Stochastic First-Order Methods with Non-smooth and Non-Euclidean Proximal Terms for Nonconvex High-Dimensional Stochastic Optimization

Abstract

When the nonconvex problem is complicated by stochasticity, the sample complexity of stochastic first-order methods may depend linearly on the problem dimension, which is undesirable for large-scale problems. In this work, we propose dimension-insensitive stochastic first-order methods (DISFOMs) to address nonconvex optimization with expected-valued objective function. Our algorithms allow for non-Euclidean and non-smooth distance functions as the proximal terms. Under mild assumptions, we show that DISFOM using minibatches to estimate the gradient enjoys sample complexity of O ( ( d) / ^4 ) to obtain an -stationary point. Furthermore, we prove that DISFOM employing variance reduction can sharpen this bound to O ( ( d)^2/3/^10/3 ), which perhaps leads to the best-known sample complexity result in terms of d. We provide two choices of the non-smooth distance functions, both of which allow for closed-form solutions to the proximal step. Numerical experiments are conducted to illustrate the dimension insensitive property of the proposed frameworks.

Tasks

Reproductions