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+# SELF-SUPERVISED LEARNING FROM A MULTI-VIEW PERSPECTIVE
+
+Yao-Hung Hubert Tsai, Yue Wu, Ruslan Salakhutdinov, Louis-Philippe Morency Machine Learning Department, Carnegie Mellon University
+
+# ABSTRACT
+
+As a subset of unsupervised representation learning, self-supervised representation learning adopts self-defined signals as supervision and uses the learned representation for downstream tasks, such as object detection and image captioning. Many proposed approaches for self-supervised learning follow naturally a multi-view perspective, where the input (e.g., original images) and the self-supervised signals (e.g., augmented images) can be seen as two redundant views of the data. Building from this multi-view perspective, this paper provides an information-theoretical framework to better understand the properties that encourage successful self-supervised learning. Specifically, we demonstrate that self-supervised learned representations can extract task-relevant information and discard task-irrelevant information. Our theoretical framework paves the way to a larger space of self-supervised learning objective design. In particular, we propose a composite objective that bridges the gap between prior contrastive and predictive learning objectives, and introduce an additional objective term to discard task-irrelevant information. To verify our analysis, we conduct controlled experiments to evaluate the impact of the composite objectives. We also explore our framework’s empirical generalization beyond the multi-view perspective, where the cross-view redundancy may not be clearly observed.
+
+# 1 INTRODUCTION
+
+Self-supervised learning (SSL) (Zhang et al., 2016; Devlin et al., 2018; Oord et al., 2018; Tian et al., 2019) learns representations using a proxy objective (i.e., SSL objective) between inputs and self-defined signals. Empirical evidence suggests that the learned representations can generalize well to a wide range of downstream tasks, even when the SSL objective has not utilize any downstream supervision during training. For example, SimCLR (Chen et al., 2020) defines a contrastive loss (i.e., an SSL objective) between images with different augmentations (i.e., one as the input and the other as the self-supervised signal). Then, one can take SimCLR as features extractor and adopt the features to various computer vision applications, spanning image classification, object detection, instance segmentation, and pose estimation (He et al., 2019). Despite success in practice, only a few work (Arora et al., 2019; Lee et al., 2020; Tosh et al., 2020) provide theoretical insights into the learning efficacy of SSL. Our work shares a similar goal to explain the success of SSL, from the perspectives of Information Theory (Cover & Thomas, 2012) and multi-view representation1.
+
+To understand (a subset2 of) SSL, we start by the following multi-view assumption. First, we regard the input and the self-supervised signals as two corresponding views of the data. Using our running example, in SimCLR (Chen et al., 2020), the augmented images (i.e., the input and the self-supervised signal) are an image with different views. Second, we adopt a common assumption in multi-view learning: either view alone is (approximately) sufficient for the downstream tasks (see Assumption 1 in prior work (Sridharan & Kakade, 2008)). The assumption suggests that the image augmentations (e.g., changing the style of an image) should not affect the labels of images, or analogously, the selfsupervised signal contains most (if not all) of the information that the input has about the downstream tasks. With this assumption, our first contribution is to formally show that the self-supervised learned representations can 1) extract all the task-relevant information (from the input) with a potential loss; and 2) discard all the task-irrelevant information (from the input) with a fixed gap. Then, using classification task as an example, we are able the quantify the smallest generalization error (Bayes error rate) given the discussed task-relevant and task-irrelevant information.
+
+![](images/8ba86b50210c1e8610db6c944c31b6b6024b12623edbd286cb4e66f5bb2426d7.jpg)  
+Figure 1: High-level takeaways for our main results using information diagrams. (a) We present to learn minimal and sufficient self-supervision: minimize $H ( Z _ { X } | \dot { S } )$ for discarding task-irrelevant information and maximize $I ( Z _ { X } ; S )$ for extracting task-relevant information. (b) The resulting learned representation $Z _ { X } { ^ * }$ contains all task relevant information from the input with a potential loss $\epsilon _ { \mathrm { i n f o } }$ and discards task-irrelevant information with a fixed gap $I ( X ; S | T )$ . (c) Our core assumption: the self-supervised signal is approximately redundant to the input for the task-relevant information.
+
+As the second contribution, our analysis 1) connects prior arts for SSL on contrastive (Oord et al., 2018; Bachman et al., 2019; Chen et al., 2020; Tian et al., 2019) and predictive learning (Zhang et al., 2016; Vondrick et al., 2016; Tulyakov et al., 2018; Devlin et al., 2018) approaches; and 2) paves the way to a larger space of composing SSL objectives to extract task-relevant and discard task-irrelevant information simultaneously. For instance, the combination between the contrastive and predictive learning approaches achieves better performance than contrastive- or predictive-alone objective and enjoys less over-fitting problem. We also present a new objective to discard task-irrelevant information. The objective can be easily incorporated with prior self-supervised learning objectives.
+
+We conduct controlled experiments on visual (the first set) and visual-textual (the second set) selfsupervised representation learning. The first set of experiments are performed when the multi-view assumption is likely to hold. The goal is to compare different compositions of SSL objectives on extracting task-relevant and discarding task-irrelevant information. The second set of experiments are performed when the input and the self-supervised signal lie in very different modalities. Under this cross-modality setting, the task-relevant information may not mostly lie in the shared information between the input and the self-supervised signal. The goal is to examine SSL objectives’ generalization, where the multi-view assumption is likely to fail.
+
+# 2 A MULTI-VIEW INFORMATION-THEORETICAL FRAMEWORK
+
+Notations. For the input, we denote its random variable as $X$ , sample space as $\mathcal { X }$ , and outcome as $x$ . We learn a representation $( Z _ { X } / \mathcal { Z } / z _ { x } )$ from the input through a deterministic mapping $F _ { X }$ : $Z _ { X } = F _ { X } ( X )$ . For the self-supervised signal, we denote its random variable/ sample space/ outcome as $S / \ S / \ s$ . Two sample spaces can be different between the input and the self-supervised signal: $\mathcal { X } \neq \mathcal { S }$ . The information required for downstream tasks is referred to as “task-relevant information”: $T / \tau / t$ . Note that SSL has no access to the task-relevant information. Lastly, we use $I ( A ; B )$ to represent mutual information, $I ( A ; B | C )$ to represent conditional mutual information, $H ( A )$ to represent the entropy, and $H ( A | B )$ to represent conditional entropy for random variables $A / B / C$ . We provide high-level takeaways for our main results in Figure 1. We defer all proofs to Supplementary.
+
+# 2.1 MULTI-VIEW ASSUMPTION
+
+In our paper, we regard the input $( X )$ and the self-supervised signals $( S )$ as two views of the data. Here, we provide a table showing different $X / S$ in various SSL frameworks:
+
+<table><tr><td>Framework</td><td>BERT(Devlin et al.,2018)</td><td>Look &amp; Listen (Arandjelovic &amp; Zisserman, 2017)</td><td>SimCLR (Chen et al., 2020)</td><td>Colorization (Zhang et al.,2016)</td></tr><tr><td>Inputs (X)</td><td>Non-masked Words</td><td>Image</td><td>Image</td><td>Image Lightness</td></tr><tr><td>Self-supervised Signals (S)</td><td>Masked Words</td><td>Audio Stream</td><td>Same Image with Augmentation</td><td>Image Color</td></tr></table>
+
+We note that not all SSL frameworks realize the inputs and the self-supervised signals as corresponding views. For instance, Jigsaw puzzle (Noroozi & Favaro, 2016) considers (shuffled) image patches as the input and the positions of the patches as the self-supervised signals. Another example is Learning by Predicting Rotations (Gidaris et al., 2018), which considers an image (rotating with a specific angle) as the input and the rotation angle of the image as the self-supervised signal. We point out that the frameworks that regard $X / S$ as two corresponding views (Chen et al.; 2020; He et al., 2019) have a much better empirical downstream performance than the frameworks that do not (Noroozi & Favaro, 2016; Gidaris et al., 2018). Our paper hence focuses on the multi-view setting between $X / S$ .
+
+Next, we adopt the common assumption (i.e., multi-view assumption (Sridharan & Kakade, 2008; Xu et al., 2013)) in the multi-view learning between the input and the self-supervised signal:
+
+Assumption 1 (Multi-view, restating Assumption 1 in prior work (Sridharan & Kakade, 2008)). The self-supervised signal is approximately redundant to the input for the task-relevant information. In other words, there exist an $\epsilon _ { \mathrm { i n f o } } > 0$ such that $I ( X ; T | S ) \le \epsilon _ { \mathrm { i n f o } }$ .
+
+Assumption 1 states that, when $\epsilon _ { \mathrm { i n f o } }$ is small, the task-relevant information lies mostly in the shared information between the input and the self-supervised signals. We argue this assumption is mild with the following example. For self-supervised visual contrastive learning (Hjelm et al., 2018; Chen et al., 2020), the input and the self-supervised signal are the same image with different augmentations. Using image augmentations can be seen as changing the style of an image while not affecting the content. And we argue that the information required for downstream tasks should only be retained in the content but not the style. Next, we point out the failure cases of the assumption (or have large $\epsilon _ { \mathrm { i n f o . } }$ ): the input and the self-supervised signal contain very different task-relevant information. For instance, a drastic image augmentation (e.g., adding large noise) may change the content of the image (e.g., the noise completely occludes the objects). Another example is BERT (Devlin et al., 2018), with too much masking, downstream information may exist differently in the masked (i.e., the self-supervised signals) and the non-masked (i.e., the input) words. Analogously, too much masking makes the non-masked words have insufficient context to predict the masked words.
+
+# 2.2 LEARNING MINIMAL AND SUFFICIENT REPRESENTATIONS FOR SELF-SUPERVISION
+
+We start by discussing the supervised representation learning. The Information Bottleneck (IB) method (Tishby et al., 2000; Achille & Soatto, 2018) generalizes minimal sufficient statistics to the representations that are minimal (i.e., less complexity) and sufficient (i.e., better fidelity). To learn such representations for downstream supervision, we consider the following objectives:
+
+Definition 1 (Minimal and Sufficient Representations for Downstream Supervision). Let sup $Z _ { X } ^ { \mathrm { s u p } }$ be the sufficient supervised representation and $Z _ { X } ^ { \mathrm { s u p } _ { \mathrm { m i n } } }$ X be the minimal and sufficient representation:
+
+To reduce the complexity of the representation $Z _ { X }$ , the prior methods (Tishby et al., 2000; Achille & Soatto, 2018) presented to minimize $I ( Z _ { X } ; X )$ while ours presents to minimize $H ( Z _ { X } | T )$ . We provide a justification: minimizing $H ( Z _ { X } | T )$ reduces the randomness from $T$ to $Z _ { X }$ , and the randomness is regarded as a form of incompressibility (Calude, 2013). Hence, minimizing $H ( Z _ { X } | T )$ leads to a more compressed representation (discarding redundant information)3. Note that we do not constrain the downstream task $T$ as classification, regression, or clustering.
+
+Then, we present SSL objectives to learn sufficient (and minimal) representations for self-supervision: Definition 2 (Minimal and Sufficient Representations for Self-supervision). Let $Z _ { X } ^ { \mathrm { s s l } }$ be the sufficient self-supervised representation and $Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ be the minimal and sufficient representation:
+
+$Z _ { X } ^ { \mathrm { s s l } } = \operatorname * { a r g m a x } _ { Z _ { X } } I ( Z _ { X } ; S )$ and $Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } = \underset { Z _ { X } } { \arg \operatorname* { m i n } } H ( Z _ { X } | S ) \mathrm { s . t . } I ( Z _ { X } ; S )$ is maximized.
+
+Definition 2 defines our self-supervised representation learning strategy. Now, we are ready to associate the supervised and self-supervised learned representations:
+
+Theorem 1 (Task-relevant information with a potential loss $\epsilon _ { \mathrm { i n f o } }$ ). The supervised learned representations (i.e., $Z _ { X } ^ { \mathrm { s u p } }$ and $Z _ { X } ^ { \mathrm { s u p } _ { \mathrm { m i n } } }$ info) contain all the task-relevant information in the input (i.e., $I ( X ; T ) )$ The self-supervised learned representations (i.e., $Z _ { X } ^ { \mathrm { s s l } }$ and $Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ ) contain all the task-relevant information in the input with a potential loss $\epsilon _ { \mathrm { i n f o } }$ . Formally,
+
+$$
+I ( X ; T ) = I ( Z _ { X } ^ { \mathrm { s u p } } ; T ) = I ( Z _ { X } ^ { \mathrm { s u p } } { \mathrm { m i n } } ; T ) \geq I ( Z _ { X } ^ { \mathrm { s s l } } ; T ) \geq I ( Z _ { X } ^ { \mathrm { s s l } } { \mathrm { m i n } } ; T ) \geq I ( X ; T ) - \epsilon _ { \mathrm { i n f o } } .
+$$
+
+![](images/26b88fdc2d35982b61cb914a62b544ad47293a5601d0d574a6bd66bec6dd85b7.jpg)  
+Figure 2: Remarks on contrastive and predictive learning objectives for self-supervised learning. Between the representation $Z _ { X }$ and the self-supervised signal $S$ , contrastive objective performs mutual information maximization and predictive objectives perform log conditional likelihood maximization. We show that the SSL objectives aim at extracting task-relevant and discarding task-irrelevant information. Last, we summarize the computational blocks for practical deployments for these objectives.
+
+When $\epsilon _ { \mathrm { i n f o } }$ is small, Theorem 1 indicates that the self-supervised learned representations can extract almost as much task-relevant information as the supervised one. While when $\epsilon _ { \mathrm { i n f o } }$ is non-trivial, the learned representations may not always lead to good downstream performance. This result has also been observed in prior work (Tschannen et al., 2019) and InfoMin (Tian et al., 2020), which claim the representations with maximal mutual information may not have the best performance.
+
+Theorem 2 (Task-irrelevant information with a fixed compression gap $I ( X ; S | T ) )$ . The sufficient self-supervised representation (i.e., $I ( Z _ { X } ^ { \mathrm { s s l } } ; T ) )$ contains more task-irrelevant information in the input than the sufficient and minimal self-supervised representation (i.e., $I ( Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; T ) )$ . The latter contains an amount of the information, $I ( X ; S | T )$ , that cannot be discarded from the input. Formally,
+
+$$
+I ( Z _ { X } ^ { \mathrm { s s l } } ; X | T ) = I ( X ; S | T ) + I ( Z _ { X } ^ { \mathrm { s s l } } ; X | S , T ) \geq I ( Z _ { X } ^ { \mathrm { s s l } _ { \operatorname* { m i n } } } ; X | T ) = I ( X ; S | T ) \geq I ( Z _ { X } ^ { \mathrm { s u p } _ { \operatorname* { m i n } } } ; X | T ) = 0 .
+$$
+
+Theorem 2 indicates that a compression gap (i.e., $I ( X ; S | T ) )$ exists when we discard the taskirrelevant information from the input. To be specific, $I ( X ; S | T )$ is the amount of the shared information between the input and the self-supervised signal excluding the task-relevant information. Hence, $I ( X ; S | T )$ would be large if the downstream tasks requires only a portion of the shared information.
+
+2.3 CONNECTIONS WITH CONTRASTIVE AND PREDICTIVE LEARNING OBJECTIVES
+
+Theorem 1 and 2 state that our self-supervised learning strategies (i.e., min $H ( Z _ { X } | S )$ and max $I ( Z _ { X } ; S )$ defined in Definition 2) can extract task-relevant and discard task-irrelevant information. A question emerges:
+
+“What are the practical aspects of the presented self-supervised learning strategies?”
+
+To answer this question, we present 1) the connections with prior SSL objectives, especially for contrastive (Oord et al., 2018; Bachman et al., 2019; Chen et al., 2020; Tian et al., 2019; Hjelm et al., 2018; He et al., 2019) and predictive (Zhang et al., 2016; Pathak et al., 2016; Vondrick et al., 2016; Tulyakov et al., 2018; Peters et al., 2018; Devlin et al., 2018) learning objectives, showing that these objectives are extracting task-relevant information; and 2) a new inverse predictive learning objective to discard task-irrelevant information. We illustrate important remarks in Figure 2.
+
+Contrastive Learning (is extracting task-relevant information). Contrastive learning objective (Oord et al., 2018) maximizes the dependency/contrastiveness between the learned representation $Z _ { X }$ and the self-supervised signal $S$ , which suggests maximizing the the mutual information $I ( Z _ { X } ; S )$ . Theorem 1 suggests that maximizing $I ( Z _ { X } ; S )$ results in $Z _ { X }$ containing (approximately) all the information required for the downstream tasks from the input $X$ . To deploy the contrastive learning objective, we suggest contrastive predictive coding (CPC) (Oord et al., $2 \dot { 0 } 1 8 )$ , which is a mutual information lower bound with low variance (Poole et al., 2019; Song & Ermon, 2019):
+
+$$
+\mathrm { L } _ { C L } : = \operatorname* { m a x } _ { z _ { S } = F _ { S } ( S ) , \atop { z _ { X } = F _ { X } ( X ) , G } } \mathbb { E } _ { ( z _ { s 1 } , z _ { X 1 } ) , \cdots , ( z _ { s n } , z _ { X n } ) \sim P ^ { n } ( Z _ { S } , Z _ { X } ) } \left[ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \log \frac { e ^ { \langle G ( z _ { x 1 } ) , G ( z _ { s i } ) \rangle } } { \frac { 1 } { n } \sum _ { j = 1 } ^ { n } e ^ { \langle G ( z _ { x i } ) , G ( z _ { s j } ) \rangle } } \right] ,
+$$
+
+where $F _ { S } : S  \mathcal { Z }$ is a deterministic mapping and $G$ is a project head that projects a representation in $\mathcal { Z }$ into a lower-dimensional vector. If the input and self-supervised signals share the same sample space, i.e., $\mathcal { X } = \mathcal { S }$ , we can impose $F _ { X } = F _ { S }$ (e.g., self-supervised visual representation learning (Chen et al., 2020)). The projection head, $G$ , can be an identity, a linear, or a non-linear mapping. Last, we note that modeling equation 1 often requires a large batch size (e.g., large $n$ in equation 1) to ensure a good downstream performance (He et al., 2019; Chen et al., 2020).
+
+Forward Predictive Learning (is extracting task-relevant information). Forward predictive learning encourages the learned representation $Z _ { X }$ to reconstruct the self-supervised signal $S$ , which suggests maximizing the log conditional likelihood $\mathbb { E } _ { P _ { S , Z _ { X } } } [ \log P ( S | Z _ { X } ) ]$ . By the chain rule, $I ( Z _ { X } ; S ) = H ( S ) - H ( S | Z _ { X } )$ , where $H ( S )$ is irrelevant to $Z _ { X }$ . Hence, maximizing $I ( Z _ { X } ; S )$ is equivalent to maximizing $- H ( S | Z _ { X } ) = \mathbb { E } _ { P _ { S , Z _ { X } } } [ \log P ( S | Z _ { X } ) ]$ , which is the predictive learning objective. Together with Theorem 1, if $z _ { x }$ can perfectly reconstruct $s$ for any $( s , z _ { x } ) \sim P _ { S , Z _ { X } }$ , then $Z _ { X }$ contains (approximately) all the information required for the downstream tasks from the input $X$ . A common approach to avoid intractability in computing $\mathbb { E } _ { P _ { S , Z _ { X } } } [ \log P ( S | Z _ { X } ) ]$ is assuming a variational distribution $Q _ { \phi } ( S | Z _ { X } )$ with $\phi$ representing the parameters in $Q _ { \phi } ( \cdot | \cdot )$ . Specifically, we present to maximize $\mathbb { E } _ { P _ { S , Z _ { X } } } [ \log Q _ { \phi } ( S | Z _ { X } ) ]$ , which is a lower bound of $\mathbb { E } _ { P _ { S , Z _ { X } } } [ \log P ( S | Z _ { X } ) ] ^ { 5 }$ . $Q _ { \phi } ( \cdot | \cdot )$ can be any distribution such as Gaussian or Laplacian and $\phi$ can be a linear model, a kernel method, or a neural network. Note that the choice of the reconstruction type of loss depends on the distribution type of $Q _ { \phi } ( \cdot | \cdot )$ , and is not fixed. For instance, if we let $Q _ { \phi } ( S | Z _ { X } )$ be Gaussian $\mathcal { N } \Big ( S | R ( Z _ { X } ) , \sigma \mathbf { I } \Big )$ with $\sigma \mathbf { I }$ as a diagonal matrix6, the objective becomes:
+
+$$
+\mathrm { L } _ { F P } : = \operatorname* { m a x } _ { Z _ { X } = F _ { X } ( X ) , R } \mathbb { E } _ { s , z _ { x } \sim P _ { S } , z _ { X } } \Bigg [ - \left\| s - R ( z _ { x } ) \right\| _ { 2 } ^ { 2 } \Bigg ] ,
+$$
+
+where $R : \mathcal { Z } \to S$ is a deterministic mapping to reconstruct $S$ from $Z$ and we ignore the constants derived from the Gaussian distribution. Last, in most real-world applications, the self-supervised signal $S$ has a much higher dimension (e.g., a $2 2 4 \times 2 2 4 \times 3$ image) than the representation $Z _ { X }$ (e.g., a 64-dim. vector). Hence, modeling a conditional generative model $Q _ { \phi } ( S | Z _ { X } )$ will be challenging.
+
+Inverse Predictive Learning (is discarding task-irrelevant information). Inverse predictive learning encourages the self-supervised signal $S$ to reconstruct the learned representation $Z _ { X }$ , which suggests maximizing the log conditional likelihood $\mathbb { E } _ { P _ { S } , z _ { X } } \left[ \log P ( Z _ { X } | S ) \right]$ . Given Theorem 2 together with $- H ( Z _ { X } | S ) = \mathbb { E } _ { P _ { S , Z _ { X } } } [ \log P ( Z _ { X } | S ) ]$ , we know if $s$ can perfectly reconstruct $z _ { x }$ for any $( s , z _ { x } ) \sim P _ { S , Z _ { X } }$ under the constraint that $I ( Z _ { X } ; S )$ is maximized, then $Z _ { X }$ discards the task-irrelevant information, excluding $I ( X ; S | T )$ . Similar to the forward predictive learning, we use $\mathbb { E } _ { P _ { S , Z _ { X } } } [ \log Q _ { \phi } ( Z _ { X } | S ) ]$ as a lower bound of $\mathbb { E } _ { P _ { S , Z _ { X } } } [ \log P ( Z _ { X } | S ) ]$ . In our deployment, we take the advantage of the design in equation 1 and let $Q _ { \phi } ( Z _ { X } | S )$ be Gaussian $\mathcal { N } \Big ( Z _ { X } | F _ { S } ( S ) , \sigma \mathbf { I } \Big )$ :
+
+$$
+\mathrm { L } _ { I P } : = \operatorname* { m a x } _ { Z _ { S } = F _ { S } ( S ) , Z _ { X } = F _ { X } ( X ) } \mathbb { E } _ { z _ { s } , z _ { x } \sim P _ { Z _ { S } } , z _ { X } } \left[ - \left. z _ { x } - z _ { s } \right. _ { 2 } ^ { 2 } \right] .
+$$
+
+Note that optimizing equation 3 alone results in a degenerated solution, e.g., learning $Z _ { X }$ and $Z _ { S }$ to be the same constant.
+
+Composing SSL Objectives (to extract task-relevant and discard task-irrelevant information simultaneously). So far, we discussed how prior self-supervised learning approaches extract taskrelevant information via the contrastive or the forward predictive learning objectives. Our analysis also inspires a new loss, the inverse predictive learning objective, to discard task-irrelevant information. Now, We present a composite loss to combine them together:
+
+$$
+\mathrm { L } _ { S S L } = \lambda _ { C L } \mathrm { L } _ { C L } + \lambda _ { F P } \mathrm { L } _ { F P } + \lambda _ { I P } \mathrm { L } _ { I P } ,
+$$
+
+where $\lambda _ { C L }$ , $\lambda _ { F P }$ , and $\lambda _ { I P }$ are hyper-parameters. This composite loss enables us to extract taskrelevant and discard task-irrelevant information simultaneously.
+
+In last subsection, we see the practical aspects of our designed SSL strategies. Now, we provide an theoretical analysis on the representations’ generalization error when $T$ is a categorical variable . We use Bayes error rate as an example, which stands for the irreducible error (smallest generalization error (Feder & Merhav, 1994)) when learning an arbitrary classifier from the representation to infer the labels. In specific, let $P _ { e }$ be the Bayes error rate of arbitrary learned representations $Z _ { X }$ and $\hat { T }$ as the estimation for $T$ from our classifier, $P _ { e } : = \mathbb { E } _ { z _ { x } \sim P _ { z _ { X } } } [ 1 - \operatorname* { m a x } _ { t \in T } P ( \hat { T } = t | z _ { x } ) ]$ .
+
+estimation using empirical samples from distribution To begin with, we present a general form of sample complexity with mutual information $P _ { Z _ { X } , S }$ . Let $P _ { Z _ { X } , S } ^ { ( n ) }$ denote the (uniformly $( I ( Z _ { X } ; S ) )$ sampled) empirical distribution of $P _ { Z _ { X } , S }$ and $\hat { I } _ { \theta } ^ { ( n ) } ( Z _ { X } ; S ) : = \mathbb { E } _ { P _ { Z _ { X } ; S } ^ { ( n ) } } [ \hat { f } _ { \theta } ( z _ { x } , s ) ]$ with $\hat { f } _ { \theta }$ being the estimated log density ratio (i.e., $\log p ( s | z _ { x } ) / p ( s ) )$ .
+
+Proposition 1 (Mutual Information Neural Estimation, restating Theorem 1 by Tsai et al. (2020)). et  i $0 < \delta < 1$ . There so that $d \in \mathbb { N }$ and a family of neuh probability at least tworks over th $\mathcal { F } : = \{ \hat { f } _ { \boldsymbol { \theta } } : \boldsymbol { \theta } \in \boldsymbol { \Theta } \subseteq \mathbb { R } ^ { d } \}$ $\Theta$ $\exists \theta ^ { * } \in \Theta$ $1 - \delta$ $\{ z _ { x i } , s _ { i } \} _ { i = 1 } ^ { n } \sim P _ { Z _ { X } , S } ^ { \otimes n }$
+
+$$
+\begin{array} { r } { \Big | \widehat { I } _ { \theta ^ { * } } ^ { ( n ) } ( Z _ { X } ; S ) - I ( Z _ { X } ; S ) \Big | \leq O \left( \sqrt { \frac { d + \log ( 1 / \delta ) } { n } } \right) . } \end{array}
+$$
+
+This proposition shows that there exists a neural network √ $\theta ^ { * }$ , with high probability, $\widehat { I } _ { \theta ^ { * } } ^ { ( n ) } ( Z _ { X } ; S )$ can approximate $I ( Z _ { X } ; S )$ with $n$ samples at rate $O ( 1 / \sqrt { n } )$ . Under this network $\theta ^ { * }$ and the same parameters $d$ and $\delta$ , we are ready to present our main results on the Bayes error rate. Formally, let $| T |$ be $T$ ’s cardinalitiy and $\mathrm { T h } ( \boldsymbol { x } ) \stackrel { - } { = } \mathrm { m i n } \left\{ \mathrm { m a x } \{ \boldsymbol { x } , 0 \} , 1 - 1 / | T | \right\}$ as a thresholding function:
+
+Theorem 3 (Bayes Error Rates for Arbitrary Learned Representations). For an arbitrary learned representations $Z _ { X }$ , $P _ { e } = \mathrm { T h } ( \bar { P } _ { e } )$ with
+
+$$
+\bar { P } _ { e } \overset { < } { \le } 1 - \exp \bigg ( - \Big ( H ( T ) + I ( X ; S | T ) + I ( Z ; X | S , T ) - \hat { I } _ { \theta ^ { * } } ^ { ( n ) } ( Z _ { X } ; S ) + O \big ( \sqrt { \frac { d + \log ( 1 / \delta ) } { n } } \big ) \Big ) \bigg ) .
+$$
+
+Given arbitrary learned representations $( Z _ { X } )$ , Theorem 3 suggests the corresponding Bayes error rate $( P _ { e } )$ is small when: 1) the estimated mutual information $\big ( \widehat { I } _ { \theta ^ { * } } ^ { ( n ) } ( Z _ { X } ; S ) \big )$ is large; 2) a larger number of samples $n$ are used for estimating the mutual information; and 3) the task-irrelevant information the compression gap $I ( X ; S | T )$ and the superfluous information $I ( Z ; X | S , T )$ , defined in Theorem 2
 is small. The first and the second results supports the claim that maximizing $I ( Z _ { X } ; S )$ may learn the representations that are beneficial to downstream tasks. The third result implies the learned representations may perform better on the downstream task when the compression gap is small. Additionally, $Z ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ is preferable than $Z ^ { \mathrm { s s l } }$ since $I ( Z ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; X | S , T ) = 0$ and $I ( Z ^ { \mathrm { s s l } } ; \bar { X } | \bar { S } , T ) \ge 0$ .
+
+Theorem 4 (Bayes Error Rates for Self-supervised Learned Representations). Let $P _ { e } ^ { \mathrm { s u p } } / P _ { e } ^ { \mathrm { s s l } } / P _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ be the Bayes error rate of the supervised or the self-supervised learned representations $Z _ { X } ^ { \mathrm { s u p } } / Z _ { X } ^ { \mathrm { s s l } } / Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ . Then, $P _ { e } ^ { \mathrm { s s l } } = \mathrm { T h } ( \bar { P } _ { e } ^ { \mathrm { s s l } } )$ and $P _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } = \mathrm { T h } ( \bar { P } _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ) ~ w$ ith
+
+$$
+- \frac { \log \left( 1 - P _ { e } ^ { \mathrm { s u p } } \right) + \log 2 } { \log \left( \vert T \vert \right) } \le \{ \bar { P } _ { e } ^ { \mathrm { s s l } } , \bar { P } _ { e } ^ { \mathrm { s s l n i n } } \} \le 1 - \exp \Big ( - \left( \log 2 + P _ { e } ^ { \mathrm { s u p } } \cdot \log \vert T \vert + \epsilon _ { \mathrm { i n f o } } \right) \Big ) .
+$$
+
+Given our self-supervised learned representations $( Z _ { X } ^ { \mathrm { s s l } }$ and $Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } .$ ), Theorem 4 suggests a smaller upper bound of $P _ { e } ^ { \mathrm { s s l } }$ (or $P _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } .$ ) when the redundancy between the input and the self-supervised signal $\scriptstyle \cdot \epsilon _ { \mathrm { i n f o } }$ , defined in Assumption 1) is small. This result implies the self-supervised learned representations may perform better on the downstream task when the multi-view redundancy is small.
+
+# 3 CONTROLLED EXPERIMENTS
+
+This section aims at providing empirical supports for Theorems 1 and 2 and comparing different SSL objectives. In particular, we present information inequalities in Theorems 1 and 2 regarding the amount of the task-relevant and the task-irrelevant information that will be extracted and discarded when learning self-supervised representations. Nonetheless, quantifying the information is notoriously hard and often leads to inaccurate quantifications in practice (McAllester & Stratos, 2020; Song &
+
+![](images/338658eaee2ba9ccd37eceb4d3014daa5c88601439dc02f3396219070ca692d2.jpg)  
+Figure 3: Comparisons for different compositions of SSL objectives on Omniglot and CIFAR10.
+
+Ermon, 2019). Not to mention the information we aim to quantify is the conditional information, which is believed to be even more challenging than quantifying the unconditional one (Póczos & Schneider, 2012). To address this concern, we instead study the generalization error of the selfsupervised learned representations, theoretically (Bayes error rate discussed in Section 2.4) and empirically (test performance discussed in this section).
+
+Another important aspect of the experimental design is examining equation 4, which can be viewed as a Lagrangian relaxation to learn representations that contain minimal and sufficient self-supervision (see Definition 2): a weighted combination between $I ( Z _ { X } ; S )$ and $- H ( Z _ { X } | S )$ . In particular, the contrastive loss $\operatorname { L } _ { C L }$ and the forward-predictive loss $\mathrm { L } _ { F P }$ represent different realizations of modeling $I ( Z _ { X } ; S )$ and the inverse-predictive loss $\mathrm { L } _ { F P }$ represents a realization of modeling $- H ( Z _ { X } | S )$ .
+
+We design two sets of experiments: The first one is when the input and self-supervised signals lie in the same modality (visual) and are likely to satisfy the multi-view redundancy assumption (Assumption 1). The second one is when the input and self-supervised signals lie in very different modalities (visual and textual), thus challenging the SSL objective’s generalization ability.
+
+Experiment I - Visual Representation Learning. We use Omniglot dataset (Lake et al., 2015) 7 in this experiment. The training set contains images from 964 characters, and the test set contains 659 characters. There are no characters overlap between the training and test set. Each character contains twenty examples drawn from twenty different people. We regard image as input $( X )$ and generate self-supervised signal $( S )$ by first sampling an image from the same character as the input image and then applying translation/ rotation to it. Furthermore, we represent task-relevant information $( T )$ by the labels of the image. Under this self-supervised signal construction, the exclusive information in $X$ or $S$ are drawing styles (i.e., by different people) and image augmentations, and only their shared information contribute to $T$ . To formally show the later, if $T$ representing the label for $X / S$ , then $P ( \boldsymbol { T } | \boldsymbol { X } )$ and $P ( \boldsymbol { T } | S )$ are Dirac. Hence, $T \perp \perp S | X$ and $T \perp \perp X | S$ , suggesting Assumption 1 holds.
+
+We train the feature mapping $F _ { X } ( \cdot )$ with SSL objectives (see eq. equation 4), set $F _ { S } ( \cdot ) = F _ { X } ( \cdot )$ , let $R ( \cdot )$ be symmetrical to $F _ { X } ( \cdot )$ , and $G ( \cdot )$ be an identity mapping. On the test set, we fix the mapping and randomly select 5 examples per character as the labeled examples. Then, we classify the rest of the examples using the 1-nearest neighbor classifier based on feature (i.e., $Z _ { X } = F _ { X } ( X ) )$ cosine similarity. The random performance on this task stands at 1659 $\begin{array} { r } { \frac { 1 } { 6 5 9 } \approx 0 . 1 5 \% } \end{array}$ . One may refer to Supplementary for more details.
+
+$\trianglerighteq$ Results & Discussions. In Figure 3, we evaluate the generalization ability on the test set for different SSL objectives. First, we examine how the introduced inverse predictive learning objective $\mathrm { L } _ { I P }$ can help improve the performance along with the contrastive learning objective $\operatorname { L } _ { C L }$ . We present the results in Figure 3 (a) and also provide experiments with SimCLR (Chen et al., 2020) on CIFAR10 (Krizhevsky et al., 2009) in Figure 3 (b), where $\lambda _ { I P } = 0$ refers to the exact same setup as in SimCLR (which considers only $\operatorname { L } _ { C L }$ ). We find that adding $\mathrm { L } _ { I P }$ in the objective can boost model performance, although being sensitive to the hyper-parameter $\lambda _ { I P }$ . According to Theorem 2, the improved performance suggests a more compressed representation results in better performance for the downstream tasks. Second, we add the discussions with the forward predictive learning objective $\mathrm { L } _ { F P }$ . We present the results in Figure 3 (c). Comparing to $\mathrm { L } _ { F P }$ , $\mathrm { L } _ { C L } \ : 1$ ) reaches better test accuracy; 2) requires shorter training epochs to reach the best performance; and 3) suffers from overfitting with long-epoch training. Combining both of them $( \mathrm { L } _ { C L } + 0 . 0 0 5 \mathrm { L } _ { F P } )$ brings their advantages together.
+
+Experiment II - Visual-Textual Representation Learning. We provide experiments using MS COCO dataset (Lin et al., 2014) that contains $3 2 8 \mathrm { k }$ multi-labeled images with 2.5 million labeled instances from 91 objects. Each image has 5 annotated captions describing the relationships between objects in the scenes. We regard image as input $( X )$ and its textual descriptions as self-supervised signal $( S )$ . Since vision and text are two very different modalities, the multi-view redundancy may not be satisfied, which means $\epsilon _ { \mathrm { i n f o } }$ may be large in Assumption 1.
+
+![](images/e5492e42c14fabe6867ed237ecb231a6593a533867eccf794d4806369967e1e9.jpg)  
+Figure 4: Comparisons for different settings on self-supervised visual-textual representation training. We report metrics on MS COCO validation set with mean and standard deviation from 5 random trials.
+
+We adopt $\operatorname { L } _ { \operatorname { C L } }$ $( + \lambda _ { I P } \mathrm { L _ { I P } } )$ as our SSL objective. We use ResNet18 (He et al., 2016) image encoder for $F _ { X } ( \cdot )$ (trained from scratch or fine-tuned on ImageNet (Deng et al., 2009) pre-trained weights), BERTuncased (Devlin et al., 2018) text encoder for $F _ { S } ( \cdot )$ (trained from scratch or BookCorpus (Zhu et al., 2015)/Wikipedia pre-trained weights), and a linear layer for $G ( \cdot )$ . After performing self-supervised visual-textual representation learning, we consider the downstream multi-label classification over 91 categories. We evaluate learned visual representation $( Z _ { X } )$ using downstream linear evaluation protocol (Oord et al., 2018; Hénaff et al., 2019; Tian et al., 2019; Hjelm et al., 2018; Bachman et al., 2019; Tschannen et al., 2019). Specifically, a linear classifier is trained from the self-supervised learned (fixed) representation to the labels on the training set. Commonly used metrics for multi-label classification are reported on MS COCO validation set: Micro ROC-AUC and Subset Accuracy. One may refer to Supplementary for more details on these metrics.
+
+. Results & Discussions. First, Figure 4 (a) suggests that the SSL strategy can still work when the input and self-supervised signals lie in different modalities. For example, pre-trained ResNet with BERT (either raw or the pre-trained one) outperforms pre-trained ResNet alone. We also see that the self-supervised learned representations benefit more if the ResNet is pre-trained but not the BERT. This result is in accord with the fact that object recognition requires more understanding in vision, and hence the pre-trained ResNet is preferrable than the pre-trained BERT. Next, Figure 4 (b) suggests that the self-supervised learned representations can be further improved by combining $\operatorname { L } _ { C L }$ and $\mathrm { L } _ { I P }$ , suggesting $\mathrm { L } _ { I P }$ may be a useful objective to discard task-irrelevant information.
+
+Remarks on $\lambda _ { I P }$ and $\mathrm { L } _ { I P }$ . As observed in the experimental results, $\lambda _ { I P }$ is a sensitive hyperparameter to the performance. We provide an optimization perspective to address this concern. Note that one of the our goals is to examine the setting when learning the minimal and sufficient representations for self-supervision (see Definition 2): minimize $H ( Z _ { X } | S )$ under the constraint that $I ( Z _ { X } ; S )$ is maximized. However, this constrained optimization is not feasible when considering gradients methods in neural networks. Hence, our approach can be seen as its Lagrangian Relaxation by a weighted combination between $\operatorname { L } _ { C L }$ (or $\mathrm { L } _ { F P }$ , representing $I ( Z _ { X } ; S ) )$ and $\mathrm { L } _ { I P }$ (representing $\bar { H } ( Z _ { X } | \bar { S } ) )$ with the $\lambda _ { I P }$ being the Lagrangian coefficient.
