations in the latent representation. Left: Mean of the pairwise entanglement scores for the correlated pair (red) and the median of the uncorrelated pairs. We see that stronger correlation leads to statistically more entangled latents compared to the baseline score without correlation (blue). Right: The same behavior can be seen for the unfairness score between the correlated pair of factors.   
+
+<table><tr><td>Correlation strength</td><td></td><td>g=0.2</td><td>g=0.4</td><td>g=0.7</td><td>σ=∞ (uc)</td></tr><tr><td rowspan="2">Shapes3d (A)</td><td>object size - azimuth</td><td>0.38</td><td>0.26</td><td>0.13</td><td>0.08</td></tr><tr><td>median uncorrelated pairs</td><td>0.09</td><td>0.09</td><td>0.09</td><td>0.08</td></tr><tr><td rowspan="2">dSprites (B)</td><td>orientation - position X</td><td>0.17</td><td>0.16</td><td>0.14</td><td>0.11</td></tr><tr><td>median uncorrelated pairs</td><td>0.13</td><td>0.13</td><td>0.13</td><td>0.13</td></tr><tr><td rowspan="2">MPI3D (C)</td><td>First DOF - Second DOF</td><td>0.2</td><td>0.19</td><td>0.17</td><td>0.16</td></tr><tr><td>median uncorrelated pairs</td><td>0.16</td><td>0.16</td><td>0.15</td><td>0.15</td></tr></table>
+
+To quantify this observation, we analyze the importance of individual latent codes in predicting the value of a given ground truth FoV. An importance weight for each pair of $\{ \mathrm { F o V } ,$ latent dimension} is computed by training a gradient boosting trees (GBT) classifier to predict the ground truth labels from the latents (10,000 samples). In the right panel of Fig. 3, we compute these importance weights for the model used to generate traversals in the left panel. The corresponding evaluation for a model trained on the same dataset with weak correlation does not reveal this feature visually (see Fig. 10 in the Appendix).
+
+To support this claim for a larger set of models, we calculate a pairwise entanglement score that allows us to measure how difficult it is to separate two factors of variation from their latent codes. This computation involves grouping FoV into pairs based on an ordering of their pairwise mutual information or GBT feature importance between latents and FoV; we defer to Appendix A for a detailed description of this procedure. Figure 4 (left) shows that across all datasets the pair of correlated FoV has a higher score than the median of all other pairs, indicating that they are harder to disentangle. This threshold decreases with weaker correlation and the pair becomes easier to disentangle for weaker correlations $( \sigma \ge 0 . 7 )$ . These findings suggest that the models still manage to disentangle correlated factors if the correlation is not too strong.
+
+Finally, correlations between variables are of crucial importance in fairness applications motivating an additional investigation on ramifications of these entangled latent spaces. In this setting we are interested in the unfairness of predicting the second correlated variable while the first correlated variable is considered being a protected or sensitive attribute. In the following, we use a variant of demographic parity (Dwork et al., 2012) that computes pairwise mutual information between latents and FoV (Locatello et al., 2019a). In Fig. 4 (right) we evaluate this score when correlations are present within the data in the case of Shapes3D (A). Unfairness tracks correlation strength in this scenario. These results suggest that we cannot expect disentangled representations learned unsupervisedly to help reduce unfairness beyond the benefits discussed in Locatello et al. (2019a). More comprehensive results, that support the finding that the correlated pair is statistically more entangled in the latent representations across all unsupervised experiments and datasets is provided in Appendix B.1.
+
+Summary. We find that existing methods fail to learn disentangled representations of correlated factors of variation, and moreover that standard disentanglement metrics are insufficient to reveal these troublesome pairwise entanglements in the latent space.
+
+# 3.3 GENERALIZATION PROPERTIES
+
+In this Section, we aim to understand how the trained models perform on unseen training data far away from the correlation line; out-of-distribution (OOD) w.r.t. the train distribution. We analyse this capability for the model from Fig. 3. In this model, the remaining factors seem to be disentangled well enough to only focus on the two latent dimensions encoding the entangled variables.
+
+As a first test, we sample observations from the FoV then set object size and azimuth to six distinct configurations of zero probability, see Fig. 5 (left). The trained model is capable of reconstructing these observations despite never having encountered these configurations or neighbors thereof during training. This suggests that the encoder maps representations to a meaningful point in the latent space from which the decoder is equally capable of generating expected observations. To test this hypothesis further, we analysed latent traversals originating from these OOD points and observe that changes in the remaining factors reliably yield the expected reconstructions. Traversals with respect to the two entangled latent codes continue to encode object size and azimuth.
+
+![](images/3a4907b6a6815be9ba92ce3f27dba690c6be01f16deae4a661172b14204a1d76.jpg)  
+Figure 5: Generalization to out-of-distribution (OOD) test data. Left: Reconstructions of observations the model has never seen during training. Right: Latent space distribution of the two entangled dimensions. Circles without edges indicates encoded data from the (correlated) training distribution, while circles with edges indicate encoded OOD data where the correlation pattern is broken.
+
+To fully understand this models’ generalization properties we visualise the occupied latent space spanned by the two identified dimensions encoding both correlated factors in Fig. 5 (right). We are particularly interested in where these points are located with respect to the ground truth value of each correlated variable, depicted via color. The two sets of depicted points are (1) latent codes sampled from the correlated training data and (2) latent codes sampled with a (object size, azimuth) configuration that has zero probability under the correlated training distribution. We observe that contours of equal color (ground truth) are not aligned with the latent axes. This indicates that the two latent dimensions encode both FoV at the same time. Likewise, we can understand the generalization capabilities of this model far away from the trained data. Extreme configurations such as small azimuth and large object size are encoded to space regions corresponding to the intersections of the manifolds with constant value for each correlated variable. This shows that all out-of-distribution points are encoded in this representation space in a way that obeys the natural ordering of each respective factor. This behaviour remains even in cases where the trained latent space does not mirror the default value ordering as stored in our ground truth table. For this we additionally trained 360 models on two additional Shapes3D variants each, where we strongly correlated object color - object size (“D”) and object color - azimuth (“E”) respectively $\sigma = 0 . 2$ and $\sigma = 0 . 4$ ). As the color values do not allow for a unique natural ordering, the trained models do often encode a different color manifold ordering into the latent space. In Appendix B.1, we show some of their characteristic latent space visualizations with similar extrapolation and generalization capabilities. We conclude from these results that disentanglement methods can generalize towards unseen FoV configurations as long as each factor value is contained in the training data within a different configuration.
+
+# 4 FINDING THE RIGHT FACTORIZATION
+
+The results from Section 3 illustrate the limitations of state-of-the-art unsupervised methods on correlated data (and thus real-world observational data). We now investigate the usefulness of several approaches for mitigating pairwise correlations in the latent code. We begin with a post-hoc procedure in Section 4.1 that uses limited label information on the ground truth factors and show that it achieves a substantial correction of the pairwise latent correlation. We then consider a recently proposed approach leveraging recent advances in weakly supervised disentanglement learning that applies directly at train time. As will be seen in Section 4.2, this method results in substantially more disentangled representations, even when applied on correlated data from different sampling scenarios.
+
+Figure 6: Fast adaption with few labels: Pairwise entanglement scores for correlated FoV pair in Shapes3D (A). The correlated pair is highlighted (red). Zero labels reflects the unsupervised baseline without any fast adaptation. Growing number of labels show that fast adaption using linear regression reduces these correlations with as little as 100 labels (blue). Reported pairwise scores are averaged over 180 models per correlation strength.   
+
+<table><tr><td>#Labels</td><td></td><td>0</td><td>100</td><td>1000</td><td>10000</td></tr><tr><td>Shapes3D (A) σ = 0.2</td><td>object size - azimuth</td><td>0.38</td><td>0.17</td><td>0.15</td><td>0.15</td></tr><tr><td>Shapes3D (A) σ= 0.2</td><td>median uncorrelated pairs</td><td>0.09</td><td>0.08</td><td>0.07</td><td>0.07</td></tr><tr><td>Shapes3D (A) σ = 0.4</td><td> object size - azimuth</td><td>0.26</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Shapes3D (A) σ = 0.4</td><td>median uncorrelated pairs</td><td>0.09</td><td>0.08</td><td>0.08</td><td>0.08</td></tr></table>
+
+# 4.1 POST-HOC ALIGNMENT CORRECTION WITH FEW LABELS
+
+When a limited number of FoV labels $\left\{ \mathbf { y } _ { i } \right\}$ can be accessed, a reasonable option for resolving entangled dimensions of the latent code is by fast adaptation. To identify the two entangled dimensions $( z _ { i } , z _ { j } )$ we look at the maximum feature importance for a given FoV from a GBT trained using these labels only. We then train a substitution function using supervised learning to replace these two dimensions with the predicted ground truth label. Crucially, both steps of this procedure rely on the same FoV labels, which should be as few as possible. In Fig. 6 we show the pairwise entanglement score of the correlated FoVs under this fast adaptation with a linear regression as the substitution function, which succeeds with as few as 100 labels, corresponding to less than $0 . 0 2 \%$ of all data points in Shapes3D. However, fast adaptation with linear regression substitution fails in some settings: when no two latent dimensions encode the applied correlation isolated from the other latent codes, or when the correlated variables do not have a unique natural ordering (e.g. color or categorical variables). To address this, a nonlinear substitution function such as a MLP can reduce this pairwise entanglement to a certain degree (see additional results in Appendix B.2).
+
+We find that the efficacy of fast adaptation depends on the level of disentanglement of the representations with respect to all the other factors. This implies that if the representation is well disentangled at the start of the fast adaption procedure, it is possible to achieve a perfectly disentangled model (according to our previous visual and quantitative evaluations). However, if all FoV are entangled at the beginning, the fast adaption method will have little effect. Finally, we note that model selection is impossible in a purely unsupervised manner based on any of the used disentanglement metrics, as they all require labeled ground truth data. These shortcomings shall be resolved by the following method capable of disentangling the correlated factors of variation much more reliably.
+
+# 4.2 ALIGNMENT DURING TRAINING USING WEAK SUPERVISION
+
+Since the unsupervised disentangling by optimising marginal likelihood in a generative model is impossible (Locatello et al., 2019b, Theorem 1), inductive biases like grouping information (Bouchacourt et al., 2018) or access to labels (Locatello et al., 2019c) is required. Changes in natural environments, which typically correspond to changes of only a few underlying factors of variation, provide a weak supervision signal for representation learning algorithms (Goyal et al., 2019; Földiák, 1991; Schmidt et al., 2007; Bengio et al., 2019). Without correlations it has been shown that this weak supervision helps in learning much more disentangled representations (Locatello et al., 2020; Shu et al., 2019). Locatello et al. (2020) showed access to observations which display differences in a known number of factors of variation (without knowing which ones specifically) is sufficient to learn disentangled representation. These additional weak assumptions render the generative model identifiable in contrast to unsupervised disentanglement. This kind of extra knowledge might be available in certain settings 4, e.g., in temporarily close frames from a video of a moving robot arm where some factors remain unchanged. Hence, we want to investigate the usefulness of such a weakly-supervised method applied in various scenarios when training data is correlated. Specifically we implement the Ada-GVAE variant of Locatello et al. (2020) that was shown to allow for model selection via the (unsupervised) reconstruction loss. The method requires a pair of observations that differs in a known number of factors, without knowing which in particular.
+
+![](images/143ef6a989097e83190d62e51d5d5cd9acff40e9f5e8469f5605bbda6b516512.jpg)  
+Figure 7: Weak supervision: Left: With weak supervision trained models on Shapes3D correlation object size and azimuth learn consistently improved, often perfect, disentangled representations across all correlation strengths. Middle: Unfairness scores between correlated FoVs are much smaller (see scale). Right: Latent dimensions of a best DCI model with strong correlation $\mathit { \check { \sigma } } = 0 . 2 $ ). Representations are axis-aligned with respect to both of the correlated variables ground truth values (right). Below: Traversals of a model trained with strong correlation $( \sigma = 0 . 2$ )
+
+Weak supervision mitigates pairwise latent entanglement. We trained the three correlated Shapes3D variants (A, D, E) with pairwise correlations between object color, object size and azimuth with the same correlation strength settings. Due to the definition of the regularizer we limit this study to the $\beta$ -VAE models with the same 6 hyperparameters and use 5 random seeds, yielding 360 additional models. For the generation of pairs we study the case where the difference in the observation pairs is present in one random FoV. Whenever we sample the difference to be in one of the correlated FoV, its respective value in each pair is drawn from the probability distribution conditioned on the other correlated FoV. This means the difference in this factor is typically very small and depends on the correlation strength. Note that this procedure assures that constructed pairs are consistent with the observational data such that the correlation is never broken. Fig. 7 summarizes the weak supervision results when imposing correlations in object size and azimuth. We consistently observe much better disentangled models, often achieving perfect DCI score irrespective of correlations in the dataset. The latent spaces tend to strongly align their coordinates with the ground truth label axis. Finally, weak-supervision reduces unfairness relative to the unsupervised baseline, and occasionally achieves zero unfairness score.
+
+These results suggest that weak supervision can provide a strong inductive bias capable of finding the right factorization and resolving spurious correlations for datasets of unknown degree of correlation. As a prominent example, this is an issue in the fairness context where real-world datasets often display unknown discriminatory correlations. Additional results on the other datasets can be found in Appendix B.2 including two additional scenarios where one has intervening capabilities to generate the pair. We consistently observe the same strong trends regarding disentangled correlations in all of the above studies using weak supervision.
+
+# 5 CONCLUSION
+
+We have presented the first large-scale empirical study examining how modern disentanglement learners cope with correlated observational data. We find that existing methods fail to learn disentangled representations of correlated factors of variation, and moreover that standard disentanglement metrics are insufficient to reveal these pairwise entanglements. We discussed practical implications for downstream tasks like fair decision making. Finally, we demonstrate how to correct for these latent correlations via various weakly supervised training scenarios. Our results thus support the importance and usefulness of learning independent mechanisms rather than independent factors of variations (Schölkopf, 2019; Parascandolo et al., 2018; Suter et al., 2019; Goyal et al., 2019). Besides the simple correlations studied in this work, future work is needed to address the open question whether these results extend to more complex nonlinear correlations and settings where many more variables are correlated simultaneously.
+
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+
+# A IMPLEMENTATION DETAILS
+
+Unsupervised Disentanglement methods. For reasons of comparison, the considered disentanglement methods in this work cover the full collection of state-of-the-art approaches in disentanglement_lib from Locatello et al. (2019b) based on representations learned by VAEs. The set contains six different methods that enforce disentanglement of the representation by equipping the loss with different regularizers that aim at enforcing the special structure of the posterior aggregate encoder distribution. A detailed description of the regularizer forms used in this work, specifically $\beta$ -VAE (Higgins et al., 2017a), FactorVAE (Kim & Mnih, 2018), AnnealedVAE (Burgess et al., 2018), DIP-VAE-I, DIP-VAE-II (Kumar et al., 2018) and $\beta$ -TC-VAE (Chen et al., 2018) is provided in (Locatello et al., 2019b). We use the same encoder architecture with 10 latent dimensions for every model.
+
+Evaluation metrics. To measure disentanglement of a learned representation, various metrics have been proposed, each requiring access to the ground truth labels. The BetaVAE score is based upon the prediction of a fixed factor from the disentangled representation using a linear classifier (Higgins et al., 2017a). The FactorVAE score is intended to correct for some failures of the former by utilizing majority vote classifiers based on a normalized variance of each latent dimension (Kim & Mnih, 2018). The SAP score represents the mean distance between the classification errors of the two latent dimensions that are most predictable (Kumar et al., 2018). For the MIG score, one computes the mutual information between the latent representation and the ground truth factors and calculates the final score using a normalized gap between the two highest MI entries for each factor. Finally, a disentanglement score proposed by Eastwood & Williams (2018), often referred to as DCI score, is calculated from a dimension-wise entropy reflecting the usefulness of the dimension to predict a single factor of variation.
+
+Unfairness between a pair of FoV: The scores reported are based on a notion of demographic parity for predicting a target variable $y$ given a protected and sensitive variable $s$ . Both $y$ and $s$ can be associated with a factor of variation here. Rather than using the global total variation average as defined in Locatello et al. (2019a), we report the individual demographic parities for the correlated factors specifically.
+
+Pairwise entanglement metric: In order to further analyze the differences between the GBT feature importance matrices between latent code and encoded factor of variation, we view them as weights on the edges of a bipartite graph encoding the statistical relation between each factor of variation and code. We can now delete all edges with weight smaller than some threshold and count (i) how many factors of variation are connected with at least a latent code and (ii) the number of connected components with size larger than one. We can then compute which factors are merged at which threshold. Factors that are merged at lower threshold are more entangled in the sense that are more statistically related to a shared latent dimension. This computation can be not only based on the GBT feature importances but likewise on weight matrices inferred from the Mutual Information.
+
+Joint distributions of correlated factors in datasets: In Fig. 8 we show the joint probability distributions of the correlated pair of FoV for all datasets and correlation strengths considered in this study. Dataset A, B and C were designed with correlated factors of variation that are ordinal for a natural visual interpretation of the traversals. In contrast, datasets D and E contain a correlated factor of variation that has no such natural ordering.
+
+# B ADDITIONAL RESULTS
+
+# B.1 SECTION 3
+
+Shortcomings of existing metrics. Following recent studies, we evaluate the trained models with the help of a broad range of disentanglement metrics that aim at quantifying overall success by the help of a single scalar measure. Perhaps surprisingly, as can be seen in Fig. 9, there is no clear trend among all implemented disentanglement scores w.r.t. correlation strength. The metrics have been evaluated by sampling from the correlated data distribution although they do not differ substantially when evaluated on the uncorrelated distribution. We thus argue that commonly used methods are insufficient to provide insight into the latent space learned.
+
+![](images/f94feea5848132b951f473cf53dfa9d1dc6ef33b49735fc543b46e24ae0ecf97.jpg)  
+Figure 8: Probability distributions for sampling training data in the correlated pair of FoVs in the respective datasets (A, B, C, D, E) considering correlation strengths of $\sigma = 0 . 2$ , $\sigma = 0 . 4$ , $\sigma = 0 . 7$ and $\sigma = \infty$ , the uncorrelated limit (from left to right).
+
+![](images/befc523d57a6f4ddbe357b7dbbffc7e45da3efd7984297017b532022ddc94e3f.jpg)  
+Figure 9: Standard disentanglement metrics evaluated on the correlated and uncorrelated (uc) training set showing no clear trend for different correlation strengths.
+
+![](images/60b117850a53daf06a30455a78996762a9ba12708937fe8953bef3fc11a32782.jpg)  
+Figure 10: We show latent traversals (left) of the best DCI score model among all 180 trained models with weak correlation $\sigma = 0 . 7$ ) in object size and azimuth. The traversals in latent code no 3 and 7/8 (highlighted in black), suggest that these dimensions encode no mixture of azimuth and object size compared to the models with stronger correlation. This is supported by the GBT feature importance matrix of this model (right).
+
+<table><tr><td>correlation strength o</td><td></td><td>0.2 0.4</td><td>0.7</td><td>8 (uc)</td></tr><tr><td>Shapes3D (A)</td><td>object size - azimuth median other pairs</td><td>0.38 0.26 0.09 0.09</td><td>0.13 0.09</td><td>0.08 0.08</td></tr><tr><td>dSprites (B)</td><td>orientation - position X median other pairs</td><td>0.17 0.16 0.13 0.13</td><td>0.14 0.13</td><td>0.11 0.13</td></tr><tr><td>MPI3D (C)</td><td>First DOF - Second DOF median other pairs</td><td>0.2 0.19 0.16 0.16</td><td>0.17 0.15</td><td>0.16 0.15</td></tr></table>
+
+<table><tr><td>0.2</td><td>0.4</td><td>0.7</td><td>8（uc)</td></tr><tr><td>0.28</td><td>0.25</td><td>0.2</td><td>0.17 0.18</td></tr><tr><td>0.2 0.34</td><td>0.2 0.31</td><td>0.19 0.24</td><td>0.14</td></tr><tr><td>0.16</td><td>0.18</td><td>0.19</td><td>0.15</td></tr><tr><td>0.54</td><td>0.52</td><td>0.5</td><td>0.49</td></tr><tr><td>0.25</td><td>0.25</td><td>0.26</td><td>0.25</td></tr></table>
+
+<table><tr><td>0.2</td><td>0.4</td></tr><tr><td>0.38 0.11</td><td>0.31 0.11</td></tr><tr><td>0.43</td><td>0.3</td></tr><tr><td>0.15</td><td>0.15</td></tr></table>
+
+Figure 11: Pairwise entanglement scores help to uncover still existent correlations in the latent representation. Left: Mean of the pairwise entanglement scores for the correlated pair (red) and the median of the uncorrelated pairs. We see that stronger correlation leads to statistically more entanglement latents across all datasets studied compared to their baseline pairwise entanglement where the data exhibits no correlations (blue). Each pairwise score is the mean across 180 models for each dataset and correlation strength. Scores in the left table are based on GBT feature importance and scores presented in the right table are based on Mutual Information.   
+
+<table><tr><td>correlation strength o</td><td></td><td>0.2</td><td>0.4</td></tr><tr><td>Shapes3D (D)</td><td>object color - object size</td><td>0.29</td><td>0.28</td></tr><tr><td></td><td>median uncorrelated pairs</td><td>0.07</td><td>0.07</td></tr><tr><td>Shapes3D (E)</td><td>object color - azimuth</td><td>0.25</td><td>0.23</td></tr><tr><td></td><td>median uncorrelated pairs</td><td>0.1</td><td>0.09</td></tr></table>
+
+Latent structure and pairwise entanglement. Our hypothesis that the latent representations are less correlated if the correlation strength is weaker is shown for a model on Shapes3D (A) with weak correlation in Fig. 10. Here the latent traversals do not mirror the major and minor axis of the correlated joint distribution. This conclusion is being backed by the thresholds of the pairwise entanglement metrics for the correlated pair vs. the median of all other pairs across all datasets, either when computing them using the GBT feature importances or the mutual information. See Fig. 11 for these respective results. Another pairwise metric that tracks the correlation strength in our scenario is the unfairness score between the correlated pair of factors that is being shown for datasets A, B and C in Fig. 12.
+
+Generalization Properties In order to support our conclusion from these results that disentanglement methods can generalize towards unseen FoV configurations we show in Fig. 13 latent traversals originating from these OOD point with smallest object size and largest azimuth. We observe that changes in the remaining factors reliably yield the expected reconstructions. Additionally, we are interested in where samples from correlated models are located with respect to the ground truth value of each correlated variable. For this, we are visualizing the latent spaces with similar extrapolation and generalization capabilities of four models from the two strongest correlation dataset variants of Shapes3d (D) and Shapes3d (E) in Fig. 14.
+
+![](images/34e189fd63cfbbe62d7bfd7f7f4115352a98f83edeef49f6dc31182e3a752e6d.jpg)  
+Figure 12: Disentangled representations trained on correlated data are anti-correlated with higher fairness properties. The plots show the mean unfairness scores between the correlated factors with decreasing correlation strength for Shapes3D (A), dSprites (B) and MPI3D-real (C).
+
+![](images/317b7f65006673a60895c3b03dc88a1dd36f82a0fe70127354c7c41ae1ed7185.jpg)  
+Figure 13: Generalization capabilities towards out-of-distribution test data. Latent traversals from an observations the model has never seen during training. The starting point corresponds to a factor configuration in point 1 from Fig. 5. Shown are the results of the model with highest DCI score among all 180 trained models on Shapes3d (A) with a very restricted correlation strength $\sigma = 0 . 2$ in object size and azimuth
+
+# B.2 SECTION 4
+
+Post-hoc alignment correction with few labels In Fig. 15, we see the axis alignment of the correlated latent space after fast adaptation using linear regression on a model trained on Shapes3D (A). Fast adaptation with linear regression substitution fails in some settings: when no two latent dimensions encode the applied correlation isolated from the other latent codes, or when the correlated variables do not have a unique natural ordering (e.g. color or categorical variables). Additionally, the functional form of the latent manifolds beyond the training distribution is unknown and in general expected to be nonlinear. We test the possibility of fast adaptation in this case using as substitution function a one-hidden layer MLP classifier of size 100 on the correlated Shapes3D variants. Under this method, we sample the FoV from a uniform independent distribution. A small number of such samples could practically be labeled manually. Using only 1000 labeled data points for our fast adaptation method shows a significant reduction in disentanglement-thresholds for the correlated pair (Fig. 16).
+
+Alignment during training using weak supervision Using the studied weakly supervision AdaGVAE method with $k = 1$ from Locatello et al. (2020), we showed that weak supervision can provide a strong inductive bias capable of finding the right factorization and resolving spurious correlations for datasets of unknown degree of correlation. Despite the results shown on Shapes3D (A) in the main paper, results across all three correlation variants in Shapes3D (A, D, E) are shown in Fig. 17 as well as some representative latent space visualizations that show strong axis alignment in Fig. 18. This study contains a total of 360 trained models.
+
+In addition to the experiment from the main paper where pairs are constructed solely from the correlated observational data, we want to study two scenarios where we have limited intervention capabilities on the FoV to generate training pairs. The resulting distribution of FoVs (still exhibiting correlations) in these pairs depends on whether the correlation between two pairs is due to a causal link or due to a common confounder.
+
+Scenario I-1: We assume there is a confounder causing a spurious correlation between the factors, such that correlation is broken whenever we constraint the change to be in one of the correlations.
+
+![](images/518e9b8bb27e4c53eed05f016e18beadb6eebf81873ce47e059542589bc0fa47.jpg)  
+Figure 14: Latent space distribution of the two entangled dimensions of the best DCI model in Shapes3d (E) with $\sigma = 0 . 2$ (first column), in Shapes3d (E) with $\sigma = 0 . 4$ (second column), in Shapes3d (D) with $\sigma = 0 . 2$ (third column) and in Shapes3d (D) with $\sigma = 0 . 4$ (fourth column). Latent codes sampled from correlated observations (circle without edge) and (2) latent codes sampled with an object size-azimuth configuration not encountered during training(squares with black edge). Each column shows the ground truth values of the two correlated factors by color.
+
+![](images/e64e0ebba8a1d2a267f6bda535faa83e124944fb4f84fe0350c180fc7cefabb0.jpg)  
+Figure 15: Latent space distribution of the two entangled dimensions of the best DCI model in Shapes3D (A). Latent codes sampled from correlated observations (circle without edge) and (2) latent codes sampled with an object size-azimuth configuration not encountered during training (squares with black edge). Left column shows the ground truth values of the two correlated factors by color. Middle and right column show the fast adapted space using linear regression and 100 or 1000 labels respectively.
+
+Figure 16: Mean of the pairwise entanglement scores for the correlated pair (red) and the median of the uncorrelated pairs (based on GBT feature importance) for all pairs of variables in Shapes3D (D) (top), Shapes3D (E) (middle) and Shapes3D (A) (bottom) all with correlation strength $\sigma = 0 . 2$ Each pairwise score is the mean across 180 models for each dataset and correlation strength. First column is the unsupervised baseline without any fast adaptation and the second column shows that fast adaption using a one-hidden layer MLP reduces these correlations with as little as 1000 labels when sampled from the uncorrelated dataset.   
+
+<table><tr><td>dataset</td><td>labels</td><td>0</td><td>1000</td></tr><tr><td>Shapes3D (D) σ = 0.2</td><td>object color - object size</td><td>0.3</td><td>0.16</td></tr><tr><td>Shapes3D (D) σ = 0.2</td><td>median uncorrelated pairs</td><td>0.07</td><td>0.07</td></tr><tr><td>Shapes3D (E)σ = 0.2</td><td>object color - azimuth</td><td>0.25</td><td>0.2</td></tr><tr><td>Shapes3D (E) σ = 0.2</td><td>median uncorrelated pairs</td><td>0.1</td><td>0.11</td></tr><tr><td>Shapes3D (A) σ = 0.2</td><td> object size - azimuth</td><td>0.37</td><td>0.26</td></tr><tr><td>Shapes3D (A) σ = 0.2</td><td>median uncorrelated pairs</td><td>0.09</td><td>0.1</td></tr></table>
+
+The value of the changing variable is then sampled uniformly in the second observation. Note that this still means that the vast majority of sampled pairs exhibit correlated FoV. This is depicted in the substantially lifted disentanglement scores shown in Fig. 19 as well as some selected latent space visualizations that show strong axis alignment in Fig. 20. We consistently observe much better disentangled models, often achieving perfect DCI score irrespective of correlations in the data set. The latent spaces tend to strongly align their coordinates with the ground truth label axis. We chose 10 random seeds per configuration in this study, yielding 720 models in total.
+
+Scenario I-2: Let us assume $C _ { 1 }$ causes $C _ { 2 }$ in our examples, which manifests as the studied linear correlations. Within this setting we cannot sample uniformly in $C _ { 2 }$ if we intervene (or “fix”) all factors except for this causal factor. Intervening on all factors but $C _ { 1 }$ , however, allows us to sample any value in $C _ { 1 }$ as it is not causally affected by $C _ { 2 }$ . To test the hypothesis that this constraint still allows for disentangling the correlation, we trained on Shapes3D (A) and sample pairs consistent with this causal model. Besides observing visually disentangled factors in the latent traversals, we show a summary of our results in Fig. 21 with the same significant improvements regarding disentangling the correlated FoVs. Besides the above correlation strengths, we additionally trained the same models using a very strong correlation of $\sigma = 0 . 1$ , yielding 300 models trained in this study.
+
+![](images/55a02791c8d0b50de5188e3389cffb5031ed780b7e45e63f8426086856db4b98.jpg)  
+Figure 17: Standard disentanglement metrics evaluated for the weakly supervised scenario using correlated observational data.
+
+![](images/f997e0d82272cc859cc5242bc563cb88283e962db14a373f621b6f0f37e42519.jpg)  
+Figure 18: Left: For the weakly supervised scenario using correlated observational data trained models on Shapes3D (A), (D) and (E) correlating object color and azimuth learn consistently improved, often perfect, disentangled representation across all correlation strengths. Right: Latent dimensions of a best DCI model trained on strongly correlated observational data. Representations are perfectly axis-aligned with respect to both of the correlated variables ground truth values (right).
+
+![](images/a25e94ba9291585646c3768650c399555dbe873fdf0046bcbe0de8f4bd961538.jpg)  
+Figure 19: Standard disentanglement metrics evaluated on the correlated training sets for the weakly supervised scenario with intervening capabilities (I-1).
+
+![](images/0a38d07ea4f3e624025b25e402e0f7d956eb38708636435fdc8e5dcdc86b34fd.jpg)  
+Figure 20: Left: For the weakly supervised scenario with intervening capabilities (I-1) trained models on Shapes3D (A), (D) and (E) correlating object color and azimuth learn consistently improved, often perfect, disentangled representation across all correlation strengths. Right: Latent dimensions of a best DCI model with strong correlation (0.2). Representations are perfectly axis-aligned with respect to both of the correlated variables ground truth values (right).
+
+![](images/c22f1e0add00466036990823304d24372b6bfc1093ac60cd57568df61d0bf979.jpg)  
+Figure 21: Disentanglement metrics (top), unfairness scores (middle) and latent spaces (bottom) show strong disentanglement using weak supervision with intervening capabilities (I-2) - even under the stronger assumption that sampling of observation pairs follow its causal generative model. We show the learnt latent space encoding of the two correlated factors of variation for a model with $\sigma = 0 . 1$ and low reconstruction loss.

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+        "text": "Despite impressive progress in the last decade, it still remains an open challenge to build models that generalize well across multiple tasks and datasets. One path to achieve this is to learn meaningful and compact representations, in which different semantic aspects of data are structurally disentangled. The focus of disentanglement approaches has been on separating independent factors of variation despite the fact that real-world observations are often not structured into meaningful independent causal variables. In this work, we bridge the gap to real-world scenarios by analyzing the behavior of most prominent methods and disentanglement scores on correlated data in a large scale empirical study (including 4260 models). We show that systematically induced correlations in the dataset are being learned and reflected in the latent representations, while widely used disentanglement scores fall short of capturing these latent correlations. Finally, we demonstrate how to disentangle these latent correlations using weak supervision, even if we constrain this supervision to be causally plausible. Our results thus support the argument to learn independent mechanisms rather than independent factors of variations. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Due to the induced structure, disentangled representations promise generalization to unseen scenarios (Higgins et al., 2017b), increased interpretability (Adel et al., 2018; Higgins et al., 2018) and faster learning on downstream tasks (van Steenkiste et al., 2019; Locatello et al., 2019a). While the advantages of disentangled representations have been well established, they generally assume the existence of natural factors that vary independently within the given dataset, which is rarely the case in real-world settings. As an example, consider a scene with a table and some chairs (see Fig. 1). The higher-level factors of this representation are in fact correlated and what we actually want to infer are independent (causal) mechanisms (Peters et al., 2017; Parascandolo et al., 2018; Suter et al., 2019; Goyal et al., 2019). ",
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+        "type": "text",
+        "text": "A complex generative model can be thought of as the composition of independent mechanisms or “causal” modules, which generate highdimensional observations (such as images or videos). In the causality community, this is often considered a prerequisite to achieve representations which are robust to interventions upon variables determined by such models (Peters et al., 2017). One particular instantiation of this idea in the machine learning community is the notion of disentangled representations (Bengio et al., 2013). The goal of disentanglement learning is to find a representation of the data which captures all the ground-truth factors of variation (FoV) independently. ",
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+        "image_caption": [
+            "Figure 1: While in principle we consider the presence of the objects (coffee cup, table, chair) to be independent mechanisms, they tend to appear together in observed data. "
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+        "text": "Despite the recent growth of the field, the performance of state-of-the-art disentanglement learners remains unknown for more realistic settings where FoV are correlated during training. Given the potential societal impact in the medical domain (Chartsias et al., 2018) or fair decision making (Locatello et al., $2 0 1 9 \\mathrm { a }$ ; Madras et al., 2018; Creager et al., 2019), the evaluation of the usefulness of disentangled representations trained on correlated data is of high importance. ",
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+        "type": "text",
+        "text": "To go beyond the highly idealized settings considered thus far, we conducted a large scale empirical study to systematically assess the effect of induced correlations between pairs of factors of variation in training data on the learned representations. To provide a qualitative and quantitative evaluation, we investigate multiple datasets with access to ground-truth labels. Moreover, we study the generalization abilities of the representations learned on correlated data as well as their performance in particular for the downstream task of fair decision making. ",
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+        "text": "Contributions. Our main contributions can be summarized as follows: ",
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+        "text": "• We present the first large-scale empirical study (4260 models)1 that examines how modern disentanglement learners perform when ground truth factors of the observational data are correlated.   \n• We find that factorization-based inductive biases are insufficient to learn disentangled representations from observational data. Existing methods fail to disentangle correlated factors of variation, resulting in correlated latent space dimensions. Moreover, standard disentanglement metrics do not reveal these persisting correlations.   \n• We investigate the usefulness of semi-supervised and weakly-supervised approaches to resolve latent entanglement. For the latter setting, we focus on multiple observational and interventional distributions. ",
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+        "text": "2 BACKGROUND AND RELATED WORK ",
+        "text_level": 1,
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+        "text": "Disentanglement. Current state-of-the-art disentanglement approaches use the framework of variational auto-encoders (VAEs) (Kingma & Welling, 2014; Rezende et al., 2014). The (high-dimensional) observations $_ { \\textbf { \\em x } }$ are modelled as being generated from some latent features $_ { z }$ with chosen prior $p ( z )$ according to the probabilistic model $p _ { \\theta } ( \\pmb { x } | \\pmb { z } ) p ( \\pmb { z } )$ . The generative model $p _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } | \\boldsymbol { z } )$ as well as the proxy posterior $q _ { \\phi } ( \\pmb { z } | \\pmb { x } )$ can be parameterized by neural networks, which are optimized by maximizing the variational lower bound (ELBO) of $\\log p ( \\pmb { x } _ { 1 } , \\ldots , \\pmb { x } _ { N } )$ . ",
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+        "img_path": "images/693f1f177d04eee7f59d20593a9c4ed0b62faa512ca5309428478a6b55c667e9.jpg",
+        "text": "$$\n\\mathcal { L } _ { V A E } = \\sum _ { i = 1 } ^ { N } \\mathbb { E } _ { q _ { \\phi } ( z | \\mathbf { x } ^ { ( i ) } ) } [ \\log p _ { \\theta } ( \\mathbf { x } ^ { ( i ) } | z ) ] - D _ { K L } \\big ( q _ { \\phi } ( z | \\mathbf { x } ^ { ( i ) } ) \\| p ( z ) \\big )\n$$",
+        "text_format": "latex",
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+        "text": "The above objective does not enforce any structure on the latent space, except for similarity (in KL-divergence) to the prior $p ( z )$ (typically chosen as an isotropic Gaussian). However, the structure and semantic meaning of latent representations can be relevant to study generation properties. Consequently, various proposals for structure-imposing regularization and commonly used evaluation metrics measuring different notions of disentanglement of the learned representations have been made (Higgins et al., 2017a; Kim & Mnih, 2018; Burgess et al., 2018; Kumar et al., 2018; Chen et al., 2018; Eastwood & Williams, 2018; Mathieu et al., 2018). Recently, it has been shown that unsupervised disentangling by optimising marginal likelihood in a generative model is impossible without further inductive bias (Locatello et al., 2019b). To address this theoretical limitation, methods have been proposed that do not require explicitly labelled data but only some weak labeling information (Locatello et al., 2020; Shu et al., 2019). Ideas related to disentangling the factors of variations date back to the non-linear ICA literature (Bach & Jordan, 2002; Comon, 1994; Jutten & Karhunen, 2003; Hyvärinen & Pajunen, 1999; Hyvarinen et al., 2019; Hyvarinen & Morioka, 2016; Gresele et al., 2019). Recent work combines non-linear ICA with disentanglement (Khemakhem et al., 2020; Sorrenson et al., 2020; Klindt et al., 2020) ",
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+        "type": "text",
+        "text": "Correlations. A set of random variables $X _ { i = 1 , \\ldots , n }$ is not independent, if and only if their joint distribution does not factorize ",
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+        "text": "$$\nP ( X _ { 1 } , X _ { 2 } , \\ldots , X _ { n } ) \\neq \\prod _ { i = 1 } ^ { n } P ( X _ { i } ) .\n$$",
+        "text_format": "latex",
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+        "text": "In this case, we speak of dependence between the random variables, also commonly referred to as correlation.2 Correlation between two variables can either stem from a direct causal relationship (one variable affects the other), but can also be due to different unobserved circumstances (confounders) affecting both. Real-world datasets display many of these (“spurious” and often a priori unknown) correlations (Geirhos et al., 2020). However, most work on learning disentangled representations assumes that there is an underlying set of independent ground truth variables that govern the generative process of observable data. These methods are hence predominantly evaluated on data that obey independence in the true factors of variation, which we then consider to be the correct factorization. In the real world, the observation generating process is likely not always as clearly “disentangled” and we do expect correlations in the collected datasets. It is thus an open question to what degree existing inductive biases from the encoder/decoder architecture, but more importantly the dataset biases, affect the learned representation. In our experiments, we introduce dataset correlations in a controlled manner to understand to what degree state-of-the-art approaches can cope with such correlations. We believe these correlations to reflect a major feature of more realistic environments. ",
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+        "img_path": "images/8434d800ad72d18ee910812aa691bbcbaf1629e87d4199312f4297e9514c68d5.jpg",
+        "image_caption": [
+            "Figure 2: Disentanglement metrics show no clear trend along different correlation strengths (each violin represents 180 models, higher $\\sigma$ indicates less correlation). "
+        ],
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+        "text": "Other Related Work. Most popular datasets in the disentanglement literature exhibit perfect independence in their FoV. At some level this is sensible as it reflects the underlying assumption in the inductive bias being studied. However, this assumption is unlikely to hold in practice as shown by Li et al. (2019), who propose methods based on a pairwise independence assumption instead. The literature so far has not thoroughly measured how popular inductive biases such as factorized priors behave when learning from correlated datasets, although several smaller experiments along these lines can be acknowledged. Chen et al. (2018) studied correlated 3DFaces (Paysan et al., 2009) by fixing all except three factors in which the authors conclude that the $\\beta$ -TC-VAE regularizer can help to disentangle imposed correlations. Brekelmans et al. (2019) show that Echo noise results in superior disentanglement compared to standard betaVAE in a small experiment on a downsampled dSprites variant where randomly selected factor pairs are excluded. However, the latent structure was not studied in detail; our findings suggest that global disentanglement metrics are insufficient to diagnose issues when models learn from correlated data. Creager et al. (2019) based some of the evaluations of a proposed new autoencoder architecture in the fairness context on a biased dSprites variant and Yang et al. (2020) study a linear SCM in a VAE architecure on datasets with dependent variables. However, their studies focused on representation learners that require strong supervision via FoV labels at train time. ",
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+    {
+        "type": "text",
+        "text": "3 THE EFFECT OF CORRELATED DATA ON DISENTANGLEMENT LEARNERS ",
+        "text_level": 1,
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+        "text": "In this section, we want to present the key findings from our empirical study of unsupervised disentanglement learning on a particular variant of correlated data. We start by outlining the experimental design of our study in Section 3.1. Based on this, we analyze the latent spaces in Section 3.2 and find that factorization-based inductive biases are insufficient to learn disentangled representations from observational data. Persisting pairwise correlations in the latent space are not sufficiently revealed by standard disentanglement metrics that might be particularly relevant and problematic for fairness applications. Finally, in Section 3.3, we show extrapolation and generalization capabilities of the learned representations towards unseen factor combinations due to the induced correlations. ",
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+        "type": "text",
+        "text": "3.1 EXPERIMENTAL DESIGN ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "For our first experiments we introduce correlations between single pairs of factors of variation on the following three datasets: Shapes3D with object size and azimuth (denoted “A”), dSprites with orientation and X-position (“B”) and finally the real-world observations dataset MPI-3D with first and second degree of freedom $\\left( \\ \\stackrel { 6 6 } { \\ C } \\right)$ . We focus on linear correlations with Gaussian noise between the two variables, which we denote by $c 1 , c 2$ . The ground truth factors for $c 1 , c 2$ take values $z _ { c 1 } \\in \\{ 0 , \\dots , z _ { c 1 } ^ { \\operatorname* { m a x } } \\}$ and $z _ { c 2 } \\in \\{ 0 , \\dots , z _ { c 2 } ^ { \\operatorname* { m a x } } \\}$ respectively. We then parameterize correlations by sampling the training dataset from the joint $P ( z _ { c 1 } , z _ { c 2 } ) \\propto \\mathcal { N } ( z _ { c 2 } - \\alpha z _ { c 1 } , \\sigma )$ where $\\alpha = z _ { c 2 } ^ { m a x } / z _ { c 1 } ^ { m a \\bar { x } }$ The strength of the correlations can be tuned by $\\sigma$ , for which we choose 0.2, 0.4, 0.7 in normalized units with respect to the range of values in $z _ { c 1 , c 2 }$ . Lower $\\sigma$ indicates stronger correlation. See Fig. 5 for an example of $P ( z _ { c 1 } , z _ { c 2 } )$ for correlating azimuth and object size in Shapes3D with $\\sigma = 0 . 2$ . Additionally, we study the uncorrelated limit $\\sigma = \\infty$ ), which amounts to the case typically studied in the literature. We train the same six VAE methods as discussed in Locatello et al. (2019b), including $\\beta$ -VAE, FactorVAE, AnnealedVAE, DIP-VAE-I, DIP-VAE-II and $\\beta$ -TC-VAE, each with six hyperparameter settings. Each method has been trained using five different random seeds. All remaining factors of variation are sampled uniformly at random. This first study sums up to a total of 2160 trained models, or 180 models per dataset and correlation strength3. Appendix A describes additional implementation details. ",
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+        "image_caption": [
+            "Figure 3: We show latent traversals (left) of the best DCI score model among all 180 trained models with strongest correlation $\\sigma = 0 . 2$ ) on Shapes3D (A). The traversals in latent code dimensions 4 and 8 (highlighted in black), suggest that these dimensions encode a mixture of azimuth and object size, reflecting one major axis along the correlation line of the joint distribution and one smaller, locally orthogonal axis. This is supported by a heat map of the GBT feature importance matrix of this model (right) indicating an entanglement of azimuth and object size encoded into both latent codes. "
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+        "text": "3.2 CAN UNSUPERVISED METHODS ACHIEVE DISENTANGLEMENT OF CORRELATED DATA? ",
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+        "text": "Shortcomings of existing metrics. Following recent studies, we evaluate the trained models with the help of a broad range of disentanglement metrics that aim at quantifying overall success by a single scalar measure. Perhaps surprisingly, Fig. 2 shows no clear trend among all implemented disentanglement scores w.r.t. correlation strength (see Fig. 9 in the Appendix for the full result across all datasets and metrics). The metrics have been evaluated by sampling from the correlated data distribution although they do not differ substantially when evaluated on the uncorrelated distribution. However, as we will demonstrate along the following analysis, the latent spaces in this setting show some characteristic differences when trained on a strongly correlated pair of FoVs. We thus argue that common disentanglement metrics are limited when correlations are introduced into the training data. To conduct a more careful analysis of the inductive data bias applied on the learned representations we will instead evaluate pairwise metrics. Note that regarding BetaVAE and FactorVAE this observed trend is to some degree expected as they would yield perfect disentanglement scores even if we would take the correlated ground truth factors or a linear transformation in the case of BetaVAE as the representation. ",
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+        "text": "Latent structure and pairwise entanglement. We start by analysing latent traversals of some trained models on Shapes3D (A). For strong correlations $\\sigma = 0 . 2$ and $\\sigma = 0 . 4$ ), we typically observe trained models with two latent codes encoding the two correlated variables simultaneously. In these cases, one of the latent codes corresponds to data along the major axis of the correlation line whereas the other latent code dimension manifests in an orthogonal change of the two variables along the minor axis. Still, a full traversal of the code corresponding to the minor axis often seems to cover only observations within the correlation line. Fig. 3 (left) shows this effect for the latent space of a model trained on Shapes3D (A) with strongest correlation $\\sigma = 0 . 2$ ). ",
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+        "table_caption": [
+            "Figure 4: Left: Pairwise entanglement scores help to uncover still existent correlations in the latent representation. Left: Mean of the pairwise entanglement scores for the correlated pair (red) and the median of the uncorrelated pairs. We see that stronger correlation leads to statistically more entangled latents compared to the baseline score without correlation (blue). Right: The same behavior can be seen for the unfairness score between the correlated pair of factors. "
+        ],
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+        "table_body": "<table><tr><td>Correlation strength</td><td></td><td>g=0.2</td><td>g=0.4</td><td>g=0.7</td><td>σ=∞ (uc)</td></tr><tr><td rowspan=\"2\">Shapes3d (A)</td><td>object size - azimuth</td><td>0.38</td><td>0.26</td><td>0.13</td><td>0.08</td></tr><tr><td>median uncorrelated pairs</td><td>0.09</td><td>0.09</td><td>0.09</td><td>0.08</td></tr><tr><td rowspan=\"2\">dSprites (B)</td><td>orientation - position X</td><td>0.17</td><td>0.16</td><td>0.14</td><td>0.11</td></tr><tr><td>median uncorrelated pairs</td><td>0.13</td><td>0.13</td><td>0.13</td><td>0.13</td></tr><tr><td rowspan=\"2\">MPI3D (C)</td><td>First DOF - Second DOF</td><td>0.2</td><td>0.19</td><td>0.17</td><td>0.16</td></tr><tr><td>median uncorrelated pairs</td><td>0.16</td><td>0.16</td><td>0.15</td><td>0.15</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "To quantify this observation, we analyze the importance of individual latent codes in predicting the value of a given ground truth FoV. An importance weight for each pair of $\\{ \\mathrm { F o V } ,$ latent dimension} is computed by training a gradient boosting trees (GBT) classifier to predict the ground truth labels from the latents (10,000 samples). In the right panel of Fig. 3, we compute these importance weights for the model used to generate traversals in the left panel. The corresponding evaluation for a model trained on the same dataset with weak correlation does not reveal this feature visually (see Fig. 10 in the Appendix). ",
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+    {
+        "type": "text",
+        "text": "To support this claim for a larger set of models, we calculate a pairwise entanglement score that allows us to measure how difficult it is to separate two factors of variation from their latent codes. This computation involves grouping FoV into pairs based on an ordering of their pairwise mutual information or GBT feature importance between latents and FoV; we defer to Appendix A for a detailed description of this procedure. Figure 4 (left) shows that across all datasets the pair of correlated FoV has a higher score than the median of all other pairs, indicating that they are harder to disentangle. This threshold decreases with weaker correlation and the pair becomes easier to disentangle for weaker correlations $( \\sigma \\ge 0 . 7 )$ . These findings suggest that the models still manage to disentangle correlated factors if the correlation is not too strong. ",
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+    {
+        "type": "text",
+        "text": "Finally, correlations between variables are of crucial importance in fairness applications motivating an additional investigation on ramifications of these entangled latent spaces. In this setting we are interested in the unfairness of predicting the second correlated variable while the first correlated variable is considered being a protected or sensitive attribute. In the following, we use a variant of demographic parity (Dwork et al., 2012) that computes pairwise mutual information between latents and FoV (Locatello et al., 2019a). In Fig. 4 (right) we evaluate this score when correlations are present within the data in the case of Shapes3D (A). Unfairness tracks correlation strength in this scenario. These results suggest that we cannot expect disentangled representations learned unsupervisedly to help reduce unfairness beyond the benefits discussed in Locatello et al. (2019a). More comprehensive results, that support the finding that the correlated pair is statistically more entangled in the latent representations across all unsupervised experiments and datasets is provided in Appendix B.1. ",
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+        "type": "text",
+        "text": "Summary. We find that existing methods fail to learn disentangled representations of correlated factors of variation, and moreover that standard disentanglement metrics are insufficient to reveal these troublesome pairwise entanglements in the latent space. ",
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+    {
+        "type": "text",
+        "text": "3.3 GENERALIZATION PROPERTIES",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this Section, we aim to understand how the trained models perform on unseen training data far away from the correlation line; out-of-distribution (OOD) w.r.t. the train distribution. We analyse this capability for the model from Fig. 3. In this model, the remaining factors seem to be disentangled well enough to only focus on the two latent dimensions encoding the entangled variables. ",
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+    {
+        "type": "text",
+        "text": "As a first test, we sample observations from the FoV then set object size and azimuth to six distinct configurations of zero probability, see Fig. 5 (left). The trained model is capable of reconstructing these observations despite never having encountered these configurations or neighbors thereof during training. This suggests that the encoder maps representations to a meaningful point in the latent space from which the decoder is equally capable of generating expected observations. To test this hypothesis further, we analysed latent traversals originating from these OOD points and observe that changes in the remaining factors reliably yield the expected reconstructions. Traversals with respect to the two entangled latent codes continue to encode object size and azimuth. ",
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+        "page_idx": 4
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+    {
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+        "img_path": "images/3a4907b6a6815be9ba92ce3f27dba690c6be01f16deae4a661172b14204a1d76.jpg",
+        "image_caption": [
+            "Figure 5: Generalization to out-of-distribution (OOD) test data. Left: Reconstructions of observations the model has never seen during training. Right: Latent space distribution of the two entangled dimensions. Circles without edges indicates encoded data from the (correlated) training distribution, while circles with edges indicate encoded OOD data where the correlation pattern is broken. "
+        ],
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+    {
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+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "To fully understand this models’ generalization properties we visualise the occupied latent space spanned by the two identified dimensions encoding both correlated factors in Fig. 5 (right). We are particularly interested in where these points are located with respect to the ground truth value of each correlated variable, depicted via color. The two sets of depicted points are (1) latent codes sampled from the correlated training data and (2) latent codes sampled with a (object size, azimuth) configuration that has zero probability under the correlated training distribution. We observe that contours of equal color (ground truth) are not aligned with the latent axes. This indicates that the two latent dimensions encode both FoV at the same time. Likewise, we can understand the generalization capabilities of this model far away from the trained data. Extreme configurations such as small azimuth and large object size are encoded to space regions corresponding to the intersections of the manifolds with constant value for each correlated variable. This shows that all out-of-distribution points are encoded in this representation space in a way that obeys the natural ordering of each respective factor. This behaviour remains even in cases where the trained latent space does not mirror the default value ordering as stored in our ground truth table. For this we additionally trained 360 models on two additional Shapes3D variants each, where we strongly correlated object color - object size (“D”) and object color - azimuth (“E”) respectively $\\sigma = 0 . 2$ and $\\sigma = 0 . 4$ ). As the color values do not allow for a unique natural ordering, the trained models do often encode a different color manifold ordering into the latent space. In Appendix B.1, we show some of their characteristic latent space visualizations with similar extrapolation and generalization capabilities. We conclude from these results that disentanglement methods can generalize towards unseen FoV configurations as long as each factor value is contained in the training data within a different configuration. ",
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+    {
+        "type": "text",
+        "text": "4 FINDING THE RIGHT FACTORIZATION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The results from Section 3 illustrate the limitations of state-of-the-art unsupervised methods on correlated data (and thus real-world observational data). We now investigate the usefulness of several approaches for mitigating pairwise correlations in the latent code. We begin with a post-hoc procedure in Section 4.1 that uses limited label information on the ground truth factors and show that it achieves a substantial correction of the pairwise latent correlation. We then consider a recently proposed approach leveraging recent advances in weakly supervised disentanglement learning that applies directly at train time. As will be seen in Section 4.2, this method results in substantially more disentangled representations, even when applied on correlated data from different sampling scenarios. ",
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+    {
+        "type": "table",
+        "img_path": "images/f65dbd59fd8e99fc5f75f7c34b471caa2596a6b42e1ea9fe5ad6e417196228e7.jpg",
+        "table_caption": [
