Original: G RADIENT D ESCENT R ESISTS C OMPOSITIONALITY
Revised: G RADIENT D ESCENT R ESISTS C OMPOSITIONALITY

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: In this paper, we argue that gradient descent is one of the reasons that make compositionality learning hard during neural network optimization.
Revised: In this paper, we argue that gradient descent is one of the reasons that make compositionality learning hard during neural network optimization.

Original: We ﬁnd that the optimization process imposes a bias toward non-compositional solutions.
Revised: We ﬁnd that the optimization process imposes a bias toward non-compositional solutions.

Original: This is caused by gradient descent, trying to use all available and redundant information from input, violating the conditional independence property of compositionality.
Revised: This is caused by gradient descent, trying to use all available and redundant information from input, violating the conditional independence property of compositionality.

Original: Based on this ﬁnding, we suggest that compositionality learning approaches considering only model architecture design are unlikely to achieve complete compositionality.
Revised: Based on this ﬁnding, we suggest that compositionality learning approaches considering only model architecture design are unlikely to achieve complete compositionality.

Original: This is the ﬁrst work to investigate the relation between compositional learning and gradient descent.
Revised: This is the ﬁrst work to investigate the relation between compositional learning and gradient descent.

Original: We hope this study provides novel insights into compositional generalization, and forms a basis for new research directions to equip machine learning models with such skills for human-level intelligence.
Revised: We hope this study provides novel insights into compositional generalization, and forms a basis for new research directions to equip machine learning models with such skills for human-level intelligence.

Original: The source code is included in supplementary material.
Revised: The source code is included in supplementary material.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Compositional generalization is the algebraic capacity to understand and produce many novel combinations from known components (Chomsky, 1957; Montague, 1970), and it is a key element of human intelligence (Minsky, 1986;
Revised: Compositional generalization is the algebraic capacity to understand and produce many novel combinations from known components (Chomsky, 1957; Montague, 1970), and it is a key element of human intelligence (Minsky, 1986;

Original: Lake et al., 2017) to recognize the world efﬁciently and create imagination.
Revised: Lake et al., 2017) to recognize the world efﬁciently and create imagination.

Original: Broadly speaking, compositional generalization is a class of out-of-distribution generalization (Bengio, 2017), where the training and test distributions are different.
Revised: Broadly speaking, compositional generalization is a class of out-of-distribution generalization (Bengio, 2017), where the training and test distributions are different.

Original: A sample in such a setting is a combination of several components, and the generalization is enabled by recombining the seen components of the unseen combination during inference.
Revised: A sample in such a setting is a combination of several components, and the generalization is enabled by recombining the seen components of the unseen combination during inference.

Original: For example, in the image domain, an object is a combination of multiple parts or properties.
Revised: For example, in the image domain, an object is a combination of multiple parts or properties.

Original: In the language domain, a sentence is a combination of syntax and semantics.
Revised: In the language domain, a sentence is a combination of syntax and semantics.

Original: Each component of an output depends only on the corresponding input component, but not on other variables.
Revised: Each component of an output depends only on the corresponding input component, but not on other variables.

Original: We call this the conditional independence property , and will formally introduce in Section 3.
Revised: We call this the conditional independence property , and will formally introduce in Section 3.

Original: We hope to design machine learning algorithms with compositional generalization skills.
Revised: People hope to design machine learning algorithms with compositional generalization skills.

Original: Current neural network models, however, generally lack such ability.
Revised: However, conventional neural network models generally lack such ability.

Original: There have been many attempts to equip models with compositionality (Fodor & Pylyshyn, 1988; Bahdanau et al., 2019), and most efforts focus on designing neural network architectures (Graves et al., 2014; Andreas et al., 2016;Henaff et al., 2016;
Revised: There have been many attempts to equip models with compositionality (Fodor & Pylyshyn, 1988; Bahdanau et al., 2019), and most efforts focus on designing neural network architectures (Graves et al., 2014; Andreas et al., 2016;

Original:  Shazeer et al., 2017; Li et al., 2018; Santoro et al., 2018; Kirsch et al., 2018;
Revised: Henaff et al., 2016; Shazeer et al., 2017; Li et al., 2018; Santoro et al., 2018; Kirsch et al., 2018;

Original: Rosenbaum et al., 2019; Goyal et al., 2019).
Revised: Rosenbaum et al., 2019; Goyal et al., 2019).

Original: Recently, multiple approaches showed progress in speciﬁc tasks (Li et al., 2019; 2020; Lake, 2019; Russin et al., 2019), but we still do not know why standard approaches seldom achieve good compositionality in general.
Revised: Recently, multiple approaches showed progress in speciﬁc tasks (Li et al., 2019; 2020; Lake, 2019; Russin et al., 2019), but we still do not know why standard approaches seldom achieve good compositionality in general.

Original: In this paper, we argue that there is a bias to prevent parameters from reaching compositional solutions, when we use gradient descent in optimization (please see Figure 1 for illustrations).
Revised: In this paper, we argue that there is a bias to prevent parameters from reaching compositional solutions, when we use gradient descent in optimization (please see Figure 1 for illustrations).

Original: This is because gradient seeks the steepest direction, so that it uses all possible and redundant input information, which contradicts to the conditional independence property of compositionality.
Revised: This is because gradient seeks the steepest direction, so that it uses all possible and redundant input information, which contradicts to the conditional independence property of compositionality.

Original: This problem is not due to how gradient is computed, such as back propagation, but caused by the essential property of gradient.
Revised: This problem is not due to how gradient is computed, such as back propagation, but caused by the essential property of gradient.

Original: We derive theoretical relation between gradient descent and compositionality with information theory.
Revised: We derive theoretical relation between gradient descent and compositionality with information theory.

Original: We also provide examples and visualization to show the detailed process of how gradient resists compositionality.
Revised: We also provide examples and visualization to show the detailed process of how gradient resists compositionality.

Original: Based on the ﬁnding, we propose that compositionality learning approaches with model structure design (manual or searching) alone are not likely to achieve complete compositionality.
Revised: Based on the ﬁnding, we propose that compositionality learning approaches with model structure design (manual or searching) alone are not likely to achieve complete compositionality.

Original: [Figure]
Revised: [Figure]

Original: We hope this research provides new insights and forms a basis for new research directions in compositional generalization, and helps to improve machine intelligence towards human-level.
Revised: We hope this research provides new insights and forms a basis for new research directions in compositional generalization, and helps to improve machine intelligence towards human-level.

Original: The contributions of this paper can be summarized as follows.
Revised: The contributions of this paper can be summarized as follows.

Original: • The novelty of this work is to ﬁnd the relation between compositional learning and gradient descent in optimization process, i.e., gradient descent resists compositionality.
Revised: • The novelty of this work is to ﬁnd the relation between compositional learning and gradient descent in optimization process, i.e., gradient descent resists compositionality.