+
+The optimal $\lambda _ { I P }$ can be obtained by solving the Lagrangian dual, which depends on the parametrization of $\operatorname { L } _ { C L }$ (or $\operatorname { L } _ { F P } ,$ ) and $\mathrm { L } _ { I P }$ . Different parameterizations lead to different loss and gradient landscapes, and hence the optimal $\lambda _ { I P }$ differs across experiments. This conclusion is verified by the results presented in Figure 3 (a) and (b) and Figure 4 (b). Lastly, we point out that even not solving the Lagrangian dual, an empirical observation across experiments is that $\lambda _ { I P }$ which leads to the best performance is when the scale of $\mathrm { L } _ { I P }$ is one-tenth to the scale of $\operatorname { L } _ { C L }$ (or $\operatorname { L } _ { F P }$ ).
+
+# 4 RELATED WORK
+
+Prior work by Arora et al. (2019) and the recent concurrent work (Lee et al., 2020; Tosh et al., 2020) are landmarks for theoretically understanding the success of SSL. In particular, Arora et al. (2019); Lee et al. (2020) showed a decreased sample complexity for downstream supervised tasks when adopting contrastive learning objectives (Arora et al., 2019) or predicting the known information in the data (Lee et al., 2020). Tosh et al. (2020) showed that the linear functions of the learned representations are nearly optimal on downstream prediction tasks. By viewing the input and the self-supervised signal as two corresponding views of the data, we discuss the differences among these works and ours. On the one hand, the work by Arora et al. (2019); Lee et al. (2020) assume strong independence between the views conditioning on the downstream tasks , i.e., $I ( X ; S | T ) \approx 0$ .
+
+On the other hand, the work by Tosh et al. (2020) and ours assume strong independence between the downstream task and one view conditioning on the other view, i.e., $I ( T ; X | S ) \approx 0$ . Prior work (Balcan et al., 2005; Du et al., 2010) have compared these two assumptions and pointed out the former one $I ( X ; S | T ) \approx 0 )$ is too strong and not likely to hold in practice. We note that all these related work and ours have shown that the self-supervised learning methods are learning to extract task-relevant information. Our work additionally presents to discard task-irrelevant information and quantifies the amount of information that cannot be discarded.
+
+Our method also resembles the InfoMax principle (Linsker, 1988; Hjelm et al., 2018) and the Multiview Information Bottleneck method (Federici et al., 2020). The InfoMax principle aims at preserving the information of itself, while ours aims at extracting the information in the self-supervised signal. On the other hand, to reduce the redundant information across views, the Multi-view Information Bottleneck method proposed to minimize the conditional mutual information $I ( Z _ { X } ; X | S )$ , while ours propose to minimize the conditional entropy $H ( Z _ { X } | S )$ . The conditional entropy minimization problem can be easily optimized via our proposed inversed predictive learning objective.
+
+Another related work is InfoMin (Tian et al., 2020), where both InfoMin and our method suggest to learn the representations that contain “not” too much information. In particular, InfoMin presents to augment the data (i.e., by constructing learnable data augmentations) such that the shared information between augmented variants is as minimal as possible, followed by the mutual information maximization between the learned features from the augmented variants. Our method instead considers standard augmentations (e.g., rotations and translations), followed by learning representations that contain no more than the shared information between the augmented variants of the data.
+
+On the empirical side, we explain why contrastive (Oord et al., 2018; Bachman et al., 2019; Chen et al., 2020) and predictive learning (Zhang et al., 2016; Pathak et al., 2016; Vondrick et al., 2016; Chen et al.) approaches can unsupervised extract task-relevant information. Different from these work, we present an objective to discard task-irrelevant information and show its combination with existing contrastive or predictive objectives benefits the performance.
+
+# 5 CONCLUSION
+
+This work studies both theoretical and empirical perspectives on self-supervised learning. We show that the self-supervised learned representations could extract task-relevant information (with a potential loss) and discard task-irrelevant information (with a fixed gap), along with their practical deployments such as contrastive and predictive learning objectives. We believe this work sheds light on the advantages of self-supervised learning and may help better understand when and why self-supervised learning is likely to work. In the future, we plan to connect our framework and recent SSL methods that cannot be easily fit into our analysis: e.g., BYOL (Grill et al., 2020), SWAV (Caron et al., 2020), and Unifromality-Alignment (Wang & Isola, 2020).
+
+# ACKNOWLEDGEMENT
+
+This work was supported in part by the NSF IIS1763562, NSF Awards #1750439 #1722822, National Institutes of Health, IARPA D17PC00340, ONR Grant N000141812861, and Facebook PhD Fellowship. We would also like to acknowledge NVIDIA’s GPU support.
+
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+
+# A REMARKS ON LEARNING MINIMAL AND SUFFICIENT REPRESENTATIONS
+
+In the main text, we discussed the objectives to learn minimal and sufficient representations (Definition 1). Here, we discuss the similarities and differences between the prior methods (Tishby et al., 2000; Achille & Soatto, 2018) and ours. First, to obtain sufficient representations (for the downstream task $T$ ), all the methods presented to maximize $I ( Z _ { X } ; T )$ . Then, to maintain minimal amount of information in the representations, the prior methods (Tishby et al., 2000; Achille & Soatto, 2018) presented to minimize $I ( Z _ { X } ; X )$ and the ours presents to minimize $H ( Z _ { X } | T )$ . Our goal is to relate $I ( Z _ { X } ; X )$ minimization and $H ( Z _ { X } | T )$ minimization in our framework.
+
+To begin with, under the constraint $I ( Z _ { X } ; T )$ is maximized, we see that minimizing $I ( Z _ { X } ; X )$ is equivalent to minimizing $I ( Z _ { X } ; X | T )$ . The reason is that $I ( Z _ { X } ; X ) = I ( Z _ { X } ; X | T ) \dot { + } I ( Z _ { X } ; X ; T )$ , where $I ( Z _ { X } ; X ; T ) = I ( Z _ { X } ; T )$ due to the determinism from $X$ to $Z _ { X }$ (our framework learns a deterministic function from $X$ to $Z _ { X }$ ) and $I ( Z _ { X } ; T )$ is maximized in our constraint. Then, $I ( Z _ { X } ; X | T ) = H ( Z _ { X } | T ) - H ( Z _ { X } | X , T ) .$ , where $H ( Z _ { X } | T )$ contains no randomness (no information) as $Z _ { X }$ being deterministic from $X$ . Hence, $I ( \vec { Z } _ { X } ; X | T )$ minimization and $H ( Z _ { X } | T )$ minimization are interchangeable.
+
+The same claim can be made from the downstream task $T$ to the self-supervised signal $S$ . In specific, when $X$ to $Z _ { X }$ is deterministic, $I ( Z _ { X } ; X | S )$ minimization and ${ \bar { H ( Z _ { X } | S ) } }$ minimization are interchangeable. As discussed in the related work section, for reducing the amount of the redundant information, Federici et al. (2020) presented to use $I ( Z _ { X } ; X | S )$ minimization and ours presented to use $H ( Z _ { X } | T )$ minimization. We also note that directly minimizing the conditional mutual information (i.e., $I ( Z _ { X } ; X | S ) )$ requires a min-max optimization (Mukherjee et al., 2020), which may cause instability in practice. To overcome the issue, Federici et al. (2020) assumes a Gaussian encoder for $X  Z _ { X }$ and presents an upper bound of the original objective.
+
+# B PROOFS FOR THEOREM 1 AND 2
+
+We start by presenting a useful lemma from the fact that $F _ { X } ( \cdot )$ is a deterministic function:
+
+Lemma 1 (Determinism). If $P ( Z _ { X } | X )$ is Dirac, then the following conditional independence holds: $T$ ⊥⊥ $Z _ { X } | X$ and $S$ ⊥⊥ $Z _ { X } | X$ , inducing a Markov chain $S  T  X  Z _ { X }$ .
+
+Proof. When $Z _ { X }$ is a deterministic function of $X$ , for any $A$ in the sigma-algebra induced by $Z _ { X }$ we have $\mathbb { E } [ { \mathbf 1 } _ { [ Z _ { X } \in A ] } | X , \{ T , S \} ] = \mathbb { E } [ { \mathbf 1 } _ { [ Z _ { X } \in A ] } | X , S ] = \mathbb { E } [ { \mathbf 1 } _ { [ Z _ { X } \in A ] } | X ]$ , which implies $T$ ⊥⊥ $Z _ { X } | X$ and $S$ ⊥⊥ $Z _ { X } | X$ . □
+
+Theorem 1 and 2 in the main text restated:
+
+Theorem 5 (Task-relevant information with a potential loss $\epsilon _ { \mathrm { i n f o } }$ , restating Theorem 1 in the main text). The supervised learned representations (i.e., $I ( Z _ { X } ^ { \mathrm { s u p } } ; T )$ and $I ( Z _ { X } ^ { \mathrm { s u p } _ { \mathrm { m i n } } } ; T ) )$ contain all the task-relevant information in the input (i.e., $I ( X ; T ) ,$ ). The self-supervised learned representations (i.e., $I ( Z _ { X } ^ { \mathrm { s s l } } ; T )$ and  Form $I ( Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; T ) )$ contain all the task-relevant information in the input with a potential $\epsilon _ { \mathrm { i n f o } }$
+
+$$
+I ( X ; T ) = I ( Z _ { X } ^ { \mathrm { s u p } } ; T ) = I ( Z _ { X } ^ { \mathrm { s u p } _ { \mathrm { m i n } } } ; T ) \geq I ( Z _ { X } ^ { \mathrm { s s l } } ; T ) \geq I ( Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; T ) \geq I ( X ; T ) - \epsilon _ { \mathrm { i n f o } } .
+$$
+
+Proof. The proofs contain two parts. The first one is showing the results for the supervised learned representations and the second one is for the self-supervised learned representations.
+
+Supervised Learned Representations: Adopting Data Processing Inequality (DPI by Cover & Thomas (2012)) in the Markov chain ${ \overline { { S  T } } }  X \to Z _ { X }$ (Lemma 1), $I ( Z _ { X } ; T )$ is maximized at $I ( X ; T )$ . Since both supervised learned representations $( Z _ { X } ^ { \mathrm { s u p } }$ and $Z _ { X } ^ { \mathrm { s u p } _ { \mathrm { m i n } } }$ ) maximize $I ( Z _ { X } ; T )$ , we conclude $I ( Z _ { X } ^ { \mathrm { s u p } } ; T ) \stackrel { \cdot } { = } I ( Z _ { X } ^ { \mathrm { s u p } _ { \mathrm { m i n } } } ; T ) = \dot { I } ( X ; T ) .$ .
+
+Self-supervised Learned Representations: First, we have
+
+$$
+I ( Z _ { X } ; S ) = I ( Z _ { X } ; T ) - I ( Z _ { X } ; T | S ) + I ( Z _ { X } ; S | T ) = I ( Z _ { X } ; T ; S ) + I ( Z _ { X } ; S | T )
+$$
+
+and
+
+By DPI in the Markov chain $S  T  X  Z _ { X }$ (Lemma 1), we know • $I ( Z _ { X } ; S )$ is maximized at $I ( X ; S )$ • $I ( Z _ { X } ; S ; T )$ is maximized at $I ( X ; S ; T )$ • $I ( Z _ { X } ; S | T )$ is maximized at $I ( X ; S | T )$
+
+Since both self-supervised learned representations $( Z _ { X } ^ { \mathrm { s s l } }$ and $Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ ) maximize $I ( Z _ { X } ; S )$ , we have $I ( Z _ { X } ^ { \mathrm { s s l } } ; S ) = I ( Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; S ) = I ( X ; S )$ . Hence, $I ( Z _ { X } ^ { \mathrm { s s l } } ; S ; T ) = I ( Z _ { X } ^ { \mathrm { s s l _ { m i n } } } ; S ; T ) = I ( X ; S ; T )$ and $I ( Z _ { X } ^ { \mathrm { s s l } } ; S | T ) = I ( Z _ { X } ^ { \mathrm { s s l _ { m i n } } } ; S | T ) = I ( X ; S | T )$ . Using the result $I ( Z _ { X } ^ { \mathrm { s s l . } } ; S ; T ) = I ( Z _ { X } ^ { \mathrm { s s l . } } ; S ; T ) =$ $I ( X ; S ; T )$ , we get
+
+$$
+I ( Z _ { X } ^ { \mathrm { s s l } } ; T ) = I ( X ; T ) - I ( X ; T | S ) + I ( Z _ { X } ^ { \mathrm { s s l } } ; T | S )
+$$
+
+and
+
+$$
+I ( Z _ { X } ^ { \mathrm { s s l _ { m i n } } } ; T ) = I ( X ; T ) - I ( X ; T | S ) + I ( Z _ { X } ^ { \mathrm { s s l _ { m i n } } } ; T | S ) .
+$$
+
+Now, we are ready to present the inequalities:
+
+1. $I ( X ; T ) \ge I ( Z _ { X } ^ { \mathrm { s s l } } ; T )$ due to $I ( X ; T | S ) \ge I ( Z _ { X } ^ { \mathrm { s s l . } } ; T | S )$ by DPI.
+
+2. $I ( Z _ { X } ^ { \mathrm { s s l } } ; T ) \ge I ( Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; T )$ due to $I ( Z _ { X } ^ { \mathrm { s s l } } ; T | S ) \ge I ( Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; T | S ) = 0$ . Since $H ( Z _ { X } | S )$ is minimized at $Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ , $I ( Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; T | S ) = 0$ .
+
+3. $I ( Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; T ) \ge I ( X ; T ) - \epsilon _ { \mathrm { i n f o } }$ due to
+
+$I ( X ; T ) - I ( X ; T | S ) + I ( Z _ { X } ^ { \smash { \mathrm { s l } _ { \mathrm { m i n } } } } ; T | S ) \geq I ( X ; T ) - I ( X ; T | S ) \geq I ( X ; T ) - \epsilon _ { \mathrm { i n f o } } ,$ where $I ( X ; T | S ) \le \epsilon _ { \mathrm { i n f o } }$ by the redundancy assumption.
+
+Theorem 6 (Task-irrelevant information with a fixed compression gap $I ( X ; S | T )$ , restating Theorem 2 in the main text). The sufficient self-supervised representation (i.e., $I ( Z _ { X } ^ { \mathrm { s s l } } ; T ) )$ contains more taskirrelevant information in the input than then the sufficient and minimal self-supervised representation (i.e., $I ( Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; T ) )$ . The later contains an amount of the information, $I ( X ; S | T )$ , that cannot be discarded from the input. Formally,
+
+$$
+\begin{array} { r } { \mathbf { \tilde { \rho } } ( Z _ { X } ^ { \mathrm { s x l } } ; X | T ) = I ( X ; S | T ) + I ( Z _ { X } ^ { \mathrm { s x l } } ; X | S , T ) \ge I ( Z _ { X } ^ { \mathrm { s s l n } _ { \mathrm { m i n } } } ; X | T ) = I ( X ; S | T ) \ge I ( Z _ { X } ^ { \mathrm { s u p } _ { \mathrm { m i n } } } ; X | T ) = \alpha _ { 1 } ( X ; X | T ) , } \end{array}
+$$
+
+Proof. First, we see that
+
+$I ( Z _ { X } ; X | T ) = I ( Z _ { X } ; X ; S | T ) + I ( Z _ { X } ; X | S , T ) = I ( Z _ { X } ; S | T ) + I ( Z _ { X } ; X | S , T ) ,$ where $I ( Z _ { X } ; X ; S | T ) = I ( Z _ { X } ; S | T )$ by DPI in the Markov chain $S  T  X  Z _ { X }$ .
+
+We conclude the proof by combining the following:
+
+• From the proof in Theorem 5, we showed $I ( Z _ { X } ^ { \mathrm { s s l . } } ; S | T ) = I ( Z _ { X } ^ { \mathrm { s s l . n i n } } ; S | T ) = I ( X ; S | T ) .$ • Since $H ( Z _ { X } | S )$ is minimized at $Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ , $I ( Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; X | S , T ) = 0$ . • Since $H ( Z _ { X } | T )$ is minimized at $Z _ { X } ^ { \mathrm { s u p } _ { \mathrm { m i n } } }$ , $I ( Z _ { X } ^ { \mathrm { s u p m i n } } ; X | T ) = 0$ .
+
+# C PROOF FOR PROPOSITION 1
+
+Proposition 2 (Mutual Information Neural Estimation, restating Proposition 1 in the main text). Let $0 < \delta < 1$ . Thereso that $d \in \mathbb { N }$ and a family of neuralth probability at least rks ove $\mathcal { F } : = \{ \hat { f } _ { \boldsymbol { \theta } } : \boldsymbol { \theta } \in \boldsymbol { \Theta } \subseteq \mathbb { R } ^ { d } \}$ $\Theta$ $\exists \theta ^ { * } \in \Theta$ $1 - \delta$ $\{ z _ { x i } , s _ { i } \} _ { i = 1 } ^ { n } \sim P _ { Z _ { X } , S } ^ { \otimes n }$
+
+$$
+\begin{array} { r } { \Big | \widehat { I } _ { \theta ^ { * } } ^ { ( n ) } ( Z _ { X } ; S ) - I ( Z _ { X } ; S ) \Big | \leq O \left( \sqrt { \frac { d + \log ( 1 / \delta ) } { n } } \right) . } \end{array}
+$$
+
+Sketch of Proof. The proof is a standard instance of uniform convergence bound. First, we assume the boundness and the Lipschitzness of $\hat { f } _ { \theta }$ . Then, we use the universal approximation lemma of neural networks (Hornik et al.). Last, combing all these two along with the uniform convergence in terms of the covering number (Bartlett, 1998), we complete the proof. □
+
+We note that the complete proof can be found in the prior work (Tsai et al., 2020). An alternative but similar proof can be found in another prior work (Belghazi et al., 2018), which gives us $\begin{array} { r } { | \widehat { I } _ { \theta ^ { * } } ^ { ( n ) } ( Z _ { X } ; S )  - I ( Z _ { X } ; S ) | \leq O ( \sqrt { \frac { d \log d + \log ( 1 / \delta ) } { n } } ) } \end{array}$ . The subtle difference between them is that, given a neural network function space $\Theta \ \subseteq \ \mathbb { R } ^ { d }$ and its covering number $\mathcal { N } ( \Theta , \eta )$ , Tsai et al. (2020) has $\mathcal { N } ( \Theta , \eta ) ~ = ~ O \Big ( ( \eta ) ^ { - d } \Big )$ by Bartlett (1998) and Belghazi et al. (2018) has $\mathcal { N } ( \Theta , \eta ) = O \Big ( ( \eta / \sqrt { d } ) ^ { - d } \Big )$ by Shalev-Shwartz & Ben-David (2014). Both are valid and the one used by Tsai et al. (2020) is tighter.
+
+# D PROOFS FOR THEOREM 3 AND 4
+
+To begin with, we see that
+
+$$
+\begin{array} { r l } & { I ( Z _ { X } ; T ) = I ( Z _ { X } ; X ) - I ( Z _ { X } ; X | T ) + I ( Z _ { X } ; T | X ) = I ( Z _ { X } ; X ) - I ( Z _ { X } ; X | T ) } \\ & { \qquad = I ( Z _ { X } ; S ) - I ( Z _ { X } ; S | X ) + I ( Z _ { X } ; X | S ) - I ( Z _ { X } ; X | T ) } \\ & { \qquad = I ( Z _ { X } ; S ) + I ( Z _ { X } ; X | S ) - I ( Z _ { X } ; X | T ) } \\ & { \qquad \geq I ( Z _ { X } ; S ) - I ( Z _ { X } ; X | T ) , } \end{array}
+$$
+
+where $I ( Z _ { X } ; T | X ) = I ( Z _ { X } ; S | X ) = 0$ due to the determinism from $X$ to $Z _ { X }$ . Then, in the proof of Theorem 6, we have shown $I ( Z _ { X } ; X | T ) = I ( Z _ { X } ; S | T ) + I ( Z _ { X } ; X | S , T )$ . Hence,
+
+$$
+\begin{array} { r l r } & { } & { I ( Z _ { X } ; T ) \ge I ( Z _ { X } ; S ) - I ( Z _ { X } ; S | T ) - I ( Z _ { X } ; X | S , T ) } \\ & { } & { \ge I ( Z _ { X } ; S ) - I ( X ; S | T ) - I ( Z _ { X } ; X | S , T ) , } \end{array}
+$$
+
+where $I ( Z _ { X } ; S | T ) \le I ( X ; S | T )$ by DPI.
+
+Theorem 3 and 4 in the main text restated:
+
+Theorem 7 (Bayes Error Rates for Arbitrary Learned Representations, restating Theorem 3 in the main text). For an arbitrary learned representations $Z _ { X }$ , $\begin{array} { r } { P _ { e } = \mathrm { T h } ( \bar { P } _ { e } ) } \end{array}$ with
+
+$$
+\begin{array} { r } { \bar { P } _ { e } \overset { _ { \sim } } { \le } 1 - \exp ^ { - \left( H ( T ) + I ( X ; S | T ) + I ( Z ; X | S , T ) - \hat { I } _ { \theta ^ { * } } ^ { ( n ) } ( Z _ { X } ; S ) + O \left( \sqrt { \frac { d + \log ( 1 / \delta ) } { n } } \right) \right) } . } \end{array}
+$$
+
+Proof. We use the inequality between $P _ { e }$ and $H ( T | Z _ { X } )$ indicated by Feder & Merhav (1994):
+
+$$
+- \mathrm { l o g } ( 1 - P _ { e } ) \leq H ( T | Z _ { X } ) .
+$$
+
+Combining with $I ( Z _ { X } ; T ) = H ( T ) - H ( T | Z _ { X } )$ and $I ( Z _ { X } ; T ) \ge I ( Z _ { X } ; S ) - I ( X ; S | T ) -$ $I ( Z _ { X } ; X | S , T )$ , we have
+
+$$
+\begin{array} { r } { \log ( 1 - P _ { e } ) \geq - H ( T ) + I ( Z _ { X } ; S ) - I ( X ; S | T ) - I ( Z _ { X } ; X | S , T ) . } \end{array}
+$$
+
+Hence,
+
+$$
+P _ { e } \leq 1 - \exp ^ { - \bigg ( H ( T ) + I ( X ; S | T ) + I ( Z ; X | S , T ) - I ( Z _ { X } ; S ) \bigg ) } .
+$$
+
+Next, by definition of the Bayes error rate, we know $\begin{array} { r } { 0 \leq P _ { e } \leq 1 - \frac { 1 } { | T | } } \end{array}$ .
+
+We conclude the proof by combining Proposition 2, Ib(n)θ∗ (ZX ; S) − I(ZX ; S) ≤ $O \left( \sqrt { \frac { d + \log ( 1 / \delta ) } { n } } \right) .$
+
+Theorem 8 (Bayes Error Rates for Self-supervised Learned Representations, restating Theorem 4 in the main text). Let $P _ { e } ^ { \mathrm { s u p } } / P _ { e } ^ { \mathrm { s s l } } / P _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ be the Bayes error rate of the supervised or the self-supervised learned representations $Z _ { X } ^ { \mathrm { s u p } } / Z _ { X } ^ { \mathrm { s s l } } / Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ . Then, $P _ { e } ^ { \mathrm { s s l } } = \mathrm { T h } ( \bar { P } _ { e } ^ { \mathrm { s s l } } )$ and $P _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } = \mathrm { T h } ( \bar { P } _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } )$ with
+
+$$
+- \frac { \log \left( 1 - P _ { e } ^ { \mathrm { s u p } } \right) + \log 2 } { \log \left( | T | \right) } \leq \{ \bar { P } _ { e } ^ { \mathrm { s s l } } , \bar { P } _ { e } ^ { \mathrm { s s l _ { m i n } } } \} \leq 1 - \exp ^ { - ( \log 2 + P _ { e } ^ { \mathrm { s u p } } \cdot \log | T | + \epsilon _ { \mathrm { i n f o } } ) } .
+$$
+
+Proof. We use the two inequalities between $P _ { e }$ and $H ( T | Z _ { X } )$ by Feder & Merhav (1994) and Cover & Thomas (2012):
+
+$$
+- \mathrm { l o g } ( 1 - P _ { e } ) \leq H ( T | Z _ { X } )
+$$
+
+and
+
+$$
+H ( T | Z _ { X } ) \leq \log 2 + P _ { e } \mathrm { l o g } | T | .
+$$
+
+Combining the results from Theorem 5:
+
+$$
+I ( Z _ { X } ^ { \mathrm { s u p } } ; T ) \ge I ( Z _ { X } ^ { \mathrm { s s l } } ; T ) \ge I ( Z _ { X } ^ { \mathrm { s s l _ { m i n } } } ; T ) \ge I ( Z _ { X } ^ { \mathrm { s u p } } ; T ) - \epsilon _ { \mathrm { i n f o } } ,
+$$
+
+we have
+
+• the upper bound of the self-supervised learned representations’ Bayes error rate:
+
+$$
+\begin{array} { r l } { \{ { - \log ( 1 - P _ { e } ^ { \mathrm { s s l } } ) } , { - \log ( 1 - P _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ) } \} \le \{ H ( T | Z _ { X } ^ { \mathrm { s s l } } ) , H ( T | Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ) \} } & { } \\ { \le H ( T | Z _ { X } ^ { \mathrm { s u p } } ) + \epsilon _ { \mathrm { i n f o } } } & { } \\ { \le \log 2 + P _ { e } ^ { \mathrm { s u p } } \mathrm { l o g } | T | + \epsilon _ { \mathrm { i n f o } } , } \end{array}
+$$
+
+which suggests $\{ P _ { e } ^ { \mathrm { s s l } } , P _ { e } ^ { \mathrm { s s l _ { m i n } } } \} \le 1 - \exp ^ { - ( \log 2 + P _ { e } ^ { \mathrm { s u p } } \cdot \log | T | + \epsilon _ { \mathrm { i n f o } } ) }$
+
+• the lower bound of the self-supervised learned representations’ Bayes error rate:
+
+$$
+\begin{array} { r l } & { - \mathrm { l o g } ( 1 - P _ { e } ^ { \mathrm { s u p } } ) \leq H ( T | Z _ { X } ^ { \mathrm { s u p } } ) } \\ & { \qquad \leq \{ H ( T | Z _ { X } ^ { \mathrm { s s l } } ) , H ( T | Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ) \} } \\ & { \qquad \leq \{ \log 2 + P _ { e } ^ { \mathrm { s s l } } \mathrm { l o g } | T | , \leq \{ \log 2 + P _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } \mathrm { l o g } | T | \} , } \\ & { \mathrm { g e s t s } - \frac { \log \left( 1 - P _ { e } ^ { \mathrm { s u p } } \right) + \log 2 } { \log \left( | T | \right) } \leq \{ P _ { e } ^ { \mathrm { s s l } } , P _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } \} . } \end{array}
+$$
+
+We conclude the proof by having $P _ { e }$ lie in the feasible range: $\begin{array} { r } { 0 \leq P _ { e } \leq 1 - \frac { 1 } { | T | } } \end{array}$ .
+
+# E TIGHTER BOUNDS FOR THE BAYES ERROR RATES
+
+We note that the bound used in Theorems 7 and 8: $- \mathrm { l o g } ( 1 - P _ { e } ) \le H ( T | Z _ { X } ) \le \mathrm { l o g } 2 + P _ { e } \mathrm { l o g } | T |$ is not tight. A tighter bound is $H ^ { - } ( P _ { e } ) \leq H ( T | Z _ { X } ) \leq H ^ { + } ( P _ { e } )$ with
+
+$$
+H ^ { - } ( P _ { e } ) : = H { \Big ( } k ( 1 - P _ { e } ) { \Big ) } + k ( 1 - P _ { e } ) \log k { \mathrm { ~ w h e n ~ } } { \frac { k - 1 } { k } } \leq P _ { e } \leq { \frac { k } { k + 1 } } , 1 \leq k \leq | T | - 1 ,
+$$
+
+$$
+H ^ { + } ( P _ { e } ) : = H ( P _ { e } ) + P _ { e } { \log { ( | T | - 1 ) } } ,
+$$
+
+where $H ( x ) = - x \mathrm { l o g } ( x ) - ( 1 - x ) \mathrm { l o g } ( 1 - x )$
+
+It is clear that $- \mathrm { l o g } ( 1 - P _ { e } ) \le H ^ { - } ( P _ { e } )$ and $H ^ { + } ( P _ { e } ) \leq \log 2 + P _ { e } \mathrm { l o g } ( | T | )$
+
+Hence, Theorem 7 and 8 can be improved as follows:
+
+Theorem 9 (Tighter Bayes Error Rates for Arbitrary Learned Representations). For an arbitrary learned representations $Z _ { X }$ , $P _ { e } = \mathrm { T h } ( \bar { P } _ { e } )$ with $\bar { P } _ { e } \overset { \cdot } { \leq } P _ { e \mathrm { u p p e r } }$ . $P _ { e \mathrm { u p p e r } }$ is derived from the program
+
+$$
+\displaystyle \operatorname* { r g m a x } _ { P _ { e } } H ^ { - } ( P _ { e } ) \leq H ( T ) - \hat { I } _ { \boldsymbol { \theta } } ^ { ( n ) } ( Z _ { X } ^ { \mathrm { s } \bot } ; S ) + I ( X ; S | T ) + I ( Z _ { X } ; X | S , T ) + O \big ( \sqrt { \frac { d + \log ( 1 / \delta ) } { n } } \big ) .
+$$
+
+Theorem 10 (Tighter Bayes Error Rates for Self-supervised Learned Representations). Let $P _ { e } ^ { \mathrm { s u p } } / P _ { e } ^ { \mathrm { s s l } } / P _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ be the Bayes error rate of the supervised or the self-supervised learned representations $Z _ { X } ^ { \mathrm { s u p } } / Z _ { X } ^ { \mathrm { s s l } } / Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ . Then, $P _ { e } ^ { \mathrm { s s l } } = \mathrm { T h } ( \bar { P } _ { e } ^ { \mathrm { s s l } } )$ and $P _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } = \mathrm { T h } ( \bar { P } _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } )$ with
+
+$$
+P _ { e \mathrm { l o w e r } } ^ { \mathrm { \scriptsize ~ s s l } } \le \{ \bar { P } _ { e } ^ { \mathrm { s s l } } , \bar { P } _ { e } ^ { \mathrm { s s l _ { m i n } } } \} \le P _ { e \mathrm { u p p e r } } ^ { \mathrm { \scriptsize ~ s s l } } .
+$$
+
+$P _ { e \mathrm { l o w e r } } ^ { \mathrm { \ s s l } }$ is derived from the following program
+
+$$
+\arg \operatorname* { m i n } _ { P _ { e } ^ { \mathrm { s s l } } } H ^ { - } ( P _ { e } ^ { \mathrm { s u p } } ) \leq H ^ { + } ( P _ { e } ^ { \mathrm { s s l } } )
+$$
+
+$P _ { e \mathrm { u p p e r } } ^ { \mathrm { s s l } }$
+
+$$
+\underset { P _ { e } ^ { \mathrm { s s } 1 } } { \arg \operatorname* { m a x } } H ^ { - } ( P _ { e } ^ { \mathrm { s s l } } ) \leq H ^ { + } ( P _ { e } ^ { \mathrm { s u p } } ) + \epsilon _ { \mathrm { i n f o } } .
+$$
+
+# F MORE ON VISUAL REPRESENTATION LEARNING EXPERIMENTS
+
+In the main text, we design controlled experiments on self-supervised visual representation learning to empirically support our theorem and examine different compositions of SSL objectives. In this section, we will discuss 1) the architecture design; 2) different deployments of contrastive/ forward predictive learning; and 3) different self-supervised signal construction strategy. We argue that these three additional set of experiments may be interesting future work.
+
+# F.1 ARCHITECTURE DESIGN
+
+The input image has size $1 0 5 \times 1 0 5$ . For image augmentations, we adopt 1) rotation with degrees from $- 1 0 ^ { \circ }$ to $+ 1 0 ^ { \circ }$ ; 2) translation from $- 1 5$ pixels to $+ 1 5$ pixels; 3) scaling both width and height from 0.85 to 1.0; 4) scaling width from 0.85 to 1.25 while fixing the height; and 5) resizing the image to $2 8 \times 2 8$ . Then, a deep network takes a $2 8 \times 2 8$ image and outputs a 1024−dim. feature vector. The deep network has the structure: Conv − BN − ReLU − Conv − BN − ReLU − MaxPool − Conv − BN − ReLU − MaxPool − Conv −BN − ReLU − MaxPool − Flatten − Linear − L2Norm. Conv has 3x3 kernel size with 128 output channels, $\mathsf { M a x P o o l }$ has $2 \mathbf { x } 2$ kernel size, and Linear is a 1152 to 1024 weight matrix. $R ( \cdot )$ is symmetric to $F _ { X } ( \cdot )$ , which has Linear − BN − ReLU − UnFlatten $- \mathsf { D e } \mathsf { C o n v } - \mathsf { B N } - \mathsf { R e } \mathsf { L U } - \mathsf { D e } \mathsf { C o n v } - \mathsf { B N } - \mathsf { R e } \mathsf { L U } - \mathsf { D e } \mathsf { C o n v }$ −BN − ReLU − DeConv. $R ( \cdot )$ has the exact same number of parameters as $F _ { X } ( \cdot )$ . Note that we use the same network designs in $I ( \cdot , \cdot )$ and $H ( \cdot | \cdot )$ estimations. To reproduce the results in our experimental section, please refer to our released code8.
+
+![](images/f72d16a1c6578fdb097f9e36c35cbb050ff7d0aca73763d22cd3195de60233f3.jpg)  
+Figure 5: Comparisons for different objectives/compositions of SSL objectives on self-supervised visual representation training. We report mean and its standard error from 5 random trials.
+
+# F.2 DIFFERENT DEPLOYMENTS FOR CONTRASTIVE AND PREDICTIVE LEARNING OBJECTIVES
+
+In the main text, for practical deployments, we suggest Contrastive Predictive Coding (CPC) Oord et al. (2018) for $\operatorname { L } _ { C L }$ and assume Gaussian distribution for the variational distributions in $\mathrm { L } _ { F P } / \mathrm { L } _ { I P }$ The practical deployments can be abundant by using different mutual information approximations for $\operatorname { L } _ { C L }$ and having different distribution assumptions for $\mathrm { L } _ { F P } / \mathrm { L } _ { I P }$ . In the following, we discuss a few examples.
+
+Contrastive Learning. Other than CPC Oord et al. (2018), another popular contrastive learning objective is JS Bachman et al. (2019), which is the lower bound of Jensen-Shannon divergence between $P ( Z _ { S } , Z _ { X } )$ and $P ( Z _ { S } ) P ( Z _ { X } )$ (a variational bound of mutual information). Its objective can be written as
+
+$$
+\operatorname* { m a x } _ { \substack { \tau _ { S } = F _ { S } ( S ) , Z _ { X } = F _ { X } ( X ) , G } } \mathbb { E } _ { P ( Z _ { S } , Z _ { X } ) } [ - \mathrm { s o f t p l u s } ( - \langle G ( z _ { x } ) , G ( z _ { s } ) \rangle ) ] - \mathbb { E } _ { P ( Z _ { S } ) P ( Z _ { X } ) } [ \mathrm { s o f t p l u s } ( \langle G ( z _ { x } ) , G ( z _ { x } ) \rangle ) ] .
+$$
+
+where we use softplus to denote softplus $\left( x \right) = \log \left( 1 + \exp \left( x \right) \right)$ .
+
+Predictive Learning. Gaussian distribution may be the simplest distribution form that we can imagine, which leads to Mean Square Error (MSE) reconstruction loss. Here, we use forward predictive learning as an example, and we discuss the case when $s$ lies in discrete $\{ 0 , 1 \}$ sample space. Specifically, we let $Q _ { \phi } ( S | Z _ { X } )$ be factorized multivariate Bernoulli:
+
+$$
+\operatorname* { m a x } _ { Z _ { X } = F _ { X } ( X ) , R } \mathbb { E } _ { P _ { S } , z _ { X } } \left[ \sum _ { i = 1 } ^ { p } s _ { i } \cdot \log \left[ R ( z _ { x } ) \right] _ { i } + ( 1 - s _ { i } ) \cdot \log \left[ 1 - R ( z _ { x } ) \right] _ { i } \right] .
+$$
+
+This objective leads to Binary Cross Entropy (BCE) reconstruction loss.
+
+If we assume each reconstruction loss corresponds to a particular distribution form, then by ignoring which variatioinal distribution we choose, we are free to choose arbitrary reconstruction loss. For instance, by switching $s$ and $z$ in eq. equation 5, the objective can be regarded as Reverse Binary Cross Entropy Loss (RevBCE) reconstruction loss. In our experiments, we find RevBCE works the best among {MSE, BCE, and RevBCE}. Therefore, in the main text, we choose RevBCE as the example reconstruction loss as $\mathrm { L } _ { F P }$ .
+
+More Experiments. We provide an additional set of experiments by having {CPC, JS} for $\operatorname { L } _ { C L }$ and {MSE, BCE, RevBCE} reconstruction loss for $\mathrm { L } _ { F P }$ in Figure 5. From the results, we find different formulation of objectives bring very different test generalization performance. We argue that, given a particular task, it is challenging but important to find the best deployments for contrastive and predictive learning objectives.
+
+# F.3 DIFFERENT SELF-SUPERVISED SIGNAL CONSTRUCTION STRATEGY
+
+In the main text, we design a self-supervised signal construction strategy that the input $( X )$ and the self-supervised signal $( S )$ differ in {drawing styles, image augmentations}. This self-supervised signal construction strategy is different from the one that is commonly adopted in most self-supervised visual representation learning work Tian et al. (2019); Bachman et al. (2019); Chen et al. (2020). Specifically, prior work consider the difference between input and the self-supervised signal only in image augmentations. We provide additional experiments in Fig. 6 to compare these two different self-supervised signal construction strategies.
+
+We see that, comparing to the common self-supervised signal construction strategy Tian et al. (2019); Bachman et al. (2019); Chen et al. (2020), the strategy introduced in our controlled experiments has much better generalization ability to test set. It is worth noting that, although our construction strategy has access to the label information (i.e., we sample the self-supervised signal image from the same character with the input image), our SSL objectives do not train with the labels. Nonetheless, since we implicitly utilize the label information in our self-supervised construction strategy, it will be unfair to directly compare our strategy and prior one. An interesting future research direction is examining different self-supervised signal construction strategy and even combine full/part of label information into self-supervised learning.
+
+![](images/975a718af39fb59a40227f9ceb2bbbde8478c83db9a9589e3a5e7d29592fe385.jpg)  
+Figure 6: Comparisons for different self-supervised signal construction strategies. The differences between the input and the self-supervised signals are {drawing styles, image augmentations} for our construction strategy and only {image augmentations} for SimCLR Chen et al. (2020)’s strategy. We choose $\operatorname { L } _ { C L }$ as our objective, reporting mean and its standard error from 5 random trials.
+
+# G METRICS IN VISUAL-TEXTUAL REPRESENTATION LEARNING
+
+• Subset Accuracy $( A )$ Sorower, also know as the Exact Match Ratio (MR), ignores all partially correct (consider them incorrect) outputs and extend accuracy from the single label case to the multi-label setting.