+            "Figure 6: Fast adaption with few labels: Pairwise entanglement scores for correlated FoV pair in Shapes3D (A). The correlated pair is highlighted (red). Zero labels reflects the unsupervised baseline without any fast adaptation. Growing number of labels show that fast adaption using linear regression reduces these correlations with as little as 100 labels (blue). Reported pairwise scores are averaged over 180 models per correlation strength. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>#Labels</td><td></td><td>0</td><td>100</td><td>1000</td><td>10000</td></tr><tr><td>Shapes3D (A) σ = 0.2</td><td>object size - azimuth</td><td>0.38</td><td>0.17</td><td>0.15</td><td>0.15</td></tr><tr><td>Shapes3D (A) σ= 0.2</td><td>median uncorrelated pairs</td><td>0.09</td><td>0.08</td><td>0.07</td><td>0.07</td></tr><tr><td>Shapes3D (A) σ = 0.4</td><td> object size - azimuth</td><td>0.26</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Shapes3D (A) σ = 0.4</td><td>median uncorrelated pairs</td><td>0.09</td><td>0.08</td><td>0.08</td><td>0.08</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 POST-HOC ALIGNMENT CORRECTION WITH FEW LABELS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "When a limited number of FoV labels $\\left\\{ \\mathbf { y } _ { i } \\right\\}$ can be accessed, a reasonable option for resolving entangled dimensions of the latent code is by fast adaptation. To identify the two entangled dimensions $( z _ { i } , z _ { j } )$ we look at the maximum feature importance for a given FoV from a GBT trained using these labels only. We then train a substitution function using supervised learning to replace these two dimensions with the predicted ground truth label. Crucially, both steps of this procedure rely on the same FoV labels, which should be as few as possible. In Fig. 6 we show the pairwise entanglement score of the correlated FoVs under this fast adaptation with a linear regression as the substitution function, which succeeds with as few as 100 labels, corresponding to less than $0 . 0 2 \\%$ of all data points in Shapes3D. However, fast adaptation with linear regression substitution fails in some settings: when no two latent dimensions encode the applied correlation isolated from the other latent codes, or when the correlated variables do not have a unique natural ordering (e.g. color or categorical variables). To address this, a nonlinear substitution function such as a MLP can reduce this pairwise entanglement to a certain degree (see additional results in Appendix B.2). ",
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+    {
+        "type": "text",
+        "text": "We find that the efficacy of fast adaptation depends on the level of disentanglement of the representations with respect to all the other factors. This implies that if the representation is well disentangled at the start of the fast adaption procedure, it is possible to achieve a perfectly disentangled model (according to our previous visual and quantitative evaluations). However, if all FoV are entangled at the beginning, the fast adaption method will have little effect. Finally, we note that model selection is impossible in a purely unsupervised manner based on any of the used disentanglement metrics, as they all require labeled ground truth data. These shortcomings shall be resolved by the following method capable of disentangling the correlated factors of variation much more reliably. ",
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+    {
+        "type": "text",
+        "text": "4.2 ALIGNMENT DURING TRAINING USING WEAK SUPERVISION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Since the unsupervised disentangling by optimising marginal likelihood in a generative model is impossible (Locatello et al., 2019b, Theorem 1), inductive biases like grouping information (Bouchacourt et al., 2018) or access to labels (Locatello et al., 2019c) is required. Changes in natural environments, which typically correspond to changes of only a few underlying factors of variation, provide a weak supervision signal for representation learning algorithms (Goyal et al., 2019; Földiák, 1991; Schmidt et al., 2007; Bengio et al., 2019). Without correlations it has been shown that this weak supervision helps in learning much more disentangled representations (Locatello et al., 2020; Shu et al., 2019). Locatello et al. (2020) showed access to observations which display differences in a known number of factors of variation (without knowing which ones specifically) is sufficient to learn disentangled representation. These additional weak assumptions render the generative model identifiable in contrast to unsupervised disentanglement. This kind of extra knowledge might be available in certain settings 4, e.g., in temporarily close frames from a video of a moving robot arm where some factors remain unchanged. Hence, we want to investigate the usefulness of such a weakly-supervised method applied in various scenarios when training data is correlated. Specifically we implement the Ada-GVAE variant of Locatello et al. (2020) that was shown to allow for model selection via the (unsupervised) reconstruction loss. The method requires a pair of observations that differs in a known number of factors, without knowing which in particular. ",
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+    {
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+        "img_path": "images/143ef6a989097e83190d62e51d5d5cd9acff40e9f5e8469f5605bbda6b516512.jpg",
+        "image_caption": [
+            "Figure 7: Weak supervision: Left: With weak supervision trained models on Shapes3D correlation object size and azimuth learn consistently improved, often perfect, disentangled representations across all correlation strengths. Middle: Unfairness scores between correlated FoVs are much smaller (see scale). Right: Latent dimensions of a best DCI model with strong correlation $\\mathit { \\check { \\sigma } } = 0 . 2 $ ). Representations are axis-aligned with respect to both of the correlated variables ground truth values (right). Below: Traversals of a model trained with strong correlation $( \\sigma = 0 . 2$ ) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Weak supervision mitigates pairwise latent entanglement. We trained the three correlated Shapes3D variants (A, D, E) with pairwise correlations between object color, object size and azimuth with the same correlation strength settings. Due to the definition of the regularizer we limit this study to the $\\beta$ -VAE models with the same 6 hyperparameters and use 5 random seeds, yielding 360 additional models. For the generation of pairs we study the case where the difference in the observation pairs is present in one random FoV. Whenever we sample the difference to be in one of the correlated FoV, its respective value in each pair is drawn from the probability distribution conditioned on the other correlated FoV. This means the difference in this factor is typically very small and depends on the correlation strength. Note that this procedure assures that constructed pairs are consistent with the observational data such that the correlation is never broken. Fig. 7 summarizes the weak supervision results when imposing correlations in object size and azimuth. We consistently observe much better disentangled models, often achieving perfect DCI score irrespective of correlations in the dataset. The latent spaces tend to strongly align their coordinates with the ground truth label axis. Finally, weak-supervision reduces unfairness relative to the unsupervised baseline, and occasionally achieves zero unfairness score. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "These results suggest that weak supervision can provide a strong inductive bias capable of finding the right factorization and resolving spurious correlations for datasets of unknown degree of correlation. As a prominent example, this is an issue in the fairness context where real-world datasets often display unknown discriminatory correlations. Additional results on the other datasets can be found in Appendix B.2 including two additional scenarios where one has intervening capabilities to generate the pair. We consistently observe the same strong trends regarding disentangled correlations in all of the above studies using weak supervision. ",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have presented the first large-scale empirical study examining how modern disentanglement learners cope with correlated observational data. We find that existing methods fail to learn disentangled representations of correlated factors of variation, and moreover that standard disentanglement metrics are insufficient to reveal these pairwise entanglements. We discussed practical implications for downstream tasks like fair decision making. Finally, we demonstrate how to correct for these latent correlations via various weakly supervised training scenarios. Our results thus support the importance and usefulness of learning independent mechanisms rather than independent factors of variations (Schölkopf, 2019; Parascandolo et al., 2018; Suter et al., 2019; Goyal et al., 2019). Besides the simple correlations studied in this work, future work is needed to address the open question whether these results extend to more complex nonlinear correlations and settings where many more variables are correlated simultaneously. ",
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+    {
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+        "text": "Sjoerd van Steenkiste, Francesco Locatello, Jürgen Schmidhuber, and Olivier Bachem. Are disentangled representations helpful for abstract visual reasoning? arXiv preprint arXiv:1905.12506, 2019. ",
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+        "text": "Mengyue Yang, Furui Liu, Zhitang Chen, Xinwei Shen, Jianye Hao, and Jun Wang. Causalvae: Structured causal disentanglement in variational autoencoder. arXiv preprint arXiv:2004.08697, 2020. ",
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+    {
+        "type": "text",
+        "text": "A IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            106,
+            436,
+            121
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Unsupervised Disentanglement methods. For reasons of comparison, the considered disentanglement methods in this work cover the full collection of state-of-the-art approaches in disentanglement_lib from Locatello et al. (2019b) based on representations learned by VAEs. The set contains six different methods that enforce disentanglement of the representation by equipping the loss with different regularizers that aim at enforcing the special structure of the posterior aggregate encoder distribution. A detailed description of the regularizer forms used in this work, specifically $\\beta$ -VAE (Higgins et al., 2017a), FactorVAE (Kim & Mnih, 2018), AnnealedVAE (Burgess et al., 2018), DIP-VAE-I, DIP-VAE-II (Kumar et al., 2018) and $\\beta$ -TC-VAE (Chen et al., 2018) is provided in (Locatello et al., 2019b). We use the same encoder architecture with 10 latent dimensions for every model. ",
+        "bbox": [
+            174,
+            136,
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+            275
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Evaluation metrics. To measure disentanglement of a learned representation, various metrics have been proposed, each requiring access to the ground truth labels. The BetaVAE score is based upon the prediction of a fixed factor from the disentangled representation using a linear classifier (Higgins et al., 2017a). The FactorVAE score is intended to correct for some failures of the former by utilizing majority vote classifiers based on a normalized variance of each latent dimension (Kim & Mnih, 2018). The SAP score represents the mean distance between the classification errors of the two latent dimensions that are most predictable (Kumar et al., 2018). For the MIG score, one computes the mutual information between the latent representation and the ground truth factors and calculates the final score using a normalized gap between the two highest MI entries for each factor. Finally, a disentanglement score proposed by Eastwood & Williams (2018), often referred to as DCI score, is calculated from a dimension-wise entropy reflecting the usefulness of the dimension to predict a single factor of variation. ",
+        "bbox": [
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+            457
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Unfairness between a pair of FoV: The scores reported are based on a notion of demographic parity for predicting a target variable $y$ given a protected and sensitive variable $s$ . Both $y$ and $s$ can be associated with a factor of variation here. Rather than using the global total variation average as defined in Locatello et al. (2019a), we report the individual demographic parities for the correlated factors specifically. ",
+        "bbox": [
+            174,
+            464,
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+            534
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Pairwise entanglement metric: In order to further analyze the differences between the GBT feature importance matrices between latent code and encoded factor of variation, we view them as weights on the edges of a bipartite graph encoding the statistical relation between each factor of variation and code. We can now delete all edges with weight smaller than some threshold and count (i) how many factors of variation are connected with at least a latent code and (ii) the number of connected components with size larger than one. We can then compute which factors are merged at which threshold. Factors that are merged at lower threshold are more entangled in the sense that are more statistically related to a shared latent dimension. This computation can be not only based on the GBT feature importances but likewise on weight matrices inferred from the Mutual Information. ",
+        "bbox": [
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+            541,
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+            666
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Joint distributions of correlated factors in datasets: In Fig. 8 we show the joint probability distributions of the correlated pair of FoV for all datasets and correlation strengths considered in this study. Dataset A, B and C were designed with correlated factors of variation that are ordinal for a natural visual interpretation of the traversals. In contrast, datasets D and E contain a correlated factor of variation that has no such natural ordering. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B ADDITIONAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.1 SECTION 3 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            801,
+            294,
+            816
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Shortcomings of existing metrics. Following recent studies, we evaluate the trained models with the help of a broad range of disentanglement metrics that aim at quantifying overall success by the help of a single scalar measure. Perhaps surprisingly, as can be seen in Fig. 9, there is no clear trend among all implemented disentanglement scores w.r.t. correlation strength. The metrics have been evaluated by sampling from the correlated data distribution although they do not differ substantially when evaluated on the uncorrelated distribution. We thus argue that commonly used methods are insufficient to provide insight into the latent space learned. ",
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+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/f94feea5848132b951f473cf53dfa9d1dc6ef33b49735fc543b46e24ae0ecf97.jpg",
+        "image_caption": [
+            "Figure 8: Probability distributions for sampling training data in the correlated pair of FoVs in the respective datasets (A, B, C, D, E) considering correlation strengths of $\\sigma = 0 . 2$ , $\\sigma = 0 . 4$ , $\\sigma = 0 . 7$ and $\\sigma = \\infty$ , the uncorrelated limit (from left to right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/befc523d57a6f4ddbe357b7dbbffc7e45da3efd7984297017b532022ddc94e3f.jpg",
+        "image_caption": [
+            "Figure 9: Standard disentanglement metrics evaluated on the correlated and uncorrelated (uc) training set showing no clear trend for different correlation strengths. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
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+        "img_path": "images/60b117850a53daf06a30455a78996762a9ba12708937fe8953bef3fc11a32782.jpg",
+        "image_caption": [
+            "Figure 10: We show latent traversals (left) of the best DCI score model among all 180 trained models with weak correlation $\\sigma = 0 . 7$ ) in object size and azimuth. The traversals in latent code no 3 and 7/8 (highlighted in black), suggest that these dimensions encode no mixture of azimuth and object size compared to the models with stronger correlation. This is supported by the GBT feature importance matrix of this model (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/92fb788e2b4ef4bef433d594e9007b2ccdfd2f4923a5d55f801cfb86dd94c508.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>correlation strength o</td><td></td><td>0.2 0.4</td><td>0.7</td><td>8 (uc)</td></tr><tr><td>Shapes3D (A)</td><td>object size - azimuth median other pairs</td><td>0.38 0.26 0.09 0.09</td><td>0.13 0.09</td><td>0.08 0.08</td></tr><tr><td>dSprites (B)</td><td>orientation - position X median other pairs</td><td>0.17 0.16 0.13 0.13</td><td>0.14 0.13</td><td>0.11 0.13</td></tr><tr><td>MPI3D (C)</td><td>First DOF - Second DOF median other pairs</td><td>0.2 0.19 0.16 0.16</td><td>0.17 0.15</td><td>0.16 0.15</td></tr></table>",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/f65aa70925e357e2a8cd57ef3b28b2b239a67e235a5ec882aaaef5b256a05e4b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>0.2</td><td>0.4</td><td>0.7</td><td>8（uc)</td></tr><tr><td>0.28</td><td>0.25</td><td>0.2</td><td>0.17 0.18</td></tr><tr><td>0.2 0.34</td><td>0.2 0.31</td><td>0.19 0.24</td><td>0.14</td></tr><tr><td>0.16</td><td>0.18</td><td>0.19</td><td>0.15</td></tr><tr><td>0.54</td><td>0.52</td><td>0.5</td><td>0.49</td></tr><tr><td>0.25</td><td>0.25</td><td>0.26</td><td>0.25</td></tr></table>",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/848bea5784a5a31bf08dca0ec1214457ecf3bdc1b7c3d3e4f6c4759d131b7758.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>0.2</td><td>0.4</td></tr><tr><td>0.38 0.11</td><td>0.31 0.11</td></tr><tr><td>0.43</td><td>0.3</td></tr><tr><td>0.15</td><td>0.15</td></tr></table>",
+        "bbox": [
+            735,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/223fc1196af2fe691974ba8a606eb694152476c0f7eecea5147c866b452e4ea7.jpg",
+        "table_caption": [
+            "Figure 11: Pairwise entanglement scores help to uncover still existent correlations in the latent representation. Left: Mean of the pairwise entanglement scores for the correlated pair (red) and the median of the uncorrelated pairs. We see that stronger correlation leads to statistically more entanglement latents across all datasets studied compared to their baseline pairwise entanglement where the data exhibits no correlations (blue). Each pairwise score is the mean across 180 models for each dataset and correlation strength. Scores in the left table are based on GBT feature importance and scores presented in the right table are based on Mutual Information. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>correlation strength o</td><td></td><td>0.2</td><td>0.4</td></tr><tr><td>Shapes3D (D)</td><td>object color - object size</td><td>0.29</td><td>0.28</td></tr><tr><td></td><td>median uncorrelated pairs</td><td>0.07</td><td>0.07</td></tr><tr><td>Shapes3D (E)</td><td>object color - azimuth</td><td>0.25</td><td>0.23</td></tr><tr><td></td><td>median uncorrelated pairs</td><td>0.1</td><td>0.09</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Latent structure and pairwise entanglement. Our hypothesis that the latent representations are less correlated if the correlation strength is weaker is shown for a model on Shapes3D (A) with weak correlation in Fig. 10. Here the latent traversals do not mirror the major and minor axis of the correlated joint distribution. This conclusion is being backed by the thresholds of the pairwise entanglement metrics for the correlated pair vs. the median of all other pairs across all datasets, either when computing them using the GBT feature importances or the mutual information. See Fig. 11 for these respective results. Another pairwise metric that tracks the correlation strength in our scenario is the unfairness score between the correlated pair of factors that is being shown for datasets A, B and C in Fig. 12. ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Generalization Properties In order to support our conclusion from these results that disentanglement methods can generalize towards unseen FoV configurations we show in Fig. 13 latent traversals originating from these OOD point with smallest object size and largest azimuth. We observe that changes in the remaining factors reliably yield the expected reconstructions. Additionally, we are interested in where samples from correlated models are located with respect to the ground truth value of each correlated variable. For this, we are visualizing the latent spaces with similar extrapolation and generalization capabilities of four models from the two strongest correlation dataset variants of Shapes3d (D) and Shapes3d (E) in Fig. 14. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/34e189fd63cfbbe62d7bfd7f7f4115352a98f83edeef49f6dc31182e3a752e6d.jpg",
+        "image_caption": [
+            "Figure 12: Disentangled representations trained on correlated data are anti-correlated with higher fairness properties. The plots show the mean unfairness scores between the correlated factors with decreasing correlation strength for Shapes3D (A), dSprites (B) and MPI3D-real (C). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/317b7f65006673a60895c3b03dc88a1dd36f82a0fe70127354c7c41ae1ed7185.jpg",
+        "image_caption": [
+            "Figure 13: Generalization capabilities towards out-of-distribution test data. Latent traversals from an observations the model has never seen during training. The starting point corresponds to a factor configuration in point 1 from Fig. 5. Shown are the results of the model with highest DCI score among all 180 trained models on Shapes3d (A) with a very restricted correlation strength $\\sigma = 0 . 2$ in object size and azimuth "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "B.2 SECTION 4 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            505,
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+            518
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Post-hoc alignment correction with few labels In Fig. 15, we see the axis alignment of the correlated latent space after fast adaptation using linear regression on a model trained on Shapes3D (A). Fast adaptation with linear regression substitution fails in some settings: when no two latent dimensions encode the applied correlation isolated from the other latent codes, or when the correlated variables do not have a unique natural ordering (e.g. color or categorical variables). Additionally, the functional form of the latent manifolds beyond the training distribution is unknown and in general expected to be nonlinear. We test the possibility of fast adaptation in this case using as substitution function a one-hidden layer MLP classifier of size 100 on the correlated Shapes3D variants. Under this method, we sample the FoV from a uniform independent distribution. A small number of such samples could practically be labeled manually. Using only 1000 labeled data points for our fast adaptation method shows a significant reduction in disentanglement-thresholds for the correlated pair (Fig. 16). ",
+        "bbox": [
+            173,
+            531,
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+            699
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Alignment during training using weak supervision Using the studied weakly supervision AdaGVAE method with $k = 1$ from Locatello et al. (2020), we showed that weak supervision can provide a strong inductive bias capable of finding the right factorization and resolving spurious correlations for datasets of unknown degree of correlation. Despite the results shown on Shapes3D (A) in the main paper, results across all three correlation variants in Shapes3D (A, D, E) are shown in Fig. 17 as well as some representative latent space visualizations that show strong axis alignment in Fig. 18. This study contains a total of 360 trained models. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "In addition to the experiment from the main paper where pairs are constructed solely from the correlated observational data, we want to study two scenarios where we have limited intervention capabilities on the FoV to generate training pairs. The resulting distribution of FoVs (still exhibiting correlations) in these pairs depends on whether the correlation between two pairs is due to a causal link or due to a common confounder. ",
+        "bbox": [
+            174,
+            820,
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+            890
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Scenario I-1: We assume there is a confounder causing a spurious correlation between the factors, such that correlation is broken whenever we constraint the change to be in one of the correlations. ",
+        "bbox": [
+            173,
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+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/518e9b8bb27e4c53eed05f016e18beadb6eebf81873ce47e059542589bc0fa47.jpg",
+        "image_caption": [
+            "Figure 14: Latent space distribution of the two entangled dimensions of the best DCI model in Shapes3d (E) with $\\sigma = 0 . 2$ (first column), in Shapes3d (E) with $\\sigma = 0 . 4$ (second column), in Shapes3d (D) with $\\sigma = 0 . 2$ (third column) and in Shapes3d (D) with $\\sigma = 0 . 4$ (fourth column). Latent codes sampled from correlated observations (circle without edge) and (2) latent codes sampled with an object size-azimuth configuration not encountered during training(squares with black edge). Each column shows the ground truth values of the two correlated factors by color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "img_path": "images/e64e0ebba8a1d2a267f6bda535faa83e124944fb4f84fe0350c180fc7cefabb0.jpg",
+        "image_caption": [
+            "Figure 15: Latent space distribution of the two entangled dimensions of the best DCI model in Shapes3D (A). Latent codes sampled from correlated observations (circle without edge) and (2) latent codes sampled with an object size-azimuth configuration not encountered during training (squares with black edge). Left column shows the ground truth values of the two correlated factors by color. Middle and right column show the fast adapted space using linear regression and 100 or 1000 labels respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            494,
+            823,
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/738969d2d7b5843e87cb113fa4eb91b59ae7d4b2a3eb21149bee77cf127386a8.jpg",
+        "table_caption": [
+            "Figure 16: Mean of the pairwise entanglement scores for the correlated pair (red) and the median of the uncorrelated pairs (based on GBT feature importance) for all pairs of variables in Shapes3D (D) (top), Shapes3D (E) (middle) and Shapes3D (A) (bottom) all with correlation strength $\\sigma = 0 . 2$ Each pairwise score is the mean across 180 models for each dataset and correlation strength. First column is the unsupervised baseline without any fast adaptation and the second column shows that fast adaption using a one-hidden layer MLP reduces these correlations with as little as 1000 labels when sampled from the uncorrelated dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>dataset</td><td>labels</td><td>0</td><td>1000</td></tr><tr><td>Shapes3D (D) σ = 0.2</td><td>object color - object size</td><td>0.3</td><td>0.16</td></tr><tr><td>Shapes3D (D) σ = 0.2</td><td>median uncorrelated pairs</td><td>0.07</td><td>0.07</td></tr><tr><td>Shapes3D (E)σ = 0.2</td><td>object color - azimuth</td><td>0.25</td><td>0.2</td></tr><tr><td>Shapes3D (E) σ = 0.2</td><td>median uncorrelated pairs</td><td>0.1</td><td>0.11</td></tr><tr><td>Shapes3D (A) σ = 0.2</td><td> object size - azimuth</td><td>0.37</td><td>0.26</td></tr><tr><td>Shapes3D (A) σ = 0.2</td><td>median uncorrelated pairs</td><td>0.09</td><td>0.1</td></tr></table>",
+        "bbox": [
+            266,
+            103,
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+            233
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "The value of the changing variable is then sampled uniformly in the second observation. Note that this still means that the vast majority of sampled pairs exhibit correlated FoV. This is depicted in the substantially lifted disentanglement scores shown in Fig. 19 as well as some selected latent space visualizations that show strong axis alignment in Fig. 20. We consistently observe much better disentangled models, often achieving perfect DCI score irrespective of correlations in the data set. The latent spaces tend to strongly align their coordinates with the ground truth label axis. We chose 10 random seeds per configuration in this study, yielding 720 models in total. ",
+        "bbox": [
+            173,
+            366,
+            825,
+            464
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Scenario I-2: Let us assume $C _ { 1 }$ causes $C _ { 2 }$ in our examples, which manifests as the studied linear correlations. Within this setting we cannot sample uniformly in $C _ { 2 }$ if we intervene (or “fix”) all factors except for this causal factor. Intervening on all factors but $C _ { 1 }$ , however, allows us to sample any value in $C _ { 1 }$ as it is not causally affected by $C _ { 2 }$ . To test the hypothesis that this constraint still allows for disentangling the correlation, we trained on Shapes3D (A) and sample pairs consistent with this causal model. Besides observing visually disentangled factors in the latent traversals, we show a summary of our results in Fig. 21 with the same significant improvements regarding disentangling the correlated FoVs. Besides the above correlation strengths, we additionally trained the same models using a very strong correlation of $\\sigma = 0 . 1$ , yielding 300 models trained in this study. ",
+        "bbox": [
+            173,
+            470,
+            825,
+            597
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/55a02791c8d0b50de5188e3389cffb5031ed780b7e45e63f8426086856db4b98.jpg",
+        "image_caption": [
+            "Figure 17: Standard disentanglement metrics evaluated for the weakly supervised scenario using correlated observational data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            342,
+            820,
+            651
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/f997e0d82272cc859cc5242bc563cb88283e962db14a373f621b6f0f37e42519.jpg",
+        "image_caption": [
+            "Figure 18: Left: For the weakly supervised scenario using correlated observational data trained models on Shapes3D (A), (D) and (E) correlating object color and azimuth learn consistently improved, often perfect, disentangled representation across all correlation strengths. Right: Latent dimensions of a best DCI model trained on strongly correlated observational data. Representations are perfectly axis-aligned with respect to both of the correlated variables ground truth values (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            217,
+            825,
+            736
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "image",
+        "img_path": "images/a25e94ba9291585646c3768650c399555dbe873fdf0046bcbe0de8f4bd961538.jpg",
+        "image_caption": [
+            "Figure 19: Standard disentanglement metrics evaluated on the correlated training sets for the weakly supervised scenario with intervening capabilities (I-1). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            289,
+            820,
+            704
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "image",
+        "img_path": "images/0a38d07ea4f3e624025b25e402e0f7d956eb38708636435fdc8e5dcdc86b34fd.jpg",
+        "image_caption": [
+            "Figure 20: Left: For the weakly supervised scenario with intervening capabilities (I-1) trained models on Shapes3D (A), (D) and (E) correlating object color and azimuth learn consistently improved, often perfect, disentangled representation across all correlation strengths. Right: Latent dimensions of a best DCI model with strong correlation (0.2). Representations are perfectly axis-aligned with respect to both of the correlated variables ground truth values (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            220,
+            825,
+            736
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "image",
+        "img_path": "images/c22f1e0add00466036990823304d24372b6bfc1093ac60cd57568df61d0bf979.jpg",
+        "image_caption": [
+            "Figure 21: Disentanglement metrics (top), unfairness scores (middle) and latent spaces (bottom) show strong disentanglement using weak supervision with intervening capabilities (I-2) - even under the stronger assumption that sampling of observation pairs follow its causal generative model. We show the learnt latent space encoding of the two correlated factors of variation for a model with $\\sigma = 0 . 1$ and low reconstruction loss. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            232,
+            820,
+            703
+        ],
+        "page_idx": 23
+    }
+]

parse/train/BdKxQp0iBi8/BdKxQp0iBi8.md

@@ -0,0 +1,270 @@
+# Counterfactual Invariance to Spurious Correlations: Why and How to Pass Stress Tests
+
+Victor Veitch1,2, Alexander D’Amour1, Steve Yadlowsky1, and Jacob Eisenstein1
+
+1Google Research   
+2University of Chicago
+
+# Abstract
+
+Informally, a ‘spurious correlation’ is the dependence of a model on some aspect of the input data that an analyst thinks shouldn’t matter. In machine learning, these have a know-it-when-you-see-it character; e.g., changing the gender of a sentence’s subject changes a sentiment predictor’s output. To check for spurious correlations, we can ‘stress test’ models by perturbing irrelevant parts of input data and seeing if model predictions change. In this paper, we study stress testing using the tools of causal inference. We introduce counterfactual invariance as a formalization of the requirement that changing irrelevant parts of the input shouldn’t change model predictions. We connect counterfactual invariance to outof-domain model performance, and provide practical schemes for learning (approximately) counterfactual invariant predictors (without access to counterfactual examples). It turns out that both the means and implications of counterfactual invariance depend fundamentally on the true underlying causal structure of the data—in particular, whether the label causes the features or the features cause the label. Distinct causal structures require distinct regularization schemes to induce counterfactual invariance. Similarly, counterfactual invariance implies different domain shift guarantees depending on the underlying causal structure. This theory is supported by empirical results on text classification.
+
+# 1 Introduction
+
+Our focus in this paper is the sort of spurious correlations revealed by “poke it and see what happens” testing procedures for machine-learning models. For example, we might test a sentiment analysis tool by changing one proper noun for another (“tasty Mexican food” to “tasty Indian food”), with the expectation that the predicted sentiment should not change. This kind of perturbative stress testing is increasingly popular: it is straightforward to understand and offers a natural way to test the behavior of models against the expectations of practitioners [Rib $+ 2 0$ ; $\mathrm { W u } { + } 1 9$ ; Nai+18].
+
+Intuitively, models that pass such stress tests are preferable to those that do not. However, fundamental questions about the use and meaning of perturbative stress tests remain open. For instance, what is the connection between passing stress tests and model performance on prediction? Eliminating predictor dependence on a spurious correlation should help with domain shifts that affect the spurious correlation—but how do we make this precise? And, how should we develop models that pass stress tests when our ability to generate perturbed examples is limited? For example, automatically perturbing the sentiment of a document in a general fashion is difficult.
+
+The ad hoc nature of stress testing makes it difficult to give general answers to such questions. In this paper, we will use the tools of causal inference to formalize what it means for models to pass stress tests, and use this formalization to answer the questions above. We will formalize passing stress tests as counterfactual invariance, a condition on how a predictor should behave when given certain (unobserved) counterfactual input data. We will then derive implications of counterfactual invariance that can be measured in the observed data. Regularizing predictors to satisfy these observable implications provides a means for achieving (partial) counterfactual invariance. Then, we will connect counterfactual invariance to robust prediction under certain domain shifts, with the aim of clarifying what counterfactual invariance buys and when it is desirable.
+
+An important insight that emerges from the formalization is that the true underlying causal structure of the data has fundamental implications for both model training and guarantees. Methods for handing ‘spurious correlations’ in data with a given causal structure need not perform well when blindly translated to another causal structure.
+
+Counterfactual Invariance Consider the problem of learning a predictor $f$ that predicts a label $Y$ from covariates $X$ . In this paper, we’re interested in constructing predictors whose predictions are invariant to certain perturbations on $X$ . Our first task is to formalize the invariance requirement.
+
+To that end, assume that there is an additional variable $Z$ that captures information that should not influence predictions. However, $Z$ may causally influence the covariates $X$ . Using the potential outcomes notation, let $X ( z )$ to denote the counterfactual $X$ we would have seen had $Z$ been set to $z$ , leaving all else fixed. Informally, we can understand perturbative stress tests as a way of producing particular realizations of counterfactual pairs $X ( z )$ , $X ( z ^ { \prime } )$ that differ by an intervention on $z$ . Then, we formalize the requirement that an arbitrary change to $z$ does not change predictions:
+
+Definition 1.1. A predictor $f$ is counterfactually invariant to $Z$ if $f ( X ( z ) ) \ : = \ : f ( X ( z ^ { \prime } ) )$ almost everywhere, for all $z , z ^ { \prime }$ in the sample space of $Z$ . When $Z$ is clear from context, we’ll just say the predictor is counterfactually invariant.
+
+# 2 Causal Structure
+
+Counterfactual invariance is a condition on how the predicted label behaves under interventions on parts of the input data. However, intuitions about stress testing are based on how the true label behaves under interventions on parts of the input data. We will see that the true causal structure fundamentally affects both the implications of counterfactual invariance, and the techniques we use to achieve it. To study this phenomenon, we’ll use two causal structures that are commonly encountered in applications; see Figure 1.
+
+# 2.1 Prediction in the Causal Direction
+
+We begin with the case where $X$ is a cause of $Y$
+
+Example 2.1. We want to automatically classify the quality of product reviews. Each review has a number of “helpful” votes $Y$ (from site users). We predict $Y$ using the text of the product review $X$ . However, we find interventions on the sentiment $Z$ of the text change our prediction; changing “Great shoes!” to “Bad shoes!” changes the prediction.
+
+In the examples in this paper, the covariate $X$ is text data. Usually, the causal relationship between the text and $Y$ and $Z$ will be complex— e.g., the relationships may depend on abstract, unlabeled, parts of the text such as topic, writing quality, or tone. In principle, we could enumerate all such latent variables, construct a causal graph capturing the relationships between these variables and $Y , Z$ , and use this causal structure to study counterfactual invariance. For instance, if we think that topic causally influences the helpfulness $Y$ , but is not influenced by sen
+
+![](images/db6f289cfcf6978a6de564bcb64d3637aad50130eac9708db787c9ff88ca27f4.jpg)  
+Figure 1: Causal models for the data generating process. We decompose the observed covariate $X$ into latent parts defined by their causal relationships with $Z$ and $Y$ . Solid arrows denote causal relationships, while dashed lines denote non-causal associations. The differences between these causal structures will turn out to be key for understanding counterfactual invariance.
+
+timent $Z$ , then we could build a counterfactually invariant predictor by extracting the topic and predicting $Y$ using topic alone. However, exhaustively articulating all possible such variables is a herculean task.
+
+Instead, notice that the only thing that’s relevant about these latent variables is their causal relationship with $Y$ and $Z$ . Accordingly, we’ll decompose the observed variable $X$ into parts defined by their causal relationships with $Y$ and $Z$ . We remain agnostic to the semantic interpretation of these parts. Namely, we define $X _ { Z } ^ { \perp }$ as the part of $X$ that is not causally influenced by $Z$ (but may influence $Y )$ , $X _ { Y } ^ { \perp }$ as the part that does not causally influence $Y$ (but may be influenced by $Z$ ), and $X _ { Y \wedge Z }$ is the remaining part that is both influenced by $Z$ and that influences $Y$ . The causal structure is shown in Figure 1a.
+
+We see there are two paths that lead to $Y$ and $Z$ being associated. The first is when $Z$ affects $X _ { Y \wedge Z }$ which, in turn, affects $Y$ . For example, a very enthusiastic reviewer might write a longer, more detailed review, which will in turn be more helpful. The second is when a common cause or selection effect in the data generating process induces an association between $Z$ and $Y$ , which we denote with a dashed arrow. For example, if books tend to get more positive reviews, and also people who buy books are more likely to flag reviews as helpful, then the product type would be a common cause of sentiment and helpfulness.
+
+# 2.2 Prediction in the Anti-Causal Direction
+
+We also consider the case where $Y$ causes $X$
+
+Example 2.2. We want to predict the star rating $Y$ of movie reviews from the text $X$ . However, we find that predictions are influenced by the movie genre $Z$ ; e.g., changing “Adam Sandler” (a comedy actor) to “Hugh Grant” (a romance actor) changes the predictions.
+
+Figure 1b shows the causal structure. Here, the observed $X$ is influenced by both $Y$ and $Z$ . Again, we decompose $X$ into parts defined by their causal relationship with $Z$ and $Y$ . Here, $Z$ (and thus $X _ { Y } ^ { \perp }$ ) can be associated with $Y$ through two paths. First, if $X _ { Y \wedge Z }$ is non-trivial, then conditioning on it causes a dependence between $Z$ and $Y$ (because $X _ { Y \wedge Z }$ is a collider). For example, if Adam Sandler tends to appear in good comedy movies but bad movies of other genres then seeing “Sandler” in the text induces a dependency between sentiment and genre. Second, $Z$ and $Y$ may be associated due to a common cause, or due to selection effects in the data collection protocol—this is represented by the dashed line between $Z$ and $Y$ . For example, fans of romantic comedies may tend to give higher reviews (to all films) than fans of horror movies.
+
+# 2.3 Non-Causal Associations
+
+Frequently, a predictor trained to predict $Y$ from $X$ will rely on $X _ { Y } ^ { \perp }$ , even though there is no causal connection between $Y$ and $X _ { Y } ^ { \perp }$ , and therefore will fail counterfactual invariance. The reason is that $X _ { Y } ^ { \perp }$ serves as a proxy for $Z$ , and $Z$ is predictive of $Y$ due to the non-causal (dashed line) association.
+
+There are two mechanisms that can induce such associations. First, $Y$ and $Z$ may be confounded: they are both influenced by an unobserved common cause $U$ . For example, people who review books may be more upbeat than people who review clothing. This leads to positive sentiments and high helpfulness votes for books, creating an association between sentiment and helpfulness. Second, $Y$ and $Z$ may be subject to selection: there is some condition (event) $S$ that depends on $Y$ and $Z$ , such that a data point from the population is included in the sample only if $S = 1$ holds. For example, our training data might only include movies with at least 100 reviews. If only excellent horror movies have so many reviews (but most rom-coms get that many), then this selection would induce an association between genre and score. Formally, the dashed-line causal graphs mean our sample is distributed according to $\begin{array} { r } { P ( X , Y , Z ) = \int \mathrm { P } ( \dot { X } , Y , Z , u \mid S = 1 ) \mathrm { d } \mathrm { P } ( u ) } \end{array}$ where $Y , Z$ are caused by $U$ and are causes of $S$ , and $( X , Y , Z )$ are causally related according to the graph.
+
+In addition to the non-causal dashed-line relationship, there is also dependency induced by between $Y$ and $Z$ by $X _ { Y \wedge Z }$ . Whether or not each of these dependencies is “spurious” is a problem-specific judgement that must be made by each analyst based on their particular use case. E.g., using genre to predict sentiment may or may not be reasonable, depending on the actual application in mind. However, there is a special case that captures a common intuition for purely spurious association.
+
+Definition 2.3. We say that the association between $Y$ and $Z$ is purely spurious if $Y \perp X \mid X _ { Z } ^ { \perp } , Z$
+
+That is, if the dashed-line association did not exist (removed by conditioning on $Z$ ) then the part of $X$ that is not influenced by $Z$ would suffice to estimate $Y$ .
+
+# 3 Observable Signatures of Counterfactually Invariant Predictors
+
+We now consider the question of how to achieve counterfactual invariance in practice. The challenge is that counterfactual invariance is defined by the behavior of the predictor on counterfactual data that is never actually observed. This makes checking counterfactual invariance impossible. Instead, we’ll derive a signature of counterfactual invariance that actually can be measured—and enforced— using ordinary datasets where $Z$ (or a proxy) is measured. For example, the star rating of a review as a proxy for sentiment, or genre labels in the movie review case.
+
+Intuitively, a predictor $f$ is counterfactually invariant if it depends only on $X _ { Z } ^ { \perp }$ , the part of $X$ that is not affected by $Z$ . To formalize this, we need to show that such a $X _ { Z } ^ { \perp }$ is well defined:
+
+Lemma 3.1. Let $X _ { Z } ^ { \perp }$ be a $X$ -measurable random variable such that, for all measurable functions $f$ , we have that $f$ is counterfactually invariant if and only if $f ( X )$ is $X _ { Z } ^ { \perp }$ -measurable. If $Z$ is discrete1 then such a $X _ { Z } ^ { \perp }$ exists.
+
+Accordingly, we’d like to construct a predictor that is a function of $X _ { Z } ^ { \perp }$ only (i.e., is $X _ { Z } ^ { \perp }$ measurable). The key insight is that we can use the causal graphs to read off a set of conditional independence relationships that are satisfied by $Z , X _ { Z } ^ { \perp } , Y$ . Critically, these conditional independence relationships are testable from the observed data. Thus, they provide a signature of counterfactual invariance:
+
+Theorem 3.2. If $f$ is a counterfactually invariant predictor:
+
+1. Under the anti-causal graph, $f ( X ) \perp Z \mid Y$ .   
+2. Under the causal-direction graph, if $Y$ and $Z$ are not subject to selection (but possibly confounded), $f ( X ) \perp \perp Z$ .   
+3. Under the causal-direction graph, if the association is purely spurious, $Y \perp X \mid X _ { Z } ^ { \perp } , Z$ , and $Y$ and $Z$ are not confounded (but possibly selected), $f ( X ) \perp Z \mid Y$ .
+
+Remark 3.3 (Connection to Fairness). This result has an interesting interpretation when $Z$ is a protected attribute (e.g., sex or race) that we’d like to be fair with respect to. There are many ways of formalizing fairness, which are usually mutually incompatible. In the fairness setting, counterfactual invariance is equivalent to counterfactual fairness [Kus+17; Gar+19], the condition $f ( X ) \ \perp \ Z$ is equivalent to demographic parity, and the condition $f ( X ) \perp Z \mid Y$ is equivalent to equalized odds [Meh+19]. Theorem 3.2 then says that counterfactual fairness implies either demographic parity or equalized odds, depending on the true underlying causal structure of the problem. Hence, the relationship between disparate fairness notions is clarified by taking the underlying causal structure into account. This also suggests we can take counterfactual fairness as the fundamental notion, then use demographic parity in the causal-confounding case and equalized odds otherwise. However, we leave the (ethical) question of whether this is a sound strategy to future work.
+
+Causal Regularization Without access to counterfactual examples, we cannot directly enforce counterfactual invariance. However, we can require a trained model to satisfy the counterfactual invariance signature of Theorem 3.2. The hope is that enforcing the signature will lead the model to be counterfactually invariant. To do this, we regularize the model to satisfy the appropriate conditional independence condition. For simplicity of exposition, we restrict to binary $Y$ and $Z$ . The (infinite data) regularization terms are
+
+$$
+\begin{array} { r l } & { = \mathrm { M M D } ( \mathrm { P } ( f ( X ) \mid Z = 0 ) , \mathrm { P } ( f ( X ) \mid Z = 1 ) ) } \\ & { = \mathrm { M M D } ( \mathrm { P } ( f ( X ) \mid Z = 0 , Y = 0 ) , \mathrm { P } ( f ( X ) \mid Z = 1 , Y = 0 ) ) } \\ & { \quad + \mathrm { M M D } ( \mathrm { P } ( f ( X ) \mid Z = 0 , Y = 1 ) , \mathrm { P } ( f ( X ) \mid Z = 1 , Y = 1 ) ) . } \end{array}
+$$
+
+Maximum mean discrepancy (MMD) is a metric on probability measures.2 The marginal independence condition is equivalent to (3.1) equal 0, and the conditional independence is equivalent to (3.2)
+
+equal 0. In practice, we can estimate the MMD with finite data samples $[ \mathrm { G r e } { + } 1 2 ]$ . When training with stochastic gradient descent, we compute the penalty on each minibatch.
+
+The procedure is then: if the data has causal-direction structure and the $Y  Z$ association is due to confounding, add the marginal regularization term to the the training objective. If the data has anti-causal structure, or the association is due to selection, add the conditional regularization term instead. In this way, we regularize towards models that satisfy the counterfactual invariance signature.
+
+A key point is that the regularizer we must use depends on the true causal structure. The conditional and marginal independence conditions are generally incompatible. Enforcing the condition that is mismatched to the true underlying causal structure may fail to encourage counterfactual invariance, or may throw away more information than is required.
+
+Gap to Counterfactual Invariance The conditional independence signature of Theorem 3.2 is necessary but not sufficient for counterfactual invariance. This is for two reasons. First, counterfactual invariance applies to individual datapoint realizations, but the signature is distributional. In particular, the invariance $\mathrm { P } ( f ( X ) \mid \mathrm { d o } ( Z = z ) ) = \mathrm { P } ( f ( X ) \mid \mathrm { d o } ( Z = \bar { z } ^ { \prime } ) )$ for all $z , z ^ { \prime }$ would also imply the conditional independence signature. But, this invariance is weaker than counterfactual invariance, since it doesn’t require access to counterfactual realizations. Second, $f ( X ) \ \perp \perp Z$ does not imply, in general, that $Z$ is not a cause of $f ( X )$ . This (unusual) behavior can happen if, e.g., there are levels of $Z$ that we do not observe in the training data, or there are variables omitted from the causal graph that are a common cause of $Z$ and $X$ .
+
+Unfortunately, the gap between the signature and counterfactual invariance is a fundamental restriction of using observational data. The conditional independence signature is in some sense the closest proxy for counterfactual invariance we can hope for. In section 5, we’ll see that enforcing the signature does a good job of enforcing counterfactual invariance in practice.
+
+# 4 Performance Out of Domain
+
+Counterfactual invariance is an intuitively desirable property for a predictor to have. However, it’s not immediately clear how it relates to model performance as measured by, e.g., accuracy. Intuitively, eliminating predictor dependence on a spurious $Z$ may help with domain shift, where the data distribution in the target domain differs from the distribution of the training data. We now turn to formalizing this idea.
+
+First, we must articulate the set of domain shifts to be considered. In our setting, the natural thing is to hold the causal relationships fixed across domains, but to allow the non-causal (“spurious”) dependence between $Y$ and $Z$ to vary. Demanding that the causal relationships stay fixed reflects the requirement that the causal structure describes the dynamics of an underlying real-world process— e.g., the author’s sentiment is always a cause (not an effect) of the text in all domains. On the other hand, the dependency between $Y$ and $Z$ induced by either confounding or selection can vary without changing the underlying causal structure. For confounding, the distribution of the confounder may differ between domains—e.g., books are rare in training, but common in deployment. For selection, the selection criterion may differ between domains—e.g., we include only frequently reviewed movies in training, but make predictions for all movies in deployment.
+
+We want to capture spurious domain shifts by considering domain shifts induced by selection or confounding. However, there is an additional nuance. Changes to the marginal distribution of $Y$ will affect the risk of a predictor, even in the absence of any spurious association between $Y$ and $Z$ . Therefore, we restrict to shifts that preserve the marginal distribution of $Y$ .
+
+Definition 4.1. We say that distributions $P , Q$ are causally compatible if both obey the same causal graph, $P ( Y ) = Q ( Y )$ , and there is a confounder $U$ and/or selection conditions $S , { \tilde { S } }$ such that $\begin{array} { r } { P { = } \int \operatorname { P } ( X , Y , Z \mid U , S { = } 1 ) \mathrm { d } \tilde { P } ( U ) } \end{array}$ and $\begin{array} { r } { Q { = } \int \operatorname { P } ( X , Y , Z \mid U , \tilde { S } { = } 1 ) \mathrm { d } \tilde { Q } ( U ) } \end{array}$ for some $\tilde { P } ( U ) , \tilde { Q } ( U )$ .
+
+We can now connect counterfactual invariance and robustness to domain shift.
+
+Theorem 4.2. Let ${ \mathcal { F } } ^ { \mathrm { i n v a r } }$ be the set of all counterfactually invariant predictors. Let $L$ be either square error or cross entropy loss. And, let $f ^ { * } : = \operatorname { a r g m i n } _ { \underline { { f } } \in \mathcal { F } ^ { \mathrm { i n v a } } }$ r $\mathbb { E } _ { P } [ L ( Y , f ( X ) ) ]$ be the counterfactually invariant risk minimizer. Suppose that the target distribution $Q$ is causally compatible with the training distribution $P$ . Suppose that any of the following conditions hold:
+
+1. the data obeys the anti-causal graph   
+2. the data obeys the causal-direction graph, there is no confounding (but possibly selection), and the association is purely spurious, $Y \perp X \mid X _ { Z } ^ { \perp } , Z$ , or   
+3. the data obeys the causal-direction graph, there is no selection (but possibly confounding), the association is purely spurious and the causal effect of $X _ { Z } ^ { \perp }$ on $Y$ is additive, i.e., the true data generating process is
+
+$$
+Y  g ( X _ { Z } ^ { \perp } ) + \tilde { g } ( U ) + \xi w h e r e \operatorname { \mathbb { E } } [ \xi \mid X _ { Z } ^ { \perp } ] = 0 ,
+$$
+
+for some functions $g , \tilde { g }$
+
+Then, the training domain counterfactually invariant risk minimizer is also the target domain counterfactually invariant risk minimizer, $f ^ { * } = \mathrm { a r g m i n } _ { f \in \mathcal { F } ^ { \mathrm { i n } } }$ var $\mathbb { E } _ { Q } [ L ( Y , f ( X ) ) ]$ .
+
+Remark 4.3. The causal case with confounding requires an additional assumption (additive structure) because, e.g., an interaction between confounder and $X _ { Z } ^ { \perp }$ can yield a case where $X _ { Z } ^ { \perp }$ and $Y$ have a different relationship in each domain (whence, out-of-domain learning is impossible).
+
+This result gives a recipe for finding a good predictor in the target domain even without access to any target domain examples at training time. Namely, find the counterfactually invariant risk minimizer in the training domain. In practice, we can use the regularization scheme of section 3 to (approximately) achieve this. We’ll see in section 5 that this works well in practice.
+
+Optimality Theorem 4.2 begs the question: if the only thing we know about the target setting is that it’s causally compatible with the training data, is the best predictor the counterfactually invariant predictor with lowest training risk? A natural way to formalize this question is to study the predictor with the best performance in the worst case target distribution. We define $ \bar { \mathcal { Q } } ~ = ~ \{ \bar { Q } ~ : ~ Q$ causally compatible with $P \}$ and the $\mathcal { Q }$ -minimax predictor $f _ { \mathrm { m i n i m a x } } ^ { * } ~ =$ $\operatorname { \arg m i n } _ { f \in { \mathcal { F } } } \operatorname* { m a x } _ { Q \in { \mathcal { Q } } } \mathbb { E } _ { Q } [ L ( Y , f ( X ) ]$ . The question is then: what’s the relationship between the counterfactually invariant risk minimizer and the minimax predictor?
+
+Theorem 4.4. The counterfactually invariant risk minimizer is not $\mathcal { Q }$ -minimax in general. However, under the conditions of Theorem 4.2, if the association is purely spurious, $X _ { Y \wedge Z } \ { \bar { \perp } } \ Y \mid X _ { Z } ^ { \perp } , Z$ , and $\mathrm { P } ( Z , Y )$ satisfies overlap, then the two predictors are the same. By overlap we mean that $\mathrm { P } ( Z , Y )$ is a discrete distribution such that for all $( z , y )$ , $i f \mathrm { P } ( z , y ) > 0$ then there is some $y ^ { \prime } \ne y$ such that also $\mathrm { P } ( z , y ^ { \prime } ) > 0$ .
+
+Conceptually, Theorem 4.4 just says that the counterfactually invariant predictor excludes $X _ { Y \wedge Z }$ , even when this information is useful in every domain. In the purely spurious case, $X _ { Y \wedge Z }$ carries no useful information, so counterfactual invariance is optimal.
+
+# 5 Experiments
+
+The main claims of the paper are:
+
+1. Stress test violations can be reduced by suitable conditional independence regularization.   
+2. This reduction will improve out-of-domain prediction performance.   
+3. To get the full effect, the imposed penalty must match the causal structure of the data.
+
+Setup To assess these claims, we’ll examine the behavior of predictors trained with the marginal or conditinal regularization on multiple text datasets that have either causal or anti-causal structure. We expect to see that marginal regularization improves stress test and out-of-domain performance on data with causal-confounded structure, and conditional regularization improves these on data with anti-causal structure.
+
+For each experiment, we use BERT [Dev+19] finetuned to predict a label $Y$ from the text as our base model. We train multiple causally-regularized models on the each dataset. The training varies by whether we use the conditional or marginal penalty, and by the strength of the regularization term. That is, we train identical architectures using CrossEntropy $+ \lambda \cdot$ Regularizer as the objective function, where we vary $\lambda$ and take Regularizer as either the marginal penalty, (3.1), or conditional penalty, (3.2). We compare these models’ predictions on data with causal and anti-causal structure.
+
+See supplement for experimental details.
+
+![](images/53efb42512085a29a0de7812344b3d241b39e4349e1f4adf7ba004e54ec1e32c.jpg)  
+Figure 2: Regularizing conditional MMD improves counterfactual invariance on synthetic anti-causal data. Sufficiently high regularization of marginal MMD also improves invariance, but impairs accuracy. Dashed lines show baseline performance of an unregularized predictor. Left: lower conditional MMD implies that predictive probabilities are invariant to perturbation. Although marginal MMD penalization can result in low conditional MMD and good stress test performance, this comes at the cost of very low in-domain accuracy. Right: MMD regularization reduces the rate of predicted label flips on perturbed data, with little affect on indomain accuracy. Conditional MMD regularization reduces predicted label flips to $1 . 4 \%$ , while the best result for marginal MMD is $2 . 8 \%$ .
+
+# 5.1 Robustness to Stress Tests
+
+First, we examine whether enforcing the causal regularization actually helps to enforce counterfactual invariance. We create counterfactual (stress test) examples by perturbing the input data and compare the prediction on these. We build the experimental datasets using Amazon reviews from the product category “Clothing, Shoes, and Jewelry” [NLM19].
+
+Synthetic To study the relationship between counterfactual invariance and the distributional signature of Theorem 3.2, we construct a synthetic confound. For each review, we draw a Bernoulli random $Z$ , and then perturb the text $X$ so that the common words “the” and “a” carry information about $Z$ : for example, we replace “the” with the token “thexxxxx” when $Z = 1$ . We take $Y$ to be the review score, and subsample so $Y$ is balanced. This data has anti-causal structure: the text $X$ is written to explain the score $Y$ . Further, we expect that the $Y , Z$ association is purely spurious, because “the” and “a” carry little information about the label.
+
+We train the models on data where $\operatorname { P } ( Y = Z ) = 0 . 3$ . We then create perturbed stress-test datasets by changing each example $X _ { i } ( z )$ to the counterfactual $X _ { i } ( 1 - z )$ (using the synthetic model). By measuring the performance of each model on the perturbed data, we can test whether the distributional properties enforced by the regularizers result in counterfactual invariance at the instance level. Figure 2 shows that conditional regularization (matching the anti-causal structure) reduces checklist failures, as measured by the frequency that the predicted label changes due to perturbation as well as the mean absolute difference in predictive probabilities that is induced by perturbation.
+
+Natural To study the relationship in real data, we use the review data in a different way. We now take $Z$ to be the score, binarized as $Z \in { }$ $\{ 1$ or 2 stars, 4 or 5 stars}. We use this $Z$ as a proxy for sentiment, and consider problems where sentiment should (plausibly) not have a causal effect on $Y$ . For the causal prediction problem, we take $Y$ to be the helpfulness score of the review (binarized as described below). This is causal because readers decide whether the review is helpful based on the text. For the anti-causal prediction problem, we take $Y$ to be whether “Clothing” is included as a category tag for the product under review (e.g., boots typically do not have this tag). This is anti-causal because the product category affects the text.
+
+![](images/25d13abb0d1a94526c03a9edcbbe9537be776b40dc5768ab7b2869bd796e53c3.jpg)  
+Figure 3: Penalizing the MMD matching the causal structure improves stress test performance on natural product review data. Note that penalizing the wrong MMD may not help: the marginal MMD hurts on the anticausal dataset. Perturbations are generated by swapping positive and negative sentiment adjectives in examples.
+
+We control the strength of the spurious association between $Y$ and $Z$ . In the anti-causal case, this is done by selection: we randomly subset the data to enforce a target level of dependence between $Y$ and $Z$ . The causal-direction case with confounding is more complicated. To manipulate confounding strength, we binarize the number of helpfulness votes $V$ in a manner determined by the target level of association. We take $Y = 1 [ V > T _ { Z } ]$ where $T _ { Z }$ is a $Z$ -dependent threshold, chosen to induce a target association. We choose $\operatorname { P } ( Y = { \bar { 0 } } \mid Z = 0 ) = \operatorname { P } ( Y = { \bar { 1 } } \mid Z = 1 ) = 0 . 3$ . We balance $Z$ by subsampling, which also balances $Y$ .
+
+Now, we create stress test perturbations of these datasets by randomly changing adjectives in the examples. Using predefined lists of postive sentiment adjectives and negative sentiment adjectives, we swap any adjective that shows up on a list with a randomly sampled adjective from the other list. This preserves basic sentence structure, and thus creates a limited set of counterfactual pairs that differ on sentiment.
