Linearly Convergent Algorithms for Nonsmooth Problems with Unknown Smooth Pieces

TL;DR

This paper introduces the first linearly convergent algorithms for optimizing piecewise smooth functions with unknown partitions, extending to weakly-convex cases and providing a practical termination criterion.

Contribution

It develops a bundle-level method with global linear convergence for PWS functions satisfying quadratic growth, and introduces a parameter-free termination criterion.

Findings

Achieves global linear convergence for PWS functions with quadratic growth.

Extends algorithms to approximately PWS and weakly-convex problems, matching smooth non-convex complexity.

Provides a verifiable termination criterion that estimates optimality gap without problem parameters.

Abstract

We develop efficient algorithms for optimizing piecewise smooth (PWS) functions where the underlying partition of the domain into smooth pieces is \emph{unknown}. For PWS functions satisfying a quadratic growth (QG) condition, we propose a bundle-level (BL) type method that achieves global linear convergence -- to our knowledge, the first such result for any algorithm for this problem class. We extend this method to handle approximately PWS functions and to solve weakly-convex PWS problems, improving the state-of-the-art complexity to match the benchmark for smooth non-convex optimization. Furthermore, we introduce the first verifiable and accurate termination criterion for PWS optimization. Similar to the gradient norm in smooth optimization, this certificate tightly characterizes the optimality gap under the QG condition, and can moreover be evaluated without knowledge of any problem parameters. We develop a search subroutine for this certificate and embed it within a guess-and-check framework, resulting in an almost parameter-free algorithm for both the convex QG and weakly-convex settings.