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

TL;DR

This paper introduces dimension-insensitive stochastic first-order methods for nonconvex stochastic optimization, achieving improved sample complexity bounds and accommodating non-smooth, non-Euclidean proximal terms, with practical numerical validation.

Contribution

The work develops novel DISFOM algorithms that are dimension-insensitive and handle non-smooth, non-Euclidean proximal functions, with enhanced theoretical sample complexity bounds.

Findings

Sample complexity of O((log d)/ε^4) for minibatch-based DISFOM.

Variance reduction improves complexity to O((log d)^{2/3}/ε^{10/3}).

Numerical experiments confirm the dimension-insensitive property.

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 $ \mathcal{O} ( (\log d) / \epsilon^4 ) $ to obtain an $\epsilon$-stationary point. Furthermore, we prove that DISFOM employing variance reduction can sharpen this bound to $\mathcal{O} ( (\log d)^{2/3}/\epsilon^{10/3} )$, 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.