Tight Convergence Rate in Subgradient Norm of the Proximal Point Algorithm

TL;DR

This paper establishes the exact convergence rate of the subgradient norm for the proximal point algorithm, confirming a conjecture and enhancing understanding of its efficiency in optimization tasks.

Contribution

It provides the first tight convergence rate in subgradient norm for the proximal point algorithm, resolving a conjecture from prior research.

Findings

Proximal point algorithm's subgradient norm convergence rate is now precisely characterized.

The results have implications for dual methods and primal feasibility analysis.

The convergence rate matches the conjectured bounds, confirming their optimality.

Abstract

Proximal point algorithm has found many applications, and it has been playing fundamental roles in the understanding, design, and analysis of many first-order methods. In this paper, we derive the tight convergence rate in subgradient norm of the proximal point algorithm, which was conjectured by Taylor, Hendrickx and Glineur [SIAM J.~Optim., 27 (2017), pp.~1283--1313]. This sort of convergence results in terms of the residual (sub)gradient norm is particularly interesting when considering dual methods, where the dual residual gradient norm corresponds to the primal distance to feasibility.