  We study the training of regularized neural networks where the regularizer
can be non-smooth and non-convex. We propose a unified framework for stochastic
proximal gradient descent, which we term ProxGen, that allows for arbitrary
positive preconditioners and lower semi-continuous regularizers. Our framework
encompasses standard stochastic proximal gradient methods without
preconditioners as special cases, which have been extensively studied in
various settings. Not only that, we present two important update rules beyond
the well-known standard methods as a byproduct of our approach: (i) the first
closed-form proximal mappings of $\ell_q$ regularization ($0 \leq q \leq 1$)
for adaptive stochastic gradient methods, and (ii) a revised version of
ProxQuant that fixes a caveat of the original approach for
quantization-specific regularizers. We analyze the convergence of ProxGen and
show that the whole family of ProxGen enjoys the same convergence rate as
stochastic proximal gradient descent without preconditioners. We also
empirically show the superiority of proximal methods compared to
subgradient-based approaches via extensive experiments. Interestingly, our
results indicate that proximal methods with non-convex regularizers are more
effective than those with convex regularizers.
