# A New Exact Worst-Case Linear Convergence Rate of the Proximal Gradient Method

## TL;DR

This paper establishes a new exact worst-case linear convergence rate for the proximal gradient method based on the proximal gradient norm, refining existing results and improving convergence rate estimates under the Polyak-Lojasiewicz inequality.

## Contribution

It introduces a new exact worst-case linear convergence rate for the proximal gradient method, enhancing understanding of its convergence behavior.

## Key findings

- New exact worst-case linear convergence rate established
- Improved convergence rate of objective function under Polyak-Lojasiewicz inequality
- Refined descent lemma for the proximal gradient method

## Abstract

In this note, we establish a new exact worst-case linear convergence rate of the proximal gradient method in terms of the proximal gradient norm, which complements the recent results in [1] and implies a refined descent lemma.descent lemma. Based on the new lemma, we improve the linear convergence rate of the objective function accuracy under the Polyak-Lojasiewicz inequality.