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We consider multi-level composite optimization problems where each mapping inthe composition is the expectation over a family of random smooth mappings orthe sum of some finite number of smooth mappings. We present a normalizedproximal approximate gradient (NPAG) method where the approximate gradients areobtained via nested stochastic variance reduction. In order to find anapproximate stationary point where the expected norm of its gradient mapping isless than $\epsilon$, the total sample complexity of our method is$O(\epsilon^{-3})$ in the expectation case, and $O(N+\sqrt{N}\epsilon^{-2})$ inthe finite-sum case where $N$ is the total number of functions across allcomposition levels. In addition, the dependence of our total sample complexityon the number of composition levels is polynomial, rather than exponential asin previous work.

Multi-Level Composite Stochastic Optimization via Nested Variance Reduction