+
+Results for differences in predicted probabilities between original and perturbed data are shown in Figure 3. Each point is a trained model, which vary in measured MMD on the test data and on sensitivity to perturbations. Recall that the conditional independence signature of Theorem 3.2 are necessary but not sufficient for counterfactual invariance, so it’s not certain that regularizing to reduce the MMD will reduce perturbation sensitivity. Happily, we see that regularizing to reduce the MMD that matches the causal structure does indeed reduce sensitivity to perturbations.
+
+Notice that regularizing the causally mismatched MMD can have strange effects. Regularizing marginal MMD in the anti-causal case actually makes the model more sensitive to perturbations!
+
+# 5.2 Domain Shift
+
+Next, we study the effect of causal regularization on model performance under domain shift.
+
+Natural Product Review We again use the natural Amazon review data described above. For both the causal and anticausal data, we create multiple test sets with variable spurious correlation strength. This is done in the manner described above, varying $\operatorname { P } ( Y = 0 \mid Z =$ $0 ) = \operatorname { P } ( Y = 1 \mid Z = 1 ) =$ $\gamma$ . Here, $\gamma$ is the strength of spurious association. The test sets are out-of-domain samples. By design, $Y$ is balanced in each dataset, so these samples are causally compatible with the training data. For both the causal and anti-causal datasets, the training data has $\begin{array} { r l } { \mathrm { P } ( Y } & { { } = } \end{array}$ $0 \mid Z = 0 \bar { ) } = \mathrm { P } ( Y = 1 \mid Z =$ $1 ) ~ = ~ 0 . 3$ . We train a classifier for each regularization type and regularization strength, and measure the accuracy on each test domain. The results are shown in Figure 4.
+
+![](images/cc50159fcb6943974ab865fe3f6e0e57889243fedb2eb828a12ef8b80e7b30b1.jpg)  
+Anti-Causal Data: conditional regularization improves domain-shift robustness.
+
+![](images/1aad0d1d99d5ec2ccf3cc08ff60ebd5e9382fde6d09560b6780ee4953b6c184c.jpg)  
+Causal-Direction Data: marginal regularization improves domain-shift robustness.   
+Figure 4: The best domain-shift robustness is obtained by using the regularizer that matches the underlying causal structure of the data. The plots show out-of-domain accuracy for models trained on the (natural) review data. In each row, the left figure shows out-of-domain accuracies (lines are models), with the $X$ -axis showing the level of spurious correlation in the test data (0.3 is the training condition); the right figure shows worst out-of-domain accuracy versus in-domain test accuracy (dots are models).
+
+First, the unregularized predictors do indeed learn to rely on the spurious association between sentiment and the label. The accuracy of these predictors decays dramatically as the spurious assocation moves from negative (0.3) to positive—in the causal case, the unregularized predictor is worse than chance in the 0.8 domain.
+
+Following section 3, the regularization that matches the underlying causal structure should yield a predictor that is (approximately) counterfactually invariant. Following Theorem 4.2, we expect that good performance of a counterfactually-invariant predictor in the training domain should imply good performance in each of the other domains. Indeed, we see that this is so. Models that are regularized to have small values of the appropriate MMD do indeed have better out-of-domain performance.
+
+![](images/a10a5b2248012b6f81c4cccbf8cd45aa1b0e5f41578a88117ad325f1b0dfe2e9.jpg)  
+Figure 5: Conditional MMD penalization improves robustness in anti-causal MNLI data. Marginal regularization does not improve over the baseline unregularized model, shown with dashed lines. Left: Conditional regularization improves minimum accuracy across $( Y , Z )$ groups. When overregularized, the predictor returns the same $\hat { Y }$ for all inputs, yielding a worst-group accuracy of 0. Right: Conditional MMD regularization significantly improves worst $( Y , Z )$ group accuracy $y$ -axis) while only mildly reducing overall accuracy ( $x$ -axis).
+
+Such models have somewhat worse in-domain performance, because they no longer exploit the spurious correlation.
+
+MNLI Data For an additional test on naturally-occurring confounds, we use the multigenre natural language inference (MNLI) dataset [WNB18]. Instances are concatenations of two sentences, and the label describes the semantic relationship between them, $Y \in \mathbf { \Sigma }$ {contradiction, entailment, neutral}. There is a well-known confound in this dataset: examples where the second sentence contain a negation word (e.g., “not”) are much more likely to be labeled as contradictions $[ \mathrm { G u r } { + } 1 8 ]$ . Following Sagawa et al. $[ \mathrm { S a g } + 2 0 ]$ , we set $Z$ to indicate whether one of a small set of negation words is present. Although $Z$ is derived from the text $X$ , it can be viewed as a proxy for a latent variable indicating whether the author intended to use negation in the text. This is an anti-causal prediction problem: the annotators were instructed to write text to reflect the desired label [WNB18].
+
+Following Sagawa et al. $[ \mathrm { S a g } + 2 0 ]$ , we divide the MNLI data into groups by $( Y , Z )$ and compute the “worst group accuracy” across all such groups. Because this is an anti-causal problem, we predict that the conditional MMD is a more appropriate penalty than the marginal MMD. As shown in Figure 5, this prediction holds: conditional MMD regularization dramatically improves performance on the worst group, while only lightly impacting the overall accuracy across groups.
+
+# 6 Related work
+
+Several papers draw a connection between causality and domain shifts [SS18; SCS19; $\mathrm { \bf A r j + } 2 0$ ; Mei18; PBM16; $\mathrm { R C } { + 1 8 }$ ; Zha+13; $\operatorname { S c h } { + 1 2 } ]$ . Typically, this work considers a prediction setting where the covariates $X$ include both causes and effects of $Y$ , and it is unknown which is which. The goal is to learn to predict $Y$ using only its causal parents. Zhang et al. $\left[ Z \mathrm { h a } + 1 3 \right]$ considers anti-causal domain shift induced by changing $P ( Y )$ and proposes a data reweighting scheme. By contrast, counterfactual invariance is an example-wise notion of invariance, not an invariance across environments. In particular, learning a counterfactually invariant predictor only requires access to data from a single environment. Theorem 4.2 establishes that counterfactual invariance does imply a certain domain generalization guarantee over causally-compatible domains. Note though that the notion of causal compatibility is not generally the same as class of domain shifts previously considered. For example, we have invariant performance in the anti-causal setting, but this is ruled out by Arjovsky et al. $[ \mathrm { A r j } { + } 2 0 ]$ .
+
+A related body of work focuses on “causal representation learning” [Bes+19; $\mathrm { L o c } { + 2 0 }$ ; $\operatorname { S c h } { + 2 1 }$ ; $\mathrm { A r j } { + } 2 0 ]$ . Our approach follows this tradition, but focuses on splitting $X$ into components defined by their causal relationships with the label $Y$ and an additional covariate $Z$ . Rather than attempting to infer the causal relationship between $X$ and $Y$ , we show that domain knowledge of this relationship is essential for obtaining counterfactually-invariant predictors. The role of causal vs anti-causal data generation in semi-supervised learning and transfer learning has also been studied $[ \mathrm { S c h } \cdot 2 1 $ ; $\operatorname { S c h + } 1 2 ]$ . In this paper we focus on a different implication of the causal vs anti-causal distinction.
+
+Another line of work considers the case where either the counterfactuals $X ( z )$ , $X ( z ^ { \prime } )$ are observed for at least some data points, or where it’s assumed that there’s enough structural knowledge to (learn to) generate counterfactual examples [RPH21; $\mathrm { W u } { + } 2 1$ ; Gar+19; $\mathrm { M i t } { + } 2 0$ ; WZ19; KCC20; KHL20; TAH20]. Kusner et al. [Kus+17] and Garg et al. [Gar $^ +$ 19] in particular examine a notion of counterfactual fairness that can be seen as equivalent to counterfactual invariance. In these papers, approximate counterfactuals are produced by direct manipulation of the features (e.g., change male to female names), generative models (e.g., style transfer of images), or crowdsourcing. Then, these counterfactuals can either be used as additional training data or the predictor can be regularized such that it cannot distinguish between $X _ { i } ( z )$ and $X _ { i } ( z ^ { \prime } )$ . This strategy can be viewed as enforcing counterfactual invariance directly; an advantage is that it avoids the necessary-but-not-sufficient nuance of Theorem 3.2. However, counterfactual examples can be difficult to obtain for language data in many realistic problem domains, and it may be difficult to learn to generalize from such examples [HLB20].
+
+The marginal and conditional independencies of Theorem 3.2 have appeared in other contexts. As discussed in remark 3.3, if we think of $Z$ as a protected attribute and $f$ as a ‘fair’ classifier, then counterfactual invariance is counterfactual fairness, the marginal independence is demographic parity, and the conditional independence is equalized odds [Meh $^ +$ 19]. In another setting, one approach to domain adaptation is to seek representations $\phi$ such that either $\phi ( X )$ [e.g., MBS13; Bak $+ 1 3$ ; Gan $+ 1 6$ ; Tze+14] or $\phi ( X ) \mid Y$ [e.g., MLM19; $\mathrm { Y a n } { + } 1 7 ]$ have the same distribution in each environment. The connection here is subtler; it is inappropriate to interpret $Z$ as a domain label (we only have access to one domain at train time, and all values of $Z$ are present in each domain). To clarify the connection, consider introducing an extra variable $E$ that labels the domain. For concreteness, consider the anti-causal case where the spurious association between $Y$ and $Z$ is due to an observed confounder $U$ . Now, suppose that $E$ is a cause of $U$ . Then, $E$ labels the distribution of the unobserved confounder, and thus different values of $E$ correspond to different causally compatible domains. Now, if we draw the causal graph we can read off that $X _ { z } ^ { \perp } E | Y$ . That is, we arrive at the same conditional independence criterion. However, it’s essentially a coincidence that this matches the criterion we’d use if we’d observed $Z$ —the two variables have totally different causal meanings and interpretations. For example, in the review data case, we might take $Z$ to be review score and $E$ to be the website the review is collected from. The review quality doesn’t need to be counterfactually invariant to the website, and indeed we wouldn’t expect it to be!
+
+Finally, this work can be viewed as part of a recent line of work using counterfactuals to examine the connection between example-wise robustness (stress testing) and distributional-level robustness (domain shift) [e.g., TAH20; RPH21; Kau+21]. Teney et al. [TAH20] use near counterfactuals with different labels in a modified training objective, and show improved performance out-of-domain empirically. Robey et al. [RPH21] use image models to generate counterfactuals, and incorporate the model into training, showing out-of-domain improvement. They assume a somewhat different causal structure—articulating the precise connection is an interesting future direction. Kaushik et al. [Kau+21] study a linear-Gaussian model and argue that out-of-domain performance can be used as a signal to distinguish ‘causal’ and ‘spurious’ features.
+
+# 7 Discussion
+
+We used the tools of causal inference to formalize and study perturbative stress tests. A main insight of the paper is that counterfactual desiderata can be linked to observationally-testable conditional independence criteria. This requires consideration of the true underlying causal structure of the data. Done correctly, the link yields a simple procedure for enforcing the counterfactual desiderata, and mitigating the effects of domain shift.
+
+The main limitation of the paper is the restrictive causal structures we consider. In particular, we require that $X _ { Z } ^ { \perp }$ , the part of $X$ not causally affected by $Z$ , is also statistically independent of $Z$ in the observed data. However, in practice these may be dependent due to a common cause. In this case, the procedure here will be overly conservative, throwing away more information than required. Additionally, it is not obvious how to apply the ideas described here to more complicated causal situations, which can occur in structured prediction (e.g., question answering). Extending the ideas to handle richer causal structures is an important direction for future work. The work described here can provide a template for this research program.
+
+# 8 Acknowledgements
+
+Thanks to Kevin Murphy, Been Kim, Molly Roberts, Justin Grimmer, and Brandon Stewart for feedback on an earlier version.
+
+# 9 Funding Transparency
+
+This paper was not supported by any third party funding, and the authors have no competing interests to declare.
+
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+$[ \mathrm { W u } + 2 1 ]$ T. Wu, M. T. Ribeiro, J. Heer, and D. S. Weld. “Polyjuice: generating counterfactuals for explaining, evaluating, and improving models”. In: Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics. 2021.   
+[Yan+17] H. Yan, Y. Ding, P. Li, Q. Wang, Y. Xu, and W. Zuo. Mind the Class Weight Bias: Weighted Maximum Mean Discrepancy for Unsupervised Domain Adaptation. 2017. arXiv: 1705.00609 [cs.CV].   
+[Zha+13] K. Zhang, B. Scholkopf, K. Muandet, and Z. Wang. “Domain adaptation under target ¨ and conditional shift”. In: Proceedings of the 30th International Conference on Machine Learning. 2013.

parse/train/BdKxQp0iBi8/BdKxQp0iBi8_content_list.json

@@ -0,0 +1,1313 @@
+[
+    {
+        "type": "text",
+        "text": "Counterfactual Invariance to Spurious Correlations: Why and How to Pass Stress Tests ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Victor Veitch1,2, Alexander D’Amour1, Steve Yadlowsky1, and Jacob Eisenstein1 ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1Google Research   \n2University of Chicago ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Informally, a ‘spurious correlation’ is the dependence of a model on some aspect of the input data that an analyst thinks shouldn’t matter. In machine learning, these have a know-it-when-you-see-it character; e.g., changing the gender of a sentence’s subject changes a sentiment predictor’s output. To check for spurious correlations, we can ‘stress test’ models by perturbing irrelevant parts of input data and seeing if model predictions change. In this paper, we study stress testing using the tools of causal inference. We introduce counterfactual invariance as a formalization of the requirement that changing irrelevant parts of the input shouldn’t change model predictions. We connect counterfactual invariance to outof-domain model performance, and provide practical schemes for learning (approximately) counterfactual invariant predictors (without access to counterfactual examples). It turns out that both the means and implications of counterfactual invariance depend fundamentally on the true underlying causal structure of the data—in particular, whether the label causes the features or the features cause the label. Distinct causal structures require distinct regularization schemes to induce counterfactual invariance. Similarly, counterfactual invariance implies different domain shift guarantees depending on the underlying causal structure. This theory is supported by empirical results on text classification. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our focus in this paper is the sort of spurious correlations revealed by “poke it and see what happens” testing procedures for machine-learning models. For example, we might test a sentiment analysis tool by changing one proper noun for another (“tasty Mexican food” to “tasty Indian food”), with the expectation that the predicted sentiment should not change. This kind of perturbative stress testing is increasingly popular: it is straightforward to understand and offers a natural way to test the behavior of models against the expectations of practitioners [Rib $+ 2 0$ ; $\\mathrm { W u } { + } 1 9$ ; Nai+18]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Intuitively, models that pass such stress tests are preferable to those that do not. However, fundamental questions about the use and meaning of perturbative stress tests remain open. For instance, what is the connection between passing stress tests and model performance on prediction? Eliminating predictor dependence on a spurious correlation should help with domain shifts that affect the spurious correlation—but how do we make this precise? And, how should we develop models that pass stress tests when our ability to generate perturbed examples is limited? For example, automatically perturbing the sentiment of a document in a general fashion is difficult. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The ad hoc nature of stress testing makes it difficult to give general answers to such questions. In this paper, we will use the tools of causal inference to formalize what it means for models to pass stress tests, and use this formalization to answer the questions above. We will formalize passing stress tests as counterfactual invariance, a condition on how a predictor should behave when given certain (unobserved) counterfactual input data. We will then derive implications of counterfactual invariance that can be measured in the observed data. Regularizing predictors to satisfy these observable implications provides a means for achieving (partial) counterfactual invariance. Then, we will connect counterfactual invariance to robust prediction under certain domain shifts, with the aim of clarifying what counterfactual invariance buys and when it is desirable. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "An important insight that emerges from the formalization is that the true underlying causal structure of the data has fundamental implications for both model training and guarantees. Methods for handing ‘spurious correlations’ in data with a given causal structure need not perform well when blindly translated to another causal structure. ",
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+    {
+        "type": "text",
+        "text": "Counterfactual Invariance Consider the problem of learning a predictor $f$ that predicts a label $Y$ from covariates $X$ . In this paper, we’re interested in constructing predictors whose predictions are invariant to certain perturbations on $X$ . Our first task is to formalize the invariance requirement. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To that end, assume that there is an additional variable $Z$ that captures information that should not influence predictions. However, $Z$ may causally influence the covariates $X$ . Using the potential outcomes notation, let $X ( z )$ to denote the counterfactual $X$ we would have seen had $Z$ been set to $z$ , leaving all else fixed. Informally, we can understand perturbative stress tests as a way of producing particular realizations of counterfactual pairs $X ( z )$ , $X ( z ^ { \\prime } )$ that differ by an intervention on $z$ . Then, we formalize the requirement that an arbitrary change to $z$ does not change predictions: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Definition 1.1. A predictor $f$ is counterfactually invariant to $Z$ if $f ( X ( z ) ) \\ : = \\ : f ( X ( z ^ { \\prime } ) )$ almost everywhere, for all $z , z ^ { \\prime }$ in the sample space of $Z$ . When $Z$ is clear from context, we’ll just say the predictor is counterfactually invariant. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Causal Structure ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Counterfactual invariance is a condition on how the predicted label behaves under interventions on parts of the input data. However, intuitions about stress testing are based on how the true label behaves under interventions on parts of the input data. We will see that the true causal structure fundamentally affects both the implications of counterfactual invariance, and the techniques we use to achieve it. To study this phenomenon, we’ll use two causal structures that are commonly encountered in applications; see Figure 1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1 Prediction in the Causal Direction ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We begin with the case where $X$ is a cause of $Y$ ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Example 2.1. We want to automatically classify the quality of product reviews. Each review has a number of “helpful” votes $Y$ (from site users). We predict $Y$ using the text of the product review $X$ . However, we find interventions on the sentiment $Z$ of the text change our prediction; changing “Great shoes!” to “Bad shoes!” changes the prediction. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the examples in this paper, the covariate $X$ is text data. Usually, the causal relationship between the text and $Y$ and $Z$ will be complex— e.g., the relationships may depend on abstract, unlabeled, parts of the text such as topic, writing quality, or tone. In principle, we could enumerate all such latent variables, construct a causal graph capturing the relationships between these variables and $Y , Z$ , and use this causal structure to study counterfactual invariance. For instance, if we think that topic causally influences the helpfulness $Y$ , but is not influenced by sen",
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/db6f289cfcf6978a6de564bcb64d3637aad50130eac9708db787c9ff88ca27f4.jpg",
+        "image_caption": [
+            "Figure 1: Causal models for the data generating process. We decompose the observed covariate $X$ into latent parts defined by their causal relationships with $Z$ and $Y$ . Solid arrows denote causal relationships, while dashed lines denote non-causal associations. The differences between these causal structures will turn out to be key for understanding counterfactual invariance. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "timent $Z$ , then we could build a counterfactually invariant predictor by extracting the topic and predicting $Y$ using topic alone. However, exhaustively articulating all possible such variables is a herculean task. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Instead, notice that the only thing that’s relevant about these latent variables is their causal relationship with $Y$ and $Z$ . Accordingly, we’ll decompose the observed variable $X$ into parts defined by their causal relationships with $Y$ and $Z$ . We remain agnostic to the semantic interpretation of these parts. Namely, we define $X _ { Z } ^ { \\perp }$ as the part of $X$ that is not causally influenced by $Z$ (but may influence $Y )$ , $X _ { Y } ^ { \\perp }$ as the part that does not causally influence $Y$ (but may be influenced by $Z$ ), and $X _ { Y \\wedge Z }$ is the remaining part that is both influenced by $Z$ and that influences $Y$ . The causal structure is shown in Figure 1a. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We see there are two paths that lead to $Y$ and $Z$ being associated. The first is when $Z$ affects $X _ { Y \\wedge Z }$ which, in turn, affects $Y$ . For example, a very enthusiastic reviewer might write a longer, more detailed review, which will in turn be more helpful. The second is when a common cause or selection effect in the data generating process induces an association between $Z$ and $Y$ , which we denote with a dashed arrow. For example, if books tend to get more positive reviews, and also people who buy books are more likely to flag reviews as helpful, then the product type would be a common cause of sentiment and helpfulness. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Prediction in the Anti-Causal Direction ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We also consider the case where $Y$ causes $X$ ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Example 2.2. We want to predict the star rating $Y$ of movie reviews from the text $X$ . However, we find that predictions are influenced by the movie genre $Z$ ; e.g., changing “Adam Sandler” (a comedy actor) to “Hugh Grant” (a romance actor) changes the predictions. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Figure 1b shows the causal structure. Here, the observed $X$ is influenced by both $Y$ and $Z$ . Again, we decompose $X$ into parts defined by their causal relationship with $Z$ and $Y$ . Here, $Z$ (and thus $X _ { Y } ^ { \\perp }$ ) can be associated with $Y$ through two paths. First, if $X _ { Y \\wedge Z }$ is non-trivial, then conditioning on it causes a dependence between $Z$ and $Y$ (because $X _ { Y \\wedge Z }$ is a collider). For example, if Adam Sandler tends to appear in good comedy movies but bad movies of other genres then seeing “Sandler” in the text induces a dependency between sentiment and genre. Second, $Z$ and $Y$ may be associated due to a common cause, or due to selection effects in the data collection protocol—this is represented by the dashed line between $Z$ and $Y$ . For example, fans of romantic comedies may tend to give higher reviews (to all films) than fans of horror movies. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 Non-Causal Associations ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Frequently, a predictor trained to predict $Y$ from $X$ will rely on $X _ { Y } ^ { \\perp }$ , even though there is no causal connection between $Y$ and $X _ { Y } ^ { \\perp }$ , and therefore will fail counterfactual invariance. The reason is that $X _ { Y } ^ { \\perp }$ serves as a proxy for $Z$ , and $Z$ is predictive of $Y$ due to the non-causal (dashed line) association. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "There are two mechanisms that can induce such associations. First, $Y$ and $Z$ may be confounded: they are both influenced by an unobserved common cause $U$ . For example, people who review books may be more upbeat than people who review clothing. This leads to positive sentiments and high helpfulness votes for books, creating an association between sentiment and helpfulness. Second, $Y$ and $Z$ may be subject to selection: there is some condition (event) $S$ that depends on $Y$ and $Z$ , such that a data point from the population is included in the sample only if $S = 1$ holds. For example, our training data might only include movies with at least 100 reviews. If only excellent horror movies have so many reviews (but most rom-coms get that many), then this selection would induce an association between genre and score. Formally, the dashed-line causal graphs mean our sample is distributed according to $\\begin{array} { r } { P ( X , Y , Z ) = \\int \\mathrm { P } ( \\dot { X } , Y , Z , u \\mid S = 1 ) \\mathrm { d } \\mathrm { P } ( u ) } \\end{array}$ where $Y , Z$ are caused by $U$ and are causes of $S$ , and $( X , Y , Z )$ are causally related according to the graph. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In addition to the non-causal dashed-line relationship, there is also dependency induced by between $Y$ and $Z$ by $X _ { Y \\wedge Z }$ . Whether or not each of these dependencies is “spurious” is a problem-specific judgement that must be made by each analyst based on their particular use case. E.g., using genre to predict sentiment may or may not be reasonable, depending on the actual application in mind. However, there is a special case that captures a common intuition for purely spurious association. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.3. We say that the association between $Y$ and $Z$ is purely spurious if $Y \\perp X \\mid X _ { Z } ^ { \\perp } , Z$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "That is, if the dashed-line association did not exist (removed by conditioning on $Z$ ) then the part of $X$ that is not influenced by $Z$ would suffice to estimate $Y$ . ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 Observable Signatures of Counterfactually Invariant Predictors ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We now consider the question of how to achieve counterfactual invariance in practice. The challenge is that counterfactual invariance is defined by the behavior of the predictor on counterfactual data that is never actually observed. This makes checking counterfactual invariance impossible. Instead, we’ll derive a signature of counterfactual invariance that actually can be measured—and enforced— using ordinary datasets where $Z$ (or a proxy) is measured. For example, the star rating of a review as a proxy for sentiment, or genre labels in the movie review case. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Intuitively, a predictor $f$ is counterfactually invariant if it depends only on $X _ { Z } ^ { \\perp }$ , the part of $X$ that is not affected by $Z$ . To formalize this, we need to show that such a $X _ { Z } ^ { \\perp }$ is well defined: ",
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+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.1. Let $X _ { Z } ^ { \\perp }$ be a $X$ -measurable random variable such that, for all measurable functions $f$ , we have that $f$ is counterfactually invariant if and only if $f ( X )$ is $X _ { Z } ^ { \\perp }$ -measurable. If $Z$ is discrete1 then such a $X _ { Z } ^ { \\perp }$ exists. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Accordingly, we’d like to construct a predictor that is a function of $X _ { Z } ^ { \\perp }$ only (i.e., is $X _ { Z } ^ { \\perp }$ measurable). The key insight is that we can use the causal graphs to read off a set of conditional independence relationships that are satisfied by $Z , X _ { Z } ^ { \\perp } , Y$ . Critically, these conditional independence relationships are testable from the observed data. Thus, they provide a signature of counterfactual invariance: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2. If $f$ is a counterfactually invariant predictor: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1. Under the anti-causal graph, $f ( X ) \\perp Z \\mid Y$ .   \n2. Under the causal-direction graph, if $Y$ and $Z$ are not subject to selection (but possibly confounded), $f ( X ) \\perp \\perp Z$ .   \n3. Under the causal-direction graph, if the association is purely spurious, $Y \\perp X \\mid X _ { Z } ^ { \\perp } , Z$ , and $Y$ and $Z$ are not confounded (but possibly selected), $f ( X ) \\perp Z \\mid Y$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Remark 3.3 (Connection to Fairness). This result has an interesting interpretation when $Z$ is a protected attribute (e.g., sex or race) that we’d like to be fair with respect to. There are many ways of formalizing fairness, which are usually mutually incompatible. In the fairness setting, counterfactual invariance is equivalent to counterfactual fairness [Kus+17; Gar+19], the condition $f ( X ) \\ \\perp \\ Z$ is equivalent to demographic parity, and the condition $f ( X ) \\perp Z \\mid Y$ is equivalent to equalized odds [Meh+19]. Theorem 3.2 then says that counterfactual fairness implies either demographic parity or equalized odds, depending on the true underlying causal structure of the problem. Hence, the relationship between disparate fairness notions is clarified by taking the underlying causal structure into account. This also suggests we can take counterfactual fairness as the fundamental notion, then use demographic parity in the causal-confounding case and equalized odds otherwise. However, we leave the (ethical) question of whether this is a sound strategy to future work. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Causal Regularization Without access to counterfactual examples, we cannot directly enforce counterfactual invariance. However, we can require a trained model to satisfy the counterfactual invariance signature of Theorem 3.2. The hope is that enforcing the signature will lead the model to be counterfactually invariant. To do this, we regularize the model to satisfy the appropriate conditional independence condition. For simplicity of exposition, we restrict to binary $Y$ and $Z$ . The (infinite data) regularization terms are ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/86ce0fc659886f708a5869aef0be03c74b34df0057a1a477df5f58b2916a67d7.jpg",
+        "text": "$$\n\\begin{array} { r l } & { = \\mathrm { M M D } ( \\mathrm { P } ( f ( X ) \\mid Z = 0 ) , \\mathrm { P } ( f ( X ) \\mid Z = 1 ) ) } \\\\ & { = \\mathrm { M M D } ( \\mathrm { P } ( f ( X ) \\mid Z = 0 , Y = 0 ) , \\mathrm { P } ( f ( X ) \\mid Z = 1 , Y = 0 ) ) } \\\\ & { \\quad + \\mathrm { M M D } ( \\mathrm { P } ( f ( X ) \\mid Z = 0 , Y = 1 ) , \\mathrm { P } ( f ( X ) \\mid Z = 1 , Y = 1 ) ) . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Maximum mean discrepancy (MMD) is a metric on probability measures.2 The marginal independence condition is equivalent to (3.1) equal 0, and the conditional independence is equivalent to (3.2) ",
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+    },
+    {
+        "type": "text",
+        "text": "equal 0. In practice, we can estimate the MMD with finite data samples $[ \\mathrm { G r e } { + } 1 2 ]$ . When training with stochastic gradient descent, we compute the penalty on each minibatch. ",
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+    {
+        "type": "text",
+        "text": "The procedure is then: if the data has causal-direction structure and the $Y  Z$ association is due to confounding, add the marginal regularization term to the the training objective. If the data has anti-causal structure, or the association is due to selection, add the conditional regularization term instead. In this way, we regularize towards models that satisfy the counterfactual invariance signature. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A key point is that the regularizer we must use depends on the true causal structure. The conditional and marginal independence conditions are generally incompatible. Enforcing the condition that is mismatched to the true underlying causal structure may fail to encourage counterfactual invariance, or may throw away more information than is required. ",
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+    {
+        "type": "text",
+        "text": "Gap to Counterfactual Invariance The conditional independence signature of Theorem 3.2 is necessary but not sufficient for counterfactual invariance. This is for two reasons. First, counterfactual invariance applies to individual datapoint realizations, but the signature is distributional. In particular, the invariance $\\mathrm { P } ( f ( X ) \\mid \\mathrm { d o } ( Z = z ) ) = \\mathrm { P } ( f ( X ) \\mid \\mathrm { d o } ( Z = \\bar { z } ^ { \\prime } ) )$ for all $z , z ^ { \\prime }$ would also imply the conditional independence signature. But, this invariance is weaker than counterfactual invariance, since it doesn’t require access to counterfactual realizations. Second, $f ( X ) \\ \\perp \\perp Z$ does not imply, in general, that $Z$ is not a cause of $f ( X )$ . This (unusual) behavior can happen if, e.g., there are levels of $Z$ that we do not observe in the training data, or there are variables omitted from the causal graph that are a common cause of $Z$ and $X$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Unfortunately, the gap between the signature and counterfactual invariance is a fundamental restriction of using observational data. The conditional independence signature is in some sense the closest proxy for counterfactual invariance we can hope for. In section 5, we’ll see that enforcing the signature does a good job of enforcing counterfactual invariance in practice. ",
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+    {
+        "type": "text",
+        "text": "4 Performance Out of Domain ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Counterfactual invariance is an intuitively desirable property for a predictor to have. However, it’s not immediately clear how it relates to model performance as measured by, e.g., accuracy. Intuitively, eliminating predictor dependence on a spurious $Z$ may help with domain shift, where the data distribution in the target domain differs from the distribution of the training data. We now turn to formalizing this idea. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "First, we must articulate the set of domain shifts to be considered. In our setting, the natural thing is to hold the causal relationships fixed across domains, but to allow the non-causal (“spurious”) dependence between $Y$ and $Z$ to vary. Demanding that the causal relationships stay fixed reflects the requirement that the causal structure describes the dynamics of an underlying real-world process— e.g., the author’s sentiment is always a cause (not an effect) of the text in all domains. On the other hand, the dependency between $Y$ and $Z$ induced by either confounding or selection can vary without changing the underlying causal structure. For confounding, the distribution of the confounder may differ between domains—e.g., books are rare in training, but common in deployment. For selection, the selection criterion may differ between domains—e.g., we include only frequently reviewed movies in training, but make predictions for all movies in deployment. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We want to capture spurious domain shifts by considering domain shifts induced by selection or confounding. However, there is an additional nuance. Changes to the marginal distribution of $Y$ will affect the risk of a predictor, even in the absence of any spurious association between $Y$ and $Z$ . Therefore, we restrict to shifts that preserve the marginal distribution of $Y$ . ",
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+    {
+        "type": "text",
+        "text": "Definition 4.1. We say that distributions $P , Q$ are causally compatible if both obey the same causal graph, $P ( Y ) = Q ( Y )$ , and there is a confounder $U$ and/or selection conditions $S , { \\tilde { S } }$ such that $\\begin{array} { r } { P { = } \\int \\operatorname { P } ( X , Y , Z \\mid U , S { = } 1 ) \\mathrm { d } \\tilde { P } ( U ) } \\end{array}$ and $\\begin{array} { r } { Q { = } \\int \\operatorname { P } ( X , Y , Z \\mid U , \\tilde { S } { = } 1 ) \\mathrm { d } \\tilde { Q } ( U ) } \\end{array}$ for some $\\tilde { P } ( U ) , \\tilde { Q } ( U )$ . ",
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+    {
+        "type": "text",
+        "text": "We can now connect counterfactual invariance and robustness to domain shift. ",
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+    {
+        "type": "text",
+        "text": "Theorem 4.2. Let ${ \\mathcal { F } } ^ { \\mathrm { i n v a r } }$ be the set of all counterfactually invariant predictors. Let $L$ be either square error or cross entropy loss. And, let $f ^ { * } : = \\operatorname { a r g m i n } _ { \\underline { { f } } \\in \\mathcal { F } ^ { \\mathrm { i n v a } } }$ r $\\mathbb { E } _ { P } [ L ( Y , f ( X ) ) ]$ be the counterfactually invariant risk minimizer. Suppose that the target distribution $Q$ is causally compatible with the training distribution $P$ . Suppose that any of the following conditions hold: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1. the data obeys the anti-causal graph   \n2. the data obeys the causal-direction graph, there is no confounding (but possibly selection), and the association is purely spurious, $Y \\perp X \\mid X _ { Z } ^ { \\perp } , Z$ , or   \n3. the data obeys the causal-direction graph, there is no selection (but possibly confounding), the association is purely spurious and the causal effect of $X _ { Z } ^ { \\perp }$ on $Y$ is additive, i.e., the true data generating process is ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8cec76378561abb8947c6c9ec4fdd61d4fcc15bf3f528b3e86048cc86f46e240.jpg",
+        "text": "$$\nY  g ( X _ { Z } ^ { \\perp } ) + \\tilde { g } ( U ) + \\xi w h e r e \\operatorname { \\mathbb { E } } [ \\xi \\mid X _ { Z } ^ { \\perp } ] = 0 ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "for some functions $g , \\tilde { g }$ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Then, the training domain counterfactually invariant risk minimizer is also the target domain counterfactually invariant risk minimizer, $f ^ { * } = \\mathrm { a r g m i n } _ { f \\in \\mathcal { F } ^ { \\mathrm { i n } } }$ var $\\mathbb { E } _ { Q } [ L ( Y , f ( X ) ) ]$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark 4.3. The causal case with confounding requires an additional assumption (additive structure) because, e.g., an interaction between confounder and $X _ { Z } ^ { \\perp }$ can yield a case where $X _ { Z } ^ { \\perp }$ and $Y$ have a different relationship in each domain (whence, out-of-domain learning is impossible). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This result gives a recipe for finding a good predictor in the target domain even without access to any target domain examples at training time. Namely, find the counterfactually invariant risk minimizer in the training domain. In practice, we can use the regularization scheme of section 3 to (approximately) achieve this. We’ll see in section 5 that this works well in practice. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Optimality Theorem 4.2 begs the question: if the only thing we know about the target setting is that it’s causally compatible with the training data, is the best predictor the counterfactually invariant predictor with lowest training risk? A natural way to formalize this question is to study the predictor with the best performance in the worst case target distribution. We define $ \\bar { \\mathcal { Q } } ~ = ~ \\{ \\bar { Q } ~ : ~ Q$ causally compatible with $P \\}$ and the $\\mathcal { Q }$ -minimax predictor $f _ { \\mathrm { m i n i m a x } } ^ { * } ~ =$ $\\operatorname { \\arg m i n } _ { f \\in { \\mathcal { F } } } \\operatorname* { m a x } _ { Q \\in { \\mathcal { Q } } } \\mathbb { E } _ { Q } [ L ( Y , f ( X ) ]$ . The question is then: what’s the relationship between the counterfactually invariant risk minimizer and the minimax predictor? ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 4.4. The counterfactually invariant risk minimizer is not $\\mathcal { Q }$ -minimax in general. However, under the conditions of Theorem 4.2, if the association is purely spurious, $X _ { Y \\wedge Z } \\ { \\bar { \\perp } } \\ Y \\mid X _ { Z } ^ { \\perp } , Z$ , and $\\mathrm { P } ( Z , Y )$ satisfies overlap, then the two predictors are the same. By overlap we mean that $\\mathrm { P } ( Z , Y )$ is a discrete distribution such that for all $( z , y )$ , $i f \\mathrm { P } ( z , y ) > 0$ then there is some $y ^ { \\prime } \\ne y$ such that also $\\mathrm { P } ( z , y ^ { \\prime } ) > 0$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Conceptually, Theorem 4.4 just says that the counterfactually invariant predictor excludes $X _ { Y \\wedge Z }$ , even when this information is useful in every domain. In the purely spurious case, $X _ { Y \\wedge Z }$ carries no useful information, so counterfactual invariance is optimal. ",
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+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The main claims of the paper are: ",
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+    {
+        "type": "text",
+        "text": "1. Stress test violations can be reduced by suitable conditional independence regularization.   \n2. This reduction will improve out-of-domain prediction performance.   \n3. To get the full effect, the imposed penalty must match the causal structure of the data. ",
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+    {
+        "type": "text",
+        "text": "Setup To assess these claims, we’ll examine the behavior of predictors trained with the marginal or conditinal regularization on multiple text datasets that have either causal or anti-causal structure. We expect to see that marginal regularization improves stress test and out-of-domain performance on data with causal-confounded structure, and conditional regularization improves these on data with anti-causal structure. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For each experiment, we use BERT [Dev+19] finetuned to predict a label $Y$ from the text as our base model. We train multiple causally-regularized models on the each dataset. The training varies by whether we use the conditional or marginal penalty, and by the strength of the regularization term. That is, we train identical architectures using CrossEntropy $+ \\lambda \\cdot$ Regularizer as the objective function, where we vary $\\lambda$ and take Regularizer as either the marginal penalty, (3.1), or conditional penalty, (3.2). We compare these models’ predictions on data with causal and anti-causal structure. ",
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+    {
+        "type": "text",
+        "text": "See supplement for experimental details. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/53efb42512085a29a0de7812344b3d241b39e4349e1f4adf7ba004e54ec1e32c.jpg",
+        "image_caption": [
+            "Figure 2: Regularizing conditional MMD improves counterfactual invariance on synthetic anti-causal data. Sufficiently high regularization of marginal MMD also improves invariance, but impairs accuracy. Dashed lines show baseline performance of an unregularized predictor. Left: lower conditional MMD implies that predictive probabilities are invariant to perturbation. Although marginal MMD penalization can result in low conditional MMD and good stress test performance, this comes at the cost of very low in-domain accuracy. Right: MMD regularization reduces the rate of predicted label flips on perturbed data, with little affect on indomain accuracy. Conditional MMD regularization reduces predicted label flips to $1 . 4 \\%$ , while the best result for marginal MMD is $2 . 8 \\%$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 Robustness to Stress Tests ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "First, we examine whether enforcing the causal regularization actually helps to enforce counterfactual invariance. We create counterfactual (stress test) examples by perturbing the input data and compare the prediction on these. We build the experimental datasets using Amazon reviews from the product category “Clothing, Shoes, and Jewelry” [NLM19]. ",
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+    },
+    {
+        "type": "text",
+        "text": "Synthetic To study the relationship between counterfactual invariance and the distributional signature of Theorem 3.2, we construct a synthetic confound. For each review, we draw a Bernoulli random $Z$ , and then perturb the text $X$ so that the common words “the” and “a” carry information about $Z$ : for example, we replace “the” with the token “thexxxxx” when $Z = 1$ . We take $Y$ to be the review score, and subsample so $Y$ is balanced. This data has anti-causal structure: the text $X$ is written to explain the score $Y$ . Further, we expect that the $Y , Z$ association is purely spurious, because “the” and “a” carry little information about the label. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We train the models on data where $\\operatorname { P } ( Y = Z ) = 0 . 3$ . We then create perturbed stress-test datasets by changing each example $X _ { i } ( z )$ to the counterfactual $X _ { i } ( 1 - z )$ (using the synthetic model). By measuring the performance of each model on the perturbed data, we can test whether the distributional properties enforced by the regularizers result in counterfactual invariance at the instance level. Figure 2 shows that conditional regularization (matching the anti-causal structure) reduces checklist failures, as measured by the frequency that the predicted label changes due to perturbation as well as the mean absolute difference in predictive probabilities that is induced by perturbation. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Natural To study the relationship in real data, we use the review data in a different way. We now take $Z$ to be the score, binarized as $Z \\in { }$ $\\{ 1$ or 2 stars, 4 or 5 stars}. We use this $Z$ as a proxy for sentiment, and consider problems where sentiment should (plausibly) not have a causal effect on $Y$ . For the causal prediction problem, we take $Y$ to be the helpfulness score of the review (binarized as described below). This is causal because readers decide whether the review is helpful based on the text. For the anti-causal prediction problem, we take $Y$ to be whether “Clothing” is included as a category tag for the product under review (e.g., boots typically do not have this tag). This is anti-causal because the product category affects the text. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/25d13abb0d1a94526c03a9edcbbe9537be776b40dc5768ab7b2869bd796e53c3.jpg",
+        "image_caption": [
+            "Figure 3: Penalizing the MMD matching the causal structure improves stress test performance on natural product review data. Note that penalizing the wrong MMD may not help: the marginal MMD hurts on the anticausal dataset. Perturbations are generated by swapping positive and negative sentiment adjectives in examples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We control the strength of the spurious association between $Y$ and $Z$ . In the anti-causal case, this is done by selection: we randomly subset the data to enforce a target level of dependence between $Y$ and $Z$ . The causal-direction case with confounding is more complicated. To manipulate confounding strength, we binarize the number of helpfulness votes $V$ in a manner determined by the target level of association. We take $Y = 1 [ V > T _ { Z } ]$ where $T _ { Z }$ is a $Z$ -dependent threshold, chosen to induce a target association. We choose $\\operatorname { P } ( Y = { \\bar { 0 } } \\mid Z = 0 ) = \\operatorname { P } ( Y = { \\bar { 1 } } \\mid Z = 1 ) = 0 . 3$ . We balance $Z$ by subsampling, which also balances $Y$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Now, we create stress test perturbations of these datasets by randomly changing adjectives in the examples. Using predefined lists of postive sentiment adjectives and negative sentiment adjectives, we swap any adjective that shows up on a list with a randomly sampled adjective from the other list. This preserves basic sentence structure, and thus creates a limited set of counterfactual pairs that differ on sentiment. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results for differences in predicted probabilities between original and perturbed data are shown in Figure 3. Each point is a trained model, which vary in measured MMD on the test data and on sensitivity to perturbations. Recall that the conditional independence signature of Theorem 3.2 are necessary but not sufficient for counterfactual invariance, so it’s not certain that regularizing to reduce the MMD will reduce perturbation sensitivity. Happily, we see that regularizing to reduce the MMD that matches the causal structure does indeed reduce sensitivity to perturbations. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Notice that regularizing the causally mismatched MMD can have strange effects. Regularizing marginal MMD in the anti-causal case actually makes the model more sensitive to perturbations! ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 Domain Shift ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Next, we study the effect of causal regularization on model performance under domain shift. ",
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+    {
+        "type": "text",
+        "text": "Natural Product Review We again use the natural Amazon review data described above. For both the causal and anticausal data, we create multiple test sets with variable spurious correlation strength. This is done in the manner described above, varying $\\operatorname { P } ( Y = 0 \\mid Z =$ $0 ) = \\operatorname { P } ( Y = 1 \\mid Z = 1 ) =$ $\\gamma$ . Here, $\\gamma$ is the strength of spurious association. The test sets are out-of-domain samples. By design, $Y$ is balanced in each dataset, so these samples are causally compatible with the training data. For both the causal and anti-causal datasets, the training data has $\\begin{array} { r l } { \\mathrm { P } ( Y } & { { } = } \\end{array}$ $0 \\mid Z = 0 \\bar { ) } = \\mathrm { P } ( Y = 1 \\mid Z =$ $1 ) ~ = ~ 0 . 3$ . We train a classifier for each regularization type and regularization strength, and measure the accuracy on each test domain. The results are shown in Figure 4. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/cc50159fcb6943974ab865fe3f6e0e57889243fedb2eb828a12ef8b80e7b30b1.jpg",
+        "image_caption": [
+            "Anti-Causal Data: conditional regularization improves domain-shift robustness. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/1aad0d1d99d5ec2ccf3cc08ff60ebd5e9382fde6d09560b6780ee4953b6c184c.jpg",
+        "image_caption": [
+            "Causal-Direction Data: marginal regularization improves domain-shift robustness. ",
+            "Figure 4: The best domain-shift robustness is obtained by using the regularizer that matches the underlying causal structure of the data. The plots show out-of-domain accuracy for models trained on the (natural) review data. In each row, the left figure shows out-of-domain accuracies (lines are models), with the $X$ -axis showing the level of spurious correlation in the test data (0.3 is the training condition); the right figure shows worst out-of-domain accuracy versus in-domain test accuracy (dots are models). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "First, the unregularized predictors do indeed learn to rely on the spurious association between sentiment and the label. The accuracy of these predictors decays dramatically as the spurious assocation moves from negative (0.3) to positive—in the causal case, the unregularized predictor is worse than chance in the 0.8 domain. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Following section 3, the regularization that matches the underlying causal structure should yield a predictor that is (approximately) counterfactually invariant. Following Theorem 4.2, we expect that good performance of a counterfactually-invariant predictor in the training domain should imply good performance in each of the other domains. Indeed, we see that this is so. Models that are regularized to have small values of the appropriate MMD do indeed have better out-of-domain performance. ",
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+        "img_path": "images/a10a5b2248012b6f81c4cccbf8cd45aa1b0e5f41578a88117ad325f1b0dfe2e9.jpg",
+        "image_caption": [
+            "Figure 5: Conditional MMD penalization improves robustness in anti-causal MNLI data. Marginal regularization does not improve over the baseline unregularized model, shown with dashed lines. Left: Conditional regularization improves minimum accuracy across $( Y , Z )$ groups. When overregularized, the predictor returns the same $\\hat { Y }$ for all inputs, yielding a worst-group accuracy of 0. Right: Conditional MMD regularization significantly improves worst $( Y , Z )$ group accuracy $y$ -axis) while only mildly reducing overall accuracy ( $x$ -axis). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Such models have somewhat worse in-domain performance, because they no longer exploit the spurious correlation. ",
+        "bbox": [
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+        "type": "text",
+        "text": "MNLI Data For an additional test on naturally-occurring confounds, we use the multigenre natural language inference (MNLI) dataset [WNB18]. Instances are concatenations of two sentences, and the label describes the semantic relationship between them, $Y \\in \\mathbf { \\Sigma }$ {contradiction, entailment, neutral}. There is a well-known confound in this dataset: examples where the second sentence contain a negation word (e.g., “not”) are much more likely to be labeled as contradictions $[ \\mathrm { G u r } { + } 1 8 ]$ . Following Sagawa et al. $[ \\mathrm { S a g } + 2 0 ]$ , we set $Z$ to indicate whether one of a small set of negation words is present. Although $Z$ is derived from the text $X$ , it can be viewed as a proxy for a latent variable indicating whether the author intended to use negation in the text. This is an anti-causal prediction problem: the annotators were instructed to write text to reflect the desired label [WNB18]. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Following Sagawa et al. $[ \\mathrm { S a g } + 2 0 ]$ , we divide the MNLI data into groups by $( Y , Z )$ and compute the “worst group accuracy” across all such groups. Because this is an anti-causal problem, we predict that the conditional MMD is a more appropriate penalty than the marginal MMD. As shown in Figure 5, this prediction holds: conditional MMD regularization dramatically improves performance on the worst group, while only lightly impacting the overall accuracy across groups. ",
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+    {
+        "type": "text",
+        "text": "6 Related work ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Several papers draw a connection between causality and domain shifts [SS18; SCS19; $\\mathrm { \\bf A r j + } 2 0$ ; Mei18; PBM16; $\\mathrm { R C } { + 1 8 }$ ; Zha+13; $\\operatorname { S c h } { + 1 2 } ]$ . Typically, this work considers a prediction setting where the covariates $X$ include both causes and effects of $Y$ , and it is unknown which is which. The goal is to learn to predict $Y$ using only its causal parents. Zhang et al. $\\left[ Z \\mathrm { h a } + 1 3 \\right]$ considers anti-causal domain shift induced by changing $P ( Y )$ and proposes a data reweighting scheme. By contrast, counterfactual invariance is an example-wise notion of invariance, not an invariance across environments. In particular, learning a counterfactually invariant predictor only requires access to data from a single environment. Theorem 4.2 establishes that counterfactual invariance does imply a certain domain generalization guarantee over causally-compatible domains. Note though that the notion of causal compatibility is not generally the same as class of domain shifts previously considered. For example, we have invariant performance in the anti-causal setting, but this is ruled out by Arjovsky et al. $[ \\mathrm { A r j } { + } 2 0 ]$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A related body of work focuses on “causal representation learning” [Bes+19; $\\mathrm { L o c } { + 2 0 }$ ; $\\operatorname { S c h } { + 2 1 }$ ; $\\mathrm { A r j } { + } 2 0 ]$ . Our approach follows this tradition, but focuses on splitting $X$ into components defined by their causal relationships with the label $Y$ and an additional covariate $Z$ . Rather than attempting to infer the causal relationship between $X$ and $Y$ , we show that domain knowledge of this relationship is essential for obtaining counterfactually-invariant predictors. The role of causal vs anti-causal data generation in semi-supervised learning and transfer learning has also been studied $[ \\mathrm { S c h } \\cdot 2 1 $ ; $\\operatorname { S c h + } 1 2 ]$ . In this paper we focus on a different implication of the causal vs anti-causal distinction. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Another line of work considers the case where either the counterfactuals $X ( z )$ , $X ( z ^ { \\prime } )$ are observed for at least some data points, or where it’s assumed that there’s enough structural knowledge to (learn to) generate counterfactual examples [RPH21; $\\mathrm { W u } { + } 2 1$ ; Gar+19; $\\mathrm { M i t } { + } 2 0$ ; WZ19; KCC20; KHL20; TAH20]. Kusner et al. [Kus+17] and Garg et al. [Gar $^ +$ 19] in particular examine a notion of counterfactual fairness that can be seen as equivalent to counterfactual invariance. In these papers, approximate counterfactuals are produced by direct manipulation of the features (e.g., change male to female names), generative models (e.g., style transfer of images), or crowdsourcing. Then, these counterfactuals can either be used as additional training data or the predictor can be regularized such that it cannot distinguish between $X _ { i } ( z )$ and $X _ { i } ( z ^ { \\prime } )$ . This strategy can be viewed as enforcing counterfactual invariance directly; an advantage is that it avoids the necessary-but-not-sufficient nuance of Theorem 3.2. However, counterfactual examples can be difficult to obtain for language data in many realistic problem domains, and it may be difficult to learn to generalize from such examples [HLB20]. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The marginal and conditional independencies of Theorem 3.2 have appeared in other contexts. As discussed in remark 3.3, if we think of $Z$ as a protected attribute and $f$ as a ‘fair’ classifier, then counterfactual invariance is counterfactual fairness, the marginal independence is demographic parity, and the conditional independence is equalized odds [Meh $^ +$ 19]. In another setting, one approach to domain adaptation is to seek representations $\\phi$ such that either $\\phi ( X )$ [e.g., MBS13; Bak $+ 1 3$ ; Gan $+ 1 6$ ; Tze+14] or $\\phi ( X ) \\mid Y$ [e.g., MLM19; $\\mathrm { Y a n } { + } 1 7 ]$ have the same distribution in each environment. The connection here is subtler; it is inappropriate to interpret $Z$ as a domain label (we only have access to one domain at train time, and all values of $Z$ are present in each domain). To clarify the connection, consider introducing an extra variable $E$ that labels the domain. For concreteness, consider the anti-causal case where the spurious association between $Y$ and $Z$ is due to an observed confounder $U$ . Now, suppose that $E$ is a cause of $U$ . Then, $E$ labels the distribution of the unobserved confounder, and thus different values of $E$ correspond to different causally compatible domains. Now, if we draw the causal graph we can read off that $X _ { z } ^ { \\perp } E | Y$ . That is, we arrive at the same conditional independence criterion. However, it’s essentially a coincidence that this matches the criterion we’d use if we’d observed $Z$ —the two variables have totally different causal meanings and interpretations. For example, in the review data case, we might take $Z$ to be review score and $E$ to be the website the review is collected from. The review quality doesn’t need to be counterfactually invariant to the website, and indeed we wouldn’t expect it to be! ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Finally, this work can be viewed as part of a recent line of work using counterfactuals to examine the connection between example-wise robustness (stress testing) and distributional-level robustness (domain shift) [e.g., TAH20; RPH21; Kau+21]. Teney et al. [TAH20] use near counterfactuals with different labels in a modified training objective, and show improved performance out-of-domain empirically. Robey et al. [RPH21] use image models to generate counterfactuals, and incorporate the model into training, showing out-of-domain improvement. They assume a somewhat different causal structure—articulating the precise connection is an interesting future direction. Kaushik et al. [Kau+21] study a linear-Gaussian model and argue that out-of-domain performance can be used as a signal to distinguish ‘causal’ and ‘spurious’ features. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 Discussion ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            665,
+            292,
+            683
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We used the tools of causal inference to formalize and study perturbative stress tests. A main insight of the paper is that counterfactual desiderata can be linked to observationally-testable conditional independence criteria. This requires consideration of the true underlying causal structure of the data. Done correctly, the link yields a simple procedure for enforcing the counterfactual desiderata, and mitigating the effects of domain shift. ",
+        "bbox": [
+            174,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The main limitation of the paper is the restrictive causal structures we consider. In particular, we require that $X _ { Z } ^ { \\perp }$ , the part of $X$ not causally affected by $Z$ , is also statistically independent of $Z$ in the observed data. However, in practice these may be dependent due to a common cause. In this case, the procedure here will be overly conservative, throwing away more information than required. Additionally, it is not obvious how to apply the ideas described here to more complicated causal situations, which can occur in structured prediction (e.g., question answering). Extending the ideas to handle richer causal structures is an important direction for future work. The work described here can provide a template for this research program. ",
+        "bbox": [
+            173,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "8 Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            89,
+            367,
+            107
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Thanks to Kevin Murphy, Been Kim, Molly Roberts, Justin Grimmer, and Brandon Stewart for feedback on an earlier version. ",
+        "bbox": [
+            174,
+            119,
+            825,
+            148
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "9 Funding Transparency ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            167,
+            397,
+            185
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This paper was not supported by any third party funding, and the authors have no competing interests to declare. ",
+        "bbox": [
+            174,
+            199,
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+        "page_idx": 10
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
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+            263
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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Larochelle, F. Laviolette, M. Marchand, and V. Lempitsky. “Domain-adversarial training of neural networks”. In: J. Mach. Learn. Res. 1 (2016).   \n[Gar+19] S. Garg, V. Perot, N. Limtiaco, A. Taly, E. H. Chi, and A. Beutel. “Counterfactual fairness in text classification through robustness”. In: Proceedings of the 2019 AAAI/ACM Conference on AI, Ethics, and Society. 2019.   \n[Gre+12] A. Gretton, K. M. Borgwardt, M. J. Rasch, B. Scholkopf, and A. Smola. “A kernel ¨ two-sample test”. In: The Journal of Machine Learning Research 1 (2012).   \n[Gur+18] S. Gururangan, S. Swayamdipta, O. Levy, R. Schwartz, S. Bowman, and N. A. Smith. “Annotation artifacts in natural language inference data”. In: Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Short Papers). 2018.   \n[HLB20] W. Huang, H. Liu, and S. R. Bowman. 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Rose, and G. Neubig. “Stress test evaluation for natural language inference”. In: Proceedings of the 27th International Conference on Computational Linguistics. 2018.   \n[NLM19] J. Ni, J. Li, and J. McAuley. “Justifying recommendations using distantly-labeled reviews and fined-grained aspects”. In: Empirical Methods in Natural Language Processing (EMNLP) (2019).   \n[PBM16] J. Peters, P. Buhlmann, and N. Meinshausen. “Causal inference by using invariant pre- ¨ diction: identification and confidence intervals”. In: Journal of the Royal Statistical Society. Series B (Statistical Methodology) 5 (2016).   \n$[ \\mathrm { R i b } { + } 2 0 ]$ 1 M. T. Ribeiro, T. Wu, C. Guestrin, and S. Singh. “Beyond accuracy: behavioral testing of nlp models with checklist”. In: Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. 2020.   \n[RPH21] A. Robey, G. J. Pappas, and H. Hassani. Model-Based Domain Generalization. 2021. arXiv: 2102.11436 [stat.ML].   \n$[ \\mathrm { R C } \\substack { + 1 8 } ]$ 1 M. Rojas-Carulla, B. Scholkopf, R. Turner, and J. Peters. “Invariant models for causal ¨ transfer learning”. In: Journal of Machine Learning Research 36 (2018).   \n$[ \\mathrm { S a g } + 2 0 ]$ S. Sagawa, P. W. Koh, T. B. Hashimoto, and P. Liang. “Distributionally robust neural networks for group shifts: on the importance of regularization for worst-case generalization”. In: International Conference on Learning Representations. 2020.   \n$[ \\mathrm { S c h } \\cdot 1 2 ]$ B. Scholkopf, D. Janzing, J. Peters, E. Sgouritsa, K. Zhang, and J. Mooij. “On causal ¨ and anticausal learning”. In: Proceedings of the 29th International Coference on International Conference on Machine Learning. 2012.   \n$[ \\mathrm { S c h } + 2 1 ]$ B. Scholkopf, F. Locatello, S. Bauer, N. R. Ke, N. Kalchbrenner, A. Goyal, and Y. ¨ Bengio. “Toward causal representation learning”. 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In: Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). 2019.   \n[WNB18] A. Williams, N. Nangia, and S. Bowman. “A broad-coverage challenge corpus for sentence understanding through inference”. In: Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers). 2018.   \n$[ \\mathrm { W u } + 1 9 ]$ 1 T. Wu, M. T. Ribeiro, J. Heer, and D. Weld. “Errudite: scalable, reproducible, and testable error analysis”. In: Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics. 2019.   \n$[ \\mathrm { W u } + 2 1 ]$ T. Wu, M. T. Ribeiro, J. Heer, and D. S. Weld. “Polyjuice: generating counterfactuals for explaining, evaluating, and improving models”. 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parse/train/ByfXe2C5tm/ByfXe2C5tm.md

@@ -0,0 +1,312 @@
+# NLPROLOG: REASONING WITH WEAK UNIFICATION FOR NATURAL LANGUAGE QUESTION ANSWERING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Symbolic logic allows practitioners to build systems that perform rule-based reasoning which is interpretable and which can easily be augmented with prior knowledge. However, such systems are traditionally difficult to apply to problems involving natural language due to the large linguistic variability of language. Currently, most work in natural language processing focuses on neural networks which learn distributed representations of words and their composition, thereby performing well in the presence of large linguistic variability. We propose to reap the benefits of both approaches by applying a combination of neural networks and logic programming to natural language question answering. We propose to employ an external, nondifferentiable Prolog prover which utilizes a similarity function over pretrained sentence encoders. We fine-tune these representations via Evolution Strategies with the goal of multi-hop reasoning on natural language. This allows us to create a system that can apply rule-based reasoning to natural language and induce domainspecific natural language rules from training data. We evaluate the proposed system on two different question answering tasks, showing that it complements two very strong baselines – BIDAF (Seo et al., 2016a) and FASTQA (Weissenborn et al., 2017) – and outperforms both when used in an ensemble.
