Exact worst-case convergence rates of the proximal gradient method for composite convex minimization

TL;DR

This paper precisely characterizes the worst-case convergence rates of the proximal gradient method for composite convex minimization, providing exact guarantees for various performance measures and extending existing results.

Contribution

It introduces a novel semidefinite programming-based methodology to obtain exact, non-asymptotic worst-case convergence rates for the proximal gradient method under various conditions.

Findings

Exact worst-case convergence rates for objective, distance, and residual measures.

The same fixed step size is optimal across different performance metrics.

Extension of gradient descent line search results to the proximal setting.

Abstract

We study the worst-case convergence rates of the proximal gradient method for minimizing the sum of a smooth strongly convex function and a non-smooth convex function whose proximal operator is available. We establish the exact worst-case convergence rates of the proximal gradient method in this setting for any step size and for different standard performance measures: objective function accuracy, distance to optimality and residual gradient norm. The proof methodology relies on recent developments in performance estimation of first-order methods based on semidefinite programming. In the case of the proximal gradient method, this methodology allows obtaining exact and non-asymptotic worst-case guarantees that are conceptually very simple, although apparently new. On the way, we discuss how strong convexity can be replaced by weaker assumptions, while preserving the corresponding convergence rates. We also establish that the same fixed step size policy is optimal for all three performance measures. Finally, we extend recent results on the worst-case behavior of gradient descent with exact line search to the proximal case.