  In this paper, we propose robust stochastic algorithms for solving convex
compositional problems of the form $f(\E_\xi g(\cdot; \xi)) + r(\cdot)$ by
establishing {\bf sub-Gaussian confidence bounds} under weak assumptions about
the tails of noise distribution, i.e., {\bf heavy-tailed noise} with bounded
second-order moments. One can achieve this goal by using an existing boosting
strategy that boosts a low probability convergence result into a high
probability result. However, piecing together existing results for solving
compositional problems suffers from several drawbacks: (i) the boosting
technique requires strong convexity of the objective; (ii) it requires a
separate algorithm to handle non-smooth $r$; (iii) it also suffers from an
additional polylogarithmic factor of the condition number. To address these
issues, we directly develop a single-trial stochastic algorithm for minimizing
optimal strongly convex compositional objectives, which has a nearly optimal
high probability convergence result matching the lower bound of stochastic
strongly convex optimization up to a logarithmic factor. To the best of our
knowledge, this is the first work that establishes nearly optimal sub-Gaussian
confidence bounds for compositional problems under heavy-tailed assumptions.
