On Randomized Fictitious Play for Approximating Saddle Points over Convex Sets

Abstract

Given two bounded convex sets X ⊆ ℝm and Y ⊆ ℝn, specified by membership oracles, and a continuous convex-concave function F:X×Y → ℝ, we consider the problem of computing an ε-approximate saddle point, that is, a pair (x *,y *) ∈ X×Y such that \(\sup_{y\in Y} F(x^*,y)\le \inf_{x\in X}F(x,y^*)+\varepsilon .\) Grigoriadis and Khachiyan (1995), based on a randomized variant of fictitious play, gave a simple algorithm for computing an ε-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 ε-approximate saddle point can be computed using O *(n + m) random samples from log-concave distributions over the convex sets X and Y. 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 ε ∈ (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. A full version of this paper can be found at http://arxiv.org/abs/1301.5290 .KeywordsSaddle PointFractional PackingMatrix GameFictitious PlayMultiplicative WeightThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.