Optimal Methods for Unknown Piecewise Smooth Problems I: Convex Optimization

TL;DR

This paper presents APEX, an optimal, nearly parameter-free algorithm for minimizing unknown piecewise smooth convex functions, achieving the best theoretical guarantees and providing verifiable termination certificates.

Contribution

Introduces APEX, the first algorithm to achieve optimal convergence and certificate guarantees for unknown PWS convex optimization problems.

Findings

Achieves tight oracle complexity bounds matching lower bounds.

Provides a verifiable termination certificate for the optimization process.

First to combine optimal convergence with certificate guarantees in PWS optimization.

Abstract

We introduce an optimal and nearly parameter-free algorithm for minimizing piecewise smooth (PWS) convex functions under the quadratic growth (QG) condition, where the locations and structure of the smooth regions are entirely \textit{unknown}. Our algorithm, \apex{} (Accelerated Prox-Level method for Exploring Piecewise Smoothness), is an accelerated bundle-level method designed to adaptively exploit the underlying PWS structure. APEX enjoys optimal theoretical guarantees, achieving a tight oracle complexity bound that matches the lower bound established in this work for convex PWS optimization. Furthermore, APEX generates a verifiable and accurate termination certificate, enabling a robust, almost parameter-free implementation. To the best of our knowledge, APEX is the first algorithm to simultaneously achieve the optimal convergence rate for PWS optimization and provide certificate guarantees.