  We present $\textit{Free-MESSAGE}^{p}$, the first zeroth-order algorithm for
(weakly-)convex mean-semideviation-based risk-aware learning, which is also the
first three-level zeroth-order compositional stochastic optimization algorithm
whatsoever. Using a non-trivial extension of Nesterov's classical results on
Gaussian smoothing, we develop the $\textit{Free-MESSAGE}^{p}$ algorithm from
first principles, and show that it essentially solves a smoothed surrogate to
the original problem, the former being a uniform approximation of the latter,
in a useful, convenient sense. We then present a complete analysis of the
$\textit{Free-MESSAGE}^{p}$ algorithm, which establishes convergence in a
user-tunable neighborhood of the optimal solutions of the original problem for
convex costs, as well as explicit convergence rates for convex, weakly convex,
and strongly convex costs, and in a unified way. Orderwise, and for fixed
problem parameters, our results demonstrate no sacrifice in convergence speed
as compared to existing first-order methods, while striking a certain balance
among the condition of the problem, its dimensionality, as well as the accuracy
of the obtained results, naturally extending previous results in zeroth-order
risk-neutral learning.
