Stochastic compositional optimization arises in many important machine learning applications. The objective function is the composition of two expectations of stochastic functions, and is more challenging to optimize than vanilla stochastic optimization problems. In this paper, we investigate the stochastic compositional optimization in the general smooth non-convex setting. We employ a recently developed idea of Stochastic Recursive Gradient Descent to design a novel algorithm named SARAH-Compositional, and prove a sharp Incremental First-order Oracle (IFO) complexity upper bound for stochastic compositional optimization: O((n + m)~(1/2)ε~(-2)) in the finite-sum case and O(ε~(-3)) in the online case. Such a complexity is known to be the best one among IFO complexity results for non-convex stochastic compositional optimization. Numerical experiments on risk-adverse portfolio management validate the superiority of SARAH-Compositional over a few rival algorithms.
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