An accelerated proximal bundle method for convex optimization

TL;DR

This paper introduces the first accelerated proximal bundle method that achieves optimal convergence rates for smooth convex optimization, improving upon traditional PBM methods.

Contribution

It presents a simple, structurally similar acceleration of the classical PBM that attains the optimal iteration complexity for smooth convex functions.

Findings

Achieves the optimal $oxed{O(1/\sqrt{ ext{ extepsilon}})}$ iteration complexity.

Retains key structural properties of classical PBM.

Numerical experiments confirm the theoretical acceleration.

Abstract

The proximal bundle method (PBM) is a powerful and widely used approach for minimizing nonsmooth convex functions. However, for smooth objectives, its best-known convergence rate remains suboptimal, and whether PBM can be accelerated remains open. In this work, we present the first accelerated proximal bundle method that achieves the optimal $\mathscr{O}(1/\sqrt{\epsilon})$ iteration complexity for obtaining an $\epsilon$-accurate solution in smooth convex optimization. The proposed method is conceptually simple, which differs from Nesterov's accelerated gradient descent by only a single line and retains all key structural properties of the classical PBM. In particular, it relies on the same minimal assumptions on model approximations and preserves the standard bundle testing criterion. Numerical experiments confirm the accelerated $\mathscr{O}(1/\sqrt{\epsilon})$ convergence rate predicted by our theory.