+
+# 1 INTRODUCTION
+
+We consider the problem of multi-hop reasoning on natural language input. For instance, consider the statements Socrates was born in Athens and Athens belongs to Greece, together with the question Where was Socrates born? There are two obvious answers following from the given statements: Athens and Greece. While Athens follows directly from the single statement Socrates was born in Athens, deducing Greece requires a reader to combine both provided statements using the knowledge that a person that was born in a city, which is part of a country, was also born in the respective country.
+
+Most recent work that addresses such challenges leverages deep learning based methods (Sukhbaatar et al., 2015; Peng et al., 2015; Seo et al., 2016b; Raison et al., 2018; Henaff et al., 2016; Kumar et al., 2016; Graves et al., 2016; Dhingra et al., 2018), capable of dealing with the linguistic variability and ambiguity of natural language text. However, the black-box nature of neural networks makes it hard to interpret the exact reasoning steps leading to a prediction (local interpretation), as well as the induced model (global interpretation).
+
+Logic programming languages like Prolog (Wielemaker et al., 2012), on the other hand, are built on the idea of using symbolic rules to reason about entities, which makes them highly interpretable both locally and globally. The capability to use user-defined logic rules allows users to incorporate external knowledge in a straightforward manner. Unfortunately, because of their reliance on symbolic logic, systems built on logic programming need extensive preprocessing to account for the linguistic variability that comes with natural language (Moldovan et al., 2003).
+
+We introduce NLPROLOG, a system which combines a symbolic reasoner and a rule-learning method with pretrained sentence representations to perform rule-based multi-hop reasoning on natural language input.1 Like inductive logic programming methods, it facilitates both global as well as local interpretation, and allows for straightforward integration of prior knowledge. Similarly to deep learning based approaches, it can be applied to natural language text without the need to transforming it to formal logic.
+
+At the core of the proposed method is an external non-differentiable theorem prover which can take similarities between symbols into account. Specifically, we modify a Prolog interpreter to support weak-unification as proposed by Sessa (2002). To obtain similarities between symbols, we utilize sentence encoders initialized with pretrained sentence embeddings (Pagliardini et al., 2017) and then fine-tune these for a downstream question answering task via gradient-based optimization methods. Since the resulting system contains non-differentiable components, we propose using Evolution Strategies (ES) (Eiben et al., 2003) as a gradient estimator (Williams, 1992) for training the system – enabling us to fine-tune the sentence encoders and to learn domain-specific logic rules (e.g. that the relation is in is transitive) from natural language training data. This results in a system where training can be trivially parallelized, and which allows to change the logic formalism by simply exchanging the external prover without the need for an intricate re-implementation as an end-to-end differentiable function.
+
+In summary, our main contributions are: a) we show how Prolog-like reasoning can be applied to natural language input by employing a combination of pretrained sentence embeddings, an external logic prover, and fine-tuning using Evolution Strategies, $b _ { . }$ ) we extend a Prolog interpreter with weak unification based on distributed representations, $c _ { . }$ ) we present Gradual Rule Learning (GRL), a training algorithm that allows the proposed system to learn First-Order Logic (FOL) rules from entailment, and $d$ ) we evaluate the proposed system on two different Question Answering (QA) datasets and demonstrate that its performance is on par with state-of-the-art neural QA models in many cases, while having different failure modes. This allows to build an ensemble of NLPROLOG and a neural QA model that outperforms all individual models.
+
+# 2 RELATED WORK
+
+Our work touches in general on weak-unification based fuzzy logic (Sessa, 2002) and focuses on multi-hop reasoning for QA, the combination of logic and distributed representations, and theorem proving for question answering.
+
+Multi-hop Reasoning for QA. One prominent approach for enabling multi-hop reasoning in neural QA models is to iteratively update a query embedding by integrating information from embeddings of context sentences, usually using an attention mechanism and some form of recurrency (Sukhbaatar et al., 2015; Peng et al., 2015; Seo et al., 2016b; Raison et al., 2018). These models have achieved state-of-the-art results in a number of reasoning-focused QA tasks. Henaff et al. (2016) employ a differentiable memory structure that is updated each time a new piece of information (usually a sentence) is processed. The memory slots can be used to track the state of various entities, which can be considered as a form of temporal reasoning. Similarly, the Neural Turing Machine (Graves et al., 2016) and the Dynamic Memory Network (Kumar et al., 2016), which are built on differentiable memory structures, have been used to solve synthetic QA problems requiring multi-hop reasoning. Dhingra et al. (2018) modify an existing neural QA model to additionally incorporate coreference information provided by a coreference resolution model in a preprocessing step, which improves performance on QA problems requiring multi-hop reasoning.
+
+All of the methods above perform reasoning more or less implicitly by updating latent vector representations, which makes an unambiguous interpretation of the exact reasoning steps difficult. Additionally, it is not obvious how a strong prior, like user-defined inference rules, could be imposed on the respective reasoning procedures. Besides, many of them have been evaluated only on artificially generated data sets and thus it is unclear how they perform when on data that involves natural linguistic variability.
+
+Combination of FOL and Distributed Representations. Investigating the combination of formal logic and distributed representations has a long tradition, which is reviewed by Besold et al. (2017). Strongly related to our approach is the combination of Markov Logic Networks (Richardson & Domingos, 2006), Probabilistic Soft Logic (Bach et al., 2017), and word embeddings, which has been applied to Recognizing Textual Entailment (RTE) and Semantic Textual Similarity (STS) (Garrette et al., 2011; 2014; Beltagy et al., 2013; 2014), and improves upon baselines utilizing either only logic or only distributed representations.
+
+An area in which neural multi-hop reasoning models have been thoroughly investigated is Knowledge Base Completion (Das et al., 2016; Cohen, 2016; Neelakantan et al., 2015; Das et al., 2017). While QA could be in principle modeled as a KB-completion task, the construction of a densely connected KB from text is far from trivial, due to the inherent ambiguity of natural language. Without any preprocessing, even the moderately sized QA tasks considered in this work would produce a very large and sparsely connected KB.
+
+Closest to our approach is the Natural Theorem Prover (NTP) (Rocktäschel & Riedel, 2017), which obtains the final proof score for a statement by constructing a neural network that represents all possible proofs. The model is trained end-to-end using backward chaining and a differentiable unification operator. Since the number of possible proofs grows exponentially with the number of facts and rules, NTPs cannot scale even to moderately sized knowledge bases, and are thus not applicable to natural language problems in its current form. We circumvent this issue by using a non-differentiable prover and fine-tune the model using Evolution Strategies.
+
+Theorem Proving for Question Answering. Our work is not the first to apply theorem proving to QA problems. Angeli et al. (2016) employ a system based on Natural Logic to search a large KB for a single statement that entails the candidate answer. This is somewhat orthogonal to our approach, as we aim to learn rules that combine multiple statements to answer a question.
+
+More traditional systems like Watson (Ferrucci et al., 2010) or COGEX (Moldovan et al., 2003) utilize an integrated theorem prover, but require a transformation of the natural language input to logical form. In the case of COGEX, this improves the accuracy of the underlying system by $30 \%$ , and increases its interpretability. While this work is similar in spirit, we greatly simplify the preprocessing step by replacing the transformation of natural language to logic with the simpler approach of taking Open Information Extraction (Open IE) (Etzioni et al., 2008) textual patterns.
+
+Fader et al. (2014) propose the OPENQA system that utilizes a mixture of handwritten and automatically obtained operators that are able to parse, paraphrase and rewrite queries, which allows them to perform large-scale QA on KBs that include Open IE triples. While this work shares the same goal – answering questions using facts extracted by Open IE – we choose a completely different approach to address the problem of linguistic variablity and focus on the combination of multiple facts by learning logical rules.
+
+# 3 BACKGROUND
+
+In the following, we review the background relevant for the custom Prolog engine employed by our method. Specifically, we briefly introduce Prolog’s backward chaining algorithm and unification procedure (Russell & Norvig, 2016). We assume basic knowledge of formal logic and logic programming.
+
+In a nutshell, a Prolog program consists of a set of rules in the form of Horn clauses
+
+$$
+h ( f _ { 1 } ^ { h } , \ldots , f _ { n } ^ { h } ) \ :  \ : p _ { 1 } ( f _ { 1 } ^ { 1 } , \ldots , f _ { m } ^ { 1 } ) \land \ldots \land p _ { B } ( f _ { 1 } ^ { B } , \ldots , f _ { l } ^ { B } ) ,
+$$
+
+where $h , p _ { i }$ are predicate symbols and $f _ { j } ^ { i }$ are either function (denoted in lower case) or variable (upper case) symbols. The domain of function symbols is denoted by $\mathcal { F }$ and the domain of predicate symbols by $\mathcal { P }$ . $h ( f _ { 1 } ^ { h } , \ldots , f _ { n } ^ { h } )$ is called the head and $p _ { 1 } \bigl ( f _ { 1 } ^ { 1 } , \ldots , f _ { m } ^ { 1 } \bigr ) \ \wedge \ \ldots \ \wedge \ p _ { B } \bigl ( f _ { 1 } ^ { B } , \ldots , \bar { f } _ { l } ^ { B } \bigr )$ the body of the rule. We call $B$ the body size of the rule and rules with a body size of zero are named atoms (short for atomic formula).2 If an atom does not contain any variable symbols it is termed fact. As in related work (Sessa, 2002; Julián-Iranzo et al., 2009), we disregard negation and disjunction.
+
+The central procedure of Prolog is unification. It receives two atoms and attempts to find variable substitutions that make both atoms syntactically equal. For example, the input atoms country(Greece, Socrates) and country $( X , Y )$ result in the following variable substitution after unification: $\{ X / G r e e c e , Y / S o c r a t e s \}$ .
+
+The proof algorithm of Prolog is called backward-chaining. It starts from a goal atom $g$ and attempts to prove it by applying suitable rules, thereby generating subgoals that are proved next. To find applicable rules, it attempts to unify $g$ with the heads of all available rules. If this unification succeeds, the resulting variable substitutions are applied to the atoms in the rule body and each of those atoms becomes a new subgoal. For instance, the application of the rule countr $\prime ( X , Y ) \Leftarrow b o r n \_ i n ( Y , X )$ to the goal country(Greece, Socrates) would yield the subgoal born_in(Socrates, Greece). Then the process is repeated for all subgoals until no subgoal is left, i.e., until all subgoals have been unified with a fact. The result of this procedure is a set of rule applications and variable substitutions called proof. Note, that the number of possible proofs is exponential in the number of predicate and entity symbols, as every rule might be used in the proof of each subgoal. Pseudo code for weak unification can be found in Appendix A.1 and we refer the reader to Russell & Norvig (2010) for more details. To apply standard Prolog to a NLP problem like QA, one would have to account for semantic similarities and ambiguities with extensive and error-prone preprocessing, e.g. when transforming the natural language input to logical form.
+
+# 4 NLPROLOG
+
+Our aim is to apply Prolog to natural language question answering where the same entity or relation can have different natural language surface forms. Thus, we replace the equality-based unification operator with similarity-based weak unification (Sessa, 2002), which allows to unify two symbols $x , y$ if they are sufficiently similar, as judged by a similarity function $\sim \theta$ parameterized by $\theta$ . Then, the result of unification also contains a proof success score $S$ that is the result of the symbols’ similarity and the previous success score $S ^ { \prime }$ : $S = \top ( x \sim y , S ^ { \prime } )$ , where $\top \in \{ \mathrm { m i n } , \cdot \}$ is an aggregation function. The result of backward-chaining with weak unification are (possibly) multiple proofs, each with an associated proof success score. NLPROLOG combines inference based on weak unification and distributed representations to allow reasoning on natural language statements. The natural language statements are first transformed to triples using Open IE (Etzioni et al., 2008). The symbols occurring in these triples and in the rules are then embedded into a vector space, which in turn is used to estimate similarities between symbols. The resulting similarity function is subsequently used to perform a proof and consequently obtain a proof success score $S$ . The proof success score is then utilized as a training signal for ES. An illustration of the process can be found in Fig. 1, where we visualize the interplay of the different components for our running example.
+
+# 4.1 SIMILARITY COMPUTATION
+
+We embed symbols using an encoder $\mathcal { E } _ { \theta } : \mathcal { F } \cup \mathcal { P } \mapsto \mathbb { R } ^ { d }$ parametrized by $\theta$ for entity and predicate symbols, where $d$ denotes the embedding size. The resulting embeddings are used to induce $\sim \theta$ $( \mathcal { \dot { F } } \cup \mathcal { P } ) ^ { 2 } \mapsto \mathbb { R }$ :
+
+$$
+x \sim _ { \theta } y = \frac { 1 + c o s ( \mathcal { E } _ { \theta } ( x ) , \mathcal { E } _ { \theta } ( y ) ) } { 2 }
+$$
+
+where cos denotes the cosine similarity between two vectors. There are alternative similarity functions such as Euclidean distance or RBF kernel, but in preliminary experiments we found cosine simlarity to work more robustly.
+
+We use an encoder function that uses a embedding lookup table for predicate symbols and a different one for entities. All embeddings representing natural language phrases are populated with sentence vectors that were pretrained with SENT2VEC (Pagliardini et al., 2017). Additionally, we introduce a third lookup table for the predicate symbols of rules and goals, because semantics of goal or rule predicates might differ from the semantics of fact predicates even if they share the same surface form. For instance, the query $( X , p a r e n t , Y )$ could be interpreted either as (X, is the parent of, $Y$ ) or as (X, has the parent, Y) which are fundamentally different things.
+
+# 4.2 FINE-TUNING THE ENCODER TO A DOWNSTREAM TASK
+
+We propose to fine-tune the similarity function on a downstream task by updating the symbol embeddings. As NLPROLOG involves the non-differentiable proof search step, we cannot apply backpropagation for optimization. Instead, we propose to employ Evolution Strategies in conjunction with Adam (Kingma & Ba, 2014) to estimate the weight updates. ES recently showed good results for Reinforcement Learning problems in Salimans et al. (2017); Mania et al. (2018). More formally, the parameter update is computed as:
+
+![](images/e6c463178bdb0f3faaa5be7c882e363fdc3e665790d2ed68cdbfb26410667e64.jpg)  
+Figure 1: Overview of NLPROLOG. The components of NLPROLOG are depicted as ellipses, while inputs and outputs are drawn as squares. Phrases with red background are entities and blue ones are predicates.
+
+$$
+\theta _ { t + 1 } = \theta _ { t } + \frac { \alpha _ { t } } { N \sigma _ { J } ^ { t } } \sum _ { \stackrel { k = 1 } { \epsilon _ { k } \sim \mathcal { N } ( 0 , \sigma ) } } ^ { N } \left[ J ( \theta _ { t } + \epsilon _ { k } ) - J ( \theta _ { t } - \epsilon _ { k } ) \right] \epsilon _ { k }
+$$
+
+where $J ( \theta )$ is the reward obtained by $\theta$ , $\boldsymbol { \sigma } _ { J } ^ { t }$ is the standard deviation of all rewards obtained at time $t$ as proposed by Mania et al. (2018), and $\alpha _ { t }$ are adaptive learning rates selected by ADAM (Kingma & Ba, 2014). The standard deviation $\sigma$ of the distribution that is generating the perturbations is treated as a hyperparameter.
+
+We train NLPROLOG with ES using a learning from entailment setting (Muggleton & Raedt, 1994), in which the model is trained to decide whether a Prolog program $\mathcal { R }$ entails the truth of a candidate triple $c$ . The objective of the model is to assign high probabilities $p ( c | \mathcal { R } ; \theta )$ to true candidate triples and low probabilities to false triples. To achieve this, we model the reward as $J ( \theta ) = y p ( c | \mathcal { R } ; \theta )$ , where $\bar { y \in \{ - 1 , 1 \} }$ is the gold label. To estimate $p ( c | \mathcal { R } ; \theta )$ , we exhaustively search for all proofs for the triple $c$ , up to a given depth $D$ which we treat as a hyperparameter. This search yields a number of proofs each with a success score $S _ { i }$ . We set $p ( c | \mathcal { R } ; \theta )$ to be the maximum of these scores $S _ { \operatorname* { m a x } } = \operatorname* { m a x } _ { i } S _ { i }$ .
+
+# 4.3 GRADUAL RULE LEARNING
+
+The reasoning process of NLPROLOG crucially depends on rules that describe the relations between predicates. While it is possible to write down rules in natural language, this approach is hardly scalable. Thus, we follow Rocktäschel & Riedel (2017) and use rule templates to perform Inductive Logic Programming (ILP) (Muggleton, 1991) which allows NLPROLOG to learn rules from training data. For this, a user has to define a set of rules with a given structure as input. Then, NLPROLOG randomly initializes the predicates of these rules. For instance, to induce a rule that can model transitivity, one would add a rule template of the form $p _ { 1 } ( X , Z )  p _ { 2 } ( X , Y ) \land p _ { 3 } ( Y , Z )$ , and NLPROLOG would instantiate multiple rules with randomly initialized embeddings for $p _ { 1 } , p _ { 2 }$ , and $p _ { 3 }$ . The exact number and structure of the rule templates is treated as a hyper-parameter. If not explicitly stated otherwise, the experiments were performed with the same set of rule templates containing two rules for each of the forms $p _ { 1 } ( X , { \bar { Y } } ) \Leftarrow p _ { 2 } ( X , Y )$ , $p _ { 1 } ( X , Y ) \Leftarrow p _ { 2 } ( Y , \bar { X _ { \lambda } } )$ and $p _ { 1 } ( X , Z ) { \overset { \vartriangle } { \Leftarrow } } p _ { 2 } ( X , Y ) \land p _ { 3 } ( Y , Z )$ .
+
+In preliminary experiments, we observed that unification with such randomly initialized embeddings always leads to a stark drop of the proof success score. This implies in turn that proofs involving rule templates rarely yield the highest score $S _ { \mathrm { m a x } }$ and thus, might have no impact on the value of the reward function. Due to that, the expected gradient estimate for the rule embeddings is zero, and they will remain close to their initialization.
+
+The main reason for this behavior is that the monotonicity of the aggregation functions implies that each sub-goal created by an application of a rule will only decrease the success score of the proof. Thus, all other things being equal, rules with a small number of body atoms (and facts in particular) will be preferred over more complex rules with a higher number of body atoms. Note that this problem is particularly severe in our setting where rules are initialized randomly while the remaining predicate symbols are instantiated using pretrained sentence embeddings.
+
+We propose a Gradual Rule Learning (GRL) algorithm which counteracts this effect during training. GRL segments the training process into $B _ { \mathrm { m a x } } + 1$ phases, where $B _ { \mathrm { m a x } }$ is the maximum body size of all available rules. In the $k$ -th phase of GRL, only proofs which employ at least one rule with a body size of $B _ { \mathrm { m a x } } + 1 - k$ are considered to estimate $p ( t | F ; \theta )$ . Thus, it is guaranteed that in each training step in phase $k$ at least one rule with a body size of $B _ { \mathrm { m a x } } + 1 - k$ receives a training signal.
+
+# 5 EVALUATION
+
+We evaluate our method on two different QA data sets: BABI-1K-STATIC (Weston et al., 2015) and different subsets of WIKIHOP (Welbl et al., 2017). The used hyperparameter configurations can be found in Section B.
+
+# 5.1 SUBSETS OF WIKIHOP
+
+We evaluate on different subsets of WIKIHOP (Welbl et al., 2017), each containing a single query predicate. We consider the predicates publisher, developer, and country, because their semantics ensure that the annotated answer is unique and they contain a relatively high amount of questions that are annotated as requiring true multi-hop reasoning. For publisher, this yields 509 training and 54 validation questions, for developer 267 and 29, and for country 742 and 194. As the test set of WIKIHOP is not publicly available, and splitting the small train set would lead to a far too small validation set, we report scores for the validation set and refrain from hyperparameter optimization and early stopping.
+
+Each data point consists of a query $p ( q , X )$ where $q$ is some query entity, $X$ is the entity that has to be predicted, $C$ is a list of candidates entities, $a \in C$ is an answer entity and $p \in$ {publisher, developer, country} is the query predicate. In addition, every query is accompanied by a set of support documents which can be used to decide which candidate is the correct answer. To transform the support documents to natural language triples, we use the Open IE system MINIE (Gashteovski et al., 2017). We use the publicly available version of $\mathbf { M N I E } ^ { 3 }$ in the dictionary mode, and use a list of all WIKIHOP candidate entities as our dictionary of multi-token entities.
+
+Following Welbl et al. (2017), we use the two neural QA models BIDAF (Seo et al., 2016a) and FASTQA (Weissenborn et al., 2017) as baselines for the selected predicates of WIKIHOP. We employ the implementation provided by the QA framework $\mathrm { J A C K } ^ { 4 }$ (Weissenborn et al., 2018) with the same hyperparameters as used by Welbl et al. (2017) and train a separate model for each predicate. In order to compensate for the fact that both models are extractive QA models which cannot make use of the candidate entities, we additionally evaluate modified versions which transform both the predicted answer and all candidates to vectors using the wiki-unigrams model5 of SENT2VEC (Pagliardini et al., 2017). Consequently, we return the candidate entity which has the highest cosine similarity to the predicted entity.
+
+# 5.2 BABI-1K-STATIC
+
+We utilize a subset of BABI-1K to study the behavior of NLPROLOG in a very controlled environment. Note however, that the experiments on WIKIHOP are much more significant, as they involve natural linguistic variability. BABI-1K-STATIC consists of the tasks QA4,QA15,QA17, and QA18 from the BABI suite (Weston et al., 2015), each containing 1,000 training and 1,000 testing questions. These tasks were selected because, unlike the other tasks of BABI, they do not require reasoning about a dynamically changing world state, which is not supported by the current implementation of NLPROLOG. We automatically transform all statements and queries of the respective tasks to triples and use the resulting KB as input to NLPROLOG. We train and evaluate on each problem individually. The tasks QA4 and QA15 require entities as an output, thus we consider every entity that occurs at least once in any problem of the task as a candidate for all problems. Tasks QA17 and QA18 are binary classification tasks, and thus we determine the optimal threshold on the training set, after the training of NLPROLOG has finished.
+
+We refrain from systematically comparing results on the individual BABI tasks to competing methods like Seo et al. (2016b); Peng et al. (2015); Dhingra et al. (2018); Henaff et al. (2016); Sukhbaatar et al. (2015), since our non-negligible preprocessing and evaluation on only four out of 20 tasks does not allow us to match the relevant evaluation protocols. We therefore utilize BABI-1K-STATIC only for ablation experiments, but note that NLPROLOG achieves similar or better accuracy values than the mentioned methods in all instances we studied, except on $Q A 4$ .
+
+# 5.3 DEALING WITH CANDIDATES
+
+All questions of WIKIHOP and some of BABI-1K-STATIC include a set of answer candidates $C$ . For those cases, we modify our reward function to leverage this information, taking inspiration from Bayesian Personalized Ranking (Rendle et al., 2009):
+
+$$
+J ( \theta ) = p ( a | \mathcal { R } ; \theta ) - \operatorname* { m a x } _ { c \in C \setminus \{ a \} } p ( c | \mathcal { R } ; \theta ) ,
+$$
+
+where $a \in C$ is the true answer.
+
+We observed that this reward function does not work well with the minimum aggregation function. Therefore, we employ this modified reward only when using the product aggregation and utilize the reward described in Section 4.2 with the minimum aggregation.
+
+# 5.4 RESULTS
+
+The results for the selected predicates of WIKIHOP can be found in Table 1. While there is no single best performing model, NLPROLOG is outperformed by at least one neural QA model on every predicate. For country, this only holds true when considering the versions of the neural models that have been augmented to consider candidate entities. For all three predicates, only a single transitive rule is utilized across all validation questions. Since we observe a more diverse set of induced rules on BABI-1K, we partly attributed this lack of diverse rules to the multi-hop characteristic of the WIKIHOP data. It seems that NLPROLOG struggles to find meaningful rules for the predicates developer and publisher, which leads to very few proofs involving rules on the development set: 1 out of 54 for publisher and 2 out of 29 for developer, compared with 159 out of 194 for country. We partially attribute this to the fact, that the semantics of country suggest a straightforward rule (transitivity of location), which is not true for developer or publisher. Additionally, the annotations regarding the neccessity of multi-hop reasoning provided for the validation set (Welbl et al., 2017) suggest that publisher and developer contain significantly fewer training examples involving multi-hop reasoning.
+
+Exemplary proofs generated by NLPROLOG on the predicates developer and country can be found in Fig. 3. As we are especially interested in assessing the capability for multi-hop reasoning, we additionally evaluate on a subset of problems which have been unanimously labelled as requiring multi-hop reasoning. On this subset of the development data, which is denoted as Countryhop (mh) in Table 1, NLPROLOG outperforms all other single models.
+
+# 5.5 NEURAL QA AND NLPROLOG ENSEMBLE
+
+If the proof of NLPROLOG producing the prediction does not employ any rules, the prediction is essentially the result of performing a simple nearest neighbor search among the embeddings of all fact triples. We hypothesize that the neural QA models FASTQA and BIDAF are better suited for finding the most predictive single support statement, which motivates an ensemble of a neural QA model and NLPROLOG. We built a system which predicts the output of NLPROLOG when it used at least one rule and the output of the neural QA model otherwise. This allows to employ the multihop reasoning of NLPROLOG when possible and to utilize the pattern matching capabilities of the neural QA models in the remaining cases. The results for these ensembles are given in the bottom part of Table 1. In all instances, ensembling a neural QA model with NLPROLOG improved upon all single models, indicating that they complement each other well. We analyze reasons for the success of this ensembling strategy in Section 5.6.
+
+Table 1: Accuracy scores in percent for different predicates on the development set of WIKIHOP. country (mh) only considers the problems that have been unanimously judged to require multi-hop reasoning. The upper part lists single models and the bottom part ensembles.   
+
+<table><tr><td>Model</td><td>publisher</td><td>developer</td><td>country</td><td>country (mh)</td></tr><tr><td>BIDAF</td><td>66.67</td><td>65.52</td><td>53.09</td><td>53.19</td></tr><tr><td>FastQA</td><td>62.96</td><td>62.07</td><td>57.21</td><td>57.44</td></tr><tr><td>BIDAF-Sent2Vec</td><td>75.93</td><td>68.97</td><td>61.86</td><td>65.96</td></tr><tr><td>FastQA-Sent2Vec</td><td>75.93</td><td>58.62</td><td>64.95</td><td>65.96</td></tr><tr><td>NLProlog (ours)</td><td>51.85</td><td>48.28</td><td>63.92</td><td>68.09</td></tr><tr><td>BIDAF-Sent2Vec + NLProlog</td><td>77.78</td><td>72.41</td><td>65.46</td><td>70.21</td></tr><tr><td>FastQA-Sent2Vec + NLProlog</td><td>77.78</td><td>62.07</td><td>69.07</td><td>70.21</td></tr></table>
+
+# 5.6 ERROR ANALYSIS
+
+We conducted an extensive error analysis, in which we manually classified all errors of NLPROLOG on the selected WIKIHOP predicates into predefined categories. The results are depicted in Figure 2.
+
+For the developer predicate, in the majority of cases, errors were caused by OPENIE during the fact extraction step: in all but one case, OPENIE did not produce the necessary facts, or a necessary facts was not stated in the support text, or there were multiple correct candidates and NLPROLOG selected the wrong one. As a consequence, for both the publisher and developer predicates, the majority of queries would not answerable, even when the necessary rules were correctly induced. The predictive accuracy was significantly higher for the country predicate, where errors were mostly due to entitities not having SENT2VEC embeddings and a few missing rules.
+
+Fig. 2 indicates that FASTQA can produce the correct answer, even if crucial information is missing from the supporting documents. To analyze this further, we evaluated FASTQA and NLPROLOG on a test set of the country predicate in which all documents mentioning the query entity were removed. Remarkably, the accuracy of FASTQA increased by approximately $1 \%$ , while the accuracy of NLPROLOG decreased by approx. $1 1 \%$ .
+
+Furthermore, we evaluated both FASTQA and NLPROLOG on the hard subset of country as defined by Sugawara et al. (2018): on these 62 problems which cannot be solved with a simple heuristic, NLPROLOG achieved an accuracy of $5 1 . { \bar { 6 } } 1 \%$ , as opposed to $4 6 . 7 7 \%$ by FASTQA.
+
+We conjecture that – besides NLPROLOG’s multi-hop reasoning capability – this is one reason why the neural models and NLPROLOG complement each other nicely: neural models can compensate for missing information, while NLPROLOG is less susceptible for spurious correlations between the query and supporting texts. The complementary nature of both approaches is further supported by the error analysis, described in Fig. 4.
+
+# 5.7 ABLATION EXPERIMENTS
+
+We perform experiments on BABI-1K-STATIC to investigate the effect of the GRL training procedure. Specifically, we perform experiments on BABI-1K-STATIC with only the last phase of GRL (i.e. training without GRL), with the last and the penultimate phase, and with all three phases, corresponding to full GRL as we limit the rule complexity to two body atoms in rule templates. To maintain comparability between runs, we keep the number of SGD steps constant across all experiments.
+
+![](images/84a090321be673203bf5af7c84917979a70a09bb28ca2bb215b164e77262c9b5.jpg)  
+Figure 2: Results of the error analysis. The y-axis shows the number of errors per category: O(penIE) refers to a crucial information missing due to an error of the OpenIE tool, A(mbiguous) means that multiple candidates were correct and NLProlog chose one which was not labeled as the correct one, while I(nfo) refers to problems that are not answerable with the provided support texts. $R ( u l e )$ means that a required rule was not induced, and E(mbedding) implies that the answer was correctly deduced but received a lower score than an erroneous deduction of another candidate.
+
+<table><tr><td></td><td>QA4</td><td>QA15</td><td>QA17</td><td>QA18</td></tr><tr><td>GRL phase 3 (no)</td><td>56.4</td><td>37.1</td><td>62.8</td><td>90.9</td></tr><tr><td>GRL phase 2+3</td><td>88.8</td><td>33.5</td><td>66.7</td><td>94.7</td></tr><tr><td>GRL phase 1+2+3 (full)</td><td>88.0</td><td>100.0</td><td>67.6</td><td>97.6</td></tr><tr><td>Gold Rules (Untrained)</td><td>100.0</td><td>100.0</td><td>73.2</td><td>92.0</td></tr><tr><td>Gold Rules (Trained)</td><td>100.0</td><td>100.0</td><td>84.8</td><td>98.1</td></tr><tr><td>No Rules (Trained)</td><td>60.0</td><td>32.8</td><td>63.0</td><td>90.4</td></tr><tr><td>Ten Templates</td><td>88.1</td><td>100.0</td><td>61.9</td><td>89.4</td></tr></table>
+
+Table 2: Accuracies in percent of ablation experiments on BABI-1K-STATIC. The upper half examines the effectiveness of GRL, while the bottom results concern the effect of training and different choices of rules.
+
+Additionally, we experiment with manually defined rules, which we deem sufficient for solving each of the four tasks. For these, we report results before and after training, as well as for a run without any rule templates. The accuracy scores for all experiments on BABI are provided in Table 2.
+
+To assess the impact of the choice of rule templates, we evaluate NLPROLOG on bAbI-1k-static with a different set of rule templates containing two rules of the form $p _ { 1 } ( X , Y ) \Leftarrow p _ { 2 } ( X , Y )$ , four with the form $p _ { 1 } ( X , Y ) \Leftarrow p _ { 2 } ( Y , X )$ and another four for $p _ { 1 } ( X , Z ) \in p _ { 2 } ( X , Y ) \land p _ { 3 } ( Y , Z )$ .
+
+Clearly, the full GRL is necessary to obtain acceptable results in any of the problems. Interestingly, phase 1 of GRL does not contribute anything for QA4, which is perfectly solvable using only rules of body size 1. On the other hand, QA15 and QA18 both require a rule of body size 2, which makes phase 1 strongly improve the results. Only for $Q A 1 7$ the results are inconclusive. Nevertheless, this indicates that GRL works as intended, with the earlier phases encouraging the induction of rules with a higher number of conjuncts in the body.
+
+The results using manually defined rules suggest that even when sufficient rules are provided, training with ES is helpful nevertheless. Interestingly, the model using no rules is able to solve over $9 0 \%$ of the problems in $Q A I \&$ , indicating that this problem is not well suited for evaluating reasoning capabilities of QA models.
+
+Using ten instead of six templates leads to worse performance on all BABI problems but QA15, which is solved perfectly with a much faster convergence rate. This result indicates that the choice of rule templates might be an important hyperparameter which should be optimized for a given problem.
+
+# 6 DISCUSSION AND FUTURE WORK
+
+We have developed NLPROLOG, a system that is able to perform rule-based reasoning on natural language input, and can learn domain-specific natural language rules from training data. To this end, we have proposed to combine a symbolic prover with pretrained sentence embeddings and to train the resulting system with Evolution Strategies. We have evaluated NLPROLOG on two different QA tasks, showing that it can learn domain-specific rules and produce predictions which complement those of the two strong baselines BIDAF and FASTQA. This allows to build an ensemble of a baseline and NLPROLOG which outperforms all single models.
+
+![](images/7723a7462c403a1c25c15c96c367976e9d871a3b0467a3e7ebbd6acdeb635581.jpg)  
+Figure 3: Example proof trees generated by NLPROLOG. Each of the two trees shows an application of a transitive rule, the first for the predicate developer and the second for the predicate country. The rule templates are displayed with the most similar predicate. Note the noise introduced by the Open IE process, e.g. QUANT_0_1 and that entities and predicates do not need to match exactly.
+
+While we have focused on a subset of First Order Logic in this work, the expressiveness of NLPROLOG could be extended by incorporating a different symbolic prover. For instance, a prover for temporal logic (Orgun & Ma, 1994) would allow to model temporal dynamics in natural language and enable us to evaluate NLPROLOG on the full set of BABI tasks. We are also interested in incorporating future improvements of symbolic provers, Open IE systems and pretrained sentence representations to further enhance the performance of NLPROLOG. To study the performance of the proposed method without the noise introduced by the Open IE step, it would be useful to evaluate it on tasks like knowledge graph reasoning. Additionally, it would be interesting to study the behavior of NLPROLOG in the presence of multiple WIKIHOP query predicates.
+
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+
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+
+A ALGORITHMS
+
+# A.1 WEAK UNIFICATION
+
+fun unify $( x , y , \theta , S )$
+
+#
+
+$x$ : function $f ( \dots ) \mid$ atom $p ( \dots ) \mid$ variable $\mid$ list $x _ { 1 } : : x _ { 2 } : : \ldots : : x _ { n }$ $y$ : function $f ^ { \prime } ( \dots ) \mid$ atom $p ^ { \prime } ( \ldots ) \mid$ variable $|$ list $y _ { 1 } : : y _ { 2 } : : \dots : : y _ { m }$ $\theta$ : current substitutions, default $= \{ \}$   
+$S$ : current success score, default $= 1 . 0$
+
+Output: (Unifying substitution $\theta ^ { \prime }$ or failure, Updated success score $S ^ { \prime }$ ) if $\theta =$ failure then return (failure, 0) else if $S < \lambda$ then return (failure, 0) else if $x = y$ then return $( \theta , S )$ else if $x$ is Var then return unify $_ - \nu a r ( x , y , \theta , S )$ else if $y$ is Var then return unify_ $\nu a r ( y , x , \theta , S )$ else if $x$ is $f ( x _ { 1 } , \ldots , x _ { n } )$ , y is $f ^ { \prime } ( y _ { 1 } , \ldots , y _ { n } )$ , and $f \sim f ^ { \prime } \geq \lambda$ then $S ^ { \prime } : = S \wedge f \sim f ^ { \prime }$ return uni $\mathfrak { f } ( x _ { 1 } : : \dots : : x _ { n } , y _ { 1 } : : \dots : : y _ { n } , \theta , S ^ { \prime } )$ end else if $x$ is $p ( x _ { 1 } , \ldots , x _ { n } )$ , y is $p ^ { \prime } ( y _ { 1 } , \ldots , y _ { n } )$ , and $p \sim p ^ { \prime } \ge \lambda$ then $S ^ { \prime } : = S \wedge f \sim f ^ { \prime }$ return unif $\nu ( x _ { 1 } : : \ldots : : x _ { n } , y _ { 1 } : : \ldots : : y _ { n } , \theta , S ^ { \prime } )$ end else if $x$ is $x _ { 1 } : : \ldots : : x _ { n }$ and $y$ is $y _ { 1 } : : \ldots : : y _ { n }$ then $( \theta ^ { \prime } , S ^ { \prime } ) : = u n i f y ( x _ { 1 } , y _ { 1 } , \theta , S )$ $\mathbf { r e t u r n } u n i f y ( x _ { 2 } : : \dots : : x _ { n } , y _ { 2 } : : \dots : : y _ { n } , \theta ^ { \prime } , S ^ { \prime } )$ end else if $x$ is empty list and $y$ is empty list then return $( \theta , S )$ else return (failure, 0) un unify_va $r ( v , o , \theta , S )$ if $\{ \nu / \nu a l \} \in \theta$ then return unify $( \nu a l , o , \theta , S )$ else if $\{ o / \nu a l \} \in \theta$ then return unif $\dot { y } ( \nu a r , \nu a l , \theta , S )$ else return $( \bar { \{ } v / o \} + \theta , S )$
+
+Algorithm 1: The weak unification algorithm in Spyrolog without occurs check
+
+# A.2 RUNTIME OF PROOF SEARCH
+
+The worst case complexity vanilla logic programming is exponential in the depth of the proof (Russell & Norvig, 2010). However, in our case this is a particular problem because weak unification requires the prover to attempt unification between all entity/predicate symbols.
+
+To keep things tractable, NLPROLOG only attempts to unify symbols with a similarity greater than some user-defined threshold $\lambda$ . Furthermore, in the search step for one statement $q$ , for the rest of the search, $\lambda$ is set to $\lambda : = \operatorname* { m a x } ( \lambda , S )$ whenever a proof for $q$ with success score $S$ is found. Due to the monotonicity of the employed aggregation functions, this allows to prune the search tree without losing the guarantee to find the proof yielding the maximum success score. We found this optimization to be crucial to make the proof search scale for the studied wikihop predicates.
+
+# B HYPERPARAMETER CONFIGURATION
+
+On BABI-1K we optimize the embeddings of predicate symbols of rules and query triples, as well as of entities. WIKIHOP has a large number of unique entity symbols and thus, optimizing their embeddings is prohibitive. Thus, we only train the predicate symbols of rules and query triples on this data set. The embeddings for entities and predicate symbols of fact and query triples are initialized using the WIKI-UNIGRAMS model of SENT2VEC, while the embeddings of rule predicates are intialized by uniformly sampling from the interval $[ - \frac { 1 } { \sqrt { 6 0 0 } } , \frac { 1 } { \sqrt { 6 0 0 } } ]$ . All experiments were performed with the same set of rule templates containing two rules for each of the forms $p ( X , Y ) \Leftarrow q ( X , Y )$ , $p ( X , Y ) \Leftarrow q ( Y , X )$ and $p ( X , Z ) \Leftarrow q ( X , Y ) \wedge r ( Y , Z )$ and set the similarity threshold $\lambda$ to 0.3. At each optimization step, we evaluate 100 perturbations sampled from $\mathcal { N } ( 0 , 0 . 0 4 )$ on a mini-batch of 16 training problems and use all of the directions in the generation of the next weight vector. If not stated otherwise, we use GRL with three phases training for 500 mini-batches in each phase. For the predicates publisher and developer, we used 1,000 mini-batches in the final phase of GRL. To further encourage rule usage, we use the minimum aggregation function in all but the last phases of GRL, in which we switch to product.
+
+# C ERROR SETS
+
+![](images/8ed0a056d6d6697c5278a4b2b4a83f050042fb18571dfeb6c97d3c56fb6ab74a.jpg)  
+Figure 4: Venn diagram of the error sets per predicate.

