  In this paper, we explore a specific optimization problem that involves the
combination of a differentiable nonconvex function and a nondifferentiable
function. The differentiable component lacks a global Lipschitz continuous
gradient, posing challenges for optimization. To address this issue and
accelerate the convergence, we propose a Bregman proximal stochastic gradient
method with extrapolation (BPSGE), which only requires smooth adaptivity of the
differentiable part. Under the variance reduction framework, we not only
analyze the subsequential and global convergence of the proposed algorithm
under certain conditions, but also analyze the sublinear convergence rate of
the subsequence, and the complexity of the algorithm, revealing that the BPSGE
algorithm requires at most O(epsilon\^\,(-2)) iterations in expectation to
attain an epsilon-stationary point. To validate the effectiveness of our
proposed algorithm, we conduct numerical experiments on three real-world
applications: graph regularized nonnegative matrix factorization (NMF), matrix
factorization with weakly-convex regularization, and NMF with nonconvex
sparsity constraints. These experiments demonstrate that BPSGE is faster than
the baselines without extrapolation.