+
+$$
+M R = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \mathbb { 1 } _ { [ Y _ { i } = H _ { i } ] }
+$$
+
+• Micro AUC ROC score Fawcett (2006) computes the AUC (Area under the curve) of a receiver operating characteristic (ROC) curve.

parse/train/HJz05o0qK7/HJz05o0qK7.md

@@ -0,0 +1,404 @@
+# MEASURING COMPOSITIONALITY IN REPRESENTATION LEARNING
+
+Jacob Andreas Computer Science Division University of California, Berkeley jda@cs.berkeley.edu
+
+# ABSTRACT
+
+Many machine learning algorithms represent input data with vector embeddings or discrete codes. When inputs exhibit compositional structure (e.g. objects built from parts or procedures from subroutines), it is natural to ask whether this compositional structure is reflected in the the inputs’ learned representations. While the assessment of compositionality in languages has received significant attention in linguistics and adjacent fields, the machine learning literature lacks general-purpose tools for producing graded measurements of compositional structure in more general (e.g. vector-valued) representation spaces. We describe a procedure for evaluating compositionality by measuring how well the true representation-producing model can be approximated by a model that explicitly composes a collection of inferred representational primitives. We use the procedure to provide formal and empirical characterizations of compositional structure in a variety of settings, exploring the relationship between compositionality and learning dynamics, human judgments, representational similarity, and generalization.
+
+# 1 INTRODUCTION
+
+The success of modern representation learning techniques has been accompanied by an interest in understanding the structure of learned representations. One feature shared by many humandesigned representation systems is compositionality: the capacity to represent complex concepts (from objects to procedures to beliefs) by combining simple parts (Fodor & Lepore, 2002). While many machine learning approaches make use of human-designed compositional analyses for representation and prediction (Socher et al., 2013; Dong & Lapata, 2016), it is also natural to ask whether (and how) compositionality arises in learning problems where compositional structure has not been built in from the start. Consider the example in Figure 1, which shows a hypothetical character-based encoding scheme learned for a simple communication task (similar to the one studied by Lazaridou et al., 2016).
+
+![](images/30fd3900a44f2bd0b90a20e9472129db34cffa7608df1209a8d0634c96c73047.jpg)  
+Figure 1: Representations arising from a communication game. In this game, an observation (b) is presented to a learned speaker model (c), which encodes it as a discrete character sequence (d) to be consumed by a listener model for some downstream task. The space of inputs has known compositional structure (a). We want to measure the extent to which this structure is reflected (perhaps imperfectly) in the structure of the learned codes.
+
+Is this encoding scheme compositional? That is, to what extent can we analyze the agents’ messages as being built from smaller pieces (e.g. pieces xx meaning blue and bb meaning triangle)?
+
+A large body of work, from early experiments on language evolution to recent deep learning models (Kirby, 1998; Lazaridou et al., 2017), aims to answer questions like this one. But existing solutions rely on manual (and often subjective) analysis of model outputs (Mordatch & Abbeel, 2017), or at best automated procedures tailored to the specifics of individual problem domains (Brighton & Kirby, 2006). They are difficult to compare and difficult to apply systematically.
+
+We are left with a need for a standard, formal, automatable and quantitative technique for evaluating claims about compositional structure in learned representations. The present work aims at first steps toward meeting that need. We focus on an oracle setting where the compositional structure of model inputs is known, and where the only question is whether this structure is reflected in model outputs. This oracle evaluation paradigm covers most of the existing representation learning problems in which compositionality has been studied.
+
+The first contribution of this paper is a simple formal framework for measuring how well a collection of representations (discrete- or continuous-valued) reflects an oracle compositional analysis of model inputs. We propose an evaluation metric called TRE, which provides graded judgments of compositionality for a given set of (input, representation) pairs. The core of our proposal is to treat a set of primitive meaning representations as hidden, and optimize over them to find an explicitly compositional model that approximates the true model as well as possible. For example, if the compositional structure that describes an object is a simple conjunction of attributes, we can search for a collection of “attribute vectors” that sum together to produce the observed object representations; if it is a sparse combination of (attribute, value) pairs we can additionally search for “value vectors” and parameters of a binding operation; and so on for more complex compositions.
+
+Having developed a tool for assessing the compositionality of representations, the second contribution of this paper is a survey of applications. We present experiments and analyses aimed at answering four questions about the relationship between compositionality and learning:
+
+• How does compositionality of representations evolve in relation to other measurable model properties over the course of the learning process? (Section 4)   
+• How well does compositionality of representations track human judgments about the compositionality of model inputs? (Section 5)   
+• How does compositionality constrain distances between representations, and how does TRE relate to other methods that analyze representations based on similarity? (Section 6)   
+• Are compositional representations necessary for generalization to out-of-distribution inputs? (Section 7)
+
+We conclude with a discussion of possible applications and generalizations of TRE-based analysis.
+
+# 2 RELATED WORK
+
+Arguments about whether distributed (and other non-symbolic) representations could model compositional phenomena were a staple of 1980s-era connectionist–classicist debates. Smolensky (1991) provides an overview of this discussion and its relation to learnability, as well as a concrete implementation of a compositional encoding scheme with distributed representations. Since then, numerous other approaches for compositional representation learning have been proposed, with (Mitchell & Lapata, 2008; Socher et al., 2012) and without (Dircks & Stoness, 1999; Havrylov & Titov, 2017) the scaffolding of explicit composition operations built into the model.
+
+The main experimental question is thus when and how compositionality arises “from scratch” in the latter class of models. In order to answer this question it is first necessary to determine whether compositional structure is present at all. Most existing proposals come from linguistics and and philosophy, and offer evaluations of compositionality targeted at analysis of formal and natural languages (Carnap, 1937; Lewis, 1976). Techniques from this literature are specialized to the details of linguistic representations—particularly the algebraic structure of grammars (Montague, 1970). It is not straightforward to apply these techniques in more general settings, particularly those featuring non-string-valued representation spaces. We are not aware of existing work that describes a procedure suitable for answering questions about compositionality in the general case.
+
+Machine learning research has responded to this absence in several ways. One class of evaluations (Mordatch & Abbeel, 2017; Choi et al., 2018) derives judgments from ad-hoc manual analyses of representation spaces. These analyses provide insight into the organization of representations but are time-consuming and non-reproducible. Another class of evaluations (Brighton, 2002; Andreas & Klein, 2017; Bogin et al., 2018) exploits task-specific structure (e.g. the ability to elicit pairs of representations known to feature particular relationships) to give evidence of compositionality. Our work aims to provide a standard and scalable alternative to these model- and task-specific evaluations.
+
+Other authors refrain from measuring compositionality directly, and instead base analysis on measurement of related phenomena, for which more standardized evaluations exist. Examples include correlation between representation similarity and similarity of oracle compositional analyses (Brighton & Kirby, 2006) and generalization to structurally novel inputs (Kottur et al., 2017). Our approach makes it possible to examine the circumstances under which these surrogate measures in fact track stricter notions of compositionality; similarity is discussed in Sec. 6 and generalization in Sec. 7.
+
+A long line of work in natural language processing (Coecke et al., 2010; Baroni & Zamparelli, 2010; Clark, 2012; Fyshe et al., 2015) focuses on learning composition functions to produce distributed representations of phrases and sentences—that is, for purposes of modeling rather than evaluation. We use one experiment from this literature to validate our own approach (Section 5). On the whole, we view work on compositional representation learning in NLP as complementary to the framework presented here: our approach is agnostic to the particular choice of composition function, and the aforementioned references provide well-motivated choices suitable for evaluating data from language and other sources. Indeed, one view of the present work is simply as a demonstration that we can take existing NLP techniques for compositional representation learning, fit them to representations produced by other models (even in non-linguistic settings), and view the resulting training loss as a measure of the compositionality of the representation system in question.
+
+# 3 EVALUATING COMPOSITIONALITY
+
+Consider again the communication task depicted in Figure 1. Here, a speaker model observes a target object described by a feature vector. The speaker sends a message to a listener model, which uses the message to complete a downstream task—for example, identifying the referent from a collection of distractors based on the content of the message (Enquist & Arak, 1994; Lazaridou et al., 2017). Messages produced by the speaker model serve as representations of input objects; we want to know if these representations are compositional. Crucially, we may already know something about the structure of the inputs themselves. In this example, inputs can be identified via composition of categorical shape and color attributes. How might we determine whether this oracle analysis of input structure is reflected in the structure of representations? This section proposes an automated procedure for answering the question.
+
+Representations A representation learning problem is defined by a dataset $\mathcal { X }$ of observations $x$ (Figure 1b); a space $\Theta$ of representations $\theta$ (Figure 1d); and a model $f : \mathcal { X }  \Theta$ (Figure 1c). We assume that the representations produced by $f$ are used in a larger system to accomplish some concrete task, the details of which are not important for our analysis.
+
+Derivations The technique we propose additionally assumes we have prior knowledge about the compositional structure of inputs. In particular, we assume that inputs can be labeled with treestructured derivations $d$ (Figure 1a), defined by a finite set $\mathcal { D } _ { 0 }$ of primitives and a binary bracketing operation $\langle \cdot , \cdot \rangle$ , such that if $d _ { i }$ and $d _ { j }$ are derivations, $\langle d _ { i } , d _ { j } \rangle$ is a derivation. Derivations are produced by a derivation oracle $D : \mathcal { X }  \mathcal { D }$ .
+
+Compositionality In intuitive terms, the representations computed by $f$ are compositional if each $f ( x )$ is determined by the structure of $D ( x )$ . Most discussions of compositionality, following Montague (1970), make this precise by defining a composition operation $\theta _ { a } * \theta _ { b } \mapsto \theta$ in the space of representations. Then the model $f$ is compositional if it is a homomorphism from inputs to representations: we require that for any $x$ with $D ( x ) = \langle D ( x _ { a } ) , D ( x _ { b } ) \rangle$ ,
+
+$$
+f ( x ) = f ( x _ { a } ) * f ( x _ { b } ) .
+$$
+
+In the linguistic contexts for which this definition was originally proposed, it is straightforward to apply. Inputs $x$ are natural language strings. Their associated derivations $D ( x )$ are syntax trees, and composition of derivations is syntactic composition. Representations $\theta$ are logical representations of meaning (for an overview see van Benthem $\&$ ter Meulen, 1996). To argue that a particular fragment of language is compositional, it is sufficient to exhibit a lexicon $\mathcal { D } _ { 0 }$ mapping words to their associated meaning representations, and a grammar for composing meanings where licensed by derivations. Algorithms for learning grammars and lexicons from data are a mainstay of semantic parsing approaches to language understanding problems like question answering and instruction following (Zettlemoyer & Collins, 2005; Chen, 2012; Artzi et al., 2014).
+
+But for questions of compositionality involving more general representation spaces and more general analyses, the above definition presents two difficulties: (1) In the absence of a clearly-defined syntax of the kind available in natural language, how do we identify lexicon entries: the primitive parts from which representations are constructed? (2) What do we do with languages like the one in Figure 1d, which seem to exhibit some kind of regular structure, but for which the homomorphism condition given in Equation 1 cannot be made to hold exactly?
+
+Consider again the example in Figure 1. The oracle derivations tell us to identify primitive representations for dark, blue, green, square, and triangle. The derivations then suggest a process for composing these primitives (e.g. via string concatenation) to produce full representations. The speaker model is compositional (in the sense of Equation 1) as long as there is some assignment of representations to primitives such that for each model input, composing primitive representations according to the oracle derivation reproduces the speaker’s prediction.
+
+In Figure 1 there is no assignment of strings to primitives that reproduces model predictions exactly. But predictions can be reproduced approximately—by taking $\times \times$ to mean blue, aa to mean square, etc. The quality of the approximation serves as a measure of the compositionality of the true predictor: predictors that are mostly compositional but for a few exceptions, or compositional but for the addition of some noise, will be well-approximated on average, while arbitrary mappings from inputs to representations will not. This suggests that we should measure compositionality by searching for representations that allow an explicitly compositional model to approximate the true $f$ as closely as possible. We define our evaluation procedure as follows:
+
+# Tree Reconstruction Error (TRE)
+
+First choose :
+
+• a distance function $\delta : \Theta \times \Theta \to [ 0 , \infty )$ satisfying $\delta ( \theta , \theta ^ { \prime } ) = 0 \Leftrightarrow \theta = \theta ^ { \prime }$ • a composition function $\ast : \Theta \times \Theta \to \Theta$
+
+Define $\hat { f } _ { \eta } ( d )$ , a compositional approximation to $f$ with parameters $\eta$ , as:
+
+$$
+\begin{array} { c } { { \hat { f } _ { \eta } ( d _ { i } ) = \eta _ { i } } } \\ { { \hat { f } _ { \eta } \big ( \langle d , d ^ { \prime } \rangle \big ) = \hat { f } _ { \eta } ( d ) * \hat { f } _ { \eta } ( d ^ { \prime } ) } } \end{array}
+$$
+
+$$
+\begin{array} { l } { \mathrm { f o r } d _ { i } \in \mathcal { D } _ { 0 } } \\ { \mathrm { f o r } \mathrm { a l l } \mathrm { o t h e r } d } \end{array}
+$$
+
+$\hat { f } _ { \eta }$ has one parameter vector $\eta _ { i }$ for every $d _ { i }$ in $\mathcal { D } _ { 0 }$ ; these vectors are members of the representation space $\Theta$ .
+
+Given a dataset $\mathcal { X }$ of inputs $x _ { i }$ with derivations $d _ { i } = D ( x _ { i } )$ , compute:
+
+$$
+\eta ^ { * } = \arg \operatorname* { m i n } _ { \eta } \sum _ { i } \delta { \big ( } f ( x _ { i } ) , { \hat { f } } _ { \eta } ( d _ { i } ) { \big ) }
+$$
+
+Then we can define datum- and dataset-level evaluation metrics:
+
+$$
+\begin{array} { c } { { \mathrm { T R E } ( x ) = \displaystyle \delta \big ( f ( x ) , \hat { f } _ { \eta ^ { * } } ( d ) \big ) } } \\ { { \mathrm { T R E } ( \mathcal { X } ) = \displaystyle \frac { 1 } { n } \sum _ { i } \mathrm { T R E } ( x _ { i } ) } } \end{array}
+$$
+
+TRE and compositionality How well does the evaluation metric $\mathrm { T R E } ( \mathcal { X } )$ capture the intuition behind Equation 1? The definition above uses parameters $\eta _ { i }$ to witness the constructability of representations from parts, in this case by explicitly optimizing over those parts rather than taking them to be given by $f$ . Each term in Equation 2 is analogous to an instance of Equation 1, measuring how well $\hat { f } _ { \eta ^ { * } } ( x _ { i } )$ , the best compositional prediction, matches the true model prediction $f ( x _ { i } )$ . In the case of models that are homomorphisms in the sense of Equation 1, TRE reduces to the familiar case:
+
+Remark 1. $\mathrm { T R E } ( x ) = 0$ for all $x$ if and only if Equation 1 holds exactly (that is, $f ( x ) = f ( x _ { a } ) * f ( x _ { b } )$ for any $x , x _ { a } , x _ { b }$ with $D ( x ) = \langle D ( x _ { a } ) , D ( x _ { b } ) \rangle \rangle$ .
+
+Proof. One direction follows immediately from defining ${ \hat { f } } _ { \eta ^ { * } } ( x ) = f ( x )$ . For the other, $f ( x ) =$ ${ \hat { f } } ( D ( x ) ) = { \hat { f } } ( \langle D ( x _ { a } ) , D ( x _ { b } ) \rangle ) = { \hat { f } } ( D ( x _ { a } ) ) * { \hat { f } } ( D ( x _ { b } ) ) = f ( x _ { a } ) * f ( x _ { b } ) .$
+
+Learnable composition operators The definition of TRE leaves the choice of $\delta$ and $* \ \mathrm { u p }$ to the evaluator. Indeed, if the exact form of the composition function is not known a priori, it is natural to define $^ *$ with free parameters (as in e.g. Baroni & Zamparelli, 2010), treat these as another learned part of $\hat { f }$ , and optimize them jointly with the $\eta _ { i }$ . However, some care must be taken when choosing $^ *$ (especially when learning it) to avoid trivial solutions:
+
+Remark 2. Suppose $D$ is injective; that is, every $x \in \mathcal { X }$ is assigned a unique derivation. Then there is always some $^ *$ that achieves $\mathrm { T R E } ( \mathcal { X } ) = 0$ : simply define $f ( x _ { a } ) * f ( x _ { b } ) = f ( x ) .$ for any $x , x _ { a } , x _ { b }$ as in the preceding definition, and set ${ \hat { f } } = f$ .
+
+In other words, some pre-commitment to a restricted composition function is essentially inevitable: if we allow the evaluation procedure to select an arbitrary composition function, the result will be trivial. This paper features experiments with $^ *$ in both a fixed functional form and a learned parametric one.
+
+Implementation details For models with continuous $\Theta$ and differentiable $\delta$ and $^ *$ , $\mathrm { T R E } ( \mathcal { X } )$ is also differentiable. Equation 2 can be solved using gradient descent. We use this strategy in Sections 4 and 5. For discrete $\Theta$ , it may be possible to find a continuous relaxation with respect to which $\delta ( \theta , \cdot )$ and $^ *$ are differentiable, and gradient descent again employed. We use this strategy in Section 7 (discussed further there). An implementation of an SGD-based TRE solver is provided in the accompanying software release. For other problems, task-specific optimizers (e.g. machine translation alignment models; Bogin et al., 2018) or general-purpose discrete optimization toolkits can be applied to Equation 2.
+
+The remainder of the paper highlights ways of using TRE to answer questions about compositionality that arise in machine learning problems of various kinds.
+
+# 4 COMPOSITIONALITY AND LEARNING DYNAMICS
+
+We begin by studying the relationship between compositionality and learning dynamics, focusing on the information bottleneck theory of representation learning proposed by Tishby & Zaslavsky (2015). This framework proposes that learning in deep models consists of an error minimization phase followed by a compression phase, and that compression is characterized by a decrease in the mutual information between inputs and their computed representations. We investigate the hypothesis that the compression phase finds a compositional representation of the input distribution, isolating decision-relevant attributes and discarding irrelevant information.
+
+Data comes from a few-shot classification task. Because our analysis focuses on compositional hypothesis classes, we use visual concepts from the Color MNIST dataset of Seo et al. (2017) (Figure 2). We predict classifiers in a meta-learning framework (Schmidhuber, 1987; Santoro et al., 2016): for each sub-task, the learner is presented with two images corresponding to some compositional visual concept (e.g. “digit 8 on a black background” or “green with heavy stroke”) and must determine whether a held-out image is an example of the same visual concept.
+
+![](images/e2079b0cfcc80dba025b391c6de4d28344306c749460b8f6d19f3d68b71209f6.jpg)  
+Figure 2: Meta-learning task: learners are presented with two example images depicting a visual concept (a), and must determine whether a third image (b) is an example of the same concept (c).
+
+![](images/22d41ccd6dcfd0a29fab0ebd9d505f674866a7361329c81645c87376dd97c62e.jpg)  
+Figure 3: Relationship between reconstruction error TRE and mutual information $I ( \theta ; X )$ between inputs and representations. (a) Evolution of the two quantities over the course of a single run. Both initially increase, then decrease. The color bar shows the training epoch. (b) Values from ten training runs. (c) Values from the second half of each training run, taken to begin when $I ( \theta ; X )$ reaches a maximum. In (b) and (c), the observed correlation is significant: respectively $\mathrm { \Delta } r = 0 . 7 0$ , $p < 1 e { - } 1 0 )$ ) and $\mathrm { \Delta } r = 0 . 7 1$ , $p < 1 e { - 8 } )$ .
+
+Given example images $x _ { 1 }$ and $x _ { 2 }$ , a test image $x ^ { * }$ , and label $y ^ { * }$ , the model computes:
+
+$$
+\begin{array} { r l } & { z _ { i } =  { \mathtt { C N N } } ( x _ { i } ) \mathrm { ~ f o r ~ } i \in \{ 1 , 2 , * \} } \\ & { \theta = \operatorname { t a n h } ( W ( z _ { 1 } + z _ { 2 } ) ) } \\ & { \hat { y } = \theta ^ { \top } z _ { t } } \end{array}
+$$
+
+We use $\theta$ as the representation of a classifier for analysis. The model is trained to minimize the logistic loss between logits $\hat { y }$ and ground-truth labels $y ^ { * }$ . More details are given in Appendix A.
+
+Compositional structure Visual concepts used in this task are all single attributes or conjunctions of attributes; i.e. their associated derivations are of the form attr or $\langle \arctan _ { \mathsf { 1 } } , \mathsf { a t t r } _ { \mathsf { 2 } } \rangle$ . Attributes include background color, digit color, digit identity and stroke type. The composition function $^ *$ is addition and the distance $\delta ( \theta , \theta ^ { \prime } )$ is cosine similarity $1 - \theta ^ { \top } \theta ^ { \prime } / ( \lVert \theta \rVert \lVert \theta ^ { \prime } \rVert )$ .
+
+Evaluation The training dataset consists of 9000 image triplets, evenly balanced between positive and negative classes, with a validation set of 500 examples. At convergence, the model achieves validation accuracy of $7 5 . 2 \%$ on average over ten training runs. (Perfect accuracy is not possible because the true classifier is not fully determined by two training examples). We explore the relationship between the information bottleneck and compositionality by comparing $\mathrm { T R E } ( \mathcal { X } )$ to the mutual information $I ( \theta ; x )$ between representations and inputs over the course of training. Both quantities are computed on the validation set, calculating $\mathrm { T R E } ( \mathcal { X } )$ as described in Section 3 and $I ( \theta ; X )$ as described in Shwartz-Ziv & Tishby (2017). (For discussion of limitations of this approach to computing mutual information between inputs and representations, see Saxe et al., 2018.)
+
+Figure 3 shows the relationship between $\mathrm { T R E } ( \mathcal { X } )$ and $I ( \theta ; X )$ . Recall that small TRE is indicative of a high degree of compositionality. It can be seen that both mutual information and reconstruction error are initially low (because representations initially encode little about distinctions between inputs). Both increase over the course of training, and decrease together after mutual information reaches a maximum (Figure 3a). This pattern holds if we plot values from multiple training runs at the same time (Figure 3b), or if we consider only the postulated compression phase (Figure 3c). These results are consistent with the hypothesis that compression in the information bottleneck framework is associated with the discovery of compositional representations.
+
+# 5 COMPOSITIONALITY AND HUMAN JUDGMENTS
+
+Next we investigate a more conventional representation learning task. High-dimensional embeddings of words and phrases are useful for many natural language processing applications (Turian et al., 2010), and many techniques exist to learn them from unlabeled text (Deerwester et al., 1990; Mikolov et al., 2013). The question we wish to explore is not whether phrase vectors are compositional in aggregate, but rather how compositional individual phrase representations are. Our hypothesis is that bigrams whose representations have low TRE are those whose meaning is essentially compositional, and well-explained by the constituent words, while bigrams with large reconstruction error will correspond to non-compositional multi-word expressions (Nattinger & DeCarrico, 1992).
+
+This task is already well-studied in the natural language processing literature (Salehi et al., 2015), and the analysis we present differs only in the use of TRE to search for atomic representations rather than taking them to be given by pre-trained word representations. Our goal is to validate our approach in a language processing context, and show how existing work on compositionality (and representations of natural language in particular) fit into the more general framework proposed in the current paper.
+
+We train embeddings for words and bigrams using the CBOW objective of Mikolov et al. (2013) using the implementation provided in FastText (Bojanowski et al., 2017) with 100-dimensional vectors and a context size of 5. Vectors are estimated from a 250M-word subset of the Gigaword dataset (Parker et al., 2011). More details are provided in Appendix A.
+
+Compositional structure We want to know how close phrase embeddings are to the composition of their constituent word embeddings. We define derivations for words and phrases in the natural way: single words $w$ have primitive derivations $d = w$ ; bigrams $w _ { 1 } w _ { 2 }$ have derivations of the form $\langle w _ { 1 } , w _ { 2 } \rangle$ . The composition function is again vector addition and distance is cosine distance. (Future work might explore learned composition functions as in e.g. Grefenstette et al., 2013, for future work.) We compare bigram-level judgments of compositionality computed by TRE with a dataset of human judgments about noun–noun compounds (Reddy et al., 2011). In this dataset, humans rate bigrams as compositional on a scale from 0 to 5, with highly conventionalized phrases like gravy train assigned low scores and graduate student assigned high ones.
+
+Results We reproduce the results of Salehi et al. (2015) within the tree reconstruction error framework: for a given $x$ , $\mathrm { T R E } ( x )$ is anticorrelated with human judgments of compositionality $\zeta = - 0 . 3 4$ , $p < 0 . 0 1 $ ). Collocations rated “most compositional” by our approach (i.e. with lowest TRE) are: application form, polo shirt, research project; words rated “least compositional” are fine line, lip service, and nest egg.
+
+# 6 COMPOSITIONALITY AND SIMILARITY
+
+The next section aims at providing a formal, rather than experimental, characterization of the relationship between TRE and another perspective on the analysis of representations with help from oracle derivations. Brighton & Kirby (2006) introduce a notion of topographic similarity, arguing that a learned representation captures relevant domain structure if distances between learned representations are correlated with distances between their associated derivations. This can be viewed as providing a weak form of evidence for compositionality—if the distance function rewards pairs of representations that share overlapping substructure (as might be the case with e.g. string edit distance), edit distance will be expected to correlate with some notion of derivational similarity (Lazaridou et al., 2018).
+
+In this section we aim to clarify the relationship between the two evaluations. To do this we first need to equip the space of derivations described in Section 3 with a distance function. As the derivations considered in this paper are all tree-structured, it is natural to use a simple tree edit distance (Bille, 2005) for this purpose. We claim the following:
+
+Proposition 1. Let $\hat { f } = \hat { f } _ { \eta ^ { * } }$ be an approximation to $f$ estimated as in Equation 2, with all $\mathrm { T R E } ( x ) \leq \epsilon$ for some . Let $\Delta$ be the tree edit distance (defined formally in Appendix B, Definition 2), and let δ be any distance on $\Theta$ satisfying the following properties:
+
+$$
+\delta ( \hat { f } ( d _ { i } ) , \hat { f } ( d _ { j } ) ) \leq 1 f o r d _ { i } , d _ { j } \in \mathcal { D } _ { 0 }
+$$
+
+2. $\delta ( { \hat { f } } ( d ) , 0 ) \leq 1$ for $d \in \mathcal { D } _ { 0 }$ , where 0 is the identity element for $^ *$
+
+3. $\delta ( \theta _ { i } \ast \theta _ { j } , \theta _ { k } \ast \theta _ { \ell } ) \leq \delta ( \theta _ { i } , \theta _ { k } ) + \delta ( \theta _ { j } , \theta _ { \ell } ) .$ . (This condition is satisfied by any translation-invariant metric.)
+
+Then $\Delta$ is an approximate upper bound on $\delta$ : for any $x$ , $x ^ { \prime }$ with $d = D ( x )$ , $d ^ { \prime } = D ( x ^ { \prime } )$
+
+$$
+\delta ( f ( x ) , f ( x ^ { \prime } ) ) \leq \Delta ( d , d ^ { \prime } ) + 2 \epsilon .
+$$
+
+In other words, representations cannot be much farther apart than the derivations that produce them.   
+Proof is provided in Appendix B.
+
+We emphasize that small TRE is not a sufficient condition for topographic similarity as defined by Brighton & Kirby (2006): very different derivations might be associated with the same representation (e.g. when representing arithmetic expressions by their results). But this result does demonstrate that compositionality imposes some constraints on the inferences that can be drawn from similarity judgments between representations.
+
+# 7 COMPOSITIONALITY AND GENERALIZATION
+
+In our final set of experiments, we investigate the relationship between compositionality and generalization. Here we focus on communication games like the one depicted in Figure 1 and in more detail in Figure 4. As in the previous section, existing work argues for a relationship between compositionality and generalization, claiming that agents need compositional communication protocols to generalize to unseen referents (Kottur et al., 2017; Choi et al., 2018). Here we are able to evaluate this claim empirically by training a large number of agents from random initial conditions, measuring the compositional structure of the language that emerges, and seeing how this relates to their performance on both familiar and novel objects.
+
+Our experiment focuses on a reference game (Gatt et al., 2007). Two policies are trained: a speaker and a listener. The speaker observes pair of target objects represented with a feature vector. The speaker then sends a message (coded as a discrete character sequence) to the listener model. The listener observes this message and attempts to reconstruct the target objects by predicting a sequence of attribute sets. If all objects are predicted correctly, both the speaker and the listener receive a reward of 1 (partial credit is awarded for partly-correct objects; Figure 4).
+
+Because the communication protocol is discrete, policies are jointly trained using a policy gradient objective (Williams, 1992). The speaker and listener are implemented with RNNs; details are provided in Appendix A.
+
+Compositional structure Every target referent consists of two objects; each object has two attributes. The derivation associated with each communicative task thus has the tree structure $\langle \langle \mathsf { a t t r } _ { 1 a } , \mathsf { a t t r } _ { 1 b } \rangle$ , $ \mathsf { a t t r } _ { 2 a } , \mathsf { a t t r } _ { 2 b }  $ . We hold out a subset of these object pairs at training time to evaluate generalization: in each training run, 1/3 of possible reference candidates are never presented to the agent at training time.
+
+![](images/d78ad82d3d03a6516b645718844166c74e09fb61482aa4d425065d389bdcf188.jpg)  
+Figure 4: The communication task: A speaker model observes a pair of target objects, and sends a description of the objects (as a discrete code) to a listener model. The listener attempts to reconstruct the targets, receiving fractional reward for partially-correct predictions.
+
+Where the previous examples involved a representation space of real embeddings, here representations are fixed-length discrete codes. Moreover, the derivations themselves have a more complicated semantics than in Sections 4 and 5: order matters, and a commutative operation like addition cannot capture the distinction between hhgreen, squarei, $\left. \mathrm { b 1 u e } , \mathrm { t r i a n g 1 e } \right. \rangle$ and $\langle \langle { \mathsf { g r e e n } } , { \mathsf { t r i a n g l e } } \rangle$ , $\langle \mathtt { b l u e } , \mathtt { s q u a r e } \rangle \rangle$ . We thus need a different class of composition and distance operations. We represent each agent message as a sequence of one-hot vectors, and take the error function $\delta$ to be the $\ell _ { 1 }$ distance between vectors. The composition function has the form:
+
+$$
+\theta * \theta ^ { \prime } = A \theta + B \theta ^ { \prime }
+$$
+
+with free composition parameters $\eta _ { * } = \{ A , B \}$ in Equation 2. These matrices can redistribute the tokens in $\theta$ and $\theta ^ { \prime }$ across different positions of the input string, but cannot affect the choice of the tokens themselves; this makes it possible to model non-commutative aspects of string production. To compute TRE via gradient descent, we allow the elements of $\mathcal { D } _ { 0 }$ to be arbitrary vectors (intuitively assigning fractional token counts to string indices) rather than restricting them to one-hot indicators. With this change, both $\delta$ and $^ *$ have subgradients and can be optimized using the same procedure as in preceding sections.
+
+![](images/a583b3354405f8fca05ffef241f58682de4171ec038521fdb8bcc174710c75bb.jpg)  
+Figure 5: Relationship between TRE and reward. (a) Compositional languages exhibit lower generalization error, measured as the difference between train and test reward $r = 0 . 5 0$ , $p < 1 e { - } 6 )$ . (b) However, compositional languages also exhibit lower absolute performance ${ \mathrm { ' } r = 0 . 5 7 }$ , $p < 1 e { - 9 } ,$ ). Both facts remain true even if we restrict analysis to “successful” training runs in which agents achieve a reward $> 0 . 5$ on held-out referents $r = 0 . 6$ , $p < 1 e { - 3 }$ and $r = 0 . 3 8$ , $p < 0 . 0 5$ respectively).   
+Figure 6: Fragment of languages resulting from two multiagent training runs. In the first section, the left column shows the target referent, while the remaining columns show the message generated by speaker in the given training run after observing the referent. The two languages have substantially different TRE, but induce similar listener performance (Train and Test reward).
+
+<table><tr><td colspan="5">Language A</td></tr><tr><td>((red</td><td>circle)</td><td>(blue</td><td>triangle)）</td><td>jjjj</td></tr><tr><td>((red</td><td>circle)</td><td>(blue</td><td>star))</td><td>jeoo oppp jjjj</td></tr><tr><td>((red</td><td>circle)</td><td>(blue</td><td>circle))</td><td>oopp jjjj</td></tr><tr><td>((red</td><td>circle)</td><td>(blue</td><td>square))</td><td>oopp jjjb</td></tr><tr><td>((red</td><td>square)</td><td>(blue</td><td>triangle))</td><td>jjjj jbjj</td></tr><tr><td>((red</td><td>square)</td><td>(blue</td><td>star))</td><td>o00o jbjj</td></tr><tr><td>((red</td><td>square)</td><td>(blue</td><td>circle))</td><td>oo0o jbbb</td></tr><tr><td>((red</td><td>square)</td><td>(blue</td><td>square))</td><td>o00o jbbb</td></tr><tr><td colspan="3">TRE</td><td>4.30</td><td>2.96</td></tr><tr><td colspan="3">Train reward</td><td>0.78</td><td>0.75</td></tr><tr><td colspan="3">Test reward</td><td>0.61</td><td>0.59</td></tr></table>
+
+Results We train 100 speaker–listener pairs with random initial parameters and measure their performance on both training and test sets. Our results suggest a more nuanced view of the relationship between compositionality and generalization than has been argued in the existing literature. TRE is significantly correlated with generalization error (measured as the difference between training accuracies, Figure 5a). However, TRE is also significantly correlated with absolute model reward (Figure 5b)—“compositional” languages more often result from poor communication strategies than successful ones. This is largely a consequence of the fact that many languages with low TRE correspond to trivial strategies (for example, one in which the speaker sends the same message regardless of its observation) that result in poor overall performance.
+
+Moreover, despite the correlation between TRE and generalization error, low TRE is by no means a necessary condition for good generalization. We can use our technique to automatically mine a collection of training runs for languages that achieve good generalization performance at both low and high levels of compositionality. Examples of such languages are shown in Figure 6.
+
+# 8 CONCLUSIONS
+
+We have introduced a new evaluation method called TRE for generating graded judgments about compositional structure in representation learning problems where the structure of the observations is understood. TRE infers a set of primitive meaning representations that, when composed, approximate the observed representations, then measures the quality of this approximation. We have applied TRE-based analysis to four different problems in representation learning, relating compositionality to learning dynamics, linguistic compositionality, similarity and generalization.
+
+Many interesting questions regarding compositionality and representation learning remain open. The most immediate is how to generalize TRE to the setting where oracle derivations are not available; in this case Equation 2 must be solved jointly with an unsupervised grammar induction problem (Klein & Manning, 2004). Beyond this, it is our hope that this line of research opens up two different kinds of new work: better understanding of existing machine learning models, by providing a new set of tools for understanding their representational capacity; and better understanding of problems, by better understanding the kinds of data distributions and loss functions that give rise to compositionalor non-compositional representations of observations.
+
+# REPRODUCIBILITY
+
+Code and data for all experiments in this paper are provided at https://github.com/jacobandreas/tre.
+
+# ACKNOWLEDGMENTS
+
+Thanks to Daniel Fried and David Gaddy for feedback on an early draft of this paper. The author was supported by a Facebook Graduate Fellowship at the time of writing.
+
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+
+# A MODELING DETAILS
+
+Few-shot classification The CNN has the following form:
+
+Conv(out $= 6$ , kernel $= 5$ )   
+ReLU   
+MaxPool(kernel $^ { = 2 }$ )   
+Conv(out $= 1 6$ , kernel $= 5$ )   
+ReLU   
+MaxPool(kernel $^ { = 2 }$ )   
+Linear(out $= 1 2 8$ )   
+ReLU   
+Linear(out $= 6 4$ )   
+ReLU
+
+The model is trained using ADAM (Kingma & Ba, 2014) with a learning rate of .001 and a batch size of 128. Training is ended when the model stops improving on a held-out set.
+
+Word embeddings We train FastText (Bojanowski et al., 2017) on the first 250 million words of the NYT section of Gigaword (Parker et al., 2011). To acquire bigram representations, we pre-process this dataset so that each occurrence of a bigram from the Reddy et al. (2011) dataset is treated as a single word for purposes of estimating word vectors.
+
+Communication The encoder and decoder RNNs both use gated recurrent units (Cho et al., 2014) with embeddings and hidden states of size 256. The size of the discrete vocabulary is set to 16 and the maximum message length to 4. Training uses a policy gradient objective with a scalar baseline set to the running average reward; this is optimized using ADAM (Kingma & Ba, 2014) with a learning rate of .001 and a batch size of 256. Each model is trained for 500 steps. Models are trained by sampling from the decoder’s output distribution, but greedy decoding is used to evaluate performance and produce Figure 6.
+
+# B PROPOSITION 1
+
+First, some definitions:
+
+Definition 1. The size of a derivation is given by:
+
+$$
+\begin{array} { r l r } { | d | = 1 } & { { } } & { i f d \in \mathcal { D } _ { 0 } } \\ { | \langle d _ { a } , d _ { b } \rangle | = | d _ { a } | + | d _ { b } | } & { { } } & { o t h e r w i s e } \end{array}
+$$
+
+Definition 2. The tree edit distance between derivations is defined by:
+
+$$
+\Delta ( d _ { i } , d _ { j } ) = \mathbb { I } [ i = j ] \quad i f d _ { i } \in \mathcal { D } _ { 0 } a n d d _ { j } \in \mathcal { D } _ { 0 }
+$$
+
+$$
+\Delta ( d _ { i } , \langle d _ { j } , d _ { k } \rangle ) = \operatorname* { m i n } \left\{ \begin{array} { l l } { \Delta ( d _ { i } , d _ { j } ) + | d _ { k } | } \\ { \Delta ( d _ { i } , d _ { k } ) + | d _ { j } | } \end{array} \right\} \quad i f d _ { i } \in \mathcal { D } _ { 0 }
+$$
+
+$$
+\Delta ( \langle d _ { i } , d _ { j } \rangle , \langle d _ { k } , d _ { \ell } \rangle ) = \operatorname* { m i n } \left\{ \begin{array} { l l } { \Delta ( d _ { i } , d _ { k } ) + \Delta ( d _ { j } , d _ { \ell } ) } \\ { \Delta ( \langle d _ { i } , d _ { j } \rangle , d _ { k } ) + | d _ { \ell } | } & { \Delta ( \langle d _ { i } , d _ { j } \rangle , d _ { \ell } ) + | d _ { k } | } \\ { \Delta ( \langle d _ { k } , d _ { \ell } \rangle , d _ { i } ) + | d _ { j } | } & { \Delta ( \langle d _ { k } , d _ { \ell } \rangle , d _ { j } ) + | d _ { i } | } \end{array} \right\}
+$$
+
+Now, suppose we have $x$ and $x ^ { \prime }$ with derivations $d = D ( x )$ , $d ^ { \prime } = D ( x ^ { \prime } )$ and representations $\theta = f ( x )$ , $\theta ^ { \prime } = f ( x ^ { \prime } )$ . Proposition 1 claims that $\delta ( \theta , \theta ^ { \prime } ) \leq \Delta ( d , d ^ { \prime } ) + 2 \epsilon$ .
