SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator

TL;DR

This paper introduces SPIDER, a novel stochastic estimator that significantly reduces computational costs in non-convex optimization, enabling faster convergence for both first-order and zeroth-order methods.

Contribution

The paper presents SPIDER, a new stochastic estimator, and develops algorithms that achieve near-optimal convergence rates for non-convex stochastic optimization problems.

Findings

Achieves a gradient computation cost of O(min(n^{1/2} ε^{-2}, ε^{-3})) for first-order stationary points.

Outperforms existing methods in zeroth-order stochastic optimization with cost O(d min(n^{1/2} ε^{-2}, ε^{-3})).

Provides sharp convergence bounds and nearly matches lower bounds for first-order stationary point finding.

Abstract

In this paper, we propose a new technique named \textit{Stochastic Path-Integrated Differential EstimatoR} (SPIDER), which can be used to track many deterministic quantities of interest with significantly reduced computational cost. We apply SPIDER to two tasks, namely the stochastic first-order and zeroth-order methods. For stochastic first-order method, combining SPIDER with normalized gradient descent, we propose two new algorithms, namely SPIDER-SFO and SPIDER-SFO\textsuperscript{+}, that solve non-convex stochastic optimization problems using stochastic gradients only. We provide sharp error-bound results on their convergence rates. In special, we prove that the SPIDER-SFO and SPIDER-SFO\textsuperscript{+} algorithms achieve a record-breaking gradient computation cost of $\mathcal{O}\left( \min( n^{1/2} \epsilon^{-2}, \epsilon^{-3} ) \right)$ for finding an $\epsilon$-approximate first-order and $\tilde{\mathcal{O}}\left( \min( n^{1/2} \epsilon^{-2}+\epsilon^{-2.5}, \epsilon^{-3} ) \right)$ for finding an $(\epsilon, \mathcal{O}(\epsilon^{0.5}))$-approximate second-order stationary point, respectively. In addition, we prove that SPIDER-SFO nearly matches the algorithmic lower bound for finding approximate first-order stationary points under the gradient Lipschitz assumption in the finite-sum setting. For stochastic zeroth-order method, we prove a cost of $\mathcal{O}( d \min( n^{1/2} \epsilon^{-2}, \epsilon^{-3}) )$ which outperforms all existing results.