Beyond Minimax Optimality: A Subgame Perfect Gradient Method

Abstract

The study of convex optimization has historically been concerned with worst-case convergence rates. The development of the Optimized Gradient Method (OGM), due to \citet{drori2012PerformanceOF,Kim2016optimal}, marked a major milestone in this study, as OGM achieves the optimal worst-case convergence rate among all first-order methods for unconstrained smooth convex optimization. In order to examine the possibility of obtaining stronger convergence guarantees, we will consider algorithms with \emph{dynamic} convergence rates, which may improve as additional first-order information is revealed. Our main contribution is the development of an algorithm, the Subgame Perfect Gradient Method (SPGM), which refines OGM to make use of the full history of first-order information. We show that SPGM is \emph{dynamically optimal}, in the sense that in each iteration, no other algorithm can offer a strictly better convergence rate on all functions which agree with the observed first-order information up to that iteration. We formalize this notion of dynamic optimality using the game-theoretic notion of a subgame perfect equilibrium. We conclude our study with preliminary numerical experiments showing that SPGM strongly outperforms OGM.