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+        "type": "text",
+        "text": "NLPROLOG: REASONING WITH WEAK UNIFICATION FOR NATURAL LANGUAGE QUESTION ANSWERING ",
+        "text_level": 1,
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
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+        "text": "ABSTRACT ",
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+        "text": "Symbolic logic allows practitioners to build systems that perform rule-based reasoning which is interpretable and which can easily be augmented with prior knowledge. However, such systems are traditionally difficult to apply to problems involving natural language due to the large linguistic variability of language. Currently, most work in natural language processing focuses on neural networks which learn distributed representations of words and their composition, thereby performing well in the presence of large linguistic variability. We propose to reap the benefits of both approaches by applying a combination of neural networks and logic programming to natural language question answering. We propose to employ an external, nondifferentiable Prolog prover which utilizes a similarity function over pretrained sentence encoders. We fine-tune these representations via Evolution Strategies with the goal of multi-hop reasoning on natural language. This allows us to create a system that can apply rule-based reasoning to natural language and induce domainspecific natural language rules from training data. We evaluate the proposed system on two different question answering tasks, showing that it complements two very strong baselines – BIDAF (Seo et al., 2016a) and FASTQA (Weissenborn et al., 2017) – and outperforms both when used in an ensemble. ",
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+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    },
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+        "text": "We consider the problem of multi-hop reasoning on natural language input. For instance, consider the statements Socrates was born in Athens and Athens belongs to Greece, together with the question Where was Socrates born? There are two obvious answers following from the given statements: Athens and Greece. While Athens follows directly from the single statement Socrates was born in Athens, deducing Greece requires a reader to combine both provided statements using the knowledge that a person that was born in a city, which is part of a country, was also born in the respective country. ",
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+        "text": "Most recent work that addresses such challenges leverages deep learning based methods (Sukhbaatar et al., 2015; Peng et al., 2015; Seo et al., 2016b; Raison et al., 2018; Henaff et al., 2016; Kumar et al., 2016; Graves et al., 2016; Dhingra et al., 2018), capable of dealing with the linguistic variability and ambiguity of natural language text. However, the black-box nature of neural networks makes it hard to interpret the exact reasoning steps leading to a prediction (local interpretation), as well as the induced model (global interpretation). ",
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+        "text": "Logic programming languages like Prolog (Wielemaker et al., 2012), on the other hand, are built on the idea of using symbolic rules to reason about entities, which makes them highly interpretable both locally and globally. The capability to use user-defined logic rules allows users to incorporate external knowledge in a straightforward manner. Unfortunately, because of their reliance on symbolic logic, systems built on logic programming need extensive preprocessing to account for the linguistic variability that comes with natural language (Moldovan et al., 2003). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We introduce NLPROLOG, a system which combines a symbolic reasoner and a rule-learning method with pretrained sentence representations to perform rule-based multi-hop reasoning on natural language input.1 Like inductive logic programming methods, it facilitates both global as well as local interpretation, and allows for straightforward integration of prior knowledge. Similarly to deep learning based approaches, it can be applied to natural language text without the need to transforming it to formal logic. ",
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+        "text": "At the core of the proposed method is an external non-differentiable theorem prover which can take similarities between symbols into account. Specifically, we modify a Prolog interpreter to support weak-unification as proposed by Sessa (2002). To obtain similarities between symbols, we utilize sentence encoders initialized with pretrained sentence embeddings (Pagliardini et al., 2017) and then fine-tune these for a downstream question answering task via gradient-based optimization methods. Since the resulting system contains non-differentiable components, we propose using Evolution Strategies (ES) (Eiben et al., 2003) as a gradient estimator (Williams, 1992) for training the system – enabling us to fine-tune the sentence encoders and to learn domain-specific logic rules (e.g. that the relation is in is transitive) from natural language training data. This results in a system where training can be trivially parallelized, and which allows to change the logic formalism by simply exchanging the external prover without the need for an intricate re-implementation as an end-to-end differentiable function. ",
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+        "text": "In summary, our main contributions are: a) we show how Prolog-like reasoning can be applied to natural language input by employing a combination of pretrained sentence embeddings, an external logic prover, and fine-tuning using Evolution Strategies, $b _ { . }$ ) we extend a Prolog interpreter with weak unification based on distributed representations, $c _ { . }$ ) we present Gradual Rule Learning (GRL), a training algorithm that allows the proposed system to learn First-Order Logic (FOL) rules from entailment, and $d$ ) we evaluate the proposed system on two different Question Answering (QA) datasets and demonstrate that its performance is on par with state-of-the-art neural QA models in many cases, while having different failure modes. This allows to build an ensemble of NLPROLOG and a neural QA model that outperforms all individual models. ",
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+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+    },
+    {
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+        "text": "Our work touches in general on weak-unification based fuzzy logic (Sessa, 2002) and focuses on multi-hop reasoning for QA, the combination of logic and distributed representations, and theorem proving for question answering. ",
+        "bbox": [
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+        "text": "Multi-hop Reasoning for QA. One prominent approach for enabling multi-hop reasoning in neural QA models is to iteratively update a query embedding by integrating information from embeddings of context sentences, usually using an attention mechanism and some form of recurrency (Sukhbaatar et al., 2015; Peng et al., 2015; Seo et al., 2016b; Raison et al., 2018). These models have achieved state-of-the-art results in a number of reasoning-focused QA tasks. Henaff et al. (2016) employ a differentiable memory structure that is updated each time a new piece of information (usually a sentence) is processed. The memory slots can be used to track the state of various entities, which can be considered as a form of temporal reasoning. Similarly, the Neural Turing Machine (Graves et al., 2016) and the Dynamic Memory Network (Kumar et al., 2016), which are built on differentiable memory structures, have been used to solve synthetic QA problems requiring multi-hop reasoning. Dhingra et al. (2018) modify an existing neural QA model to additionally incorporate coreference information provided by a coreference resolution model in a preprocessing step, which improves performance on QA problems requiring multi-hop reasoning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "All of the methods above perform reasoning more or less implicitly by updating latent vector representations, which makes an unambiguous interpretation of the exact reasoning steps difficult. Additionally, it is not obvious how a strong prior, like user-defined inference rules, could be imposed on the respective reasoning procedures. Besides, many of them have been evaluated only on artificially generated data sets and thus it is unclear how they perform when on data that involves natural linguistic variability. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Combination of FOL and Distributed Representations. Investigating the combination of formal logic and distributed representations has a long tradition, which is reviewed by Besold et al. (2017). Strongly related to our approach is the combination of Markov Logic Networks (Richardson & Domingos, 2006), Probabilistic Soft Logic (Bach et al., 2017), and word embeddings, which has been applied to Recognizing Textual Entailment (RTE) and Semantic Textual Similarity (STS) (Garrette et al., 2011; 2014; Beltagy et al., 2013; 2014), and improves upon baselines utilizing either only logic or only distributed representations. ",
+        "bbox": [
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+        "page_idx": 1
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+        "type": "text",
+        "text": "An area in which neural multi-hop reasoning models have been thoroughly investigated is Knowledge Base Completion (Das et al., 2016; Cohen, 2016; Neelakantan et al., 2015; Das et al., 2017). While QA could be in principle modeled as a KB-completion task, the construction of a densely connected KB from text is far from trivial, due to the inherent ambiguity of natural language. Without any preprocessing, even the moderately sized QA tasks considered in this work would produce a very large and sparsely connected KB. ",
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+        "type": "text",
+        "text": "Closest to our approach is the Natural Theorem Prover (NTP) (Rocktäschel & Riedel, 2017), which obtains the final proof score for a statement by constructing a neural network that represents all possible proofs. The model is trained end-to-end using backward chaining and a differentiable unification operator. Since the number of possible proofs grows exponentially with the number of facts and rules, NTPs cannot scale even to moderately sized knowledge bases, and are thus not applicable to natural language problems in its current form. We circumvent this issue by using a non-differentiable prover and fine-tune the model using Evolution Strategies. ",
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+    {
+        "type": "text",
+        "text": "Theorem Proving for Question Answering. Our work is not the first to apply theorem proving to QA problems. Angeli et al. (2016) employ a system based on Natural Logic to search a large KB for a single statement that entails the candidate answer. This is somewhat orthogonal to our approach, as we aim to learn rules that combine multiple statements to answer a question. ",
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+        "type": "text",
+        "text": "More traditional systems like Watson (Ferrucci et al., 2010) or COGEX (Moldovan et al., 2003) utilize an integrated theorem prover, but require a transformation of the natural language input to logical form. In the case of COGEX, this improves the accuracy of the underlying system by $30 \\%$ , and increases its interpretability. While this work is similar in spirit, we greatly simplify the preprocessing step by replacing the transformation of natural language to logic with the simpler approach of taking Open Information Extraction (Open IE) (Etzioni et al., 2008) textual patterns. ",
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+        "type": "text",
+        "text": "Fader et al. (2014) propose the OPENQA system that utilizes a mixture of handwritten and automatically obtained operators that are able to parse, paraphrase and rewrite queries, which allows them to perform large-scale QA on KBs that include Open IE triples. While this work shares the same goal – answering questions using facts extracted by Open IE – we choose a completely different approach to address the problem of linguistic variablity and focus on the combination of multiple facts by learning logical rules. ",
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+        "type": "text",
+        "text": "3 BACKGROUND ",
+        "text_level": 1,
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+        "text": "In the following, we review the background relevant for the custom Prolog engine employed by our method. Specifically, we briefly introduce Prolog’s backward chaining algorithm and unification procedure (Russell & Norvig, 2016). We assume basic knowledge of formal logic and logic programming. ",
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+        "text": "In a nutshell, a Prolog program consists of a set of rules in the form of Horn clauses ",
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+        "text": "$$\nh ( f _ { 1 } ^ { h } , \\ldots , f _ { n } ^ { h } ) \\ :  \\ : p _ { 1 } ( f _ { 1 } ^ { 1 } , \\ldots , f _ { m } ^ { 1 } ) \\land \\ldots \\land p _ { B } ( f _ { 1 } ^ { B } , \\ldots , f _ { l } ^ { B } ) ,\n$$",
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+        "type": "text",
+        "text": "where $h , p _ { i }$ are predicate symbols and $f _ { j } ^ { i }$ are either function (denoted in lower case) or variable (upper case) symbols. The domain of function symbols is denoted by $\\mathcal { F }$ and the domain of predicate symbols by $\\mathcal { P }$ . $h ( f _ { 1 } ^ { h } , \\ldots , f _ { n } ^ { h } )$ is called the head and $p _ { 1 } \\bigl ( f _ { 1 } ^ { 1 } , \\ldots , f _ { m } ^ { 1 } \\bigr ) \\ \\wedge \\ \\ldots \\ \\wedge \\ p _ { B } \\bigl ( f _ { 1 } ^ { B } , \\ldots , \\bar { f } _ { l } ^ { B } \\bigr )$ the body of the rule. We call $B$ the body size of the rule and rules with a body size of zero are named atoms (short for atomic formula).2 If an atom does not contain any variable symbols it is termed fact. As in related work (Sessa, 2002; Julián-Iranzo et al., 2009), we disregard negation and disjunction. ",
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+        "text": "The central procedure of Prolog is unification. It receives two atoms and attempts to find variable substitutions that make both atoms syntactically equal. For example, the input atoms country(Greece, Socrates) and country $( X , Y )$ result in the following variable substitution after unification: $\\{ X / G r e e c e , Y / S o c r a t e s \\}$ . ",
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+    {
+        "type": "text",
+        "text": "The proof algorithm of Prolog is called backward-chaining. It starts from a goal atom $g$ and attempts to prove it by applying suitable rules, thereby generating subgoals that are proved next. To find applicable rules, it attempts to unify $g$ with the heads of all available rules. If this unification succeeds, the resulting variable substitutions are applied to the atoms in the rule body and each of those atoms becomes a new subgoal. For instance, the application of the rule countr $\\prime ( X , Y ) \\Leftarrow b o r n \\_ i n ( Y , X )$ to the goal country(Greece, Socrates) would yield the subgoal born_in(Socrates, Greece). Then the process is repeated for all subgoals until no subgoal is left, i.e., until all subgoals have been unified with a fact. The result of this procedure is a set of rule applications and variable substitutions called proof. Note, that the number of possible proofs is exponential in the number of predicate and entity symbols, as every rule might be used in the proof of each subgoal. Pseudo code for weak unification can be found in Appendix A.1 and we refer the reader to Russell & Norvig (2010) for more details. To apply standard Prolog to a NLP problem like QA, one would have to account for semantic similarities and ambiguities with extensive and error-prone preprocessing, e.g. when transforming the natural language input to logical form. ",
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+        "type": "text",
+        "text": "4 NLPROLOG ",
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+        "text": "Our aim is to apply Prolog to natural language question answering where the same entity or relation can have different natural language surface forms. Thus, we replace the equality-based unification operator with similarity-based weak unification (Sessa, 2002), which allows to unify two symbols $x , y$ if they are sufficiently similar, as judged by a similarity function $\\sim \\theta$ parameterized by $\\theta$ . Then, the result of unification also contains a proof success score $S$ that is the result of the symbols’ similarity and the previous success score $S ^ { \\prime }$ : $S = \\top ( x \\sim y , S ^ { \\prime } )$ , where $\\top \\in \\{ \\mathrm { m i n } , \\cdot \\}$ is an aggregation function. The result of backward-chaining with weak unification are (possibly) multiple proofs, each with an associated proof success score. NLPROLOG combines inference based on weak unification and distributed representations to allow reasoning on natural language statements. The natural language statements are first transformed to triples using Open IE (Etzioni et al., 2008). The symbols occurring in these triples and in the rules are then embedded into a vector space, which in turn is used to estimate similarities between symbols. The resulting similarity function is subsequently used to perform a proof and consequently obtain a proof success score $S$ . The proof success score is then utilized as a training signal for ES. An illustration of the process can be found in Fig. 1, where we visualize the interplay of the different components for our running example. ",
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+        "text": "4.1 SIMILARITY COMPUTATION ",
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+        "text": "We embed symbols using an encoder $\\mathcal { E } _ { \\theta } : \\mathcal { F } \\cup \\mathcal { P } \\mapsto \\mathbb { R } ^ { d }$ parametrized by $\\theta$ for entity and predicate symbols, where $d$ denotes the embedding size. The resulting embeddings are used to induce $\\sim \\theta$ $( \\mathcal { \\dot { F } } \\cup \\mathcal { P } ) ^ { 2 } \\mapsto \\mathbb { R }$ : ",
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+        "img_path": "images/4cfde273aac8112af1ac6fcf5b0bf2b506871a8cb3d9ee7cbbf94d5a9168e57a.jpg",
+        "text": "$$\nx \\sim _ { \\theta } y = \\frac { 1 + c o s ( \\mathcal { E } _ { \\theta } ( x ) , \\mathcal { E } _ { \\theta } ( y ) ) } { 2 }\n$$",
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+        "type": "text",
+        "text": "where cos denotes the cosine similarity between two vectors. There are alternative similarity functions such as Euclidean distance or RBF kernel, but in preliminary experiments we found cosine simlarity to work more robustly. ",
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+        "type": "text",
+        "text": "We use an encoder function that uses a embedding lookup table for predicate symbols and a different one for entities. All embeddings representing natural language phrases are populated with sentence vectors that were pretrained with SENT2VEC (Pagliardini et al., 2017). Additionally, we introduce a third lookup table for the predicate symbols of rules and goals, because semantics of goal or rule predicates might differ from the semantics of fact predicates even if they share the same surface form. For instance, the query $( X , p a r e n t , Y )$ could be interpreted either as (X, is the parent of, $Y$ ) or as (X, has the parent, Y) which are fundamentally different things. ",
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+    {
+        "type": "text",
+        "text": "4.2 FINE-TUNING THE ENCODER TO A DOWNSTREAM TASK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We propose to fine-tune the similarity function on a downstream task by updating the symbol embeddings. As NLPROLOG involves the non-differentiable proof search step, we cannot apply backpropagation for optimization. Instead, we propose to employ Evolution Strategies in conjunction with Adam (Kingma & Ba, 2014) to estimate the weight updates. ES recently showed good results for Reinforcement Learning problems in Salimans et al. (2017); Mania et al. (2018). More formally, the parameter update is computed as: ",
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+        "image_caption": [
+            "Figure 1: Overview of NLPROLOG. The components of NLPROLOG are depicted as ellipses, while inputs and outputs are drawn as squares. Phrases with red background are entities and blue ones are predicates. "
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+        "text": "$$\n\\theta _ { t + 1 } = \\theta _ { t } + \\frac { \\alpha _ { t } } { N \\sigma _ { J } ^ { t } } \\sum _ { \\stackrel { k = 1 } { \\epsilon _ { k } \\sim \\mathcal { N } ( 0 , \\sigma ) } } ^ { N } \\left[ J ( \\theta _ { t } + \\epsilon _ { k } ) - J ( \\theta _ { t } - \\epsilon _ { k } ) \\right] \\epsilon _ { k }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $J ( \\theta )$ is the reward obtained by $\\theta$ , $\\boldsymbol { \\sigma } _ { J } ^ { t }$ is the standard deviation of all rewards obtained at time $t$ as proposed by Mania et al. (2018), and $\\alpha _ { t }$ are adaptive learning rates selected by ADAM (Kingma & Ba, 2014). The standard deviation $\\sigma$ of the distribution that is generating the perturbations is treated as a hyperparameter. ",
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+        "type": "text",
+        "text": "We train NLPROLOG with ES using a learning from entailment setting (Muggleton & Raedt, 1994), in which the model is trained to decide whether a Prolog program $\\mathcal { R }$ entails the truth of a candidate triple $c$ . The objective of the model is to assign high probabilities $p ( c | \\mathcal { R } ; \\theta )$ to true candidate triples and low probabilities to false triples. To achieve this, we model the reward as $J ( \\theta ) = y p ( c | \\mathcal { R } ; \\theta )$ , where $\\bar { y \\in \\{ - 1 , 1 \\} }$ is the gold label. To estimate $p ( c | \\mathcal { R } ; \\theta )$ , we exhaustively search for all proofs for the triple $c$ , up to a given depth $D$ which we treat as a hyperparameter. This search yields a number of proofs each with a success score $S _ { i }$ . We set $p ( c | \\mathcal { R } ; \\theta )$ to be the maximum of these scores $S _ { \\operatorname* { m a x } } = \\operatorname* { m a x } _ { i } S _ { i }$ . ",
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+    {
+        "type": "text",
+        "text": "4.3 GRADUAL RULE LEARNING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The reasoning process of NLPROLOG crucially depends on rules that describe the relations between predicates. While it is possible to write down rules in natural language, this approach is hardly scalable. Thus, we follow Rocktäschel & Riedel (2017) and use rule templates to perform Inductive Logic Programming (ILP) (Muggleton, 1991) which allows NLPROLOG to learn rules from training data. For this, a user has to define a set of rules with a given structure as input. Then, NLPROLOG randomly initializes the predicates of these rules. For instance, to induce a rule that can model transitivity, one would add a rule template of the form $p _ { 1 } ( X , Z )  p _ { 2 } ( X , Y ) \\land p _ { 3 } ( Y , Z )$ , and NLPROLOG would instantiate multiple rules with randomly initialized embeddings for $p _ { 1 } , p _ { 2 }$ , and $p _ { 3 }$ . The exact number and structure of the rule templates is treated as a hyper-parameter. If not explicitly stated otherwise, the experiments were performed with the same set of rule templates containing two rules for each of the forms $p _ { 1 } ( X , { \\bar { Y } } ) \\Leftarrow p _ { 2 } ( X , Y )$ , $p _ { 1 } ( X , Y ) \\Leftarrow p _ { 2 } ( Y , \\bar { X _ { \\lambda } } )$ and $p _ { 1 } ( X , Z ) { \\overset { \\vartriangle } { \\Leftarrow } } p _ { 2 } ( X , Y ) \\land p _ { 3 } ( Y , Z )$ . ",
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+        "type": "text",
+        "text": "In preliminary experiments, we observed that unification with such randomly initialized embeddings always leads to a stark drop of the proof success score. This implies in turn that proofs involving rule templates rarely yield the highest score $S _ { \\mathrm { m a x } }$ and thus, might have no impact on the value of the reward function. Due to that, the expected gradient estimate for the rule embeddings is zero, and they will remain close to their initialization. ",
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+        "type": "text",
+        "text": "The main reason for this behavior is that the monotonicity of the aggregation functions implies that each sub-goal created by an application of a rule will only decrease the success score of the proof. Thus, all other things being equal, rules with a small number of body atoms (and facts in particular) will be preferred over more complex rules with a higher number of body atoms. Note that this problem is particularly severe in our setting where rules are initialized randomly while the remaining predicate symbols are instantiated using pretrained sentence embeddings. ",
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+        "type": "text",
+        "text": "We propose a Gradual Rule Learning (GRL) algorithm which counteracts this effect during training. GRL segments the training process into $B _ { \\mathrm { m a x } } + 1$ phases, where $B _ { \\mathrm { m a x } }$ is the maximum body size of all available rules. In the $k$ -th phase of GRL, only proofs which employ at least one rule with a body size of $B _ { \\mathrm { m a x } } + 1 - k$ are considered to estimate $p ( t | F ; \\theta )$ . Thus, it is guaranteed that in each training step in phase $k$ at least one rule with a body size of $B _ { \\mathrm { m a x } } + 1 - k$ receives a training signal. ",
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+        "type": "text",
+        "text": "5 EVALUATION ",
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+        "text": "We evaluate our method on two different QA data sets: BABI-1K-STATIC (Weston et al., 2015) and different subsets of WIKIHOP (Welbl et al., 2017). The used hyperparameter configurations can be found in Section B. ",
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+        "type": "text",
+        "text": "5.1 SUBSETS OF WIKIHOP ",
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+        "type": "text",
+        "text": "We evaluate on different subsets of WIKIHOP (Welbl et al., 2017), each containing a single query predicate. We consider the predicates publisher, developer, and country, because their semantics ensure that the annotated answer is unique and they contain a relatively high amount of questions that are annotated as requiring true multi-hop reasoning. For publisher, this yields 509 training and 54 validation questions, for developer 267 and 29, and for country 742 and 194. As the test set of WIKIHOP is not publicly available, and splitting the small train set would lead to a far too small validation set, we report scores for the validation set and refrain from hyperparameter optimization and early stopping. ",
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+        "type": "text",
+        "text": "Each data point consists of a query $p ( q , X )$ where $q$ is some query entity, $X$ is the entity that has to be predicted, $C$ is a list of candidates entities, $a \\in C$ is an answer entity and $p \\in$ {publisher, developer, country} is the query predicate. In addition, every query is accompanied by a set of support documents which can be used to decide which candidate is the correct answer. To transform the support documents to natural language triples, we use the Open IE system MINIE (Gashteovski et al., 2017). We use the publicly available version of $\\mathbf { M N I E } ^ { 3 }$ in the dictionary mode, and use a list of all WIKIHOP candidate entities as our dictionary of multi-token entities. ",
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+        "type": "text",
+        "text": "Following Welbl et al. (2017), we use the two neural QA models BIDAF (Seo et al., 2016a) and FASTQA (Weissenborn et al., 2017) as baselines for the selected predicates of WIKIHOP. We employ the implementation provided by the QA framework $\\mathrm { J A C K } ^ { 4 }$ (Weissenborn et al., 2018) with the same hyperparameters as used by Welbl et al. (2017) and train a separate model for each predicate. In order to compensate for the fact that both models are extractive QA models which cannot make use of the candidate entities, we additionally evaluate modified versions which transform both the predicted answer and all candidates to vectors using the wiki-unigrams model5 of SENT2VEC (Pagliardini et al., 2017). Consequently, we return the candidate entity which has the highest cosine similarity to the predicted entity. ",
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+    {
+        "type": "text",
+        "text": "5.2 BABI-1K-STATIC ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We utilize a subset of BABI-1K to study the behavior of NLPROLOG in a very controlled environment. Note however, that the experiments on WIKIHOP are much more significant, as they involve natural linguistic variability. BABI-1K-STATIC consists of the tasks QA4,QA15,QA17, and QA18 from the BABI suite (Weston et al., 2015), each containing 1,000 training and 1,000 testing questions. These tasks were selected because, unlike the other tasks of BABI, they do not require reasoning about a dynamically changing world state, which is not supported by the current implementation of NLPROLOG. We automatically transform all statements and queries of the respective tasks to triples and use the resulting KB as input to NLPROLOG. We train and evaluate on each problem individually. The tasks QA4 and QA15 require entities as an output, thus we consider every entity that occurs at least once in any problem of the task as a candidate for all problems. Tasks QA17 and QA18 are binary classification tasks, and thus we determine the optimal threshold on the training set, after the training of NLPROLOG has finished. ",
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+        "type": "text",
+        "text": "We refrain from systematically comparing results on the individual BABI tasks to competing methods like Seo et al. (2016b); Peng et al. (2015); Dhingra et al. (2018); Henaff et al. (2016); Sukhbaatar et al. (2015), since our non-negligible preprocessing and evaluation on only four out of 20 tasks does not allow us to match the relevant evaluation protocols. We therefore utilize BABI-1K-STATIC only for ablation experiments, but note that NLPROLOG achieves similar or better accuracy values than the mentioned methods in all instances we studied, except on $Q A 4$ . ",
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+    {
+        "type": "text",
+        "text": "5.3 DEALING WITH CANDIDATES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "All questions of WIKIHOP and some of BABI-1K-STATIC include a set of answer candidates $C$ . For those cases, we modify our reward function to leverage this information, taking inspiration from Bayesian Personalized Ranking (Rendle et al., 2009): ",
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+        "type": "equation",
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+        "text": "$$\nJ ( \\theta ) = p ( a | \\mathcal { R } ; \\theta ) - \\operatorname* { m a x } _ { c \\in C \\setminus \\{ a \\} } p ( c | \\mathcal { R } ; \\theta ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $a \\in C$ is the true answer. ",
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+    {
+        "type": "text",
+        "text": "We observed that this reward function does not work well with the minimum aggregation function. Therefore, we employ this modified reward only when using the product aggregation and utilize the reward described in Section 4.2 with the minimum aggregation. ",
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+    {
+        "type": "text",
+        "text": "5.4 RESULTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "The results for the selected predicates of WIKIHOP can be found in Table 1. While there is no single best performing model, NLPROLOG is outperformed by at least one neural QA model on every predicate. For country, this only holds true when considering the versions of the neural models that have been augmented to consider candidate entities. For all three predicates, only a single transitive rule is utilized across all validation questions. Since we observe a more diverse set of induced rules on BABI-1K, we partly attributed this lack of diverse rules to the multi-hop characteristic of the WIKIHOP data. It seems that NLPROLOG struggles to find meaningful rules for the predicates developer and publisher, which leads to very few proofs involving rules on the development set: 1 out of 54 for publisher and 2 out of 29 for developer, compared with 159 out of 194 for country. We partially attribute this to the fact, that the semantics of country suggest a straightforward rule (transitivity of location), which is not true for developer or publisher. Additionally, the annotations regarding the neccessity of multi-hop reasoning provided for the validation set (Welbl et al., 2017) suggest that publisher and developer contain significantly fewer training examples involving multi-hop reasoning. ",
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+    {
+        "type": "text",
+        "text": "Exemplary proofs generated by NLPROLOG on the predicates developer and country can be found in Fig. 3. As we are especially interested in assessing the capability for multi-hop reasoning, we additionally evaluate on a subset of problems which have been unanimously labelled as requiring multi-hop reasoning. On this subset of the development data, which is denoted as Countryhop (mh) in Table 1, NLPROLOG outperforms all other single models. ",
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+    {
+        "type": "text",
+        "text": "5.5 NEURAL QA AND NLPROLOG ENSEMBLE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "If the proof of NLPROLOG producing the prediction does not employ any rules, the prediction is essentially the result of performing a simple nearest neighbor search among the embeddings of all fact triples. We hypothesize that the neural QA models FASTQA and BIDAF are better suited for finding the most predictive single support statement, which motivates an ensemble of a neural QA model and NLPROLOG. We built a system which predicts the output of NLPROLOG when it used at least one rule and the output of the neural QA model otherwise. This allows to employ the multihop reasoning of NLPROLOG when possible and to utilize the pattern matching capabilities of the neural QA models in the remaining cases. The results for these ensembles are given in the bottom part of Table 1. In all instances, ensembling a neural QA model with NLPROLOG improved upon all single models, indicating that they complement each other well. We analyze reasons for the success of this ensembling strategy in Section 5.6. ",
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+        "table_caption": [
+            "Table 1: Accuracy scores in percent for different predicates on the development set of WIKIHOP. country (mh) only considers the problems that have been unanimously judged to require multi-hop reasoning. The upper part lists single models and the bottom part ensembles. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>publisher</td><td>developer</td><td>country</td><td>country (mh)</td></tr><tr><td>BIDAF</td><td>66.67</td><td>65.52</td><td>53.09</td><td>53.19</td></tr><tr><td>FastQA</td><td>62.96</td><td>62.07</td><td>57.21</td><td>57.44</td></tr><tr><td>BIDAF-Sent2Vec</td><td>75.93</td><td>68.97</td><td>61.86</td><td>65.96</td></tr><tr><td>FastQA-Sent2Vec</td><td>75.93</td><td>58.62</td><td>64.95</td><td>65.96</td></tr><tr><td>NLProlog (ours)</td><td>51.85</td><td>48.28</td><td>63.92</td><td>68.09</td></tr><tr><td>BIDAF-Sent2Vec + NLProlog</td><td>77.78</td><td>72.41</td><td>65.46</td><td>70.21</td></tr><tr><td>FastQA-Sent2Vec + NLProlog</td><td>77.78</td><td>62.07</td><td>69.07</td><td>70.21</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5.6 ERROR ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We conducted an extensive error analysis, in which we manually classified all errors of NLPROLOG on the selected WIKIHOP predicates into predefined categories. The results are depicted in Figure 2. ",
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+    {
+        "type": "text",
+        "text": "For the developer predicate, in the majority of cases, errors were caused by OPENIE during the fact extraction step: in all but one case, OPENIE did not produce the necessary facts, or a necessary facts was not stated in the support text, or there were multiple correct candidates and NLPROLOG selected the wrong one. As a consequence, for both the publisher and developer predicates, the majority of queries would not answerable, even when the necessary rules were correctly induced. The predictive accuracy was significantly higher for the country predicate, where errors were mostly due to entitities not having SENT2VEC embeddings and a few missing rules. ",
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+    {
+        "type": "text",
+        "text": "Fig. 2 indicates that FASTQA can produce the correct answer, even if crucial information is missing from the supporting documents. To analyze this further, we evaluated FASTQA and NLPROLOG on a test set of the country predicate in which all documents mentioning the query entity were removed. Remarkably, the accuracy of FASTQA increased by approximately $1 \\%$ , while the accuracy of NLPROLOG decreased by approx. $1 1 \\%$ . ",
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+    {
+        "type": "text",
+        "text": "Furthermore, we evaluated both FASTQA and NLPROLOG on the hard subset of country as defined by Sugawara et al. (2018): on these 62 problems which cannot be solved with a simple heuristic, NLPROLOG achieved an accuracy of $5 1 . { \\bar { 6 } } 1 \\%$ , as opposed to $4 6 . 7 7 \\%$ by FASTQA. ",
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+    {
+        "type": "text",
+        "text": "We conjecture that – besides NLPROLOG’s multi-hop reasoning capability – this is one reason why the neural models and NLPROLOG complement each other nicely: neural models can compensate for missing information, while NLPROLOG is less susceptible for spurious correlations between the query and supporting texts. The complementary nature of both approaches is further supported by the error analysis, described in Fig. 4. ",
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+    {
+        "type": "text",
+        "text": "5.7 ABLATION EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We perform experiments on BABI-1K-STATIC to investigate the effect of the GRL training procedure. Specifically, we perform experiments on BABI-1K-STATIC with only the last phase of GRL (i.e. training without GRL), with the last and the penultimate phase, and with all three phases, corresponding to full GRL as we limit the rule complexity to two body atoms in rule templates. To maintain comparability between runs, we keep the number of SGD steps constant across all experiments. ",
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+    {
+        "type": "image",
+        "img_path": "images/84a090321be673203bf5af7c84917979a70a09bb28ca2bb215b164e77262c9b5.jpg",
+        "image_caption": [
+            "Figure 2: Results of the error analysis. The y-axis shows the number of errors per category: O(penIE) refers to a crucial information missing due to an error of the OpenIE tool, A(mbiguous) means that multiple candidates were correct and NLProlog chose one which was not labeled as the correct one, while I(nfo) refers to problems that are not answerable with the provided support texts. $R ( u l e )$ means that a required rule was not induced, and E(mbedding) implies that the answer was correctly deduced but received a lower score than an erroneous deduction of another candidate. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/fc806b26f3b425e539cfbc4cd0786e585248dc3a1f8d62bd86caa650c4cbb87d.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>QA4</td><td>QA15</td><td>QA17</td><td>QA18</td></tr><tr><td>GRL phase 3 (no)</td><td>56.4</td><td>37.1</td><td>62.8</td><td>90.9</td></tr><tr><td>GRL phase 2+3</td><td>88.8</td><td>33.5</td><td>66.7</td><td>94.7</td></tr><tr><td>GRL phase 1+2+3 (full)</td><td>88.0</td><td>100.0</td><td>67.6</td><td>97.6</td></tr><tr><td>Gold Rules (Untrained)</td><td>100.0</td><td>100.0</td><td>73.2</td><td>92.0</td></tr><tr><td>Gold Rules (Trained)</td><td>100.0</td><td>100.0</td><td>84.8</td><td>98.1</td></tr><tr><td>No Rules (Trained)</td><td>60.0</td><td>32.8</td><td>63.0</td><td>90.4</td></tr><tr><td>Ten Templates</td><td>88.1</td><td>100.0</td><td>61.9</td><td>89.4</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 2: Accuracies in percent of ablation experiments on BABI-1K-STATIC. The upper half examines the effectiveness of GRL, while the bottom results concern the effect of training and different choices of rules. ",
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+    {
+        "type": "text",
+        "text": "Additionally, we experiment with manually defined rules, which we deem sufficient for solving each of the four tasks. For these, we report results before and after training, as well as for a run without any rule templates. The accuracy scores for all experiments on BABI are provided in Table 2. ",
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+    },
+    {
+        "type": "text",
+        "text": "To assess the impact of the choice of rule templates, we evaluate NLPROLOG on bAbI-1k-static with a different set of rule templates containing two rules of the form $p _ { 1 } ( X , Y ) \\Leftarrow p _ { 2 } ( X , Y )$ , four with the form $p _ { 1 } ( X , Y ) \\Leftarrow p _ { 2 } ( Y , X )$ and another four for $p _ { 1 } ( X , Z ) \\in p _ { 2 } ( X , Y ) \\land p _ { 3 } ( Y , Z )$ . ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Clearly, the full GRL is necessary to obtain acceptable results in any of the problems. Interestingly, phase 1 of GRL does not contribute anything for QA4, which is perfectly solvable using only rules of body size 1. On the other hand, QA15 and QA18 both require a rule of body size 2, which makes phase 1 strongly improve the results. Only for $Q A 1 7$ the results are inconclusive. Nevertheless, this indicates that GRL works as intended, with the earlier phases encouraging the induction of rules with a higher number of conjuncts in the body. ",
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+    {
+        "type": "text",
+        "text": "The results using manually defined rules suggest that even when sufficient rules are provided, training with ES is helpful nevertheless. Interestingly, the model using no rules is able to solve over $9 0 \\%$ of the problems in $Q A I \\&$ , indicating that this problem is not well suited for evaluating reasoning capabilities of QA models. ",
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+    {
+        "type": "text",
+        "text": "Using ten instead of six templates leads to worse performance on all BABI problems but QA15, which is solved perfectly with a much faster convergence rate. This result indicates that the choice of rule templates might be an important hyperparameter which should be optimized for a given problem. ",
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION AND FUTURE WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We have developed NLPROLOG, a system that is able to perform rule-based reasoning on natural language input, and can learn domain-specific natural language rules from training data. To this end, we have proposed to combine a symbolic prover with pretrained sentence embeddings and to train the resulting system with Evolution Strategies. We have evaluated NLPROLOG on two different QA tasks, showing that it can learn domain-specific rules and produce predictions which complement those of the two strong baselines BIDAF and FASTQA. This allows to build an ensemble of a baseline and NLPROLOG which outperforms all single models. ",
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+        "img_path": "images/7723a7462c403a1c25c15c96c367976e9d871a3b0467a3e7ebbd6acdeb635581.jpg",
+        "image_caption": [
+            "Figure 3: Example proof trees generated by NLPROLOG. Each of the two trees shows an application of a transitive rule, the first for the predicate developer and the second for the predicate country. The rule templates are displayed with the most similar predicate. Note the noise introduced by the Open IE process, e.g. QUANT_0_1 and that entities and predicates do not need to match exactly. "
+        ],
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+    {
+        "type": "text",
+        "text": "While we have focused on a subset of First Order Logic in this work, the expressiveness of NLPROLOG could be extended by incorporating a different symbolic prover. For instance, a prover for temporal logic (Orgun & Ma, 1994) would allow to model temporal dynamics in natural language and enable us to evaluate NLPROLOG on the full set of BABI tasks. We are also interested in incorporating future improvements of symbolic provers, Open IE systems and pretrained sentence representations to further enhance the performance of NLPROLOG. To study the performance of the proposed method without the noise introduced by the Open IE step, it would be useful to evaluate it on tasks like knowledge graph reasoning. Additionally, it would be interesting to study the behavior of NLPROLOG in the presence of multiple WIKIHOP query predicates. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    },
+    {
+        "type": "text",
+        "text": "A ALGORITHMS ",
+        "bbox": [
+            176,
+            102,
+            326,
+            117
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.1 WEAK UNIFICATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            361,
+            147
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "fun unify $( x , y , \\theta , S )$ ",
+        "bbox": [
+            173,
+            167,
+            310,
+            181
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "text_level": 1,
+        "bbox": [
+            199,
+            183,
+            245,
+            194
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "$x$ : function $f ( \\dots ) \\mid$ atom $p ( \\dots ) \\mid$ variable $\\mid$ list $x _ { 1 } : : x _ { 2 } : : \\ldots : : x _ { n }$ $y$ : function $f ^ { \\prime } ( \\dots ) \\mid$ atom $p ^ { \\prime } ( \\ldots ) \\mid$ variable $|$ list $y _ { 1 } : : y _ { 2 } : : \\dots : : y _ { m }$ $\\theta$ : current substitutions, default $= \\{ \\}$   \n$S$ : current success score, default $= 1 . 0$ ",
+        "bbox": [
+            199,
+            194,
+            645,
+            250
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Output: (Unifying substitution $\\theta ^ { \\prime }$ or failure, Updated success score $S ^ { \\prime }$ ) if $\\theta =$ failure then return (failure, 0) else if $S < \\lambda$ then return (failure, 0) else if $x = y$ then return $( \\theta , S )$ else if $x$ is Var then return unify $_ - \\nu a r ( x , y , \\theta , S )$ else if $y$ is Var then return unify_ $\\nu a r ( y , x , \\theta , S )$ else if $x$ is $f ( x _ { 1 } , \\ldots , x _ { n } )$ , y is $f ^ { \\prime } ( y _ { 1 } , \\ldots , y _ { n } )$ , and $f \\sim f ^ { \\prime } \\geq \\lambda$ then $S ^ { \\prime } : = S \\wedge f \\sim f ^ { \\prime }$ return uni $\\mathfrak { f } ( x _ { 1 } : : \\dots : : x _ { n } , y _ { 1 } : : \\dots : : y _ { n } , \\theta , S ^ { \\prime } )$ end else if $x$ is $p ( x _ { 1 } , \\ldots , x _ { n } )$ , y is $p ^ { \\prime } ( y _ { 1 } , \\ldots , y _ { n } )$ , and $p \\sim p ^ { \\prime } \\ge \\lambda$ then $S ^ { \\prime } : = S \\wedge f \\sim f ^ { \\prime }$ return unif $\\nu ( x _ { 1 } : : \\ldots : : x _ { n } , y _ { 1 } : : \\ldots : : y _ { n } , \\theta , S ^ { \\prime } )$ end else if $x$ is $x _ { 1 } : : \\ldots : : x _ { n }$ and $y$ is $y _ { 1 } : : \\ldots : : y _ { n }$ then $( \\theta ^ { \\prime } , S ^ { \\prime } ) : = u n i f y ( x _ { 1 } , y _ { 1 } , \\theta , S )$ $\\mathbf { r e t u r n } u n i f y ( x _ { 2 } : : \\dots : : x _ { n } , y _ { 2 } : : \\dots : : y _ { n } , \\theta ^ { \\prime } , S ^ { \\prime } )$ end else if $x$ is empty list and $y$ is empty list then return $( \\theta , S )$ else return (failure, 0) un unify_va $r ( v , o , \\theta , S )$ if $\\{ \\nu / \\nu a l \\} \\in \\theta$ then return unify $( \\nu a l , o , \\theta , S )$ else if $\\{ o / \\nu a l \\} \\in \\theta$ then return unif $\\dot { y } ( \\nu a r , \\nu a l , \\theta , S )$ else return $( \\bar { \\{ } v / o \\} + \\theta , S )$ ",
+        "bbox": [
+            186,
+            243,
+            678,
+            584
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: The weak unification algorithm in Spyrolog without occurs check ",
+        "bbox": [
+            236,
+            583,
+            761,
+            598
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.2 RUNTIME OF PROOF SEARCH ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            628,
+            421,
+            642
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The worst case complexity vanilla logic programming is exponential in the depth of the proof (Russell & Norvig, 2010). However, in our case this is a particular problem because weak unification requires the prover to attempt unification between all entity/predicate symbols. ",
+        "bbox": [
+            174,
+            654,
+            821,
+            696
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To keep things tractable, NLPROLOG only attempts to unify symbols with a similarity greater than some user-defined threshold $\\lambda$ . Furthermore, in the search step for one statement $q$ , for the rest of the search, $\\lambda$ is set to $\\lambda : = \\operatorname* { m a x } ( \\lambda , S )$ whenever a proof for $q$ with success score $S$ is found. Due to the monotonicity of the employed aggregation functions, this allows to prune the search tree without losing the guarantee to find the proof yielding the maximum success score. We found this optimization to be crucial to make the proof search scale for the studied wikihop predicates. ",
+        "bbox": [
+            174,
+            703,
+            825,
+            787
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B HYPERPARAMETER CONFIGURATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            808,
+            516,
+            824
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "On BABI-1K we optimize the embeddings of predicate symbols of rules and query triples, as well as of entities. WIKIHOP has a large number of unique entity symbols and thus, optimizing their embeddings is prohibitive. Thus, we only train the predicate symbols of rules and query triples on this data set. The embeddings for entities and predicate symbols of fact and query triples are initialized using the WIKI-UNIGRAMS model of SENT2VEC, while the embeddings of rule predicates are intialized by uniformly sampling from the interval $[ - \\frac { 1 } { \\sqrt { 6 0 0 } } , \\frac { 1 } { \\sqrt { 6 0 0 } } ]$ . All experiments were performed with the same set of rule templates containing two rules for each of the forms $p ( X , Y ) \\Leftarrow q ( X , Y )$ , $p ( X , Y ) \\Leftarrow q ( Y , X )$ and $p ( X , Z ) \\Leftarrow q ( X , Y ) \\wedge r ( Y , Z )$ and set the similarity threshold $\\lambda$ to 0.3. At each optimization step, we evaluate 100 perturbations sampled from $\\mathcal { N } ( 0 , 0 . 0 4 )$ on a mini-batch of 16 training problems and use all of the directions in the generation of the next weight vector. If not stated otherwise, we use GRL with three phases training for 500 mini-batches in each phase. For the predicates publisher and developer, we used 1,000 mini-batches in the final phase of GRL. To further encourage rule usage, we use the minimum aggregation function in all but the last phases of GRL, in which we switch to product. ",
+        "bbox": [
+            174,
+            839,
+            825,
+            928
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            826,
+            215
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C ERROR SETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            234,
+            315,
+            251
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/8ed0a056d6d6697c5278a4b2b4a83f050042fb18571dfeb6c97d3c56fb6ab74a.jpg",
+        "image_caption": [
+            "Figure 4: Venn diagram of the error sets per predicate. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            270,
+            808,
+            405
+        ],
+        "page_idx": 14
+    }
+]

parse/train/UoaQUQREMOs/UoaQUQREMOs.md

@@ -0,0 +1,311 @@
+# CT-NET: CHANNEL TENSORIZATION NETWORK FOR VIDEO CLASSIFICATION
+
+Kunchang $\mathbf { L i } ^ { 1 2 }$ ∗, Xianhang $\mathbf { L i ^ { \alpha } }$ ∗, Yali Wang13∗, Jun Wang 4& Yu Qiao13†
+
+1Guangdong-Hong Kong-Macao Joint Laboratory of Human-Machine   
+Intelligence-Synergy Systems, Shenzhen Institutes of Advanced Technology,   
+Chinese Academy of Sciences, Shenzhen, 518055, China   
+2University of Chinese Academy of Sciences   
+3SIAT Branch, Shenzhen Institute of Artificial Intelligence and Robotics for Society   
+4University of Central Florida   
+{kc.li, yl.wang, yu.qiao}@siat.ac.cn   
+xianhangli@knights.ucf.edu, Jun.Wang@ucf.edu
+
+# ABSTRACT
+
+3D convolution is powerful for video classification but often computationally expensive, recent studies mainly focus on decomposing it on spatial-temporal and/or channel dimensions. Unfortunately, most approaches fail to achieve a preferable balance between convolutional efficiency and feature-interaction sufficiency. For this reason, we propose a concise and novel Channel Tensorization Network (CTNet), by treating the channel dimension of input feature as a multiplication of $K$ sub-dimensions. On one hand, it naturally factorizes convolution in a multiple dimension way, leading to a light computation burden. On the other hand, it can effectively enhance feature interaction from different channels, and progressively enlarge the 3D receptive field of such interaction to boost classification accuracy. Furthermore, we equip our CT-Module with a Tensor Excitation (TE) mechanism. It can learn to exploit spatial, temporal and channel attention in a high-dimensional manner, to improve the cooperative power of all the feature dimensions in our CT-Module. Finally, we flexibly adapt ResNet as our CT-Net. Extensive experiments are conducted on several challenging video benchmarks, e.g., Kinetics-400, Something-Something V1 and V2. Our CT-Net outperforms a number of recent SOTA approaches, in terms of accuracy and/or efficiency.
+
+# 1 INTRODUCTION
+
+3D convolution has been widely used to learn spatial-temporal representation for video classification (Tran et al., 2015; Carreira & Zisserman, 2017). However, over parameterization often makes it computationally expensive and hard to train. To alleviate such difficulty, recent studies mainly focus on decomposing 3D convolution (Tran et al., 2018; 2019). One popular approach is spatial-temporal factorization (Qiu et al., 2017; Tran et al., 2018; Xie et al., 2018), which can reduce overfitting by replacing 3D convolution with 2D spatial convolution and 1D temporal convolution. But it still introduces unnecessary computation burden, since both spatial convolution and temporal convolution are performed over all the feature channels. To further decrease such computation cost, channel separation has been recently developed via operating 3D convolution in the depth-wise manner (Tran et al., 2019). However, it inevitably loses accuracy due to the lack of feature interaction between different channels. For compensation, it has to introduce point-wise convolution to preserve interaction with extra computation. So there is a natural question: How to construct effective 3D convolution to achieve a preferable trade-off between efficiency and accuracy for video classification?
+
+![](images/33180bbaf52b7732efb27fed5479943b88d9cbd561ea5a770ea36421515293a0.jpg)  
+Interaction (no.of channels,size of receptive field): $( C _ { 1 }$ channels $3 \times 3 \times 3 )$ $( C _ { 1 } \times C _ { 2 } = C$ channels,5 × 5 × 5)
+
+Figure 1: Simple illustration of channel tensorization $K = 2$ ). We tensorize the channel dimension of input feature as a multiplication of $K$ sub-dimensions. Via performing spatial/temporal tensor separable convolution along each sub-dimension, we can achieve a preferable balance between convolutional efficiency and feature-interaction sufficiency. Introduction shows more explanations.   
+
+<table><tr><td rowspan=2 colspan=1>Method</td><td rowspan=2 colspan=1>3D Convolution(t ×hxw)</td><td rowspan=1 colspan=2>Convolutional Efficiency</td><td rowspan=1 colspan=1>Feature-Interact</td><td rowspan=1 colspan=1>onSufficiency</td></tr><tr><td rowspan=1 colspan=1>Spatial-temporal</td><td rowspan=1 colspan=1>Channel</td><td rowspan=1 colspan=1>Interact Manner</td><td rowspan=1 colspan=1>InteractField1</td></tr><tr><td rowspan=1 colspan=1>C3D(Tran et al., 2015)</td><td rowspan=1 colspan=1>Full:3×3×3</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>STC</td><td rowspan=1 colspan=1>33</td></tr><tr><td rowspan=1 colspan=1>R(2+1)D(Tran et al., 2018)</td><td rowspan=1 colspan=1>Full:1×3×3Full:3×1×1</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>SCTC</td><td rowspan=1 colspan=1>3</td></tr><tr><td rowspan=1 colspan=1>CSN(Tran et al., 2019)</td><td rowspan=1 colspan=1>Full:1×1×1DW:3×3×3</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>CST</td><td rowspan=1 colspan=1>33</td></tr><tr><td rowspan=1 colspan=1>Our CT-Net(C=C1x..xCK)</td><td rowspan=1 colspan=1>C1:C1×.x1x(1x3×3+3×1×1)Ck:1xxCkx(1x3×3+3×1×1)</td><td rowspan=1 colspan=1>&lt;</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>STC1STCK</td><td rowspan=1 colspan=1>(2K + 1)3</td></tr></table>
+
+1 Interact Field means the receptive field for feature interaction.
+
+Table 1: Two design principles to build effective video representation and efficient convolution.
+
+This paper attempts to address this question by investigating two design principles. (1) Convolutional Efficiency. As shown in Table 1, current designs of spatial-temporal convolution mainly focus on decomposition from either spatial-temporal (Tran et al., 2018) or channel dimension (Tran et al., 2019). To enhance convolutional efficiency, we consider decomposing convolution in a higher dimension with a novel representation of feature tensor. (2) Feature-Interaction Sufficiency. Table 1 clearly shows that, for current decomposition approaches (Tran et al., 2018; 2019), feature interaction only contains one or two of spatial, temporal and channel dimensions at each sub-operation. Such a partial interaction manner would reduce classification accuracy. On one hand, it decreases the discriminative power of video representation, due to the lack of joint learning on all the dimensions. On the other hand, it restricts feature interaction in a limited receptive field, which ignores rich context from a larger 3D region. Hence, to boost classification accuracy, each sub-operation should achieve feature interaction on all the dimensions, and the receptive field of such interaction should be progressively enlarged as the number of sub-operations increases.
+
+Based on these desirable principles, we design a novel and concise Channel Tensorization Module (CT-Module). Specifically, we propose to tensorize the channel dimension of input feature as a multiplication of $K$ sub-dimensions, i.e., $C = C _ { 1 } \times C _ { 2 } \times \cdot \cdot \cdot \times C _ { K }$ . Via performing spatial/temporal separable convolution along each sub-dimension, we can effectively achieve convolutional efficiency and feature-interaction sufficiency. For better understanding, we use the case of $K = 2$ as a simple illustration in Figure 1. First, we tensorize the input channel into $C = C _ { 1 } \times C _ { 2 }$ . Naturally, we separate the convolution into distinct ones along each sub-dimension, e.g., for the $1 ^ { s t }$ sub-dimension, we apply our spatial-temporal tensor separable convolution with the size $C _ { 1 } \times 1 \times t \times h \times w$ , which allows us to achieve convolutional efficiency on all the spatial, temporal and channel dimensions. After that, we sequentially perform the tensor separable convolution sub-dimension by sub-dimension. As a result, we can progressively achieve feature interaction on all the channels, and enlarge the spatialtemporal receptive field. For example, after operating $1 ^ { s t }$ tensor separable convolution on the $1 ^ { s t }$ sub-dimension, $C _ { 1 }$ channels interact, and 3D receptive field of such interaction is $3 \times 3 \times 3$ . Via further operating $2 ^ { n d }$ tensor separable convolution on the $2 ^ { n d }$ sub-dimension, all $C _ { 1 } \times C _ { 2 } = C$ channels have feature interaction, and 3D receptive field of such interaction becomes $5 \times 5 \times 5$ . This clearly satisfies our principle of feature-interaction sufficiency.
+
+We summarize our contributions in the following. First, we design a novel Channel Tensorized Module (CT-Module), which can achieve convolutional efficiency and feature-interaction sufficiency, via progressively performing spatial/temporal tensor separable convolution along each sub-dimension of the tensorized channel. Second, we equip CT-Module with a distinct Tensor Excitation (TE) mechanism, which can further activate the video features of each sub-operation by spatial, temporal and channel attention in a tensor-wise manner. Subsequently, we apply this full module in a residual block, and flexibly adopt 2D ResNet as our Channel Tensorized Network (CT-Net). In this case, we can gradually enhance feature interaction from a broader 3D receptive field, and learn the key spatial-temporal representation with light computation. Finally, we conduct extensive experiments on a number of popular and challenging benchmarks, e.g., Kinetics (Carreira & Zisserman, 2017), Something-Something V1 and V2 (Goyal et al., 2017b). Our CT-Net outperforms the state-of-the-art methods in terms of classification accuracy and/or computation cost.
+
+# 2 RELATED WORKS
+
+2D CNN for video classification. 2D CNN is a straightforward but useful method for video classification (Karpathy et al., 2014; Simonyan & Zisserman, 2014; Wang et al., 2016; Liu et al., 2020; Jiang et al., 2019). For example, Two-stream methods (Simonyan & Zisserman, 2014) learn video representations by fusing the features from RGB and optical flow respectively. Instead of sampling a single RGB frame, TSN (Wang et al., 2016) proposes a sparse temporal sampling strategy to learn video representations. To further improve accuracy, TSM (Lin et al., 2019) proposes a zeroparameter temporal shift module to exchange information with adjacent frames. However, these methods may lack the capacity of learning spatial-temporal interaction comprehensively, which often reduces their discriminative power to recognize complex human actions.
+
+3D CNN for video classification. 3D CNN has been widely used to learn a rich spatial-temporal context better (Tran et al., 2015; Carreira & Zisserman, 2017; Feichtenhofer et al., 2019; Sudhakaran et al., 2020; Feichtenhofer, 2020). However, it introduces a lot of parameters, which leads to a difficult optimization problem and large computational load. To resolve this issue, I3D (Carreira & Zisserman, 2017) inflates all the 2D convolution kernels pre-trained on ImageNet, which is helpful for optimizing. Other works also try to factorize 3D convolution kernel to reduce complexity, such as P3D (Qiu et al., 2017) and $\mathrm { R } ( 2 { + } 1 ) \mathrm { D }$ (Tran et al., 2018). Recently, CSN (Tran et al., 2019) operates 3D convolution in the depth-wise manner. Nevertheless, all these methods still do not achieve a good trade-off between accuracy and efficiency. To tackle this challenge, we propose CT-Net which learns on spatial-temporal and channel dimensions jointly with lower computation than previous methods.
+
+# 3 METHODS
+
+In this section, we describe our Channel Tensorization Network (CT-Net) in detail. First, we formally introduce our CT-Module in a generic manner. Second, we design a Tensor Excitation (TE) mechanism to enhance CT-Module. Finally, we flexibly adapt ResNet as our CT-Net to achieve a preferable trade-off between accuracy and efficiency for video classification.
+
+# 3.1 CHANNEL TENSORIZATION MODULE
+
+As discussed in the introduction, the previous approaches have problems in convolutional efficiency or feature-interaction sufficiency. To tackle such a problem, we introduce a generic Channel Tensorization Module (CT-Module), by treating the channel dimension of input feature as a multiplication of $K$ sub-dimensions, i.e., $C = C _ { 1 } \times C _ { 2 } \times \cdot \cdot \cdot \times C _ { K }$ . Naturally, this tensor representation allows to tensorize the kernel size of convolution TConv() as a multiplication of $K$ sub-dimensions, too. To simplify the notation, the channel dimension of the output is omitted by default. The output $X _ { o u t }$ can be calculated as follows:
+
+$$
+{ \bf X } _ { o u t } = T C o n v \left( { \bf X } _ { i n } , { \cal W } ^ { C _ { 1 } \times C _ { 2 } \times \dots \times C _ { K } \times t \times h \times w } \right)
+$$
+
+where ${ \bf X } _ { i n }$ and $W$ denote the tensorized input and kernel respectively. However, such an operation requires large computation, so we introduce the tensor separable convolution to alleviate the issue.
+
+Tensor Separable Convolution. We propose to factorize TConv() along $K$ channel sub-dimensions. Specifically, we decompose $T C o n \nu ( )$ as $K$ tensor separable convolutions $T S C o n \nu ( )$ , and apply TSConv() sub-dimension by sub-dimension as follows:
+
+$$
+{ { \bf { X } } _ { k } } = T S C o n \nu \left( { { \bf { X } } _ { k - 1 } } , { W ^ { 1 \times \cdots \times { C _ { k } } \times \cdots \times 1 \times t \times h \times w } } \right)
+$$
+
+where ${ \bf X } _ { 0 } ~ = ~ { \bf X } _ { i n }$ and ${ \bf { X } } _ { o u t } ~ = { \bf { X } } _ { K }$ . On one hand, the kernel size of the $k ^ { t h }$ TSConv() is $( 1 \times \cdot \cdot \cdot \times C _ { k } \times \cdot \cdot \cdot \times 1 \times t \times h \times w )$ ). It illustrates that only $C _ { k }$ channels interact in the $k ^ { t h }$ suboperation, which leads to convolution efficiency. On the other hand, as we stack the TSConv(), each convolution performs on the output features of the previous convolution. Therefore, the spatialtemporal receptive field is enlarged. Besides, interactions first occur in $C _ { 1 }$ channels, second in $C _ { 1 } \times C _ { 2 }$ channels and so on. Finally, $C _ { 1 } \times C _ { 2 } \times \cdot \cdot \cdot \times C _ { K } = C$ channels can progressively interact. This clearly satisfies our principle of feature-interaction sufficiency.
+
+Spatial-Temporal Tensor Separable Convolution. To further improve convolution efficiency, we factorize the 3D TSConv() into 2D spatial TSConv() and 1D temporal TSConv(). Thus, we can obtain the output features $\mathbf { X } _ { k } ^ { S }$ and $\mathbf { X } _ { k } ^ { T }$ as follows:
+
+$$
+\begin{array} { r l } & { \mathbf { X } _ { k } ^ { S } = { S } { - } T { S } C o n \nu \left( \mathbf { X } _ { k - 1 } , W ^ { 1 \times \cdots \times C _ { k } \times \cdots \times 1 \times 1 \times h \times w } \right) } \\ & { } \\ & { \mathbf { X } _ { k } ^ { T } = { T } { - } T { S } C o n \nu \left( \mathbf { X } _ { k - 1 } , W ^ { 1 \times \cdots \times C _ { k } \times \cdots \times 1 \times t \times 1 \times 1 } \right) } \end{array}
+$$
+
+where $S _ { - } T S C o n \nu ( \ j$ and T-TSConv() represent spatial and temporal tensor separable convolution respectively. Finally, we attempt to aggregate spatial and temporal convolution. There are various connection types of spatial and temporal tensor separable convolution, e.g., parallel and serial types. According to the results of the experiments in Section 4, we utilize the parallel method, which illustrates that we sum the spatial feature $\mathbf { X } _ { k } ^ { S }$ and temporal feature $\mathbf { X } _ { k } ^ { T }$ :
+
+$$
+\mathbf { X } _ { k } = \mathbf { X } _ { k } ^ { S } + \mathbf { X } _ { k } ^ { T }
+$$
+
+# 3.2 TENSOR EXCITATION
+
+Our CT-Module separates feature along spatial, temporal and channel dimensions. To make full use of their cooperative power to learn distinct video features, we design a concise Tensor Excitation (TE) mechanism for each dimension. First of all, we attempt to utilize the TE mechanism to enhance spatial and temporal features respectively. For the spatial feature $\mathbf { X } _ { k } ^ { S }$ obtained by Equation 5, our corresponding spatial TE mechanism can be formulated as:
+
+$$
+\mathbf { U } _ { k } = \mathbf { X } _ { k } ^ { S } \otimes S i g m o d ( S \mathbf { - } T S C o n \nu ( T \mathbf { - } P o o l ( \mathbf { X } _ { k } ^ { S } ) ) )
+$$
+
+where T-Pool() represents global temporal pooling, i.e., $T \times 1 \times 1$ average pooling. By performing it on $\mathbf { X } _ { k } ^ { S }$ , we obtain the feature with the size $\left( C _ { 1 } \times C _ { 2 } \times \cdots \times C _ { K } \times 1 \times H \times W \right)$ , which gathers spatial contexts along temporal dimension. Subsequently, the spatial tensor separable convolution $s$ - TSConv() and the activate function Sigmod() are performed to generate the spatial attention heatmap. Finally, the element-wise multiplication $\otimes$ broadcasts the spatial attention along the temporal dimension. Similarly, we perform the temporal TE mechanism for the temporal feature $\mathbf { X } _ { k } ^ { T }$ :
+
+$$
+\mathbf { V } _ { k } = \mathbf { X } _ { k } ^ { T } \otimes S i g m o d ( T \mathbf { \cdot } T S C o n \nu ( S \mathbf { \cdot } P o o l ( \mathbf { X } _ { k } ^ { T } ) ) )
+$$
+
+where S-Pool() and $T – T S C o n \nu ( )$ are global spatial pooling and temporal tensor separable convolution correspondingly. At last, after aggregating the spatial and temporal features by addition, i.e., ${ \bf R } _ { k } = { \bf \Phi }$ $\mathbf { U } _ { k } + \mathbf { V } _ { k }$ , we perform a channel-wise TE mechanism as follows:
+
+$$
+\mathbf { X } _ { k } = \mathbf { R } _ { k } \otimes S i g m o d ( P W \mathbf { - } T S C o n \nu ( S \mathbf { - } P o o l ( \mathbf { R } _ { k } ) ) )
+$$
+
+We adopt point-wise tensor separable convolution $P W  – T S C o n \nu ( )$ to learn the weights for aggregating distinctive channels. The rest follows the previous design. Note that all tensor separable convolutions are performed on the same sub-dimension as the previous convolution, which is essentially different from the SE mechanism (Hu et al., 2020). Through the cooperation of the TE mechanism along three different dimensions, the spatial-temporal features can be significantly enhanced.
+
+![](images/3123f128bfea1d046977bc9497a4d16414806dba1459d1342333046e084a8d08.jpg)  
+Figure 2: The pipelines of CT-Blocks and the overall architecture of CT-Net. We replace one of every two ResBlocks in ResNet with our CT-Block and the extra point-wise convolution in the last sub-dimension $k = K$ ) is ignored. More details can be found in Section3.3.
+
+# 3.3 CHANNEL TENSORIZATIONN NETWORK
+
+We regard ResNet as an exemplar and build up our CT-Net from ResNet50 (or ResNet101). First, we design a simple CT-Block in Figure 2(a), which adapts the $3 \times 3$ convolutional layer in Residual Block (ResBlock) into our CT-Module. It can achieve both convolutional efficiency and featureinteraction sufficiency. Second, we equip our simple CT-Block with the TE mechanism in Figure 2(b), forming a full CT-Block that can improve the cooperative power of all the feature dimensions. Besides, extra point-wise convolutions are added between different sub-operations, which are beneficial for more sufficient feature interaction. At last, we build up a novel CT-Net with CT-Block. As shown in Figure 2(c), we replace one of every two ResBlocks with our CT-Block in every stage. Such a method guarantees a better balance between efficiency and accuracy in our experiments.
+
+Discussion. In fact, the popular methods in video classification like C3D, $\mathrm { R } ( 2 { + } 1 ) \mathrm { D }$ and CSN (Tran et al., 2015; 2018; 2019) can be viewed as special cases of our CT-Net. We can generate different forms by adjusting three hyper-parameters: the number of sub-dimensions $\mathbf { \eta } ^ { ( K ) }$ , the corresponding dimension size $( C _ { k } )$ and the spatial-temporal kernel size $( K e r n e l _ { k } )$ . To degenerate into C3D, we can set $K = 1$ and $K e r n e l _ { 1 } = 3 \times 3 \times 3$ . When $K = 2$ , $K e r n e l _ { 1 } = 1 \times 3 \times 3$ , $K e r n e l _ { 2 } = 3 \times 1 \times 1$ and $C _ { 1 } = C _ { 2 } = C$ without channel tensorization, it becomes $\mathrm { R } ( 2 { + } 1 ) \mathrm { D }$ . Unfortunately, because of lacking the decomposition of channels, C3D and $\mathrm { R } ( 2 { + } 1 ) \mathrm { D }$ still have a large computational load. When $\bar { K ^ { { \prime } } } = 2 , C = \bar { C } _ { 1 } \times C _ { 2 } = C \times 1 , K e r n e l _ { 1 } = 1 \times 1 \times 1 , K e r n e l _ { 2 } = \bar { 3 } \times 3 \times \bar { 3 }$ , obviously it is equivalent to CSN. However, CSN has a limited receptive field of spatial-temporal interaction. In our CT-Net, we utilize channel tensorization and perform tensor separable convolution along each sub-dimension in turn. Such design can not only preserve interaction among spatial, temporal and channel dimensions but also enlarge the receptive field of feature interaction progressively.
