Stochastic Bregman Subgradient Methods for Nonsmooth Nonconvex Optimization Problems

Kuangyu Ding () and Kim-Chuan Toh ()
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Kuangyu Ding: National University of Singapore
Kim-Chuan Toh: National University of Singapore

Journal of Optimization Theory and Applications, 2025, vol. 206, issue 3, No 11, 36 pages

Abstract: Abstract This paper focuses on the problem of minimizing a locally Lipschitz continuous function. Motivated by the effectiveness of Bregman gradient methods in training nonsmooth deep neural networks and the recent progress in stochastic subgradient methods for nonsmooth nonconvex optimization problems [11, 12, 58], we investigate the long-term behavior of stochastic Bregman subgradient methods in such context, especially when the objective function lacks Clarke regularity. We begin by exploring a general framework for Bregman-type methods, establishing their convergence by a differential inclusion approach. For practical applications, we develop a stochastic Bregman subgradient method that allows the subproblems to be solved inexactly. Furthermore, we demonstrate how a single timescale momentum can be integrated into the Bregman subgradient method with slight modifications to the momentum update. Additionally, we introduce a Bregman proximal subgradient method for solving composite optimization problems possibly with constraints, whose convergence can be guaranteed based on the general framework. Numerical experiments on training nonsmooth neural networks are conducted to validate the effectiveness of our proposed methods.

Keywords: Nonsmooth nonconvex optimization; Clarke regularity; Conservative field; Bregman subgradient methods; Deep learning; 90C05; 90C06; 90C25 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-025-02749-7

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