Competitive Gradient Optimization

TL;DR

This paper introduces Competitive Gradient Optimization (CGO), a novel gradient-based method for zero-sum games that improves convergence analysis and proposes optimistic variants with convergence guarantees to saddle points.

Contribution

The paper develops CGO, a new optimization method for zero-sum games, with continuous-time analysis, convergence rates, and an optimistic variant for better saddle point convergence.

Findings

CGO converges to stationary points with a provable rate.

For strictly α-coherent functions, CGO converges to saddle points.

The optimistic CGO variant achieves convergence to saddle points in α-coherent functions.

Abstract

We study the problem of convergence to a stationary point in zero-sum games. We propose competitive gradient optimization (CGO ), a gradient-based method that incorporates the interactions between the two players in zero-sum games for optimization updates. We provide continuous-time analysis of CGO and its convergence properties while showing that in the continuous limit, CGO predecessors degenerate to their gradient descent ascent (GDA) variants. We provide a rate of convergence to stationary points and further propose a generalized class of $\alpha$-coherent function for which we provide convergence analysis. We show that for strictly $\alpha$-coherent functions, our algorithm convergences to a saddle point. Moreover, we propose optimistic CGO (OCGO), an optimistic variant, for which we show convergence rate to saddle points in $\alpha$-coherent class of functions.