+
+Lemma 1. $\delta ( { \hat { f } } ( d ) , 0 ) \leq | d |$
+
+Proof. For $d \in \mathcal { D } _ { 0 }$ this follows immediately from Condition 2 in the proposition. For composed derivations it follows from Condition 3 taking ${ \theta _ { k } } = { \theta _ { \ell } } = 0$ and induction on $| d |$ . □
+
+Lemma 2. $\delta ( { \hat { f } } ( d ) , { \hat { f } } ( d ^ { \prime } ) ) \leq \Delta ( d , d ^ { \prime } )$
+
+Proof. By induction on the structure of $d$ and $d ^ { \prime }$ :
+
+Base case Both $d , d ^ { \prime } \in \mathcal { D } _ { 0 }$ .
+
+If $= d ^ { \prime } , \delta ( \hat { f } ( d ) , \hat { f } ( d ^ { \prime } ) ) = \delta ( \hat { f } ( d ) , \hat { f } ( d ) ) = 0 = \Delta ( d , d ^ { \prime } ) .$ If $d \neq d ^ { \prime }$ $d ^ { \prime } , \delta ( \hat { f } ( d ) , \hat { f } ( d ^ { \prime } ) ) \leq 1 = \Delta ( d , d ^ { \prime } )$ from Condition 1.
+
+Inductive case Consider the arrangement of derivations that minimizes Equation 8 for derivation $d$ and $d ^ { \prime }$ . There are two possibilities:
+
+Case 1: $\Delta ( d , d ^ { \prime } )$ has the form $\Delta ( d _ { i } , d _ { k } ) + \Delta ( d _ { j } , d _ { \ell } )$ for some $d _ { i , j , k , \ell }$ . W.l.o.g. let $d = \langle d _ { i } , d _ { j } \rangle$ and $d ^ { \prime } = \langle d _ { k } , d _ { \ell } \rangle$ . Then,
+
+$$
+\begin{array} { r l } & { \delta ( \hat { f } ( d ) , \hat { f } ( d ^ { \prime } ) ) = \delta ( \hat { f } ( d _ { i } ) * \hat { f } ( d _ { j } ) , \hat { f } ( d _ { k } ) * \hat { f } d _ { \ell } ) } \\ & { \hphantom { \delta ( \hat { f } ( d ) , \hat { f } ( d _ { k } ) ) = } \leq \delta ( \hat { f } ( d _ { i } ) , \hat { f } ( d _ { k } ) ) + \delta ( \hat { f } ( d _ { j } ) , \hat { f } d _ { \ell } ) } \\ & { \hphantom { \delta ( \hat { f } ( d ) , \hat { f } ( d _ { k } ) ) = } \leq \Delta ( d _ { i } , d _ { k } ) + \Delta ( d _ { j } , d _ { \ell } ) } \\ & { \hphantom { \delta ( \hat { f } ( d ) , \hat { f } ( d ) ) = } = \Delta ( d , d ^ { \prime } ) } \end{array}
+$$
+
+Case 2: $\Delta ( d , d ^ { \prime } )$ has the form $\Delta ( d _ { i } , d _ { k } ) + | d _ { j } |$ for some $d _ { i , j , k }$ . W.l.o.g. let $d = \langle d _ { i } , d _ { j } \rangle$ and $d ^ { \prime } = d _ { k }$ . Abusing notation slightly, let us define $\Delta ( \bar { d } , 0 ) = { \left| { d } \right| }$ . If we let $d _ { \ell } = 0$ this case reduces to the previous one. □
+
+Finally,
+
+Proof of Proposition $^ { l }$ .
+
+$$
+\begin{array} { c } { { \delta ( \theta , \theta ^ { \prime } ) \leq \delta ( \hat { f } ( d ) , \hat { f } ( d ^ { \prime } ) ) + 2 \epsilon } } \\ { { \leq \Delta ( d , d ^ { \prime } ) + 2 \epsilon } } \end{array}
+$$

parse/train/HJz05o0qK7/HJz05o0qK7_content_list.json

@@ -0,0 +1,2074 @@
+[
+    {
+        "type": "text",
+        "text": "MEASURING COMPOSITIONALITY IN REPRESENTATION LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            99,
+            612,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jacob Andreas Computer Science Division University of California, Berkeley jda@cs.berkeley.edu ",
+        "bbox": [
+            184,
+            170,
+            408,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            262,
+            544,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Many machine learning algorithms represent input data with vector embeddings or discrete codes. When inputs exhibit compositional structure (e.g. objects built from parts or procedures from subroutines), it is natural to ask whether this compositional structure is reflected in the the inputs’ learned representations. While the assessment of compositionality in languages has received significant attention in linguistics and adjacent fields, the machine learning literature lacks general-purpose tools for producing graded measurements of compositional structure in more general (e.g. vector-valued) representation spaces. We describe a procedure for evaluating compositionality by measuring how well the true representation-producing model can be approximated by a model that explicitly composes a collection of inferred representational primitives. We use the procedure to provide formal and empirical characterizations of compositional structure in a variety of settings, exploring the relationship between compositionality and learning dynamics, human judgments, representational similarity, and generalization. ",
+        "bbox": [
+            233,
+            296,
+            766,
+            489
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            522,
+            336,
+            537
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The success of modern representation learning techniques has been accompanied by an interest in understanding the structure of learned representations. One feature shared by many humandesigned representation systems is compositionality: the capacity to represent complex concepts (from objects to procedures to beliefs) by combining simple parts (Fodor & Lepore, 2002). While many machine learning approaches make use of human-designed compositional analyses for representation and prediction (Socher et al., 2013; Dong & Lapata, 2016), it is also natural to ask whether (and how) compositionality arises in learning problems where compositional structure has not been built in from the start. Consider the example in Figure 1, which shows a hypothetical character-based encoding scheme learned for a simple communication task (similar to the one studied by Lazaridou et al., 2016). ",
+        "bbox": [
+            174,
+            556,
+            485,
+            818
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/30fd3900a44f2bd0b90a20e9472129db34cffa7608df1209a8d0634c96c73047.jpg",
+        "image_caption": [
+            "Figure 1: Representations arising from a communication game. In this game, an observation (b) is presented to a learned speaker model (c), which encodes it as a discrete character sequence (d) to be consumed by a listener model for some downstream task. The space of inputs has known compositional structure (a). We want to measure the extent to which this structure is reflected (perhaps imperfectly) in the structure of the learned codes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            506,
+            559,
+            812,
+            671
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Is this encoding scheme compositional? That is, to what extent can we analyze the agents’ messages as being built from smaller pieces (e.g. pieces xx meaning blue and bb meaning triangle)? ",
+        "bbox": [
+            174,
+            819,
+            821,
+            847
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A large body of work, from early experiments on language evolution to recent deep learning models (Kirby, 1998; Lazaridou et al., 2017), aims to answer questions like this one. But existing solutions rely on manual (and often subjective) analysis of model outputs (Mordatch & Abbeel, 2017), or at best automated procedures tailored to the specifics of individual problem domains (Brighton & Kirby, 2006). They are difficult to compare and difficult to apply systematically. ",
+        "bbox": [
+            174,
+            854,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We are left with a need for a standard, formal, automatable and quantitative technique for evaluating claims about compositional structure in learned representations. The present work aims at first steps toward meeting that need. We focus on an oracle setting where the compositional structure of model inputs is known, and where the only question is whether this structure is reflected in model outputs. This oracle evaluation paradigm covers most of the existing representation learning problems in which compositionality has been studied. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The first contribution of this paper is a simple formal framework for measuring how well a collection of representations (discrete- or continuous-valued) reflects an oracle compositional analysis of model inputs. We propose an evaluation metric called TRE, which provides graded judgments of compositionality for a given set of (input, representation) pairs. The core of our proposal is to treat a set of primitive meaning representations as hidden, and optimize over them to find an explicitly compositional model that approximates the true model as well as possible. For example, if the compositional structure that describes an object is a simple conjunction of attributes, we can search for a collection of “attribute vectors” that sum together to produce the observed object representations; if it is a sparse combination of (attribute, value) pairs we can additionally search for “value vectors” and parameters of a binding operation; and so on for more complex compositions. ",
+        "bbox": [
+            174,
+            194,
+            825,
+            333
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Having developed a tool for assessing the compositionality of representations, the second contribution of this paper is a survey of applications. We present experiments and analyses aimed at answering four questions about the relationship between compositionality and learning: ",
+        "bbox": [
+            176,
+            340,
+            821,
+            382
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• How does compositionality of representations evolve in relation to other measurable model properties over the course of the learning process? (Section 4)   \n• How well does compositionality of representations track human judgments about the compositionality of model inputs? (Section 5)   \n• How does compositionality constrain distances between representations, and how does TRE relate to other methods that analyze representations based on similarity? (Section 6)   \n• Are compositional representations necessary for generalization to out-of-distribution inputs? (Section 7) ",
+        "bbox": [
+            215,
+            392,
+            825,
+            516
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We conclude with a discussion of possible applications and generalizations of TRE-based analysis. ",
+        "bbox": [
+            173,
+            525,
+            815,
+            540
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            559,
+            344,
+            575
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Arguments about whether distributed (and other non-symbolic) representations could model compositional phenomena were a staple of 1980s-era connectionist–classicist debates. Smolensky (1991) provides an overview of this discussion and its relation to learnability, as well as a concrete implementation of a compositional encoding scheme with distributed representations. Since then, numerous other approaches for compositional representation learning have been proposed, with (Mitchell & Lapata, 2008; Socher et al., 2012) and without (Dircks & Stoness, 1999; Havrylov & Titov, 2017) the scaffolding of explicit composition operations built into the model. ",
+        "bbox": [
+            174,
+            589,
+            825,
+            688
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The main experimental question is thus when and how compositionality arises “from scratch” in the latter class of models. In order to answer this question it is first necessary to determine whether compositional structure is present at all. Most existing proposals come from linguistics and and philosophy, and offer evaluations of compositionality targeted at analysis of formal and natural languages (Carnap, 1937; Lewis, 1976). Techniques from this literature are specialized to the details of linguistic representations—particularly the algebraic structure of grammars (Montague, 1970). It is not straightforward to apply these techniques in more general settings, particularly those featuring non-string-valued representation spaces. We are not aware of existing work that describes a procedure suitable for answering questions about compositionality in the general case. ",
+        "bbox": [
+            174,
+            694,
+            825,
+            819
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Machine learning research has responded to this absence in several ways. One class of evaluations (Mordatch & Abbeel, 2017; Choi et al., 2018) derives judgments from ad-hoc manual analyses of representation spaces. These analyses provide insight into the organization of representations but are time-consuming and non-reproducible. Another class of evaluations (Brighton, 2002; Andreas & Klein, 2017; Bogin et al., 2018) exploits task-specific structure (e.g. the ability to elicit pairs of representations known to feature particular relationships) to give evidence of compositionality. Our work aims to provide a standard and scalable alternative to these model- and task-specific evaluations. ",
+        "bbox": [
+            174,
+            827,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Other authors refrain from measuring compositionality directly, and instead base analysis on measurement of related phenomena, for which more standardized evaluations exist. Examples include correlation between representation similarity and similarity of oracle compositional analyses (Brighton & Kirby, 2006) and generalization to structurally novel inputs (Kottur et al., 2017). Our approach makes it possible to examine the circumstances under which these surrogate measures in fact track stricter notions of compositionality; similarity is discussed in Sec. 6 and generalization in Sec. 7. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A long line of work in natural language processing (Coecke et al., 2010; Baroni & Zamparelli, 2010; Clark, 2012; Fyshe et al., 2015) focuses on learning composition functions to produce distributed representations of phrases and sentences—that is, for purposes of modeling rather than evaluation. We use one experiment from this literature to validate our own approach (Section 5). On the whole, we view work on compositional representation learning in NLP as complementary to the framework presented here: our approach is agnostic to the particular choice of composition function, and the aforementioned references provide well-motivated choices suitable for evaluating data from language and other sources. Indeed, one view of the present work is simply as a demonstration that we can take existing NLP techniques for compositional representation learning, fit them to representations produced by other models (even in non-linguistic settings), and view the resulting training loss as a measure of the compositionality of the representation system in question. ",
+        "bbox": [
+            174,
+            194,
+            826,
+            347
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 EVALUATING COMPOSITIONALITY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            368,
+            491,
+            385
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consider again the communication task depicted in Figure 1. Here, a speaker model observes a target object described by a feature vector. The speaker sends a message to a listener model, which uses the message to complete a downstream task—for example, identifying the referent from a collection of distractors based on the content of the message (Enquist & Arak, 1994; Lazaridou et al., 2017). Messages produced by the speaker model serve as representations of input objects; we want to know if these representations are compositional. Crucially, we may already know something about the structure of the inputs themselves. In this example, inputs can be identified via composition of categorical shape and color attributes. How might we determine whether this oracle analysis of input structure is reflected in the structure of representations? This section proposes an automated procedure for answering the question. ",
+        "bbox": [
+            174,
+            400,
+            825,
+            540
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Representations A representation learning problem is defined by a dataset $\\mathcal { X }$ of observations $x$ (Figure 1b); a space $\\Theta$ of representations $\\theta$ (Figure 1d); and a model $f : \\mathcal { X }  \\Theta$ (Figure 1c). We assume that the representations produced by $f$ are used in a larger system to accomplish some concrete task, the details of which are not important for our analysis. ",
+        "bbox": [
+            174,
+            556,
+            825,
+            612
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Derivations The technique we propose additionally assumes we have prior knowledge about the compositional structure of inputs. In particular, we assume that inputs can be labeled with treestructured derivations $d$ (Figure 1a), defined by a finite set $\\mathcal { D } _ { 0 }$ of primitives and a binary bracketing operation $\\langle \\cdot , \\cdot \\rangle$ , such that if $d _ { i }$ and $d _ { j }$ are derivations, $\\langle d _ { i } , d _ { j } \\rangle$ is a derivation. Derivations are produced by a derivation oracle $D : \\mathcal { X }  \\mathcal { D }$ . ",
+        "bbox": [
+            174,
+            628,
+            825,
+            699
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Compositionality In intuitive terms, the representations computed by $f$ are compositional if each $f ( x )$ is determined by the structure of $D ( x )$ . Most discussions of compositionality, following Montague (1970), make this precise by defining a composition operation $\\theta _ { a } * \\theta _ { b } \\mapsto \\theta$ in the space of representations. Then the model $f$ is compositional if it is a homomorphism from inputs to representations: we require that for any $x$ with $D ( x ) = \\langle D ( x _ { a } ) , D ( x _ { b } ) \\rangle$ , ",
+        "bbox": [
+            174,
+            715,
+            825,
+            786
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/903406126230284da900b80bdc9856dd27cd46a748e672d286285205eaccd050.jpg",
+        "text": "$$\nf ( x ) = f ( x _ { a } ) * f ( x _ { b } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            792,
+            578,
+            811
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the linguistic contexts for which this definition was originally proposed, it is straightforward to apply. Inputs $x$ are natural language strings. Their associated derivations $D ( x )$ are syntax trees, and composition of derivations is syntactic composition. Representations $\\theta$ are logical representations of meaning (for an overview see van Benthem $\\&$ ter Meulen, 1996). To argue that a particular fragment of language is compositional, it is sufficient to exhibit a lexicon $\\mathcal { D } _ { 0 }$ mapping words to their associated meaning representations, and a grammar for composing meanings where licensed by derivations. Algorithms for learning grammars and lexicons from data are a mainstay of semantic parsing approaches to language understanding problems like question answering and instruction following (Zettlemoyer & Collins, 2005; Chen, 2012; Artzi et al., 2014). ",
+        "bbox": [
+            173,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "But for questions of compositionality involving more general representation spaces and more general analyses, the above definition presents two difficulties: (1) In the absence of a clearly-defined syntax of the kind available in natural language, how do we identify lexicon entries: the primitive parts from which representations are constructed? (2) What do we do with languages like the one in Figure 1d, which seem to exhibit some kind of regular structure, but for which the homomorphism condition given in Equation 1 cannot be made to hold exactly? ",
+        "bbox": [
+            174,
+            138,
+            825,
+            223
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Consider again the example in Figure 1. The oracle derivations tell us to identify primitive representations for dark, blue, green, square, and triangle. The derivations then suggest a process for composing these primitives (e.g. via string concatenation) to produce full representations. The speaker model is compositional (in the sense of Equation 1) as long as there is some assignment of representations to primitives such that for each model input, composing primitive representations according to the oracle derivation reproduces the speaker’s prediction. ",
+        "bbox": [
+            174,
+            229,
+            825,
+            314
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In Figure 1 there is no assignment of strings to primitives that reproduces model predictions exactly. But predictions can be reproduced approximately—by taking $\\times \\times$ to mean blue, aa to mean square, etc. The quality of the approximation serves as a measure of the compositionality of the true predictor: predictors that are mostly compositional but for a few exceptions, or compositional but for the addition of some noise, will be well-approximated on average, while arbitrary mappings from inputs to representations will not. This suggests that we should measure compositionality by searching for representations that allow an explicitly compositional model to approximate the true $f$ as closely as possible. We define our evaluation procedure as follows: ",
+        "bbox": [
+            174,
+            319,
+            826,
+            431
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Tree Reconstruction Error (TRE) ",
+        "text_level": 1,
+        "bbox": [
+            196,
+            465,
+            429,
+            481
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "First choose : ",
+        "bbox": [
+            196,
+            493,
+            287,
+            507
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• a distance function $\\delta : \\Theta \\times \\Theta \\to [ 0 , \\infty )$ satisfying $\\delta ( \\theta , \\theta ^ { \\prime } ) = 0 \\Leftrightarrow \\theta = \\theta ^ { \\prime }$ • a composition function $\\ast : \\Theta \\times \\Theta \\to \\Theta$ ",
+        "bbox": [
+            238,
+            511,
+            736,
+            546
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Define $\\hat { f } _ { \\eta } ( d )$ , a compositional approximation to $f$ with parameters $\\eta$ , as: ",
+        "bbox": [
+            194,
+            551,
+            674,
+            568
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5d7ca556a4c4a888d2e60700ffb115f0174746be7b47c3bd678dafb07bc5eb0e.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\hat { f } _ { \\eta } ( d _ { i } ) = \\eta _ { i } } } \\\\ { { \\hat { f } _ { \\eta } \\big ( \\langle d , d ^ { \\prime } \\rangle \\big ) = \\hat { f } _ { \\eta } ( d ) * \\hat { f } _ { \\eta } ( d ^ { \\prime } ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            573,
+            496,
+            617
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d77873fe6619cee7f541a805a0e45be60f467422f7019df823e7927e50d0d075.jpg",
+        "text": "$$\n\\begin{array} { l } { \\mathrm { f o r } d _ { i } \\in \\mathcal { D } _ { 0 } } \\\\ { \\mathrm { f o r } \\mathrm { a l l } \\mathrm { o t h e r } d } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            598,
+            575,
+            692,
+            616
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\hat { f } _ { \\eta }$ has one parameter vector $\\eta _ { i }$ for every $d _ { i }$ in $\\mathcal { D } _ { 0 }$ ; these vectors are members of the representation space $\\Theta$ . ",
+        "bbox": [
+            196,
+            625,
+            800,
+            654
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given a dataset $\\mathcal { X }$ of inputs $x _ { i }$ with derivations $d _ { i } = D ( x _ { i } )$ , compute: ",
+        "bbox": [
+            197,
+            666,
+            656,
+            681
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e6afa6beac2ff92b0d9d84c58036f43b6afcc8344eab384c642578c73bb77466.jpg",
+        "text": "$$\n\\eta ^ { * } = \\arg \\operatorname* { m i n } _ { \\eta } \\sum _ { i } \\delta { \\big ( } f ( x _ { i } ) , { \\hat { f } } _ { \\eta } ( d _ { i } ) { \\big ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            688,
+            635,
+            722
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Then we can define datum- and dataset-level evaluation metrics: ",
+        "bbox": [
+            194,
+            732,
+            617,
+            747
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dc414b522036ba0ee3549612d79095c6d695422a1235122d051cb0ec63c67fa3.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\mathrm { T R E } ( x ) = \\displaystyle \\delta \\big ( f ( x ) , \\hat { f } _ { \\eta ^ { * } } ( d ) \\big ) } } \\\\ { { \\mathrm { T R E } ( \\mathcal { X } ) = \\displaystyle \\frac { 1 } { n } \\sum _ { i } \\mathrm { T R E } ( x _ { i } ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            753,
+            545,
+            814
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "TRE and compositionality How well does the evaluation metric $\\mathrm { T R E } ( \\mathcal { X } )$ capture the intuition behind Equation 1? The definition above uses parameters $\\eta _ { i }$ to witness the constructability of representations from parts, in this case by explicitly optimizing over those parts rather than taking them to be given by $f$ . Each term in Equation 2 is analogous to an instance of Equation 1, measuring how well $\\hat { f } _ { \\eta ^ { * } } ( x _ { i } )$ , the best compositional prediction, matches the true model prediction $f ( x _ { i } )$ . In the case of models that are homomorphisms in the sense of Equation 1, TRE reduces to the familiar case: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Remark 1. $\\mathrm { T R E } ( x ) = 0$ for all $x$ if and only if Equation 1 holds exactly (that is, $f ( x ) = f ( x _ { a } ) * f ( x _ { b } )$ for any $x , x _ { a } , x _ { b }$ with $D ( x ) = \\langle D ( x _ { a } ) , D ( x _ { b } ) \\rangle \\rangle$ . ",
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+    {
+        "type": "text",
+        "text": "Proof. One direction follows immediately from defining ${ \\hat { f } } _ { \\eta ^ { * } } ( x ) = f ( x )$ . For the other, $f ( x ) =$ ${ \\hat { f } } ( D ( x ) ) = { \\hat { f } } ( \\langle D ( x _ { a } ) , D ( x _ { b } ) \\rangle ) = { \\hat { f } } ( D ( x _ { a } ) ) * { \\hat { f } } ( D ( x _ { b } ) ) = f ( x _ { a } ) * f ( x _ { b } ) .$ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Learnable composition operators The definition of TRE leaves the choice of $\\delta$ and $* \\ \\mathrm { u p }$ to the evaluator. Indeed, if the exact form of the composition function is not known a priori, it is natural to define $^ *$ with free parameters (as in e.g. Baroni & Zamparelli, 2010), treat these as another learned part of $\\hat { f }$ , and optimize them jointly with the $\\eta _ { i }$ . However, some care must be taken when choosing $^ *$ (especially when learning it) to avoid trivial solutions: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Remark 2. Suppose $D$ is injective; that is, every $x \\in \\mathcal { X }$ is assigned a unique derivation. Then there is always some $^ *$ that achieves $\\mathrm { T R E } ( \\mathcal { X } ) = 0$ : simply define $f ( x _ { a } ) * f ( x _ { b } ) = f ( x ) .$ for any $x , x _ { a } , x _ { b }$ as in the preceding definition, and set ${ \\hat { f } } = f$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In other words, some pre-commitment to a restricted composition function is essentially inevitable: if we allow the evaluation procedure to select an arbitrary composition function, the result will be trivial. This paper features experiments with $^ *$ in both a fixed functional form and a learned parametric one. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Implementation details For models with continuous $\\Theta$ and differentiable $\\delta$ and $^ *$ , $\\mathrm { T R E } ( \\mathcal { X } )$ is also differentiable. Equation 2 can be solved using gradient descent. We use this strategy in Sections 4 and 5. For discrete $\\Theta$ , it may be possible to find a continuous relaxation with respect to which $\\delta ( \\theta , \\cdot )$ and $^ *$ are differentiable, and gradient descent again employed. We use this strategy in Section 7 (discussed further there). An implementation of an SGD-based TRE solver is provided in the accompanying software release. For other problems, task-specific optimizers (e.g. machine translation alignment models; Bogin et al., 2018) or general-purpose discrete optimization toolkits can be applied to Equation 2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The remainder of the paper highlights ways of using TRE to answer questions about compositionality that arise in machine learning problems of various kinds. ",
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+    {
+        "type": "text",
+        "text": "4 COMPOSITIONALITY AND LEARNING DYNAMICS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We begin by studying the relationship between compositionality and learning dynamics, focusing on the information bottleneck theory of representation learning proposed by Tishby & Zaslavsky (2015). This framework proposes that learning in deep models consists of an error minimization phase followed by a compression phase, and that compression is characterized by a decrease in the mutual information between inputs and their computed representations. We investigate the hypothesis that the compression phase finds a compositional representation of the input distribution, isolating decision-relevant attributes and discarding irrelevant information. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Data comes from a few-shot classification task. Because our analysis focuses on compositional hypothesis classes, we use visual concepts from the Color MNIST dataset of Seo et al. (2017) (Figure 2). We predict classifiers in a meta-learning framework (Schmidhuber, 1987; Santoro et al., 2016): for each sub-task, the learner is presented with two images corresponding to some compositional visual concept (e.g. “digit 8 on a black background” or “green with heavy stroke”) and must determine whether a held-out image is an example of the same visual concept. ",
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+    {
+        "type": "image",
+        "img_path": "images/e2079b0cfcc80dba025b391c6de4d28344306c749460b8f6d19f3d68b71209f6.jpg",
+        "image_caption": [
+            "Figure 2: Meta-learning task: learners are presented with two example images depicting a visual concept (a), and must determine whether a third image (b) is an example of the same concept (c). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "img_path": "images/22d41ccd6dcfd0a29fab0ebd9d505f674866a7361329c81645c87376dd97c62e.jpg",
+        "image_caption": [
+            "Figure 3: Relationship between reconstruction error TRE and mutual information $I ( \\theta ; X )$ between inputs and representations. (a) Evolution of the two quantities over the course of a single run. Both initially increase, then decrease. The color bar shows the training epoch. (b) Values from ten training runs. (c) Values from the second half of each training run, taken to begin when $I ( \\theta ; X )$ reaches a maximum. In (b) and (c), the observed correlation is significant: respectively $\\mathrm { \\Delta } r = 0 . 7 0$ , $p < 1 e { - } 1 0 )$ ) and $\\mathrm { \\Delta } r = 0 . 7 1$ , $p < 1 e { - 8 } )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Given example images $x _ { 1 }$ and $x _ { 2 }$ , a test image $x ^ { * }$ , and label $y ^ { * }$ , the model computes: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/f3ffa76c1fe8e1a28daf391cb72c2059dc0b6ccdfab37df2c05db94068e71bcf.jpg",
+        "text": "$$\n\\begin{array} { r l } & { z _ { i } =  { \\mathtt { C N N } } ( x _ { i } ) \\mathrm { ~ f o r ~ } i \\in \\{ 1 , 2 , * \\} } \\\\ & { \\theta = \\operatorname { t a n h } ( W ( z _ { 1 } + z _ { 2 } ) ) } \\\\ & { \\hat { y } = \\theta ^ { \\top } z _ { t } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We use $\\theta$ as the representation of a classifier for analysis. The model is trained to minimize the logistic loss between logits $\\hat { y }$ and ground-truth labels $y ^ { * }$ . More details are given in Appendix A. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Compositional structure Visual concepts used in this task are all single attributes or conjunctions of attributes; i.e. their associated derivations are of the form attr or $\\langle \\arctan _ { \\mathsf { 1 } } , \\mathsf { a t t r } _ { \\mathsf { 2 } } \\rangle$ . Attributes include background color, digit color, digit identity and stroke type. The composition function $^ *$ is addition and the distance $\\delta ( \\theta , \\theta ^ { \\prime } )$ is cosine similarity $1 - \\theta ^ { \\top } \\theta ^ { \\prime } / ( \\lVert \\theta \\rVert \\lVert \\theta ^ { \\prime } \\rVert )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Evaluation The training dataset consists of 9000 image triplets, evenly balanced between positive and negative classes, with a validation set of 500 examples. At convergence, the model achieves validation accuracy of $7 5 . 2 \\%$ on average over ten training runs. (Perfect accuracy is not possible because the true classifier is not fully determined by two training examples). We explore the relationship between the information bottleneck and compositionality by comparing $\\mathrm { T R E } ( \\mathcal { X } )$ to the mutual information $I ( \\theta ; x )$ between representations and inputs over the course of training. Both quantities are computed on the validation set, calculating $\\mathrm { T R E } ( \\mathcal { X } )$ as described in Section 3 and $I ( \\theta ; X )$ as described in Shwartz-Ziv & Tishby (2017). (For discussion of limitations of this approach to computing mutual information between inputs and representations, see Saxe et al., 2018.) ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Figure 3 shows the relationship between $\\mathrm { T R E } ( \\mathcal { X } )$ and $I ( \\theta ; X )$ . Recall that small TRE is indicative of a high degree of compositionality. It can be seen that both mutual information and reconstruction error are initially low (because representations initially encode little about distinctions between inputs). Both increase over the course of training, and decrease together after mutual information reaches a maximum (Figure 3a). This pattern holds if we plot values from multiple training runs at the same time (Figure 3b), or if we consider only the postulated compression phase (Figure 3c). These results are consistent with the hypothesis that compression in the information bottleneck framework is associated with the discovery of compositional representations. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "5 COMPOSITIONALITY AND HUMAN JUDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Next we investigate a more conventional representation learning task. High-dimensional embeddings of words and phrases are useful for many natural language processing applications (Turian et al., 2010), and many techniques exist to learn them from unlabeled text (Deerwester et al., 1990; Mikolov et al., 2013). The question we wish to explore is not whether phrase vectors are compositional in aggregate, but rather how compositional individual phrase representations are. Our hypothesis is that bigrams whose representations have low TRE are those whose meaning is essentially compositional, and well-explained by the constituent words, while bigrams with large reconstruction error will correspond to non-compositional multi-word expressions (Nattinger & DeCarrico, 1992). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This task is already well-studied in the natural language processing literature (Salehi et al., 2015), and the analysis we present differs only in the use of TRE to search for atomic representations rather than taking them to be given by pre-trained word representations. Our goal is to validate our approach in a language processing context, and show how existing work on compositionality (and representations of natural language in particular) fit into the more general framework proposed in the current paper. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We train embeddings for words and bigrams using the CBOW objective of Mikolov et al. (2013) using the implementation provided in FastText (Bojanowski et al., 2017) with 100-dimensional vectors and a context size of 5. Vectors are estimated from a 250M-word subset of the Gigaword dataset (Parker et al., 2011). More details are provided in Appendix A. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Compositional structure We want to know how close phrase embeddings are to the composition of their constituent word embeddings. We define derivations for words and phrases in the natural way: single words $w$ have primitive derivations $d = w$ ; bigrams $w _ { 1 } w _ { 2 }$ have derivations of the form $\\langle w _ { 1 } , w _ { 2 } \\rangle$ . The composition function is again vector addition and distance is cosine distance. (Future work might explore learned composition functions as in e.g. Grefenstette et al., 2013, for future work.) We compare bigram-level judgments of compositionality computed by TRE with a dataset of human judgments about noun–noun compounds (Reddy et al., 2011). In this dataset, humans rate bigrams as compositional on a scale from 0 to 5, with highly conventionalized phrases like gravy train assigned low scores and graduate student assigned high ones. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results We reproduce the results of Salehi et al. (2015) within the tree reconstruction error framework: for a given $x$ , $\\mathrm { T R E } ( x )$ is anticorrelated with human judgments of compositionality $\\zeta = - 0 . 3 4$ , $p < 0 . 0 1 $ ). Collocations rated “most compositional” by our approach (i.e. with lowest TRE) are: application form, polo shirt, research project; words rated “least compositional” are fine line, lip service, and nest egg. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6 COMPOSITIONALITY AND SIMILARITY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "The next section aims at providing a formal, rather than experimental, characterization of the relationship between TRE and another perspective on the analysis of representations with help from oracle derivations. Brighton & Kirby (2006) introduce a notion of topographic similarity, arguing that a learned representation captures relevant domain structure if distances between learned representations are correlated with distances between their associated derivations. This can be viewed as providing a weak form of evidence for compositionality—if the distance function rewards pairs of representations that share overlapping substructure (as might be the case with e.g. string edit distance), edit distance will be expected to correlate with some notion of derivational similarity (Lazaridou et al., 2018). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section we aim to clarify the relationship between the two evaluations. To do this we first need to equip the space of derivations described in Section 3 with a distance function. As the derivations considered in this paper are all tree-structured, it is natural to use a simple tree edit distance (Bille, 2005) for this purpose. We claim the following: ",
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+    {
+        "type": "text",
+        "text": "Proposition 1. Let $\\hat { f } = \\hat { f } _ { \\eta ^ { * } }$ be an approximation to $f$ estimated as in Equation 2, with all $\\mathrm { T R E } ( x ) \\leq \\epsilon$ for some \u000f. Let $\\Delta$ be the tree edit distance (defined formally in Appendix B, Definition 2), and let δ be any distance on $\\Theta$ satisfying the following properties: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/82d1d1bcab656faccb00f7638d2592ebfd889be8174e4442b9e7a78c57353504.jpg",
+        "text": "$$\n\\delta ( \\hat { f } ( d _ { i } ) , \\hat { f } ( d _ { j } ) ) \\leq 1 f o r d _ { i } , d _ { j } \\in \\mathcal { D } _ { 0 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2. $\\delta ( { \\hat { f } } ( d ) , 0 ) \\leq 1$ for $d \\in \\mathcal { D } _ { 0 }$ , where 0 is the identity element for $^ *$ ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3. $\\delta ( \\theta _ { i } \\ast \\theta _ { j } , \\theta _ { k } \\ast \\theta _ { \\ell } ) \\leq \\delta ( \\theta _ { i } , \\theta _ { k } ) + \\delta ( \\theta _ { j } , \\theta _ { \\ell } ) .$ . (This condition is satisfied by any translation-invariant metric.) ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Then $\\Delta$ is an approximate upper bound on $\\delta$ : for any $x$ , $x ^ { \\prime }$ with $d = D ( x )$ , $d ^ { \\prime } = D ( x ^ { \\prime } )$ ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5c893b007f4194b7e3f9dc829efb9682c363152ef9ce2db77037093904020f99.jpg",
+        "text": "$$\n\\delta ( f ( x ) , f ( x ^ { \\prime } ) ) \\leq \\Delta ( d , d ^ { \\prime } ) + 2 \\epsilon .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In other words, representations cannot be much farther apart than the derivations that produce them.   \nProof is provided in Appendix B. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We emphasize that small TRE is not a sufficient condition for topographic similarity as defined by Brighton & Kirby (2006): very different derivations might be associated with the same representation (e.g. when representing arithmetic expressions by their results). But this result does demonstrate that compositionality imposes some constraints on the inferences that can be drawn from similarity judgments between representations. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "7 COMPOSITIONALITY AND GENERALIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "In our final set of experiments, we investigate the relationship between compositionality and generalization. Here we focus on communication games like the one depicted in Figure 1 and in more detail in Figure 4. As in the previous section, existing work argues for a relationship between compositionality and generalization, claiming that agents need compositional communication protocols to generalize to unseen referents (Kottur et al., 2017; Choi et al., 2018). Here we are able to evaluate this claim empirically by training a large number of agents from random initial conditions, measuring the compositional structure of the language that emerges, and seeing how this relates to their performance on both familiar and novel objects. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our experiment focuses on a reference game (Gatt et al., 2007). Two policies are trained: a speaker and a listener. The speaker observes pair of target objects represented with a feature vector. The speaker then sends a message (coded as a discrete character sequence) to the listener model. The listener observes this message and attempts to reconstruct the target objects by predicting a sequence of attribute sets. If all objects are predicted correctly, both the speaker and the listener receive a reward of 1 (partial credit is awarded for partly-correct objects; Figure 4). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Because the communication protocol is discrete, policies are jointly trained using a policy gradient objective (Williams, 1992). The speaker and listener are implemented with RNNs; details are provided in Appendix A. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Compositional structure Every target referent consists of two objects; each object has two attributes. The derivation associated with each communicative task thus has the tree structure $\\langle \\langle \\mathsf { a t t r } _ { 1 a } , \\mathsf { a t t r } _ { 1 b } \\rangle$ , $ \\mathsf { a t t r } _ { 2 a } , \\mathsf { a t t r } _ { 2 b }  $ . We hold out a subset of these object pairs at training time to evaluate generalization: in each training run, 1/3 of possible reference candidates are never presented to the agent at training time. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/d78ad82d3d03a6516b645718844166c74e09fb61482aa4d425065d389bdcf188.jpg",
+        "image_caption": [
