(Local) Non-Asymptotic Analysis of Logistic Fictitious Play for Two-Player Zero-Sum Games and Its Deterministic Variant

We conduct a local non-asymptotic analysis of the logistic fictitious play (LFP) algorithm, and show that with high probability, this algorithm converges locally at rate $O(1/t)$. To achieve this, we first develop a global non-asymptotic analysis of the deterministic variant of LFP, which we call DLFP, and derive a class of convergence rates based on different step-sizes. We then incorporate a particular form of stochastic noise to the analysis of DLFP, and obtain the local convergence rate of LFP. As a result of independent interest, we extend DLFP to solve a class of strongly convex composite optimization problems. We show that although the resulting algorithm is a simple variant of the generalized Frank-Wolfe method in Nesterov [1,Section 5], somewhat surprisingly, it enjoys significantly improved convergence rate.