Universal Adaptive Proximal Gradient Methods via Gradient Mapping Accumulation

TL;DR

This paper introduces an adaptive proximal gradient method with gradient mapping accumulation that converges across multiple problem classes without prior knowledge of parameters, achieving near-optimal rates.

Contribution

It presents a novel adaptive step size scheme that works universally for nonconvex, convex nonsmooth, and convex smooth problems, with new techniques for stochastic noise control.

Findings

Converges across all three problem classes under mild assumptions.

Achieves near-optimal convergence rates in deterministic and stochastic settings.

Develops new techniques for controlling stochastic noise effects.

Abstract

We propose an adaptive proximal gradient method for minimizing the sum of two functions, where one is a simple convex function, and the other belongs to one of the three classes: nonconvex smooth, convex nonsmooth, or convex smooth. The key feature of the method is an adaptive step size that accumulates historical gradient mapping norms in the denominator. Without any modification or knowledge of problem parameters, the method converges across all three problem classes under mild bounded-iterates and bounded-variance assumptions, with rates matching those of the proximal gradient method up to logarithmic factors, in both deterministic and stochastic settings. For the convex setting, we further propose an accelerated variant. It retains a similar near-optimal convergence rate for the nonsmooth case and achieves an improved rate of order $\widetilde{O}\big(1/t^2 + \sigma/\sqrt{t}\big)$ for the smooth case, which is optimal up to logarithmic factors. Notably, we develop new techniques for controlling the effect of stochastic noise, which are applicable across all three problem classes in the stochastic setting and enable simplified analysis.