Survey Descent: A Multipoint Generalization of Gradient Descent for Nonsmooth Optimization

TL;DR

This paper introduces a multipoint generalization of gradient descent tailored for nonsmooth optimization, achieving linear convergence for max-of-smooth objectives and offering a new theoretical framework for such problems.

Contribution

It proposes a novel multipoint descent method that extends gradient descent to nonsmooth settings and proves linear convergence for max-of-smooth objectives.

Findings

Proves linear convergence for max-of-smooth objectives.

Demonstrates the method's effectiveness through experiments.

Addresses challenges in nonsmooth optimization theory.

Abstract

For strongly convex objectives that are smooth, the classical theory of gradient descent ensures linear convergence relative to the number of gradient evaluations. An analogous nonsmooth theory is challenging. Even when the objective is smooth at every iterate, the corresponding local models are unstable and the number of cutting planes invoked by traditional remedies is difficult to bound, leading to convergences guarantees that are sublinear relative to the cumulative number of gradient evaluations. We instead propose a multipoint generalization of the gradient descent iteration for local optimization. While designed with general objectives in mind, we are motivated by a ``max-of-smooth'' model that captures the subdifferential dimension at optimality. We prove linear convergence when the objective is itself max-of-smooth, and experiments suggest a more general phenomenon.