Stablizing Adversarial Invariance Induction by Discriminator Matching

1 INTRODUCTION . Extensive studies have demonstrated that deep neural networks ( DNNs ) can uncover complicated variations in data to provide powerful representations that are useful for classification tasks ( Hinton et al. , 2006 ; Krizhevsky et al. , 2012 ) . However , in some scenarios , the learned representation should be invariant to some attribute of the input data . A motivating example is a task called domain generalization ( Blanchard et al. , 2011 ) , which requires learning a domain-invariant representation that applies to unseen domains ( e.g. , the data of an unseen user or different image sources ) . When practitioners apply DNNs to data that include a large amount of user information ( such as images with usernames ( Edwards & Storkey , 2016 ) or data from wearables ( Iwasawa et al. , 2017 ) ) , the desired representations should not include user identifying information . For legal and ethical reasons , machine learning algorithms must make fair decisions that are independent of sensitive variables such as gender , age , or race ( Louizos et al. , 2016 ) . Therefore , this study aims to answer the following question : how can we systematically incorporate the desired invariance into representation learning ? Invariance induction is a systematic solution to this problem , which often introduces an additional regularization term that measures the level of invariance ( Edwards & Storkey , 2016 ; Iwasawa et al. , 2017 ; Xie et al. , 2017 ) . One theoretically sound metric is the conditional entropy of attributes given representations of data H ( a|z ) = Ep ( z , a ) [ − log p ( a|z ) ] , where a and z denote the random variables of attributes and representations , as it is maximized if and only if the representations are invariant to attributes . However , exact calculation is intractable as p ( a|z ) is unknown in general . Recently , Xie et al . ( 2017 ) introduced a sensible way to approximate the conditional entropy ( Figure 1- ( a ) ) , which uses probabilistic attribute classifier qφ , where φ represents the parameter of the classifier , to approximate the conditional probability distribution p ( a|z ) . One can estimate the desired conditional entropy by simply substituting p ( a|z ) to the approximated conditional distribution qφ ( a|z ) . The estimated conditional entropy is then used to update the weights of the encoder fθ so that the updated representations increase the approximated conditional entropy . Since fθ and qφ is an adversarial relationship , this framework is called adversarial invariance induction ( AII ) . A similar approach was extensively used in domain generalization , fair-prediction , and privacy-protection contexts ( Edwards & Storkey , 2016 ; Motiian et al. , 2017 ; Xie et al. , 2017 ; Iwasawa et al. , 2017 ) . Build upon the above achievements , this paper examines a way to improve the stability of the adversarial invariance induction framework . Note that , Xie et al . ( 2017 ) already demonstrated that the above procedure ( alternatively optimizing attribute classifier and feature extractor ) possesses an equilibrium where the encoder maximizes the true conditional entropy . However , their analysis impractically assumes qφ ( a|z ) = p ( a|z ) , so little is known about the practical behavior of AII and how to improve it . Indeed , several prior works report unsuccessful results of AII ( Xie et al. , 2017 ; Moyer et al. , 2018 ) . To further examine the practical behavior of AII , we first analyze the behavior of AII from the lens of the variational approximation . Our analysis shows that AII maximizes the variational upper bound of the conditional entropy H ( a|z ) , and AII objective itself is not upperbounded . This result suggests that the optimization of AII , without the assumption of the optimality of qφ ( a|z ) , does not need to maximize the conditional entropy , as the maximizing upper bound does not give any guarantees . Empirical validations show that AII catastrophically fails to induce invariance even in satisfactory simple situations , as suggested by the above finding . We then argue that the simple modification to AII attains better property from the optimization perspective while achieving the same goal asymptotically . Our modification is based on the property of the conditional entropy , i.e. , it is maximized if and only if the divergence between all pairs of marginal distributions over z between different attributes are minimized ( under the uniform assumption of p ( a ) ) . This property suggests that the invariance induction algorithm should also minimize the divergence , which is not considered on AII and possibly induce unstable behavior . In contrast , our proposed method , invariance induction by discriminator matching ( IIDM ) , explicitly considers the divergence minimization requirements by minimizing the proxy of the divergence between the marginals , which push the representations with different attributes are recognized similarly by the discriminator ( as shown in Figure 1- ( b ) ) . We discuss the relationship among our proxy , the divergence between the marginals , and non-saturating heuristics used in different but related community ( Goodfellow et al. , 2014 ) . The main contributions of this paper can be summarized as follows . ( 1 ) We highlight the practical issues of AII , which is a state-of-the-art framework for invariance induction . ( 2 ) We propose a modification to AII by explicitly consider the property of the maximum conditional entropy . Empirical validations on the toy dataset and four real-world datasets demonstrate that the proposed method provides superior performance and faster convergence at the level of invariance induction . For example , in the experiments on toy dataset , the proposed method converges to mostly the maximum conditional entropy , while the optimization of AII is catastrophically unstable . Gif visualizations available at https : //drive.google.com/open ? id= 1N2kuTCfwjQBGgv3dxTSD9oBklAenLj-J ( see Section 4.3 for more detail ) . 