Mixed Optimization for Smooth Functions

Abstract

It is well known that the optimal convergence rate for stochastic optimization of smooth functions is O(1= p T), which is same as stochastic optimization of Lips-chitz continuous convex functions. This is in contrast to optimizing smooth func-tions using full gradients, which yields a convergence rate of O(1=T 2). In this work, we consider a new setup for optimizing smooth functions, termed asMixed Optimization, which allows to access both a stochastic oracle and a full gradient oracle. Our goal is to significantly improve the convergence rate of stochastic op-timization of smooth functions by having an additional small number of accesses to the full gradient oracle. We show that, with an O(lnT) calls to the full gradient oracle and anO(T) calls to the stochastic oracle, the proposed mixed optimization algorithm is able to achieve an optimization error of O(1=T). 1