标题： 一种在局部Kurdyka-Łojasiewicz条件下用于非凸非凹极小极大问题的一阶方法 显示英文标题

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

摘要： 我们研究一类非凸-非凹的极小极大问题，其中内部最大化问题满足一个局部Kurdyka-{\L }ojasiewicz (KL) 条件，该条件可能随着外部最小化变量而变化。 与文献中通常假设的全局KL或Polyak-{\L }ojasiewicz (PL) 条件不同——这些条件要强得多，并且在实践中往往过于严格——这种局部KL条件可以适应更广泛的实际情况。 然而，它也带来了新的分析挑战。 特别是，当优化算法向问题的平稳点推进时，KL条件成立的区域可能会缩小，从而导致更加复杂且可能病态的景观。 为了解决这一挑战，我们证明了相关最大函数是局部Hölder光滑的。 利用这一关键性质，我们开发了一种求解极小极大问题的不精确邻近梯度方法，其中最大函数的不精确梯度是通过对一个KL结构的子问题应用邻近梯度方法来计算的。 在较弱的假设下，我们建立了计算极小极大问题近似平稳点的复杂度保证。 显示英文摘要

摘要： We study a class of nonconvex-nonconcave minimax problems in which the inner maximization problem satisfies a local Kurdyka-{\L}ojasiewicz (KL) condition that may vary with the outer minimization variable. In contrast to the global KL or Polyak-{\L}ojasiewicz (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 H\"older 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.