  Distributionally robust optimization (DRO) is a widely-used approach to learn
models that are robust against distribution shift. Compared with the standard
optimization setting, the objective function in DRO is more difficult to
optimize, and most of the existing theoretical results make strong assumptions
on the loss function. In this work we bridge the gap by studying DRO algorithms
for general smooth non-convex losses. By carefully exploiting the specific form
of the DRO objective, we are able to provide non-asymptotic convergence
guarantees even though the objective function is possibly non-convex,
non-smooth and has unbounded gradient noise. In particular, we prove that a
special algorithm called the mini-batch normalized gradient descent with
momentum, can find an $\epsilon$ first-order stationary point within $O(
\epsilon^{-4} )$ gradient complexity. We also discuss the conditional
value-at-risk (CVaR) setting, where we propose a penalized DRO objective based
on a smoothed version of the CVaR that allows us to obtain a similar
convergence guarantee. We finally verify our theoretical results in a number of
tasks and find that the proposed algorithm can consistently achieve prominent
acceleration.