Original: • We theoretically derive the result and explain why standard approaches with architecture design alone do not address compositionality.
Revised: • We theoretically derive the result and explain why standard approaches with architecture design alone do not address compositionality.

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: Compositionality Humans learn language and recognize the world in a ﬂexible way by leveraging systematic compositionality .
Revised: Compositionality Humans learn language and recognize the world in a ﬂexible way by leveraging systematic compositionality .

Original: The compositional generalization is critical in human cognition (Minsky, 1986;
Revised: The compositional generalization is critical in human cognition (Minsky, 1986;

Original: Lake et al., 2017), and it helps humans to connect limited amount of learned concepts for unseen combinations.
Revised: Lake et al., 2017), and it helps humans to connect limited amount of learned concepts for unseen combinations.

Original: Though deep learning has many achievements in recent years (LeCun et al., 2015; Krizhevsky et al., 2012; Yu & Deng, 2012;
Revised: Though deep learning has many achievements in recent years (LeCun et al., 2015; Krizhevsky et al., 2012; Yu & Deng, 2012;

Original: He et al., 2016; Wu & et al, 2016), compositional generalization has not been well addressed (Fodor & Pylyshyn, 1988; Marcus, 1998; Fodor & Lepore, 2002; Marcus, 2003; Calvo & Symons, 2014).
Revised: He et al., 2016; Wu & et al, 2016), compositional generalization has not been well addressed (Fodor & Pylyshyn, 1988; Marcus, 1998; Fodor & Lepore, 2002; Marcus, 2003; Calvo & Symons, 2014).

Original: There are observations that current neural network models do not learn compositionality (Bahdanau et al., 2019).
Revised: There are observations that current neural network models do not learn compositionality (Bahdanau et al., 2019).

Original: Most recently, multiple approaches are proposed to address compositionality in neural networks (Li et al., 2019; 2020; Lake, 2019; Russin et al., 2019) for speciﬁc tasks.
Revised: Most recently, multiple approaches are proposed to address compositionality in neural networks (Li et al., 2019; 2020; Lake, 2019; Russin et al., 2019) for speciﬁc tasks.

Original: However, we are still not sure why compositionality is hard to achieve in general cases, and this work discusses about this problem from optimization perspective.
Revised: However, we are still not sure why compositionality is hard to achieve in general cases, and this work discusses about this problem from optimization perspective.

Original: Another line of related work is independent disentangled representation learning (Higgins et al., 2017; Locatello et al., 2019).
Revised: Another line of related work is independent disentangled representation learning (Higgins et al., 2017; Locatello et al., 2019).

Original: Its main assumption is that the expected components are statistically independent in training data.
Revised: Its main assumption is that the expected components are statistically independent in training data.

Original: This setting does not have transferring problem in test, because all combinations have positive joint probabilities in training (please refer to Section 3).
Revised: This setting does not have transferring problem in test, because all combinations have positive joint probabilities in training (please refer to Section 3).

Original: Compositionality is applied in different areas such as continual learning (Jin et al., 2020; Li et al., 2020), question answering (Andreas et al., 2016; Hudson & Manning, 2019; Keysers et al., 2020), and reasoning (Talmor et al., 2020).
Revised: Compositionality is applied in different areas such as continual learning (Jin et al., 2020; Li et al., 2020), question answering (Andreas et al., 2016; Hudson & Manning, 2019; Keysers et al., 2020), and reasoning (Talmor et al., 2020).

Original: Gradient Descent Gradient descent is a powerful and general purpose optimization tool for solving large scale problems in deep neural networks.
Revised: Gradient Descent Gradient descent is a powerful and general purpose optimization tool for solving large scale problems in deep neural networks.

Original: It is usually used in a stochastic way (Stochastic
Revised: It is usually used in a stochastic way (Stochastic

Original: Gradient Descent) with mini-batches, and has many variations such as Momentum (Rumelhart et al., 1986), averaging (Polyak & Juditsky, 1992), AdaGrad (Duchi et al., 2011), AdaDelta (Zeiler, 2012),
Revised: Gradient Descent) with mini-batches, and has many variations such as Momentum (Rumelhart et al., 1986), averaging (Polyak & Juditsky, 1992), AdaGrad (Duchi et al., 2011), AdaDelta (Zeiler, 2012),

Original: RMSProp (Tieleman & Hinton, 2012), Adam (Kingma & Ba, 2014).
Revised: RMSProp (Tieleman & Hinton, 2012), Adam (Kingma & Ba, 2014).

Original: Most of the previous work focus on faster reduction of loss and theoretical convergence analysis of
Revised: Most of the previous work focus on faster reduction of loss and theoretical convergence analysis of

Original: SGD (Bottou et al., 2018; Luo, 1991; Reddi et al., 2018; Chen et al., 2018;
Revised: SGD (Bottou et al., 2018; Luo, 1991; Reddi et al., 2018; Chen et al., 2018;

Original: Zhou et al., 2018; Zou &
Revised: Zhou et al., 2018; Zou &

Original: Shen, 2018; De et al., 2018; Zou et al., 2018; Ward et al., 2018; Barakat & Bianchi, 2019).
Revised: Shen, 2018; De et al., 2018; Zou et al., 2018; Ward et al., 2018; Barakat & Bianchi, 2019).

Original: In particular, this work focuses on investigating why standard neural network training only achieves limited level of compositionality by studying the relationship between gradient descent and compositionality.
Revised: In particular, this work focuses on investigating why standard neural network training only achieves limited level of compositionality by studying the relationship between gradient descent and compositionality.

Original:  C ONCEPTS FOR C OMPOSITIONALITY AND G RADIENT D ESCENT
Revised:  C ONCEPTS FOR C OMPOSITIONALITY AND G RADIENT D ESCENT

Original: We ﬁrst formulate compositionality using the conditional independence property, and deﬁne compositional generalization.
Revised: We ﬁrst formulate compositionality using the conditional independence property, and deﬁne compositional generalization.

Original: We then review properties of gradient for the derivation in the next section.
Revised: We then review properties of gradient for the derivation in the next section.

Original: Conditional Independence Property for Compositionality When multiple hidden variables live in the same representation, and cannot be separated by simply splitting the representation, then these variables are entangled in the representation.
Revised: Conditional Independence Property for Compositionality When multiple hidden variables live in the same representation, and cannot be separated by simply splitting the representation, then these variables are entangled in the representation.

Original: For example, color and shape are two hidden variables and they share the same representation of image.
Revised: For example, color and shape are two hidden variables and they share the same representation of image.

Original: Also, syntax and semantics are two hidden variables and they share the same representation of sentence.
Revised: Also, syntax and semantics are two hidden variables and they share the same representation of sentence.

Original: When we extract the hidden variables from their shared representation, we disentangle them.
Revised: When we extract the hidden variables from their shared representation, we disentangle them.

