Accelerated stochastic approximation with state-dependent noise

TL;DR

This paper introduces two accelerated stochastic approximation algorithms, SAGD and SGE, that achieve optimal convergence rates for convex optimization problems with state-dependent noise, especially in statistical estimation contexts.

Contribution

The paper proposes two novel accelerated stochastic approximation methods, SAGD and SGE, that attain optimal convergence under more general noise assumptions than classical methods.

Findings

Both SAGD and SGE achieve optimal iteration and sample complexities.

SGE handles heavy tail noises and discontinuous score functions effectively.

Algorithms demonstrate strong performance in high-dimensional simulations.

Abstract

We consider a class of stochastic smooth convex optimization problems under rather general assumptions on the noise in the stochastic gradient observation. As opposed to the classical problem setting in which the variance of noise is assumed to be uniformly bounded, herein we assume that the variance of stochastic gradients is related to the "sub-optimality" of the approximate solutions delivered by the algorithm. Such problems naturally arise in a variety of applications, in particular, in the well-known generalized linear regression problem in statistics. However, to the best of our knowledge, none of the existing stochastic approximation algorithms for solving this class of problems attain optimality in terms of the dependence on accuracy, problem parameters, and mini-batch size. We discuss two non-Euclidean accelerated stochastic approximation routines--stochastic accelerated gradient descent (SAGD) and stochastic gradient extrapolation (SGE)--which carry a particular duality relationship. We show that both SAGD and SGE, under appropriate conditions, achieve the optimal convergence rate, attaining the optimal iteration and sample complexities simultaneously. However, corresponding assumptions for the SGE algorithm are more general; they allow, for instance, for efficient application of the SGE to statistical estimation problems under heavy tail noises and discontinuous score functions. We also discuss the application of the SGE to problems satisfying quadratic growth conditions, and show how it can be used to recover sparse solutions. Finally, we report on some simulation experiments to illustrate numerical performance of our proposed algorithms in high-dimensional settings.