+            "Figure 4: The communication task: A speaker model observes a pair of target objects, and sends a description of the objects (as a discrete code) to a listener model. The listener attempts to reconstruct the targets, receiving fractional reward for partially-correct predictions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Where the previous examples involved a representation space of real embeddings, here representations are fixed-length discrete codes. Moreover, the derivations themselves have a more complicated semantics than in Sections 4 and 5: order matters, and a commutative operation like addition cannot capture the distinction between hhgreen, squarei, $\\left. \\mathrm { b 1 u e } , \\mathrm { t r i a n g 1 e } \\right. \\rangle$ and $\\langle \\langle { \\mathsf { g r e e n } } , { \\mathsf { t r i a n g l e } } \\rangle$ , $\\langle \\mathtt { b l u e } , \\mathtt { s q u a r e } \\rangle \\rangle$ . We thus need a different class of composition and distance operations. We represent each agent message as a sequence of one-hot vectors, and take the error function $\\delta$ to be the $\\ell _ { 1 }$ distance between vectors. The composition function has the form: ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "img_path": "images/5f2f07a46340a68c1ceced863b6032c2fca63d86eca5b768eb4b4fe6ab1bf0c2.jpg",
+        "text": "$$\n\\theta * \\theta ^ { \\prime } = A \\theta + B \\theta ^ { \\prime }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "with free composition parameters $\\eta _ { * } = \\{ A , B \\}$ in Equation 2. These matrices can redistribute the tokens in $\\theta$ and $\\theta ^ { \\prime }$ across different positions of the input string, but cannot affect the choice of the tokens themselves; this makes it possible to model non-commutative aspects of string production. To compute TRE via gradient descent, we allow the elements of $\\mathcal { D } _ { 0 }$ to be arbitrary vectors (intuitively assigning fractional token counts to string indices) rather than restricting them to one-hot indicators. With this change, both $\\delta$ and $^ *$ have subgradients and can be optimized using the same procedure as in preceding sections. ",
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+        "image_caption": [
+            "Figure 5: Relationship between TRE and reward. (a) Compositional languages exhibit lower generalization error, measured as the difference between train and test reward $r = 0 . 5 0$ , $p < 1 e { - } 6 )$ . (b) However, compositional languages also exhibit lower absolute performance ${ \\mathrm { ' } r = 0 . 5 7 }$ , $p < 1 e { - 9 } ,$ ). Both facts remain true even if we restrict analysis to “successful” training runs in which agents achieve a reward $> 0 . 5$ on held-out referents $r = 0 . 6$ , $p < 1 e { - 3 }$ and $r = 0 . 3 8$ , $p < 0 . 0 5$ respectively). ",
+            "Figure 6: Fragment of languages resulting from two multiagent training runs. In the first section, the left column shows the target referent, while the remaining columns show the message generated by speaker in the given training run after observing the referent. The two languages have substantially different TRE, but induce similar listener performance (Train and Test reward). "
+        ],
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+        "type": "table",
+        "img_path": "images/a5eca68a8c3adb6bcdfbcfe43ee5fa28d4b6272f493b95b9f2d03af663543bff.jpg",
+        "table_caption": [],
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+        "table_body": "<table><tr><td colspan=\"5\">Language A</td></tr><tr><td>((red</td><td>circle)</td><td>(blue</td><td>triangle)）</td><td>jjjj</td></tr><tr><td>((red</td><td>circle)</td><td>(blue</td><td>star))</td><td>jeoo oppp jjjj</td></tr><tr><td>((red</td><td>circle)</td><td>(blue</td><td>circle))</td><td>oopp jjjj</td></tr><tr><td>((red</td><td>circle)</td><td>(blue</td><td>square))</td><td>oopp jjjb</td></tr><tr><td>((red</td><td>square)</td><td>(blue</td><td>triangle))</td><td>jjjj jbjj</td></tr><tr><td>((red</td><td>square)</td><td>(blue</td><td>star))</td><td>o00o jbjj</td></tr><tr><td>((red</td><td>square)</td><td>(blue</td><td>circle))</td><td>oo0o jbbb</td></tr><tr><td>((red</td><td>square)</td><td>(blue</td><td>square))</td><td>o00o jbbb</td></tr><tr><td colspan=\"3\">TRE</td><td>4.30</td><td>2.96</td></tr><tr><td colspan=\"3\">Train reward</td><td>0.78</td><td>0.75</td></tr><tr><td colspan=\"3\">Test reward</td><td>0.61</td><td>0.59</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "Results We train 100 speaker–listener pairs with random initial parameters and measure their performance on both training and test sets. Our results suggest a more nuanced view of the relationship between compositionality and generalization than has been argued in the existing literature. TRE is significantly correlated with generalization error (measured as the difference between training accuracies, Figure 5a). However, TRE is also significantly correlated with absolute model reward (Figure 5b)—“compositional” languages more often result from poor communication strategies than successful ones. This is largely a consequence of the fact that many languages with low TRE correspond to trivial strategies (for example, one in which the speaker sends the same message regardless of its observation) that result in poor overall performance. ",
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+    {
+        "type": "text",
+        "text": "Moreover, despite the correlation between TRE and generalization error, low TRE is by no means a necessary condition for good generalization. We can use our technique to automatically mine a collection of training runs for languages that achieve good generalization performance at both low and high levels of compositionality. Examples of such languages are shown in Figure 6. ",
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+    {
+        "type": "text",
+        "text": "8 CONCLUSIONS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We have introduced a new evaluation method called TRE for generating graded judgments about compositional structure in representation learning problems where the structure of the observations is understood. TRE infers a set of primitive meaning representations that, when composed, approximate the observed representations, then measures the quality of this approximation. We have applied TRE-based analysis to four different problems in representation learning, relating compositionality to learning dynamics, linguistic compositionality, similarity and generalization. ",
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+    {
+        "type": "text",
+        "text": "Many interesting questions regarding compositionality and representation learning remain open. The most immediate is how to generalize TRE to the setting where oracle derivations are not available; in this case Equation 2 must be solved jointly with an unsupervised grammar induction problem (Klein & Manning, 2004). Beyond this, it is our hope that this line of research opens up two different kinds of new work: better understanding of existing machine learning models, by providing a new set of tools for understanding their representational capacity; and better understanding of problems, by better understanding the kinds of data distributions and loss functions that give rise to compositionalor non-compositional representations of observations. ",
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+    {
+        "type": "text",
+        "text": "REPRODUCIBILITY ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Code and data for all experiments in this paper are provided at https://github.com/jacobandreas/tre. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
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+        "type": "text",
+        "text": "Thanks to Daniel Fried and David Gaddy for feedback on an early draft of this paper. The author was supported by a Facebook Graduate Fellowship at the time of writing. ",
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+        "text": "Richard Socher, John Bauer, Christopher D. Manning, and Andrew Y. Ng. Parsing with compositional vector grammars. In Proceedings of the Annual Meeting of the Association for Computational Linguistics, 2013. ",
+        "bbox": [
+            171,
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+        "text": "Naftali Tishby and Noga Zaslavsky. Deep learning and the information bottleneck principle. In Information Theory Workshop, 2015. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Joseph Turian, Lev Ratinov, and Yoshua Bengio. Word representations: a simple and general method for semi-supervised learning. In Proceedings of the 48th Annual Meeting of the Association for Computational Linguistics, pp. 384–394. Proceedings of the Annual Meeting of the Association for Computational Linguistics, 2010. ",
+        "bbox": [
+            174,
+            103,
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+            160
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "JFAK van Benthem and Alice ter Meulen. Handbook of logic and language. 1996. ",
+        "bbox": [
+            174,
+            169,
+            715,
+            184
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 1992. ",
+        "bbox": [
+            169,
+            193,
+            821,
+            220
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Luke S. Zettlemoyer and Michael Collins. Learning to map sentences to logical form: Structured classification with probabilistic categorial grammars. In Proceedings of the Conference on Uncertainty in Artificial Intelligence, pp. 658–666, 2005. ",
+        "bbox": [
+            174,
+            231,
+            825,
+            272
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A MODELING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            383,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Few-shot classification The CNN has the following form: ",
+        "bbox": [
+            173,
+            132,
+            568,
+            148
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Conv(out $= 6$ , kernel $= 5$ )   \nReLU   \nMaxPool(kernel $^ { = 2 }$ )   \nConv(out $= 1 6$ , kernel $= 5$ )   \nReLU   \nMaxPool(kernel $^ { = 2 }$ )   \nLinear(out $= 1 2 8$ )   \nReLU   \nLinear(out $= 6 4$ )   \nReLU ",
+        "bbox": [
+            173,
+            160,
+            387,
+            297
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The model is trained using ADAM (Kingma & Ba, 2014) with a learning rate of .001 and a batch size of 128. Training is ended when the model stops improving on a held-out set. ",
+        "bbox": [
+            174,
+            310,
+            823,
+            338
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Word embeddings We train FastText (Bojanowski et al., 2017) on the first 250 million words of the NYT section of Gigaword (Parker et al., 2011). To acquire bigram representations, we pre-process this dataset so that each occurrence of a bigram from the Reddy et al. (2011) dataset is treated as a single word for purposes of estimating word vectors. ",
+        "bbox": [
+            174,
+            353,
+            825,
+            410
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Communication The encoder and decoder RNNs both use gated recurrent units (Cho et al., 2014) with embeddings and hidden states of size 256. The size of the discrete vocabulary is set to 16 and the maximum message length to 4. Training uses a policy gradient objective with a scalar baseline set to the running average reward; this is optimized using ADAM (Kingma & Ba, 2014) with a learning rate of .001 and a batch size of 256. Each model is trained for 500 steps. Models are trained by sampling from the decoder’s output distribution, but greedy decoding is used to evaluate performance and produce Figure 6. ",
+        "bbox": [
+            173,
+            424,
+            825,
+            522
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B PROPOSITION 1 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            541,
+            338,
+            558
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "First, some definitions: ",
+        "bbox": [
+            174,
+            574,
+            325,
+            588
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Definition 1. The size of a derivation is given by: ",
+        "bbox": [
+            173,
+            606,
+            501,
+            621
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0ec8e02b652e70b0ac1c11cd7eb57d83294bb24b16c67a13c33cf1e386096c41.jpg",
+        "text": "$$\n\\begin{array} { r l r } { | d | = 1 } & { { } } & { i f d \\in \\mathcal { D } _ { 0 } } \\\\ { | \\langle d _ { a } , d _ { b } \\rangle | = | d _ { a } | + | d _ { b } | } & { { } } & { o t h e r w i s e } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            626,
+            629,
+            662
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Definition 2. The tree edit distance between derivations is defined by: ",
+        "bbox": [
+            169,
+            667,
+            637,
+            683
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/39d315d1c6c6f195a793ea77a6f2a95d7bfa8f23dae68b5ada3ec0dbfb825d5a.jpg",
+        "text": "$$\n\\Delta ( d _ { i } , d _ { j } ) = \\mathbb { I } [ i = j ] \\quad i f d _ { i } \\in \\mathcal { D } _ { 0 } a n d d _ { j } \\in \\mathcal { D } _ { 0 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/61ccbb42c8e4fca79b227c2a73fb5521b49579a701fc13c66358b174ac044a17.jpg",
+        "text": "$$\n\\Delta ( d _ { i } , \\langle d _ { j } , d _ { k } \\rangle ) = \\operatorname* { m i n } \\left\\{ \\begin{array} { l l } { \\Delta ( d _ { i } , d _ { j } ) + | d _ { k } | } \\\\ { \\Delta ( d _ { i } , d _ { k } ) + | d _ { j } | } \\end{array} \\right\\} \\quad i f d _ { i } \\in \\mathcal { D } _ { 0 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            263,
+            710,
+            656,
+            752
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b6fd615050dbe9761495f2e85bfca290115b0ad5755433ab5cf05f2c36aace98.jpg",
+        "text": "$$\n\\Delta ( \\langle d _ { i } , d _ { j } \\rangle , \\langle d _ { k } , d _ { \\ell } \\rangle ) = \\operatorname* { m i n } \\left\\{ \\begin{array} { l l } { \\Delta ( d _ { i } , d _ { k } ) + \\Delta ( d _ { j } , d _ { \\ell } ) } \\\\ { \\Delta ( \\langle d _ { i } , d _ { j } \\rangle , d _ { k } ) + | d _ { \\ell } | } & { \\Delta ( \\langle d _ { i } , d _ { j } \\rangle , d _ { \\ell } ) + | d _ { k } | } \\\\ { \\Delta ( \\langle d _ { k } , d _ { \\ell } \\rangle , d _ { i } ) + | d _ { j } | } & { \\Delta ( \\langle d _ { k } , d _ { \\ell } \\rangle , d _ { j } ) + | d _ { i } | } \\end{array} \\right\\}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            227,
+            756,
+            767,
+            814
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Now, suppose we have $x$ and $x ^ { \\prime }$ with derivations $d = D ( x )$ , $d ^ { \\prime } = D ( x ^ { \\prime } )$ and representations $\\theta = f ( x )$ , $\\theta ^ { \\prime } = f ( x ^ { \\prime } )$ . Proposition 1 claims that $\\delta ( \\theta , \\theta ^ { \\prime } ) \\leq \\Delta ( d , d ^ { \\prime } ) + 2 \\epsilon$ . ",
+        "bbox": [
+            173,
+            827,
+            825,
+            856
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. $\\delta ( { \\hat { f } } ( d ) , 0 ) \\leq | d |$ ",
+        "bbox": [
+            174,
+            861,
+            362,
+            878
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Proof. For $d \\in \\mathcal { D } _ { 0 }$ this follows immediately from Condition 2 in the proposition. For composed derivations it follows from Condition 3 taking ${ \\theta _ { k } } = { \\theta _ { \\ell } } = 0$ and induction on $| d |$ . □ ",
+        "bbox": [
+            171,
+            892,
+            823,
+            921
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2. $\\delta ( { \\hat { f } } ( d ) , { \\hat { f } } ( d ^ { \\prime } ) ) \\leq \\Delta ( d , d ^ { \\prime } )$ ",
+        "bbox": [
+            173,
+            102,
+            426,
+            119
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Proof. By induction on the structure of $d$ and $d ^ { \\prime }$ : ",
+        "bbox": [
+            173,
+            133,
+            495,
+            148
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Base case Both $d , d ^ { \\prime } \\in \\mathcal { D } _ { 0 }$ . ",
+        "bbox": [
+            174,
+            162,
+            366,
+            178
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "If $= d ^ { \\prime } , \\delta ( \\hat { f } ( d ) , \\hat { f } ( d ^ { \\prime } ) ) = \\delta ( \\hat { f } ( d ) , \\hat { f } ( d ) ) = 0 = \\Delta ( d , d ^ { \\prime } ) .$ If $d \\neq d ^ { \\prime }$ $d ^ { \\prime } , \\delta ( \\hat { f } ( d ) , \\hat { f } ( d ^ { \\prime } ) ) \\leq 1 = \\Delta ( d , d ^ { \\prime } )$ from Condition 1. ",
+        "bbox": [
+            173,
+            167,
+            563,
+            228
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Inductive case Consider the arrangement of derivations that minimizes Equation 8 for derivation $d$ and $d ^ { \\prime }$ . There are two possibilities: ",
+        "bbox": [
+            171,
+            239,
+            825,
+            270
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Case 1: $\\Delta ( d , d ^ { \\prime } )$ has the form $\\Delta ( d _ { i } , d _ { k } ) + \\Delta ( d _ { j } , d _ { \\ell } )$ for some $d _ { i , j , k , \\ell }$ . W.l.o.g. let $d = \\langle d _ { i } , d _ { j } \\rangle$ and $d ^ { \\prime } = \\langle d _ { k } , d _ { \\ell } \\rangle$ . Then, ",
+        "bbox": [
+            173,
+            275,
+            825,
+            305
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0339ffbdc9f7b7361cd2ee1807c1550fe0271181c4cd69305e3021dc9cfd3a95.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\delta ( \\hat { f } ( d ) , \\hat { f } ( d ^ { \\prime } ) ) = \\delta ( \\hat { f } ( d _ { i } ) * \\hat { f } ( d _ { j } ) , \\hat { f } ( d _ { k } ) * \\hat { f } d _ { \\ell } ) } \\\\ & { \\hphantom { \\delta ( \\hat { f } ( d ) , \\hat { f } ( d _ { k } ) ) = } \\leq \\delta ( \\hat { f } ( d _ { i } ) , \\hat { f } ( d _ { k } ) ) + \\delta ( \\hat { f } ( d _ { j } ) , \\hat { f } d _ { \\ell } ) } \\\\ & { \\hphantom { \\delta ( \\hat { f } ( d ) , \\hat { f } ( d _ { k } ) ) = } \\leq \\Delta ( d _ { i } , d _ { k } ) + \\Delta ( d _ { j } , d _ { \\ell } ) } \\\\ & { \\hphantom { \\delta ( \\hat { f } ( d ) , \\hat { f } ( d ) ) = } = \\Delta ( d , d ^ { \\prime } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            330,
+            310,
+            666,
+            392
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Case 2: $\\Delta ( d , d ^ { \\prime } )$ has the form $\\Delta ( d _ { i } , d _ { k } ) + | d _ { j } |$ for some $d _ { i , j , k }$ . W.l.o.g. let $d = \\langle d _ { i } , d _ { j } \\rangle$ and $d ^ { \\prime } = d _ { k }$ . Abusing notation slightly, let us define $\\Delta ( \\bar { d } , 0 ) = { \\left| { d } \\right| }$ . If we let $d _ { \\ell } = 0$ this case reduces to the previous one. □ ",
+        "bbox": [
+            173,
+            402,
+            828,
+            446
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Finally, ",
+        "bbox": [
+            173,
+            462,
+            225,
+            476
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Proof of Proposition $^ { l }$ . ",
+        "bbox": [
+            174,
+            491,
+            325,
+            507
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ea042366a763be71fcd0c23b34aa46976d566fe2a3885e116a32d3bb4697dfc5.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\delta ( \\theta , \\theta ^ { \\prime } ) \\leq \\delta ( \\hat { f } ( d ) , \\hat { f } ( d ^ { \\prime } ) ) + 2 \\epsilon } } \\\\ { { \\leq \\Delta ( d , d ^ { \\prime } ) + 2 \\epsilon } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            512,
+            602,
+            553
+        ],
+        "page_idx": 14
+    }
+]

parse/train/HyUmbjsiz/HyUmbjsiz.md

@@ -0,0 +1,185 @@
+# Mitral Valve Leaflets Segmentation in Echocardiography using Convolutional Neural Networks
+
+Eva Costa Neadvance, Machine Vision, SA Braga, Portugal ecosta@neadvance.com
+
+Nelson Martins Neadvance, Machine Vision, SA Braga, Portugal nmartins@neadvance.com
+
+Malik Saad Sultan   
+Instituto de Telecomunicações FCUP∗, Porto, Portugal   
+engr.saadsultan@gmail.com
+
+Diana Veiga Neadvance, Machine Vision, SA Braga, Portugal dveiga@neadvance.com
+
+Manuel Ferreira   
+Neadvance, Machine Vision, SA Braga, Portugal   
+mferreira@neadvance.com   
+Sandra Mattos   
+UCMF †   
+Recife PE, Brazil   
+ssmattos@gmail.com Miguel Coimbra   
+Instituto de Telecomunicações FCUP∗, Porto, Portugal   
+mcoimbra@dcc.fc.up.pt
+
+# Abstract
+
+Rheumatic heart disease remains a major burden in the developing countries. The World Heart Federation proposed guidelines for the echocardiographic detection of the disease, in which the mitral leaflets’ morphology assessment is a key indicator. The drawback is that these guidelines are dependent on the clinician experience. To overcome this limitation, we propose an automatic segmentation of the mitral leaflets using a new method based on convolutional neural network, specifically the UNet architecture. The results indicate a median DICE coefficient of 0.74 in $P L A X$ and 0.79 in $A 4 C$ for the anterior mitral leaflet segmentation, while median DICE of 0.60 in $P L A X$ and $0 . 6 9 \ A 4 C$ are met for the posterior leaflet. A visual evaluation of this segmentation approach by two cardiologists is in line with the numerical results. The false detection due to overestimation and artifacts remains an issue to be addressed in the future.
+
+# 1 Introduction
+
+# 1.1 Motivation
+
+Rheumatic heart disease (RHD) is a preventable chronic sequel of acute rheumatic fever (ARF), an autoimmune response to group A streptococcal infection. Although being almost eradicated in high-income countries, it remains a major burden in the developing countries, where it causes most of the cardiovascular mortality and morbidity in the young [1]. RHD can be definite (clinically diagnosed) or borderline/sub-clinical (detected only by echocardiography). In a recent prevalence study, the RHD was followed globally over a period of 25-years, [2], and it was estimated that in
+
+![](images/2fbbfe8e29929f2179b71c705a03092b92570cbc85d73c2ea87c5d591b6d5785.jpg)  
+Figure 1: Brightness mode echocardiography. (a) parasternal long-axis view. (b) apical four-chamber view.
+
+2015 alone, there were 33.4 million cases of RHD, 10.5 million disability-adjusted life-years related to RHD and 319400 RHD-related deaths. However, the estimations may fall short due to missing data in some regions of the globe, misidentification in the causes of death and due to sub-clinical RHD not being included in the prevalence study. This last aspect should not be disregarded since screening studies [3] point that for each case of clinical RHD, 3 to 10 cases of sub-clinical disease exist. It is important to note that even though sub-clinical cases may not develop into definite RHD, it is at this stage that the treatment is most effective with milder health repercussions.
+
+With the advent of portable echocardiography and the increasing detection rates of sub-clinical RHD, an evidence-based set of guidelines was defined by the World Heart Federation (WHF) for the echocardiographic assessment of RHD [4]. RHD mostly affects heart valves, especially the mitral valve and, therefore, the WHF echocardiographic criteria are generally based on the morphology and functionality of this valve. The mitral leaflets’ morphology and mobility is assessed through brightness mode echocardiography, usually in the parasternal long-axis view $( P L A X )$ , and in some cases using the apical four-chamber view $( A 4 C )$ . These echo views are shown in Fig. 1, with the anterior mitral valve leaflet depicted as AMVL and the posterior mitral valve leaflet as PMVL. Morphological assessment is usually done for AMVL instead of PMVL, solely because higher inter-observer agreement is met [4]. Clinical observation suggests the tip of the leaflet is the most commonly part to be affected [5]. Echocardiography assessment requires highly experienced operators, which is a scarce resource in developing countries. The use of image processing tools has the potential to reduce the operator dependency in screening settings, reduce the subjectivity and, in this way, improve the diagnosis.
+
+# 1.2 State of the art
+
+Ultrasound images are affected by several acquisition artifacts such as attenuation, speckle, shadows and signal dropout. Apart from that, the quality of the acquisition is strongly dependent on the human operator and the machine settings. The intensity and texture differences between structures and the contrast between structures and blood pool are low. These conditions raise several problems for classic image processing methods [6]. In [7], the authors proposed a combination of active contours and optical flow for the AMVL segmentation. The algorithm is semi-automatic and fails when the leaflet’s displacement between frames is large and irregular. Another semi-automatic approach for the AMVL segmentation was proposed in [8]. Two connected active contours identify the cardiac muscle and the leaflet. The manual initialization and the parameters selection highly affect the method’s performance. In [9], the authors propose a semi-automatic segmentation strategy, with a single point input from the user. The input point defines a set of scanning lines for a virtual motionmode (M-mode) reconstruction. The posterior aorta wall’s motion pattern is obtained by applying open-ended active contours to the virtual M-mode, using prior knowledge to establish constraints. The pattern provides a seed for each frame to segment the AMVL with localizing region-based active contours. Although it delivers the middle part of the leaflet, it sometimes fails to segment its tip, which is the most relevant part of the structure diagnosis-wise. An approach based on outlier detection in low-rank matrix was proposed in [10]. The authors aim to overcome the shortfalls of the previous methods, with a fully automatic unsupervised method. However, this solution still requires an extensive parameter fine-tuning and cropping of the images around the region of interest. Also, the method does not discriminate between AMVL and PMVL. In [11] the authors claim to prevent tracking drifts caused by motion ambiguities by constraining the outlier pursuit, and refining the segmentation with region-scalable active contours. Significant parameter fine-tuning remains as a drawback.
+
+Most of the literature approaches are highly sensitive to initialization, image quality and acquisition parameters. None of them segments both leaflets consistently. The complexity of the problem calls for supervised learning methods such as Convolutional Neural Networks (CNN). CNNs have become the state of the art solution for image recognition problems, even outperforming human operators in some tasks. This approach will shift the burden of manual input from the final user to the training phase, and also will not rely on hand-crafted image features, making segmentation a fully automatic and robust process. To our knowledge there are no works in the literature on semantic segmentation of the mitral leaflets using $C N N s$ . In [12], a partial segmentation of the mitral leaflets was needed to segment the left ventricle. The authors propose a network for patches’ classification and then a second network for segmentation of the ventricle. However, they were not able to detect the contours of fast moving structures such as the mitral leaflets. The amount of data, and the respective manual annotations required for training a $C N N$ is a major point to take into account. The UNet architecture, proposed in [13], claims to produce accurate results, with a small number of observations. This trait of the U N et makes it an interesting contender for application in the present work, since the available dataset is also limited. The architecture allows for a multi-scale representation, with coarser information being collected in the bottom layers and finer information at the top ones. The architecture is composed by two paths: one of contraction, with convolutional layers and another of expansion with deconvolution layers. The paths are connected by skip layers before each max-pooling operation. This ensures that both local and global information is captured.
+
+# 2 Proposed Work
+
+In this work the U N et will be used for the mitral leaflets’ segmentation in the $A 4 C$ and $P L A X$ views. Each view produces distinct representations of the heart structures, thus, the model’s development will be adapted for each one separately.
+
+# 2.1 The UNet model
+
+The most favourable aspects of the U N et architecture are that it does not require a large training set, and that only the image is needed as input. The least favourable trait of the U N et is transversal to all $C N N$ architectures: the parameterization of the network requires a training phase. Depending on the complexity of the network, the training phase may require high computational power and time.
+
+# 2.1.1 Model implementation
+
+The UNet architecture was recursively implemented in TensorFlow’s [14] front-end TFlearn [15] (Python), allowing expansion of the depth $D$ of the architecture (number of steps on the paths). In Fig. 2, the implemented $U N e t$ architecture is shown.
+
+Taking into account the specificity of the problem, some simplifications and changes were made to the UNet model proposed in [13].
+
+The first architectural change is the use of zero padding in the convolutions instead of the valid values. The authors of [13] proposed the use of the valid values with a mirror padding pre-processing, so the final outputs have the same spatial dimensions as the original input image. They argue that this accelerates the training, however this was not observed during preliminary tests and therefore, it was decided to use zero padding in all convolutions.
+
+The second change is the use of batch normalization layers before the concatenation steps. This adds a regularization effect by ensuring that the concatenated feature maps have the same order of magnitude.
+
+Since the present work is focused on the evaluation of the potential of $D N N$ architectures in segmenting the mitral leaflets’, no extensive studies were made for the hyper-parameter optimization.
+
+![](images/6ac81e3becae434a1ead79036966743a48ac0a6f6f3248e2850fe4936a5fb35a.jpg)  
+Figure 2: Proposed UNet model, adapted from [13]. Normalizing layers were added before every $m a x - p o o l$ operation; same padding in all Conv layers, instead of valid. $N _ { x }$ stands for the number of filters, $D$ for depth and $S$ for the spatial size of the feature map.
+
+Most of the parameters were empirically chosen and maintained unchanged throughout the training stages: the learning rate was set to 0.001, the optimization algorithm was the Adam, a batch size of 4 was set, as loss function the mean square difference was selected, and as activation function the Sigmoid was used with threshold of 0.5.
+
+The samples were randomly shuffled and divided into a training and validation set ( $7 5 \%$ for training and $2 5 \%$ for validation). Due to memory management and taking into account the down-sampling process, samples’ dimensions were set to a 416 pixel height and 512 pixel width.
+
+To achieve more accurate results, the architecture’s depth $D$ and the number of incoming neurons $N _ { 0 }$ were subject to a greed search optimization. Preliminary tests have shown that lower depths result in less accurate results.
+
+# 3 Materials and Methods
+
+The datasets used for this work contain videos from different patients, as described in Table 1.
+
+Table 1: Distribution of the datasets for Training and Validation, Test and Application phases.   
+
+<table><tr><td>Echo view</td><td>Train and Validation</td><td>Test</td><td>Application</td></tr><tr><td>PLAX</td><td>21 videos (2163 frames）6 videos (520 frames)</td><td></td><td>23 videos</td></tr><tr><td>A4C</td><td>22 videos (2400 frames)</td><td>）6 videos (526 frames)</td><td>23 videos</td></tr></table>
+
+At the application phase, the tested U N et models were applied to the dataset for clinical assessment. Two cardiologists were asked to evaluate the segmentation quality based on 6 parameters for the two views: overall detection of the each leaflet’ tip pixels, overall estimate of each leaflet’ thickness, amount of false positives and repeatability of the segmentation quality along the video. Each case was graded with scores of 0, 1 and 2 (0 connotes failure and 2 success).
+
+The echocardiography sequences were acquired during the Heart Caravan of 2016, a health care provision initiative which took place in the State of Paraíba, Brazil. All images were acquired using a Vivid I, by GE and/or a CX-50, by Philips and from children with ages between 4 and 16 years old. The sets used for training, validation and test were manually annotated in each frame as depicted in Fig. 3.These annotations of the mitral leaflets (AMVL and PMVL) were made by an experienced user and validated by two pediatric cardiologists.
+
+For the models’ performance assessment during parameterization, the dice similarity coefficient $( D I C E )$ was used. This metric measures the similarity between the model’s prediction and the manual annotation. For the evaluation of the UNet’s segmentation results, the $D I C E$ , precision and recall were used. For the assessment of inter-rater agreement on the evaluation of the results in clinical context, the Bennett’s Sscore [16] was applied. The Sscore estimates the agreement assuming that the likelihood of random agreement (both rater agree, when both select a category randomly) is solely dependent on the number of categories. For $q$ categories, $r$ raters and $n$ rated items, the Sscore is defined as:
+
+![](images/7ede108114bd8d896cc2d4c7553ad6b5a4715192c40840765f5f9b03cbfeedc9.jpg)  
+Figure 3: Manual annotations of the AMVL, PMVL and CT. (a) parasternal long-axis view. (b) apical four-chamber view.
+
+$$
+S s c o r e = \frac { p _ { o } - p _ { c } } { 1 - p _ { c } } , w i t h :
+$$
+
+$$
+p _ { c } = \frac { 1 } { q ^ { 2 } } \sum _ { k , l } w _ { k l } , \qquad p _ { o } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \sum _ { k = 1 } ^ { q } \frac { r _ { i k } ( r _ { i k } ^ { * } - 1 ) } { r _ { i } ( r _ { i } - 1 ) } , \qquad r _ { i k } ^ { * } \sum _ { l = 1 } ^ { q } w _ { k l } r _ { i l }
+$$
+
+where, $r _ { i l }$ and $r _ { i k }$ are the number of raters assigning item $i$ to category $l$ or $k$ , respectively. Since in our case the categorization is a grading process, ordinal weighting was applied, thus when the two raters agree total credit is given $w _ { k l } = 1 ,$ ), when raters disagree by choosing immediate neighbor categories partial credit is given ${ w _ { k l } = 0 . 3 3 } $ ), and when raters disagree by choosing non-neighbor categories no credit is given $\mathbf { \boldsymbol { w } } _ { k l } = 0$ ).
+
+The UNet models were trained in a Desktop PC with Intel Core i7 Processor $( 3 . 4 0 \mathrm { G H z }$ ), 16 GB RAM, and NVIDIA GeForce GTX 970 GPU with 4 GB RAM.
+
+# 4 Results and Discussion
+
+In this section we elaborate on the results gathered in this work, and on their evaluation. The following subsection covers the results of the $D$ and $N _ { 0 }$ parameters grid search, the results on the test dataset and the results of the clinical evaluation of the proposed method.
+
+# 4.1 Parameterization
+
+The parameterization of the architecture’s depth $D$ and number of incoming neurons $N _ { 0 }$ was made for each echo view by evaluating the $D I C E$ in the validation stage. The training stages were all stopped at 30 epochs and the model with higher $D I C E$ in validation was saved. The highest result was always found before the $3 0 ^ { t h }$ epoch. Depths higher than 5 resulted in GPU memory overflow and, because of that, only depths of 4 and 5 were tested. The number of incoming neurons $N _ { 0 }$ was studied in the range from 4 to 32 with base 2 steps.
+
+# 4.1.1 Parasternal long-axis
+
+The results obtained in the validation stage are summarized in Table 2.
+
+The highest $D I C E$ was obtained for $D = 5$ and $N _ { 0 } = 8$ . Even though the model lies in the border of $D$ , further exploration was not made due to GPU overflow for depths higher than 5.
+
+Table 2: Mean DICE Coefficient results in the validation set. In bold is the highest DICE.   
+
+<table><tr><td>D</td><td>No 4</td><td>8</td><td>16</td></tr><tr><td>4</td><td>&lt;0.710</td><td>0.762</td><td>0.770</td></tr><tr><td>5</td><td>&lt;0.710</td><td>0.791</td><td>0.786</td></tr></table>
+
+# 4.1.2 Apical four-chamber
+
+The results obtained in the validation stage are summarized in Table 3.
+
+Table 3: Mean DICE Coefficient results in the validation set. In bold is the highest $D I C E$   
+
+<table><tr><td></td><td>No</td><td rowspan="2">4</td><td rowspan="2">8</td><td rowspan="2">16</td><td rowspan="2">32</td></tr><tr><td>D</td><td></td></tr><tr><td colspan="2">4</td><td>&lt;0.710</td><td>0.757</td><td>0.757</td><td>0.760</td></tr><tr><td colspan="2">5</td><td>&lt;0.710</td><td>0.756</td><td>0.762</td><td>0.771</td></tr></table>
+
+Concerning $N _ { 0 }$ , contrary to what happened with the PLAX view, it was verified the results improved with higher values. Thus, the search was expanded. It was not possible to test $N _ { 0 } = 6 4$ due to hardware resource exhaustion. The model with highest $D I C E$ has $D = 5$ and $N _ { 0 } = 3 2$ .
+
+# 4.2 Results on test dataset
+
+In this section the U N et segmentation quality is evaluated in the test set. From the $5 2 6 A 4 C$ frames, 3 were excluded due to motion artifacts or probe mispositioning, which impeded the user from annotating the structures. The same happened with 12 of the $5 2 0 ~ P L A X$ frames.
+
+In Fig. 4 the evaluation metrics’ distributions are shown. An immediate assertion to be made is that the results are better in the AMVL than in the PMVL segmentation, in both views. This is in line with what happens with human observers, who have higher inter-observer agreement for the AMVL. Concerning the AMVL segmentation, the $D I C E$ values are above 0.5 in $\bar { P } L A X$ and above 0.6 in $A 4 C$ , with a median of 0.742 in $P L A X$ and 0.795 in $A 4 C$ . High recall values (0.903 median in $P L A X$ and 0.927 in $A 4 C$ ) indicate that most of the leaflets’ pixels were correctly detected as such, so false rejection is not a significant issue. On the other hand, precision presents lower scores (0.688 in $P L A X$ and 0.710 in $A 4 C$ ), which might indicate that some false detection is happening. Post processing techniques may have a positive effect removing false positives. In what concerns PMVL segmentation, the same trends obtained in the AMVL are observed, yet the metrics present wider distributions. This denotes for higher variability in the results, with more false rejection and false detection. In $P L A X$ , median values are $D I C E$ of 0.600, recall of 0.787 and precision of 0.512. In $A 4 C$ , median values are $D I C E$ of 0.690, recall of 0.817 and precision of 0.615. Examples of the obtained segmentation results are shown in Fig. 5. The best $( a , d )$ and worst $( b , e )$ results are displayed for PLAX and A4C. This selection takes into account the average $D I C E$ of the two classes (AMVL and PMVL). The best average result for PLAX is 0.848 and the worst is 0.354. The best average result for A4C is 0.869 and the worst is 0.260. Two examples of false detection errors are also shown: $( c )$ demonstrates overestimation of the structures’ borders and $( f )$ demonstrates the presence of false positives due to reflection artifacts on the US response. In some cases, overestimation may not be a significant error, since some limits of the leaflets (AMVL - posterior wall of the Aorta boundary and PMVL - left atrium wall boundary) are almost arbitrarily chosen when manually annotating. These frames present high recall values, while precision is low.
+
+# 4.3 Results on application dataset
+
+The clinical evaluation results of the application dataset are summarized in Table 4.
+
+The confusion matrices in Table 4 show that both raters assigned score 2 more often than 1, and 1 more often than 0. The pooled Sscore is 0.781, which means a substantial agreement between raters, which reinforces the assigned scores. From all the evaluated parameters, the amount of false positives
+
+![](images/43c4256566d40417560ee57d06f74e9028b12d5ffd4898690b7a43b8b863a0c5.jpg)  
+Figure 4: Boxplot of the evaluation metrics (DICE, recall and precision) obtained from the test images using the $U N e t$ method. The $\times$ are the outliers. (a) AMVL segmentation in A4C view. (b) PMVL segmentation in A4C view. (c) AMVL segmentation in PLAX view. (d) PMVL segmentation in PLAX view.
+
+Table 4: Results of the clinical evaluation of the segmentations’ quality by two raters $R 1$ and $R 2$ ). P1: AMVL tip detection; P2: AMVL thickness; P3: PMVL tip detection; P4: PMVL thickness; P5: amount of false positives; P6: repeatability along the video. S2, S1 and S0 stands for the grading scores.   
+
+<table><tr><td></td><td>P1</td><td colspan="2">P2</td><td colspan="2"></td><td colspan="2">P4</td><td colspan="2">P5</td><td colspan="2"></td><td colspan="2">P6</td></tr><tr><td>R2</td><td></td><td>S2 S1 S0</td><td>S2 S1 S0</td><td></td><td> S2 S1 S0</td><td></td><td> S2 S1 S0</td><td></td><td></td><td> S2 S1 S0</td><td></td><td> S2 S1 S0</td></tr><tr><td>R1 S2</td><td>20</td><td>1 0</td><td>19 3</td><td>0</td><td>21 0</td><td>0</td><td>19 1</td><td>0</td><td>6</td><td>3</td><td>1 18</td><td>2 0</td></tr><tr><td>S1</td><td>1</td><td>1 0</td><td>0 0</td><td>1</td><td>0 1</td><td>1</td><td>1 1</td><td>1</td><td>4</td><td>7</td><td>0 2</td><td>0 0</td></tr><tr><td>S0</td><td>0</td><td>0 0</td><td>0 0</td><td>0</td><td>0 0</td><td>0</td><td>0 0</td><td>0</td><td>0</td><td>1</td><td>1 0</td><td>0 1</td></tr><tr><td>IR Agreement</td><td>0.888</td><td></td><td>0.776</td><td></td><td>0.944</td><td></td><td>0.832</td><td></td><td></td><td>0.469</td><td></td><td>0.776</td></tr></table>
+
+(P5) is the one with lower scores assigned, which is in line with the numerical results that were discussed in the previous section. While most of the parameters met substantial or almost perfect inter-rater agreement, the amount of false positives only met moderate agreement.
+
+# 5 Conclusion
+
+A new method based on the U N et architecture was proposed for the segmentation of the mitral valve leaflets. The architecture was parameterized and trained for each one of the target echocardiographic views, resulting in two models. Results show that both models perform in a similar way, with slight superior performance in the $A 4 C$ model. Moreover, they indicate a median DICE coefficient of 0.74 in $P L A X$ and 0.79 in $A 4 C$ for the anterior mitral leaflet segmentation, while median DICE of 0.60 in $P L A X$ and 0.69 $A 4 C$ are met for the posterior leaflet. By analyzing the recall and precision scores it is possible to understand that the most significant source of error is the false detection. Visual inspection of the results allows to identify two kinds of false detection: overestimation of the structures’ borders and false structures detection caused by imaging artifacts. Future developments should include application of post-processing techniques, which may have a significant impact on the false positives elimination.
+
+![](images/01f9352e7b3c299114d79cdf6dbe13b735c7861757e367ec03c9a4c8d9041bc5.jpg)  
+Figure 5: Example of segmentation results. White contours correspond to manual annotations and red-green to AMVL and PMVL automatic segmentations respectively. $1 ^ { s t }$ row: best (a) and worst (b) results for PLAX view; (c) is an example of error by overestimation. $2 ^ { n d }$ row: best (d) and worst (e) results for A4C view; (f) is an example of false detection.
+
+The clinical evaluation of the segmentation results is in agreement with the quantitative results. The parameter with the lowest scores is the amount of false positives, although the agreement is only moderate enforcing the challenge of this task.
+
+In the future, further model optimization should be tested, as well as include data augmentation to simulate different acquisition settings. The database should also be expanded with representative examples. The clinical evaluation of the results should be continued with more cases to assess real world applicability of the proposed method.
+
+# Acknowledgments
+
+This article is a result of the project (NORTE-01-0247-FEDER-003507-RHDecho), co-funded by Norte Portugal Regional Operational Program (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement, through the European Regional Development Fund (ERDF). This work also had the collaboration of the Fundação para a Ciência a e Tecnologia (FCT) grant no: PD/BD/105761/2014 and has contributions from the project NanoSTIMA, NORTE-01-0145-FEDER-000016, supported by Norte Portugal Regional Operational Program (NORTE 2020), through Portugal 2020 and the European Regional Development Fund (ERDF). GE and PHILLIPS for providing the equipment. Health professionals from Círculo do Coração for their volunteer work and data collection. The Health Secretary of Paraíba for their support to the actualization of the Heart Caravan.
