  Composition optimization is widely-applied in nonconvex machine learning.
Various advanced stochastic algorithms that adopt momentum and variance
reduction techniques have been developed for composition optimization. However,
these algorithms do not fully exploit both techniques to accelerate the
convergence and are lack of convergence guarantee in nonconvex optimization.
This paper complements the existing literature by developing various momentum
schemes with SPIDER-based variance reduction for non-convex composition
optimization. In particular, our momentum design requires less number of
proximal mapping evaluations per-iteration than that required by the existing
Katyusha momentum. Furthermore, our algorithm achieves near-optimal sample
complexity results in both non-convex finite-sum and online composition
optimization and achieves a linear convergence rate under the gradient dominant
condition. Numerical experiments demonstrate that our algorithm converges
significantly faster than existing algorithms in nonconvex composition
optimization.
