Strengthening the finite characterizations of smooth min-max games

TL;DR

This paper develops new interpolation conditions for smooth min-max games, improving the understanding of their properties and convergence, which aids in designing optimal algorithms for machine learning applications.

Contribution

It introduces necessary and sufficient interpolation conditions for smooth convex-concave functions and analyzes the convergence of algorithms like Alt-GDA using PEP-like techniques.

Findings

Derived tighter interpolation conditions for smooth min-max games

Analyzed linear convergence of Alt-GDA with new constraints

Provided improved complexity estimates for algorithms

Abstract

In this paper, we address the problem of interpolation of smooth convex-concave functions. Interpolation is a key step for computer-assisted estimation of worst-case performance via PEP-like techniques, and smooth convex-concave functions are frequently used to model min-max problems arising in machine learning. We address the challenges associated with deriving conditions that are necessary and sufficient for the interpolation of smooth min-max games and show how existing approaches can be adapted to this setting. As part of this effort, we study the smoothing properties of Moreau-Yosida-like approximations of convex-concave functions. Next, we derive interpolation conditions for several key special cases of smooth min-max games. Finally, we obtain improved (i.e., tighter) characterizations for smooth strongly monotone convex-concave functions. We analyze the linear convergence of Alt-GDA using a PEP-like technique with novel constraints and (numerically) obtain a new estimate of its complexity. We are confident that the results of this paper provide meaningful progress toward establishing optimal worst-case guarantees for algorithms in the setting of smooth min-max games.