On the convergence of adaptive first order methods: proximal gradient and alternating minimization algorithms

TL;DR

This paper introduces a unified adaptive proximal gradient framework with larger stepsizes and improved bounds, analyzes its convergence in a general setting, and extends adaptivity to alternating minimization algorithms.

Contribution

It proposes adaPG^{q,r}, a flexible adaptive proximal gradient method with theoretical convergence guarantees and extends adaptivity to alternating minimization beyond strongly convex cases.

Findings

Demonstrates efficacy of the methods through numerical simulations.

Establishes convergence in a general, time-varying parameter setting.

Introduces an adaptive alternating minimization algorithm with broader applicability.

Abstract

Building upon recent works on linesearch-free adaptive proximal gradient methods, this paper proposes adaPG$^{q,r}$, a framework that unifies and extends existing results by providing larger stepsize policies and improved lower bounds. Different choices of the parameters $q$ and $r$ are discussed and the efficacy of the resulting methods is demonstrated through numerical simulations. In an attempt to better understand the underlying theory, its convergence is established in a more general setting that allows for time-varying parameters. Finally, an adaptive alternating minimization algorithm is presented by exploring the dual setting. This algorithm not only incorporates additional adaptivity, but also expands its applicability beyond standard strongly convex settings.