2 RELATED WORKS . The goal of the invariance induction is to learn attribute-invariant spaces given a training dataset made of tuples of { ( xn , yn , an ) } N n=1 ( supervised setting ) or pairs of { ( xn , yn ) } N n=1 ( referred to as unsupervised setting ( Jaiswal et al. , 2018 ) ) , where xn is an observation and yn is a target of xn , an ∈ A is a realization of a categorical random variable a . This setting arises in many application areas . For example , Iwasawa et al . ( 2017 ) considers the privacy-preserving activity recognition problem using the data of wearables , where one wants to learn user invariant representations while keeping the information about activity into the representations . In this case , the attribute a corresponds to some sensitive user information , and y is the activity label . In the context of fairness-aware decision making , one needs to ensure the algorithm to make fair decisions , which often done by statistical away the sensitive attributes a ( such as gender , age , or race ) from the representations ( Edwards & Storkey , 2016 ) . To build a classifier robust to the domain shifts ( called domain generalization Blanchard et al . ( 2011 ) ) , learning representations invariant to domain shifts is a popular approach ( the domain label is the attribute in this case ) ( Muandet et al. , 2013 ; Ghifary et al. , 2015 ) . Assume fθ is a function ( encoder ) that parameterized by a neural network , which maps observations x to representations z ∈ Z . Typically , the invariance induction algorithm use differentiable dependency measurements and use it as a regularization term to learn fθ : E [ L ( fθ ( xn ) , yn ) + λV ( fθ ( xn ) , an ) ] , where a λ is the weighting parameter , V somehow measures invariance of the representations regarding the attribute , and L is a loss function that represents how much information about y is present in the representations . Adversarial invariance induction ( AII ) is a recently proposed approach for measuring the V by an external neural network . That is , if the external network can accurately predict a from z = fθ ( x ) , AII considers z to have considerable information about a . The external neural network is often called as a discriminator or adversary in this context . Information from the discriminator is used to update the weights of the encoder fθ so that the updated representations have less information about a . Mathematically , the method optimizes following min–max game : min θ , ψ max φ Ep ( x , a ) [ − log qψ ( yn|zn=fθ ( xn ) ) + λ log qφ ( an|zn=fθ ( xn ) ) ] , ( 1 ) where qψ ( y| . ) and qφ ( a| . ) is a conditional probability distribution approximated by a categorical classifier parameterized by ψ ( for y ) and φ ( for a ) respectively . By alternatingly ( or jointly with the gradient reversal layer ( Gan et al. , 2016 ) ) optimizing θ and φ , this framework ensures that there is little or no information about the nuisance attributes in the representations . As Xie et al . ( 2017 ) demonstrated , the above min–max game can be regarded as a way to estimate the conditional entropyH ( a|z ) and use the estimates to induce the invariance . The key advantage of AII is that this framework does not depend on the pre-defined metrics such as L2 distance and maximum mean discrepancy ( Zemel et al. , 2013 ; Li et al. , 2014 ; Louizos et al. , 2016 ) ) , which is often easy to compute yet less powerful . Formally , Xie et al . ( 2017 ) demonstrated that the above adversarial game has an equilibrium where the encoder maximizes the true conditional entropy , under the assumption that qφ ( a|z ) correctly estimates the true p ( a|z ) . However , their analysis assumes the optimality of qφ ( a|z ) , which is rarely held in practice . This paper analyzes the behavior of AII without such an assumption to examine the practical behavior of AII . It is noteworthy that the above formulation resembles the original formulation of GAN ( Goodfellow et al. , 2014 ) and domain adversarial networks ( DAN ) ( Ajakan et al. , 2014 ; Gan et al. , 2016 ) , which is never used practically . Instead , they often incorporate several heuristics , e.g. , non-saturating heuristics in GAN , and asymmetric mapping in domain adaptation ( Tzeng et al. , 2017 ) . These works motivate us to replace the min–max game of the adversarial invariance induction problem . Although no prior works in the invariance induction community have been explicitly considered yet , one can transfer the non-saturating heuristic used in the GAN via label flipping : min θ Epθ ( z , a ) − ∑ aj 6=an log qφ ( aj |zn = fθ ( xn ) ) , ( 2 ) which also enhances the discriminator to misclassify the attribute of the data . In the remainder of this paper , we refer to this version as the non-saturating version and denote it by NS . As discussed in the GAN community , this objective provides meaningful gradients because it not saturated even if the discriminator is a supremum . Later , we discuss the connection between the proposal and this objective , and how our proposal attains better properties while inheriting the non-saturating nature of the NS objective .

Summary: The paper points out that the practical behavior of AII assumes the optimality of the attribute classifier, which is rarely held in practice. And claims that the paper analyzes the practical behavior of AII both theoretically and empirically, indicating that AII has theoretical difficulty as it maximizes variational upper bound of the actual conditional entropy. Then it argues an ugly modification based on a wrong property of conditional entropy.