Projected gradient methods for nonconvex and stochastic smooth optimization: new complexities and auto-conditioned stepsizes

TL;DR

This paper introduces new projected gradient methods for smooth optimization, including auto-conditioned stepsizes and stochastic variants, achieving improved iteration complexity without requiring prior knowledge of Lipschitz constants.

Contribution

The paper proposes auto-conditioned projected gradient methods that adaptively estimate Lipschitz constants, providing improved complexity bounds for both deterministic and stochastic smooth optimization.

Findings

Auto-conditioned PG achieves optimal iteration complexity without line search.

Stochastic PG methods with variance reduction improve convergence bounds.

Auto-conditioned stepsizes work effectively in stochastic settings, matching deterministic guarantees.

Abstract

We present a novel class of projected gradient (PG) methods for minimizing a smooth but not necessarily convex function over a convex compact set. We first provide a novel analysis of the constant-stepsize PG method, achieving the best-known iteration complexity for finding an approximate stationary point of the problem. We then develop an "auto-conditioned" projected gradient (AC-PG) variant that achieves the same iteration complexity without requiring the input of the Lipschitz constant of the gradient or any line search procedure. The key idea is to estimate the Lipschitz constant using first-order information gathered from the previous iterations, and to show that the error caused by underestimating the Lipschitz constant can be properly controlled. We then generalize the PG methods to the stochastic setting, by proposing a stochastic projected gradient (SPG) method and a variance-reduced stochastic gradient (VR-SPG) method, achieving new complexity bounds in different oracle settings. We also present auto-conditioned stepsize policies for both stochastic PG methods and establish comparable convergence guarantees.