Original: We then consider a prediction problem, where input X and output Y have multiple entangled components that are not labeled in data, and they are aligned.
Revised: We then consider a prediction problem, where input X and output Y have multiple entangled components that are not labeled in data, and they are aligned.

Original: For example in machine translation,
Revised: For example in machine translation,

Original: X 1 is input syntax, and X 2 is input semantics.
Revised: X 1 is input syntax, and X 2 is input semantics.

Original: Y 1 is output syntax, and Y 2 is output semantics.
Revised: Y 1 is output syntax, and Y 2 is output semantics.

Original: The syntax of output Y 1 depends only on the syntax of input X 1 , and the semantics of output Y 2 depends only on the semantics of X 2 .
Revised: The syntax of output Y 1 depends only on the syntax of input X 1 , and the semantics of output Y 2 depends only on the semantics of X 2 .

Original: We can formalize the alignments as conditional independence property :
Revised: We can formalize the alignments as conditional independence property :

Original: Y i depends only on X i .
Revised: Y i depends only on X i .

Original: [Equation]
Revised: [Equation]

Original: When a model ﬁts this property, we say it has compositionality .
Revised: When a model ﬁts this property, we say it has compositionality .

Original: Note that this can be understood as a kind of sparseness property (Bengio, 2017), because it restricts effective connection between input and output components.
Revised: Note that this can be understood as a kind of sparseness property (Bengio, 2017), because it restricts effective connection between input and output components.

Original: Compositional Generalization In compositional generalization, each sample in either training or test is a combination of several components.
Revised: Compositional Generalization In compositional generalization, each sample in either training or test is a combination of several components.

Original: A test sample has a combination that does not appear in training, but each component of the test sample appears in training.
Revised: A test sample has a combination that does not appear in training, but each component of the test sample appears in training.

Original: We need to recombine the seen components to generalize to the test sample.
Revised: We need to recombine the seen components to generalize to the test sample.

Original: We can deﬁne compositional generalization probabilistically as follows.
Revised: We can deﬁne compositional generalization probabilistically as follows.

Original: In train,
Revised: In train,

Original: In test,
Revised: In test,

Original: [Equation]
Revised: [Equation]

Original: ∀
Revised: ∀

Original: i : P ( Y i | X i ) is high.
Revised: i : P ( Y i | X i ) is high.

Original: [Equation]
Revised: [Equation]

Original: Compositionality bridges the gap between training and test distributions to achieve compositional generalization.
Revised: Compositionality bridges the gap between training and test distributions to achieve compositional generalization.

Original: We ﬁrst apply chain rule, and then use compositionality as follows.
Revised: We ﬁrst apply chain rule, and then use compositionality as follows.

Original: [Equation]
Revised: [Equation]

Original: When P ( Y i | X i ) are all high, their product should also be high.
Revised: When P ( Y i | X i ) are all high, their product should also be high.

Original: Therefore, a model with compositionality—satisfying this conditional independence property—addresses compositional generalization.
Revised: Therefore, a model with compositionality—satisfying this conditional independence property—addresses compositional generalization.

Original: Property of Gradient For a function f ( x 1 , . . .
Revised: Property of Gradient For a function f ( x 1 , . . .

Original: , x K ) , the gradient ∇ f is the steepest direction to change the function’s value.
Revised: , x K ) , the gradient ∇ f is the steepest direction to change the function’s value.

Original: Generally, gradient descent methods estimate ∇ f using low-order local estimation.
Revised: Generally, gradient descent methods estimate ∇ f using low-order local estimation.

Original: By deﬁnition, it is the vector of partial derivatives with respective to the inputs.
Revised: By deﬁnition, it is the vector of partial derivatives with respective to the inputs.

Original: [Equation]
Revised: [Equation]

Original: We will use the following deﬁnition in later arguments.
Revised: We will use the following deﬁnition in later arguments.

Original: Deﬁnition 1 (Partial derivative) .
Revised: Deﬁnition 1 (Partial derivative) .

Original: Partial derivative for an input is the derivative assuming other inputs are constant.
Revised: Partial derivative for an input is the derivative assuming other inputs are constant.

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original:  G RADIENT D ESCENT R ESISTS C OMPOSITIONALITY
Revised:  G RADIENT D ESCENT R ESISTS C OMPOSITIONALITY

Original: We focus on the early phase of training to show that gradient descent causes a model to use the redundant information to compute output when it has information to reduce the loss.
Revised: We focus on the early phase of training to show that gradient descent causes a model to use the redundant information to compute output when it has information to reduce the loss.

Original: We develop the arguments step by step.
Revised: We develop the arguments step by step.

Original: We ﬁrst analyze the inﬂuence of the input on an output variable.
Revised: We ﬁrst analyze the inﬂuence of the input on an output variable.

Original: We then consider the case of entangled inputs and one output.
Revised: We then consider the case of entangled inputs and one output.

Original: Finally, we discuss the case with entangled inputs and entangled outputs.
Revised: Finally, we discuss the case with entangled inputs and entangled outputs.

Original: The gradient is used to reduce loss, so we aim to relate loss reduction and the inﬂuence from input to output.
Revised: The gradient is used to reduce loss, so we aim to relate loss reduction and the inﬂuence from input to output.

Original: To do that, we use mutual information to describe the inﬂuence, and use knowledge from information theory (Theorem 1 and Theorem 2).
Revised: To do that, we use mutual information to describe the inﬂuence, and use knowledge from information theory (Theorem 1 and Theorem 2).

Original: We also study the impact of gradient descent to the inﬂuence, so we relate mutual information with gradient with Proposition 1.
Revised: We also study the impact of gradient descent to the inﬂuence, so we relate mutual information with gradient with Proposition 1.

Original: Theorem 1 (Data-processing inequality (Cover, 1999) p.34) .
Revised: Theorem 1 (Data-processing inequality (Cover, 1999) p.34) .

Original: For random variables X, Y, Z , if the conditional distribution of Z depends only on Y and is conditionally independent of X , then
Revised: For random variables X, Y, Z , if the conditional distribution of Z depends only on Y and is conditionally independent of X , then

Original: I ( X ; Y ) ≥ I ( X ; Z ) .
Revised: I ( X ; Y ) ≥ I ( X ; Z ) .

Original: Theorem 2 (Chain rule for information (Cover, 1999) p.24) .
Revised: Theorem 2 (Chain rule for information (Cover, 1999) p.24) .

Original: For random variables X, Y, Z ,
Revised: For random variables X, Y, Z ,

Original: I ( X, Y ; Z ) =
Revised: I ( X, Y ; Z ) =

Original: I ( Y ; Z | X )
Revised: I ( Y ; Z | X )

Original: + I ( X ; Z ) .
Revised: + I ( X ; Z ) .

Original: Proposition 1.
Revised: Proposition 1.

