Global min-max Computation for α-Hölder Games

Abstract

min-max optimization problems recently arose in various settings. From Generative Adversarial Networks (GANs) to aerodynamic optimization through Game Theory, the assumptions on the objective function vary. Motivated by the applications to deep learning and especially GANs, most recent works assume differentiability to design local search algorithms such as Gradient Descend Ascent (GDA). In contrast, this work will only require α-Hölder properties to tackle general gametheoretic problems with poor continuity assumptions. Focusing on the example of problems in which max and min optimization variables live in simplices, we provide a simple algorithm, based on Deterministic Optimistic Optimization (DOO), relying on an outer min-optimization using the solutions of an inner maxoptimization. The algorithm is shown to converge in finite time to an ϵ-global optimum. Experimental validations are given and the time complexity of our algorithm is studied.