  The widely used stochastic gradient methods for minimizing nonconvex
composite objective functions require the Lipschitz smoothness of the
differentiable part. But the requirement does not hold true for problem classes
including quadratic inverse problems and training neural networks. To address
this issue, we investigate a family of stochastic Bregman proximal gradient
(SBPG) methods, which only require smooth adaptivity of the differentiable
part. SBPG replaces the upper quadratic approximation used in SGD with the
Bregman proximity measure, resulting in a better approximation model that
captures the non-Lipschitz gradients of the nonconvex objective. We formulate
the vanilla SBPG and establish its convergence properties under nonconvex
setting without finite-sum structure. Experimental results on quadratic inverse
problems testify the robustness of SBPG. Moreover, we propose a momentum-based
version of SBPG (MSBPG) and prove it has improved convergence properties. We
apply MSBPG to the training of deep neural networks with a polynomial kernel
function, which ensures the smooth adaptivity of the loss function.
Experimental results on representative benchmarks demonstrate the effectiveness
and robustness of MSBPG in training neural networks. Since the additional
computation cost of MSBPG compared with SGD is negligible in large-scale
optimization, MSBPG can potentially be employed as an universal open-source
optimizer in the future.