Original: When ∂Y∂X is deﬁned, I ( X ; Y ) > 0 ⇐⇒ ∂Y∂X
Revised: When ∂Y∂X is deﬁned, I ( X ; Y ) > 0 ⇐⇒ ∂Y∂X

Original: (cid:54) =
Revised: (cid:54) =

Original: Proof.
Revised: Proof.

Original: I ( X ; Y ) > 0 means X
Revised: I ( X ; Y ) > 0 means X

Original: and Y are not independent, which means Y is not invariant to X .
Revised: and Y are not independent, which means Y is not invariant to X .

Original: [Equation]
Revised: [Equation]

Original: AND O NE O UTPUT
Revised: AND O NE O UTPUT

Original: We ﬁrst consider a basic setting that the data has a single input X and output Y .
Revised: We ﬁrst consider a basic setting that the data has a single input X and output Y .

Original: A model f with parameters θ has input X and output
Revised: A model f with parameters θ has input X and output

Original: ˆ Y (Figure 2 middle).
Revised: ˆ Y (Figure 2 middle).

Original: We optimize a loss function L .
Revised: We optimize a loss function L .

Original: Applying the gradient ∇ θ L ( Y, ˆ Y ) reduces L ( Y, ˆ Y ) , bringing Y and ˆ Y closer.
Revised: Applying the gradient ∇ θ L ( Y, ˆ Y ) reduces L ( Y, ˆ Y ) , bringing Y and ˆ Y closer.

Original: Since Y changes according to
Revised: Since Y changes according to

Original: X , ˆ Y is encouraged to change according to X .
Revised: X , ˆ Y is encouraged to change according to X .

Original: We look into details as follows.
Revised: We look into details as follows.

Original: In the common supervised learning setting, given X , the ground truth Y does not depend on prediction ˆ
Revised: In the common supervised learning setting, given X , the ground truth Y does not depend on prediction ˆ

Original: Y .
Revised: Y .

Original: We do not require speciﬁc form of the loss function, L , but we assume that when it is reduced, ˆ Y moves closer to Y , and increases the mutual information I ( ˆ Y , Y ) .
Revised: We do not require speciﬁc form of the loss function, L , but we assume that when it is reduced, ˆ Y moves closer to Y , and increases the mutual information I ( ˆ Y , Y ) .

Original: Many widely used loss functions encourage increased mutual information between model output and dataset labels.
Revised: Many widely used loss functions encourage increased mutual information between model output and dataset labels.

Original: Also, training algorithms are designed to reduce loss when the input has information to do so.
Revised: Also, training algorithms are designed to reduce loss when the input has information to do so.

Original: These assumptions are likely to hold especially in the early part of training.
Revised: These assumptions are likely to hold especially in the early part of training.

Original: We also use local linear approximation when discussing gradients.
Revised: We also use local linear approximation when discussing gradients.

Original: We derive the proof by studying relations between random variables, and show that gradient descent increases the lower bound of mutual information between model input and output, hence the output is dependent on the input.
Revised: We derive the proof by studying relations between random variables, and show that gradient descent increases the lower bound of mutual information between model input and output, hence the output is dependent on the input.

Original: Proposition 2.
Revised: Proposition 2.

Original: If an inﬁnitesimal change of parameters dθ increases I ( Y ; ˆ Y θ ) , then I ( X ; ˆ Y θ + dθ ) is positive with parameters θ + dθ .
Revised: If an inﬁnitesimal change of parameters dθ increases I ( Y ; ˆ Y θ ) , then I ( X ; ˆ Y θ + dθ ) is positive with parameters θ + dθ .

Original: ∀ dθ : dI ( Y ; ˆ Y θ ) > 0 = ⇒
Revised: ∀ dθ : dI ( Y ; ˆ Y θ ) > 0 = ⇒

Original: I ( X ; ˆ Y θ + dθ ) > 0 .
Revised: I ( X ; ˆ Y θ + dθ ) > 0 .

Original: Proof.
Revised: Proof.

Original: Since Y and ˆ Y are conditionally independent given X , with data-processing inequality (Theorem 1), I ( ˆ Y ; X ) ≥ I ( ˆ Y ; Y ) .
Revised: Since Y and ˆ Y are conditionally independent given X , with data-processing inequality (Theorem 1), I ( ˆ Y ; X ) ≥ I ( ˆ Y ; Y ) .

Original: So I ( Y ; ˆ Y θ + dθ ) is a lower bound of I ( X ; ˆ Y θ + dθ ) , and it is the sum of I ( Y ; ˆ Y θ ) and dI ( Y ; ˆ Y θ ) .
Revised: So I ( Y ; ˆ Y θ + dθ ) is a lower bound of I ( X ; ˆ Y θ + dθ ) , and it is the sum of I ( Y ; ˆ Y θ ) and dI ( Y ; ˆ Y θ ) .

Original: Also, I ( Y ; ˆ Y θ ) ≥ 0 by deﬁnition.
Revised: Also, I ( Y ; ˆ Y θ ) ≥ 0 by deﬁnition.

Original: Therefore I ( Y ; ˆ Y θ + dθ ) > 0 .
Revised: Therefore I ( Y ; ˆ Y θ + dθ ) > 0 .

Original: I
Revised: I

Original: [Equation]
Revised: [Equation]

Original: θ + dθ ) =
Revised: θ + dθ ) =

Original: I ( Y ; ˆ Y θ ) + dI ( Y ; ˆ Y θ ) >
Revised: I ( Y ; ˆ Y θ ) + dI ( Y ; ˆ Y θ ) >

Original: Proposition 3.
Revised: Proposition 3.

Original: If X has information to reduce loss L ( Y, ˆ Y ) , ∂ ˆ Y∂X (cid:54) = 0 for updated parameters.
Revised: If X has information to reduce loss L ( Y, ˆ Y ) , ∂ ˆ Y∂X (cid:54) = 0 for updated parameters.

Original: Proof.
Revised: Proof.

Original: Since training algorithm reduces loss, and hence increases mutual information, Propositionapplies.
Revised: Since training algorithm reduces loss, and hence increases mutual information, Propositionapplies.

Original: With local linear approximation, we can use ∆ θ for dθ .
Revised: With local linear approximation, we can use ∆ θ for dθ .

Original: So we have I ( X ; ˆ Y θ +∆ θ ) > 0 .
Revised: So we have I ( X ; ˆ Y θ +∆ θ ) > 0 .

Original: With
Revised: With

Original: Proposition 1, we have ∂ ˆ Y∂X (cid:54) = 

Revised: Proposition 1, we have ∂ ˆ Y∂X (cid:54) =

Original: for the updated parameters θ + ∆ θ .
Revised: for the updated parameters θ + ∆ θ .

Original: 4. E NTANGLED I NPUT AND O NE O UTPUT
Revised: 4. E NTANGLED I NPUT AND O NE O UTPUT

Original: Then, we study the case where input X is entanglement of multiple hidden input components
Revised: Then, we study the case where input X is entanglement of multiple hidden input components

Original: X 1 , . . .
Revised: X 1 , . . .

