  We consider a distributionally robust stochastic optimization problem and
formulate it as a stochastic two-level composition optimization problem with
the use of the mean--semideviation risk measure. In this setting, we consider a
single time-scale algorithm, involving two versions of the inner function value
tracking: linearized tracking of a continuously differentiable loss function,
and SPIDER tracking of a weakly convex loss function. We adopt the norm of the
gradient of the Moreau envelope as our measure of stationarity and show that
the sample complexity of $\mathcal{O}(\varepsilon^{-3})$ is possible in both
cases, with only the constant larger in the second case. Finally, we
demonstrate the performance of our algorithm with a robust learning example and
a weakly convex, non-smooth regression example.
