Universality of AdaGrad Stepsizes for Stochastic Optimization: Inexact Oracle, Acceleration and Variance Reduction

We present adaptive gradient methods (both basic and accelerated) for solving convex composite optimization problems in which the main part is approximately smooth (a.k.a. $(\delta, L)$-smooth) and can be accessed only via a (potentially biased) stochastic gradient oracle. This setting covers many interesting examples including Hölder smooth problems and various inexact computations of the stochastic gradient. Our methods use AdaGrad stepsizes and are adaptive in the sense that they do not require knowing any problem-dependent constants except an estimate of the diameter of the feasible set but nevertheless achieve the best possible convergence rates as if they knew the corresponding constants. We demonstrate that AdaGrad stepsizes work in a variety of situations by proving, in a unified manner, three types of new results. First, we establish efficiency guarantees for our methods in the classical setting where the oracle's variance is uniformly bounded. We then show that, under more refined assumptions on the variance, the same methods without any modifications enjoy implicit variance reduction properties allowing us to express their complexity estimates in terms of the variance only at the minimizer. Finally, we show how to incorporate explicit SVRG-type variance reduction into our methods and obtain even faster algorithms. In all three cases, we present both basic and accelerated algorithms achieving state-of-the-art complexity bounds. As a direct corollary of our results, we obtain universal stochastic gradient methods for Hölder smooth problems which can be used in all situations.