Original: , X K , and output is a single variable ˆ Y that depends only on X i .
Revised: , X K , and output is a single variable ˆ Y that depends only on X i .

Original: We hope to make ˆ Y invariant to X j , ∀ j (cid:54)
Revised: We hope to make ˆ Y invariant to X j , ∀ j (cid:54)

Original: =
Revised: =

Original: i .
Revised: i .

Original: For example, in a parsing task (Li & Eisner, 2019), output parse tree Y depends only on the input syntax X 1 , but not on input semantics X 2 .
Revised: For example, in a parsing task (Li & Eisner, 2019), output parse tree Y depends only on the input syntax X 1 , but not on input semantics X 2 .

Original: For the convenience of analysis, we assume we have a ﬁxed differentiable oracle encoder network g and decoder network g − 1 .
Revised: For the convenience of analysis, we assume we have a ﬁxed differentiable oracle encoder network g and decoder network g − 1 .

Original: g − 1 maps X to X 1 , . . .
Revised: g − 1 maps X to X 1 , . . .

Original: , X K , and g maps them back.
Revised: , X K , and g maps them back.

Original: [Equation]
Revised: [Equation]

Original: We extend the model structure with g and g − 1 and use the input to g − 1 as model input (Figure 2 left and middle).
Revised: We extend the model structure with g and g − 1 and use the input to g − 1 as model input (Figure 2 left and middle).

Original: [Equation]
Revised: [Equation]

Original: This model is exactly the same as the original one, because there is no additional trainable parameters, and g ◦ g − 1 does not change X .
Revised: This model is exactly the same as the original one, because there is no additional trainable parameters, and g ◦ g − 1 does not change X .

Original: Proposition 4.
Revised: Proposition 4.

Original: If X i has information to reduce loss L ( Y, ˆ Y ) , ∂ ˆ Y ∂X i (cid:54) = 0 for updated parameters.
Revised: If X i has information to reduce loss L ( Y, ˆ Y ) , ∂ ˆ Y ∂X i (cid:54) = 0 for updated parameters.

Original: Proof.
Revised: Proof.

Original: With the property of gradient (Deﬁnition 1), we can regard X j , ∀ j (cid:54) =
Revised: With the property of gradient (Deﬁnition 1), we can regard X j , ∀ j (cid:54) =

Original: i as constant values when computing the gradient w.r.t.
Revised: i as constant values when computing the gradient w.r.t.

Original: X i .
Revised: X i .

Original: So Proposition 3 applies.
Revised: So Proposition 3 applies.

Original: 4. E NTANGLED I NPUT AND E NTANGLED O UTPUT
Revised: 4. E NTANGLED I NPUT AND E NTANGLED O UTPUT

Original: We then discuss the case that output Y is also the entanglement of Y 1 , . . .
Revised: We then discuss the case that output Y is also the entanglement of Y 1 , . . .

Original: , Y K .
Revised: , Y K .

Original: Y i depends only on
Revised: Y i depends only on

Original: X i for all i = 1 , . . .
Revised: X i for all i = 1 , . . .

Original: , K .
Revised: , K .

Original: This corresponds to the example of machine translation.
Revised: This corresponds to the example of machine translation.

Original: We assume we have a ﬁxed differentiable oracle encoder network h and decoder network h − 1 .
Revised: We assume we have a ﬁxed differentiable oracle encoder network h and decoder network h − 1 .

Original: h takes ˆ Y as input and produce K outputs ˆ Y 1 , . . .
Revised: h takes ˆ Y as input and produce K outputs ˆ Y 1 , . . .

Original: , ˆ Y K
Revised: , ˆ Y K

Original: =
Revised: =

Original: h
Revised: h

Original: ( ˆ Y ) , and h maps them back.
Revised: ( ˆ Y ) , and h maps them back.

Original: We extend the model structure with h and h − 1 (Figure 2).
Revised: We extend the model structure with h and h − 1 (Figure 2).

Original: [Equation]
Revised: [Equation]

Original: This model is the same as the original one, because there is no additional trainable parameters and h ◦ h − 1 does not change ˆ Y .
Revised: This model is the same as the original one, because there is no additional trainable parameters and h ◦ h − 1 does not change ˆ Y .

Original: We derive proof with this extended model.
Revised: We derive proof with this extended model.

Original: The intuitive idea is that the reduction of loss will make each Y i contain more information of Y , and for each Y i , we can apply previous discussion.
Revised: The intuitive idea is that the reduction of loss will make each Y i contain more information of Y , and for each Y i , we can apply previous discussion.

Original: We denote ˆ Y (cid:54)
Revised: We denote ˆ Y (cid:54)

Original: =
Revised: =

Original: i =
Revised: i =

Original: ˆ Y 1 , . .
Revised: ˆ Y 1 , . .

Original: , ˆ Y i − 1 , ˆ Y
Revised: , ˆ Y i − 1 , ˆ Y

Original: i +1 , . .
Revised: i +1 , . .

Original: , ˆ Y K .
Revised: , ˆ Y K .

Original: Proposition 5.
Revised: Proposition 5.

Original: If X j has information to reduce loss L ( Y, ˆ Y ) , then ∂ ˆ Y i ∂X j (cid:54) = 0 , ∀ j (cid:54)
Revised: If X j has information to reduce loss L ( Y, ˆ Y ) , then ∂ ˆ Y i ∂X j (cid:54) = 0 , ∀ j (cid:54)

Original: =
Revised: =

Original: i for updated parameters.
Revised: i for updated parameters.

Original: Proof.
Revised: Proof.

Original: To study gradient of output ˆ Y i , we regard other outputs ˆ Y (cid:54)
Revised: To study gradient of output ˆ Y i , we regard other outputs ˆ Y (cid:54)

Original: =
Revised: =

Original: i as ﬁxed, and look at the conditional mutual information
Revised: i as ﬁxed, and look at the conditional mutual information

Original: I ( ˆ Y i ; Y | ˆ Y (cid:54) = i ) .
Revised: I ( ˆ Y i ; Y | ˆ Y (cid:54) = i ) .

Original: With chain rule for information (Theorem 2), we have I ( ˆ Y ; Y ) =
Revised: With chain rule for information (Theorem 2), we have I ( ˆ Y ; Y ) =

Original: I ( ˆ Y i ; Y | ˆ Y (cid:54) = i )
Revised: I ( ˆ Y i ; Y | ˆ Y (cid:54) = i )

Original: + I ( ˆ Y (cid:54)
Revised: + I ( ˆ Y (cid:54)

Original: =
Revised: =

Original: i ; Y ) .
Revised: i ; Y ) .

Original: Since Y and ˆ Y (cid:54) = i are both ﬁxed, I ( ˆ Y (cid:54)
Revised: Since Y and ˆ Y (cid:54) = i are both ﬁxed, I ( ˆ Y (cid:54)

Original: =
Revised: =

Original: i ; Y ) is ﬁxed.
Revised: i ; Y ) is ﬁxed.

