Accuracy Certificates for Convex Minimization with Inexact Oracle

TL;DR

This paper extends accuracy certificates to convex minimization with inexact oracles, enabling online verification, primal recovery, and effective use with noisy oracles, demonstrated through numerical experiments.

Contribution

It generalizes accuracy certificates for inexact first-order oracles and proposes explicit constructions for cutting plane methods, including noisy oracle scenarios.

Findings

Certificates provide tight upper bounds on objective residuals.

Cutting plane methods can be used effectively with noisy oracles.

The approach enables online verification and primal solution recovery.

Abstract

Accuracy certificates for convex minimization problems allow for online verification of the accuracy of approximate solutions and provide a theoretically valid online stopping criterion. When solving the Lagrange dual problem, accuracy certificates produce a simple way to recover an approximate primal solution and estimate its accuracy. In this paper, we generalize accuracy certificates for the setting of inexact first-order oracle, including the setting of primal and Lagrange dual pair of problems. We further propose an explicit way to construct accuracy certificates for a large class of cutting plane methods based on polytopes. As a by-product, we show that the considered cutting plane methods can be efficiently used with a noisy oracle even thought they were originally designed to be equipped with an exact oracle. Finally, we illustrate the work of the proposed certificates in the numerical experiments highlighting that our certificates provide a tight upper bound on the objective residual.