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In this paper, we propose robust stochastic algorithms for solving convexcompositional problems of the form $f(\E_\xi g(\cdot; \xi)) + r(\cdot)$ byestablishing {\bf sub-Gaussian confidence bounds} under weak assumptions aboutthe tails of noise distribution, i.e., {\bf heavy-tailed noise} with boundedsecond-order moments. One can achieve this goal by using an existing boostingstrategy that boosts a low probability convergence result into a highprobability result. However, piecing together existing results for solvingcompositional problems suffers from several drawbacks: (i) the boostingtechnique requires strong convexity of the objective; (ii) it requires aseparate algorithm to handle non-smooth $r$; (iii) it also suffers from anadditional polylogarithmic factor of the condition number. To address theseissues, we directly develop a single-trial stochastic algorithm for minimizingoptimal strongly convex compositional objectives, which has a nearly optimalhigh probability convergence result matching the lower bound of stochasticstrongly convex optimization up to a logarithmic factor. To the best of ourknowledge, this is the first work that establishes nearly optimal sub-Gaussianconfidence bounds for compositional problems under heavy-tailed assumptions.

Nearly Optimal Robust Method for Convex Compositional Problems with Heavy-Tailed Noise