Original: So the change of
Revised: So the change of

Original: I ( ˆ Y ; Y ) equals to the change of I ( ˆ Y i ; Y | ˆ Y (cid:54)
Revised: I ( ˆ Y ; Y ) equals to the change of I ( ˆ Y i ; Y | ˆ Y (cid:54)

Original: = i ) .
Revised: = i ) .

Original: Therefore, the reduction of loss L increases the mutual information for each component.
Revised: Therefore, the reduction of loss L increases the mutual information for each component.

Original: We can then apply Proposition 4.
Revised: We can then apply Proposition 4.

Original: In Proposition 4 and Proposition 5, the gradient is not zero, so an output depends on redundant input (Proposition 2) in both cases.
Revised: In Proposition 4 and Proposition 5, the gradient is not zero, so an output depends on redundant input (Proposition 2) in both cases.

Original: Therefore, gradient descent resists conditional independence property of compositionality.
Revised: Therefore, gradient descent resists conditional independence property of compositionality.

Original: [Figure]
Revised: [Figure]

Original:  E XAMPLES
Revised:  E XAMPLES

Original: In this section, we show example cases to emphasize that the theoretical result occurs practically.
Revised: In this section, we show example cases to emphasize that the theoretical result occurs practically.

Original: We focus on the conditional independence property, and we design the experiments in the following way.
Revised: We focus on the conditional independence property, and we design the experiments in the following way.

Original: To test the compositionality, we use different training and test distributions, and the test prediction requires compositional generalization.
Revised: To test the compositionality, we use different training and test distributions, and the test prediction requires compositional generalization.

Original: We measure training and test accuracy to evaluate the ability for compositional generalization.
Revised: We measure training and test accuracy to evaluate the ability for compositional generalization.

Original: We use two settings in each experiment.
Revised: We use two settings in each experiment.

Original: In the ﬁrst setting (A), we use both X 1 and X 2 as input to the model.
Revised: In the ﬁrst setting (A), we use both X 1 and X 2 as input to the model.

Original: In the second setting (B), we only use X 1 , and remove information of X 2 by setting it to be a random input.
Revised: In the second setting (B), we only use X 1 , and remove information of X 2 by setting it to be a random input.

Original: The model architecture and other settings are the same.
Revised: The model architecture and other settings are the same.

Original: By comparing the test performance in the two settings, we show that a model can be trained faster with X 2 , but it does not hold compositionality.
Revised: By comparing the test performance in the two settings, we show that a model can be trained faster with X 2 , but it does not hold compositionality.

Original: We run experiments for 5 times, and plot the mean and variance at each training step.
Revised: We run experiments for 5 times, and plot the mean and variance at each training step.

Original: 5. I MAGE C LASSIFICATION
Revised: 5. I MAGE C LASSIFICATION

Original: We use MNIST dataset (LeCun et al., 1998) in this experiment.
Revised: We use MNIST dataset (LeCun et al., 1998) in this experiment.

Original: The dataset contains pairs of input image and output label.
Revised: The dataset contains pairs of input image and output label.

Original: An image is gray scale with ﬁxed size, and a label is in the set of ten possible values { 0 , 1 , . . .
Revised: An image is gray scale with ﬁxed size, and a label is in the set of ten possible values { 0 , 1 , . . .

Original: We use two original samples ( X 1 , Y 1 ) , ( X 2 , Y 2 ) to make one sample ( X, Y ) .
Revised: We use two original samples ( X 1 , Y 1 ) , ( X 2 , Y 2 ) to make one sample ( X, Y ) .

Original: X is the horizontal concatenation of X 1 and X 2 , and Y =
Revised: X is the horizontal concatenation of X 1 and X 2 , and Y =

Original: Y 1 .
Revised: Y 1 .

Original: Note that a generated sample does not directly use the label of the second sample Y 2 .
Revised: Note that a generated sample does not directly use the label of the second sample Y 2 .

Original: We have two different settings in the experiment.
Revised: We have two different settings in the experiment.

Original: In the ﬁrst setting, we have the following data distribution.
Revised: In the ﬁrst setting, we have the following data distribution.

Original: In training, the data are generated from the original training dataset.
Revised: In training, the data are generated from the original training dataset.

Original: The samples are chosen uniformly at random in the corresponding conditions.
Revised: The samples are chosen uniformly at random in the corresponding conditions.

Original: Y 1 is chosen from all possible labels, and X 1 with the label is chosen.
Revised: Y 1 is chosen from all possible labels, and X 1 with the label is chosen.

Original: Y 2 is chosen from { Y, Y + 1 } (we use modular for labels), and Xwith the label is chosen.
Revised: Y 2 is chosen from { Y, Y + 1 } (we use modular for labels), and Xwith the label is chosen.

Original: In test, the data are generated from the original test dataset.
Revised: In test, the data are generated from the original test dataset.

Original: Y 1 , X 1 are chosen in the same way as in training, but Y 2 is chosen from the other eight classes { Y +2 , Y +3 , . . .
Revised: Y 1 , X 1 are chosen in the same way as in training, but Y 2 is chosen from the other eight classes { Y +2 , Y +3 , . . .

Original: , Y +9 } , and then X 2 with the label is chosen.
Revised: , Y +9 } , and then X 2 with the label is chosen.

Original: This means, in training, X 2 contains a part of information for
Revised: This means, in training, X 2 contains a part of information for

Original: Y , and X 1 contain all information for Y .
Revised: Y , and X 1 contain all information for Y .

Original: This is because Y 2 is Y with half chance, and Y 1 is always
Revised: This is because Y 2 is Y with half chance, and Y 1 is always

Original: Y .
Revised: Y .

Original: We hope the model is trained to make Y not dependent on X 2 .
Revised: We hope the model is trained to make Y not dependent on X 2 .

Original: The second setting has the same test distribution as the ﬁrst setting, but the training distributions are different.
Revised: The second setting has the same test distribution as the ﬁrst setting, but the training distributions are different.

Original: In training of the second setting, Y 2 is chosen from all possible labels, so that X 2 does not have information for Y .
Revised: In training of the second setting, Y 2 is chosen from all possible labels, so that X 2 does not have information for Y .

Original: [Table]
Revised: [Table]

Original: For both the ﬁrst andthe second settings, we use a standard convolutional neural network model with three convolution layers and two fully-connected layers.
Revised: For both the ﬁrst and second settings, we use a standard convolutional neural network model with three convolution layers and two fully-connected layers.

Original: The details of model design and optimization can be found in Appendix A.
Revised: The details of model design and optimization can be found in Appendix A.

Original: The results are shown in Figure 3a.
Revised: The results are shown in Figure 3a.

