On proximal gradient mapping and its minimization in norm via potential function-based acceleration

TL;DR

This paper extends a potential function-based acceleration framework from gradient descent to proximal gradient descent by leveraging properties of proximal gradient mapping, including norm monotonicity and refined descent.

Contribution

It introduces a novel extension of the potential function framework to proximal gradient methods, utilizing new insights into proximal gradient mapping.

Findings

Proximal gradient mapping exhibits norm monotonicity.

Refined descent properties are established for proximal gradient methods.

The framework enables improved convergence analysis for composite optimization.

Abstract

The proximal gradient descent method, well-known for composite optimization, can be completely described by the concept of proximal gradient mapping. In this paper, we highlight our previous two discoveries of proximal gradient mapping--norm monotonicity and refined descent, with which we are able to extend the recently proposed potential function-based framework from gradient descent to proximal gradient descent.