Domain Adversarial Training: A Game Perspective

1 INTRODUCTION . Unsupervised domain adaptation ( UDA ) deals with the lack of labeled data in a target domain by transferring knowledge from a labeled source domain ( i.e. , a related dataset with different distribution where abundant labeled data already exists ) . The paramount importance of this paradigm has led to remarkable advances in the field in terms of both theory and algorithms ( Ben-David et al. , 2007 ; 2010a ; b ; Mansour et al. , 2009 ) . Several state-of-the-art algorithms tackle UDA by learning domaininvariant representations in an adversarial fashion ( Shu et al. , 2018 ; Long et al. , 2018 ; Saito et al. , 2018 ; Hoffman et al. , 2018 ; Zhang et al. , 2019 ; Acuna et al. , 2021 ) . Their goal is to fool an auxiliary classifier that operates in a representation space and aims to classify whether the datapoint belongs to either the source or the target domain . This idea , called Domain-Adversarial Learning ( DAL ) , was introduced by Ganin et al . ( 2016 ) and can be more formally understood as minimizing the discrepancy between source and target domain in a representation space ( Acuna et al. , 2021 ) . Despite DAL being a dominant approach for UDA , alternative solutions have been sought as DAL is noticeably unstable and difficult to train in practice ( Sener et al. , 2016 ; Sun et al. , 2019 ; Chang et al. , 2019 ) . One major cause of instability is the adversarial nature of the learning algorithm which results from the introduction of the Gradient Reversal Layer ( GRL , Ganin et al. , 2016 ) ( Figure 1 ) . GRL flips the sign of the gradient during the backward pass , which has profound implications on the training dynamics and asymptotic behavior of the learning algorithm . Indeed , GRL transforms gradient descent into a competitive gradient-based algorithm which may converge to periodic orbits and other non-trivial limiting behavior that arise for instance in chaotic systems ( Mazumdar et al. , 2020 ) . Surprisingly , little attention has been paid to this fact , and specifically to the adversarial component and interaction among the three different networks in the algorithm . In particular , three fundamental questions have not been answered from an algorithmic point of view , 1 ) What is optimality in DAL ? 2 ) What makes DAL difficult to train and 3 ) How can we mitigate this problem ? In this work , we aim to answer these questions by interpreting the DAL framework through the lens of game theory . Specifically , we use tools developed by the game theoretical community in Başar & Olsder ( 1998 ) ; Letcher et al . ( 2019 ) ; Mazumdar et al . ( 2020 ) and draw inspiration from the existing two-player zero-sum game interpretations of Generative Adversarial Networks ( GANs ) ( Goodfellow et al. , 2014 ) . We emphasize that in DAL , however , we have three rather than two networks interacting with