Original: The training accuracy improves faster in the ﬁrst setting than in the second one, indicating that X 2 helps to train the model quickly.
Revised: The training accuracy improves faster in the ﬁrst setting than in the second one, indicating that X 2 helps to train the model quickly.

Original: In the ﬁrst setting, the gap of training and test accuracy is signiﬁcantly larger than that in the second setting, meaning the model does not learn compositionality in the ﬁrst setting.
Revised: In the ﬁrst setting, the gap of training and test accuracy is signiﬁcantly larger than that in the second setting, meaning the model does not learn compositionality in the ﬁrst setting.

Original: Therefore, this experiment shows that the redundant information X 2 is used to help training model quickly, but the model does not learn compositionality.
Revised: Therefore, this experiment shows that the redundant information X 2 is used to help training model quickly, but the model does not learn compositionality.

Original: 5. L ANGUAGE L EARNING
Revised: 5. L ANGUAGE L EARNING

Original: We also run an experiment of instruction language learning with SCAN dataset (Lake & Baroni, 2018).
Revised: We also run an experiment of instruction language learning with SCAN dataset (Lake & Baroni, 2018).

Original: We focus on Jump task, the most difﬁcult task in the dataset.
Revised: We focus on Jump task, the most difﬁcult task in the dataset.

Original: The input is a command instruction, and the output is a corresponding action sequence.
Revised: The input is a command instruction, and the output is a corresponding action sequence.

Original: The training data include a one-word command “jump”, but other training data do not contain the word.
Revised: The training data include a one-word command “jump”, but other training data do not contain the word.

Original: In test data, the word “jump” appears in a multiple-words sentence with other words.
Revised: In test data, the word “jump” appears in a multiple-words sentence with other words.

Original: Please see Table 1 for examples.
Revised: Please see Table 1 for examples.

Original: In this task, syntax and semantics are two entangled components, and it requires compositional generalization to new combinations.
Revised: In this task, syntax and semantics are two entangled components, and it requires compositional generalization to new combinations.

Original: Syntax is the way the actions are organized, and semantic is the mapping from word to action.
Revised: Syntax is the way the actions are organized, and semantic is the mapping from word to action.

Original: We also design two settings in this experiment.
Revised: We also design two settings in this experiment.

Original: In the ﬁrst setting, we use the original training and test data, which are from different distributions.
Revised: In the ﬁrst setting, we use the original training and test data, which are from different distributions.

Original: In the second setting, we remove the dependency of input semantics to output syntax by using action words uniformly at random, but we still keep the correspondence between input words and output actions.
Revised: In the second setting, we remove the dependency of input semantics to output syntax by using action words uniformly at random, but we still keep the correspondence between input words and output actions.

Original: We use a standard sequence-to-sequence model with LSTM and attention.
Revised: We use a standard sequence-to-sequence model with LSTM and attention.

Original: More details can be found in Appendix B.
Revised: More details can be found in Appendix B.

Original: Following previous works on SCAN dataset, we use sentence accuracy as the evaluation metric.
Revised: Following previous works on SCAN dataset, we use sentence accuracy as the evaluation metric.

Original: The results are shown in Figure 3b.
Revised: The results are shown in Figure 3b.

Original: We can observe that the training accuracy increases faster during the early training (steps 10-25) in the ﬁrst setting than in the second setting.
Revised: We can observe that the training accuracy increases faster during the early training (steps 10-25) in the ﬁrst setting than in the second setting.

Original: Also, the gap of training and test accuracy is signiﬁcantly larger in the ﬁrst setting than in the second setting, so the model does not learn compositionality in the ﬁrst setting.
Revised: Also, the gap of training and test accuracy is signiﬁcantly larger in the ﬁrst setting than in the second setting, so the model does not learn compositionality in the ﬁrst setting.

Original:  D ISCUSSION
Revised:  D ISCUSSION

Original: We use a simple case to visualize the process for better understanding.
Revised: We use a simple case to visualize the process for better understanding.

Original: We consider two random variables Y 1 , Y 2 ∈ { 0 , 1 } .
Revised: We consider two random variables Y 1 , Y 2 ∈ { 0 , 1 } .

Original: In training, their joint distribution is uniform on combinations ( Y 1 , Y 2 ) ∈ { (0 , 0) , (0 , 1) , (1 , 0) } , and zero on (1 , 1) .
Revised: In training, their joint distribution is uniform on combinations ( Y 1 , Y 2 ) ∈ { (0 , 0) , (0 , 1) , (1 , 0) } , and zero on (1 , 1) .

Original: In test, ( Y 1 , Y 2 )
Revised: In test, ( Y 1 , Y 2 )

Original: = (1 , 1) .
Revised: = (1 , 1) .

Original: We use uniform noises with range of 1, by subtracting them from the values (Figure 4).
Revised: We use uniform noises with range of 1, by subtracting them from the values (Figure 4).

Original: [Equation]
Revised: [Equation]

Original: We study a prediction problem with two inputs X 1 , X 2 and one output Y 1 .
Revised: We study a prediction problem with two inputs X 1 , X 2 and one output Y 1 .

Original: This problem has a property that X 1 contains entire information of Y 1 , but X 2 only contains part of information of Y 1 .
Revised: This problem has a property that X 1 contains entire information of Y 1 , but X 2 only contains part of information of Y 1 .

Original: Y 1 can be predicted from X 1 alone, and X 2 is redundant for Y 1 .
Revised: Y 1 can be predicted from X 1 alone, and X 2 is redundant for Y 1 .

Original: We want to train a model f with parameters θ .
Revised: We want to train a model f with parameters θ .

Original: f has X 1 , X 2 as input and ˆ Y as output: ˆ Y = f
Revised: f has X 1 , X 2 as input and ˆ Y as output: ˆ Y = f

Original: ( X 1 , X 2 ; θ ) .
Revised: ( X 1 , X 2 ; θ ) .

Original: This problem requires compositional generalization, and the model needs to have compositionality.
Revised: This problem requires compositional generalization, and the model needs to have compositionality.

Original: The model has three fully-connected hidden layers with ReLU activations, and each hidden layer has eight nodes.
Revised: The model has three fully-connected hidden layers with ReLU activations, and each hidden layer has eight nodes.

Original: More details can be found in Appendix C.
Revised: More details can be found in Appendix C.

Original: [Figure]
Revised: [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: Figure 5 shows the decision boundary for each training step.
Revised: Figure 5 shows the decision boundary for each training step.

Original: We see that the initialized parameters at step 0 output wrong predictions for samples in the upper left area, so that the samples in this area are useful to reduce loss, and they push the updated boundary in step 1 to the middle of upper right area.
Revised: We see that the initialized parameters at step 0 output wrong predictions for samples in the upper left area, so that the samples in this area are useful to reduce loss, and they push the updated boundary in step 1 to the middle of upper right area.

Original: In the following steps the loss is low and do not change much, so that the boundary remains stable.
Revised: In the following steps the loss is low and do not change much, so that the boundary remains stable.