+
+# 4 EXPERIMENTS AND RESULTS
+
+Datasets and implementation details. We conduct experiments on three large video benchmarks: Kinetics-400 (Carreira & Zisserman, 2017), Something-Something V1 and V2 (Goyal et al., 2017b). We choose ResNet50 and ResNet101 (He et al., 2016) pre-trained on ImageNet as the backbone and the parameters of CT-Module are randomly initialized. For training, we utilize the dense sampling strategy (Wang et al., 2018) for Kinetics and sparse sampling strategy (Wang et al., 2016) for Something-Something. Random scaling and cropping are applied for data argumentation. Finally, we resize the cropped regions to $2 5 6 \times 2 5 6$ . For testing, we sample multiple clips per video (4 for Kinetics, 2 for others) for pursuing high accuracy, and average all scores for the final prediction.
+
+(a) Effectiveness of CT-Module. CT-Module outperforms the recent modules for video modeling.   
+
+<table><tr><td>Method</td><td>3D Convolution (t × h × w)</td><td>GFLOPs</td><td>Top-1</td><td>Top-5</td></tr><tr><td>C3D-Module (Tran et al.,2015)</td><td>Full:3×3×3</td><td>59.9</td><td>46.1</td><td>75.0</td></tr><tr><td>R(2+1)D-Module (Tran et al., 2018)</td><td>Full:1×3×3+Full:3×1×1</td><td>45.8</td><td>47.0</td><td>76.1</td></tr><tr><td>CSN-Module (Tran et al., 2019)</td><td>Full:1×1×1+DW:3×3×3</td><td>35.6</td><td>46.8</td><td>75.7</td></tr><tr><td>Our CT-Module</td><td>C,…,Cκ:1×3×3+3×1×1</td><td>36.3</td><td>47.3</td><td>76.2</td></tr></table>
+
+<table><tr><td>Number</td><td>GFLOPs</td><td>Top-1</td><td>Top-5</td></tr><tr><td>1D</td><td>45.8</td><td>46.5</td><td>75.6</td></tr><tr><td>2D</td><td>36.3</td><td>47.3</td><td>76.2</td></tr><tr><td>3D</td><td>35.7</td><td>47.1</td><td>75.8</td></tr><tr><td>4D</td><td>35.6</td><td>46.5</td><td>75.3</td></tr></table>
+
+<table><tr><td>Type</td><td>Top-1</td><td>Top-5</td></tr><tr><td>coupling</td><td>47.1</td><td>76.0</td></tr><tr><td>serial</td><td>47.2</td><td>76.1</td></tr><tr><td>parallel</td><td>47.3</td><td>76.2</td></tr></table>
+
+<table><tr><td>C</td><td>GFLOPs</td><td>Top-1</td><td>Top-5</td></tr><tr><td>1</td><td>45.9</td><td>47.0</td><td>76.1</td></tr><tr><td>4</td><td>37.6</td><td>46.8</td><td>76.0</td></tr><tr><td>16</td><td>36.4</td><td>47.2</td><td>76.0</td></tr><tr><td>[√C]</td><td>36.3</td><td>47.3</td><td>76.2</td></tr></table>
+
+(b) Number of sub-dimensions. The higher the dimension, the smaller the GFLOPs. 2D channel tensorization achieves the best trade-off.
+
+(c) Connection type of spatiotemporal convolution. The parallel connection between spatial and temporal convolution is the best choice.
+
+(d) Dimension size. $C = C _ { 1 } \times C _ { 2 }$ and the best trade-off is achieved when adopting the rounded middle size $\lfloor { \sqrt { C } } \rfloor$ .
+
+<table><tr><td>Number</td><td>GFLOPs</td><td>Top-1</td><td>Top-5</td></tr><tr><td>+0 (TSN)</td><td>43.0</td><td>16.9</td><td>42.0</td></tr><tr><td>+1 stage5</td><td>41.9</td><td>42.3</td><td>71.1</td></tr><tr><td>+4 stage4-5</td><td>38.9</td><td>46.9</td><td>75.7</td></tr><tr><td>+6 stage3-5</td><td>37.1</td><td>47.2</td><td>76.1</td></tr><tr><td>+7 stage2-5</td><td>36.3</td><td>47.3</td><td>76.2</td></tr><tr><td>+12 stage2-5</td><td>31.5</td><td>45.9</td><td>76.1</td></tr></table>
+
+(e) Number and location of CT-Blocks. Simply replacing 1 block in stage5 can bring significant performance improvement. As we replace more blocks from the bottom up, the GFLOPs continues to decrease. Replacing 7 blocks achieves the best trade-off between accuracy and GFLOPs.
+
+(f) Kernel sizes along different dimensions. The larger kernel size brings improvement with more calculation.   
+
+<table><tr><td>Kernel size</td><td></td><td>GFLOPs</td><td>Top-1</td><td>Top-5</td></tr><tr><td>C1 1 ×1× 1 C 1 1x3×3</td><td>1×1×1 3×1×1</td><td>35.5</td><td>46.1</td><td>75.1</td></tr><tr><td>C1 1×1×1 C2 1×5×5</td><td>1×1×1 5×1×1</td><td>36.6</td><td>47.2</td><td>76.2</td></tr><tr><td>C1 1 1x3×3 C 1×3×3</td><td>3×1×1 3×1×1</td><td>36.3</td><td>47.3</td><td>76.2</td></tr><tr><td>C 1×3×3</td><td>3×1×1</td><td>37.4</td><td>47.5</td><td>76.4</td></tr><tr><td>C2 1×5×5 C1 1 ×5×5</td><td>5×1×1 5×1×1</td><td>38.9</td><td>47.6</td><td>76.5</td></tr></table>
+
+<table><tr><td>Model</td><td>GFLOPs</td><td>Top-1</td><td>Top-5</td></tr><tr><td>Baseline (TSN)</td><td>43.0</td><td>16.9</td><td>42.0</td></tr><tr><td>+CT-Module</td><td>36.3</td><td>47.3</td><td>76.2</td></tr><tr><td>+CT-Module+PWConv</td><td>37.2</td><td>48.0</td><td>76.7</td></tr><tr><td>+CT-Module+PWConv+SE</td><td>37.2</td><td>48.8</td><td>77.4</td></tr><tr><td>+CT-Module+PWConv+TE</td><td>37.3</td><td>50.1</td><td>78.8</td></tr></table>
+
+(g) Impact of different modules. CT-Module is essential for temporal modeling and TE mechanism is also beneficial.
+
+<table><tr><td></td><td>Input</td><td>GFLOPs</td><td>Top-1</td><td>Top-5</td></tr><tr><td>Train: Test:</td><td>224×224 224× 224</td><td>28.6</td><td>49.1</td><td>77.4</td></tr><tr><td>Train: Test:</td><td>224×224 256×256</td><td>37.3</td><td>49.7</td><td>77.7</td></tr><tr><td>Train:</td><td>256×256</td><td>37.3</td><td></td><td>78.8</td></tr><tr><td>Test:</td><td>256×256</td><td></td><td>50.1</td><td></td></tr></table>
+
+(h) Impact of different spatial resolution.
+
+Table 2: Ablation studies on Something-Something V1. All models use ResNet50 as the backbone
+
+We follow the same strategy in Non-local (Wang et al., 2018) to pre-process the frames and take 3 crops of $2 5 6 \times 2 5 6$ as input. Because some multi-clip models in Table 3 and Table 4 sample crops of $2 5 6 \times 2 5 6$ , we simply multiply the GFLOPs reported in the corresponding papers by (256/224)2 for a fair comparison. When considering efficiency, we use just 1 clip per video and the final crop is scaled to $2 5 6 \times 2 5 6$ to ensure comparable GFLOPs.
+
+# 4.1 ABLATION STUDIES
+
+Table 2 shows our ablation studies on Something-Something V1, which is a challenging dataset that requires video architecture to have a robust spatial-temporal representation ability and is suitable to verify the effectiveness of our method. All models use ResNet50 as the backbone.
+
+Effectiveness of CT-Module. In Table 2a, we replace the $3 \times 3$ convolutional layer in ResNet50 with different modules in recent methods (Tran et al., 2015; 2018; 2019). Compared with CSN-Module, our module achieves a better result with similar computation, which reflects the importance of sufficient feature interaction. Besides, it is slightly better than $\mathrm { R } ( 2 { + } 1 ) \mathrm { D }$ -Module with much lower calculation, showing the necessity of efficient convolution. Such results demonstrate the effectiveness of our CT-Module. It illustrates our two design principles give preferable guidance for designing an efficient module for temporal modeling.
+
+Table 3: Comparison with the state-of-the-art on Something-Something V1&V2. Our CT$\mathrm { N e t } _ { 1 6 f }$ outperforms all the single-clip models in Something-Something and even better than most of the multi-clip models. And our $\mathrm { C T - N e t } _ { E N }$ outperforms all methods with very lower computation.   
+
+<table><tr><td>Method</td><td>Backbone</td><td>#Frame</td><td>GFLOPs</td><td colspan="2">SomethingV1 Top-1 Top-5</td><td colspan="2">SomethingV2 Top-1 Top-5</td></tr><tr><td rowspan="2">ECOENLite (Zolfaghari et al.,2018) NL I3D + GCN (Wang &amp; Gupta,2018)</td><td>Incep+3D R18</td><td>92 32×3×2</td><td rowspan="2">267</td><td>46.4</td><td>=</td><td>■</td><td>=</td></tr><tr><td>3D R50</td><td>1818</td><td>46.1</td><td>76.8</td><td>■</td><td></td></tr><tr><td rowspan="2">ir-CSN (Tran et al.,2019) ir-CSN (Tran et al., 2019) CorrNet (Wang et al.,2020)</td><td>3D R101</td><td>32×1×10</td><td>738</td><td>48.4</td><td>-</td><td>■</td><td>■</td></tr><tr><td>3D R152</td><td>32×1×10</td><td>967 1150</td><td>49.3</td><td>-</td><td>-</td><td>■</td></tr><tr><td rowspan="4">TSN (Wang et al.,2016) TSM (Lin et al., 2019)</td><td>3D R50 2DR50</td><td>32×1×10</td><td></td><td>49.3</td><td>-</td><td>-</td><td>-</td></tr><tr><td></td><td>8</td><td>33</td><td>19.7</td><td>46.6</td><td>27.8</td><td>57.6</td></tr><tr><td>2D R50</td><td>8</td><td>33</td><td>45.6</td><td>74.2</td><td>59.1</td><td>85.6</td></tr><tr><td>bLR50 2D R50</td><td>8×2 8</td><td>24 33</td><td>46.4 47.4</td><td>76.6</td><td>59.1 61.3</td><td>86.0</td></tr><tr><td rowspan="4">TEINet (Liu et al.,2020) TEA (Li et al.,2020b)</td><td>2D Res2Net50</td><td>8</td><td>35</td><td></td><td>■ 78.1</td><td></td><td>-</td></tr><tr><td></td><td></td><td>33</td><td>48.9</td><td></td><td>-</td><td>-</td></tr><tr><td>2D R50+TIM 2D R50+TIM</td><td>8 8×3×2</td><td>259</td><td>49.8 50.4</td><td>- -</td><td>62.6 63.5</td><td>-</td></tr><tr><td>2DR50</td><td>8</td><td>37</td><td>50.1</td><td>78.8</td><td>62.5</td><td>- 87.7</td></tr><tr><td>Our CT-Net</td><td>2D R50</td><td>8×3×2</td><td>224</td><td>51.7</td><td>80.1</td><td>63.9</td><td>88.8</td></tr><tr><td>TSN (Wang et al.,2016) TSM (Lin et al.,2019)</td><td>2D R50</td><td>16</td><td>66</td><td>19.9</td><td>47.3</td><td>30.0</td><td>60.5</td></tr><tr><td rowspan="9">bLVNet-TAM (Fan et al.,2019) TEINet (Liu et al., 2020)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>2D R50</td><td>16</td><td>66</td><td>47.2</td><td>77.1</td><td>63.4</td><td>88.5</td></tr><tr><td>bLR50</td><td>16×2</td><td>48</td><td>48.4</td><td>78.8</td><td>61.7</td><td>88.1</td></tr><tr><td>2D R50</td><td>16</td><td>66</td><td>49.9</td><td>-</td><td>62.1</td><td>-</td></tr><tr><td>TEA (Li et al.,2020b) 2D Res2Net50</td><td>16</td><td>70</td><td>51.9</td><td>80.3</td><td>-</td><td>-</td></tr><tr><td>PEM+TDLoss (Weng et al.,2020) 2D R50+TIM</td><td>16</td><td>66</td><td>50.9</td><td>-</td><td>63.8</td><td>-</td></tr><tr><td>2D R50+TIM</td><td>16×3×2</td><td>517</td><td>52.0</td><td>1</td><td>65.0</td><td>-</td></tr><tr><td>2D R50</td><td>16</td><td>75</td><td>52.5</td><td>80.9</td><td>64.5</td><td></td></tr><tr><td></td><td>16×3×2</td><td>447</td><td>53.4</td><td>81.7</td><td>65.9</td><td>89.3</td></tr><tr><td>Our CT-Net</td><td>2D R50</td><td></td><td></td><td></td><td></td><td></td><td>90.1</td></tr><tr><td>Our CT-NetEN</td><td>2D (R50)×4</td><td>8+12+16+24</td><td>280</td><td>56.6</td><td>83.9</td><td>67.8</td><td>91.1</td></tr></table>
+
+Number of sub-dimensions. Increasing the number of sub-dimensions saves a lot of computation, but the corresponding accuracy first increases and then decreases as shown in Table 2b. Compared with the 1D method, the 4D method significantly reduces GFLOPs, achieving comparable accuracy. As for the decrease of accuracy when $K$ is too large, we argue that the number of channel in the shallow layer is small (64/128), thus there are too few channels in a single dimension, leading to insufficient feature-interaction. Since the 2D method obtains the best trade-off, we set $K = 2$ in all the following experiments.
+
+Connection type of spatiotemporal convolution. The coupling $3 \times 3 \times 3$ convolution can be decomposed into serial or parallel spatial/temporal convolution. Table 2c reveals that factorizing the 3D kernel can boost results as expected. Besides, the parallel connection is better, thus we adopt parallel connection as the default.
+
+Dimension size. As we set $K = 2$ , it is essential to explore the impact of changing the dimension size $C _ { 2 }$ . We can demonstrate that the computation is the lowest when $C _ { 1 } = C _ { 2 } = { \sqrt { C } }$ . Since $C$ is not always a perfect square number, we adopt the rounded middle size √ $\lfloor { \sqrt { C } } \rfloor$ . Table 2d shows that when $\dot { C _ { 2 } } = \dot { [ \sqrt { C } ] }$ , the model not only requires the fewest computation cost but also achieves the best performance. Hence, we set $C _ { 2 } = \lfloor { \sqrt { C } } \rfloor$ naturally.
+
+Number and location of CT-Blocks. Table 2e illustrates that simply replacing 1 block in stage5 can bring significant performance improvement ( $1 6 . 9 \%$ vs. $4 2 . 3 \%$ ). As we replace more blocks from the bottom up, the GFLOPs continues to decrease. Moreover, the bottom blocks seem to be more beneficial to temporal modeling, since replacing the extra 3 blocks in stage2 and stage3 only improve the accuracy by $0 . 4 \%$ $4 6 . 9 \%$ vs. $4 7 . 3 \%$ ). Since replacing 7 blocks achieves the highest accuracy, we replace 7 blocks by default.
+
+Kernel sizes along different dimensions. In Table 2f, we can observe that two concatenated $3 ^ { 3 }$ convolution kernels are slightly better than those with the same receptive field $( 1 ^ { 3 } + 5 ^ { 3 } )$ . Furthermore, the larger kernel size can bring performance improvement but more calculation. It reveals that our CT-Module avoids the limited receptive field of feature interaction, and it can progressively enlarge the receptive field of such interaction on all the dimensions. Considering a better trade-off between accuracy and computation, we choose two concatenated $3 ^ { 3 }$ convolution kernels.
+
+Table 4: Comparison with the state-of-the-art on Kinetics-400. It shows that CT-Net- $R 5 0 _ { 1 6 f }$ can surpass all existing lightweight models and even SlowFast- $R 5 0 _ { 4 0 f }$ . When fusing different models, our model is $\mathbf { 2 . 4 } \times$ faster than SlowFast- $\operatorname { R } 1 0 1 _ { 8 0 f }$ and shows an $0 . { \dot { 9 } } \%$ performance gain.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Backbone</td><td rowspan=1 colspan=1>#Frame</td><td rowspan=1 colspan=1>GFLOPs</td><td rowspan=1 colspan=1>Top-1 Top-5</td></tr><tr><td rowspan=1 colspan=1>R(2+1)D (Tran et al.,2018)TSN (Wang et al., 2016)I3D (Carreira &amp; Zisserman,2017)</td><td rowspan=1 colspan=1>2DR34InceptionInception</td><td rowspan=1 colspan=1>32×1×1025×10×164×N/A×N/A</td><td rowspan=1 colspan=1>1520=152×10800=80×10108×N/A</td><td rowspan=1 colspan=1>72.0  91.472.5  90.271.1  89.3</td></tr><tr><td rowspan=2 colspan=1>TSM (Lin et al.,2019)TEINet (Liu et al., 2020)bLVNet-TAM (Fan et al., 2019)TEA (Li et al., 2020b)PEM+TDLoss (Weng et al., 2020)CorrNet (Wang et al., 2020)SlowFast (Feichtenhofer et al.,2019)SlowFast (Feichtenhofer et al., 2019)</td><td rowspan=2 colspan=1>2DR502D R50bLR502DRes2Net502D R50+TIM3D R503D R50+R503D R50+R50</td><td rowspan=2 colspan=1>16×3��1016×3×10(16×2)×3×316×3×1016×3×1032×1×1036=(4+32)×3×1040=(8+32)×3×10</td><td rowspan=1 colspan=1>2580=86×302580=86×30</td><td rowspan=1 colspan=1>74.7    -76.2   92.5</td></tr><tr><td rowspan=1 colspan=1>561=62.3×92730=91×302580=86×301150=115×101083=36.1×301971=65.7×30</td><td rowspan=1 colspan=1>72.0  90.676.1   92.576.9  93.077.2    -75.2  91.576.4  92.2</td></tr><tr><td rowspan=1 colspan=1>Our CT-Net</td><td rowspan=1 colspan=1>2D R50</td><td rowspan=1 colspan=1>16×3×4</td><td rowspan=1 colspan=1>895=74.6×12</td><td rowspan=1 colspan=1>77.3  92.7</td></tr><tr><td rowspan=4 colspan=1>X3D-XL (Feichtenhofer,2020)SmallBigNet (Li et al.,2020a)ip-CSN (Tran et al., 2019)ip-CSN (Tran et al.,2019)CorrNet (Wang et al.,2020)SlowFast (Feichtenhofer et al.,2019)SlowFast (Feichtenhofer et al.,2019)NL I3D (Wang &amp; Gupta,2018)</td><td rowspan=1 colspan=1>-2D R101</td><td rowspan=1 colspan=1>16×3×1032×3×4</td><td rowspan=1 colspan=1>1452=48.4×306552=546×12</td><td rowspan=4 colspan=1>79.1   93.977.4  93.376.8  92.577.8  92.879.277.9   93.278.9  93.577.7   93.3</td></tr><tr><td rowspan=1 colspan=1>3D R1013D R1523D R101</td><td rowspan=3 colspan=1>32×3×1032×3×1032×3×1040=(8+32)×3×1080=(16+64)×3×10128×3×10</td><td rowspan=2 colspan=1>2490=83.0×303264=108.8×306720=224×303180=106×306390=213×30</td></tr><tr><td rowspan=2 colspan=1>3D R101+R1013D R101+R1013D R101</td></tr><tr><td rowspan=1 colspan=1>10770=359×30</td></tr><tr><td rowspan=1 colspan=1>Our CT-NetOur CT-NetEN</td><td rowspan=1 colspan=1>2D R1012D R50+R101</td><td rowspan=1 colspan=1>16×3×4(16+16)×3×4</td><td rowspan=1 colspan=1>1746=145.5×122641=220.1×12</td><td rowspan=1 colspan=1>78.8  93.779.8  94.2</td></tr></table>
+
+Impact of different modules and different spatial resolution. In Table $2 \mathrm { g }$ , our CT-Module can significantly boost its baseline ( $1 6 . 9 \%$ vs. $4 7 . 3 \%$ ) and the TE mechanism can further improve the accuracy by $2 . 1 \%$ $4 8 . 0 \%$ vs. $5 0 . 1 \%$ ). The extra point-wise convolution also boosts performance, which demonstrates that it is beneficial for sufficient feature interaction. Compared with the SE mechanisms, our TE mechanism focuses more on features in different sub-dimensions individually, thus effectively enhancing spatial-temporal features. In our experiments, to ensure GFLOPs is comparable with other methods, we crop the input to $2 5 6 \times 2 5 6$ during testing. Table 2h shows both training and testing with a larger spatial resolution of input bring clear performance improvement.
+
+# 4.2 COMPARISONS WITH THE STATE-OF-THE-ARTS
+
+Something-Something V1&V2. We make a comprehensive comparison in Table 3. Compared with NL $\mathrm { I } 3 \mathrm { D } { + } \mathrm { G C N } _ { 3 2 f }$ , our $\mathrm { C T - N e t } _ { 8 f }$ gains $4 . 0 \%$ top-1 accuracy improvement with ${ \bf 4 9 . 1 \times }$ fewer GFLOPs in Something-Something V1. Besides, our ${ \mathrm { C T - N e t } } _ { 8 f }$ $( 5 1 . 7 \% )$ is better than the ir- $\mathrm { C S N _ { 3 2 f } }$ $( 4 9 . 3 \% )$ which adopts ResNet-152 as the backbone. Moreover, our $\mathrm { C T - N e t } _ { 1 6 f }$ outperforms all the single-clip models in Something-Something V1&V2 and even better than most of the multiclip models. It illustrates that our CT-Net is preferable to capture temporal contextual information efficiently. Surprisingly, with only 280 GFLOPs, our ensemble model $\mathrm { C T - N e t } _ { E N }$ achieves $5 6 . 6 \% ( 6 7 . 8 \% )$ top-1 accuracy in Something-Something V1(V2), which outperforms all methods.
+
+Kinetics-400. Kinetics-400 is a large-scale sence-related dataset, and the lightweight 2D models are usually inferior to the 3D models on it. Table 4 shows our CT-Net- $R 5 0 _ { 1 6 f }$ can surpass all existing lightweight models based on 2D backbone. Even compared with SlowFast- $R 5 0 _ { 4 0 f }$ , our CT-Net$\mathsf { R } 5 0 _ { 1 6 f }$ also achieves higher accuracy $7 7 . 3 \%$ vs. $7 6 . 4 \%$ ). Note that our reproduced SlowFastR50 performs worse than that in the paper (Feichtenhofer et al., 2019), which may result from the missing videos in Kinetics-400. As for the deeper model, compared with SlowFast- $\mathbf { R } 1 0 1 _ { 8 0 f }$ , our CT-Net- $\mathbf { R } 1 0 1 _ { 1 6 f }$ requires $\mathbf { 3 . 7 \times }$ fewer GFLOPs but gains comparable results $7 8 . 8 \%$ vs. $7 8 . 9 \%$ . Besides, it achieves comparable top-1 accuracy with X3D-XL $7 8 . 8 \%$ vs. $7 9 . 1 \%$ ) under a similar GFLOPs. However, X3D requires extensive model searching with an expensive GPU setting, while our CT-Net can be trained traditionally with feasible computation. We perform score fusion over CT-Net- $\mathsf { R } 5 0 _ { 1 6 f }$ and ${ \mathrm { C T - N e t - R } } 1 0 1 _ { 1 6 f }$ , which mimics two-steam fusion with two temporal rates. In this setting, our model is $\mathbf { 2 . 4 } \times$ faster than SlowFast- $\operatorname { R } 1 0 1 _ { 8 0 f }$ and shows an $0 . 9 \%$ performance gain $7 9 . 8 \%$ vs. $7 8 . 9 \%$ ) but only uses 32 frames.
+
+![](images/e9e0e87428692253ba228bd386f54e25af6ab14d47236004693285bf8da2671f.jpg)  
+Figure 3: Comparison of visualization. Videos are sampled from Something-Something V1. Compared with $\mathrm { R } ( 2 { + } 1 ) \mathrm { D }$ and CSN, our CT-Net can localize the action and object better both in space and time thanks to the larger spatial-temporal receptive field.
+
+# 4.3 VISUALIZATION
+
+We use Saliency Tubes (Stergiou et al., 2019) to generate the visualization, for it can show the most discriminative features that the network locates. In Figure 3, we sample two videos from SomethingSomething V1 which requires complex temporal modeling. In the left example, our CT-Net focuses on a larger area around the towel, especially in the fourth and fifth frames, thus predicting that someone is twisting it. In contrast, $\mathrm { R } ( 2 { + } 1 ) \mathrm { D }$ only concentrates on one side of the towel and gives the wrong judgment. The same situation can be seen in the right example. We argue that CT-Net can localize the action and object accurately thanks to the larger spatial-temporal receptive field. As for CSN, the regions of interest seem to be scattered, because it lacks sufficient spatial-temporal interaction, thus ignoring the rich context both in space and time.
+
+# 5 CONCLUSIONS
+
+In this paper, we construct an efficient tensor separable convolution to learn the discriminative video representation. We view the channel dimension of the input feature as a multiplication of K subdimensions and stack spatial/temporal tensor separable convolution along each of K sub-dimensions. Moreover, CT-Module is cooperated with the Tensor Excitation mechanism to further improve performance. All experiments demonstrate that our concise and novel CT-Net obtains a preferable balance between accuracy and efficiency on large-scale video datasets. Our proposed principles are preferable guidance for designing an efficient module for temporal modeling.
+
+# 6 ACKNOWLEDGEMENT
+
+This work is partially supported by National Natural Science Foundation of China (6187617, U1713208), the National Key Research and Development Program of China (No. 2020YFC2004800), Science and Technology Service Network Initiative of Chinese Academy of Sciences (KFJ-STS-QYZX-092), Shenzhen Institute of Artificial Intelligence and Robotics for Society.
+
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+
+# A APPENDIX
+
+# A.1 MORE TRAINING DETAILS
+
+We use SGD with momentum 0.9 and cosine learning rate schedule (Loshchilov & Hutter, 2017) to train the entire network. The first 10 epochs are used for warm-up (Goyal et al., 2017a) to overcome early optimization difficulty. For kinetics, the batch, total epochs, initial learning rate, dropout and weight decay are set to 64, 110, 0.01, 0.5 and 1e-4 respectively. All these hyper-parameters are set to 64, 45, 0.02, 0.3 and 5e-4 respectively for Something-Something.
+
+# A.2 TENSOR EXCITATION MECHANISM
+
+The implementation of our Tensor Excitation is shown in Figure 4. Different from the SE module, we use tensor separable convolution in the TE mechanism. Moreover, when obtaining the spatial attention, we squeeze the temporal dimension and perform spatial tensor separable convolution, because temporal information is insignificant for spatial attention and vice versa. We add the Batch Normalization (BN) layer for better optimization.
+
+# A.3 RESULTS ON UCF101 AND HMDB51
+
+To verify the generation ability of our CT-Net on smaller datasets, we conduct transfer learning experiments from Kinetics400 to UCF101 (Soomro et al., 2012) and HMDB-51 (Kuehne et al., 2011). We test CT-Net with 16 input frames and evaluate it over three splits and report the averaged results. As shown in Table 5, our $\mathrm { C T - N e t } _ { 1 6 f }$ achieves competitive performance when compared with the recent methods, which demonstrates the generation ability of our CT-Net.
+
+![](images/1d22d1e7f98cf1eda2d2109a7687480c59fdd392ebb5fd9b50f5dcead5888ea9.jpg)
+
+Figure 4: The implementation of our Tensor Excitation (TE) mechanism.   
+Table 5: Comparison results on UCF101 and HMDB51.   
+
+<table><tr><td>Method</td><td>Backbone</td><td>Pretrain</td><td>UCF101</td><td>HMDB51</td></tr><tr><td>C3D(Tran et al.,2015)</td><td>3D VGG-11</td><td>Sports-1M</td><td>82.3</td><td>51.6</td></tr><tr><td>I3D(Carreira &amp; Zisserman, 2017)</td><td>3D Inception</td><td>ImageNet+Kinetics</td><td>95.1</td><td>74.3</td></tr><tr><td>ECO(Zolfaghari et al.,2018)</td><td>Inception+3D R18</td><td>ImageNet+Kinetics</td><td>94.8</td><td>72.4</td></tr><tr><td>TSN(Wang et al., 2016)</td><td>Inception</td><td>ImageNet+Kinetics</td><td>91.1</td><td>=</td></tr><tr><td>TSM(Lin et al., 2019)</td><td>2D R50</td><td>ImageNet+Kinetics</td><td>94.5</td><td>70.7</td></tr><tr><td>STM(Jiang et al., 2019)</td><td>2D R50</td><td>ImageNet+Kinetics</td><td>96.2</td><td>72.2</td></tr><tr><td>Our CT-Net</td><td>2D R50</td><td>ImageNet+Kinetics</td><td>96.2</td><td>73.2</td></tr></table>
+
+# A.4 MORE RESULTS ON SOMETHING-SOMETHING V1&V2
+
+Table 6 shows more results on Something-Something V1&V2. We train CT-Net with a different number of input frames and then test these models with different sampling strategies. We average the prediction scores obtained from the previous models to evaluate the ensemble models. With more input frames, the corresponding accuracy becomes higher. As for the reason that $\mathrm { C T - N e t } _ { 2 4 f }$ is worse than $\mathrm { C T - N e t } _ { 1 6 f }$ , we argue that is because the model is hard to optimize with too many input frames. Sampling more clips or more crops also boosts performance. Moreover, our ensemble models gain the state-of-the-art top-1 accuracy of $5 6 . 6 \% ( 6 8 . 3 \% )$ on Something-Something V1(V2).
+
+# A.5 MORE ABLATION STUDIES ON MINI-KINETICS AND SOMETHING-SOMETHING V2
+
+To verify the effectiveness of our module comprehensively, we also conduct experiments in MiniKinetics and Something-Something V2 and report the multi-clip accuracy and single-clip accuracy respectively. Mini-Kinetics covers 200 action classes and is a subset of Kinetics-400, while Something-Something V2 covers the same action classes as Something-Something V1 but contains more videos. As shown in Table 7, the performance trend of different modules is similar to that shown in Table 2a. Since Mini-Kinetics does not highly depend on temporal modeling, the gap becomes smaller but still demonstrates the effectiveness of our CT-Module.
+
+# A.6 ADAPTING DIFFERENT PRE-TRAINED IMAGENET ARCHITECTURES AS CT-NET
+
+In fact, by directly replacing the $3 \times 3$ convolution with our CT-Module, we can easily adapt different pre-trained ImageNet architectures as CT-Net. Table 8 shows that it is also sensible to use InceptionV3 as the backbone. We believe that through more elaborate design, our CT-Net based on different backbones can achieve comparable performance.
+
+# A.7 VALIDATION PLOT
+
+In Figure 5, we plot the accuracy vs per-clip GFLOPs on Kinetics-400. It reveals that our CT-Net achieves a better trade-off than most of the existing methods on Kinetics-400.
+
+Table 6: More results on Something-Something V1&V2.   
+
+<table><tr><td>Method</td><td>#Frame</td><td>GFLOPs</td><td>#Param</td><td colspan="2">SomethingV1 Top-1 Top-5</td><td colspan="2">SomethingV2 Top-1 Top-5</td></tr><tr><td rowspan="9">Our CT-Net</td><td>8 12</td><td>37 56</td><td rowspan="3">21.0M</td><td>50.1</td><td>78.8</td><td>62.5</td><td>87.7</td></tr><tr><td></td><td></td><td>52.1</td><td>80.0</td><td>63.9</td><td>88.7</td></tr><tr><td></td><td></td><td>52.5</td><td>80.9</td><td>64.5</td><td>89.3</td></tr><tr><td>16 24</td><td>75 112</td><td rowspan="4"></td><td>52.5</td><td>80.9</td><td>64.6</td><td>89.1</td></tr><tr><td>8x1x2</td><td>75</td><td>51.6</td><td>79.7</td><td>63.5</td><td>88.5</td></tr><tr><td>12×1×2</td><td>112</td><td>52.8</td><td>80.6</td><td>64.6</td><td>89.3</td></tr><tr><td>16×1×2</td><td>151</td><td>53.2</td><td>81.3</td><td>65.2</td><td>89.7</td></tr><tr><td>24×1×2</td><td>224</td><td>52.9</td><td>81.3</td><td>65.0</td><td>89.3</td></tr><tr><td>8×3×2</td><td>224</td><td rowspan="4">21.0M</td><td>51.7</td><td>80.1</td><td>63.9</td><td>88.8</td></tr><tr><td>12×3×2</td><td>336</td><td>53.0</td><td>81.1</td><td>65.3</td><td>89.6</td></tr><tr><td>16×3×2</td><td>447</td><td>53.4</td><td>81.7</td><td>65.9</td><td>90.1</td></tr><tr><td>24×3×2</td><td></td><td>53.6</td><td>81.6</td><td>65.5</td><td>89.8</td></tr><tr><td rowspan="3">Our CT-NetEN</td><td rowspan="3">8+16 (8+12+16+24)×1× 2</td><td>672 112</td><td rowspan="3">83.8M</td><td>54.4</td><td>82.0</td><td>66.2</td><td>90.4</td></tr><tr><td>280</td><td>56.6</td><td>83.9</td><td>67.8</td><td>91.1</td></tr><tr><td>560</td><td>56.6</td><td>84.0</td><td>67.8</td><td>91.3</td></tr><tr><td></td><td>(8+12+16+24)×1 × 2 (8+12+16+24)×3×2</td><td>1679</td><td></td><td>56.6</td><td>83.9</td><td>68.3</td><td>91.3</td></tr></table>
+
+Table 7: More ablation studies on Mini-Kinetics and Something-Something V2.   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Backbone</td><td rowspan="2">GFLOPs</td><td colspan="2">Mini-Kinetics</td><td colspan="2">SomethingV2</td></tr><tr><td>Top-1</td><td>Top-5</td><td>Top-1</td><td>Top-5</td></tr><tr><td>C3D-Module (Tran et al., 2015)</td><td>2D R50</td><td>59.9</td><td>77.5</td><td>93.0</td><td>59.1</td><td>85.5</td></tr><tr><td>R(2+1)D-Module (Tran et al., 2018)</td><td>2D R50</td><td>45.8</td><td>77.8</td><td>93.2</td><td>60.0</td><td>86.0</td></tr><tr><td>CSN-Module (Tran et al.,2019)</td><td>2D R50</td><td>35.6</td><td>77.6</td><td>93.2</td><td>59.5</td><td>86.0</td></tr><tr><td>Our CT-Module</td><td>2D R50</td><td>36.3</td><td>78.0</td><td>93.6</td><td>60.3</td><td>86.4</td></tr></table>
+
+Table 8: Adapting different pre-trained ImageNet architectures as CT-Net.   
+
+<table><tr><td>Method</td><td>Backbone</td><td>GFLOPs</td><td>#Param.(M)</td><td>Top-1</td><td>Top-5</td></tr><tr><td>Baseline (TSN)</td><td>2D ResNet-50</td><td>43.0</td><td>23.9</td><td>16.9</td><td>42.0</td></tr><tr><td>Our CT-Net</td><td>2D ResNet-50</td><td>37.3</td><td>21.0</td><td>50.1</td><td>78.8</td></tr><tr><td>Baseline (TSN)</td><td>InceptionV3</td><td>45.8</td><td>22.1</td><td>18.3</td><td>43.9</td></tr><tr><td>Our CT-Net</td><td>InceptionV3</td><td>43.9</td><td>20.9</td><td>47.2</td><td>76.1</td></tr></table>
+
+![](images/b2f3ea6a4dccbaf4e10643bebf839cb1adecc4f29e2196a615abde7b18fdf557.jpg)  
+Figure 5: Accuracy vs per-clip GFLOPs on Kinetics-400.

parse/train/UoaQUQREMOs/UoaQUQREMOs_content_list.json

@@ -0,0 +1,1588 @@
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+    {
+        "type": "text",
+        "text": "CT-NET: CHANNEL TENSORIZATION NETWORK FOR VIDEO CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Kunchang $\\mathbf { L i } ^ { 1 2 }$ ∗, Xianhang $\\mathbf { L i ^ { \\alpha } }$ ∗, Yali Wang13∗, Jun Wang 4& Yu Qiao13† ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1Guangdong-Hong Kong-Macao Joint Laboratory of Human-Machine   \nIntelligence-Synergy Systems, Shenzhen Institutes of Advanced Technology,   \nChinese Academy of Sciences, Shenzhen, 518055, China   \n2University of Chinese Academy of Sciences   \n3SIAT Branch, Shenzhen Institute of Artificial Intelligence and Robotics for Society   \n4University of Central Florida   \n{kc.li, yl.wang, yu.qiao}@siat.ac.cn   \nxianhangli@knights.ucf.edu, Jun.Wang@ucf.edu ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            349,
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+            364
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "3D convolution is powerful for video classification but often computationally expensive, recent studies mainly focus on decomposing it on spatial-temporal and/or channel dimensions. Unfortunately, most approaches fail to achieve a preferable balance between convolutional efficiency and feature-interaction sufficiency. For this reason, we propose a concise and novel Channel Tensorization Network (CTNet), by treating the channel dimension of input feature as a multiplication of $K$ sub-dimensions. On one hand, it naturally factorizes convolution in a multiple dimension way, leading to a light computation burden. On the other hand, it can effectively enhance feature interaction from different channels, and progressively enlarge the 3D receptive field of such interaction to boost classification accuracy. Furthermore, we equip our CT-Module with a Tensor Excitation (TE) mechanism. It can learn to exploit spatial, temporal and channel attention in a high-dimensional manner, to improve the cooperative power of all the feature dimensions in our CT-Module. Finally, we flexibly adapt ResNet as our CT-Net. Extensive experiments are conducted on several challenging video benchmarks, e.g., Kinetics-400, Something-Something V1 and V2. Our CT-Net outperforms a number of recent SOTA approaches, in terms of accuracy and/or efficiency. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+            652,
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+            669
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "3D convolution has been widely used to learn spatial-temporal representation for video classification (Tran et al., 2015; Carreira & Zisserman, 2017). However, over parameterization often makes it computationally expensive and hard to train. To alleviate such difficulty, recent studies mainly focus on decomposing 3D convolution (Tran et al., 2018; 2019). One popular approach is spatial-temporal factorization (Qiu et al., 2017; Tran et al., 2018; Xie et al., 2018), which can reduce overfitting by replacing 3D convolution with 2D spatial convolution and 1D temporal convolution. But it still introduces unnecessary computation burden, since both spatial convolution and temporal convolution are performed over all the feature channels. To further decrease such computation cost, channel separation has been recently developed via operating 3D convolution in the depth-wise manner (Tran et al., 2019). However, it inevitably loses accuracy due to the lack of feature interaction between different channels. For compensation, it has to introduce point-wise convolution to preserve interaction with extra computation. So there is a natural question: How to construct effective 3D convolution to achieve a preferable trade-off between efficiency and accuracy for video classification? ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/33180bbaf52b7732efb27fed5479943b88d9cbd561ea5a770ea36421515293a0.jpg",
+        "image_caption": [
+            "Interaction (no.of channels,size of receptive field): $( C _ { 1 }$ channels $3 \\times 3 \\times 3 )$ $( C _ { 1 } \\times C _ { 2 } = C$ channels,5 × 5 × 5) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "table",
+        "img_path": "images/c701281a1c1bb1a9e6263575045da60c09d80d7d9e2e91b3423f600b5ba37874.jpg",
+        "table_caption": [
+            "Figure 1: Simple illustration of channel tensorization $K = 2$ ). We tensorize the channel dimension of input feature as a multiplication of $K$ sub-dimensions. Via performing spatial/temporal tensor separable convolution along each sub-dimension, we can achieve a preferable balance between convolutional efficiency and feature-interaction sufficiency. Introduction shows more explanations. "
+        ],
+        "table_footnote": [
+            "1 Interact Field means the receptive field for feature interaction. "
+        ],
+        "table_body": "<table><tr><td rowspan=2 colspan=1>Method</td><td rowspan=2 colspan=1>3D Convolution(t ×hxw)</td><td rowspan=1 colspan=2>Convolutional Efficiency</td><td rowspan=1 colspan=1>Feature-Interact</td><td rowspan=1 colspan=1>onSufficiency</td></tr><tr><td rowspan=1 colspan=1>Spatial-temporal</td><td rowspan=1 colspan=1>Channel</td><td rowspan=1 colspan=1>Interact Manner</td><td rowspan=1 colspan=1>InteractField1</td></tr><tr><td rowspan=1 colspan=1>C3D(Tran et al., 2015)</td><td rowspan=1 colspan=1>Full:3×3×3</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>STC</td><td rowspan=1 colspan=1>33</td></tr><tr><td rowspan=1 colspan=1>R(2+1)D(Tran et al., 2018)</td><td rowspan=1 colspan=1>Full:1×3×3Full:3×1×1</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>SCTC</td><td rowspan=1 colspan=1>3</td></tr><tr><td rowspan=1 colspan=1>CSN(Tran et al., 2019)</td><td rowspan=1 colspan=1>Full:1×1×1DW:3×3×3</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>CST</td><td rowspan=1 colspan=1>33</td></tr><tr><td rowspan=1 colspan=1>Our CT-Net(C=C1x..xCK)</td><td rowspan=1 colspan=1>C1:C1×.x1x(1x3×3+3×1×1)Ck:1xxCkx(1x3×3+3×1×1)</td><td rowspan=1 colspan=1>&lt;</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>STC1STCK</td><td rowspan=1 colspan=1>(2K + 1)3</td></tr></table>",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Two design principles to build effective video representation and efficient convolution. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This paper attempts to address this question by investigating two design principles. (1) Convolutional Efficiency. As shown in Table 1, current designs of spatial-temporal convolution mainly focus on decomposition from either spatial-temporal (Tran et al., 2018) or channel dimension (Tran et al., 2019). To enhance convolutional efficiency, we consider decomposing convolution in a higher dimension with a novel representation of feature tensor. (2) Feature-Interaction Sufficiency. Table 1 clearly shows that, for current decomposition approaches (Tran et al., 2018; 2019), feature interaction only contains one or two of spatial, temporal and channel dimensions at each sub-operation. Such a partial interaction manner would reduce classification accuracy. On one hand, it decreases the discriminative power of video representation, due to the lack of joint learning on all the dimensions. On the other hand, it restricts feature interaction in a limited receptive field, which ignores rich context from a larger 3D region. Hence, to boost classification accuracy, each sub-operation should achieve feature interaction on all the dimensions, and the receptive field of such interaction should be progressively enlarged as the number of sub-operations increases. ",
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+    {
+        "type": "text",
+        "text": "Based on these desirable principles, we design a novel and concise Channel Tensorization Module (CT-Module). Specifically, we propose to tensorize the channel dimension of input feature as a multiplication of $K$ sub-dimensions, i.e., $C = C _ { 1 } \\times C _ { 2 } \\times \\cdot \\cdot \\cdot \\times C _ { K }$ . Via performing spatial/temporal separable convolution along each sub-dimension, we can effectively achieve convolutional efficiency and feature-interaction sufficiency. For better understanding, we use the case of $K = 2$ as a simple illustration in Figure 1. First, we tensorize the input channel into $C = C _ { 1 } \\times C _ { 2 }$ . Naturally, we separate the convolution into distinct ones along each sub-dimension, e.g., for the $1 ^ { s t }$ sub-dimension, we apply our spatial-temporal tensor separable convolution with the size $C _ { 1 } \\times 1 \\times t \\times h \\times w$ , which allows us to achieve convolutional efficiency on all the spatial, temporal and channel dimensions. After that, we sequentially perform the tensor separable convolution sub-dimension by sub-dimension. As a result, we can progressively achieve feature interaction on all the channels, and enlarge the spatialtemporal receptive field. For example, after operating $1 ^ { s t }$ tensor separable convolution on the $1 ^ { s t }$ sub-dimension, $C _ { 1 }$ channels interact, and 3D receptive field of such interaction is $3 \\times 3 \\times 3$ . Via further operating $2 ^ { n d }$ tensor separable convolution on the $2 ^ { n d }$ sub-dimension, all $C _ { 1 } \\times C _ { 2 } = C$ channels have feature interaction, and 3D receptive field of such interaction becomes $5 \\times 5 \\times 5$ . This clearly satisfies our principle of feature-interaction sufficiency. ",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We summarize our contributions in the following. First, we design a novel Channel Tensorized Module (CT-Module), which can achieve convolutional efficiency and feature-interaction sufficiency, via progressively performing spatial/temporal tensor separable convolution along each sub-dimension of the tensorized channel. Second, we equip CT-Module with a distinct Tensor Excitation (TE) mechanism, which can further activate the video features of each sub-operation by spatial, temporal and channel attention in a tensor-wise manner. Subsequently, we apply this full module in a residual block, and flexibly adopt 2D ResNet as our Channel Tensorized Network (CT-Net). In this case, we can gradually enhance feature interaction from a broader 3D receptive field, and learn the key spatial-temporal representation with light computation. Finally, we conduct extensive experiments on a number of popular and challenging benchmarks, e.g., Kinetics (Carreira & Zisserman, 2017), Something-Something V1 and V2 (Goyal et al., 2017b). Our CT-Net outperforms the state-of-the-art methods in terms of classification accuracy and/or computation cost. ",
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+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
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+            324,
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+            340
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2D CNN for video classification. 2D CNN is a straightforward but useful method for video classification (Karpathy et al., 2014; Simonyan & Zisserman, 2014; Wang et al., 2016; Liu et al., 2020; Jiang et al., 2019). For example, Two-stream methods (Simonyan & Zisserman, 2014) learn video representations by fusing the features from RGB and optical flow respectively. Instead of sampling a single RGB frame, TSN (Wang et al., 2016) proposes a sparse temporal sampling strategy to learn video representations. To further improve accuracy, TSM (Lin et al., 2019) proposes a zeroparameter temporal shift module to exchange information with adjacent frames. However, these methods may lack the capacity of learning spatial-temporal interaction comprehensively, which often reduces their discriminative power to recognize complex human actions. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3D CNN for video classification. 3D CNN has been widely used to learn a rich spatial-temporal context better (Tran et al., 2015; Carreira & Zisserman, 2017; Feichtenhofer et al., 2019; Sudhakaran et al., 2020; Feichtenhofer, 2020). However, it introduces a lot of parameters, which leads to a difficult optimization problem and large computational load. To resolve this issue, I3D (Carreira & Zisserman, 2017) inflates all the 2D convolution kernels pre-trained on ImageNet, which is helpful for optimizing. Other works also try to factorize 3D convolution kernel to reduce complexity, such as P3D (Qiu et al., 2017) and $\\mathrm { R } ( 2 { + } 1 ) \\mathrm { D }$ (Tran et al., 2018). Recently, CSN (Tran et al., 2019) operates 3D convolution in the depth-wise manner. Nevertheless, all these methods still do not achieve a good trade-off between accuracy and efficiency. To tackle this challenge, we propose CT-Net which learns on spatial-temporal and channel dimensions jointly with lower computation than previous methods. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHODS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            646,
+            290,
+            662
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we describe our Channel Tensorization Network (CT-Net) in detail. First, we formally introduce our CT-Module in a generic manner. Second, we design a Tensor Excitation (TE) mechanism to enhance CT-Module. Finally, we flexibly adapt ResNet as our CT-Net to achieve a preferable trade-off between accuracy and efficiency for video classification. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 CHANNEL TENSORIZATION MODULE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            748,
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+            763
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As discussed in the introduction, the previous approaches have problems in convolutional efficiency or feature-interaction sufficiency. To tackle such a problem, we introduce a generic Channel Tensorization Module (CT-Module), by treating the channel dimension of input feature as a multiplication of $K$ sub-dimensions, i.e., $C = C _ { 1 } \\times C _ { 2 } \\times \\cdot \\cdot \\cdot \\times C _ { K }$ . Naturally, this tensor representation allows to tensorize the kernel size of convolution TConv() as a multiplication of $K$ sub-dimensions, too. To simplify the notation, the channel dimension of the output is omitted by default. The output $X _ { o u t }$ can be calculated as follows: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ad48e1fe8d63dd8b18c5b1bb1009bb05b521c848b9b6bb628e61bd252b63d46d.jpg",
+        "text": "$$\n{ \\bf X } _ { o u t } = T C o n v \\left( { \\bf X } _ { i n } , { \\cal W } ^ { C _ { 1 } \\times C _ { 2 } \\times \\dots \\times C _ { K } \\times t \\times h \\times w } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\bf X } _ { i n }$ and $W$ denote the tensorized input and kernel respectively. However, such an operation requires large computation, so we introduce the tensor separable convolution to alleviate the issue. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Tensor Separable Convolution. We propose to factorize TConv() along $K$ channel sub-dimensions. Specifically, we decompose $T C o n \\nu ( )$ as $K$ tensor separable convolutions $T S C o n \\nu ( )$ , and apply TSConv() sub-dimension by sub-dimension as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9ab43a75bd5e722443115b35d4a7731a8cc1bbdfc1d2ee6227978552b5d4ea90.jpg",
+        "text": "$$\n{ { \\bf { X } } _ { k } } = T S C o n \\nu \\left( { { \\bf { X } } _ { k - 1 } } , { W ^ { 1 \\times \\cdots \\times { C _ { k } } \\times \\cdots \\times 1 \\times t \\times h \\times w } } \\right)\n$$",
+        "text_format": "latex",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "where ${ \\bf X } _ { 0 } ~ = ~ { \\bf X } _ { i n }$ and ${ \\bf { X } } _ { o u t } ~ = { \\bf { X } } _ { K }$ . On one hand, the kernel size of the $k ^ { t h }$ TSConv() is $( 1 \\times \\cdot \\cdot \\cdot \\times C _ { k } \\times \\cdot \\cdot \\cdot \\times 1 \\times t \\times h \\times w )$ ). It illustrates that only $C _ { k }$ channels interact in the $k ^ { t h }$ suboperation, which leads to convolution efficiency. On the other hand, as we stack the TSConv(), each convolution performs on the output features of the previous convolution. Therefore, the spatialtemporal receptive field is enlarged. Besides, interactions first occur in $C _ { 1 }$ channels, second in $C _ { 1 } \\times C _ { 2 }$ channels and so on. Finally, $C _ { 1 } \\times C _ { 2 } \\times \\cdot \\cdot \\cdot \\times C _ { K } = C$ channels can progressively interact. This clearly satisfies our principle of feature-interaction sufficiency. ",
+        "bbox": [
+            173,
+            184,
+            825,
+            285
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Spatial-Temporal Tensor Separable Convolution. To further improve convolution efficiency, we factorize the 3D TSConv() into 2D spatial TSConv() and 1D temporal TSConv(). Thus, we can obtain the output features $\\mathbf { X } _ { k } ^ { S }$ and $\\mathbf { X } _ { k } ^ { T }$ as follows: ",
+        "bbox": [
+            174,
+            290,
+            823,
+            333
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ef59c8605018237139c876f95061ed1a6aed6afaba181a956b5ad7b8b3fa465e.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbf { X } _ { k } ^ { S } = { S } { - } T { S } C o n \\nu \\left( \\mathbf { X } _ { k - 1 } , W ^ { 1 \\times \\cdots \\times C _ { k } \\times \\cdots \\times 1 \\times 1 \\times h \\times w } \\right) } \\\\ & { } \\\\ & { \\mathbf { X } _ { k } ^ { T } = { T } { - } T { S } C o n \\nu \\left( \\mathbf { X } _ { k - 1 } , W ^ { 1 \\times \\cdots \\times C _ { k } \\times \\cdots \\times 1 \\times t \\times 1 \\times 1 } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            342,
+            669,
+            397
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $S _ { - } T S C o n \\nu ( \\ j$ and T-TSConv() represent spatial and temporal tensor separable convolution respectively. Finally, we attempt to aggregate spatial and temporal convolution. There are various connection types of spatial and temporal tensor separable convolution, e.g., parallel and serial types. According to the results of the experiments in Section 4, we utilize the parallel method, which illustrates that we sum the spatial feature $\\mathbf { X } _ { k } ^ { S }$ and temporal feature $\\mathbf { X } _ { k } ^ { T }$ : ",
+        "bbox": [
+            173,
+            401,
+            825,
+            472
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6a32a68d3611c5bee8ae6f01d311d55d1e4c2709b378e2cf817a0de23c1ae0a8.jpg",
+        "text": "$$\n\\mathbf { X } _ { k } = \\mathbf { X } _ { k } ^ { S } + \\mathbf { X } _ { k } ^ { T }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            441,
+            481,
+            557,
+            500
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 TENSOR EXCITATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            517,
+            390,
+            534
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our CT-Module separates feature along spatial, temporal and channel dimensions. To make full use of their cooperative power to learn distinct video features, we design a concise Tensor Excitation (TE) mechanism for each dimension. First of all, we attempt to utilize the TE mechanism to enhance spatial and temporal features respectively. For the spatial feature $\\mathbf { X } _ { k } ^ { S }$ obtained by Equation 5, our corresponding spatial TE mechanism can be formulated as: ",
+        "bbox": [
+            173,
+            545,
+            825,
+            616
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c724de8a9d38021636bfffc78002efebeace25a20334c37f24638942bdb70cf6.jpg",
+        "text": "$$\n\\mathbf { U } _ { k } = \\mathbf { X } _ { k } ^ { S } \\otimes S i g m o d ( S \\mathbf { - } T S C o n \\nu ( T \\mathbf { - } P o o l ( \\mathbf { X } _ { k } ^ { S } ) ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            344,
+            625,
+            651,
+            643
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where T-Pool() represents global temporal pooling, i.e., $T \\times 1 \\times 1$ average pooling. By performing it on $\\mathbf { X } _ { k } ^ { S }$ , we obtain the feature with the size $\\left( C _ { 1 } \\times C _ { 2 } \\times \\cdots \\times C _ { K } \\times 1 \\times H \\times W \\right)$ , which gathers spatial contexts along temporal dimension. Subsequently, the spatial tensor separable convolution $s$ - TSConv() and the activate function Sigmod() are performed to generate the spatial attention heatmap. Finally, the element-wise multiplication $\\otimes$ broadcasts the spatial attention along the temporal dimension. Similarly, we perform the temporal TE mechanism for the temporal feature $\\mathbf { X } _ { k } ^ { T }$ : ",
+        "bbox": [
+            173,
+            652,
+            825,
+            738
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/24e291726fba1651d65421f270e5f6067be389fea0a6de47ad9aed6b9acb6b72.jpg",
+        "text": "$$\n\\mathbf { V } _ { k } = \\mathbf { X } _ { k } ^ { T } \\otimes S i g m o d ( T \\mathbf { \\cdot } T S C o n \\nu ( S \\mathbf { \\cdot } P o o l ( \\mathbf { X } _ { k } ^ { T } ) ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            747,
+            653,
+            766
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where S-Pool() and $T – T S C o n \\nu ( )$ are global spatial pooling and temporal tensor separable convolution correspondingly. At last, after aggregating the spatial and temporal features by addition, i.e., ${ \\bf R } _ { k } = { \\bf \\Phi }$ $\\mathbf { U } _ { k } + \\mathbf { V } _ { k }$ , we perform a channel-wise TE mechanism as follows: ",
+        "bbox": [
+            174,
+            775,
+            825,
+            818
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c8f05a57834289d010919d570a9b3d53138ad31a31e98a1876d7898d5a4d2a99.jpg",
+        "text": "$$\n\\mathbf { X } _ { k } = \\mathbf { R } _ { k } \\otimes S i g m o d ( P W \\mathbf { - } T S C o n \\nu ( S \\mathbf { - } P o o l ( \\mathbf { R } _ { k } ) ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            828,
+            658,
+            844
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We adopt point-wise tensor separable convolution $P W  – T S C o n \\nu ( )$ to learn the weights for aggregating distinctive channels. The rest follows the previous design. Note that all tensor separable convolutions are performed on the same sub-dimension as the previous convolution, which is essentially different from the SE mechanism (Hu et al., 2020). Through the cooperation of the TE mechanism along three different dimensions, the spatial-temporal features can be significantly enhanced. ",
+        "bbox": [
+            173,
+            853,
+            825,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/3123f128bfea1d046977bc9497a4d16414806dba1459d1342333046e084a8d08.jpg",
+        "image_caption": [
+            "Figure 2: The pipelines of CT-Blocks and the overall architecture of CT-Net. We replace one of every two ResBlocks in ResNet with our CT-Block and the extra point-wise convolution in the last sub-dimension $k = K$ ) is ignored. More details can be found in Section3.3. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            99,