each other , with partial cooperation and competition . We propose a natural three-player game interpretation for DAL , which is not necessarily equivalent to two-player zero-sum game interpretations ( see Example 1 ) , which we coin as the Domain-Adversarial Game . We also propose to interpret and characterize optimal solutions in DAL as local Nash Equilibria ( see Section 3 ) . This characterization introduces a proper mathematical definition of algorithmic optimality for DAL . It also provides sufficient conditions for optimality that drives the algorithmic analysis . With our proposed game perspective in mind , a simple optimization solution would be to use the Gradient Descent ( GD ) algorithm , which is the de facto solution but known to be unstable . Alternatively , we could also use other popular gradient based optimizers proposed in the context of differentiable games ( e.g . Korpelevich , 1976 ; Mescheder et al. , 2017 ) . However , we notice that these do not outperform GD in practice ( see § 6 ) . To understand why , we analyze the asymptotic behavior of gradient-based algorithms in the proposed domain-adversarial game ( § 4 ) . The main result of § 4.2 ( Theorem 2 ) shows that GD with GRL ( i.e. , the existing solution for DAL ) violates the asymptotic convergence guarantees to local NE unless an upper bound is placed on the learning rate , which may explain its training instability and sensitivity to optimizer parameters . In § 4.3 , Appendix B.2 and Appendix E , we also provide a similar analysis for the popular game optimization algorithms mentioned above . We emphasize however that while some of our results may be of independent interest for learning in general games , our focus is DAL . § 4.3 and § 6 show both theoretically and experimentally that the limitations mentioned above disappear if standard optimizers are replaced with ODE solvers of at least second order . These are straightforward to implement as drop-in replacements to existing optimizers . They also lead to more stable algorithms , allow for more aggressive learning rates and provide notable performance gains . 2 PRELIMINARIES < latexit sha1_base64= '' XcLU0OQlzQn4TD3DERhz5ZsAZ/U= '' > AAAB8nicbVBNS8NAFHypX7V+VT16WSyCp5KIoseiF48VbC2moWy223bpJht2X4QS+jO8eFDEq7/Gm//GTZuDtg4sDDPvsfMmTKQw6LrfTmlldW19o7xZ2dre2d2r7h+0jUo14y2mpNKdkBouRcxbKFDyTqI5jULJH8LxTe4/PHFthIrvcZLwIKLDWAwEo2glvxtRHDEqs8dpr1pz6+4MZJl4BalBgWav+tXtK5ZGPEYmqTG+5yYYZFSjYJJPK93U8ISyMR1y39KYRtwE2SzylJxYpU8GStsXI5mpvzcyGhkziUI7mUc0i14u/uf5KQ6ugkzESYo8ZvOPBqkkqEh+P+kLzRnKiSWUaWGzEjaimjK0LVVsCd7iycukfVb3Luru3XmtcV3UUYYjOIZT8OASGnALTWgBAwXP8ApvDjovzrvzMR8tOcXOIfyB8/kDl52RdA== < /latexit > Z < latexit sha1_base64= '' 