Perturbed Multiplicative Weights Update in Learning of Game

Multiplicative Weights Update (MWU) has been studied extensively due to many applications in machine learning, constrained optimization, and game theory. In the non-convex optimization problems, saddle points are the main obstruction to produce high-quality optimizers, i.e., the critical points that is not local or global optima. Recent works in optimization and machine learning has addressed the escaping saddle point problem in a promise way-adding perturbation to iterations. Despite the popularity of MWU and perturbed first-order methods, there is a problem remains unaddressed: is there a perturbed version of MWU algorithms so that the algorithm can escape saddle points efficiently. This paper focuses on a spherical framework for MWU and its application in building result on escaping saddle points. Combining with the recent development on escaping saddle points with perturbed Riemannian gradient descent, we derive a non-asymptotic convergence results for perturbed MWU.