A first-order method for nonconvex-nonconcave minimax problems under a local Kurdyka-Łojasiewicz condition

2025-07-02

AI Summary

This work studies minimax optimization problems with a nonconvex outer objective and a nonconcave inner maximization, where the inner problem satisfies a *local* Kurdyka–Łojasiewicz (KL) condition—parameterized by the outer variable—that is weaker and more practically relevant than standard global KL or Polyak–Łojasiewicz assumptions. To address analytical challenges arising from the iteration-dependent shrinking of the local KL region, we establish, for the first time, that the inner maximum value function is locally Hölder continuously differentiable. Leveraging this property, we propose a novel algorithm based on inexact proximal gradient descent, wherein each gradient approximation is obtained by solving a KL-structured subproblem. Under mild assumptions, the algorithm converges to an approximate stationary point with an explicit computational complexity bound. This constitutes the first optimization framework for nonconvex–nonconcave games under local KL geometry that is both theoretically rigorous and computationally implementable.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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.

Problem

• Solving nonconvex-nonconcave minimax problems with local KL conditions
• Addressing shrinking KL regions during optimization progression
• Developing an inexact proximal gradient method for minimax problems

Innovation

• Local KL condition for broader applicability
• Inexact proximal gradient method for minimax
• Hölder smoothness ensures complexity guarantees