Complexity of normalized stochastic first-order methods with momentum under heavy-tailed noise

TL;DR

This paper introduces practical normalized stochastic first-order methods with momentum for unconstrained optimization under heavy-tailed noise, providing new complexity bounds without requiring problem-specific parameters.

Contribution

It proposes novel normalized stochastic methods with momentum that adaptively update parameters and establish first-order oracle complexity results under weaker assumptions.

Findings

Complexity bounds improve or match existing results.

Methods are effective in numerical experiments.

No need for explicit knowledge of Lipschitz constant or noise bound.

Abstract

In this paper, we propose practical normalized stochastic first-order methods with Polyak momentum, multi-extrapolated momentum, and recursive momentum for solving unconstrained optimization problems. These methods employ dynamically updated algorithmic parameters and do not require explicit knowledge of problem-dependent quantities such as the Lipschitz constant or noise bound. We establish first-order oracle complexity results for finding approximate stochastic stationary points under heavy-tailed noise and weakly average smoothness conditions -- both of which are weaker than the commonly used bounded variance and mean-squared smoothness assumptions. Our complexity bounds either improve upon or match the best-known results in the literature. Numerical experiments are presented to demonstrate the practical effectiveness of the proposed methods.