Lipschitz minimization and the Goldstein modulus

TL;DR

This paper introduces a new modulus based on Goldstein subgradients to measure the slope of Lipschitz functions, relating it to convergence rates of Goldstein-style minimization algorithms, and demonstrates its utility through computational experiments.

Contribution

It proposes a novel Lipschitz modulus derived from Goldstein subgradients, connecting its growth to algorithmic convergence and providing a heuristic for practical minimization.

Findings

The new modulus effectively measures the slope of Lipschitz functions.

Near-linear convergence correlates with linear growth of the modulus at minimizers.

A simple heuristic demonstrates the practical utility of the modulus.

Abstract

Goldstein's 1977 idealized iteration for minimizing a Lipschitz objective fixes a distance - the step size - and relies on a certain approximate subgradient. That "Goldstein subgradient" is the shortest convex combination of objective gradients at points within that distance of the current iterate. A recent implementable Goldstein-style algorithm allows a remarkable complexity analysis (Zhang et al. 2020), and a more sophisticated variant (Davis and Jiang, 2022) leverages typical objective geometry to force near-linear convergence. To explore such methods, we introduce a new modulus, based on Goldstein subgradients, that robustly measures the slope of a Lipschitz function. We relate near-linear convergence of Goldstein-style methods to linear growth of this modulus at minimizers. We illustrate the idea computationally with a simple heuristic for Lipschitz minimization.