+
+# References
+
+[1] E. Marijon, M. Mirabel, D. S. Celermajer, and X. Jouven, “Rheumatic heart disease,” The Lancet, vol. 379, no. 9819, pp. 953–964, 2012.   
+[2] D. A. Watkins, C. O. Johnson, S. M. Colquhoun, G. Karthikeyan, A. Beaton, G. Bukhman, M. H. Forouzanfar, C. T. Longenecker, B. M. Mayosi, G. A. Mensah et al., “Global, regional, and national burden of rheumatic heart disease, 1990–2015,” New England Journal of Medicine, vol. 377, no. 8, pp. 713–722, 2017.   
+[3] E. Marijon, D. S. Celermajer, and X. Jouven, “Rheumatic heart disease—an iceberg in tropical waters,” 2017.   
+[4] B. Reményi, N. Wilson, A. Steer, B. Ferreira, J. Kado, K. Kumar, J. Lawrenson, G. Maguire, E. Marijon, M. Mirabel et al., “World heart federation criteria for echocardiographic diagnosis of rheumatic heart disease—an evidence-based guideline,” Nature Reviews Cardiology, vol. 9, no. 5, p. 297, 2012.   
+[5] R. Cervera, “Recent advances in antiphospholipid antibody-related valvulopathies,” Journal of autoimmunity, vol. 15, no. 2, pp. 123–125, 2000.   
+[6] J. A. Noble and D. Boukerroui, “Ultrasound image segmentation: a survey,” IEEE Transactions on medical imaging, vol. 25, no. 8, pp. 987–1010, 2006.   
+[7] I. Mikic, S. Krucinski, and J. D. Thomas, “Segmentation and tracking in echocardiographic sequences: Active contours guided by optical flow estimates,” IEEE transactions on medical imaging, vol. 17, no. 2, pp. 274–284, 1998.   
+[8] S. Martin, V. Daanen, O. Chavanon, and J. Troccaz, “Fast segmentation of the mitral valve leaflet in echocardiography,” in International Workshop on Computer Vision Approaches to Medical Image Analysis. Springer, 2006, pp. 225–235.   
+[9] M. S. Sultan, N. C. Martins, E. Costa, D. Veiga, M. J. Ferreira, S. Mattos, and M. T. Coimbra, “Virtual m-mode for echocardiography: A new approach for the segmentation of the anterior mitral leaflet,” IEEE Journal of Biomedical and Health Informatics, 2018.   
+[10] X. Zhou, C. Yang, and W. Yu, “Automatic mitral leaflet tracking in echocardiography by outlier detection in the low-rank representation,” in Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Conference on. IEEE, 2012, pp. 972–979.   
+[11] X. Liu, Y.-m. Cheung, S.-J. Peng, and Q. Peng, “Automatic mitral valve leaflet tracking in echocardiography via constrained outlier pursuit and region-scalable active contours,” Neurocomputing, vol. 144, pp. 47–57, 2014.   
+[12] G. Carneiro, J. C. Nascimento, and A. Freitas, “The segmentation of the left ventricle of the heart from ultrasound data using deep learning architectures and derivative-based search methods,” IEEE Transactions on Image Processing, vol. 21, no. 3, pp. 968–982, March 2012.   
+[13] O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in International Conference on Medical image computing and computer-assisted intervention. Springer, 2015, pp. 234–241.   
+[14] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “TensorFlow: Large-scale machine learning on heterogeneous systems,” 2015, software available from tensorflow.org. [Online]. Available: https://www.tensorflow.org/   
+[15] A. Damien et al., “Tflearn,” https://github.com/tflearn/tflearn, 2016.   
+[16] E. M. Bennett, R. Alpert, and A. Goldstein, “Communications through limited-response questioning,” Public Opinion Quarterly, vol. 18, no. 3, pp. 303–308, 1954.

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+        "text": "Eva Costa Neadvance, Machine Vision, SA Braga, Portugal ecosta@neadvance.com ",
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+        "text": "Malik Saad Sultan   \nInstituto de Telecomunicações FCUP∗, Porto, Portugal   \nengr.saadsultan@gmail.com ",
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+        "text": "Manuel Ferreira   \nNeadvance, Machine Vision, SA Braga, Portugal   \nmferreira@neadvance.com   \nSandra Mattos   \nUCMF †   \nRecife PE, Brazil   \nssmattos@gmail.com Miguel Coimbra   \nInstituto de Telecomunicações FCUP∗, Porto, Portugal   \nmcoimbra@dcc.fc.up.pt ",
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+        "text": "Abstract ",
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+        "text": "Rheumatic heart disease remains a major burden in the developing countries. The World Heart Federation proposed guidelines for the echocardiographic detection of the disease, in which the mitral leaflets’ morphology assessment is a key indicator. The drawback is that these guidelines are dependent on the clinician experience. To overcome this limitation, we propose an automatic segmentation of the mitral leaflets using a new method based on convolutional neural network, specifically the UNet architecture. The results indicate a median DICE coefficient of 0.74 in $P L A X$ and 0.79 in $A 4 C$ for the anterior mitral leaflet segmentation, while median DICE of 0.60 in $P L A X$ and $0 . 6 9 \\ A 4 C$ are met for the posterior leaflet. A visual evaluation of this segmentation approach by two cardiologists is in line with the numerical results. The false detection due to overestimation and artifacts remains an issue to be addressed in the future. ",
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+        "type": "text",
+        "text": "1 Introduction ",
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+        "text": "1.1 Motivation ",
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+        "text": "Rheumatic heart disease (RHD) is a preventable chronic sequel of acute rheumatic fever (ARF), an autoimmune response to group A streptococcal infection. Although being almost eradicated in high-income countries, it remains a major burden in the developing countries, where it causes most of the cardiovascular mortality and morbidity in the young [1]. RHD can be definite (clinically diagnosed) or borderline/sub-clinical (detected only by echocardiography). In a recent prevalence study, the RHD was followed globally over a period of 25-years, [2], and it was estimated that in ",
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+            "Figure 1: Brightness mode echocardiography. (a) parasternal long-axis view. (b) apical four-chamber view. "
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+        "text": "2015 alone, there were 33.4 million cases of RHD, 10.5 million disability-adjusted life-years related to RHD and 319400 RHD-related deaths. However, the estimations may fall short due to missing data in some regions of the globe, misidentification in the causes of death and due to sub-clinical RHD not being included in the prevalence study. This last aspect should not be disregarded since screening studies [3] point that for each case of clinical RHD, 3 to 10 cases of sub-clinical disease exist. It is important to note that even though sub-clinical cases may not develop into definite RHD, it is at this stage that the treatment is most effective with milder health repercussions. ",
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+        "text": "With the advent of portable echocardiography and the increasing detection rates of sub-clinical RHD, an evidence-based set of guidelines was defined by the World Heart Federation (WHF) for the echocardiographic assessment of RHD [4]. RHD mostly affects heart valves, especially the mitral valve and, therefore, the WHF echocardiographic criteria are generally based on the morphology and functionality of this valve. The mitral leaflets’ morphology and mobility is assessed through brightness mode echocardiography, usually in the parasternal long-axis view $( P L A X )$ , and in some cases using the apical four-chamber view $( A 4 C )$ . These echo views are shown in Fig. 1, with the anterior mitral valve leaflet depicted as AMVL and the posterior mitral valve leaflet as PMVL. Morphological assessment is usually done for AMVL instead of PMVL, solely because higher inter-observer agreement is met [4]. Clinical observation suggests the tip of the leaflet is the most commonly part to be affected [5]. Echocardiography assessment requires highly experienced operators, which is a scarce resource in developing countries. The use of image processing tools has the potential to reduce the operator dependency in screening settings, reduce the subjectivity and, in this way, improve the diagnosis. ",
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+        "text": "1.2 State of the art ",
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+        "text": "Ultrasound images are affected by several acquisition artifacts such as attenuation, speckle, shadows and signal dropout. Apart from that, the quality of the acquisition is strongly dependent on the human operator and the machine settings. The intensity and texture differences between structures and the contrast between structures and blood pool are low. These conditions raise several problems for classic image processing methods [6]. In [7], the authors proposed a combination of active contours and optical flow for the AMVL segmentation. The algorithm is semi-automatic and fails when the leaflet’s displacement between frames is large and irregular. Another semi-automatic approach for the AMVL segmentation was proposed in [8]. Two connected active contours identify the cardiac muscle and the leaflet. The manual initialization and the parameters selection highly affect the method’s performance. In [9], the authors propose a semi-automatic segmentation strategy, with a single point input from the user. The input point defines a set of scanning lines for a virtual motionmode (M-mode) reconstruction. The posterior aorta wall’s motion pattern is obtained by applying open-ended active contours to the virtual M-mode, using prior knowledge to establish constraints. The pattern provides a seed for each frame to segment the AMVL with localizing region-based active contours. Although it delivers the middle part of the leaflet, it sometimes fails to segment its tip, which is the most relevant part of the structure diagnosis-wise. An approach based on outlier detection in low-rank matrix was proposed in [10]. The authors aim to overcome the shortfalls of the previous methods, with a fully automatic unsupervised method. However, this solution still requires an extensive parameter fine-tuning and cropping of the images around the region of interest. Also, the method does not discriminate between AMVL and PMVL. In [11] the authors claim to prevent tracking drifts caused by motion ambiguities by constraining the outlier pursuit, and refining the segmentation with region-scalable active contours. Significant parameter fine-tuning remains as a drawback. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Most of the literature approaches are highly sensitive to initialization, image quality and acquisition parameters. None of them segments both leaflets consistently. The complexity of the problem calls for supervised learning methods such as Convolutional Neural Networks (CNN). CNNs have become the state of the art solution for image recognition problems, even outperforming human operators in some tasks. This approach will shift the burden of manual input from the final user to the training phase, and also will not rely on hand-crafted image features, making segmentation a fully automatic and robust process. To our knowledge there are no works in the literature on semantic segmentation of the mitral leaflets using $C N N s$ . In [12], a partial segmentation of the mitral leaflets was needed to segment the left ventricle. The authors propose a network for patches’ classification and then a second network for segmentation of the ventricle. However, they were not able to detect the contours of fast moving structures such as the mitral leaflets. The amount of data, and the respective manual annotations required for training a $C N N$ is a major point to take into account. The UNet architecture, proposed in [13], claims to produce accurate results, with a small number of observations. This trait of the U N et makes it an interesting contender for application in the present work, since the available dataset is also limited. The architecture allows for a multi-scale representation, with coarser information being collected in the bottom layers and finer information at the top ones. The architecture is composed by two paths: one of contraction, with convolutional layers and another of expansion with deconvolution layers. The paths are connected by skip layers before each max-pooling operation. This ensures that both local and global information is captured. ",
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+    {
+        "type": "text",
+        "text": "2 Proposed Work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this work the U N et will be used for the mitral leaflets’ segmentation in the $A 4 C$ and $P L A X$ views. Each view produces distinct representations of the heart structures, thus, the model’s development will be adapted for each one separately. ",
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+    {
+        "type": "text",
+        "text": "2.1 The UNet model ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The most favourable aspects of the U N et architecture are that it does not require a large training set, and that only the image is needed as input. The least favourable trait of the U N et is transversal to all $C N N$ architectures: the parameterization of the network requires a training phase. Depending on the complexity of the network, the training phase may require high computational power and time. ",
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+        "type": "text",
+        "text": "2.1.1 Model implementation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The UNet architecture was recursively implemented in TensorFlow’s [14] front-end TFlearn [15] (Python), allowing expansion of the depth $D$ of the architecture (number of steps on the paths). In Fig. 2, the implemented $U N e t$ architecture is shown. ",
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+        "type": "text",
+        "text": "Taking into account the specificity of the problem, some simplifications and changes were made to the UNet model proposed in [13]. ",
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+        "type": "text",
+        "text": "The first architectural change is the use of zero padding in the convolutions instead of the valid values. The authors of [13] proposed the use of the valid values with a mirror padding pre-processing, so the final outputs have the same spatial dimensions as the original input image. They argue that this accelerates the training, however this was not observed during preliminary tests and therefore, it was decided to use zero padding in all convolutions. ",
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+    {
+        "type": "text",
+        "text": "The second change is the use of batch normalization layers before the concatenation steps. This adds a regularization effect by ensuring that the concatenated feature maps have the same order of magnitude. ",
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+        "type": "text",
+        "text": "Since the present work is focused on the evaluation of the potential of $D N N$ architectures in segmenting the mitral leaflets’, no extensive studies were made for the hyper-parameter optimization. ",
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+    {
+        "type": "image",
+        "img_path": "images/6ac81e3becae434a1ead79036966743a48ac0a6f6f3248e2850fe4936a5fb35a.jpg",
+        "image_caption": [
+            "Figure 2: Proposed UNet model, adapted from [13]. Normalizing layers were added before every $m a x - p o o l$ operation; same padding in all Conv layers, instead of valid. $N _ { x }$ stands for the number of filters, $D$ for depth and $S$ for the spatial size of the feature map. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Most of the parameters were empirically chosen and maintained unchanged throughout the training stages: the learning rate was set to 0.001, the optimization algorithm was the Adam, a batch size of 4 was set, as loss function the mean square difference was selected, and as activation function the Sigmoid was used with threshold of 0.5. ",
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+    {
+        "type": "text",
+        "text": "The samples were randomly shuffled and divided into a training and validation set ( $7 5 \\%$ for training and $2 5 \\%$ for validation). Due to memory management and taking into account the down-sampling process, samples’ dimensions were set to a 416 pixel height and 512 pixel width. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To achieve more accurate results, the architecture’s depth $D$ and the number of incoming neurons $N _ { 0 }$ were subject to a greed search optimization. Preliminary tests have shown that lower depths result in less accurate results. ",
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+    {
+        "type": "text",
+        "text": "3 Materials and Methods ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The datasets used for this work contain videos from different patients, as described in Table 1. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/1864cc2452b488408fbc19dea51618bd209a90f77f25b3592ff191a829b0e8f7.jpg",
+        "table_caption": [
+            "Table 1: Distribution of the datasets for Training and Validation, Test and Application phases. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Echo view</td><td>Train and Validation</td><td>Test</td><td>Application</td></tr><tr><td>PLAX</td><td>21 videos (2163 frames）6 videos (520 frames)</td><td></td><td>23 videos</td></tr><tr><td>A4C</td><td>22 videos (2400 frames)</td><td>）6 videos (526 frames)</td><td>23 videos</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "At the application phase, the tested U N et models were applied to the dataset for clinical assessment. Two cardiologists were asked to evaluate the segmentation quality based on 6 parameters for the two views: overall detection of the each leaflet’ tip pixels, overall estimate of each leaflet’ thickness, amount of false positives and repeatability of the segmentation quality along the video. Each case was graded with scores of 0, 1 and 2 (0 connotes failure and 2 success). ",
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+    {
+        "type": "text",
+        "text": "The echocardiography sequences were acquired during the Heart Caravan of 2016, a health care provision initiative which took place in the State of Paraíba, Brazil. All images were acquired using a Vivid I, by GE and/or a CX-50, by Philips and from children with ages between 4 and 16 years old. The sets used for training, validation and test were manually annotated in each frame as depicted in Fig. 3.These annotations of the mitral leaflets (AMVL and PMVL) were made by an experienced user and validated by two pediatric cardiologists. ",
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+    {
+        "type": "text",
+        "text": "For the models’ performance assessment during parameterization, the dice similarity coefficient $( D I C E )$ was used. This metric measures the similarity between the model’s prediction and the manual annotation. For the evaluation of the UNet’s segmentation results, the $D I C E$ , precision and recall were used. For the assessment of inter-rater agreement on the evaluation of the results in clinical context, the Bennett’s Sscore [16] was applied. The Sscore estimates the agreement assuming that the likelihood of random agreement (both rater agree, when both select a category randomly) is solely dependent on the number of categories. For $q$ categories, $r$ raters and $n$ rated items, the Sscore is defined as: ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/7ede108114bd8d896cc2d4c7553ad6b5a4715192c40840765f5f9b03cbfeedc9.jpg",
+        "image_caption": [
+            "Figure 3: Manual annotations of the AMVL, PMVL and CT. (a) parasternal long-axis view. (b) apical four-chamber view. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/0caaa68569e44e5cb9d52b023356f5161667972eaeffa41371c2a7714f4e496a.jpg",
+        "text": "$$\nS s c o r e = \\frac { p _ { o } - p _ { c } } { 1 - p _ { c } } , w i t h :\n$$",
+        "text_format": "latex",
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+    {
+        "type": "equation",
+        "img_path": "images/8a5dd196af075e8c7f4ddc6029c507b2e6467df46d7bf508c24cb088ac73e14a.jpg",
+        "text": "$$\np _ { c } = \\frac { 1 } { q ^ { 2 } } \\sum _ { k , l } w _ { k l } , \\qquad p _ { o } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\sum _ { k = 1 } ^ { q } \\frac { r _ { i k } ( r _ { i k } ^ { * } - 1 ) } { r _ { i } ( r _ { i } - 1 ) } , \\qquad r _ { i k } ^ { * } \\sum _ { l = 1 } ^ { q } w _ { k l } r _ { i l }\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where, $r _ { i l }$ and $r _ { i k }$ are the number of raters assigning item $i$ to category $l$ or $k$ , respectively. Since in our case the categorization is a grading process, ordinal weighting was applied, thus when the two raters agree total credit is given $w _ { k l } = 1 ,$ ), when raters disagree by choosing immediate neighbor categories partial credit is given ${ w _ { k l } = 0 . 3 3 } $ ), and when raters disagree by choosing non-neighbor categories no credit is given $\\mathbf { \\boldsymbol { w } } _ { k l } = 0$ ). ",
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+    {
+        "type": "text",
+        "text": "The UNet models were trained in a Desktop PC with Intel Core i7 Processor $( 3 . 4 0 \\mathrm { G H z }$ ), 16 GB RAM, and NVIDIA GeForce GTX 970 GPU with 4 GB RAM. ",
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+    {
+        "type": "text",
+        "text": "4 Results and Discussion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section we elaborate on the results gathered in this work, and on their evaluation. The following subsection covers the results of the $D$ and $N _ { 0 }$ parameters grid search, the results on the test dataset and the results of the clinical evaluation of the proposed method. ",
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+    {
+        "type": "text",
+        "text": "4.1 Parameterization ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The parameterization of the architecture’s depth $D$ and number of incoming neurons $N _ { 0 }$ was made for each echo view by evaluating the $D I C E$ in the validation stage. The training stages were all stopped at 30 epochs and the model with higher $D I C E$ in validation was saved. The highest result was always found before the $3 0 ^ { t h }$ epoch. Depths higher than 5 resulted in GPU memory overflow and, because of that, only depths of 4 and 5 were tested. The number of incoming neurons $N _ { 0 }$ was studied in the range from 4 to 32 with base 2 steps. ",
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+    {
+        "type": "text",
+        "text": "4.1.1 Parasternal long-axis ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The results obtained in the validation stage are summarized in Table 2. ",
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+    },
+    {
+        "type": "text",
+        "text": "The highest $D I C E$ was obtained for $D = 5$ and $N _ { 0 } = 8$ . Even though the model lies in the border of $D$ , further exploration was not made due to GPU overflow for depths higher than 5. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/19d17c1d8c23536bab9ecd051815c640cc0aa21fb6b200d7bde71a38e521ef5b.jpg",
+        "table_caption": [
+            "Table 2: Mean DICE Coefficient results in the validation set. In bold is the highest DICE. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>D</td><td>No 4</td><td>8</td><td>16</td></tr><tr><td>4</td><td>&lt;0.710</td><td>0.762</td><td>0.770</td></tr><tr><td>5</td><td>&lt;0.710</td><td>0.791</td><td>0.786</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.1.2 Apical four-chamber ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The results obtained in the validation stage are summarized in Table 3. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/3404de11fa4f9e9b82984cf047508822ab72727f99124dc39fc5ac00ef87334a.jpg",
+        "table_caption": [
+            "Table 3: Mean DICE Coefficient results in the validation set. In bold is the highest $D I C E$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>No</td><td rowspan=\"2\">4</td><td rowspan=\"2\">8</td><td rowspan=\"2\">16</td><td rowspan=\"2\">32</td></tr><tr><td>D</td><td></td></tr><tr><td colspan=\"2\">4</td><td>&lt;0.710</td><td>0.757</td><td>0.757</td><td>0.760</td></tr><tr><td colspan=\"2\">5</td><td>&lt;0.710</td><td>0.756</td><td>0.762</td><td>0.771</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Concerning $N _ { 0 }$ , contrary to what happened with the PLAX view, it was verified the results improved with higher values. Thus, the search was expanded. It was not possible to test $N _ { 0 } = 6 4$ due to hardware resource exhaustion. The model with highest $D I C E$ has $D = 5$ and $N _ { 0 } = 3 2$ . ",
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+    {
+        "type": "text",
+        "text": "4.2 Results on test dataset ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section the U N et segmentation quality is evaluated in the test set. From the $5 2 6 A 4 C$ frames, 3 were excluded due to motion artifacts or probe mispositioning, which impeded the user from annotating the structures. The same happened with 12 of the $5 2 0 ~ P L A X$ frames. ",
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+    {
+        "type": "text",
+        "text": "In Fig. 4 the evaluation metrics’ distributions are shown. An immediate assertion to be made is that the results are better in the AMVL than in the PMVL segmentation, in both views. This is in line with what happens with human observers, who have higher inter-observer agreement for the AMVL. Concerning the AMVL segmentation, the $D I C E$ values are above 0.5 in $\\bar { P } L A X$ and above 0.6 in $A 4 C$ , with a median of 0.742 in $P L A X$ and 0.795 in $A 4 C$ . High recall values (0.903 median in $P L A X$ and 0.927 in $A 4 C$ ) indicate that most of the leaflets’ pixels were correctly detected as such, so false rejection is not a significant issue. On the other hand, precision presents lower scores (0.688 in $P L A X$ and 0.710 in $A 4 C$ ), which might indicate that some false detection is happening. Post processing techniques may have a positive effect removing false positives. In what concerns PMVL segmentation, the same trends obtained in the AMVL are observed, yet the metrics present wider distributions. This denotes for higher variability in the results, with more false rejection and false detection. In $P L A X$ , median values are $D I C E$ of 0.600, recall of 0.787 and precision of 0.512. In $A 4 C$ , median values are $D I C E$ of 0.690, recall of 0.817 and precision of 0.615. Examples of the obtained segmentation results are shown in Fig. 5. The best $( a , d )$ and worst $( b , e )$ results are displayed for PLAX and A4C. This selection takes into account the average $D I C E$ of the two classes (AMVL and PMVL). The best average result for PLAX is 0.848 and the worst is 0.354. The best average result for A4C is 0.869 and the worst is 0.260. Two examples of false detection errors are also shown: $( c )$ demonstrates overestimation of the structures’ borders and $( f )$ demonstrates the presence of false positives due to reflection artifacts on the US response. In some cases, overestimation may not be a significant error, since some limits of the leaflets (AMVL - posterior wall of the Aorta boundary and PMVL - left atrium wall boundary) are almost arbitrarily chosen when manually annotating. These frames present high recall values, while precision is low. ",
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+    {
+        "type": "text",
+        "text": "4.3 Results on application dataset ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The clinical evaluation results of the application dataset are summarized in Table 4. ",
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+    {
+        "type": "text",
+        "text": "The confusion matrices in Table 4 show that both raters assigned score 2 more often than 1, and 1 more often than 0. The pooled Sscore is 0.781, which means a substantial agreement between raters, which reinforces the assigned scores. From all the evaluated parameters, the amount of false positives ",
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+        "img_path": "images/43c4256566d40417560ee57d06f74e9028b12d5ffd4898690b7a43b8b863a0c5.jpg",
+        "image_caption": [
+            "Figure 4: Boxplot of the evaluation metrics (DICE, recall and precision) obtained from the test images using the $U N e t$ method. The $\\times$ are the outliers. (a) AMVL segmentation in A4C view. (b) PMVL segmentation in A4C view. (c) AMVL segmentation in PLAX view. (d) PMVL segmentation in PLAX view. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/72f7a3484b913091865b73defa8127a833a447eb9021c4e70c08de97c00ee1aa.jpg",
+        "table_caption": [
+            "Table 4: Results of the clinical evaluation of the segmentations’ quality by two raters $R 1$ and $R 2$ ). P1: AMVL tip detection; P2: AMVL thickness; P3: PMVL tip detection; P4: PMVL thickness; P5: amount of false positives; P6: repeatability along the video. S2, S1 and S0 stands for the grading scores. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>P1</td><td colspan=\"2\">P2</td><td colspan=\"2\"></td><td colspan=\"2\">P4</td><td colspan=\"2\">P5</td><td colspan=\"2\"></td><td colspan=\"2\">P6</td></tr><tr><td>R2</td><td></td><td>S2 S1 S0</td><td>S2 S1 S0</td><td></td><td> S2 S1 S0</td><td></td><td> S2 S1 S0</td><td></td><td></td><td> S2 S1 S0</td><td></td><td> S2 S1 S0</td></tr><tr><td>R1 S2</td><td>20</td><td>1 0</td><td>19 3</td><td>0</td><td>21 0</td><td>0</td><td>19 1</td><td>0</td><td>6</td><td>3</td><td>1 18</td><td>2 0</td></tr><tr><td>S1</td><td>1</td><td>1 0</td><td>0 0</td><td>1</td><td>0 1</td><td>1</td><td>1 1</td><td>1</td><td>4</td><td>7</td><td>0 2</td><td>0 0</td></tr><tr><td>S0</td><td>0</td><td>0 0</td><td>0 0</td><td>0</td><td>0 0</td><td>0</td><td>0 0</td><td>0</td><td>0</td><td>1</td><td>1 0</td><td>0 1</td></tr><tr><td>IR Agreement</td><td>0.888</td><td></td><td>0.776</td><td></td><td>0.944</td><td></td><td>0.832</td><td></td><td></td><td>0.469</td><td></td><td>0.776</td></tr></table>",
+        "bbox": [
+            189,
+            642,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(P5) is the one with lower scores assigned, which is in line with the numerical results that were discussed in the previous section. While most of the parameters met substantial or almost perfect inter-rater agreement, the amount of false positives only met moderate agreement. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            851,
+            299,
+            867
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "A new method based on the U N et architecture was proposed for the segmentation of the mitral valve leaflets. The architecture was parameterized and trained for each one of the target echocardiographic views, resulting in two models. Results show that both models perform in a similar way, with slight superior performance in the $A 4 C$ model. Moreover, they indicate a median DICE coefficient of 0.74 in $P L A X$ and 0.79 in $A 4 C$ for the anterior mitral leaflet segmentation, while median DICE of 0.60 in $P L A X$ and 0.69 $A 4 C$ are met for the posterior leaflet. By analyzing the recall and precision scores it is possible to understand that the most significant source of error is the false detection. Visual inspection of the results allows to identify two kinds of false detection: overestimation of the structures’ borders and false structures detection caused by imaging artifacts. Future developments should include application of post-processing techniques, which may have a significant impact on the false positives elimination. ",
+        "bbox": [
+            173,
+            882,
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/01f9352e7b3c299114d79cdf6dbe13b735c7861757e367ec03c9a4c8d9041bc5.jpg",
+        "image_caption": [
+            "Figure 5: Example of segmentation results. White contours correspond to manual annotations and red-green to AMVL and PMVL automatic segmentations respectively. $1 ^ { s t }$ row: best (a) and worst (b) results for PLAX view; (c) is an example of error by overestimation. $2 ^ { n d }$ row: best (d) and worst (e) results for A4C view; (f) is an example of false detection. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            212,
+            102,
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+            390
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            493,
+            825,
+            617
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The clinical evaluation of the segmentation results is in agreement with the quantitative results. The parameter with the lowest scores is the amount of false positives, although the agreement is only moderate enforcing the challenge of this task. ",
+        "bbox": [
+            174,
+            626,
+            825,
+            666
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In the future, further model optimization should be tested, as well as include data augmentation to simulate different acquisition settings. The database should also be expanded with representative examples. The clinical evaluation of the results should be continued with more cases to assess real world applicability of the proposed method. ",
+        "bbox": [
+            174,
+            672,
+            825,
+            729
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            757,
+            302,
+            772
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This article is a result of the project (NORTE-01-0247-FEDER-003507-RHDecho), co-funded by Norte Portugal Regional Operational Program (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement, through the European Regional Development Fund (ERDF). This work also had the collaboration of the Fundação para a Ciência a e Tecnologia (FCT) grant no: PD/BD/105761/2014 and has contributions from the project NanoSTIMA, NORTE-01-0145-FEDER-000016, supported by Norte Portugal Regional Operational Program (NORTE 2020), through Portugal 2020 and the European Regional Development Fund (ERDF). GE and PHILLIPS for providing the equipment. Health professionals from Círculo do Coração for their volunteer work and data collection. The Health Secretary of Paraíba for their support to the actualization of the Heart Caravan. ",
+        "bbox": [
+            174,
+            786,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            89,
+            267,
+            106
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "[1] E. Marijon, M. Mirabel, D. S. Celermajer, and X. Jouven, “Rheumatic heart disease,” The Lancet, vol. 379, no. 9819, pp. 953–964, 2012.   \n[2] D. A. Watkins, C. O. Johnson, S. M. Colquhoun, G. Karthikeyan, A. Beaton, G. Bukhman, M. H. Forouzanfar, C. T. Longenecker, B. M. Mayosi, G. A. Mensah et al., “Global, regional, and national burden of rheumatic heart disease, 1990–2015,” New England Journal of Medicine, vol. 377, no. 8, pp. 713–722, 2017.   \n[3] E. Marijon, D. S. Celermajer, and X. Jouven, “Rheumatic heart disease—an iceberg in tropical waters,” 2017.   \n[4] B. Reményi, N. Wilson, A. Steer, B. Ferreira, J. Kado, K. Kumar, J. Lawrenson, G. Maguire, E. Marijon, M. Mirabel et al., “World heart federation criteria for echocardiographic diagnosis of rheumatic heart disease—an evidence-based guideline,” Nature Reviews Cardiology, vol. 9, no. 5, p. 297, 2012.   \n[5] R. Cervera, “Recent advances in antiphospholipid antibody-related valvulopathies,” Journal of autoimmunity, vol. 15, no. 2, pp. 123–125, 2000.   \n[6] J. A. Noble and D. Boukerroui, “Ultrasound image segmentation: a survey,” IEEE Transactions on medical imaging, vol. 25, no. 8, pp. 987–1010, 2006.   \n[7] I. Mikic, S. Krucinski, and J. D. Thomas, “Segmentation and tracking in echocardiographic sequences: Active contours guided by optical flow estimates,” IEEE transactions on medical imaging, vol. 17, no. 2, pp. 274–284, 1998.   \n[8] S. Martin, V. Daanen, O. Chavanon, and J. Troccaz, “Fast segmentation of the mitral valve leaflet in echocardiography,” in International Workshop on Computer Vision Approaches to Medical Image Analysis. Springer, 2006, pp. 225–235.   \n[9] M. S. Sultan, N. C. Martins, E. Costa, D. Veiga, M. J. Ferreira, S. Mattos, and M. T. Coimbra, “Virtual m-mode for echocardiography: A new approach for the segmentation of the anterior mitral leaflet,” IEEE Journal of Biomedical and Health Informatics, 2018.   \n[10] X. Zhou, C. Yang, and W. Yu, “Automatic mitral leaflet tracking in echocardiography by outlier detection in the low-rank representation,” in Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Conference on. IEEE, 2012, pp. 972–979.   \n[11] X. Liu, Y.-m. Cheung, S.-J. Peng, and Q. Peng, “Automatic mitral valve leaflet tracking in echocardiography via constrained outlier pursuit and region-scalable active contours,” Neurocomputing, vol. 144, pp. 47–57, 2014.   \n[12] G. Carneiro, J. C. Nascimento, and A. Freitas, “The segmentation of the left ventricle of the heart from ultrasound data using deep learning architectures and derivative-based search methods,” IEEE Transactions on Image Processing, vol. 21, no. 3, pp. 968–982, March 2012.   \n[13] O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in International Conference on Medical image computing and computer-assisted intervention. Springer, 2015, pp. 234–241.   \n[14] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “TensorFlow: Large-scale machine learning on heterogeneous systems,” 2015, software available from tensorflow.org. [Online]. Available: https://www.tensorflow.org/   \n[15] A. Damien et al., “Tflearn,” https://github.com/tflearn/tflearn, 2016.   \n[16] E. M. Bennett, R. Alpert, and A. Goldstein, “Communications through limited-response questioning,” Public Opinion Quarterly, vol. 18, no. 3, pp. 303–308, 1954. ",
+        "bbox": [
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+        "page_idx": 8
+    }
+]

parse/train/XN1M27T6uux/XN1M27T6uux.md

@@ -0,0 +1,282 @@
+# DECAF: Generating Fair Synthetic Data Using Causally-Aware Generative Networks
+
+Boris van Breugel∗ University of Cambridge bv292@cam.ac.uk
+
+Trent Kyono∗ University of California, Los Angeles tmkyono@ucla.edu
+
+Jeroen Berrevoets University of Cambridge jb2384@cam.ac.uk
+
+Mihaela van der Schaar University of Cambridge University of California, Los Angeles The Alan Turing Institute mv472@cam.ac.uk
+
+# Abstract
+
+Machine learning models have been criticized for reflecting unfair biases in the training data. Instead of solving for this by introducing fair learning algorithms directly, we focus on generating fair synthetic data, such that any downstream learner is fair. Generating fair synthetic data from unfair data— while remaining truthful to the underlying data-generating process $( D G P )$ —is non-trivial. In this paper, we introduce DECAF: a GAN-based fair synthetic data generator for tabular data. With DECAF we embed the DGP explicitly as a structural causal model in the input layers of the generator, allowing each variable to be reconstructed conditioned on its causal parents. This procedure enables inference-time debiasing, where biased edges can be strategically removed for satisfying user-defined fairness requirements. The DECAF framework is versatile and compatible with several popular definitions of fairness. In our experiments, we show that DECAF successfully removes undesired bias and— in contrast to existing methods —is capable of generating high-quality synthetic data. Furthermore, we provide theoretical guarantees on the generator’s convergence and the fairness of downstream models.
+
+# 1 Introduction
+
+Generative models are optimized to approximate the original data distribution as closely as possible. Most research focuses on three objectives [1]: fidelity, diversity, and privacy. The first and second are concerned with how closely synthetic samples resemble real data and how much of the real data’s distribution is covered by the new distribution, respectively. The third objective aims to avoid simply reproducing samples from the original data, which is important if the data contains privacy-sensitive information [2, 3]. We explore a much-less studied concept: synthetic data fairness.
+
+Motivation. Deployed machine learning models have been shown to reflect the bias of the data on which they are trained [4, 5, 6, 7, 8]. This has not only unfairly damaged the discriminated individuals but also society’s trust in machine learning as a whole. A large body of work has explored ways of detecting bias and creating fair predictors [9, 10, 11, 12, 13, 14, 15], while other authors propose debiasing the data itself [9, 10, 11, 16]. This work’s aim is related to the work of [17]: to generate fair synthetic data based on unfair data. Being able to generate fair data is important because end-users creating models based on publicly available data might be unaware they are inadvertently including bias or insufficiently knowledgeable to remove it from their model. Furthermore, by debiasing the data prior to public release, one can guarantee any downstream model satisfies desired fairness requirements by assigning the responsibility of debiasing to the data generating entities.
+
+Table 1: Overview of related work for synthetic data. We organize related work according to our key areas of interest: (1) Allows post-hoc distribution changes, (2) provides fairness, (3) supports causal notion of fairness, (4) allows inference-time fairness, (5) requires minimal assumptions. We highlight the key contribution, and identify non-neural approaches with “†”.   
+
+<table><tr><td>Model</td><td>Reference</td><td>(1)(2)(3)(4)(5)Goal</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="8">Standard synthetic data generation</td></tr><tr><td>VAE</td><td>[19]</td><td>X</td><td>X</td><td>X</td><td>X</td><td></td><td>Realistic synth. data.</td></tr><tr><td>GANs</td><td>[2, 3, 20, 21]</td><td>X</td><td>X</td><td>X</td><td>X</td><td></td><td>Realistic synth. data.</td></tr><tr><td colspan="8"> Methods that detect and/or remove bias</td></tr><tr><td>PSE-DD/DR†</td><td>[11]</td><td>√</td><td>√</td><td>√</td><td>×</td><td>X</td><td>Discover/Remove bias.</td></tr><tr><td>OPPDP†</td><td>[16]</td><td>X</td><td>√</td><td>xxxx&gt;</td><td>xxx</td><td>X</td><td>Remove bias.</td></tr><tr><td>DI</td><td>[10]</td><td>X</td><td>√</td><td></td><td></td><td>X</td><td>Discover/Remove bias.</td></tr><tr><td>LFR</td><td>[22]</td><td>×</td><td>一</td><td></td><td></td><td>√</td><td>Learn fair representation.</td></tr><tr><td>FairGAN</td><td>[17]</td><td>X</td><td>√</td><td></td><td>X</td><td>√</td><td>Realistic and fair synth. data.</td></tr><tr><td>CFGAN</td><td>[23]</td><td>X</td><td>√</td><td></td><td>×</td><td>√</td><td>Realistic and fair synth. data.</td></tr><tr><td>DECAF</td><td>(ours)</td><td>√</td><td>√</td><td>&lt;</td><td>√</td><td>√</td><td>Realistic and fair synth. data.</td></tr></table>
+
+Goal. From a biased dataset $\mathcal { X }$ , we are interested in learning a model $G$ , that is able to generate an equivalent synthetic unbiased dataset $\mathcal { X } ^ { \prime }$ with minimal loss of data utility. Furthermore, a downstream model trained on the synthetic data needs to make not only unbiased predictions on the synthetic data, but also on real-life datasets (as formalized in Section 4.2).
+
+Solution. We approach fairness from a causal standpoint because it provides an intuitive perspective on different definitions of fairness and discrimination [11, 13, 14, 15, 18]. We introduce DEbiasing CAusal Fairness (DECAF), a generative adversarial network (GAN) that leverages causal structure for synthesizing data. Specifically, DECAF is comprised of $d$ generators (one for each variable) that learn the causal conditionals observed in the data. At inference-time, variables are synthesized topologically starting from the root nodes in the causal graph then synthesized sequentially, terminating at the leave nodes. Because of this, DECAF can remove bias at inference-time through targeted (biased) edge removal. As a result, various datasets can be created for desired (or evolving) definitions of fairness.
+
+Contributions. We propose a framework of using causal knowledge for fair synthetic data generation. We make three main contributions: i) DECAF, a causal GAN-based model for generating synthetic data, ii) a flexible causal approach for modifying this model such that it can generate fair data, and iii) guarantees that downstream models trained on the synthetic data will also give fair predictions in other settings. Experimentally, we show how DECAF is compatible with several fairness/discrimination definitions used in literature while still maintaining high downstream utility of generated data.
+
+# 2 Related Works
+
+Here we focus on the related work concerned with data generation, in contrast to fairness definitions for which we provide a detailed overview in Section 4 and Appendix C. As an overview of how data generation methods relate to one another, we refer to Table 1 which presents all relevant related methods.
