  SARAH and SPIDER are two recently developed stochastic variance-reduced
algorithms, and SPIDER has been shown to achieve a near-optimal first-order
oracle complexity in smooth nonconvex optimization. However, SPIDER uses an
accuracy-dependent stepsize that slows down the convergence in practice, and
cannot handle objective functions that involve nonsmooth regularizers. In this
paper, we propose SpiderBoost as an improved scheme, which allows to use a much
larger constant-level stepsize while maintaining the same near-optimal oracle
complexity, and can be extended with proximal mapping to handle composite
optimization (which is nonsmooth and nonconvex) with provable convergence
guarantee. In particular, we show that proximal SpiderBoost achieves an oracle
complexity of $\mathcal{O}(\min\{n^{1/2}\epsilon^{-2},\epsilon^{-3}\})$ in
composite nonconvex optimization, improving the state-of-the-art result by a
factor of $\mathcal{O}(\min\{n^{1/6},\epsilon^{-1/3}\})$. We further develop a
novel momentum scheme to accelerate SpiderBoost for composite optimization,
which achieves the near-optimal oracle complexity in theory and substantial
improvement in experiments.
