Nonconvex minimax problems appear frequently in emerging machine learning applications, such as generative adversarial networks and adversarial learning. Simple algorithms such as the gradient descent ascent (GDA) are the common practice for solving these nonconvex games and receive lots of empirical success. Yet, it is known that these vanilla GDA algorithms with constant stepsize can potentially diverge even in the convex setting. In this work, we show that for a subclass of nonconvex-nonconcave objectives satisfying a so-called two-sided Polyak-{\L}ojasiewicz inequality, the alternating gradient descent ascent (AGDA) algorithm converges globally at a linear rate and the stochastic AGDA achieves a sublinear rate. We further develop a variance reduced algorithm that attains a provably faster rate than AGDA when the problem has the finite-sum structure.

Most existing algorithms rely on one-sided information, such as the convexity (resp.

We study a class of constrained nonconvex--nonconcave minimax problems in which the inner maximization involves potentially complex constraints. Under the assumption that the inner problem of a novel lifted minimax problem satisfies a local Kurdyka-Łojasiewicz (KL) condition, we show that the maximal function of the original problem enjoys a local Hölder smoothness property. We also propose a sequential convex programming (SCP) method for solving constrained optimization problems and establish its convergence rate under a local KL condition. Leveraging these results, we develop an inexact proximal gradient method for the original minimax problem, where the inexact gradient of the maximal function is computed via the SCP method applied to a locally KL-structured subproblem. Finally, we establish complexity guarantees for the proposed method in computing an approximate stationary point of the original minimax problem.

We study a class of nonconvex-nonconcave minimax problems in which the inner maximization problem satisfies a local Kurdyka-Łojasiewicz (KL) condition that may vary with the outer minimization variable. In contrast to the global KL or Polyak-Łojasiewicz (PL) conditions commonly assumed in the literature -- which are significantly stronger and often too restrictive in practice -- this local KL condition accommodates a broader range of practical scenarios. However, it also introduces new analytical challenges. In particular, as an optimization algorithm progresses toward a stationary point of the problem, the region over which the KL condition holds may shrink, resulting in a more intricate and potentially ill-conditioned landscape. To address this challenge, we show that the associated maximal function is locally Hölder smooth. Leveraging this key property, we develop an inexact proximal gradient method for solving the minimax problem, where the inexact gradient of the maximal function is computed by applying a proximal gradient method to a KL-structured subproblem. Under mild assumptions, we establish complexity guarantees for computing an approximate stationary point of the minimax problem.

Weaknesses: Though there are some merits of the paper, here are a bunch of major problems of the submission: 1. PL condition is a very strong assumption. Although it does not require convexity-concavity, it is a global condition, which roughly requires similar properties of strong convexity-concavity. I agree there are some applications of min-max problems under PL condition, as mentioned in the paper, but the applications are extremely limited, and I am not sure they are important applications to ML community. In general, nonconvex-nonconcave min-max problems won't satisfy PL condition. To this extend, the title of the paper is a bit misleading, and it should mention PL condition explicitly.

This paper studies AGDA/Stoc-AGDA for minimax problems that may not be nonconvex-nonconcave but obey the two-sides Polyak-Łojasiewicz (PL), Moreover, this paper proposes a variance reduction version of AGDA and achieves better complexity results. The reviewers thought the problem setting was interesting and relevant to Neurips but also had a variety of concerns. These concerns were partially mitigated based on the response but other concerns remained. The reviewers had a spirited and comprehensive technical discussion about the merits of this paper. Two reviewers raised their score R4 - 4-5 and R2 4- 7 while one reviewer slightly lowered their score 8- 7. Based on the reviews, response, discussion and my own reading the main pros and cons of this paper are as follows.