On Randomized Fictitious Play for Approximating Saddle Points Over Convex Sets

Abstract

Given two bounded convex sets $$X\subseteq \mathbb R^m$$ and $$Y\subseteq \mathbb R^n,$$ specified by membership oracles, and a continuous convex–concave function $$F:X\times Y\rightarrow \mathbb R$$ , we consider the problem of computing an $$\varepsilon $$ -approximate saddle point, that is, a pair $$(x^*,y^*)\in X\times Y$$ such that $$\sup _{y\in Y} F(x^*,y)\le \inf _{x\in X}F(x,y^*)+\varepsilon .$$ Grigoriadis and Khachiyan (Oper Res Lett 18(2):53–58, 1995) gave a simple randomized variant of fictitious play for computing an $$\varepsilon $$ -approximate saddle point for matrix games, that is, when $$F$$ is bilinear and the sets $$X$$ and $$Y$$ are simplices. In this paper, we extend their method to the general case. In particular, we show that, for functions of constant “width”, an $$\varepsilon $$ -approximate saddle point can be computed using $$O^* \big (\frac{(n+m)}{\varepsilon ^2}\ln R \big )$$ random samples from log-concave distributions over the convex sets $$X$$ and $$Y$$ . It is assumed that $$X$$ and $$Y$$ have inscribed balls of radius $$1/R$$ and circumscribing balls of radius $$R$$ . As a consequence, we obtain a simple randomized polynomial-time algorithm that computes such an approximation faster than known methods for problems with bounded width and when $$\varepsilon \in (0,1)$$ is a fixed, but arbitrarily small constant. Our main tool for achieving this result is the combination of the randomized fictitious play with the recently developed results on sampling from convex sets.