3xtdy5laA/WQWHLvq4EtMaredYI= '' > AAAB7XicbVBNS8NAEJ34WetX1aOXxSJ4Kokoeix68VjBfkAbyma7adZuNmF3IpTQ/+DFgyJe/T/e/Ddu2xy09cHA470ZZuYFqRQGXffbWVldW9/YLG2Vt3d29/YrB4ctk2Sa8SZLZKI7ATVcCsWbKFDyTqo5jQPJ28Hoduq3n7g2IlEPOE65H9OhEqFgFK3U6kUUSdSvVN2aOwNZJl5BqlCg0a989QYJy2KukElqTNdzU/RzqlEwySflXmZ4StmIDnnXUkVjbvx8du2EnFplQMJE21JIZurviZzGxozjwHbGFCOz6E3F/7xuhuG1nwuVZsgVmy8KM0kwIdPXyUBozlCOLaFMC3srYRHVlKENqGxD8BZfXiat85p3WXPvL6r1myKOEhzDCZyBB1dQhztoQBMYPMIzvMKbkzgvzrvzMW9dcYqZI/gD5/MHKgqO2w== < /latexit > ĥ < latexit sha1_base64= '' 4ZzWvs7xxQO9ik+Ene1mXfvAGHA= '' > AAAB73icbVDLSgNBEOyNrxhfUY9eBoPoKeyKosegF48RzAOSJcxOZrNDZmfXmV4hLPkJLx4U8ervePNvnDwOmljQUFR1090VpFIYdN1vp7Cyura+UdwsbW3v7O6V9w+aJsk04w2WyES3A2q4FIo3UKDk7VRzGgeSt4Lh7cRvPXFtRKIecJRyP6YDJULBKFqp3Y0okoic9soVt+pOQZaJNycVmKPeK391+wnLYq6QSWpMx3NT9HOqUTDJx6VuZnhK2ZAOeMdSRWNu/Hx675icWKVPwkTbUkim6u+JnMbGjOLAdsYUI7PoTcT/vE6G4bWfC5VmyBWbLQozSTAhk+dJX2jOUI4soUwLeythEdWUoY2oZEPwFl9eJs3zqndZde8vKrWbeRxFOIJjOAMPrqAGd1CHBjCQ8Ayv8OY8Oi/Ou/Mxay0485lD+APn8wfj5o82 < /latexit > ĥ0 < latexit sha1_base64= '' 2G0f8RUU8YWS9zeN9PTsk8ECOOY= '' > AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqsyIosuiG5dV7EPaoWQymTY0yQxJRihDv8KNC0Xc+jnu/BvT6Sy09UDgcM655N4TJJxp47rfTmlldW19o7xZ2dre2d2r7h+0dZwqQlsk5rHqBlhTziRtGWY47SaKYhFw2gnGNzO/80SVZrF8MJOE+gIPJYsYwcZKj/eDPrfhEA+qNbfu5kDLxCtIDQo0B9WvfhiTVFBpCMda9zw3MX6GlWGE02mln2qaYDLGQ9qzVGJBtZ/lC0/RiVVCFMXKPmlQrv6eyLDQeiICmxTYjPSiNxP/83qpia78jMkkNVSS+UdRypGJ0ex6FDJFieETSzBRzO6KyAgrTIztqGJL8BZPXibts7p3UXfvzmuN66KOMhzBMZyCB5fQgFtoQgsICHiGV3hzlPPivDsf82jJKWYO4Q+czx+ZdZBG < /latexit > R ( GRL ) < latexit sha1_base64= '' 1Out49/IGg3BAWnSOWbUFY4P5mo= '' > AAAB+HicdVDLSgMxFM3UV62Pjrp0EyyCG8tMHdu6K7pxWcE+oFPKnTRtQzOZIckItfRL3LhQxK2f4s6/MdNWUNEDgcM553JvThBzprTjfFiZldW19Y3sZm5re2c3b+/tN1WUSEIbJOKRbAegKGeCNjTTnLZjSSEMOG0F46vUb91RqVgkbvUkpt0QhoINGAFtpJ6dP/W5SfcB+wICDj274BQvquWSV8ZO0XEqbslNSaninXnYNUqKAlqi3rPf/X5EkpAKTTgo1XGdWHenIDUjnM5yfqJoDGQMQ9oxVEBIVXc6P3yGj43Sx4NImic0nqvfJ6YQKjUJA5MMQY/Uby8V//I6iR5Uu1Mm4kRTQRaLBgnHOsJpC7jPJCWaTwwBIpm5FZMRSCDadJUzJXz9FP9PmqWie150brxC7XJZRxYdoiN0glxUQTV0jeqogQhK0AN6Qs/WvfVovVivi2jGWs4coB+w3j4BX0GS6w== < /latexit > r < latexit sha1_base64= '' 4e7VN7IWD/ZM+123qYV7uoLuXJs= '' > AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hV1R9Bj04jGCeUCyhN7JbDJmdmaZmRVCyD948aCIV//Hm3/jJNmDJhY0FFXddHdFqeDG+v63t7K6tr6xWdgqbu/s7u2XDg4bRmWasjpVQulWhIYJLlndcitYK9UMk0iwZjS8nfrNJ6YNV/LBjlIWJtiXPOYUrZMaHYmRwG6p7Ff8GcgyCXJShhy1bumr01M0S5i0VKAx7cBPbThGbTkVbFLsZIalSIfYZ21HJSbMhOPZtRNy6pQeiZV2JS2Zqb8nxpgYM0oi15mgHZhFbyr+57UzG1+HYy7TzDJJ54viTBCryPR10uOaUStGjiDV3N1K6AA1UusCKroQgsWXl0njvBJcVvz7i3L1Jo+jAMdwAmcQwBVU4Q5qUAcKj/AMr/DmKe/Fe/c+5q0rXj5zBH/gff4AgK6PFA== < /latexit > r < latexit sha1_base64= '' 4e7VN7IWD/ZM+123qYV7uoLuXJs= '' > AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hV1R9Bj04jGCeUCyhN7JbDJmdmaZmRVCyD948aCIV//Hm3/jJNmDJhY0FFXddHdFqeDG+v63t7K6tr6xWdgqbu/s7u2XDg4bRmWasjpVQulWhIYJLlndcitYK9UMk0iwZjS8nfrNJ6YNV/LBjlIWJtiXPOYUrZMaHYmRwG6p7Ff8GcgyCXJShhy1bumr01M0S5i0VKAx7cBPbThGbTkVbFLsZIalSIfYZ21HJSbMhOPZtRNy6pQeiZV2JS2Zqb8nxpgYM0oi15mgHZhFbyr+57UzG1+HYy7TzDJJ54viTBCryPR10uOaUStGjiDV3N1K6AA1UusCKroQgsWXl0njvBJcVvz7i3L1Jo+jAMdwAmcQwBVU4Q5qUAcKj/AMr/DmKe/Fe/c+5q0rXj5zBH/gff4AgK6PFA== < /latexit > r < latexit sha1_base64= '' qmqxvw54qUUoUaz9MnxwYmJpOwM= '' > AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0WPRi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUGPbLFbfqzkFWiZeTCuSo98tfvUHM0gilYYJq3fXcxPgZVYYzgdNSL9WYUDamQ+xaKmmE2s/mh07JmVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NeGNn3GZpAYlWywKU0FMTGZfkwFXyIyYWEKZ4vZWwkZUUWZsNiUbgrf88ippXVS9q6rbuKzUbvM4inACp3AOHlxDDe6hDk1ggPAMr/DmPDovzrvzsWgtOPnMMfyB8/kDzceM7w== < /latexit > g Figure 1 : We study domainadversarial training from a game perspective . In DAL ( Ganin et al . ( 2016 ) ) , three networks interact with each other : the feature extractor ( g ) , the domain classifier ( ĥ′ ) and the classifier ( ĥ ) . During backpropagation , the GRL flips the sign of the gradient with respect to g. We focus on the UDA scenario and follow the formulation from Acuna et al . ( 2021 ) . This makes our analysis general and applicable to most state-of-the-art DAL algorithms ( e.g. , Ganin et al . ( 2016 ) ; Saito et al . ( 2018 ) ; Zhang et al . ( 2019 ) ) . We assume that the learner has access to a source dataset ( S ) with labeled examples and a target dataset ( T ) with unlabeled examples , where the source inputs xsi are sampled i.i.d . from a ( source ) distribution Ps and the target inputs xti are sampled i.i.d . from a ( target ) distribution Pt , both over X . We have Y = { 0 , 1 } for binary classification , and Y = { 1 , ... , k } in the multiclass case . The risk of a hypothesis h : X → Y w.r.t . the labeling function f , using a loss function ` : Y×Y → R+ under distribution D is defined as : R ` D ( h , f ) : = ED [ ` ( h ( x ) , f ( x ) ) ] . For simplicity , we define R ` S ( h ) : = R ` Ps ( h , fs ) and R ` T ( h ) : = R ` Pt ( h , ft ) . The hypothesis class of h is denoted byH . UDA aims to minimize the risk in the target domain while only having access to labeled data in the source domain . This risk is upper bounded in terms of the risk of the source domain , the discrepancy between the two distributions and the joint hypothesis error λ∗ : Theorem 1 . ( Acuna et al . ( 2021 ) ) Let us note ` : Y × Y → [ 0 , 1 ] , λ∗ : = minh∈HR ` S ( h ) +R ` T ( h ) , and Dφh , H ( Ps||Pt ) : = suph′∈H |Ex∼Ps [ ` ( h ( x ) , h′ ( x ) ) ] − Ex∼Pt [ φ∗ ( ` ( h ( x ) , h′ ( x ) ) ) | . We have : R ` T ( h ) ≤ R ` S ( h ) + Dφh , H ( Ps||Pt ) + λ∗ . ( 1 ) The function φ : R+ → R defines a particular f -divergence and φ∗ is its ( Fenchel ) conjugate . As is typical in UDA , we assume that the hypothesis class is complex enough and both fs and ft are similar in such a way that the non-estimable term ( λ∗ ) is negligible and can be ignored . Domain-Adversarial Training ( see Figure 1 ) aims to find a hypothesis h ∈ H that jointly minimizes the first two terms of Theorem 1 . To this end , the hypothesis h is interpreted as the