  Stochastic compositional optimization arises in many important machine
learning tasks such as value function evaluation in reinforcement learning and
portfolio management. 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 \textit{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: $\mathcal{O}((n+m)^{1/2}
\varepsilon^{-2})$ in the finite-sum case and $\mathcal{O}(\varepsilon^{-3})$
in the online case. Such a complexity is known to be the best one among IFO
complexity results for non-convex stochastic compositional optimization, and is
believed to be optimal. Our experiments validate the theoretical performance of
our algorithm.
