  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
\cite{bolte2021conservative,bolte2022subgradient,xiao2023adam}, 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.