composition of h = ĥ ◦ g with g : X → Z , and ĥ : Z → Y . Another function class Ĥ is then defined to formulate H : = { ĥ ◦ g : ĥ ∈ Ĥ , g ∈ G } . The algorithm tries to find the function g ∈ G such that ĥ ◦ g minimizes the risk of the source domain ( i.e . the first term in Theorem 1 ) , and its composition with ĥ and ĥ′ minimizes the divergence of the two distributions ( i.e . the second term in Theorem 1 ) . Algorithmically , the computation of the divergence function in Theorem 1 is estimated by a so-called domain classifier ĥ′ ∈ Ĥ whose role is to detect whether the datapoint g ( xi ) ∈ Z belongs to the source or to the target domain . When there does not exist a function ĥ′ ∈ Ĥ that can properly distinguish between g ( xsi ) and g ( x t i ) , g is said to be invariant to the domains . Learning is performed using GD and the GRL ( denoted by Rλ ) on the following objective : min ĥ∈Ĥ , g∈G , ĥ′∈Ĥ Ex∼ps [ ` ( ĥ ◦ g , y ) ] − αds , t ( ĥ , ĥ′ , Rλ ( g ) ) , ( 2 ) where ds , t ( ĥ , ĥ′ , g ) : = Ex∼ps [ ˆ̀ ( ĥ′ ◦ g , ĥ ◦ g ) ] − Ex∼pt [ ( φ∗ ◦ ˆ̀ ) ( ĥ′ ◦ g , ĥ ◦ g ) ] . Mathematically , the GRL Rλ is treated as a “ pseudo-function ” defined by two ( incompatible ) equations describing its forward and back-propagation behavior ( Ganin & Lempitsky , 2015 ; Ganin et al. , 2016 ) . Specifically , Rλ ( x ) : = x and dRλ ( x ) /dx : = −λ , ( 3 ) where λ and α are hyper-parameters that control the tradeoff between achieving small source error and learning an invariant representation . The surrogate loss ` : Y × Y → R ( e.g. , cross-entropy ) is used to minimize the empirical risk in the source domain . The choice of function ˆ̀ : Y × Y → R and of conjugate φ∗ of the f -divergence defines the particular algorithm ( Ganin et al. , 2016 ; Saito et al. , 2018 ; Zhang et al. , 2019 ; Acuna et al. , 2021 ) . From eq . 2 , we can notice that GRL introduces an adversarial scheme . We next interpret eq . 2 as a three-player game where the players are ĥ , ĥ′ and g , and study its continuous gradient dynamics .

Summary: The setting in the paper is the classic unsupervised domain adaptation problem, where we are given a labeled sample from a source distribution and an unlabeled sample from a target distribution. The goal is to minimize the risk on the target distribution. Theoretical results led to a breakthrough in practice - the Domain Adversarial Learning architecture (Ganin et al., 2016).   The paper suggests looking at the paper from a game theory perspective. This is natural, as the objective is to minimize the loss on the source distribution while maximizing the distinction between the distributions.  The optimal solutions of the game are characterized by the local NE.  Motivated by the results from game theory, the authors suggest replacing Gradient Descent (due to its limitation in this optimization problem) with other optimizers - ODE (ordinary differential equation) solvers.