A first-order method for nonconvex-nonconcave minimax problems under a local Kurdyka-Łojasiewicz condition
This work addresses a theoretical bottleneck in optimization for machine learning, offering a method for broader practical scenarios, though it appears incremental as it builds on existing proximal gradient techniques.
The authors tackled nonconvex-nonconcave minimax problems by assuming a local Kurdyka-Łojasiewicz condition, which is less restrictive than global assumptions, and developed an inexact proximal gradient method with complexity guarantees for computing approximate stationary points.
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.