Original: Therefore, the trained model does not have good compositionality.
Revised: Therefore, the trained model does not have good compositionality.

Original: Note that there can be different solutions for this problem.
Revised: Note that there can be different solutions for this problem.

Original: We use different random seeds and show that the trained model do not have compositionality (Figure 6).
Revised: We use different random seeds and show that the trained model do not have compositionality (Figure 6).

Original: [Figure]
Revised: [Figure]

Original: This visualization shows an example of process for how the gradient descent makes the model to be non-compositional.
Revised: This visualization shows an example of process for how the gradient descent makes the model to be non-compositional.

Original:  C ONCLUSIONS
Revised:  C ONCLUSIONS

Original: In this paper we investigate why standard neural network training seldom achieves compositional generalization by studying the relation between compositionality learning and gradient descent during training.
Revised: In this paper we investigate why standard neural network training seldom achieves compositional generalization by studying the relation between compositionality learning and gradient descent during training.

Original: We ﬁnd that the optimization process poses a bias towards non-compositional solutions, and this is caused by gradient descent.
Revised: We ﬁnd that the optimization process poses a bias towards non-compositional solutions, and this is caused by gradient descent.

Original: It tends to use all possible and redundant information from input, so that it violates conditional independence property of compositionality.
Revised: It tends to use all possible and redundant information from input, so that it violates conditional independence property of compositionality.

Original: Based on this study, we suggest that if only model structure design is considered in compositionality learning, it is hard to achieve good compositionality.
Revised: Based on this study, we suggest that if only model structure design is considered in compositionality learning, it is hard to achieve good compositionality.

Original: We hope this ﬁnding provides new understanding of compositional generalization mechanisms and helps to improve machine learning algorithms for higher level of  intelligence.
Revised: We hope this ﬁnding provides new understanding of compositional generalization mechanisms and helps to improve machine learning algorithms for higher level of artiﬁcial intelligence.

Original: A I MAGE C LASSIFICATION
Revised: A I MAGE C LASSIFICATION

Original: The model is a convolutional neural network with the following layers.
Revised: The model is a convolutional neural network with the following layers.

Original: The input is a gray scale image with width 56 (two times of the original width) and height 28.
Revised: The input is a gray scale image with width 56 (two times of the original width) and height 28.

Original: The ﬁrst hidden layer is a two dimensional convolutional layer with kernel size 3 × 3 , depth 32, and ReLU activation.
Revised: The ﬁrst hidden layer is a two dimensional convolutional layer with kernel size 3 × 3 , depth 32, and ReLU activation.

Original: The second hidden layer is a two dimensional max pooling layer with kernel size 2 × 2 .
Revised: The second hidden layer is a two dimensional max pooling layer with kernel size 2 × 2 .

Original: The third hidden layer is a two dimensional convolutional layer with kernel size 3 × 3 , depth 64 and ReLU activation.
Revised: The third hidden layer is a two dimensional convolutional layer with kernel size 3 × 3 , depth 64 and ReLU activation.

Original: Then the representation is ﬂatten to a one vector.
Revised: Then the representation is ﬂatten to a one vector.

Original: The fourth hidden layer is fully connected layer withnodes and ReLU activation.
Revised: The fourth hidden layer is fully connected layer withnodes and ReLU activation.

Original: The Fifth hidden layer is another fully connected layer with 64 nodes and ReLU activation.
Revised: The Fifth hidden layer is another fully connected layer with 64 nodes and ReLU activation.

Original: The output layer is a fully connected layer with 10 nodes and linear activation.
Revised: The output layer is a fully connected layer with 10 nodes and linear activation.

Original: We use mini-batch size of 64, and we run 200 steps in training.
Revised: We use mini-batch size of 64, and we run 200 steps in training.

Original: We use hinge loss, Adam optimizer (Kingma & Ba, 2014) with learning rate of 0.001.
Revised: We use hinge loss, Adam optimizer (Kingma & Ba, 2014) with learning rate of 0.001.

Original: We use TensorFlow (Abadi et al., 2015) for implementation.
Revised: We use TensorFlow (Abadi et al., 2015) for implementation.

Original: B L ANGUAGE L EARNING
Revised: B L ANGUAGE L EARNING

Original: We use a standard sequence to sequence architecture.
Revised: We use a standard sequence to sequence architecture.

Original: It has embedding layer, bidirectional LSTM encoder, and unidirectional LSTM with attention decoder.
Revised: It has embedding layer, bidirectional LSTM encoder, and unidirectional LSTM with attention decoder.

Original: The ﬁrst and last states of encoder are concatenated as initial state of decoder.
Revised: The ﬁrst and last states of encoder are concatenated as initial state of decoder.

Original: The embedding size is 32.
Revised: The embedding size is 32.

Original: The state size is 16 for encoder, and 64 for decoder.
Revised: The state size is 16 for encoder, and 64 for decoder.

Original: We run experiment on Jump task in SCAN dataset.
Revised: We run experiment on Jump task in SCAN dataset.

Original: The task contains 14, training and 7,706 test samples.
Revised: The task contains 14, training and 7,706 test samples.

Original: In the ﬁrst setting, we use the original dataset.
Revised: In the ﬁrst setting, we use the original dataset.

Original: In the second setting, we changed ’jump’ to other action word and corresponding output symbol uniformly at random.
Revised: In the second setting, we changed ’jump’ to other action word and corresponding output symbol uniformly at random.

Original: We use Adam for optimization, with cross entropy loss.
Revised: We use Adam for optimization, with cross entropy loss.

Original: We ran 2,000 training steps.
Revised: We ran 2,000 training steps.

Original: Each step has a mini-batch of 256 samples randomly and uniformly selected from training data with replacement.
Revised: Each step has a mini-batch of 256 samples randomly and uniformly selected from training data with replacement.

Original: Initial learning rate is 0.01 and it exponentially decays by a factor of 0.96 every 100 steps.
Revised: Initial learning rate is 0.01 and it exponentially decays by a factor of 0.96 every 100 steps.

Original: We use
Revised: We use

Original: TensorFlow for implementation.
Revised: TensorFlow for implementation.

Original: C V ISUALIZATION
Revised: C V ISUALIZATION

Original: The model is a fully-connected neural network, with two input nodes and two output nodes.
Revised: The model is a fully-connected neural network, with two input nodes and two output nodes.

Original: It has three hidden layers with ReLU activations, and each hidden layer has eight nodes.
Revised: It has three hidden layers with ReLU activations, and each hidden layer has eight nodes.

Original: We use mini-batch size of 10, and we run 1024 steps in training with learning rate 0.1.
Revised: We use mini-batch size of 10, and we run 1024 steps in training with learning rate 0.1.

Original: Please see the original work (Smilkov & Carter, 2016) for more information.
Revised: Please see the original work (Smilkov & Carter, 2016) for more information.