An Optimal Algorithm for Online Non-Convex Learning

Abstract

In many online learning paradigms, convexity plays a central role in the derivation and analysis of online learning algorithms. The results, however, fail to be extended to the non-convex settings, while non-convexity is necessitated by a large number of recent applications. The Online Non-Convex Learning (ønco) problem generalizes the classic Online Convex Optimization (øco) framework by relaxing the convexity assumption on the cost function (to a Lipschitz continuous function) and the decision set. The state-of-the-art result for the ønco demonstrates that the classic online exponential weighting algorithm attains a sublinear regret of $O(\sqrtTłog T )$. The regret lower bound for the øco, however, is $Ømega(\sqrtT )$, and to the best of our knowledge, there is no result in the context of the ønco problem achieving the same bound. This paper proposes the Online Recursive Weighting (\rw) algorithm with regret of $O(\sqrtT )$, matching the tight regret lower bound for the øco problem, and fills the regret gap between the state-of-the-art results in the online convex and non-convex optimization problems.