Provable Bregman-divergence based Methods for Nonconvex and Non-Lipschitz Problems

TL;DR

This paper introduces Bregman-divergence based algorithms that ensure convergence to second-order stationary points for non-Lipschitz smooth problems, broadening optimization methods beyond traditional Lipschitz smoothness assumptions.

Contribution

It generalizes the Lipschitz smoothness condition to relative smoothness and develops new algorithms with second-order convergence guarantees for non-Lipschitz problems.

Findings

Algorithms guaranteed to converge to second-order stationary points.

Applicable to a broad class of non-Lipschitz smooth problems.

Extends convergence guarantees to alternating minimization methods.

Abstract

The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to generalize the concept of Lipschitz smoothness condition to the relative smoothness condition, which is satisfied by any finite-order polynomial objective function. Further, this work develops new Bregman-divergence based algorithms that are guaranteed to converge to a second-order stationary point for any relatively smooth problem. In addition, the proposed optimization methods cover both the proximal alternating minimization and the proximal alternating linearized minimization when we specialize the Bregman divergence to the Euclidian distance. Therefore, this work not only develops guaranteed optimization methods for non-Lipschitz smooth problems but also solves an open problem of showing the second-order convergence guarantees for these alternating minimization methods.