Zeroth-Order Stochastic Compositional Algorithms for Risk-Aware Learning

Dionysios S. Kalogerias, Warren B. Powell

Abstract: We present ${\it Free-MESSAGE}^{,p}$, the first zeroth-order algorithm for (weakly) convex mean-semideviation-based risk-aware learning, which is also the first-ever three-level zeroth-order compositional stochastic optimization algorithm. Using a nontrivial extension of Nesterov's classical results on Gaussian smoothing, we develop the ${\it Free-MESSAGE}^{,p}$ 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 ${\it Free-MESSAGE}^{,p}$ 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, 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, and the accuracy of the obtained results, naturally extending previous results in zeroth-order risk-neutral learning.