+            816,
+            357
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 CHANNEL TENSORIZATIONN NETWORK ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            429,
+            544,
+            443
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We regard ResNet as an exemplar and build up our CT-Net from ResNet50 (or ResNet101). First, we design a simple CT-Block in Figure 2(a), which adapts the $3 \\times 3$ convolutional layer in Residual Block (ResBlock) into our CT-Module. It can achieve both convolutional efficiency and featureinteraction sufficiency. Second, we equip our simple CT-Block with the TE mechanism in Figure 2(b), forming a full CT-Block that can improve the cooperative power of all the feature dimensions. Besides, extra point-wise convolutions are added between different sub-operations, which are beneficial for more sufficient feature interaction. At last, we build up a novel CT-Net with CT-Block. As shown in Figure 2(c), we replace one of every two ResBlocks with our CT-Block in every stage. Such a method guarantees a better balance between efficiency and accuracy in our experiments. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Discussion. In fact, the popular methods in video classification like C3D, $\\mathrm { R } ( 2 { + } 1 ) \\mathrm { D }$ and CSN (Tran et al., 2015; 2018; 2019) can be viewed as special cases of our CT-Net. We can generate different forms by adjusting three hyper-parameters: the number of sub-dimensions $\\mathbf { \\eta } ^ { ( K ) }$ , the corresponding dimension size $( C _ { k } )$ and the spatial-temporal kernel size $( K e r n e l _ { k } )$ . To degenerate into C3D, we can set $K = 1$ and $K e r n e l _ { 1 } = 3 \\times 3 \\times 3$ . When $K = 2$ , $K e r n e l _ { 1 } = 1 \\times 3 \\times 3$ , $K e r n e l _ { 2 } = 3 \\times 1 \\times 1$ and $C _ { 1 } = C _ { 2 } = C$ without channel tensorization, it becomes $\\mathrm { R } ( 2 { + } 1 ) \\mathrm { D }$ . Unfortunately, because of lacking the decomposition of channels, C3D and $\\mathrm { R } ( 2 { + } 1 ) \\mathrm { D }$ still have a large computational load. When $\\bar { K ^ { { \\prime } } } = 2 , C = \\bar { C } _ { 1 } \\times C _ { 2 } = C \\times 1 , K e r n e l _ { 1 } = 1 \\times 1 \\times 1 , K e r n e l _ { 2 } = \\bar { 3 } \\times 3 \\times \\bar { 3 }$ , obviously it is equivalent to CSN. However, CSN has a limited receptive field of spatial-temporal interaction. In our CT-Net, we utilize channel tensorization and perform tensor separable convolution along each sub-dimension in turn. Such design can not only preserve interaction among spatial, temporal and channel dimensions but also enlarge the receptive field of feature interaction progressively. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS AND RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            779,
+            447,
+            795
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Datasets and implementation details. We conduct experiments on three large video benchmarks: Kinetics-400 (Carreira & Zisserman, 2017), Something-Something V1 and V2 (Goyal et al., 2017b). We choose ResNet50 and ResNet101 (He et al., 2016) pre-trained on ImageNet as the backbone and the parameters of CT-Module are randomly initialized. For training, we utilize the dense sampling strategy (Wang et al., 2018) for Kinetics and sparse sampling strategy (Wang et al., 2016) for Something-Something. Random scaling and cropping are applied for data argumentation. Finally, we resize the cropped regions to $2 5 6 \\times 2 5 6$ . For testing, we sample multiple clips per video (4 for Kinetics, 2 for others) for pursuing high accuracy, and average all scores for the final prediction. ",
+        "bbox": [
+            174,
+            811,
+            825,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/5ad6960e615e38cf3a0e185b9b3397f5f71846b9ac8b0d705b55eaad46a514c8.jpg",
+        "table_caption": [
+            "(a) Effectiveness of CT-Module. CT-Module outperforms the recent modules for video modeling. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>3D Convolution (t × h × w)</td><td>GFLOPs</td><td>Top-1</td><td>Top-5</td></tr><tr><td>C3D-Module (Tran et al.,2015)</td><td>Full:3×3×3</td><td>59.9</td><td>46.1</td><td>75.0</td></tr><tr><td>R(2+1)D-Module (Tran et al., 2018)</td><td>Full:1×3×3+Full:3×1×1</td><td>45.8</td><td>47.0</td><td>76.1</td></tr><tr><td>CSN-Module (Tran et al., 2019)</td><td>Full:1×1×1+DW:3×3×3</td><td>35.6</td><td>46.8</td><td>75.7</td></tr><tr><td>Our CT-Module</td><td>C,…,Cκ:1×3×3+3×1×1</td><td>36.3</td><td>47.3</td><td>76.2</td></tr></table>",
+        "bbox": [
+            178,
+            99,
+            820,
+            164
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/f5794a8615d5799763c2ef3c719186c22d2fe3f5e8f6dcf66aeebed29add648f.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Number</td><td>GFLOPs</td><td>Top-1</td><td>Top-5</td></tr><tr><td>1D</td><td>45.8</td><td>46.5</td><td>75.6</td></tr><tr><td>2D</td><td>36.3</td><td>47.3</td><td>76.2</td></tr><tr><td>3D</td><td>35.7</td><td>47.1</td><td>75.8</td></tr><tr><td>4D</td><td>35.6</td><td>46.5</td><td>75.3</td></tr></table>",
+        "bbox": [
+            176,
+            189,
+            408,
+            251
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/972d54fe7cec7fdb84fef47203ba381f459584242564e96f7eeed6b96c7a7085.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Type</td><td>Top-1</td><td>Top-5</td></tr><tr><td>coupling</td><td>47.1</td><td>76.0</td></tr><tr><td>serial</td><td>47.2</td><td>76.1</td></tr><tr><td>parallel</td><td>47.3</td><td>76.2</td></tr></table>",
+        "bbox": [
+            419,
+            189,
+            596,
+            242
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/a28d5033c0b29bf16bb79f42c9ca79415141c623ea6bcfcab3c125278cb1fa9b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>C</td><td>GFLOPs</td><td>Top-1</td><td>Top-5</td></tr><tr><td>1</td><td>45.9</td><td>47.0</td><td>76.1</td></tr><tr><td>4</td><td>37.6</td><td>46.8</td><td>76.0</td></tr><tr><td>16</td><td>36.4</td><td>47.2</td><td>76.0</td></tr><tr><td>[√C]</td><td>36.3</td><td>47.3</td><td>76.2</td></tr></table>",
+        "bbox": [
+            611,
+            188,
+            820,
+            251
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(b) Number of sub-dimensions. The higher the dimension, the smaller the GFLOPs. 2D channel tensorization achieves the best trade-off. ",
+        "bbox": [
+            173,
+            257,
+            410,
+            308
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(c) Connection type of spatiotemporal convolution. The parallel connection between spatial and temporal convolution is the best choice. ",
+        "bbox": [
+            416,
+            247,
+            599,
+            309
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(d) Dimension size. $C = C _ { 1 } \\times C _ { 2 }$ and the best trade-off is achieved when adopting the rounded middle size $\\lfloor { \\sqrt { C } } \\rfloor$ . ",
+        "bbox": [
+            607,
+            256,
+            823,
+            309
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/17ee9482ac937a85407467e39ae8c32a61b3b45f3f67e76c638e1db3d2fbb022.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Number</td><td>GFLOPs</td><td>Top-1</td><td>Top-5</td></tr><tr><td>+0 (TSN)</td><td>43.0</td><td>16.9</td><td>42.0</td></tr><tr><td>+1 stage5</td><td>41.9</td><td>42.3</td><td>71.1</td></tr><tr><td>+4 stage4-5</td><td>38.9</td><td>46.9</td><td>75.7</td></tr><tr><td>+6 stage3-5</td><td>37.1</td><td>47.2</td><td>76.1</td></tr><tr><td>+7 stage2-5</td><td>36.3</td><td>47.3</td><td>76.2</td></tr><tr><td>+12 stage2-5</td><td>31.5</td><td>45.9</td><td>76.1</td></tr></table>",
+        "bbox": [
+            176,
+            314,
+            457,
+            407
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(e) Number and location of CT-Blocks. Simply replacing 1 block in stage5 can bring significant performance improvement. As we replace more blocks from the bottom up, the GFLOPs continues to decrease. Replacing 7 blocks achieves the best trade-off between accuracy and GFLOPs. ",
+        "bbox": [
+            173,
+            414,
+            460,
+            489
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/414e865843fc5f6d49463702f399147eb8cd4e48dc3425d604f12a24e14aa76e.jpg",
+        "table_caption": [
+            "(f) Kernel sizes along different dimensions. The larger kernel size brings improvement with more calculation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Kernel size</td><td></td><td>GFLOPs</td><td>Top-1</td><td>Top-5</td></tr><tr><td>C1 1 ×1× 1 C 1 1x3×3</td><td>1×1×1 3×1×1</td><td>35.5</td><td>46.1</td><td>75.1</td></tr><tr><td>C1 1×1×1 C2 1×5×5</td><td>1×1×1 5×1×1</td><td>36.6</td><td>47.2</td><td>76.2</td></tr><tr><td>C1 1 1x3×3 C 1×3×3</td><td>3×1×1 3×1×1</td><td>36.3</td><td>47.3</td><td>76.2</td></tr><tr><td>C 1×3×3</td><td>3×1×1</td><td>37.4</td><td>47.5</td><td>76.4</td></tr><tr><td>C2 1×5×5 C1 1 ×5×5</td><td>5×1×1 5×1×1</td><td>38.9</td><td>47.6</td><td>76.5</td></tr></table>",
+        "bbox": [
+            467,
+            314,
+            820,
+            457
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/d9a12e8a633fcae92a449106c7ffe8fc65a392cfc89428e035af96c7eca5a754.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>GFLOPs</td><td>Top-1</td><td>Top-5</td></tr><tr><td>Baseline (TSN)</td><td>43.0</td><td>16.9</td><td>42.0</td></tr><tr><td>+CT-Module</td><td>36.3</td><td>47.3</td><td>76.2</td></tr><tr><td>+CT-Module+PWConv</td><td>37.2</td><td>48.0</td><td>76.7</td></tr><tr><td>+CT-Module+PWConv+SE</td><td>37.2</td><td>48.8</td><td>77.4</td></tr><tr><td>+CT-Module+PWConv+TE</td><td>37.3</td><td>50.1</td><td>78.8</td></tr></table>",
+        "bbox": [
+            176,
+            493,
+            529,
+            571
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(g) Impact of different modules. CT-Module is essential for temporal modeling and TE mechanism is also beneficial. ",
+        "bbox": [
+            174,
+            577,
+            532,
+            602
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/18b0a7cdf7649851b520257ad87b760a70c216699da56efd4da74e0aa36399b7.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(h) Impact of different spatial resolution. "
+        ],
+        "table_body": "<table><tr><td></td><td>Input</td><td>GFLOPs</td><td>Top-1</td><td>Top-5</td></tr><tr><td>Train: Test:</td><td>224×224 224× 224</td><td>28.6</td><td>49.1</td><td>77.4</td></tr><tr><td>Train: Test:</td><td>224×224 256×256</td><td>37.3</td><td>49.7</td><td>77.7</td></tr><tr><td>Train:</td><td>256×256</td><td>37.3</td><td></td><td>78.8</td></tr><tr><td>Test:</td><td>256×256</td><td></td><td>50.1</td><td></td></tr></table>",
+        "bbox": [
+            540,
+            493,
+            823,
+            578
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Ablation studies on Something-Something V1. All models use ResNet50 as the backbone ",
+        "bbox": [
+            176,
+            614,
+            820,
+            628
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We follow the same strategy in Non-local (Wang et al., 2018) to pre-process the frames and take 3 crops of $2 5 6 \\times 2 5 6$ as input. Because some multi-clip models in Table 3 and Table 4 sample crops of $2 5 6 \\times 2 5 6$ , we simply multiply the GFLOPs reported in the corresponding papers by (256/224)2 for a fair comparison. When considering efficiency, we use just 1 clip per video and the final crop is scaled to $2 5 6 \\times 2 5 6$ to ensure comparable GFLOPs. ",
+        "bbox": [
+            174,
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+            722
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "4.1 ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 2 shows our ablation studies on Something-Something V1, which is a challenging dataset that requires video architecture to have a robust spatial-temporal representation ability and is suitable to verify the effectiveness of our method. All models use ResNet50 as the backbone. ",
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+        "type": "text",
+        "text": "Effectiveness of CT-Module. In Table 2a, we replace the $3 \\times 3$ convolutional layer in ResNet50 with different modules in recent methods (Tran et al., 2015; 2018; 2019). Compared with CSN-Module, our module achieves a better result with similar computation, which reflects the importance of sufficient feature interaction. Besides, it is slightly better than $\\mathrm { R } ( 2 { + } 1 ) \\mathrm { D }$ -Module with much lower calculation, showing the necessity of efficient convolution. Such results demonstrate the effectiveness of our CT-Module. It illustrates our two design principles give preferable guidance for designing an efficient module for temporal modeling. ",
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+        "table_caption": [
+            "Table 3: Comparison with the state-of-the-art on Something-Something V1&V2. Our CT$\\mathrm { N e t } _ { 1 6 f }$ outperforms all the single-clip models in Something-Something and even better than most of the multi-clip models. And our $\\mathrm { C T - N e t } _ { E N }$ outperforms all methods with very lower computation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Backbone</td><td>#Frame</td><td>GFLOPs</td><td colspan=\"2\">SomethingV1 Top-1 Top-5</td><td colspan=\"2\">SomethingV2 Top-1 Top-5</td></tr><tr><td rowspan=\"2\">ECOENLite (Zolfaghari et al.,2018) NL I3D + GCN (Wang &amp; Gupta,2018)</td><td>Incep+3D R18</td><td>92 32×3×2</td><td rowspan=\"2\">267</td><td>46.4</td><td>=</td><td>■</td><td>=</td></tr><tr><td>3D R50</td><td>1818</td><td>46.1</td><td>76.8</td><td>■</td><td></td></tr><tr><td rowspan=\"2\">ir-CSN (Tran et al.,2019) ir-CSN (Tran et al., 2019) CorrNet (Wang et al.,2020)</td><td>3D R101</td><td>32×1×10</td><td>738</td><td>48.4</td><td>-</td><td>■</td><td>■</td></tr><tr><td>3D R152</td><td>32×1×10</td><td>967 1150</td><td>49.3</td><td>-</td><td>-</td><td>■</td></tr><tr><td rowspan=\"4\">TSN (Wang et al.,2016) TSM (Lin et al., 2019)</td><td>3D R50 2DR50</td><td>32×1×10</td><td></td><td>49.3</td><td>-</td><td>-</td><td>-</td></tr><tr><td></td><td>8</td><td>33</td><td>19.7</td><td>46.6</td><td>27.8</td><td>57.6</td></tr><tr><td>2D R50</td><td>8</td><td>33</td><td>45.6</td><td>74.2</td><td>59.1</td><td>85.6</td></tr><tr><td>bLR50 2D R50</td><td>8×2 8</td><td>24 33</td><td>46.4 47.4</td><td>76.6</td><td>59.1 61.3</td><td>86.0</td></tr><tr><td rowspan=\"4\">TEINet (Liu et al.,2020) TEA (Li et al.,2020b)</td><td>2D Res2Net50</td><td>8</td><td>35</td><td></td><td>■ 78.1</td><td></td><td>-</td></tr><tr><td></td><td></td><td>33</td><td>48.9</td><td></td><td>-</td><td>-</td></tr><tr><td>2D R50+TIM 2D R50+TIM</td><td>8 8×3×2</td><td>259</td><td>49.8 50.4</td><td>- -</td><td>62.6 63.5</td><td>-</td></tr><tr><td>2DR50</td><td>8</td><td>37</td><td>50.1</td><td>78.8</td><td>62.5</td><td>- 87.7</td></tr><tr><td>Our CT-Net</td><td>2D R50</td><td>8×3×2</td><td>224</td><td>51.7</td><td>80.1</td><td>63.9</td><td>88.8</td></tr><tr><td>TSN (Wang et al.,2016) TSM (Lin et al.,2019)</td><td>2D R50</td><td>16</td><td>66</td><td>19.9</td><td>47.3</td><td>30.0</td><td>60.5</td></tr><tr><td rowspan=\"9\">bLVNet-TAM (Fan et al.,2019) TEINet (Liu et al., 2020)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>2D R50</td><td>16</td><td>66</td><td>47.2</td><td>77.1</td><td>63.4</td><td>88.5</td></tr><tr><td>bLR50</td><td>16×2</td><td>48</td><td>48.4</td><td>78.8</td><td>61.7</td><td>88.1</td></tr><tr><td>2D R50</td><td>16</td><td>66</td><td>49.9</td><td>-</td><td>62.1</td><td>-</td></tr><tr><td>TEA (Li et al.,2020b) 2D Res2Net50</td><td>16</td><td>70</td><td>51.9</td><td>80.3</td><td>-</td><td>-</td></tr><tr><td>PEM+TDLoss (Weng et al.,2020) 2D R50+TIM</td><td>16</td><td>66</td><td>50.9</td><td>-</td><td>63.8</td><td>-</td></tr><tr><td>2D R50+TIM</td><td>16×3×2</td><td>517</td><td>52.0</td><td>1</td><td>65.0</td><td>-</td></tr><tr><td>2D R50</td><td>16</td><td>75</td><td>52.5</td><td>80.9</td><td>64.5</td><td></td></tr><tr><td></td><td>16×3×2</td><td>447</td><td>53.4</td><td>81.7</td><td>65.9</td><td>89.3</td></tr><tr><td>Our CT-Net</td><td>2D R50</td><td></td><td></td><td></td><td></td><td></td><td>90.1</td></tr><tr><td>Our CT-NetEN</td><td>2D (R50)×4</td><td>8+12+16+24</td><td>280</td><td>56.6</td><td>83.9</td><td>67.8</td><td>91.1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Number of sub-dimensions. Increasing the number of sub-dimensions saves a lot of computation, but the corresponding accuracy first increases and then decreases as shown in Table 2b. Compared with the 1D method, the 4D method significantly reduces GFLOPs, achieving comparable accuracy. As for the decrease of accuracy when $K$ is too large, we argue that the number of channel in the shallow layer is small (64/128), thus there are too few channels in a single dimension, leading to insufficient feature-interaction. Since the 2D method obtains the best trade-off, we set $K = 2$ in all the following experiments. ",
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+    },
+    {
+        "type": "text",
+        "text": "Connection type of spatiotemporal convolution. The coupling $3 \\times 3 \\times 3$ convolution can be decomposed into serial or parallel spatial/temporal convolution. Table 2c reveals that factorizing the 3D kernel can boost results as expected. Besides, the parallel connection is better, thus we adopt parallel connection as the default. ",
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+    {
+        "type": "text",
+        "text": "Dimension size. As we set $K = 2$ , it is essential to explore the impact of changing the dimension size $C _ { 2 }$ . We can demonstrate that the computation is the lowest when $C _ { 1 } = C _ { 2 } = { \\sqrt { C } }$ . Since $C$ is not always a perfect square number, we adopt the rounded middle size √ $\\lfloor { \\sqrt { C } } \\rfloor$ . Table 2d shows that when $\\dot { C _ { 2 } } = \\dot { [ \\sqrt { C } ] }$ , the model not only requires the fewest computation cost but also achieves the best performance. Hence, we set $C _ { 2 } = \\lfloor { \\sqrt { C } } \\rfloor$ naturally. ",
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+    },
+    {
+        "type": "text",
+        "text": "Number and location of CT-Blocks. Table 2e illustrates that simply replacing 1 block in stage5 can bring significant performance improvement ( $1 6 . 9 \\%$ vs. $4 2 . 3 \\%$ ). As we replace more blocks from the bottom up, the GFLOPs continues to decrease. Moreover, the bottom blocks seem to be more beneficial to temporal modeling, since replacing the extra 3 blocks in stage2 and stage3 only improve the accuracy by $0 . 4 \\%$ $4 6 . 9 \\%$ vs. $4 7 . 3 \\%$ ). Since replacing 7 blocks achieves the highest accuracy, we replace 7 blocks by default. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Kernel sizes along different dimensions. In Table 2f, we can observe that two concatenated $3 ^ { 3 }$ convolution kernels are slightly better than those with the same receptive field $( 1 ^ { 3 } + 5 ^ { 3 } )$ . Furthermore, the larger kernel size can bring performance improvement but more calculation. It reveals that our CT-Module avoids the limited receptive field of feature interaction, and it can progressively enlarge the receptive field of such interaction on all the dimensions. Considering a better trade-off between accuracy and computation, we choose two concatenated $3 ^ { 3 }$ convolution kernels. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8d802c4b38c91662b8574e070515a9fa0eebad44b6b34cc7406ca37e7101f2ef.jpg",
+        "table_caption": [
+            "Table 4: Comparison with the state-of-the-art on Kinetics-400. It shows that CT-Net- $R 5 0 _ { 1 6 f }$ can surpass all existing lightweight models and even SlowFast- $R 5 0 _ { 4 0 f }$ . When fusing different models, our model is $\\mathbf { 2 . 4 } \\times$ faster than SlowFast- $\\operatorname { R } 1 0 1 _ { 8 0 f }$ and shows an $0 . { \\dot { 9 } } \\%$ performance gain. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Backbone</td><td rowspan=1 colspan=1>#Frame</td><td rowspan=1 colspan=1>GFLOPs</td><td rowspan=1 colspan=1>Top-1 Top-5</td></tr><tr><td rowspan=1 colspan=1>R(2+1)D (Tran et al.,2018)TSN (Wang et al., 2016)I3D (Carreira &amp; Zisserman,2017)</td><td rowspan=1 colspan=1>2DR34InceptionInception</td><td rowspan=1 colspan=1>32×1×1025×10×164×N/A×N/A</td><td rowspan=1 colspan=1>1520=152×10800=80×10108×N/A</td><td rowspan=1 colspan=1>72.0  91.472.5  90.271.1  89.3</td></tr><tr><td rowspan=2 colspan=1>TSM (Lin et al.,2019)TEINet (Liu et al., 2020)bLVNet-TAM (Fan et al., 2019)TEA (Li et al., 2020b)PEM+TDLoss (Weng et al., 2020)CorrNet (Wang et al., 2020)SlowFast (Feichtenhofer et al.,2019)SlowFast (Feichtenhofer et al., 2019)</td><td rowspan=2 colspan=1>2DR502D R50bLR502DRes2Net502D R50+TIM3D R503D R50+R503D R50+R50</td><td rowspan=2 colspan=1>16×3×1016×3×10(16×2)×3×316×3×1016×3×1032×1×1036=(4+32)×3×1040=(8+32)×3×10</td><td rowspan=1 colspan=1>2580=86×302580=86×30</td><td rowspan=1 colspan=1>74.7    -76.2   92.5</td></tr><tr><td rowspan=1 colspan=1>561=62.3×92730=91×302580=86×301150=115×101083=36.1×301971=65.7×30</td><td rowspan=1 colspan=1>72.0  90.676.1   92.576.9  93.077.2    -75.2  91.576.4  92.2</td></tr><tr><td rowspan=1 colspan=1>Our CT-Net</td><td rowspan=1 colspan=1>2D R50</td><td rowspan=1 colspan=1>16×3×4</td><td rowspan=1 colspan=1>895=74.6×12</td><td rowspan=1 colspan=1>77.3  92.7</td></tr><tr><td rowspan=4 colspan=1>X3D-XL (Feichtenhofer,2020)SmallBigNet (Li et al.,2020a)ip-CSN (Tran et al., 2019)ip-CSN (Tran et al.,2019)CorrNet (Wang et al.,2020)SlowFast (Feichtenhofer et al.,2019)SlowFast (Feichtenhofer et al.,2019)NL I3D (Wang &amp; Gupta,2018)</td><td rowspan=1 colspan=1>-2D R101</td><td rowspan=1 colspan=1>16×3×1032×3×4</td><td rowspan=1 colspan=1>1452=48.4×306552=546×12</td><td rowspan=4 colspan=1>79.1   93.977.4  93.376.8  92.577.8  92.879.277.9   93.278.9  93.577.7   93.3</td></tr><tr><td rowspan=1 colspan=1>3D R1013D R1523D R101</td><td rowspan=3 colspan=1>32×3×1032×3×1032×3×1040=(8+32)×3×1080=(16+64)×3×10128×3×10</td><td rowspan=2 colspan=1>2490=83.0×303264=108.8×306720=224×303180=106×306390=213×30</td></tr><tr><td rowspan=2 colspan=1>3D R101+R1013D R101+R1013D R101</td></tr><tr><td rowspan=1 colspan=1>10770=359×30</td></tr><tr><td rowspan=1 colspan=1>Our CT-NetOur CT-NetEN</td><td rowspan=1 colspan=1>2D R1012D R50+R101</td><td rowspan=1 colspan=1>16×3×4(16+16)×3×4</td><td rowspan=1 colspan=1>1746=145.5×122641=220.1×12</td><td rowspan=1 colspan=1>78.8  93.779.8  94.2</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Impact of different modules and different spatial resolution. In Table $2 \\mathrm { g }$ , our CT-Module can significantly boost its baseline ( $1 6 . 9 \\%$ vs. $4 7 . 3 \\%$ ) and the TE mechanism can further improve the accuracy by $2 . 1 \\%$ $4 8 . 0 \\%$ vs. $5 0 . 1 \\%$ ). The extra point-wise convolution also boosts performance, which demonstrates that it is beneficial for sufficient feature interaction. Compared with the SE mechanisms, our TE mechanism focuses more on features in different sub-dimensions individually, thus effectively enhancing spatial-temporal features. In our experiments, to ensure GFLOPs is comparable with other methods, we crop the input to $2 5 6 \\times 2 5 6$ during testing. Table 2h shows both training and testing with a larger spatial resolution of input bring clear performance improvement. ",
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+    {
+        "type": "text",
+        "text": "4.2 COMPARISONS WITH THE STATE-OF-THE-ARTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Something-Something V1&V2. We make a comprehensive comparison in Table 3. Compared with NL $\\mathrm { I } 3 \\mathrm { D } { + } \\mathrm { G C N } _ { 3 2 f }$ , our $\\mathrm { C T - N e t } _ { 8 f }$ gains $4 . 0 \\%$ top-1 accuracy improvement with ${ \\bf 4 9 . 1 \\times }$ fewer GFLOPs in Something-Something V1. Besides, our ${ \\mathrm { C T - N e t } } _ { 8 f }$ $( 5 1 . 7 \\% )$ is better than the ir- $\\mathrm { C S N _ { 3 2 f } }$ $( 4 9 . 3 \\% )$ which adopts ResNet-152 as the backbone. Moreover, our $\\mathrm { C T - N e t } _ { 1 6 f }$ outperforms all the single-clip models in Something-Something V1&V2 and even better than most of the multiclip models. It illustrates that our CT-Net is preferable to capture temporal contextual information efficiently. Surprisingly, with only 280 GFLOPs, our ensemble model $\\mathrm { C T - N e t } _ { E N }$ achieves $5 6 . 6 \\% ( 6 7 . 8 \\% )$ top-1 accuracy in Something-Something V1(V2), which outperforms all methods. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Kinetics-400. Kinetics-400 is a large-scale sence-related dataset, and the lightweight 2D models are usually inferior to the 3D models on it. Table 4 shows our CT-Net- $R 5 0 _ { 1 6 f }$ can surpass all existing lightweight models based on 2D backbone. Even compared with SlowFast- $R 5 0 _ { 4 0 f }$ , our CT-Net$\\mathsf { R } 5 0 _ { 1 6 f }$ also achieves higher accuracy $7 7 . 3 \\%$ vs. $7 6 . 4 \\%$ ). Note that our reproduced SlowFastR50 performs worse than that in the paper (Feichtenhofer et al., 2019), which may result from the missing videos in Kinetics-400. As for the deeper model, compared with SlowFast- $\\mathbf { R } 1 0 1 _ { 8 0 f }$ , our CT-Net- $\\mathbf { R } 1 0 1 _ { 1 6 f }$ requires $\\mathbf { 3 . 7 \\times }$ fewer GFLOPs but gains comparable results $7 8 . 8 \\%$ vs. $7 8 . 9 \\%$ . Besides, it achieves comparable top-1 accuracy with X3D-XL $7 8 . 8 \\%$ vs. $7 9 . 1 \\%$ ) under a similar GFLOPs. However, X3D requires extensive model searching with an expensive GPU setting, while our CT-Net can be trained traditionally with feasible computation. We perform score fusion over CT-Net- $\\mathsf { R } 5 0 _ { 1 6 f }$ and ${ \\mathrm { C T - N e t - R } } 1 0 1 _ { 1 6 f }$ , which mimics two-steam fusion with two temporal rates. In this setting, our model is $\\mathbf { 2 . 4 } \\times$ faster than SlowFast- $\\operatorname { R } 1 0 1 _ { 8 0 f }$ and shows an $0 . 9 \\%$ performance gain $7 9 . 8 \\%$ vs. $7 8 . 9 \\%$ ) but only uses 32 frames. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/e9e0e87428692253ba228bd386f54e25af6ab14d47236004693285bf8da2671f.jpg",
+        "image_caption": [
+            "Figure 3: Comparison of visualization. Videos are sampled from Something-Something V1. Compared with $\\mathrm { R } ( 2 { + } 1 ) \\mathrm { D }$ and CSN, our CT-Net can localize the action and object better both in space and time thanks to the larger spatial-temporal receptive field. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3 VISUALIZATION ",
+        "text_level": 1,
+        "bbox": [
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+            338,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We use Saliency Tubes (Stergiou et al., 2019) to generate the visualization, for it can show the most discriminative features that the network locates. In Figure 3, we sample two videos from SomethingSomething V1 which requires complex temporal modeling. In the left example, our CT-Net focuses on a larger area around the towel, especially in the fourth and fifth frames, thus predicting that someone is twisting it. In contrast, $\\mathrm { R } ( 2 { + } 1 ) \\mathrm { D }$ only concentrates on one side of the towel and gives the wrong judgment. The same situation can be seen in the right example. We argue that CT-Net can localize the action and object accurately thanks to the larger spatial-temporal receptive field. As for CSN, the regions of interest seem to be scattered, because it lacks sufficient spatial-temporal interaction, thus ignoring the rich context both in space and time. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            510,
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+            526
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we construct an efficient tensor separable convolution to learn the discriminative video representation. We view the channel dimension of the input feature as a multiplication of K subdimensions and stack spatial/temporal tensor separable convolution along each of K sub-dimensions. Moreover, CT-Module is cooperated with the Tensor Excitation mechanism to further improve performance. All experiments demonstrate that our concise and novel CT-Net obtains a preferable balance between accuracy and efficiency on large-scale video datasets. Our proposed principles are preferable guidance for designing an efficient module for temporal modeling. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 ACKNOWLEDGEMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            659,
+            426,
+            675
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This work is partially supported by National Natural Science Foundation of China (6187617, U1713208), the National Key Research and Development Program of China (No. 2020YFC2004800), Science and Technology Service Network Initiative of Chinese Academy of Sciences (KFJ-STS-QYZX-092), Shenzhen Institute of Artificial Intelligence and Robotics for Society. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            570,
+            297,
+            585
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 MORE TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            602,
+            398,
+            616
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We use SGD with momentum 0.9 and cosine learning rate schedule (Loshchilov & Hutter, 2017) to train the entire network. The first 10 epochs are used for warm-up (Goyal et al., 2017a) to overcome early optimization difficulty. For kinetics, the batch, total epochs, initial learning rate, dropout and weight decay are set to 64, 110, 0.01, 0.5 and 1e-4 respectively. All these hyper-parameters are set to 64, 45, 0.02, 0.3 and 5e-4 respectively for Something-Something. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 TENSOR EXCITATION MECHANISM ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            715,
+            460,
+            729
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The implementation of our Tensor Excitation is shown in Figure 4. Different from the SE module, we use tensor separable convolution in the TE mechanism. Moreover, when obtaining the spatial attention, we squeeze the temporal dimension and perform spatial tensor separable convolution, because temporal information is insignificant for spatial attention and vice versa. We add the Batch Normalization (BN) layer for better optimization. ",
+        "bbox": [
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+            741,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.3 RESULTS ON UCF101 AND HMDB51 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            827,
+            477,
+            842
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To verify the generation ability of our CT-Net on smaller datasets, we conduct transfer learning experiments from Kinetics400 to UCF101 (Soomro et al., 2012) and HMDB-51 (Kuehne et al., 2011). We test CT-Net with 16 input frames and evaluate it over three splits and report the averaged results. As shown in Table 5, our $\\mathrm { C T - N e t } _ { 1 6 f }$ achieves competitive performance when compared with the recent methods, which demonstrates the generation ability of our CT-Net. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/1d22d1e7f98cf1eda2d2109a7687480c59fdd392ebb5fd9b50f5dcead5888ea9.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            102,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/79800b513eef8dc669dae27966d7102e9ac088d87ebb420af3a70b75c7a1dc08.jpg",
+        "table_caption": [
+            "Figure 4: The implementation of our Tensor Excitation (TE) mechanism. ",
+            "Table 5: Comparison results on UCF101 and HMDB51. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Backbone</td><td>Pretrain</td><td>UCF101</td><td>HMDB51</td></tr><tr><td>C3D(Tran et al.,2015)</td><td>3D VGG-11</td><td>Sports-1M</td><td>82.3</td><td>51.6</td></tr><tr><td>I3D(Carreira &amp; Zisserman, 2017)</td><td>3D Inception</td><td>ImageNet+Kinetics</td><td>95.1</td><td>74.3</td></tr><tr><td>ECO(Zolfaghari et al.,2018)</td><td>Inception+3D R18</td><td>ImageNet+Kinetics</td><td>94.8</td><td>72.4</td></tr><tr><td>TSN(Wang et al., 2016)</td><td>Inception</td><td>ImageNet+Kinetics</td><td>91.1</td><td>=</td></tr><tr><td>TSM(Lin et al., 2019)</td><td>2D R50</td><td>ImageNet+Kinetics</td><td>94.5</td><td>70.7</td></tr><tr><td>STM(Jiang et al., 2019)</td><td>2D R50</td><td>ImageNet+Kinetics</td><td>96.2</td><td>72.2</td></tr><tr><td>Our CT-Net</td><td>2D R50</td><td>ImageNet+Kinetics</td><td>96.2</td><td>73.2</td></tr></table>",
+        "bbox": [
+            178,
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+            422
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.4 MORE RESULTS ON SOMETHING-SOMETHING V1&V2 ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            464,
+            593,
+            479
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 6 shows more results on Something-Something V1&V2. We train CT-Net with a different number of input frames and then test these models with different sampling strategies. We average the prediction scores obtained from the previous models to evaluate the ensemble models. With more input frames, the corresponding accuracy becomes higher. As for the reason that $\\mathrm { C T - N e t } _ { 2 4 f }$ is worse than $\\mathrm { C T - N e t } _ { 1 6 f }$ , we argue that is because the model is hard to optimize with too many input frames. Sampling more clips or more crops also boosts performance. Moreover, our ensemble models gain the state-of-the-art top-1 accuracy of $5 6 . 6 \\% ( 6 8 . 3 \\% )$ on Something-Something V1(V2). ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.5 MORE ABLATION STUDIES ON MINI-KINETICS AND SOMETHING-SOMETHING V2 ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            604,
+            782,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To verify the effectiveness of our module comprehensively, we also conduct experiments in MiniKinetics and Something-Something V2 and report the multi-clip accuracy and single-clip accuracy respectively. Mini-Kinetics covers 200 action classes and is a subset of Kinetics-400, while Something-Something V2 covers the same action classes as Something-Something V1 but contains more videos. As shown in Table 7, the performance trend of different modules is similar to that shown in Table 2a. Since Mini-Kinetics does not highly depend on temporal modeling, the gap becomes smaller but still demonstrates the effectiveness of our CT-Module. ",
+        "bbox": [
+            174,
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+            728
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.6 ADAPTING DIFFERENT PRE-TRAINED IMAGENET ARCHITECTURES AS CT-NET ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            746,
+            758,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In fact, by directly replacing the $3 \\times 3$ convolution with our CT-Module, we can easily adapt different pre-trained ImageNet architectures as CT-Net. Table 8 shows that it is also sensible to use InceptionV3 as the backbone. We believe that through more elaborate design, our CT-Net based on different backbones can achieve comparable performance. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.7 VALIDATION PLOT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            844,
+            346,
+            858
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Figure 5, we plot the accuracy vs per-clip GFLOPs on Kinetics-400. It reveals that our CT-Net achieves a better trade-off than most of the existing methods on Kinetics-400. ",
+        "bbox": [
+            173,
+            871,
+            823,
+            900
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/61a4e296e7e6abe360f83e736c2d693b2aebf55da7504d345a68277b70cd5006.jpg",
+        "table_caption": [
+            "Table 6: More results on Something-Something V1&V2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>#Frame</td><td>GFLOPs</td><td>#Param</td><td colspan=\"2\">SomethingV1 Top-1 Top-5</td><td colspan=\"2\">SomethingV2 Top-1 Top-5</td></tr><tr><td rowspan=\"9\">Our CT-Net</td><td>8 12</td><td>37 56</td><td rowspan=\"3\">21.0M</td><td>50.1</td><td>78.8</td><td>62.5</td><td>87.7</td></tr><tr><td></td><td></td><td>52.1</td><td>80.0</td><td>63.9</td><td>88.7</td></tr><tr><td></td><td></td><td>52.5</td><td>80.9</td><td>64.5</td><td>89.3</td></tr><tr><td>16 24</td><td>75 112</td><td rowspan=\"4\"></td><td>52.5</td><td>80.9</td><td>64.6</td><td>89.1</td></tr><tr><td>8x1x2</td><td>75</td><td>51.6</td><td>79.7</td><td>63.5</td><td>88.5</td></tr><tr><td>12×1×2</td><td>112</td><td>52.8</td><td>80.6</td><td>64.6</td><td>89.3</td></tr><tr><td>16×1×2</td><td>151</td><td>53.2</td><td>81.3</td><td>65.2</td><td>89.7</td></tr><tr><td>24×1×2</td><td>224</td><td>52.9</td><td>81.3</td><td>65.0</td><td>89.3</td></tr><tr><td>8×3×2</td><td>224</td><td rowspan=\"4\">21.0M</td><td>51.7</td><td>80.1</td><td>63.9</td><td>88.8</td></tr><tr><td>12×3×2</td><td>336</td><td>53.0</td><td>81.1</td><td>65.3</td><td>89.6</td></tr><tr><td>16×3×2</td><td>447</td><td>53.4</td><td>81.7</td><td>65.9</td><td>90.1</td></tr><tr><td>24×3×2</td><td></td><td>53.6</td><td>81.6</td><td>65.5</td><td>89.8</td></tr><tr><td rowspan=\"3\">Our CT-NetEN</td><td rowspan=\"3\">8+16 (8+12+16+24)×1× 2</td><td>672 112</td><td rowspan=\"3\">83.8M</td><td>54.4</td><td>82.0</td><td>66.2</td><td>90.4</td></tr><tr><td>280</td><td>56.6</td><td>83.9</td><td>67.8</td><td>91.1</td></tr><tr><td>560</td><td>56.6</td><td>84.0</td><td>67.8</td><td>91.3</td></tr><tr><td></td><td>(8+12+16+24)×1 × 2 (8+12+16+24)×3×2</td><td>1679</td><td></td><td>56.6</td><td>83.9</td><td>68.3</td><td>91.3</td></tr></table>",
+        "bbox": [
+            178,
+            107,
+            823,
+            342
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/e06fabb5822b341ee6c403318776daf757b68d4b002ffeee5800bbe7af5ca5cf.jpg",
+        "table_caption": [
+            "Table 7: More ablation studies on Mini-Kinetics and Something-Something V2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Backbone</td><td rowspan=\"2\">GFLOPs</td><td colspan=\"2\">Mini-Kinetics</td><td colspan=\"2\">SomethingV2</td></tr><tr><td>Top-1</td><td>Top-5</td><td>Top-1</td><td>Top-5</td></tr><tr><td>C3D-Module (Tran et al., 2015)</td><td>2D R50</td><td>59.9</td><td>77.5</td><td>93.0</td><td>59.1</td><td>85.5</td></tr><tr><td>R(2+1)D-Module (Tran et al., 2018)</td><td>2D R50</td><td>45.8</td><td>77.8</td><td>93.2</td><td>60.0</td><td>86.0</td></tr><tr><td>CSN-Module (Tran et al.,2019)</td><td>2D R50</td><td>35.6</td><td>77.6</td><td>93.2</td><td>59.5</td><td>86.0</td></tr><tr><td>Our CT-Module</td><td>2D R50</td><td>36.3</td><td>78.0</td><td>93.6</td><td>60.3</td><td>86.4</td></tr></table>",
+        "bbox": [
+            178,
+            376,
+            820,
+            468
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/e12f2f68577c4ee75ebf9e0316bc14fa7b502b36d631278bf2767ce6c1be76f3.jpg",
+        "table_caption": [
+            "Table 8: Adapting different pre-trained ImageNet architectures as CT-Net. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Backbone</td><td>GFLOPs</td><td>#Param.(M)</td><td>Top-1</td><td>Top-5</td></tr><tr><td>Baseline (TSN)</td><td>2D ResNet-50</td><td>43.0</td><td>23.9</td><td>16.9</td><td>42.0</td></tr><tr><td>Our CT-Net</td><td>2D ResNet-50</td><td>37.3</td><td>21.0</td><td>50.1</td><td>78.8</td></tr><tr><td>Baseline (TSN)</td><td>InceptionV3</td><td>45.8</td><td>22.1</td><td>18.3</td><td>43.9</td></tr><tr><td>Our CT-Net</td><td>InceptionV3</td><td>43.9</td><td>20.9</td><td>47.2</td><td>76.1</td></tr></table>",
+        "bbox": [
+            179,
+            502,
+            820,
+            597
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/b2f3ea6a4dccbaf4e10643bebf839cb1adecc4f29e2196a615abde7b18fdf557.jpg",
+        "image_caption": [
+            "Figure 5: Accuracy vs per-clip GFLOPs on Kinetics-400. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            633,
+            816,
+            901
+        ],
+        "page_idx": 12
+    }
+]

parse/train/rkr1UDeC-/rkr1UDeC-.md

@@ -0,0 +1,222 @@
+# LARGE SCALE DISTRIBUTED NEURAL NETWORKTRAINING THROUGH ONLINE DISTILLATION
+
+Rohan Anil   
+Google   
+rohananil@google.com
+
+Gabriel Pereyra ∗ Google DeepMind pereyra@google.com
+
+Alexandre Passos Google Brain apassos@google.com
+
+Robert Ormandi Google ormandi@google.com
+
+George E. Dahl Google Brain gdahl@google.com
+
+Geoffrey E. Hinton   
+Google Brain   
+geoffhinton@google.com
+
+# ABSTRACT
+
+Techniques such as ensembling and distillation promise model quality improvements when paired with almost any base model. However, due to increased testtime cost (for ensembles) and increased complexity of the training pipeline (for distillation), these techniques are challenging to use in industrial settings. In this paper we explore a variant of distillation which is relatively straightforward to use as it does not require a complicated multi-stage setup or many new hyperparameters. Our first claim is that online distillation enables us to use extra parallelism to fit very large datasets about twice as fast. Crucially, we can still speed up training even after we have already reached the point at which additional parallelism provides no benefit for synchronous or asynchronous stochastic gradient descent. Two neural networks trained on disjoint subsets of the data can share knowledge by encouraging each model to agree with the predictions the other model would have made. These predictions can come from a stale version of the other model so they can be safely computed using weights that only rarely get transmitted. Our second claim is that online distillation is a cost-effective way to make the exact predictions of a model dramatically more reproducible. We support our claims using experiments on the Criteo Display Ad Challenge dataset, ImageNet, and the largest to-date dataset used for neural language modeling, containing $6 \times 1 0 ^ { 1 1 }$ tokens and based on the Common Crawl repository of web data.
+
+# 1 INTRODUCTION
+
+For large-scale, commercially valuable neural net training problems, practitioners would be willing to devote many more machines to training if it sped up training time dramatically or improved the quality of the final model. Currently, distributed stochastic gradient descent (SGD), in both its synchronous and asynchronous forms (Chen et al., 2016), is the dominant algorithm for large-scale neural network training across multiple interconnected machines. Unfortunately, as the number of machines increases, there are diminishing improvements to the time needed to train a high quality model, to a point where adding workers does not further improve training time. A combination of infrastructure limitations and optimization barriers constrain the scalability of distributed minibatch SGD. The overhead of communicating weight updates and the long tail of the machine and network latency distributions slow down execution and produce thorny engineering challenges. For the synchronous algorithm, there are rapidly diminishing returns from increasing the effective batch size (LeCun et al., 2012; Keskar et al., 2017). For the asynchronous algorithm, gradient interference from inconsistent weights can cause updates to thrash and even, in some cases, result in worse final accuracy or completely stall learning progress. The precise scalability limit for distributed SGD will depend on implementation details of the algorithm, specifics of the infrastructure, and the capabilities of the hardware, but in our experience it can be very difficult to scale effectively much beyond a hundred GPU workers in realistic setups. No algorithm for training neural nets will be infinitely scalable, but even scaling a bit beyond the limits of distributed SGD would be extremely valuable.
+
+Once we have reached the limits of adding workers to distributed SGD, we could instead use extra machines to train another copy of the model and create an ensemble to improve accuracy (or trade this accuracy for training time by training the members of the ensemble for fewer steps). As an added benefit, the ensemble will make more stable and reproducible predictions, which can be useful in practical applications. However, ensembling increases the cost at test time, potentially violating latency or other cost constraints. Alternatively, to get nearly the same benefits of the ensemble without increasing test time costs, we can distill (Hinton et al., 2015; Bucila et al., 2006) an $n$ -way ensemble of models into a single still-servable model using a two-phase process: first we use $n M$ machines to train an n-way ensemble with distributed SGD and then use $M$ machines to train the servable student network to mimic the $n$ -way ensemble. By adding another phase to the training process and using more machines, distillation in general increases training time and complexity in return for a quality improvement close to the larger teacher ensemble model.
+
+We believe that the additional training costs, in terms of both time and pipeline complexity, discourage practitioners from using ensemble distillation, even though it almost always would improve results. In this work, we describe a simpler online variant of distillation we call codistillation. Codistillation trains $n$ copies of a model in parallel by adding a term to the loss function of the ith model to match the average prediction of the other models.
+
+Through large-scale experiments we show that, compared to distributed SGD, codistillation improves accuracy and speeds up training by allowing the productive use of more computational resources even beyond the point where adding more workers provides no additional speedup for SGD. Specifically, codistillation provides the benefits of distilling an ensemble of models without increasing training time. Codistillation is also quite simple to use in practice compared to a multi-phase distillation training procedure. Multi-phase distillation tends to encourage human intervention between the training phases to decide when to stop training the ensemble and start distilling it into a single model. We also show that codistillation does not lose the reproducibility benefits of ensembles of neural networks, reducing churn in the predictions of different retrains of the same model. Reducing prediction churn can be essential when testing and launching new versions of a model in a non-disruptive way in an existing service, although it is not as well-studied in the academic machine learning community.
+
+Given the obvious relationship to distillation, very similar algorithms to codistillation have been independently described by multiple researchers. For example, Zhang et al. (2017) describes another simultaneous distillation algorithm but does not investigate the benefits in the distributed training case and only presents it as a potential quality improvement over regular distillation. We view the experimental validation of codistillation at scale as the primary contribution of our work. Another contribution of this work is our exploration of different design choices and implementation considerations for codistillation which we believe has produced recommendations of substantial practical utility.
+
+In general, we believe the quality gains of codistillation over well-tuned offline distillation will be minor in practice and the more interesting research direction is exploring codistillation as a distributed training algorithm that uses an additional form of communication that is far more delay tolerant.
+
+# 1.1 RELATED WORK
+
+In addition to the closely related work in Hinton et al. (2015) and Zhang et al. (2017) mentioned above, there are many different tactics for scaling up neural network training. Early work in training large distributed neural networks focused on schemes for partitioning networks over multiple cores, often referred to as model parallelism (Dean et al., 2012). As memory has increased on graphic processing units (GPUs), the majority of distributed training has shifted towards data parallelism, where the model is replicated across multiple machines and data are distributed to the different replicas, with updates being merged by parameter servers or a single allreduce step as in Goyal et al. (2017). Even without a high quality allreduce primitive, variants of centralized synchronous SGD with backup workers can scale to a large number of machines (Chen et al., 2016).
+
+Methods like ensembling and distillation are mostly orthogonal to lower level distributed training infrastructure. However, mixture of experts models have particularly natural model parallelism that can be integrated with data parallelism and a synchronous training scheme. Gross et al. (2017) and Shazeer et al. (2017) are notable examples of recent work in this area.
+
+As researchers try to scale neural network training to ever larger datasets and models, the optimization algorithm itself can be altered. For synchronous SGD there are rapidly diminishing returns (LeCun et al., 2012; Keskar et al., 2017) as the number of workers, and thus the effective batch size, increases and we might hope that algorithms like KFAC (Ba et al., 2017) would make better use of large batches. Although a promising direction for research, in this work we focus on what should hopefully be an optimization algorithm agnostic way to improve scalability and reproducibility.
+
+# 2 CODISTILLATION
+
+Distillation is a meta-algorithm which allows any algorithm to incorporate some of the model quality benefits of ensembles. The idea of distillation is to first train a teacher model, which traditionally is an ensemble or another high-capacity model, and then, once this teacher model is trained, train a student model with an additional term in the loss function which encourages its predictions to be similar to the predictions of the teacher model.
+
+There are many variants of distillation, for different types of teacher model, different types of loss function, and different choices for what dataset the student model trains on. For example, the student model could be trained on a large unlabeled dataset, on a held-out data set, or even on the original training set.
+
+Perhaps surprisingly, distillation has benefits even if the teacher model and the student model are two instances of the same neural network (see section 3 for empirical evidence), as long as they are sufficiently different (say, by having different initializations and seeing the examples in a different order). Furthermore, the teacher model predictions are still beneficial to the student model even before convergence. Finally, the distinction between teacher and student is unnecessary and two or more models all distilling from each other can also be useful.
+
+In this paper, we use codistillation to refer to distillation performed:
+
+1. using the same architecture for all the models;   
+2. using the same dataset to train all the models; and   
+3. using the distillation loss during training before any model has fully converged.
+
+For simplicity, we usually consider the case where all models have a distillation term in their loss function, but the key characteristic of codistillation is the simultaneous training of a model and its teacher.
+
+Algorithm 1 presents the codistillation algorithm. The distillation loss term $\psi$ can be the squared error between the logits of the models, the KL divergence between the predictive distributions, or some other measure of agreement between the model predictions. In this work we use the cross entropy error treating the teacher predictive distribution as soft targets. In the beginning of training, the distillation term in the loss is not very useful or may even be counterproductive, so to maintain model diversity longer and to avoid a complicated loss function schedule we only enable the distillation term in the loss function once training has gotten off the ground.
+
+# 2.1 CODISTILLATION AS A DISTRIBUTED NEURAL NETWORK TRAINING ALGORITHM
+
+In order to scale beyond the limits of distributed stochastic gradient descent we will need an algorithm that is far more communication efficient. As seen in Algorithm 1, to update the parameters of one network using codistillation one only needs the predictions of the other networks, which can be computed locally from copies of the other networks weights.
