Optimality and Stability in Non-Convex Smooth Games

TL;DR

This paper analyzes local minimax points in non-convex smooth games, exploring their properties, relation to other solution concepts, and the stability of gradient algorithms near these points, highlighting challenges and future directions.

Contribution

It provides a comprehensive analysis of local minimax points, their relation to local saddle points, and studies the stability of gradient algorithms in non-convex smooth games.

Findings

Local saddle points are a special case of local minimax points.

In quadratic games, local minimax points are equivalent to global minimax points.

Gradient algorithms often fail to converge to minimax points in general cases.

Abstract

Convergence to a saddle point for convex-concave functions has been studied for decades, while recent years has seen a surge of interest in non-convex (zero-sum) smooth games, motivated by their recent wide applications. It remains an intriguing research challenge how local optimal points are defined and which algorithm can converge to such points. An interesting concept is known as the local minimax point, which strongly correlates with the widely-known gradient descent ascent algorithm. This paper aims to provide a comprehensive analysis of local minimax points, such as their relation with other solution concepts and their optimality conditions. We find that local saddle points can be regarded as a special type of local minimax points, called uniformly local minimax points, under mild continuity assumptions. In (non-convex) quadratic games, we show that local minimax points are (in some sense) equivalent to global minimax points. Finally, we study the stability of gradient algorithms near local minimax points. Although gradient algorithms can converge to local/global minimax points in the non-degenerate case, they would often fail in general cases. This implies the necessity of either novel algorithms or concepts beyond saddle points and minimax points in non-convex smooth games.