+
+Non-parametric generative modeling. The standard models for synthetic data generation are either based on VAEs [19] or GANs [2, 3, 20, 21]. While these models are well known for their highly realistic synthetic data, they are unable to alter the synthetic data distribution to encourage fairness (except for [17, 23], discussed below). Furthermore, these methods have no causal notion, which prohibits targeted interventions for synthesizing fair data (Section 4). We explicitly leave out CausalGAN [24] and CausalVAE [25], which appear similar by incorporating causality-derived ideas but are different in both method and aim (i.e., image generation).
+
+Fair data generation. In the bottom section of Table 1, we present methods that, in some way, alter the training data of classifiers to adhere to a notion of fairness [10, 11, 16, 17, 22, 23]. While these methods have proven successful, they lack some important features. For example, none of the related methods allow for post-hoc changes of the synthetic data distribution. This is an important feature, as each situation requires a different perspective on fairness and thus requires a flexible framework for selecting protected variables. Additionally, only [11, 23] allow a causal perspective on fairness, despite causal notions underlying multiple interpretations of what should be considered fair [13]. Furthermore, only [17, 22, 23] offer a flexible framework, while the others are limited to binary [10, 11] or discrete [16] settings. Xu et al. [23] also use a causal architecture for the generator, however their method is not as flexible—e.g. it does not easily extend to multiple protected attributes. Finally, in contrast to other methods DECAF is directly concerned with fairness of the downstream model—which is dependent on the setting in which the downstream model is employed (Section 4.2). In essence, from Table 1 we learn that DECAF is the only method that combines all key areas of interest. At last, we would like to mention [26], who aim to generate data that resembles a small unbiased reference dataset, by leveraging a large but biased dataset. This is very different to our aim, as we are interested in the downstream model’s fairness and explicit notions of fairness.
+
+# 3 Preliminaries
+
+Let $X \in \mathcal { X } \subseteq \mathbb { R } ^ { d }$ denote a random variable with distribution $P _ { X } ( X )$ , with protected attributes $A \in { \mathcal { A } } \subset { \mathcal { X } }$ and target variable $Y \in \mathcal { y } \subset \mathcal { x }$ , let $\hat { Y }$ denote a prediction of $Y$ . Let the data be given by $\mathcal { D } = \{ \mathbf { x } ^ { ( k ) } \} _ { k = 1 } ^ { N }$ , where each $\mathbf { x } ^ { ( k ) } \in \mathcal { D }$ is a realization of $X$ . We assume the data generating process can be represented by a directed acyclic graph (DAG)—such that the generation of features can be written as a structural equation model (SEM) [27]—and that this DAG is causally sufficient. Let $X _ { i }$ denote the $i ^ { \mathrm { { t h } } }$ feature in $X$ with causal parents $\operatorname { P a } ( X _ { i } ) \subset \{ X _ { j } : j \neq i \}$ , the SEM is given by:
+
+$$
+X _ { i } = f _ { i } ( \operatorname { P a } ( X _ { i } ) , Z _ { i } ) , \forall i
+$$
+
+where $\{ Z _ { i } \} _ { i = 1 } ^ { d }$ are independent random noise variables, that is $\mathrm { P a } ( Z _ { i } ) = \emptyset$ , ∀i. Note that each $f _ { i }$ is a deterministic function that places all randomness of the conditional $P ( X _ { i } | \operatorname* { P a } ( X _ { i } ) )$ in the respective noise variable, $Z _ { i }$ .
+
+# 4 Fairness of Synthetic Data
+
+Algorithmic fairness is a popular topic (e.g., see [13, 28]), but fair synthetic data has been much less explored. This section highlights how the underlying graphs of the synthetic and downstream data determine whether a model trained on the synthetic data will be fair in practice. We start with the two most popular definitions of fairness, relating to the legal concepts of direct and indirect discrimination. We also explore conditional fairness [29], which is a generalization of the two. In Appendix C we discuss how the ideas in this section transfer to other independence-based definitions [30]. Throughout this section, we separate $Y$ from $X$ by defining ${ \bar { X } } { \stackrel { \cdot } { = } } X \backslash Y$ , and we will write $X  { \bar { X } }$ for ease of notation.
+
+# 4.1 Algorithmic fairness
+
+The first definition is called Fairness Through Unawareness (e.g. [31]).
+
+Definition 1. (Fairness Through Unawareness (FTU): algorithm). A predictor $f : X \mapsto { \hat { Y } }$ is fair iff protected attributes $A$ are not explicitly used by $f$ to predict $\hat { Y }$ .
+
+This definition prohibits disparate treatment [28, 32], and is related to the legal concept of direct discrimination, i.e., two equally qualified people deserve the same job opportunity independent of their race, gender, beliefs, among others.
+
+Though FTU fairness is commonly used, it might result in indirect discrimination: covariates that influence the prediction $\hat { Y }$ might not be identically distributed across different groups $a , a ^ { \prime }$ , which means an algorithm might have disparate impact on a protected group [10]. The second definition of fairness, demographic parity [32], does not allow this:
+
+Definition 2. (Demographic Parity $( D P )$ : algorithm) $A$ predictor $\hat { Y }$ is fair iff $A \perp \perp { \hat { Y } }$ , i.e. $\forall a , a ^ { \prime }$   
+$P ( \hat { Y } | A = a ) = P ( \hat { Y } | A = a ^ { \prime } )$ .
+
+Evidently, DP puts stringent constraints on the algorithm, whereas FTU might be too lenient. The third definition we include is based on the work of [29], related to unresolved discrimination [14]. The idea is that we do not allow indirect discrimination unless it runs through explanatory factors $R \subset X$ . For example, in Simpson’s paradox [33] there seems to be a bias between gender and college admissions, but this is only due to women applying to more competitive courses. In this case, one would want to regard fairness conditioned on the choice of study [14]. Let us define this as conditional fairness:
+
+Definition 3. (Conditional Fairness $( C F )$ : algorithm) $A$ predictor $\hat { Y }$ is fair iff $A \perp \perp { \hat { Y } } | R ,$ , i.e.   
+$\forall r , a , a ^ { \prime } : P ( \hat { Y } | R = r , A = a ) = P ( \hat { Y } | R = r , A = a ^ { \prime } )$ .
+
+CF generalizes FTU and DP Note that conditional fairness is a generalization of FTU and DP, by setting $R = X \backslash A$ and $R = \emptyset$ , respectively. In Appendix $\textrm { C }$ we elaborate on the connection between these, and more, definitions.
+
+# 4.2 Synthetic data fairness
+
+Algorithmic definitions can be extended to distributional fairness for synthetic data. Let $P ( X ) , P ^ { \prime } ( X )$ be probability distributions with protected attributes $A \subset X$ and labels $Y \subset X$ . Let ${ \mathcal { T } } ( A , Y )$ be a definition of algorithmic fairness (e.g., FTU). Note, that under CF, ${ \mathcal { T } } ( A , Y )$ is a function of $R$ as well. We propose $( { \mathcal { T } } ( A , Y ) , P )$ -fairness of distribution $P ^ { \prime } ( X )$ :
+
+Definition 4. (Distributional fairness) $A$ probability distribution $P ^ { \prime } ( X )$ is $( { \mathcal { T } } ( A , Y ) , P )$ -fair, iff the optimal predictor $\hat { Y } = f ^ { * } ( X )$ of $Y$ trained on $P ^ { \prime } ( X )$ satisfies ${ \mathcal { T } } ( A , Y )$ when evaluated on $P ( X )$ .
+
+In other words, when we train a predictor on $( { \mathcal { T } } ( A , Y ) , P )$ -fair distribution $P ^ { \prime } ( X )$ , we can only reach maximum performance if our model is fair. Note the explicit reference to $P ( X )$ , the distribution on which fairness is evaluated, which does not need to coincide with $P ^ { \prime } ( X )$ . This is a small but relevant detail. For example, when training a model on data ${ \mathcal { D } } ^ { \prime } \sim P ^ { \prime } ( X )$ it could seem like the model is fair when we evaluate it on a hold-out set of the data (e.g., if we simply remove the protected attribute from the data). However, when we use the model for real-world predictions of data ${ \mathcal { D } } \sim P ( X )$ , disparate impact is possibly observed due to a distributional shift.
+
+By extension, we define synthetic data as $( { \mathcal { T } } ( A , Y ) , P )$ -fair, iff it is sampled from an $( { \mathcal { T } } ( A , Y ) , P )$ - fair distribution. Defining synthetic data as fair w.r.t. an optimal predictor is especially useful when we want to publish a dataset and do not trust end-users to consider anything but performance.2
+
+Choosing $\mathbf { P } ( \mathbf { X } )$ . The setting $P ( X ) = P ^ { \prime } ( X )$ corresponds to data being fair with respect to itself. For synthetic data generation, this setting is uninteresting as any dataset can be made fair by randomly sampling or removing $A$ ; if $A$ is random, the prediction should not directly or indirectly depend on it. This ignores, however, that a downstream user might use the trained model on a real-world dataset in which other variables $B$ are correlated with $A$ , and thus their model (which is trained to use $B$ for predicting $Y$ ) will be biased. Of specific interest is the setting where $P ( X )$ corresponds to the original data distribution $P _ { X } ( X )$ that contains unfairness. In this scenario, we construct $P ^ { \prime } ( X )$ by learning $P _ { X } ( X )$ and removing the unfair characteristics. The data from $P ^ { \prime } ( X )$ can be published online, and models trained on this data can be deployed fairly in real-life scenarios where data follows $P _ { X } ( X )$ . Unless otherwise stated, henceforth, we assume ${ \dot { P } } ( X ) = P _ { X } ( X )$ .
+
+# 4.3 Graphical perspective
+
+As reflected in the widely accepted terms direct versus indirect discrimination, it is natural to define distributional fairness from a causal standpoint. Let $\mathcal { G } ^ { \prime }$ and $\mathcal { G }$ respectively denote the graphs underlying $P ^ { \prime } ( X )$ (the synthetic data distribution which we can control) and $P ( X )$ (the evaluation distribution that we cannot control). Let $\partial _ { \mathcal { G } } Y$ denote the Markov boundary of $Y$ in graph $\mathcal { G }$ . We focus on the conditional fairness definition because it subsumes the definition of DP and FTU (Section 4.1). Let $R \subset X$ be the set of explanatory features.
+
+Proposition 1. ( $C F .$ : graphical condition) If for all $B \in \partial _ { \mathcal { G } ^ { \prime } } Y , A \bot \bot _ { \mathcal { G } } B | R , ^ { 3 }$ then distribution $P ^ { \prime } ( X )$ is $C F$ fair w.r.t $P ( X )$ given explanatory factors $R$ .
+
+Proof. Without loss of generality, let us assume the label is binary.4 The optimal predictor $f ^ { * } ( X ) =$ $P ( Y | X ) = P ( Y | \partial _ { \mathcal { G } ^ { \prime } } Y )$ . Thus, if $\partial _ { \mathscr { G } ^ { \prime } } Y$ is ${ \mathrm { d } }$ -separated from $A$ in $\mathcal { G }$ given $R$ , prediction $\hat { Y } = f ^ { * } ( X )$ is independent of $A$ given $R$ and CF holds. □
+
+Corollary 1. (CF debiasing) Any distribution $P ^ { \prime } ( X )$ with graph $\mathcal { G } ^ { \prime }$ can be made CF fair w.r.t. $P ( X )$ and explanatory features $R$ by removing from $\mathcal { G } ^ { \prime }$ edges $\tilde { E } = \{ ( B \to Y )$ and $( Y \to B ) : \forall B \in \partial _ { \mathcal { G } ^ { \prime } } Y$ with $B$ 6⊥⊥ $_ { \cdot g } \ A | R \}$ .
+
+Proof. First note $\tilde { E }$ is the necessary and sufficient set of edges to remove for $\forall B \in \partial _ { \mathcal { G } ^ { \prime } } Y$ , $A$ ⊥⊥ $\mathcal { G }$ $| B | R )$ to be true, subsequently the result follows from Proposition 1. □
+
+For FTU (i.e. $R = X \backslash A )$ and DP (i.e. $R = \emptyset$ ), this corollary simplifies to:
+
+Corollary 2. (FTU debiasing) Any distribution $P ^ { \prime } ( X )$ with graph $\mathcal { G } ^ { \prime }$ can be made FTU fair w.r.t. any distribution $P ( X )$ by removing, if present, i) the edge between $A$ and $Y$ and $i i$ ) the edge $A  C$ or $Y  C$ for all shared children $C$ .
+
+Corollary 3. (DP debiasing) Any distribution $P ^ { \prime } ( X )$ with graph $\mathcal { G } ^ { \prime }$ can be made $D P$ fair w.r.t. $P ( X )$ by removing, if present, the edge between $B$ and $Y$ for any $B \in \partial _ { \mathcal { G } ^ { \prime } } Y$ with $B$ 6⊥⊥ $_ { \textit { g A } }$ .
+
+![](images/0a9fed1a05f973016276c4a64c25853270bbd1a127ff235596605576723c23e8.jpg)  
+Figure 1: Edge removal for fairness. FTU: $\pmb { \chi }$ ; DP: ✗✗✗; CF when $R = C$ : $\boldsymbol { x }$ ; CF when $B \in R$ : $\boldsymbol { \chi }$
+
+Figure 1 shows how the different fairness definitions lead to different sets of edges to be removed.
+
+Faithfulness. Usually one assumes distributions are faithful w.r.t. their respective graphs, in which case the if-statement in Proposition 1 become equivalence statements: fairness is only possible when the graphical conditions hold.
+
+Theorem 1. If $P ( X )$ and $P ^ { \prime } ( X )$ are faithful with respect to their respective graphs $\mathcal { G }$ and $\mathcal { G } ^ { \prime }$ , then Proposition $\boldsymbol { l }$ becomes an equivalence statement and Corollaries 1, 2 and 3 describe the necessary and sufficient sets of edges to remove for achieving CF, FTU and DP fairness, respectively.
+
+Proof. Faithfulness implies $A$ ⊥⊥ $P ( X ) ~ B | R \implies A$ ⊥⊥ $_ { \cdot \mathcal { G } } \ B | R$ , e.g. [34]. Thus, if $\exists B \in \partial _ { \mathcal { G } ^ { \prime } } Y$ for which $A$ 6⊥⊥ $_ { \textit { g B } | R }$ , then $A$ 6⊥⊥ $| B | R$ . Because $B \in \partial _ { \mathcal { G } ^ { \prime } } Y$ and $P ^ { \prime } ( X )$ is faithful to $\mathcal { G } ^ { \prime }$ , $\hat { Y } = f ^ { * } ( X )$ depends on $B$ , and thus $\hat { Y }$ 6⊥⊥ $A | R$ : CF does not hold. □
+
+Other definitions. Some authors define similar fairness measures in terms of directed paths (cf.   
+d-separation) [11, 14, 18], which is a milder requirement as it allows correlation via non-causal paths.   
+In Appendix C we highlight the graphical conditions for these definitions.
+
+# 5 Method: DECAF
+
+The primary design goal of DECAF is to generate fair synthetic data from unfair data. We separate DECAF into two stages. The training stage learns the causal conditionals that are observed in the data through a causally-informed GAN. At the generation (inference) stage, we intervene on the learned conditionals via Corollaries 1-3, in such a way that the generator creates fair data. We assume the underlying DGP’s graph $\mathcal { G }$ is known; otherwise, $\mathcal { G }$ needs to be approximated first using any causal discovery method, see Section 6.
+
+# 5.1 Training
+
+Overview. This stage strives to learn the causal mechanisms $\{ f _ { i } ( \mathrm { P a } ( X _ { i } ) , Z _ { i } ) \}$ . Each structural equation $f _ { i }$ (Eq. 1) is modelled by a separate generator $G _ { i } : \mathbb { R } ^ { | P a ( X _ { i } ) | + 1 }  \mathbb { R }$ . We achieve this by employing a conditional GAN framework with a causal generator. This process is illustrated in Figure 2 and detailed below.
+
+![](images/e8cf5f2fe4f5bbb028f6684371bbe8787025750f394e14eba734addfcaece58e.jpg)  
+Figure 2: Architecture. Training phase— Each component in $\hat { \bf X }$ is generated sequentially as a function (where the function is that component’s generator $G _ { i }$ ) of the component’s parents. Parental knowledge is provided by the DAG governing the data. Inference phase— As the component-wise generation of the generator network is independent of the DAG governing the data, we can easily replace (or intervene on) the DAG governing parental information. The resulting synthetic data (right) will be governed by the intervened DAG. FTU is achieved by removing: ✗; $D P$ : ✗✗✗; e.g. CF when $R = C$ : $x x$ .
+
+Features are generated sequentially following the topological ordering of the underlying causal DAG: first root nodes are generated, then their children (from generated causal parents), etc. Variable ${ \hat { X } } _ { i }$ is modelled by the associated generator $G _ { i }$ :
+
+$$
+\hat { X _ { i } } = G _ { i } ( \hat { \mathrm { P a } } ( X _ { i } ) , Z _ { i } ) \quad \forall i ,
+$$
+
+where $\hat { \operatorname { P a } } ( X _ { i } )$ denotes the generated causal parents of $X _ { i }$ (for root nodes the empty set), and each $Z _ { i }$ is independently sampled from $P ( Z )$ (e.g. standard Gaussian). We denote the full sequential generator by $G ( Z ) = [ G _ { 1 } ( Z _ { 1 } ) , . . . , G _ { d } ( \cdot , Z _ { d } ) ]$ .
+
+Subsequently, the synthetic sample $\hat { \bf x }$ is passed to a discriminator $D : \mathbb { R } ^ { d }  \mathbb { R }$ , which is trained to distinguish the generated samples from original samples. A typical minimax objective is employed for creating generated samples that confuse the discriminator most:
+
+$$
+\operatorname* { m a x } _ { \{ G _ { i } \} _ { i = 1 } ^ { d } } \operatorname* { m i n } _ { D } \mathbb { E } [ \log D ( G ( Z ) ) + \log ( 1 - D ( X ) ] ,
+$$
+
+with $X$ sampled from the original data. We optimize the discriminator and generator iteratively and add a regularization loss to both networks. Network parameters are updated using gradient descent.
+
+If we assume $P _ { X } ( X )$ is compatible with graph $\mathcal { G }$ , we can show that the sequential generator has the same theoretical convergence guarantees as standard GANs [20]:
+
+Theorem 2. (Convergence guarantee) Assuming the following three conditions hold:
+
+(i) data generating distribution $P _ { X }$ is Markov compatible with a known DAG $\mathcal { G }$ ;
+
+(ii) generator $G$ and discriminator $D$ have enough capacity; and (iii) in every training step the discriminator is trained to optimality given fixed $G$ , and $G$ is subsequently updated as to maximize the discriminator loss (Eq. 3);
+
+then generator distribution $P _ { G }$ converges to true data distribution $P _ { X }$
+
+Proof. See Appendix B
+
+Condition (i), compatibility with $\mathcal { G }$ , is a weaker assumption than assuming perfect causal knowledge. For example, suppose the Markov equivalence class of the true underlying DAG has been determined through causal discovery. In that case, any graph $\mathcal { G }$ in the equivalence class is compatible with the data and can thus be used for synthetic data generation. However, we note that debiasing can require the correct directionality for some definitions of fairness, see Discussion.
+
+Remark. The causal GAN we propose, DECAF, is simple and extendable to other generative methods, e.g., VAEs. Furthermore, from the post-processing theorem [35] it follows that DECAF can be directly used for generating private synthetic data by replacing the standard discriminator by a differentially private discriminator [2, 36].
+
+# 5.2 Inference-time Debiasing
+
+The training phase yields conditional generators $\{ G _ { i } \} _ { i = 1 } ^ { d }$ , which can be sequentially applied to generate data with the same output distribution as the original data (proof in Appendix B). The causal model allows us to go one step further: when the original data has characteristics that we do not want to propagate to the synthetic data (e.g., gender bias), individual generators can be modified to remove these characteristics. Given the generator’s graph $\mathcal { G } = ( \boldsymbol { X } , \mathbf { \bar { E } } )$ , fairness is achieved by removing edges such that the fairness criteria are met, see Section 4. Let $\tilde { E } \in E$ be the set of edges to remove for satisfying the required fairness definition. For CF, FTU and DP,5 the sets $\tilde { E }$ are given by Corollaries 1, 2 and 3, respectively.
+
+Removing an edge constitutes to what we call a “surrogate” $d o$ -operation [27] on the conditional distribution. For example, suppose we only want to remove $( i  j )$ ). For a given sample, $X _ { i }$ is generated normally (Eq. 2), but $X _ { j }$ is generated using the modified:
+
+$$
+\hat { X } _ { j } ^ { d o ( X _ { i } ) = \tilde { x } _ { i j } } = G _ { j } ( . . . , X _ { i } = \tilde { x } _ { i j } ) ,
+$$
+
+where $X _ { i } ~ = ~ \tilde { x } _ { i j }$ is the surrogate parent assignment. Value ${ \hat { X } } _ { j } ^ { d o ( X _ { i } ) }$ can be interpreted as the counterfactual value of $\hat { X _ { j } }$ , had $X _ { i }$ been equal to $\tilde { x } _ { i j }$ (see also [15]).
+
+Choosing the value of surrogate variable $\tilde { x } _ { i j }$ requires background knowledge of the task and bias at hand. For example, surrogate variable $\tilde { x } _ { i j }$ can be sampled independently from a distribution for each synthetic sample (e.g., the marginal $P ( { \bar { X } } _ { i } ) )$ , be set to a fixed value for all samples in the synthetic data (e.g., if $X _ { i }$ : gender, always set $\tilde { x } _ { i j } = m a l e$ when generating feature $X _ { j }$ : job opportunity) or be chosen as to maximize/minimize some feature (e.g. $\begin{array} { r } { \tilde { x } _ { i j } = \arg \operatorname* { m a x } _ { x } \hat { X } _ { j } ^ { d o ( X _ { i } ) = x } ) } \end{array}$ . We emphasize that we do not set $X _ { i } = \tilde { x } _ { i j }$ in the synthetic sample; $X _ { i } = \tilde { x } _ { i j }$ is only used for substitution of the removed dependence. We provide more details in Appendix E.
+
+More generally, we create surrogate variables for all edges we remove, $\{ \tilde { x } _ { i j } : ( i  j ) \in \tilde { E } \}$ . Each sample is sequentially generated by Eq. 4, with a surrogate variable for each removed incoming edge.
+
+Remark. Multiple datasets can be created based on different definitions of fairness and/or different downstream prediction targets. Because debiasing happens at inference-time, this does not require retraining the model.
+
+# 6 Experiments
+
+In this section, we validate the performance of DECAF for synthesizing bias-free data based on two datasets: i) real data with existing bias and ii) real data with synthetically injected bias. The aim of the former is to show that we can remove real, existing bias. The latter experiment provides a ground-truth unbiased target distribution, which means we can evaluate the quality of the synthetic dataset with respect to this ground truth. For example, when historically biased data is first debiased, a model trained on the synthetic data will likely create better predictions in contemporary, unbiased/less-biased settings than benchmarks that do not debias first.
+
+In both experiments, the ground-truth DAG is unknown. We use causal discovery to uncover the underlying DAG and show empirically that the performance is still good.
+
+Benchmarks. We compare DECAF against the following benchmark generative methods: a GAN, a Wasserstein GAN with gradient penalty (WGAN-GP) [21] and FairGAN [17]. FairGAN is the only benchmark designed to generate synthetic fair data,6 whereas GAN and WGAN-GP only aim to match the original data’s distribution, regardless of inherent underlying bias. For these benchmarks, fair data can be generated by naively removing the protected variable – we refer to these methods with the PR (protected removal) suffix and provide more experimental results and insight into PR in Appendix A. We benchmark DECAF debiasing in four ways: i) with no inference-time debiasing (DECAF-ND), ii) under FTU (DECAF-FTU), iii) under CF (DECAF-CF) and iv) under DP fairness (DECAF-DP). We provide $\mathrm { D E C A F ^ { 7 } }$ implementation details in Appendix D.1.
+
+Table 2: Bias removal experiment on the Adult dataset [40]. The full table with protected attribute removal can be found in Appendix A.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Data Quality</td><td colspan="2">Fairness</td></tr><tr><td>Precision↑</td><td>Recall↑</td><td>AUROC↑</td><td>FTU↓</td><td>DP↓</td></tr><tr><td>Original data D</td><td>0.920±0.006</td><td>0.936 ±0.008</td><td>0.807 ± 0.004</td><td>0.116 ± 0.028</td><td>0.180 ± 0.010</td></tr><tr><td>GAN</td><td>0.607 ± 0.080</td><td>0.439±0.037</td><td>0.567 ± 0.132</td><td>0.023 ± 0.010</td><td>0.089 ± 0.008</td></tr><tr><td>WGAN-GP</td><td>0.683 ± 0.015</td><td>0.914 ± 0.005</td><td>0.798 ± 0.009</td><td>0.120 ± 0.014</td><td>0.189 ±0.024</td></tr><tr><td>FairGAN</td><td>0.681 ± 0.023</td><td>0.814 ± 0.079</td><td>0.766 ± 0.029</td><td>0.009±0.002</td><td>0.097 ± 0.018</td></tr><tr><td>DECAF-ND</td><td>0.780 ±0.023</td><td>0.920 ± 0.045</td><td>0.781 ± 0.007</td><td>0.152 ± 0.013</td><td>0.198 ± 0.013</td></tr><tr><td>DECAF-FTU</td><td>0.763 ± 0.033</td><td>0.925 ± 0.040</td><td>0.765 ± 0.010</td><td>0.004±0.004</td><td>0.054±0.005</td></tr><tr><td>DECAF-CF</td><td>0.743 ± 0.022</td><td>0.875 ± 0.038</td><td>0.769 ± 0.004</td><td>0.003 ±0.006</td><td>0.039 ± 0.011</td></tr><tr><td>DECAF-DP</td><td>0.781± 0.018</td><td>0.881±0.050</td><td>0.672 ± 0.014</td><td>0.001± 0.002</td><td>0.001 ± 0.001</td></tr></table>
+
+Evaluation criteria. We evaluate DECAF using the following metrics:
+
+• Data quality is assessed using metrics of precision and recall [37, 38, 39]. Additionally, we evaluate all methods in terms of AUROC of predicting the target variable using a downstream classifier (MLP in these experiments) trained on synthetic data.   
+• FTU is measured by calculating the difference between the predictions of a downstream classifier for setting $A$ to 1 and 0, respectively, such that $| \bar { P } _ { A = 0 } ( \hat { Y } | X ) - P _ { A = 1 } ( \hat { Y } | X ) |$ , while keeping all other features the same. This difference measures the direct influence of $A$ on the prediction.   
+• DP is measured in terms of the Total Variation [15]: the difference between the predictions of a downstream classifier in terms of positive to negative ratio between the different classes of protected variable $A$ , i.e., $| P ( \hat { Y } | A = 0 ) - P ( \hat { Y } | A = 1 ) |$ .
+
+# 6.1 Debiasing Census Data
+
+In this experiment, we are given a biased dataset ${ \mathcal { D } } \sim P ( X )$ and wish to create a synthetic (and debiased) dataset $\mathcal { D } ^ { \prime }$ , with which a downstream classifier can be trained and subsequently be rolled out in a setting with distribution $P ( X )$ . We experiment on the Adult dataset [40], with known bias between gender and income [10, 11]. The Adult dataset contains over 65,000 samples and has 11 attributes, such as age, education, gender, income, among others. Following [11], we treat gender as the protected variable and use income as the binary target variable representing whether a person earns over $\$ 50 K$ or not. For DAG $\mathcal { G }$ , we use the graph discovered and presented by Zhang et al. [11]. In Appendix D.2, we specify edge removals for DECAF-DP, DECAF-CF, and DECAF-FTU.
+
+Synthetic data is generated using each benchmark method, after which a separate MLP is trained on each dataset for computing the metrics; see Appendix D.2 for details. We repeat this experiment 10 times for each benchmark method and report the average in Table 2. As shown, DECAF-ND (no debiasing) performs amongst the best methods in terms of data utility. Because the data utility in this experiment is measured with respect to the original (biased) dataset, we see that the methods DECAF-FTU, DECAF-CF, and DECAF-DP score lower than DECAF-ND because these methods distort the distribution – with DECAF-DP distorting the label’s conditional distribution most and thus scoring worst in terms of AUROC. Note also that a downstream user who is only focused on performance would choose the synthetic data from WGAN-GP or DECAF-ND, which are also the most biased methods. Thus, we see that there is a trade-off between fairness and data utility when the evaluation distribution $P ( X )$ is the original biased data.
+
+# 6.2 Fair Credit Approval
+
+In this experiment, direct bias, which was not previously present, is synthetically injected into a dataset $\mathcal { D }$ resulting in a biased dataset $\tilde { \mathcal { D } }$ . We show how DECAF can remove the injected bias, resulting in dataset $\mathcal { D } ^ { \prime }$ that can be used to train a downstream classifier. This is a relevant scenario if the training data $\tilde { D }$ does not follow real-world distribution $P ( X )$ , but instead a biased distribution ${ \tilde { P } } ( X )$ (due to, e.g., historical bias). In this case, we want downstream models trained on synthetic data $\mathcal { D } ^ { \prime }$ to perform well on the real-world data $\mathcal { D }$ instead of $\tilde { \mathcal { D } }$ . We show that DECAF is successful at removing the bias and how this results in higher data utility than benchmarks methods trained on $\tilde { D }$ .
+
+![](images/aa7dbcb8f119395980d64c4616f848b718969e0bbe2f5d3e309f94758656594d.jpg)  
+Figure 3: Plot of precision (a), recall (b), AUROC (c), FTU (d), and DP (e) over bias strength $\beta$ . FairGAN performs similarly in terms of DP and FTU, but DECAF-FTU and DECAF-DP have significantly better data quality as well as down stream prediction capability (AUROC).
+
+We use the Credit Approval dataset from [40], with graph $\mathcal { G }$ as discovered by the causal discovery algorithm FGES [41] using Tetrad [42] (details in Appendix D.3). We inject direct bias by decreasing the probability that a sample will have their credit approved based on the chosen $A$ .8 The credit_approval for this population was synthetically denied (set to 0) with some bias probability $\beta$ , adding a directed edge between label and protected attribute.
+
+In Figure 3, we show the results of running our experiment 10 times over various bias probabilities $\beta$ We benchmark against FairGAN, as it is the only benchmark designed for synthetic debiased data. Note that in this case, the causal DAG has only one indirect biased edge between the protected variable (see Appendix D), and thus DECAF-DP and DECAF-CF remove the same edges and are the same for this experiment. The plots show that DECAF-FTU and DECAF-DP have similar performance to FairGAN in terms of debiasing; however, all of the DECAF-\* methods have significantly better data quality metrics: precision, recall, and AUROC. DECAF-DP is one of the best performers across all 5 of the evaluation metrics and has better DP performance under higher bias. As expected, DECAF-ND (no debiasing) has the same data quality performance in terms of precision and recall as DECAF-FTU and DECAF-DP and has diminishing performance in terms of downstream AUROC, FTU, and DP as bias strength increases. See Appendix D for other benchmarks, and the same experiment under hidden confounding in Appendix G.
+
+# 7 Discussion
+
+We have proposed DECAF, a causally-aware GAN that generates fair synthetic data. DECAF’s sequential generation provides a natural way of removing these edges, with the advantage that the conditional generation of other features is left unaltered. We demonstrated on real datasets that the DECAF framework is both versatile and compatible with several popular definitions of fairness. Lastly, we provided theoretical guarantees on the generator’s convergence and fairness of downstream models. We next discuss limitations as well as applications and opportunities for future work.
+
+Definitions. DECAF achieves fairness by removing edges between features, as we have shown for the popular FTU and DP definitions. Other independence-based [30] fairness definitions can be achieved by DECAF too, as we show in Appendix C. Just like related debiasing works [10, 11, 16, 17], DECAF is not compatible with fairness definitions based on separation or sufficiency [30], as these definitions depend on the downstream model more explicitly (e.g. Equality of Opportunity [12]). More on this in Appendix C.
+
+Incorrect DAG specification. Our method relies on the provision of causal structure in the form of a DAG for i) deciding the sequential order of feature generation and ii) deciding which edges to remove to achieve fairness. This graph need not be known a priori and can be discovered instead. If discovered, the DAG needs not equal the true DAG for many definitions of fairness, including FTU and DP, but only some (in)dependence statements are required to be correct (see Proposition 1). This is shown in the Experiments, where the DAG was discovered with the PC algorithm [47] and TETRAD [42]. Furthermore, in Appendix B we prove that the causal generator converges to the right distribution for any graph that is Markov compatible with the data. We reiterate, however, that knowing (part of) the true graph is still helpful because i) it often leads to simpler functions $\{ f _ { i } \} _ { i = 1 } ^ { d }$ to approximate,9 and ii) some causal fairness definitions do require correct directionality—see Appendix
+
+C. In Appendix F, we include an ablation study on how errors in the DAG specification affect data quality and downstream fairness.
+
+Causal sufficiency. We have focused on just one type of graph: causally-sufficient directed graphs. Extending this to undirected or mixed graphs is possible as long as the generation order reflects a valid factorization of the observed distribution. This includes settings with hidden confounders. We note that for some definitions of bias, e.g., counterfactual bias, directionality is essential and hidden confounders would need to be corrected for (which is not generally possible).
+
+Time-series. We have focused on the tabular domain. The method can be extended to other domains with causal interaction between features, e.g., time-series. Application to image data is non-trivial, partly because, in this instance, the protected attribute (e.g., skin color) does not correspond to a single observed feature. DECAF might be extended to this setting in the future by first constructing a graph in a disentangled latent space (e.g., [24, 25]).
+
+Social implications. Fairness is task and context-dependent, requiring careful public debate. With that being said, DECAF empowers data issuers to take responsibility for downstream model fairness. We hope that this progresses the ubiquity of fairness in machine learning.
+
+# Acknowledgements
+
+We would like to thank the reviewers for their time and valuable feedback. This research was funded by the Office of Naval Research and the WD Armstrong Trust.
