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

Part of Advances in Neural Information Processing Systems 31 (NeurIPS 2018)

#### Authors

*Cong Fang, Chris Junchi Li, Zhouchen Lin, Tong Zhang*

#### 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 interests with significantly reduced computational cost.
Combining SPIDER with the method of normalized gradient descent, we propose SPIDER-SFO that solve non-convex stochastic optimization problems using stochastic gradients only.
We provide a few error-bound results on its convergence rates.
Specially, we prove that the SPIDER-SFO algorithm achieves a gradient computation cost of $\mathcal{O}\left( \min( n^{1/2} \epsilon^{-2}, \epsilon^{-3} ) \right)$ to find an $\epsilon$-approximate first-order stationary point.
In addition, we prove that SPIDER-SFO nearly matches the algorithmic lower bound for finding stationary point under the gradient Lipschitz assumption in the finite-sum setting.
Our SPIDER technique can be further applied to find an $(\epsilon, \mathcal{O}(\ep^{0.5}))$-approximate second-order stationary point at a gradient computation cost of $\tilde{\mathcal{O}}\left( \min( n^{1/2} \epsilon^{-2}+\epsilon^{-2.5}, \epsilon^{-3} ) \right)$.