Abstract
Non-concave maximization has been the subject of much recent study in the optimization and machine learning communities, specifically in deep learning. Recent papers (Ge et al., 2015), (Lee et al., 2017) and references therein indicate that first order methods work well and avoid saddle points. Results as in (Lee et al., 2017), however, are limited to the unconstrained case or for cases where the critical points are in the interior of the feasibility set, which fail to capture some of the most interesting applications. In this paper we focus on constrained non-concave maximization. We analyze a variant of a well-established algorithm in machine learning called Multiplicative Weights Update (MWU) for the maximization problem max(x is an element of D) P(x), where P is non-concave, twice continuously differentiable and D is a product of simplices. We show that MWU converges almost always for small enough stepsizes to critical points that satisfy the second order KKT conditions, by combining techniques from dynamical systems as well as taking advantage of a recent connection between Baum Eagon inequality and MWU (Palaiopanos et al., 2017).