+
+There are several reasons to believe that stale predictions might be much less of a problem than stale gradients for training:
+
+# Algorithm 1 Codistillation
+
+Input loss function $\phi ( 1 \mathsf { a b e 1 }$ , prediction)   
+Input distillation loss function $\psi$ (aggregated label, prediction)   
+Input prediction function F (θ, input)   
+Input learning rate $\eta$   
+for n burn in steps do for $\theta _ { i }$ in model set do y, f $=$ get train example() $\theta _ { i } = \theta _ { i } - \eta \nabla _ { \theta _ { i } } \{ \phi ( y , F ( \theta _ { i } , f ) ) \}$ end for   
+end for   
+while not converged do for $\theta _ { i }$ in model set do y, f $=$ get train example() $\begin{array} { r } { \dot { \theta _ { i } } = \theta _ { i } - \eta \nabla _ { \theta _ { i } } \{ \phi ( \mathrm { y } , F ( \theta _ { i } , f ) ) + \psi ( \{ \frac { 1 } { N - 1 } \sum _ { j \neq i } F ( \theta _ { j } , f ) \} , F ( \theta _ { i } , f ) ) \} } \end{array}$ end for   
+end while   
+1. every change in weights leads to a change in gradients, but as training progresses towards convergence, weight updates should substantially change only the predictions on a small subset of the training data;   
+2. weights (and gradients) are not statistically identifiable as different copies of the weights might have arbitrary scaling differences, permuted hidden units, or otherwise rotated or transformed hidden layer feature space so that averaging gradients does not make sense unless models are extremely similar;   
+3. sufficiently out-of-sync copies of the weights will have completely arbitrary differences that change the meaning of individual directions in feature space that are not distinguishable by measuring the loss on the training set;   
+4. in contrast, output units have a clear and consistent meaning enforced by the loss function and the training data.
+
+Furthermore, the predictive distribution of radically different models can still provide very useful information about the relationship between inputs and outputs. Empirically we’ve found that using stale predictions instead of up-to-date predictions for the other neural networks has little to no adverse effect on the quality of the final trained model produced by codistillation. We have been able to use predictions tens of thousands of updates old in the asynchronous case or $8 0 0 \mathrm { k }$ examples (i.e. 50 updates) old in the large-batch synchronous case.
+
+The tolerance of distillation for stale teacher predictions suggests a distributed training strategy which is far less communication-intensive than synchronous or asynchronous SGD.
+
+1. Each worker trains an independent version of the model on a locally available subset of the training data.   
+2. Occasionally, workers checkpoint their parameters.   
+3. Once this happens, other workers can load the freshest available checkpoints into memory and perform codistillation.
+
+Of course there is no reason not to combine this strategy with standard distributed SGD, resulting in a procedure that employs independent groups of workers that exchange checkpoints between groups and exchange gradient information within a group.
+
+In each iteration of synchronous/asynchronous distributed SGD, each worker needs to send and receive an amount of information proportional to the entire model size. When using codistillation to distribute training each worker only needs to very rarely read parameter checkpoints from the other models.1 When combining distributed SGD and codistillation, we can add workers to a group up until the point where we see diminishing returns from distributed SGD and then deploy additional workers in another group, occasionally exchanging checkpoints between the otherwise independent groups of workers.
+
+Moreover, there is no need to use high-precision floating point numbers to store the parameters used to compute the predictions for the distillation loss term as distillation is not very sensitive to the exact values of the predictions. Therefore the additional computational cost of distributed codistillation will not be much higher than the cost of independent training.
+
+Since the parameters of a model trained on a data set can be viewed as a very compressed representation of the aspects of that data set which are relevant to the learning problem at hand, it makes intuitive sense that leveraging these parameters might be more communication-efficient than sending all the data points or gradients.
+
+# 3 EXPERIMENTS AND RESULTS
+
+In order to study the scalability of distributed training using codistillation, we need a task that is representative of important large-scale neural network training problems. Neural language modeling is an ideal test bed because vast quantities of text are available on the web and because neural language models can be very expensive to train. Neural language models are representative of important problems that make common use of distributed SGD like machine translation and speech recognition, but language modeling is simpler to evaluate and uses a simpler pipeline. In order to make any potential scalability improvements as clear as possible, we selected a data set large enough that it is completely infeasible to train an expressive model to convergence on it with existing SGD parallelization strategies. In order to confirm that our results were not specific to some peculiarity of language modeling, we also validated some of our large scale codistillation results on ImageNet (Russakovsky et al., 2015) as well.
+
+To demonstrate the benefits of codistillation in reducing prediction churn and to study other properties of the algorithm we can use smaller experiments that are cheaper to perform, but it is important to actually reach the limits of distributed SGD when studying scalability.
+
+# 3.1 DATA SETS AND MODELS
+
+Common Crawl is an open repository of web crawl data. We downloaded the WET files2 and filtered them to only include English language documents that contained long paragraphs because we wanted data that allowed modeling of slightly longer range dependencies than data sets that randomize sentence order like LM1B (Chelba et al., 2013). After preprocessing, roughly 915 million documents (20TB of text) remained. We plan to release the list of document ids that remained after filtering as well as code for our invertible tokenizer for others to use this data set. The language model we trained in all of our Common Crawl experiments was an RNN language model with two LSTM layers of 1024 units each with layer normalization (Ba et al., 2016). We used 256 dimensional input embeddings and a vocabulary of 24006 word pieces (Schuster & Nakajima, 2012), including sentence and paragraph start and end tokens, out of vocabulary (generally non-English characters), and end of document. After converting to word pieces there were 673 billion tokens, which is much larger than any previous neural language modeling data set we are aware of.3 During training we constructed batches 32 word pieces long drawing tokens from $B$ different documents at a time, saving hidden state across batches. Since the hidden state never gets reset, the model has to learn to use the end of document token to reset itself. We use the ADAM optimizer Kingma & Ba (2014) for all experiments on Common Crawl.
+
+ImageNet is the most popular image classification benchmark of recent years. All of our experiments on ImageNet followed the setup of Goyal et al. (2017) as closely as possible and also used fully-synchronous SGD. We used the same learning rate scaling and schedule and used a configuration with batch size 16384 that achieves $7 5 \%$ accuracy as our primary baseline. Goyal et al. (2017) reports that increasing the batch size beyond 8192 provides rapidly diminishing returns.
+
+Criteo Display Ad Challenge dataset4 (Criteo) is a benchmark dataset for predicting click through rate for online advertising. The data contain roughly 43 million examples, each with 13 integer and 26 categorical input features. The task is formulated as binary classification and we train a feedforward fully connected neural network using ReLU activations with hidden layer sizes of 2560, 1024, 256 and a logistic output. We use the Adagrad optimizer with learning rate of 0.001 for training for all experiments on this dataset.
+
+3.2 REACHING THE LIMITS OF DISTRIBUTED SGD FOR TRAINING RNNS ON COMMON CRAWL
+
+In our first set of experiments, our goal was to approximately determine the maximum number of GPU workers that can be productively employed for SGD in our Common Crawl neural language model setup. Since our dataset is two orders of magnitude larger than English Wikipedia, there is no concern about revisiting data, which would make independent replicas more similar, even in relatively large scale experiments.
+
+We tried asynchronous SGD with 32 and 128 workers, sharding the weights across increasing numbers of parameter servers as necessary to ensure that training speed was bottlenecked by GPU computation time. We found it very difficult to keep training stable and prevent the RNNs from diverging for asynchronous SGD with large numbers of workers. We experimented with a few worker ramp up schemes and different learning rates, but ultimately decided to focus on the synchronous algorithm to make our results less dependent on the specific characteristics of our infrastructure and implementation. Gradient staleness is hard to analyze independent of the specific conditions whereas differences in implementation and infrastructure are far easier to abstract away for synchronous SGD. Although it may have been possible to make async work well with more effort, the debilitating effect of stale gradients on learning progress is a well known issue, for instance Chen et al. (2016) demonstrated that synchronous SGD can often converge to a better final accuracy than asynchronous SGD. Mitliagkas et al. (2016) argues that asynchrony can effectively increase the momentum which is part of why it tends to diverge so easily. For these and other reasons, practitioners (e.g. Goyal et al. (2017)) seem to be moving away from asynchronous SGD towards synchronous training as the default. In preliminary experiments the gains from codistillation seemed independent of the choice of asynchronous or synchronous SGD as the base algorithm.
+
+The maximum number of GPU workers that can be productively employed for synchronous SGD will depend on infrastructure limits, tail latency, and batch size effects. Fully synchronous SGD is equivalent to the single machine algorithm with a much larger batch size. Increasing the effective batch size reduces noise in the gradient estimates which allows larger step sizes with hopefully higher quality updates that result in faster convergence. Given effectively infinite training data (even with 256 GPUs we do not visit all of the Common Crawl training data) we intuitively would expect increasing the effective batch size to at worst increase the step time. We trained language models on Common Crawl with fully synchronous SGD using a per-worker batch size of 128 and 32, 64, 128, and 256 workers. Thus the effective batch size ranged from 4096 to 32768. Generally we should expect to need to increase the learning rate as we increase the effective batch size, so for each number of workers we tried learning rates of 0.1, 0.2, and 0.4. For 32 and 64 workers, 0.1 performed best and since none of the original three learning rates performed well for 256 workers, we also tried an additional intermediate learning rate of 0.3 which was the best performing learning rate for 256 workers.
+
+Figure 1a plots the validation error as a function of global steps for the different numbers of workers we tried, using the best learning rate for each number of workers. Increasing the number of workers (and thus the effective batch size) reduced the number of steps required to reach the best validation error until 128 workers, at which point there was no additional improvement. Even with idealized perfect infrastructure, 256 workers would at best result in the same end to end training time on this problem. However, because steps can take so much longer with 256 workers, going from 128 to 256 workers is highly counterproductive in practice. Figure 1b plots validation error against wall time for the same varying numbers of synchronous workers. There is a large degradation in step time, and thus learning progress, at 256 workers. Although it might be possible to improve the step time at 256 workers by using a more sophisticated scheme with backup workers (Chen et al., 2016), the operative limit to scalability on this task is the diminishing return from increasing the effective batch size, not the degradation in step times.
+
+![](images/52bb17f20408a8288b8c469d430282f12c5884bc1bf5b32f5dd8ae70d590ec1c.jpg)  
+Figure 1: Synchronous training on Common Crawl dataset.
+
+In these particular experiments, synchronous SGD with 128 workers is the strongest baseline in terms of training time and final accuracy. Therefore we focus the rest of our experiments on comparisons with 128 worker synchronous SGD and study codistillation that uses synchronous SGD as a subroutine, although it also works well with the asynchronous algorithm.
+
+# 3.3 CODISTILLATION WITH SYNCHRONOUS SGD
+
+For language modeling on Common Crawl, synchronous SGD with 128 GPUs achieved the best results for standard distributed training, at least of the configurations we tried, and we were unable to improve training time with 256 GPUs. Although the extra GPUs do not seem to help basic synchronous SGD, our hypothesis is that the extra 128 GPUs will improve training time if we use two-way codistillation with two groups of 128 GPUs using synchronous SGD that exchange checkpoints periodically.
+
+One concern would be that codistillation is merely a way of penalizing confident output distributions (Pereyra et al., 2017) or smoothing the labels, so we also compared to two label smoothing baselines. The first baseline replaces the distillation loss term with a term that matches the uniform distribution and the second uses a term that matches the unigram distribution. Trade-off hyperparameters were tuned by hand in preliminary experiments.
+
+Another important comparison is to an ensemble of two neural networks, each trained with 128 GPUs and synchronous SGD. Although we are in general interested in the regime where such an ensemble would not be practical because of the increased test time costs, given our understanding of distillation we would expect codistillation, if it achieves all of the benefits of traditional distillation, to have a training curve close to—but slightly worse than—a two-way ensemble. In the case of twoway codistillation, this would provide evidence that the gains are really coming from an ensemblinglike effect despite never explicitly averaging the predictions of multiple models as would happen when distilling an ensemble model.
+
+Figure 2a plots validation cross entropy versus step of synchronous training for codistillation using two groups of 128 GPUs along with training curves for the synchronous SGD and label smoothing baselines (each using 128 GPUs) and an ensemble of two instances of the 128 GPU baseline. All experiments in figure 2a used the learning rate found to be the best for the 128 GPU synchronous SGD baseline. Two-way codistillation successfully reduces training time substantially compared to the 128 worker baselines and almost achieves the training curve of the two-way ensemble. Measuring at the best validation error achieved by the baseline, codistillation reaches the same error in 2X fewer steps. Perhaps more importantly, codistillation reaches a lower final error so a 2X reduction in steps is likely an underestimate of the gains. In our implementation, for the model we trained, codistillation is free in terms of step time as the GPU is not fully utilized and our implementation automatically overlaps the computation of the teacher and student models. In the worst case, for a model that saturates the hardware that is implemented without quantization, prefetching predictions using the CPU, or other optimizations to compute the predictions, the extra forward pass might increase compute costs by nearly $50 \%$ . However, even with these worst case assumptions, network costs will be a substantial contributor to the total step time, easily $50 \% - 8 0 \%$ , resulting in a modest increase in time per step.
+
+![](images/974945ffbc8380f2d69a00cfae8fe969812d6a87a8fdea9e0930975268993c09.jpg)  
+Figure 2: Codistillation (abbreviated as “Cod” in the legend) results with Common Crawl.
+
+![](images/a455fe81ba4b5b80251f86c8e4d5f511b2ac2e94aeb7717e648e8f811511ba37.jpg)  
+Figure 3: Codistillation on ImageNet
+
+![](images/e6db93fc59573000cf34cce4583fd34b8eb2e9a81daebdb1c370f768ca6642e7.jpg)  
+Figure 4: Reload intervals on Common Crawl
+
+# 3.3.1 CODISTILLATION REQUIRES FEWER STEPS ON IMAGENET
+
+In order to confirm our results are not due to the vicissitudes of language modeling or the particulars of our setup, we tried two-way codistillation on ImageNet as well. As can be seen from figure 3, codistillation (enabled after 3000 steps) achieves the same $7 5 \%$ accuracy after a total of 5250 steps as the baseline does after 7250 steps. Eventually, two-way codistillation achieves a slightly better accuracy of $7 5 . 6 \%$ at 7250 steps, confirming the utility of codistillation on ImageNet in addition to language modeling.
+
+# 3.3.2 PARTITIONING THE DATASET
+
+The gains from codistillation have a variety of potential causes that we would like to tease apart. From the experiments above, we have evidence against label smoothing effects as a cause. Another potential source of the training time and quality gains from codistillation over basic SGD would be that the different codistilling models see different training data. To test this hypothesis, we ran another codistillation experiment using two groups of 128 workers, but forced the two groups to use the same training data instead of using different random subsets.5 Figure 2b compares codistillation using different subsets of the training data to codistillation using the same data. Codistillation with the same data seems to be slightly better than the baseline, but codistillation using different data gets much better results. These results show that the codistilling models are indeed successfully transmitting useful information about different parts of the training data to each other.
+
+# 3.4 PREDICTION STALENESS SENSITIVITY
+
+In general, codistillation can handle relatively stale predictions. SGD generally cannot tolerate gradients nearly as stale. The contrast is most stark with the asynchronous algorithm. We trained our two-way synchronous codistillation setup on Common Crawl with several different checkpoint exchange frequencies. We tried exchange delays of 50, 100, and 250 steps. As can be seen from figure 4, increasing the checkpoint reload interval beyond 819,200 examples or 50 steps slightly degrades the learning curve. With an interval of 50 steps, communicating checkpoints with a shared file system is still quite feasible on most problems.
+
+# 3.4.1 CODISTILLATION VS MULTI-PHASE DISTILLATION VARIANTS
+
+In our experiments and experience, the choice between multi-phase distillation and codistillation makes very little difference in terms of the quality improvements achieved by distillation, although it does affect training time. On Common Crawl, we trained an ensemble of two models for 18K steps and then trained an identical model distilling from the ensemble to reach a cross entropy error of 4.0 after 9K steps for a total of 27K steps. However, two-way codistillation reached roughly the same validation error after only 10k steps. Zhang et al. (2017) reported a benefit in quality over basic distillation, but they compare distilling model $M _ { 1 }$ into model $M _ { 2 }$ with training model $M _ { 1 }$ and model $M _ { 2 }$ using codistillation; they do not compare to distilling an ensemble of models $M _ { 1 }$ and $M _ { 2 }$ into model $M _ { 3 }$ . Furthermore, Zhang et al. (2017) only reports final validation error using a large number of training steps on two small image datasets. Overfitting in the teacher model can explain the worse performance of offline distillation Zhang et al. (2017) report on CIFAR-100. We reproduced their experiments with WideResnet-28-10 teaching Resnet-32 on CIFAR-100. When we select a checkpoint with nearly $100 \%$ training accuracy, we reproduce the $6 9 . 5 \%$ they report in table 4. However, when we select a different checkpoint, we can achieve the $7 0 . 7 \%$ they report for online distillation using traditional offline distillation. For distillation to provide value, the teacher must provide information beyond the training label. Thus a teacher network that overfits the training set will not be useful.
+
+The gains in training time of codistillation over multi-phase distillation variants are obvious, but the reduction in training pipeline complexity codistillation provides can be equally important. Using the same architecture and dataset for all the models avoids squaring the tuning problem. The codistillation protocol simplifies the choice of teacher model and restores symmetry between the various models. With traditional multi-phase distillation one must decide which teacher model or models to use and how long to train them, encouraging human intervention between the phases. If teacher models occasionally get reused across versions of the student model in an effort to save computation, rollbacks of models to deal with bugs or data corruption can be dramatically more painful. To reproduce a given final model, one needs the entire history of teacher models and everything required to reproduce them which can easily result in what Sculley et al. (2014) refers to as “pipeline jungle” and unnecessary, undesirable data dependencies.
+
+Table 1: Prediction Churn   
+
+<table><tr><td>Model</td><td>Validation Log Loss</td><td>Mean Absolute Difference6 Between Retrains</td></tr><tr><td></td><td></td><td></td></tr><tr><td>DNN</td><td>0.4480 ± 0.001</td><td>0.029 ± 0.001</td></tr><tr><td>Ensemble of Two DNNs</td><td>0.4461 ± 0.0002</td><td>0.022 ± 0.002</td></tr><tr><td>Two-way codistilled DNN</td><td>0.4458 ± 0.002</td><td>0.019 ± 0.002</td></tr></table>
+
+# 3.5 REDUCING PREDICTION CHURN WITH CODISTILLATION
+
+Unlike linear models with convex loss functions, two neural networks with the same architecture that are trained on the same data can achieve similar validation and test performance while making very different predictions, and mistakes. On large datasets with a stable training pipeline aggregate metrics can be relatively consistent, but minor changes to the model architecture or even simple retrains can cause comparatively dramatic changes in the predictions made by the network. The network will in general get different examples correct and the differences might be especially severe on atypical examples with rare features. The weights learned by stochastic gradient descent in the non-convex setting will depend on the initialization, data presentation order, and the general vicissitudes of the infrastructure, especially when parallelization is involved. It is not practical to control all these nuisance variables and, even if it was, we would still see different solutions after making slight changes to the input representation or model architecture. We will refer to the general reproducibility problem where retraining a neural network after a minor (or even no) change causes a change to the predictions as prediction churn. Prediction churn can be a serious problem when testing and launching new versions of a neural network in a non-disruptive way inside an existing service.
+
+Model averaging is a very natural solution to prediction churn. By directly averaging away the variations in the training procedure the predictions will tend to be more consistent across retrains of the ensemble and from minor modifications to the base models.
+
+Given that codistillation achieves many of the benefits of model averaging, our hypothesis is that it should similarly help reduce prediction churn. To test this hypothesis, we trained a deep neural network (DNN) on the Criteo dataset and measured the mean absolute difference between the predictions of two retrains of the same model (prediction difference, for short). The prediction differences between different versions of a model should be at least as large as the prediction differences between two retrains of the same model and serve as a way of estimating the prediction churn. We also trained an ensemble of two copies of the initial DNN and then measured the prediction difference between retrains of the ensemble. Finally, we trained the same DNN using two-way codistillation, picking one of the copies arbitrarily to make predictions and measured the reproducibility of codistillation as well. As shown in table 1, codistillation reduces prediction churn by $3 5 \%$ , achieving similar results to ensembling, but does not increase serving costs.
+
+# 4 DISCUSSION AND FUTURE WORK
+
+Distillation is a surprisingly flexible tool, especially when performed during model training instead of after. It can be used to accelerate training, improve quality, distribute training in new, more communication efficient ways, and reduce prediction churn. However, there are still many questions we would like to explore. For example, we mostly focused on pairs of models codistilling from each other. It stands to reason that if pairs are useful then so are other topologies. Fully connected graphs might make the models too similar, too quickly so ring structures might also be interesting. We also did not explore the limits of how accurate the predictions from the teacher models have to be. It might be possible to aggressively quantize the teacher model to make codistillation almost as cheap as normal training even for very large models.
+
+It is somewhat paradoxical that bad models codistilling from each other can learn faster than models training independently. Somehow the mistakes made by the teacher model carry enough information to help the student model do better than the teacher, and better than just seeing the actual label in the data. Characterizing the ideal properties of a teacher model is an exciting avenue for future work.
+
+In this work we only extract predictions from the checkpoints, as predictions are identifiable and, unlike the internal structure of the networks, have no spurious symmetries. That said, it might be possible to extract more information from a checkpoint than just predictions without hitting the same issues faced by workers communicating gradients, allowing the use of the teacher models as a stronger regularizer. Perhaps distillation-based methods could be used to augment federated learning McMahan et al. (2017) in particularly bandwidth-constrained settings.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Avital Oliver for feedback on a draft and Oriol Vinyals for many helpful discussions. We would also like to thank Mohammad Norouzi for emotional support and Dan Hurt for essential help resolving last-minute computational resources issues.
+
+# REFERENCES
+
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+Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. CoRR, abs/1607.06450, 2016. URL https://arxiv.org/abs/1607.06450.   
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+Jeffrey Dean, Greg Corrado, Rajat Monga, Kai Chen, Matthieu Devin, Mark Mao, Andrew Senior, Paul Tucker, Ke Yang, Quoc V Le, et al. Large scale distributed deep networks. In Advances in neural information processing systems, pp. 1223–1231, 2012.   
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+H. Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera y Ar- ¨ cas. Communication-efficient learning of deep networks from decentralized data. In Aarti Singh and Xiaojin (Jerry) Zhu (eds.), Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, AISTATS 2017, 20-22 April 2017, Fort Lauderdale, FL, USA, volume 54 of Proceedings of Machine Learning Research, pp. 1273–1282. PMLR, 2017. URL http://proceedings.mlr.press/v54/mcmahan17a.html.
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+Gabriel Pereyra, George Tucker, Jan Chorowski, Łukasz Kaiser, and Geoffrey Hinton. Regularizing neural networks by penalizing confident output distributions. CoRR, abs/1701.06548, 2017. URL https://arxiv.org/abs/1701.06548.
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+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision (IJCV), 115(3):211–252, 2015. doi: 10.1007/s11263-015-0816-y.
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+    {
+        "type": "text",
+        "text": "LARGE SCALE DISTRIBUTED NEURAL NETWORKTRAINING THROUGH ONLINE DISTILLATION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Rohan Anil   \nGoogle   \nrohananil@google.com ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Gabriel Pereyra ∗ Google DeepMind pereyra@google.com ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Alexandre Passos Google Brain apassos@google.com ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Robert Ormandi Google ormandi@google.com ",
+        "bbox": [
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+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "George E. Dahl Google Brain gdahl@google.com ",
+        "bbox": [
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+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Geoffrey E. Hinton   \nGoogle Brain   \ngeoffhinton@google.com ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Techniques such as ensembling and distillation promise model quality improvements when paired with almost any base model. However, due to increased testtime cost (for ensembles) and increased complexity of the training pipeline (for distillation), these techniques are challenging to use in industrial settings. In this paper we explore a variant of distillation which is relatively straightforward to use as it does not require a complicated multi-stage setup or many new hyperparameters. Our first claim is that online distillation enables us to use extra parallelism to fit very large datasets about twice as fast. Crucially, we can still speed up training even after we have already reached the point at which additional parallelism provides no benefit for synchronous or asynchronous stochastic gradient descent. Two neural networks trained on disjoint subsets of the data can share knowledge by encouraging each model to agree with the predictions the other model would have made. These predictions can come from a stale version of the other model so they can be safely computed using weights that only rarely get transmitted. Our second claim is that online distillation is a cost-effective way to make the exact predictions of a model dramatically more reproducible. We support our claims using experiments on the Criteo Display Ad Challenge dataset, ImageNet, and the largest to-date dataset used for neural language modeling, containing $6 \\times 1 0 ^ { 1 1 }$ tokens and based on the Common Crawl repository of web data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "For large-scale, commercially valuable neural net training problems, practitioners would be willing to devote many more machines to training if it sped up training time dramatically or improved the quality of the final model. Currently, distributed stochastic gradient descent (SGD), in both its synchronous and asynchronous forms (Chen et al., 2016), is the dominant algorithm for large-scale neural network training across multiple interconnected machines. Unfortunately, as the number of machines increases, there are diminishing improvements to the time needed to train a high quality model, to a point where adding workers does not further improve training time. A combination of infrastructure limitations and optimization barriers constrain the scalability of distributed minibatch SGD. The overhead of communicating weight updates and the long tail of the machine and network latency distributions slow down execution and produce thorny engineering challenges. For the synchronous algorithm, there are rapidly diminishing returns from increasing the effective batch size (LeCun et al., 2012; Keskar et al., 2017). For the asynchronous algorithm, gradient interference from inconsistent weights can cause updates to thrash and even, in some cases, result in worse final accuracy or completely stall learning progress. The precise scalability limit for distributed SGD will depend on implementation details of the algorithm, specifics of the infrastructure, and the capabilities of the hardware, but in our experience it can be very difficult to scale effectively much beyond a hundred GPU workers in realistic setups. No algorithm for training neural nets will be infinitely scalable, but even scaling a bit beyond the limits of distributed SGD would be extremely valuable. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Once we have reached the limits of adding workers to distributed SGD, we could instead use extra machines to train another copy of the model and create an ensemble to improve accuracy (or trade this accuracy for training time by training the members of the ensemble for fewer steps). As an added benefit, the ensemble will make more stable and reproducible predictions, which can be useful in practical applications. However, ensembling increases the cost at test time, potentially violating latency or other cost constraints. Alternatively, to get nearly the same benefits of the ensemble without increasing test time costs, we can distill (Hinton et al., 2015; Bucila et al., 2006) an $n$ -way ensemble of models into a single still-servable model using a two-phase process: first we use $n M$ machines to train an n-way ensemble with distributed SGD and then use $M$ machines to train the servable student network to mimic the $n$ -way ensemble. By adding another phase to the training process and using more machines, distillation in general increases training time and complexity in return for a quality improvement close to the larger teacher ensemble model. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We believe that the additional training costs, in terms of both time and pipeline complexity, discourage practitioners from using ensemble distillation, even though it almost always would improve results. In this work, we describe a simpler online variant of distillation we call codistillation. Codistillation trains $n$ copies of a model in parallel by adding a term to the loss function of the ith model to match the average prediction of the other models. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Through large-scale experiments we show that, compared to distributed SGD, codistillation improves accuracy and speeds up training by allowing the productive use of more computational resources even beyond the point where adding more workers provides no additional speedup for SGD. Specifically, codistillation provides the benefits of distilling an ensemble of models without increasing training time. Codistillation is also quite simple to use in practice compared to a multi-phase distillation training procedure. Multi-phase distillation tends to encourage human intervention between the training phases to decide when to stop training the ensemble and start distilling it into a single model. We also show that codistillation does not lose the reproducibility benefits of ensembles of neural networks, reducing churn in the predictions of different retrains of the same model. Reducing prediction churn can be essential when testing and launching new versions of a model in a non-disruptive way in an existing service, although it is not as well-studied in the academic machine learning community. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Given the obvious relationship to distillation, very similar algorithms to codistillation have been independently described by multiple researchers. For example, Zhang et al. (2017) describes another simultaneous distillation algorithm but does not investigate the benefits in the distributed training case and only presents it as a potential quality improvement over regular distillation. We view the experimental validation of codistillation at scale as the primary contribution of our work. Another contribution of this work is our exploration of different design choices and implementation considerations for codistillation which we believe has produced recommendations of substantial practical utility. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In general, we believe the quality gains of codistillation over well-tuned offline distillation will be minor in practice and the more interesting research direction is exploring codistillation as a distributed training algorithm that uses an additional form of communication that is far more delay tolerant. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1.1 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In addition to the closely related work in Hinton et al. (2015) and Zhang et al. (2017) mentioned above, there are many different tactics for scaling up neural network training. Early work in training large distributed neural networks focused on schemes for partitioning networks over multiple cores, often referred to as model parallelism (Dean et al., 2012). As memory has increased on graphic processing units (GPUs), the majority of distributed training has shifted towards data parallelism, where the model is replicated across multiple machines and data are distributed to the different replicas, with updates being merged by parameter servers or a single allreduce step as in Goyal et al. (2017). Even without a high quality allreduce primitive, variants of centralized synchronous SGD with backup workers can scale to a large number of machines (Chen et al., 2016). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Methods like ensembling and distillation are mostly orthogonal to lower level distributed training infrastructure. However, mixture of experts models have particularly natural model parallelism that can be integrated with data parallelism and a synchronous training scheme. Gross et al. (2017) and Shazeer et al. (2017) are notable examples of recent work in this area. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As researchers try to scale neural network training to ever larger datasets and models, the optimization algorithm itself can be altered. For synchronous SGD there are rapidly diminishing returns (LeCun et al., 2012; Keskar et al., 2017) as the number of workers, and thus the effective batch size, increases and we might hope that algorithms like KFAC (Ba et al., 2017) would make better use of large batches. Although a promising direction for research, in this work we focus on what should hopefully be an optimization algorithm agnostic way to improve scalability and reproducibility. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 CODISTILLATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Distillation is a meta-algorithm which allows any algorithm to incorporate some of the model quality benefits of ensembles. The idea of distillation is to first train a teacher model, which traditionally is an ensemble or another high-capacity model, and then, once this teacher model is trained, train a student model with an additional term in the loss function which encourages its predictions to be similar to the predictions of the teacher model. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "There are many variants of distillation, for different types of teacher model, different types of loss function, and different choices for what dataset the student model trains on. For example, the student model could be trained on a large unlabeled dataset, on a held-out data set, or even on the original training set. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Perhaps surprisingly, distillation has benefits even if the teacher model and the student model are two instances of the same neural network (see section 3 for empirical evidence), as long as they are sufficiently different (say, by having different initializations and seeing the examples in a different order). Furthermore, the teacher model predictions are still beneficial to the student model even before convergence. Finally, the distinction between teacher and student is unnecessary and two or more models all distilling from each other can also be useful. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we use codistillation to refer to distillation performed: ",
+        "bbox": [
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+            537,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1. using the same architecture for all the models;   \n2. using the same dataset to train all the models; and   \n3. using the distillation loss during training before any model has fully converged. ",
+        "bbox": [
+            210,
+            565,
+            753,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For simplicity, we usually consider the case where all models have a distillation term in their loss function, but the key characteristic of codistillation is the simultaneous training of a model and its teacher. ",
+        "bbox": [
+            178,
+            638,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 presents the codistillation algorithm. The distillation loss term $\\psi$ can be the squared error between the logits of the models, the KL divergence between the predictive distributions, or some other measure of agreement between the model predictions. In this work we use the cross entropy error treating the teacher predictive distribution as soft targets. In the beginning of training, the distillation term in the loss is not very useful or may even be counterproductive, so to maintain model diversity longer and to avoid a complicated loss function schedule we only enable the distillation term in the loss function once training has gotten off the ground. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 CODISTILLATION AS A DISTRIBUTED NEURAL NETWORK TRAINING ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            806,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In order to scale beyond the limits of distributed stochastic gradient descent we will need an algorithm that is far more communication efficient. As seen in Algorithm 1, to update the parameters of one network using codistillation one only needs the predictions of the other networks, which can be computed locally from copies of the other networks weights. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "There are several reasons to believe that stale predictions might be much less of a problem than stale gradients for training: ",
+        "bbox": [
+            174,
+            895,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Codistillation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            354,
+            117
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Input loss function $\\phi ( 1 \\mathsf { a b e 1 }$ , prediction)   \nInput distillation loss function $\\psi$ (aggregated label, prediction)   \nInput prediction function F (θ, input)   \nInput learning rate $\\eta$   \nfor n burn in steps do for $\\theta _ { i }$ in model set do y, f $=$ get train example() $\\theta _ { i } = \\theta _ { i } - \\eta \\nabla _ { \\theta _ { i } } \\{ \\phi ( y , F ( \\theta _ { i } , f ) ) \\}$ end for   \nend for   \nwhile not converged do for $\\theta _ { i }$ in model set do y, f $=$ get train example() $\\begin{array} { r } { \\dot { \\theta _ { i } } = \\theta _ { i } - \\eta \\nabla _ { \\theta _ { i } } \\{ \\phi ( \\mathrm { y } , F ( \\theta _ { i } , f ) ) + \\psi ( \\{ \\frac { 1 } { N - 1 } \\sum _ { j \\neq i } F ( \\theta _ { j } , f ) \\} , F ( \\theta _ { i } , f ) ) \\} } \\end{array}$ end for   \nend while   \n1. every change in weights leads to a change in gradients, but as training progresses towards convergence, weight updates should substantially change only the predictions on a small subset of the training data;   \n2. weights (and gradients) are not statistically identifiable as different copies of the weights might have arbitrary scaling differences, permuted hidden units, or otherwise rotated or transformed hidden layer feature space so that averaging gradients does not make sense unless models are extremely similar;   \n3. sufficiently out-of-sync copies of the weights will have completely arbitrary differences that change the meaning of individual directions in feature space that are not distinguishable by measuring the loss on the training set;   \n4. in contrast, output units have a clear and consistent meaning enforced by the loss function and the training data. ",
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+        "type": "text",
+        "text": "Furthermore, the predictive distribution of radically different models can still provide very useful information about the relationship between inputs and outputs. Empirically we’ve found that using stale predictions instead of up-to-date predictions for the other neural networks has little to no adverse effect on the quality of the final trained model produced by codistillation. We have been able to use predictions tens of thousands of updates old in the asynchronous case or $8 0 0 \\mathrm { k }$ examples (i.e. 50 updates) old in the large-batch synchronous case. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The tolerance of distillation for stale teacher predictions suggests a distributed training strategy which is far less communication-intensive than synchronous or asynchronous SGD. ",
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+        "type": "text",
+        "text": "1. Each worker trains an independent version of the model on a locally available subset of the training data.   \n2. Occasionally, workers checkpoint their parameters.   \n3. Once this happens, other workers can load the freshest available checkpoints into memory and perform codistillation. ",
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+    {
+        "type": "text",
+        "text": "Of course there is no reason not to combine this strategy with standard distributed SGD, resulting in a procedure that employs independent groups of workers that exchange checkpoints between groups and exchange gradient information within a group. ",
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+        "type": "text",
+        "text": "In each iteration of synchronous/asynchronous distributed SGD, each worker needs to send and receive an amount of information proportional to the entire model size. When using codistillation to distribute training each worker only needs to very rarely read parameter checkpoints from the other models.1 When combining distributed SGD and codistillation, we can add workers to a group up until the point where we see diminishing returns from distributed SGD and then deploy additional workers in another group, occasionally exchanging checkpoints between the otherwise independent groups of workers. ",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Moreover, there is no need to use high-precision floating point numbers to store the parameters used to compute the predictions for the distillation loss term as distillation is not very sensitive to the exact values of the predictions. Therefore the additional computational cost of distributed codistillation will not be much higher than the cost of independent training. ",
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+    {
+        "type": "text",
+        "text": "Since the parameters of a model trained on a data set can be viewed as a very compressed representation of the aspects of that data set which are relevant to the learning problem at hand, it makes intuitive sense that leveraging these parameters might be more communication-efficient than sending all the data points or gradients. ",
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS AND RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In order to study the scalability of distributed training using codistillation, we need a task that is representative of important large-scale neural network training problems. Neural language modeling is an ideal test bed because vast quantities of text are available on the web and because neural language models can be very expensive to train. Neural language models are representative of important problems that make common use of distributed SGD like machine translation and speech recognition, but language modeling is simpler to evaluate and uses a simpler pipeline. In order to make any potential scalability improvements as clear as possible, we selected a data set large enough that it is completely infeasible to train an expressive model to convergence on it with existing SGD parallelization strategies. In order to confirm that our results were not specific to some peculiarity of language modeling, we also validated some of our large scale codistillation results on ImageNet (Russakovsky et al., 2015) as well. ",
+        "bbox": [
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+        "type": "text",
+        "text": "To demonstrate the benefits of codistillation in reducing prediction churn and to study other properties of the algorithm we can use smaller experiments that are cheaper to perform, but it is important to actually reach the limits of distributed SGD when studying scalability. ",
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+    {
+        "type": "text",
+        "text": "3.1 DATA SETS AND MODELS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Common Crawl is an open repository of web crawl data. We downloaded the WET files2 and filtered them to only include English language documents that contained long paragraphs because we wanted data that allowed modeling of slightly longer range dependencies than data sets that randomize sentence order like LM1B (Chelba et al., 2013). After preprocessing, roughly 915 million documents (20TB of text) remained. We plan to release the list of document ids that remained after filtering as well as code for our invertible tokenizer for others to use this data set. The language model we trained in all of our Common Crawl experiments was an RNN language model with two LSTM layers of 1024 units each with layer normalization (Ba et al., 2016). We used 256 dimensional input embeddings and a vocabulary of 24006 word pieces (Schuster & Nakajima, 2012), including sentence and paragraph start and end tokens, out of vocabulary (generally non-English characters), and end of document. After converting to word pieces there were 673 billion tokens, which is much larger than any previous neural language modeling data set we are aware of.3 During training we constructed batches 32 word pieces long drawing tokens from $B$ different documents at a time, saving hidden state across batches. Since the hidden state never gets reset, the model has to learn to use the end of document token to reset itself. We use the ADAM optimizer Kingma & Ba (2014) for all experiments on Common Crawl. ",
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+    },
+    {
+        "type": "text",
+        "text": "ImageNet is the most popular image classification benchmark of recent years. All of our experiments on ImageNet followed the setup of Goyal et al. (2017) as closely as possible and also used fully-synchronous SGD. We used the same learning rate scaling and schedule and used a configuration with batch size 16384 that achieves $7 5 \\%$ accuracy as our primary baseline. Goyal et al. (2017) reports that increasing the batch size beyond 8192 provides rapidly diminishing returns. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Criteo Display Ad Challenge dataset4 (Criteo) is a benchmark dataset for predicting click through rate for online advertising. The data contain roughly 43 million examples, each with 13 integer and 26 categorical input features. The task is formulated as binary classification and we train a feedforward fully connected neural network using ReLU activations with hidden layer sizes of 2560, 1024, 256 and a logistic output. We use the Adagrad optimizer with learning rate of 0.001 for training for all experiments on this dataset. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.2 REACHING THE LIMITS OF DISTRIBUTED SGD FOR TRAINING RNNS ON COMMON CRAWL ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In our first set of experiments, our goal was to approximately determine the maximum number of GPU workers that can be productively employed for SGD in our Common Crawl neural language model setup. Since our dataset is two orders of magnitude larger than English Wikipedia, there is no concern about revisiting data, which would make independent replicas more similar, even in relatively large scale experiments. ",
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+    {
+        "type": "text",
+        "text": "We tried asynchronous SGD with 32 and 128 workers, sharding the weights across increasing numbers of parameter servers as necessary to ensure that training speed was bottlenecked by GPU computation time. We found it very difficult to keep training stable and prevent the RNNs from diverging for asynchronous SGD with large numbers of workers. We experimented with a few worker ramp up schemes and different learning rates, but ultimately decided to focus on the synchronous algorithm to make our results less dependent on the specific characteristics of our infrastructure and implementation. Gradient staleness is hard to analyze independent of the specific conditions whereas differences in implementation and infrastructure are far easier to abstract away for synchronous SGD. Although it may have been possible to make async work well with more effort, the debilitating effect of stale gradients on learning progress is a well known issue, for instance Chen et al. (2016) demonstrated that synchronous SGD can often converge to a better final accuracy than asynchronous SGD. Mitliagkas et al. (2016) argues that asynchrony can effectively increase the momentum which is part of why it tends to diverge so easily. For these and other reasons, practitioners (e.g. Goyal et al. (2017)) seem to be moving away from asynchronous SGD towards synchronous training as the default. In preliminary experiments the gains from codistillation seemed independent of the choice of asynchronous or synchronous SGD as the base algorithm. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The maximum number of GPU workers that can be productively employed for synchronous SGD will depend on infrastructure limits, tail latency, and batch size effects. Fully synchronous SGD is equivalent to the single machine algorithm with a much larger batch size. Increasing the effective batch size reduces noise in the gradient estimates which allows larger step sizes with hopefully higher quality updates that result in faster convergence. Given effectively infinite training data (even with 256 GPUs we do not visit all of the Common Crawl training data) we intuitively would expect increasing the effective batch size to at worst increase the step time. We trained language models on Common Crawl with fully synchronous SGD using a per-worker batch size of 128 and 32, 64, 128, and 256 workers. Thus the effective batch size ranged from 4096 to 32768. Generally we should expect to need to increase the learning rate as we increase the effective batch size, so for each number of workers we tried learning rates of 0.1, 0.2, and 0.4. For 32 and 64 workers, 0.1 performed best and since none of the original three learning rates performed well for 256 workers, we also tried an additional intermediate learning rate of 0.3 which was the best performing learning rate for 256 workers. ",
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+    {
+        "type": "text",
+        "text": "Figure 1a plots the validation error as a function of global steps for the different numbers of workers we tried, using the best learning rate for each number of workers. Increasing the number of workers (and thus the effective batch size) reduced the number of steps required to reach the best validation error until 128 workers, at which point there was no additional improvement. Even with idealized perfect infrastructure, 256 workers would at best result in the same end to end training time on this problem. However, because steps can take so much longer with 256 workers, going from 128 to 256 workers is highly counterproductive in practice. Figure 1b plots validation error against wall time for the same varying numbers of synchronous workers. There is a large degradation in step time, and thus learning progress, at 256 workers. Although it might be possible to improve the step time at 256 workers by using a more sophisticated scheme with backup workers (Chen et al., 2016), the operative limit to scalability on this task is the diminishing return from increasing the effective batch size, not the degradation in step times. ",
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+    {
+        "type": "image",
+        "img_path": "images/52bb17f20408a8288b8c469d430282f12c5884bc1bf5b32f5dd8ae70d590ec1c.jpg",
+        "image_caption": [
+            "Figure 1: Synchronous training on Common Crawl dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "text": "In these particular experiments, synchronous SGD with 128 workers is the strongest baseline in terms of training time and final accuracy. Therefore we focus the rest of our experiments on comparisons with 128 worker synchronous SGD and study codistillation that uses synchronous SGD as a subroutine, although it also works well with the asynchronous algorithm. ",
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+    {
+        "type": "text",
+        "text": "3.3 CODISTILLATION WITH SYNCHRONOUS SGD ",
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+        "type": "text",
+        "text": "For language modeling on Common Crawl, synchronous SGD with 128 GPUs achieved the best results for standard distributed training, at least of the configurations we tried, and we were unable to improve training time with 256 GPUs. Although the extra GPUs do not seem to help basic synchronous SGD, our hypothesis is that the extra 128 GPUs will improve training time if we use two-way codistillation with two groups of 128 GPUs using synchronous SGD that exchange checkpoints periodically. ",
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+        "type": "text",
+        "text": "One concern would be that codistillation is merely a way of penalizing confident output distributions (Pereyra et al., 2017) or smoothing the labels, so we also compared to two label smoothing baselines. The first baseline replaces the distillation loss term with a term that matches the uniform distribution and the second uses a term that matches the unigram distribution. Trade-off hyperparameters were tuned by hand in preliminary experiments. ",
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+        "type": "text",
+        "text": "Another important comparison is to an ensemble of two neural networks, each trained with 128 GPUs and synchronous SGD. Although we are in general interested in the regime where such an ensemble would not be practical because of the increased test time costs, given our understanding of distillation we would expect codistillation, if it achieves all of the benefits of traditional distillation, to have a training curve close to—but slightly worse than—a two-way ensemble. In the case of twoway codistillation, this would provide evidence that the gains are really coming from an ensemblinglike effect despite never explicitly averaging the predictions of multiple models as would happen when distilling an ensemble model. ",
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+    {
+        "type": "text",
+        "text": "Figure 2a plots validation cross entropy versus step of synchronous training for codistillation using two groups of 128 GPUs along with training curves for the synchronous SGD and label smoothing baselines (each using 128 GPUs) and an ensemble of two instances of the 128 GPU baseline. All experiments in figure 2a used the learning rate found to be the best for the 128 GPU synchronous SGD baseline. Two-way codistillation successfully reduces training time substantially compared to the 128 worker baselines and almost achieves the training curve of the two-way ensemble. Measuring at the best validation error achieved by the baseline, codistillation reaches the same error in 2X fewer steps. Perhaps more importantly, codistillation reaches a lower final error so a 2X reduction in steps is likely an underestimate of the gains. In our implementation, for the model we trained, codistillation is free in terms of step time as the GPU is not fully utilized and our implementation automatically overlaps the computation of the teacher and student models. In the worst case, for a model that saturates the hardware that is implemented without quantization, prefetching predictions using the CPU, or other optimizations to compute the predictions, the extra forward pass might increase compute costs by nearly $50 \\%$ . However, even with these worst case assumptions, network costs will be a substantial contributor to the total step time, easily $50 \\% - 8 0 \\%$ , resulting in a modest increase in time per step. ",
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+    {
+        "type": "image",
+        "img_path": "images/974945ffbc8380f2d69a00cfae8fe969812d6a87a8fdea9e0930975268993c09.jpg",
+        "image_caption": [
+            "Figure 2: Codistillation (abbreviated as “Cod” in the legend) results with Common Crawl. "
+        ],
+        "image_footnote": [],
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+    {
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+        "image_caption": [
+            "Figure 3: Codistillation on ImageNet "
+        ],
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+        "img_path": "images/e6db93fc59573000cf34cce4583fd34b8eb2e9a81daebdb1c370f768ca6642e7.jpg",
+        "image_caption": [
+            "Figure 4: Reload intervals on Common Crawl "
+        ],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.3.1 CODISTILLATION REQUIRES FEWER STEPS ON IMAGENET ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In order to confirm our results are not due to the vicissitudes of language modeling or the particulars of our setup, we tried two-way codistillation on ImageNet as well. As can be seen from figure 3, codistillation (enabled after 3000 steps) achieves the same $7 5 \\%$ accuracy after a total of 5250 steps as the baseline does after 7250 steps. Eventually, two-way codistillation achieves a slightly better accuracy of $7 5 . 6 \\%$ at 7250 steps, confirming the utility of codistillation on ImageNet in addition to language modeling. ",
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+    {
+        "type": "text",
+        "text": "3.3.2 PARTITIONING THE DATASET ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The gains from codistillation have a variety of potential causes that we would like to tease apart. From the experiments above, we have evidence against label smoothing effects as a cause. Another potential source of the training time and quality gains from codistillation over basic SGD would be that the different codistilling models see different training data. To test this hypothesis, we ran another codistillation experiment using two groups of 128 workers, but forced the two groups to use the same training data instead of using different random subsets.5 Figure 2b compares codistillation using different subsets of the training data to codistillation using the same data. Codistillation with the same data seems to be slightly better than the baseline, but codistillation using different data gets much better results. These results show that the codistilling models are indeed successfully transmitting useful information about different parts of the training data to each other. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "3.4 PREDICTION STALENESS SENSITIVITY ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In general, codistillation can handle relatively stale predictions. SGD generally cannot tolerate gradients nearly as stale. The contrast is most stark with the asynchronous algorithm. We trained our two-way synchronous codistillation setup on Common Crawl with several different checkpoint exchange frequencies. We tried exchange delays of 50, 100, and 250 steps. As can be seen from figure 4, increasing the checkpoint reload interval beyond 819,200 examples or 50 steps slightly degrades the learning curve. With an interval of 50 steps, communicating checkpoints with a shared file system is still quite feasible on most problems. ",
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+    {
+        "type": "text",
+        "text": "3.4.1 CODISTILLATION VS MULTI-PHASE DISTILLATION VARIANTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In our experiments and experience, the choice between multi-phase distillation and codistillation makes very little difference in terms of the quality improvements achieved by distillation, although it does affect training time. On Common Crawl, we trained an ensemble of two models for 18K steps and then trained an identical model distilling from the ensemble to reach a cross entropy error of 4.0 after 9K steps for a total of 27K steps. However, two-way codistillation reached roughly the same validation error after only 10k steps. Zhang et al. (2017) reported a benefit in quality over basic distillation, but they compare distilling model $M _ { 1 }$ into model $M _ { 2 }$ with training model $M _ { 1 }$ and model $M _ { 2 }$ using codistillation; they do not compare to distilling an ensemble of models $M _ { 1 }$ and $M _ { 2 }$ into model $M _ { 3 }$ . Furthermore, Zhang et al. (2017) only reports final validation error using a large number of training steps on two small image datasets. Overfitting in the teacher model can explain the worse performance of offline distillation Zhang et al. (2017) report on CIFAR-100. We reproduced their experiments with WideResnet-28-10 teaching Resnet-32 on CIFAR-100. When we select a checkpoint with nearly $100 \\%$ training accuracy, we reproduce the $6 9 . 5 \\%$ they report in table 4. However, when we select a different checkpoint, we can achieve the $7 0 . 7 \\%$ they report for online distillation using traditional offline distillation. For distillation to provide value, the teacher must provide information beyond the training label. Thus a teacher network that overfits the training set will not be useful. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The gains in training time of codistillation over multi-phase distillation variants are obvious, but the reduction in training pipeline complexity codistillation provides can be equally important. Using the same architecture and dataset for all the models avoids squaring the tuning problem. The codistillation protocol simplifies the choice of teacher model and restores symmetry between the various models. With traditional multi-phase distillation one must decide which teacher model or models to use and how long to train them, encouraging human intervention between the phases. If teacher models occasionally get reused across versions of the student model in an effort to save computation, rollbacks of models to deal with bugs or data corruption can be dramatically more painful. To reproduce a given final model, one needs the entire history of teacher models and everything required to reproduce them which can easily result in what Sculley et al. (2014) refers to as “pipeline jungle” and unnecessary, undesirable data dependencies. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c1b0d6c14b4110caf054bc68f9098953609fde6878aeeb274cd732ed80c5b92b.jpg",
+        "table_caption": [
+            "Table 1: Prediction Churn "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Validation Log Loss</td><td>Mean Absolute Difference6 Between Retrains</td></tr><tr><td></td><td></td><td></td></tr><tr><td>DNN</td><td>0.4480 ± 0.001</td><td>0.029 ± 0.001</td></tr><tr><td>Ensemble of Two DNNs</td><td>0.4461 ± 0.0002</td><td>0.022 ± 0.002</td></tr><tr><td>Two-way codistilled DNN</td><td>0.4458 ± 0.002</td><td>0.019 ± 0.002</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3.5 REDUCING PREDICTION CHURN WITH CODISTILLATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Unlike linear models with convex loss functions, two neural networks with the same architecture that are trained on the same data can achieve similar validation and test performance while making very different predictions, and mistakes. On large datasets with a stable training pipeline aggregate metrics can be relatively consistent, but minor changes to the model architecture or even simple retrains can cause comparatively dramatic changes in the predictions made by the network. The network will in general get different examples correct and the differences might be especially severe on atypical examples with rare features. The weights learned by stochastic gradient descent in the non-convex setting will depend on the initialization, data presentation order, and the general vicissitudes of the infrastructure, especially when parallelization is involved. It is not practical to control all these nuisance variables and, even if it was, we would still see different solutions after making slight changes to the input representation or model architecture. We will refer to the general reproducibility problem where retraining a neural network after a minor (or even no) change causes a change to the predictions as prediction churn. Prediction churn can be a serious problem when testing and launching new versions of a neural network in a non-disruptive way inside an existing service. ",
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+    {
+        "type": "text",
+        "text": "Model averaging is a very natural solution to prediction churn. By directly averaging away the variations in the training procedure the predictions will tend to be more consistent across retrains of the ensemble and from minor modifications to the base models. ",
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+    {
+        "type": "text",
+        "text": "Given that codistillation achieves many of the benefits of model averaging, our hypothesis is that it should similarly help reduce prediction churn. To test this hypothesis, we trained a deep neural network (DNN) on the Criteo dataset and measured the mean absolute difference between the predictions of two retrains of the same model (prediction difference, for short). The prediction differences between different versions of a model should be at least as large as the prediction differences between two retrains of the same model and serve as a way of estimating the prediction churn. We also trained an ensemble of two copies of the initial DNN and then measured the prediction difference between retrains of the ensemble. Finally, we trained the same DNN using two-way codistillation, picking one of the copies arbitrarily to make predictions and measured the reproducibility of codistillation as well. As shown in table 1, codistillation reduces prediction churn by $3 5 \\%$ , achieving similar results to ensembling, but does not increase serving costs. ",
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+    {
+        "type": "text",
+        "text": "4 DISCUSSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Distillation is a surprisingly flexible tool, especially when performed during model training instead of after. It can be used to accelerate training, improve quality, distribute training in new, more communication efficient ways, and reduce prediction churn. However, there are still many questions we would like to explore. For example, we mostly focused on pairs of models codistilling from each other. It stands to reason that if pairs are useful then so are other topologies. Fully connected graphs might make the models too similar, too quickly so ring structures might also be interesting. We also did not explore the limits of how accurate the predictions from the teacher models have to be. It might be possible to aggressively quantize the teacher model to make codistillation almost as cheap as normal training even for very large models. ",
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+    {
+        "type": "text",
+        "text": "It is somewhat paradoxical that bad models codistilling from each other can learn faster than models training independently. Somehow the mistakes made by the teacher model carry enough information to help the student model do better than the teacher, and better than just seeing the actual label in the data. Characterizing the ideal properties of a teacher model is an exciting avenue for future work. ",
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+    {
+        "type": "text",
+        "text": "In this work we only extract predictions from the checkpoints, as predictions are identifiable and, unlike the internal structure of the networks, have no spurious symmetries. That said, it might be possible to extract more information from a checkpoint than just predictions without hitting the same issues faced by workers communicating gradients, allowing the use of the teacher models as a stronger regularizer. Perhaps distillation-based methods could be used to augment federated learning McMahan et al. (2017) in particularly bandwidth-constrained settings. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Avital Oliver for feedback on a draft and Oriol Vinyals for many helpful discussions. We would also like to thank Mohammad Norouzi for emotional support and Dan Hurt for essential help resolving last-minute computational resources issues. ",
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+    {
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