+
+# References
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+        "text": "DECAF: Generating Fair Synthetic Data Using Causally-Aware Generative Networks ",
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+        "text": "Boris van Breugel∗ University of Cambridge bv292@cam.ac.uk ",
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+        "text": "Trent Kyono∗ University of California, Los Angeles tmkyono@ucla.edu ",
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+        "text": "Jeroen Berrevoets University of Cambridge jb2384@cam.ac.uk ",
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+        "text": "Mihaela van der Schaar University of Cambridge University of California, Los Angeles The Alan Turing Institute mv472@cam.ac.uk ",
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+        "text": "Abstract ",
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+        "text": "Machine learning models have been criticized for reflecting unfair biases in the training data. Instead of solving for this by introducing fair learning algorithms directly, we focus on generating fair synthetic data, such that any downstream learner is fair. Generating fair synthetic data from unfair data— while remaining truthful to the underlying data-generating process $( D G P )$ —is non-trivial. In this paper, we introduce DECAF: a GAN-based fair synthetic data generator for tabular data. With DECAF we embed the DGP explicitly as a structural causal model in the input layers of the generator, allowing each variable to be reconstructed conditioned on its causal parents. This procedure enables inference-time debiasing, where biased edges can be strategically removed for satisfying user-defined fairness requirements. The DECAF framework is versatile and compatible with several popular definitions of fairness. In our experiments, we show that DECAF successfully removes undesired bias and— in contrast to existing methods —is capable of generating high-quality synthetic data. Furthermore, we provide theoretical guarantees on the generator’s convergence and the fairness of downstream models. ",
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+        "text": "1 Introduction ",
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+        "text": "Generative models are optimized to approximate the original data distribution as closely as possible. Most research focuses on three objectives [1]: fidelity, diversity, and privacy. The first and second are concerned with how closely synthetic samples resemble real data and how much of the real data’s distribution is covered by the new distribution, respectively. The third objective aims to avoid simply reproducing samples from the original data, which is important if the data contains privacy-sensitive information [2, 3]. We explore a much-less studied concept: synthetic data fairness. ",
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+        "text": "Motivation. Deployed machine learning models have been shown to reflect the bias of the data on which they are trained [4, 5, 6, 7, 8]. This has not only unfairly damaged the discriminated individuals but also society’s trust in machine learning as a whole. A large body of work has explored ways of detecting bias and creating fair predictors [9, 10, 11, 12, 13, 14, 15], while other authors propose debiasing the data itself [9, 10, 11, 16]. This work’s aim is related to the work of [17]: to generate fair synthetic data based on unfair data. Being able to generate fair data is important because end-users creating models based on publicly available data might be unaware they are inadvertently including bias or insufficiently knowledgeable to remove it from their model. Furthermore, by debiasing the data prior to public release, one can guarantee any downstream model satisfies desired fairness requirements by assigning the responsibility of debiasing to the data generating entities. ",
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+            "Table 1: Overview of related work for synthetic data. We organize related work according to our key areas of interest: (1) Allows post-hoc distribution changes, (2) provides fairness, (3) supports causal notion of fairness, (4) allows inference-time fairness, (5) requires minimal assumptions. We highlight the key contribution, and identify non-neural approaches with “†”. "
+        ],
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+        "table_body": "<table><tr><td>Model</td><td>Reference</td><td>(1)(2)(3)(4)(5)Goal</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan=\"8\">Standard synthetic data generation</td></tr><tr><td>VAE</td><td>[19]</td><td>X</td><td>X</td><td>X</td><td>X</td><td></td><td>Realistic synth. data.</td></tr><tr><td>GANs</td><td>[2, 3, 20, 21]</td><td>X</td><td>X</td><td>X</td><td>X</td><td></td><td>Realistic synth. data.</td></tr><tr><td colspan=\"8\"> Methods that detect and/or remove bias</td></tr><tr><td>PSE-DD/DR†</td><td>[11]</td><td>√</td><td>√</td><td>√</td><td>×</td><td>X</td><td>Discover/Remove bias.</td></tr><tr><td>OPPDP†</td><td>[16]</td><td>X</td><td>√</td><td>xxxx&gt;</td><td>xxx</td><td>X</td><td>Remove bias.</td></tr><tr><td>DI</td><td>[10]</td><td>X</td><td>√</td><td></td><td></td><td>X</td><td>Discover/Remove bias.</td></tr><tr><td>LFR</td><td>[22]</td><td>×</td><td>一</td><td></td><td></td><td>√</td><td>Learn fair representation.</td></tr><tr><td>FairGAN</td><td>[17]</td><td>X</td><td>√</td><td></td><td>X</td><td>√</td><td>Realistic and fair synth. data.</td></tr><tr><td>CFGAN</td><td>[23]</td><td>X</td><td>√</td><td></td><td>×</td><td>√</td><td>Realistic and fair synth. data.</td></tr><tr><td>DECAF</td><td>(ours)</td><td>√</td><td>√</td><td>&lt;</td><td>√</td><td>√</td><td>Realistic and fair synth. data.</td></tr></table>",
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+        "text": "Goal. From a biased dataset $\\mathcal { X }$ , we are interested in learning a model $G$ , that is able to generate an equivalent synthetic unbiased dataset $\\mathcal { X } ^ { \\prime }$ with minimal loss of data utility. Furthermore, a downstream model trained on the synthetic data needs to make not only unbiased predictions on the synthetic data, but also on real-life datasets (as formalized in Section 4.2). ",
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+        "text": "Solution. We approach fairness from a causal standpoint because it provides an intuitive perspective on different definitions of fairness and discrimination [11, 13, 14, 15, 18]. We introduce DEbiasing CAusal Fairness (DECAF), a generative adversarial network (GAN) that leverages causal structure for synthesizing data. Specifically, DECAF is comprised of $d$ generators (one for each variable) that learn the causal conditionals observed in the data. At inference-time, variables are synthesized topologically starting from the root nodes in the causal graph then synthesized sequentially, terminating at the leave nodes. Because of this, DECAF can remove bias at inference-time through targeted (biased) edge removal. As a result, various datasets can be created for desired (or evolving) definitions of fairness. ",
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+        "type": "text",
+        "text": "Contributions. We propose a framework of using causal knowledge for fair synthetic data generation. We make three main contributions: i) DECAF, a causal GAN-based model for generating synthetic data, ii) a flexible causal approach for modifying this model such that it can generate fair data, and iii) guarantees that downstream models trained on the synthetic data will also give fair predictions in other settings. Experimentally, we show how DECAF is compatible with several fairness/discrimination definitions used in literature while still maintaining high downstream utility of generated data. ",
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+        "text": "2 Related Works ",
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+        "text": "Here we focus on the related work concerned with data generation, in contrast to fairness definitions for which we provide a detailed overview in Section 4 and Appendix C. As an overview of how data generation methods relate to one another, we refer to Table 1 which presents all relevant related methods. ",
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+        "text": "Non-parametric generative modeling. The standard models for synthetic data generation are either based on VAEs [19] or GANs [2, 3, 20, 21]. While these models are well known for their highly realistic synthetic data, they are unable to alter the synthetic data distribution to encourage fairness (except for [17, 23], discussed below). Furthermore, these methods have no causal notion, which prohibits targeted interventions for synthesizing fair data (Section 4). We explicitly leave out CausalGAN [24] and CausalVAE [25], which appear similar by incorporating causality-derived ideas but are different in both method and aim (i.e., image generation). ",
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+        "type": "text",
+        "text": "Fair data generation. In the bottom section of Table 1, we present methods that, in some way, alter the training data of classifiers to adhere to a notion of fairness [10, 11, 16, 17, 22, 23]. While these methods have proven successful, they lack some important features. For example, none of the related methods allow for post-hoc changes of the synthetic data distribution. This is an important feature, as each situation requires a different perspective on fairness and thus requires a flexible framework for selecting protected variables. Additionally, only [11, 23] allow a causal perspective on fairness, despite causal notions underlying multiple interpretations of what should be considered fair [13]. Furthermore, only [17, 22, 23] offer a flexible framework, while the others are limited to binary [10, 11] or discrete [16] settings. Xu et al. [23] also use a causal architecture for the generator, however their method is not as flexible—e.g. it does not easily extend to multiple protected attributes. Finally, in contrast to other methods DECAF is directly concerned with fairness of the downstream model—which is dependent on the setting in which the downstream model is employed (Section 4.2). In essence, from Table 1 we learn that DECAF is the only method that combines all key areas of interest. At last, we would like to mention [26], who aim to generate data that resembles a small unbiased reference dataset, by leveraging a large but biased dataset. This is very different to our aim, as we are interested in the downstream model’s fairness and explicit notions of fairness. ",
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+        "text": "3 Preliminaries ",
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+        "text": "Let $X \\in \\mathcal { X } \\subseteq \\mathbb { R } ^ { d }$ denote a random variable with distribution $P _ { X } ( X )$ , with protected attributes $A \\in { \\mathcal { A } } \\subset { \\mathcal { X } }$ and target variable $Y \\in \\mathcal { y } \\subset \\mathcal { x }$ , let $\\hat { Y }$ denote a prediction of $Y$ . Let the data be given by $\\mathcal { D } = \\{ \\mathbf { x } ^ { ( k ) } \\} _ { k = 1 } ^ { N }$ , where each $\\mathbf { x } ^ { ( k ) } \\in \\mathcal { D }$ is a realization of $X$ . We assume the data generating process can be represented by a directed acyclic graph (DAG)—such that the generation of features can be written as a structural equation model (SEM) [27]—and that this DAG is causally sufficient. Let $X _ { i }$ denote the $i ^ { \\mathrm { { t h } } }$ feature in $X$ with causal parents $\\operatorname { P a } ( X _ { i } ) \\subset \\{ X _ { j } : j \\neq i \\}$ , the SEM is given by: ",
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+        "text": "$$\nX _ { i } = f _ { i } ( \\operatorname { P a } ( X _ { i } ) , Z _ { i } ) , \\forall i\n$$",
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+        "text": "where $\\{ Z _ { i } \\} _ { i = 1 } ^ { d }$ are independent random noise variables, that is $\\mathrm { P a } ( Z _ { i } ) = \\emptyset$ , ∀i. Note that each $f _ { i }$ is a deterministic function that places all randomness of the conditional $P ( X _ { i } | \\operatorname* { P a } ( X _ { i } ) )$ in the respective noise variable, $Z _ { i }$ . ",
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+        "text": "4 Fairness of Synthetic Data ",
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+        "text": "Algorithmic fairness is a popular topic (e.g., see [13, 28]), but fair synthetic data has been much less explored. This section highlights how the underlying graphs of the synthetic and downstream data determine whether a model trained on the synthetic data will be fair in practice. We start with the two most popular definitions of fairness, relating to the legal concepts of direct and indirect discrimination. We also explore conditional fairness [29], which is a generalization of the two. In Appendix C we discuss how the ideas in this section transfer to other independence-based definitions [30]. Throughout this section, we separate $Y$ from $X$ by defining ${ \\bar { X } } { \\stackrel { \\cdot } { = } } X \\backslash Y$ , and we will write $X  { \\bar { X } }$ for ease of notation. ",
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+        "text": "4.1 Algorithmic fairness ",
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+        "type": "text",
+        "text": "The first definition is called Fairness Through Unawareness (e.g. [31]). ",
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+        "text": "Definition 1. (Fairness Through Unawareness (FTU): algorithm). A predictor $f : X \\mapsto { \\hat { Y } }$ is fair iff protected attributes $A$ are not explicitly used by $f$ to predict $\\hat { Y }$ . ",
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+    {
+        "type": "text",
+        "text": "This definition prohibits disparate treatment [28, 32], and is related to the legal concept of direct discrimination, i.e., two equally qualified people deserve the same job opportunity independent of their race, gender, beliefs, among others. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Though FTU fairness is commonly used, it might result in indirect discrimination: covariates that influence the prediction $\\hat { Y }$ might not be identically distributed across different groups $a , a ^ { \\prime }$ , which means an algorithm might have disparate impact on a protected group [10]. The second definition of fairness, demographic parity [32], does not allow this: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2. (Demographic Parity $( D P )$ : algorithm) $A$ predictor $\\hat { Y }$ is fair iff $A \\perp \\perp { \\hat { Y } }$ , i.e. $\\forall a , a ^ { \\prime }$   \n$P ( \\hat { Y } | A = a ) = P ( \\hat { Y } | A = a ^ { \\prime } )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Evidently, DP puts stringent constraints on the algorithm, whereas FTU might be too lenient. The third definition we include is based on the work of [29], related to unresolved discrimination [14]. The idea is that we do not allow indirect discrimination unless it runs through explanatory factors $R \\subset X$ . For example, in Simpson’s paradox [33] there seems to be a bias between gender and college admissions, but this is only due to women applying to more competitive courses. In this case, one would want to regard fairness conditioned on the choice of study [14]. Let us define this as conditional fairness: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 3. (Conditional Fairness $( C F )$ : algorithm) $A$ predictor $\\hat { Y }$ is fair iff $A \\perp \\perp { \\hat { Y } } | R ,$ , i.e.   \n$\\forall r , a , a ^ { \\prime } : P ( \\hat { Y } | R = r , A = a ) = P ( \\hat { Y } | R = r , A = a ^ { \\prime } )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "CF generalizes FTU and DP Note that conditional fairness is a generalization of FTU and DP, by setting $R = X \\backslash A$ and $R = \\emptyset$ , respectively. In Appendix $\\textrm { C }$ we elaborate on the connection between these, and more, definitions. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 Synthetic data fairness ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Algorithmic definitions can be extended to distributional fairness for synthetic data. Let $P ( X ) , P ^ { \\prime } ( X )$ be probability distributions with protected attributes $A \\subset X$ and labels $Y \\subset X$ . Let ${ \\mathcal { T } } ( A , Y )$ be a definition of algorithmic fairness (e.g., FTU). Note, that under CF, ${ \\mathcal { T } } ( A , Y )$ is a function of $R$ as well. We propose $( { \\mathcal { T } } ( A , Y ) , P )$ -fairness of distribution $P ^ { \\prime } ( X )$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 4. (Distributional fairness) $A$ probability distribution $P ^ { \\prime } ( X )$ is $( { \\mathcal { T } } ( A , Y ) , P )$ -fair, iff the optimal predictor $\\hat { Y } = f ^ { * } ( X )$ of $Y$ trained on $P ^ { \\prime } ( X )$ satisfies ${ \\mathcal { T } } ( A , Y )$ when evaluated on $P ( X )$ . ",
+        "bbox": [
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+            377,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In other words, when we train a predictor on $( { \\mathcal { T } } ( A , Y ) , P )$ -fair distribution $P ^ { \\prime } ( X )$ , we can only reach maximum performance if our model is fair. Note the explicit reference to $P ( X )$ , the distribution on which fairness is evaluated, which does not need to coincide with $P ^ { \\prime } ( X )$ . This is a small but relevant detail. For example, when training a model on data ${ \\mathcal { D } } ^ { \\prime } \\sim P ^ { \\prime } ( X )$ it could seem like the model is fair when we evaluate it on a hold-out set of the data (e.g., if we simply remove the protected attribute from the data). However, when we use the model for real-world predictions of data ${ \\mathcal { D } } \\sim P ( X )$ , disparate impact is possibly observed due to a distributional shift. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "By extension, we define synthetic data as $( { \\mathcal { T } } ( A , Y ) , P )$ -fair, iff it is sampled from an $( { \\mathcal { T } } ( A , Y ) , P )$ - fair distribution. Defining synthetic data as fair w.r.t. an optimal predictor is especially useful when we want to publish a dataset and do not trust end-users to consider anything but performance.2 ",
+        "bbox": [
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+            523,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Choosing $\\mathbf { P } ( \\mathbf { X } )$ . The setting $P ( X ) = P ^ { \\prime } ( X )$ corresponds to data being fair with respect to itself. For synthetic data generation, this setting is uninteresting as any dataset can be made fair by randomly sampling or removing $A$ ; if $A$ is random, the prediction should not directly or indirectly depend on it. This ignores, however, that a downstream user might use the trained model on a real-world dataset in which other variables $B$ are correlated with $A$ , and thus their model (which is trained to use $B$ for predicting $Y$ ) will be biased. Of specific interest is the setting where $P ( X )$ corresponds to the original data distribution $P _ { X } ( X )$ that contains unfairness. In this scenario, we construct $P ^ { \\prime } ( X )$ by learning $P _ { X } ( X )$ and removing the unfair characteristics. The data from $P ^ { \\prime } ( X )$ can be published online, and models trained on this data can be deployed fairly in real-life scenarios where data follows $P _ { X } ( X )$ . Unless otherwise stated, henceforth, we assume ${ \\dot { P } } ( X ) = P _ { X } ( X )$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.3 Graphical perspective ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            727,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As reflected in the widely accepted terms direct versus indirect discrimination, it is natural to define distributional fairness from a causal standpoint. Let $\\mathcal { G } ^ { \\prime }$ and $\\mathcal { G }$ respectively denote the graphs underlying $P ^ { \\prime } ( X )$ (the synthetic data distribution which we can control) and $P ( X )$ (the evaluation distribution that we cannot control). Let $\\partial _ { \\mathcal { G } } Y$ denote the Markov boundary of $Y$ in graph $\\mathcal { G }$ . We focus on the conditional fairness definition because it subsumes the definition of DP and FTU (Section 4.1). Let $R \\subset X$ be the set of explanatory features. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. ( $C F .$ : graphical condition) If for all $B \\in \\partial _ { \\mathcal { G } ^ { \\prime } } Y , A \\bot \\bot _ { \\mathcal { G } } B | R , ^ { 3 }$ then distribution $P ^ { \\prime } ( X )$ is $C F$ fair w.r.t $P ( X )$ given explanatory factors $R$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proof. Without loss of generality, let us assume the label is binary.4 The optimal predictor $f ^ { * } ( X ) =$ $P ( Y | X ) = P ( Y | \\partial _ { \\mathcal { G } ^ { \\prime } } Y )$ . Thus, if $\\partial _ { \\mathscr { G } ^ { \\prime } } Y$ is ${ \\mathrm { d } }$ -separated from $A$ in $\\mathcal { G }$ given $R$ , prediction $\\hat { Y } = f ^ { * } ( X )$ is independent of $A$ given $R$ and CF holds. □ ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Corollary 1. (CF debiasing) Any distribution $P ^ { \\prime } ( X )$ with graph $\\mathcal { G } ^ { \\prime }$ can be made CF fair w.r.t. $P ( X )$ and explanatory features $R$ by removing from $\\mathcal { G } ^ { \\prime }$ edges $\\tilde { E } = \\{ ( B \\to Y )$ and $( Y \\to B ) : \\forall B \\in \\partial _ { \\mathcal { G } ^ { \\prime } } Y$ with $B$ 6⊥⊥ $_ { \\cdot g } \\ A | R \\}$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof. First note $\\tilde { E }$ is the necessary and sufficient set of edges to remove for $\\forall B \\in \\partial _ { \\mathcal { G } ^ { \\prime } } Y$ , $A$ ⊥⊥ $\\mathcal { G }$ $| B | R )$ to be true, subsequently the result follows from Proposition 1. □ ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For FTU (i.e. $R = X \\backslash A )$ and DP (i.e. $R = \\emptyset$ ), this corollary simplifies to: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Corollary 2. (FTU debiasing) Any distribution $P ^ { \\prime } ( X )$ with graph $\\mathcal { G } ^ { \\prime }$ can be made FTU fair w.r.t. any distribution $P ( X )$ by removing, if present, i) the edge between $A$ and $Y$ and $i i$ ) the edge $A  C$ or $Y  C$ for all shared children $C$ . ",
+        "bbox": [
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+            273,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Corollary 3. (DP debiasing) Any distribution $P ^ { \\prime } ( X )$ with graph $\\mathcal { G } ^ { \\prime }$ can be made $D P$ fair w.r.t. $P ( X )$ by removing, if present, the edge between $B$ and $Y$ for any $B \\in \\partial _ { \\mathcal { G } ^ { \\prime } } Y$ with $B$ 6⊥⊥ $_ { \\textit { g A } }$ . ",
+        "bbox": [
+            173,
+            334,
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+            378
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/0a9fed1a05f973016276c4a64c25853270bbd1a127ff235596605576723c23e8.jpg",
+        "image_caption": [
+            "Figure 1: Edge removal for fairness. FTU: $\\pmb { \\chi }$ ; DP: ✗✗✗; CF when $R = C$ : $\\boldsymbol { x }$ ; CF when $B \\in R$ : $\\boldsymbol { \\chi }$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            692,
+            256,
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+            333
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Figure 1 shows how the different fairness definitions lead to different sets of edges to be removed. ",
+        "bbox": [
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+            388,
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+            416
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Faithfulness. Usually one assumes distributions are faithful w.r.t. their respective graphs, in which case the if-statement in Proposition 1 become equivalence statements: fairness is only possible when the graphical conditions hold. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. If $P ( X )$ and $P ^ { \\prime } ( X )$ are faithful with respect to their respective graphs $\\mathcal { G }$ and $\\mathcal { G } ^ { \\prime }$ , then Proposition $\\boldsymbol { l }$ becomes an equivalence statement and Corollaries 1, 2 and 3 describe the necessary and sufficient sets of edges to remove for achieving CF, FTU and DP fairness, respectively. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof. Faithfulness implies $A$ ⊥⊥ $P ( X ) ~ B | R \\implies A$ ⊥⊥ $_ { \\cdot \\mathcal { G } } \\ B | R$ , e.g. [34]. Thus, if $\\exists B \\in \\partial _ { \\mathcal { G } ^ { \\prime } } Y$ for which $A$ 6⊥⊥ $_ { \\textit { g B } | R }$ , then $A$ 6⊥⊥ $| B | R$ . Because $B \\in \\partial _ { \\mathcal { G } ^ { \\prime } } Y$ and $P ^ { \\prime } ( X )$ is faithful to $\\mathcal { G } ^ { \\prime }$ , $\\hat { Y } = f ^ { * } ( X )$ depends on $B$ , and thus $\\hat { Y }$ 6⊥⊥ $A | R$ : CF does not hold. □ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Other definitions. Some authors define similar fairness measures in terms of directed paths (cf.   \nd-separation) [11, 14, 18], which is a milder requirement as it allows correlation via non-causal paths.   \nIn Appendix C we highlight the graphical conditions for these definitions. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 Method: DECAF ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The primary design goal of DECAF is to generate fair synthetic data from unfair data. We separate DECAF into two stages. The training stage learns the causal conditionals that are observed in the data through a causally-informed GAN. At the generation (inference) stage, we intervene on the learned conditionals via Corollaries 1-3, in such a way that the generator creates fair data. We assume the underlying DGP’s graph $\\mathcal { G }$ is known; otherwise, $\\mathcal { G }$ needs to be approximated first using any causal discovery method, see Section 6. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.1 Training ",
+        "text_level": 1,
+        "bbox": [
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+            789,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Overview. This stage strives to learn the causal mechanisms $\\{ f _ { i } ( \\mathrm { P a } ( X _ { i } ) , Z _ { i } ) \\}$ . Each structural equation $f _ { i }$ (Eq. 1) is modelled by a separate generator $G _ { i } : \\mathbb { R } ^ { | P a ( X _ { i } ) | + 1 }  \\mathbb { R }$ . We achieve this by employing a conditional GAN framework with a causal generator. This process is illustrated in Figure 2 and detailed below. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/e8cf5f2fe4f5bbb028f6684371bbe8787025750f394e14eba734addfcaece58e.jpg",
+        "image_caption": [
+            "Figure 2: Architecture. Training phase— Each component in $\\hat { \\bf X }$ is generated sequentially as a function (where the function is that component’s generator $G _ { i }$ ) of the component’s parents. Parental knowledge is provided by the DAG governing the data. Inference phase— As the component-wise generation of the generator network is independent of the DAG governing the data, we can easily replace (or intervene on) the DAG governing parental information. The resulting synthetic data (right) will be governed by the intervened DAG. FTU is achieved by removing: ✗; $D P$ : ✗✗✗; e.g. CF when $R = C$ : $x x$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Features are generated sequentially following the topological ordering of the underlying causal DAG: first root nodes are generated, then their children (from generated causal parents), etc. Variable ${ \\hat { X } } _ { i }$ is modelled by the associated generator $G _ { i }$ : ",
+        "bbox": [
+            174,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a7d579cb3294d96251f8eea5990c8e9b1e9cc982efad27abefbc8041ffd7d7ff.jpg",
+        "text": "$$\n\\hat { X _ { i } } = G _ { i } ( \\hat { \\mathrm { P a } } ( X _ { i } ) , Z _ { i } ) \\quad \\forall i ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            395,
+            591,
+            414
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { \\operatorname { P a } } ( X _ { i } )$ denotes the generated causal parents of $X _ { i }$ (for root nodes the empty set), and each $Z _ { i }$ is independently sampled from $P ( Z )$ (e.g. standard Gaussian). We denote the full sequential generator by $G ( Z ) = [ G _ { 1 } ( Z _ { 1 } ) , . . . , G _ { d } ( \\cdot , Z _ { d } ) ]$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Subsequently, the synthetic sample $\\hat { \\bf x }$ is passed to a discriminator $D : \\mathbb { R } ^ { d }  \\mathbb { R }$ , which is trained to distinguish the generated samples from original samples. A typical minimax objective is employed for creating generated samples that confuse the discriminator most: ",
+        "bbox": [
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+            469,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/825a23bbfd260af7c371b7e802b484341d08d7b21c9e24f008361b3f59fd48ca.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\{ G _ { i } \\} _ { i = 1 } ^ { d } } \\operatorname* { m i n } _ { D } \\mathbb { E } [ \\log D ( G ( Z ) ) + \\log ( 1 - D ( X ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            516,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "with $X$ sampled from the original data. We optimize the discriminator and generator iteratively and add a regularization loss to both networks. Network parameters are updated using gradient descent. ",
+        "bbox": [
+            176,
+            546,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "If we assume $P _ { X } ( X )$ is compatible with graph $\\mathcal { G }$ , we can show that the sequential generator has the same theoretical convergence guarantees as standard GANs [20]: ",
+        "bbox": [
+            173,
+            580,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. (Convergence guarantee) Assuming the following three conditions hold: ",
+        "bbox": [
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+            613,
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+            627
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(i) data generating distribution $P _ { X }$ is Markov compatible with a known DAG $\\mathcal { G }$ ; ",
+        "bbox": [
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+            636,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(ii) generator $G$ and discriminator $D$ have enough capacity; and (iii) in every training step the discriminator is trained to optimality given fixed $G$ , and $G$ is subsequently updated as to maximize the discriminator loss (Eq. 3); ",
+        "bbox": [
+            179,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            178,
+            683,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "then generator distribution $P _ { G }$ converges to true data distribution $P _ { X }$ ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proof. See Appendix B ",
+        "bbox": [
+            176,
+            750,
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+            765
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Condition (i), compatibility with $\\mathcal { G }$ , is a weaker assumption than assuming perfect causal knowledge. For example, suppose the Markov equivalence class of the true underlying DAG has been determined through causal discovery. In that case, any graph $\\mathcal { G }$ in the equivalence class is compatible with the data and can thus be used for synthetic data generation. However, we note that debiasing can require the correct directionality for some definitions of fairness, see Discussion. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark. The causal GAN we propose, DECAF, is simple and extendable to other generative methods, e.g., VAEs. Furthermore, from the post-processing theorem [35] it follows that DECAF can be directly used for generating private synthetic data by replacing the standard discriminator by a differentially private discriminator [2, 36]. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.2 Inference-time Debiasing ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The training phase yields conditional generators $\\{ G _ { i } \\} _ { i = 1 } ^ { d }$ , which can be sequentially applied to generate data with the same output distribution as the original data (proof in Appendix B). The causal model allows us to go one step further: when the original data has characteristics that we do not want to propagate to the synthetic data (e.g., gender bias), individual generators can be modified to remove these characteristics. Given the generator’s graph $\\mathcal { G } = ( \\boldsymbol { X } , \\mathbf { \\bar { E } } )$ , fairness is achieved by removing edges such that the fairness criteria are met, see Section 4. Let $\\tilde { E } \\in E$ be the set of edges to remove for satisfying the required fairness definition. For CF, FTU and DP,5 the sets $\\tilde { E }$ are given by Corollaries 1, 2 and 3, respectively. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Removing an edge constitutes to what we call a “surrogate” $d o$ -operation [27] on the conditional distribution. For example, suppose we only want to remove $( i  j )$ ). For a given sample, $X _ { i }$ is generated normally (Eq. 2), but $X _ { j }$ is generated using the modified: ",
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+        "type": "equation",
+        "img_path": "images/e3f5357434282a405e1f82b2f6890bda1e04d7b383ea3fc6c696850c25376598.jpg",
+        "text": "$$\n\\hat { X } _ { j } ^ { d o ( X _ { i } ) = \\tilde { x } _ { i j } } = G _ { j } ( . . . , X _ { i } = \\tilde { x } _ { i j } ) ,\n$$",
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+    {
+        "type": "text",
+        "text": "where $X _ { i } ~ = ~ \\tilde { x } _ { i j }$ is the surrogate parent assignment. Value ${ \\hat { X } } _ { j } ^ { d o ( X _ { i } ) }$ can be interpreted as the counterfactual value of $\\hat { X _ { j } }$ , had $X _ { i }$ been equal to $\\tilde { x } _ { i j }$ (see also [15]). ",
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+    {
+        "type": "text",
+        "text": "Choosing the value of surrogate variable $\\tilde { x } _ { i j }$ requires background knowledge of the task and bias at hand. For example, surrogate variable $\\tilde { x } _ { i j }$ can be sampled independently from a distribution for each synthetic sample (e.g., the marginal $P ( { \\bar { X } } _ { i } ) )$ , be set to a fixed value for all samples in the synthetic data (e.g., if $X _ { i }$ : gender, always set $\\tilde { x } _ { i j } = m a l e$ when generating feature $X _ { j }$ : job opportunity) or be chosen as to maximize/minimize some feature (e.g. $\\begin{array} { r } { \\tilde { x } _ { i j } = \\arg \\operatorname* { m a x } _ { x } \\hat { X } _ { j } ^ { d o ( X _ { i } ) = x } ) } \\end{array}$ . We emphasize that we do not set $X _ { i } = \\tilde { x } _ { i j }$ in the synthetic sample; $X _ { i } = \\tilde { x } _ { i j }$ is only used for substitution of the removed dependence. We provide more details in Appendix E. ",
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+    {
+        "type": "text",
+        "text": "More generally, we create surrogate variables for all edges we remove, $\\{ \\tilde { x } _ { i j } : ( i  j ) \\in \\tilde { E } \\}$ . Each sample is sequentially generated by Eq. 4, with a surrogate variable for each removed incoming edge. ",
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+    {
+        "type": "text",
+        "text": "Remark. Multiple datasets can be created based on different definitions of fairness and/or different downstream prediction targets. Because debiasing happens at inference-time, this does not require retraining the model. ",
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+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we validate the performance of DECAF for synthesizing bias-free data based on two datasets: i) real data with existing bias and ii) real data with synthetically injected bias. The aim of the former is to show that we can remove real, existing bias. The latter experiment provides a ground-truth unbiased target distribution, which means we can evaluate the quality of the synthetic dataset with respect to this ground truth. For example, when historically biased data is first debiased, a model trained on the synthetic data will likely create better predictions in contemporary, unbiased/less-biased settings than benchmarks that do not debias first. ",
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+    {
+        "type": "text",
+        "text": "In both experiments, the ground-truth DAG is unknown. We use causal discovery to uncover the underlying DAG and show empirically that the performance is still good. ",
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+        "type": "text",
+        "text": "Benchmarks. We compare DECAF against the following benchmark generative methods: a GAN, a Wasserstein GAN with gradient penalty (WGAN-GP) [21] and FairGAN [17]. FairGAN is the only benchmark designed to generate synthetic fair data,6 whereas GAN and WGAN-GP only aim to match the original data’s distribution, regardless of inherent underlying bias. For these benchmarks, fair data can be generated by naively removing the protected variable – we refer to these methods with the PR (protected removal) suffix and provide more experimental results and insight into PR in Appendix A. We benchmark DECAF debiasing in four ways: i) with no inference-time debiasing (DECAF-ND), ii) under FTU (DECAF-FTU), iii) under CF (DECAF-CF) and iv) under DP fairness (DECAF-DP). We provide $\\mathrm { D E C A F ^ { 7 } }$ implementation details in Appendix D.1. ",
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+        "type": "table",
+        "img_path": "images/20b9b11b3d53e42e118f688c0e61082e6a5fe008fbf05e81794bec0354efcc07.jpg",
+        "table_caption": [
+            "Table 2: Bias removal experiment on the Adult dataset [40]. The full table with protected attribute removal can be found in Appendix A. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">Data Quality</td><td colspan=\"2\">Fairness</td></tr><tr><td>Precision↑</td><td>Recall↑</td><td>AUROC↑</td><td>FTU↓</td><td>DP↓</td></tr><tr><td>Original data D</td><td>0.920±0.006</td><td>0.936 ±0.008</td><td>0.807 ± 0.004</td><td>0.116 ± 0.028</td><td>0.180 ± 0.010</td></tr><tr><td>GAN</td><td>0.607 ± 0.080</td><td>0.439±0.037</td><td>0.567 ± 0.132</td><td>0.023 ± 0.010</td><td>0.089 ± 0.008</td></tr><tr><td>WGAN-GP</td><td>0.683 ± 0.015</td><td>0.914 ± 0.005</td><td>0.798 ± 0.009</td><td>0.120 ± 0.014</td><td>0.189 ±0.024</td></tr><tr><td>FairGAN</td><td>0.681 ± 0.023</td><td>0.814 ± 0.079</td><td>0.766 ± 0.029</td><td>0.009±0.002</td><td>0.097 ± 0.018</td></tr><tr><td>DECAF-ND</td><td>0.780 ±0.023</td><td>0.920 ± 0.045</td><td>0.781 ± 0.007</td><td>0.152 ± 0.013</td><td>0.198 ± 0.013</td></tr><tr><td>DECAF-FTU</td><td>0.763 ± 0.033</td><td>0.925 ± 0.040</td><td>0.765 ± 0.010</td><td>0.004±0.004</td><td>0.054±0.005</td></tr><tr><td>DECAF-CF</td><td>0.743 ± 0.022</td><td>0.875 ± 0.038</td><td>0.769 ± 0.004</td><td>0.003 ±0.006</td><td>0.039 ± 0.011</td></tr><tr><td>DECAF-DP</td><td>0.781± 0.018</td><td>0.881±0.050</td><td>0.672 ± 0.014</td><td>0.001± 0.002</td><td>0.001 ± 0.001</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Evaluation criteria. We evaluate DECAF using the following metrics: ",
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+        "text": "• Data quality is assessed using metrics of precision and recall [37, 38, 39]. Additionally, we evaluate all methods in terms of AUROC of predicting the target variable using a downstream classifier (MLP in these experiments) trained on synthetic data.   \n• FTU is measured by calculating the difference between the predictions of a downstream classifier for setting $A$ to 1 and 0, respectively, such that $| \\bar { P } _ { A = 0 } ( \\hat { Y } | X ) - P _ { A = 1 } ( \\hat { Y } | X ) |$ , while keeping all other features the same. This difference measures the direct influence of $A$ on the prediction.   \n• DP is measured in terms of the Total Variation [15]: the difference between the predictions of a downstream classifier in terms of positive to negative ratio between the different classes of protected variable $A$ , i.e., $| P ( \\hat { Y } | A = 0 ) - P ( \\hat { Y } | A = 1 ) |$ . ",
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+    {
+        "type": "text",
+        "text": "6.1 Debiasing Census Data ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this experiment, we are given a biased dataset ${ \\mathcal { D } } \\sim P ( X )$ and wish to create a synthetic (and debiased) dataset $\\mathcal { D } ^ { \\prime }$ , with which a downstream classifier can be trained and subsequently be rolled out in a setting with distribution $P ( X )$ . We experiment on the Adult dataset [40], with known bias between gender and income [10, 11]. The Adult dataset contains over 65,000 samples and has 11 attributes, such as age, education, gender, income, among others. Following [11], we treat gender as the protected variable and use income as the binary target variable representing whether a person earns over $\\$ 50 K$ or not. For DAG $\\mathcal { G }$ , we use the graph discovered and presented by Zhang et al. [11]. In Appendix D.2, we specify edge removals for DECAF-DP, DECAF-CF, and DECAF-FTU. ",
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+        "type": "text",
+        "text": "Synthetic data is generated using each benchmark method, after which a separate MLP is trained on each dataset for computing the metrics; see Appendix D.2 for details. We repeat this experiment 10 times for each benchmark method and report the average in Table 2. As shown, DECAF-ND (no debiasing) performs amongst the best methods in terms of data utility. Because the data utility in this experiment is measured with respect to the original (biased) dataset, we see that the methods DECAF-FTU, DECAF-CF, and DECAF-DP score lower than DECAF-ND because these methods distort the distribution – with DECAF-DP distorting the label’s conditional distribution most and thus scoring worst in terms of AUROC. Note also that a downstream user who is only focused on performance would choose the synthetic data from WGAN-GP or DECAF-ND, which are also the most biased methods. Thus, we see that there is a trade-off between fairness and data utility when the evaluation distribution $P ( X )$ is the original biased data. ",
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+    {
+        "type": "text",
+        "text": "6.2 Fair Credit Approval ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this experiment, direct bias, which was not previously present, is synthetically injected into a dataset $\\mathcal { D }$ resulting in a biased dataset $\\tilde { \\mathcal { D } }$ . We show how DECAF can remove the injected bias, resulting in dataset $\\mathcal { D } ^ { \\prime }$ that can be used to train a downstream classifier. This is a relevant scenario if the training data $\\tilde { D }$ does not follow real-world distribution $P ( X )$ , but instead a biased distribution ${ \\tilde { P } } ( X )$ (due to, e.g., historical bias). In this case, we want downstream models trained on synthetic data $\\mathcal { D } ^ { \\prime }$ to perform well on the real-world data $\\mathcal { D }$ instead of $\\tilde { \\mathcal { D } }$ . We show that DECAF is successful at removing the bias and how this results in higher data utility than benchmarks methods trained on $\\tilde { D }$ . ",
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+        "img_path": "images/aa7dbcb8f119395980d64c4616f848b718969e0bbe2f5d3e309f94758656594d.jpg",
+        "image_caption": [
+            "Figure 3: Plot of precision (a), recall (b), AUROC (c), FTU (d), and DP (e) over bias strength $\\beta$ . FairGAN performs similarly in terms of DP and FTU, but DECAF-FTU and DECAF-DP have significantly better data quality as well as down stream prediction capability (AUROC). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "We use the Credit Approval dataset from [40], with graph $\\mathcal { G }$ as discovered by the causal discovery algorithm FGES [41] using Tetrad [42] (details in Appendix D.3). We inject direct bias by decreasing the probability that a sample will have their credit approved based on the chosen $A$ .8 The credit_approval for this population was synthetically denied (set to 0) with some bias probability $\\beta$ , adding a directed edge between label and protected attribute. ",
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+    {
+        "type": "text",
+        "text": "In Figure 3, we show the results of running our experiment 10 times over various bias probabilities $\\beta$ We benchmark against FairGAN, as it is the only benchmark designed for synthetic debiased data. Note that in this case, the causal DAG has only one indirect biased edge between the protected variable (see Appendix D), and thus DECAF-DP and DECAF-CF remove the same edges and are the same for this experiment. The plots show that DECAF-FTU and DECAF-DP have similar performance to FairGAN in terms of debiasing; however, all of the DECAF-\\* methods have significantly better data quality metrics: precision, recall, and AUROC. DECAF-DP is one of the best performers across all 5 of the evaluation metrics and has better DP performance under higher bias. As expected, DECAF-ND (no debiasing) has the same data quality performance in terms of precision and recall as DECAF-FTU and DECAF-DP and has diminishing performance in terms of downstream AUROC, FTU, and DP as bias strength increases. See Appendix D for other benchmarks, and the same experiment under hidden confounding in Appendix G. ",
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+    {
+        "type": "text",
+        "text": "7 Discussion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We have proposed DECAF, a causally-aware GAN that generates fair synthetic data. DECAF’s sequential generation provides a natural way of removing these edges, with the advantage that the conditional generation of other features is left unaltered. We demonstrated on real datasets that the DECAF framework is both versatile and compatible with several popular definitions of fairness. Lastly, we provided theoretical guarantees on the generator’s convergence and fairness of downstream models. We next discuss limitations as well as applications and opportunities for future work. ",
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+    {
+        "type": "text",
+        "text": "Definitions. DECAF achieves fairness by removing edges between features, as we have shown for the popular FTU and DP definitions. Other independence-based [30] fairness definitions can be achieved by DECAF too, as we show in Appendix C. Just like related debiasing works [10, 11, 16, 17], DECAF is not compatible with fairness definitions based on separation or sufficiency [30], as these definitions depend on the downstream model more explicitly (e.g. Equality of Opportunity [12]). More on this in Appendix C. ",
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+    {
+        "type": "text",
+        "text": "Incorrect DAG specification. Our method relies on the provision of causal structure in the form of a DAG for i) deciding the sequential order of feature generation and ii) deciding which edges to remove to achieve fairness. This graph need not be known a priori and can be discovered instead. If discovered, the DAG needs not equal the true DAG for many definitions of fairness, including FTU and DP, but only some (in)dependence statements are required to be correct (see Proposition 1). This is shown in the Experiments, where the DAG was discovered with the PC algorithm [47] and TETRAD [42]. Furthermore, in Appendix B we prove that the causal generator converges to the right distribution for any graph that is Markov compatible with the data. We reiterate, however, that knowing (part of) the true graph is still helpful because i) it often leads to simpler functions $\\{ f _ { i } \\} _ { i = 1 } ^ { d }$ to approximate,9 and ii) some causal fairness definitions do require correct directionality—see Appendix ",
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+        "type": "text",
+        "text": "C. In Appendix F, we include an ablation study on how errors in the DAG specification affect data quality and downstream fairness. ",
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+        "type": "text",
+        "text": "Causal sufficiency. We have focused on just one type of graph: causally-sufficient directed graphs. Extending this to undirected or mixed graphs is possible as long as the generation order reflects a valid factorization of the observed distribution. This includes settings with hidden confounders. We note that for some definitions of bias, e.g., counterfactual bias, directionality is essential and hidden confounders would need to be corrected for (which is not generally possible). ",
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+    {
+        "type": "text",
+        "text": "Time-series. We have focused on the tabular domain. The method can be extended to other domains with causal interaction between features, e.g., time-series. Application to image data is non-trivial, partly because, in this instance, the protected attribute (e.g., skin color) does not correspond to a single observed feature. DECAF might be extended to this setting in the future by first constructing a graph in a disentangled latent space (e.g., [24, 25]). ",
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+    {
+        "type": "text",
+        "text": "Social implications. Fairness is task and context-dependent, requiring careful public debate. With that being said, DECAF empowers data issuers to take responsibility for downstream model fairness. We hope that this progresses the ubiquity of fairness in machine learning. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We would like to thank the reviewers for their time and valuable feedback. This research was funded by the Office of Naval Research and the WD Armstrong Trust. ",
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+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+        "text": "[1] Ahmed M Alaa, Boris van Breugel, Evgeny Saveliev, and Mihaela van der Schaar. How faithful is your synthetic data? sample-level metrics for evaluating and auditing generative models. arXiv preprint arXiv:2102.08921, 2021.   \n[2] Liyang Xie, Kaixiang Lin, Shu Wang, Fei Wang, and Jiayu Zhou. Differentially private generative adversarial network. CoRR, abs/1802.06739, 2018. URL http://arxiv.org/ abs/1802.06739.   \n[3] Jinsung Yoon, L. Drumright, and M. van der Schaar. Anonymization through data synthesis using generative adversarial networks (ads-gan). IEEE Journal of Biomedical and Health Informatics, 24:2378–2388, 2020.   \n[4] Jason Tashea. Courts are using ai to sentence criminals. that must stop now. WIRED, Apr 2017. URL https://www.wired.com/2017/04/ courts-using-ai-sentence-criminals-must-stop-now/.   \n[5] Jeffrey Dastin. Amazon scraps secret AI recruiting tool that showed bias against women. 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