What is a Good Metric to Study Generalization of Minimax Learners?

TL;DR

This paper introduces the primal gap as a new metric for evaluating the generalization of minimax learners, addressing limitations of primal risk, and provides theoretical bounds and comparisons of algorithms in nonconvex-concave settings.

Contribution

The paper proposes the primal gap as a novel metric for minimax generalization, deriving error bounds and comparing GDA and GDMax algorithms without strong concavity assumptions.

Findings

Primal gap effectively measures generalization in minimax problems.

Derived error bounds for primal gap in nonconvex-concave settings.

Compared GDA and GDMax algorithms using the new metric.

Abstract

Minimax optimization has served as the backbone of many machine learning (ML) problems. Although the convergence behavior of optimization algorithms has been extensively studied in the minimax settings, their generalization guarantees in stochastic minimax optimization problems, i.e., how the solution trained on empirical data performs on unseen testing data, have been relatively underexplored. 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, which has also been adopted recently to study generalization in minimax ones, fails in simple examples. We thus propose a new metric to study generalization of minimax learners: the primal gap, defined as the difference between the primal risk and its minimum over all models, to circumvent the issues. Next, we derive generalization error bounds for the primal gap in nonconvex-concave settings. As byproducts of our analysis, we also solve two open questions: establishing generalization error bounds for primal risk and primal-dual risk, another existing metric that is only well-defined when the global saddle-point exists, in the strong sense, i.e., without strong concavity or assuming that the maximization and expectation can be interchanged, while either of these assumptions was needed in the literature. 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 stochastic minimax optimization.