An Absolute-Error Proximal Bundle Method through the Lens of Frank-Wolf

Abstract

The proximal bundle method (PBM) is a fundamental and computationally effective algorithm for solving optimization problems with nonsmooth components. In this paper, we conduct a theoretical investigation of a modified proximal bundle method, which we call the Modified Proximal Bundle with Fixed Absolute Accuracy (MPB-FA). MPB-FA modifies the serious-step test of the PBM serious-step test. In MPB-FA, it is based on an absolute accuracy criterion, and the accuracy is fixed over iterations, while the standard PBM uses a relative accuracy in the serious-step test, which changes with iterations. Also, similarly to multiple PBM analyses, the proximal parameter in MPB-FA is also fixed over iterations, while it is permitted to change in the standard PBM. These modifications allow us to build the first link between a proximal bundle method and a Frank-Wolfe algorithm on the Moreau envelope of the dual problem. In light of this correspondence, we first extend the linear convergence of Kelley's method on the sum of a smooth strongly convex function and a convex piecewise linear function from the positive homogeneous to the general case. Building on this result, we propose a novel complexity analysis of MPB-FA when the objective is the sum of a smooth and a piecewise function and derive an $\mathcal{O}(\hat{I}{\mu }^{-8/9})$ iteration complexity, improving upon the best known $\mathcal{O}(\hat{I}{\mu }^{-2})$ guarantee on a related variant of PBM. It is worth-noting that the best known complexity bound for the classical PBM in the general case is $\mathcal{O}(\hat{I}{\mu }^{-3})$ and $\mathcal{O}(\hat{I}{\mu }^{-2})$ when the proximal parameter is fixed. Our approach also reveals new insights on bundle management and empirical behavior of the proximal bundle methods.