Minimax optimization has served as the backbone of many machine learning problems. Although the convergence behavior of optimization algorithms has been extensively studied in minimax settings, their generalization guarantees, i.e., how the model trained on empirical data performs on the unseen testing data, have been relatively under-explored. A fundamental question remains elusive: What is a good metric to study generalization of minimax learners? In this paper, we aim to answer this question by first showing that primal risk, a universal metric to study generalization in minimization problems, fails in simple examples of minimax problems. Furthermore, another popular metric, the primal-dual risk, also fails to characterize the generalization behavior for minimax problems with nonconvexity, due to non-existence of saddle points. We thus propose a new metric to study generalization of minimax learners: the primal gap, to circumvent these issues. Next, we derive generalization bounds for the primal gap in nonconvex-concave settings. As byproducts of our analysis, we also solve two open questions: establishing generalization bounds for primal risk and primal-dual risk in this setting, and in the strong sense, i.e., without assuming that the maximization and expectation can be interchanged. Finally, we leverage this new metric to compare the generalization behavior of two popular algorithms - gradient descent-ascent (GDA) and gradient descent-max (GDMax) in minimax optimization.

The idea of studying GANs from the game theory perspective is not new; however, using the duality gap as a performance metric (some sort of divergence between the generated data distribution and the real data distribution) is original to the best of my knowledge. The paper is written clearly. In terms of significance, while the idea of the duality gap is "natural" when considering the game theory perspective for GANs, it is not clear why this is a good metric for _any_ domain. The authors imply that it is a good idea to find a metric that does not depend on the domain of the data, but given all the parallels between GANs and the different divergences between probability distributions (JS, Wasserstein, etc.) I think the main problem is to find a metric that can be thought as correctly modeling the distance between high-dimensional datasets such as the ones given by images. In that case, modeling this aspect (which is highly domain-dependent) is crucial for understanding what a GAN is capturing about the data distribution.

Minimax optimization has served as the backbone of many machine learning problems. Although the convergence behavior of optimization algorithms has been extensively studied in minimax settings, their generalization guarantees, i.e., how the model trained on empirical data performs on the unseen testing data, have been relatively under-explored. A fundamental question remains elusive: What is a good metric to study generalization of minimax learners? In this paper, we aim to answer this question by first showing that primal risk, a universal metric to study generalization in minimization problems, fails in simple examples of minimax problems. Furthermore, another popular metric, the primal-dual risk, also fails to characterize the generalization behavior for minimax problems with nonconvexity, due to non-existence of saddle points. We thus propose a new metric to study generalization of minimax learners: the primal gap, to circumvent these issues.

Many algorithms rely critically on being given a good metric over their inputs. For instance, data can often be clustered in many "plausible" ways, and if a clustering algorithm such as K-means initially fails to find one that is meaningful to a user, the only recourse may be for the user to manually tweak the metric until sufficiently good clusters are found. For these and other applications requiring good metrics, it is desirable that we provide a more systematic way for users to indicate what they con- sider "similar." For instance, we may ask them to provide examples. In this paper, we present an algorithm that, given examples of similar (and,, learns a distance metric over if desired, dissimilar) pairs of points in that respects these relationships.

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Many algorithms rely critically on being given a good metric over their inputs. For instance, data can often be clustered in many "plausible" ways, and if a clustering algorithm such as K-means initially fails to find one that is meaningful to a user, the only recourse may be for the user to manually tweak the metric until sufficiently good clusters are found. For these and other applications requiring good metrics, it is desirable that we provide a more systematic way for users to indicate what they consider "similar." For instance, we may ask them to provide examples.

Many algorithms rely critically on being given a good metric over their inputs. For instance, data can often be clustered in many "plausible" ways, and if a clustering algorithm such as K-means initially fails to find one that is meaningful to a user, the only recourse may be for the user to manually tweak the metric until sufficiently good clusters are found. For these and other applications requiring good metrics, it is desirable that we provide a more systematic way for users to indicate what they consider "similar." For instance, we may ask them to provide examples.

Many algorithms rely critically on being given a good metric over their inputs. For instance, data can often be clustered in many "plausible" ways, and if a clustering algorithm such as K-means initially fails to find one that is meaningful to a user, the only recourse may be for the user to manually tweak the metric until sufficiently good clusters are found. For these and other applications requiring good metrics, it is desirable that we provide a more systematic way for users to indicate what they consider "similar."For