Stochastic Recursive Variance-Reduced Cubic Regularization Methods

Abstract

Stochastic Variance-Reduced Cubic regularization (SVRC) algorithms have received increasing attention due to its improved gradient/Hessian complexities (i.e., number of queries to stochastic gradient/Hessian oracles) to find local minima for nonconvex finite-sum optimization. However, it is unclear whether existing SVRC algorithms can be further improved. Moreover, the semi-stochastic Hessian estimator adopted in existing SVRC algorithms prevents the use of Hessian-vector product-based fast cubic subproblem solvers, which makes SVRC algorithms computationally intractable for high-dimensional problems. In this paper, we first present a Stochastic Recursive Variance-Reduced Cubic regularization method (SRVRC) using a recursively updated semi-stochastic gradient and Hessian estimators. It enjoys improved gradient and Hessian complexities to find an $(\epsilon, \sqrt{\epsilon})$-approximate local minimum, and outperforms the state-of-the-art SVRC algorithms. Built upon SRVRC, we further propose a Hessian-free SRVRC algorithm, namely SRVRC$_{\text{free}}$, which only needs $\tilde O(n\epsilon^{-2} \land \epsilon^{-3})$ stochastic gradient and Hessian-vector product computations, where $n$ is the number of component functions in the finite-sum objective and $\epsilon$ is the optimization precision. This outperforms the best-known result $\tilde O(\epsilon^{-3.5})$ achieved by stochastic cubic regularization algorithm proposed in \cite{tripuraneni2018stochastic}.