SpiderBoost and Momentum: Faster Stochastic Variance Reduction Algorithms

TL;DR

This paper introduces SpiderBoost, an improved stochastic variance reduction algorithm that uses larger stepsizes, handles nonsmooth regularizers, and accelerates convergence in nonconvex optimization, with theoretical guarantees and practical improvements.

Contribution

SpiderBoost extends SPIDER with larger stepsizes and proximal mapping, achieving near-optimal complexity and improved practical performance in nonsmooth nonconvex optimization.

Findings

SpiderBoost achieves an oracle complexity of O(min{n^{1/2}ε^{-2}, ε^{-3}}).

Proposed momentum scheme accelerates convergence in experiments.

Improves state-of-the-art complexity bounds for composite nonconvex optimization.

Abstract

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.