Stabilized SVRG: Simple Variance Reduction for Nonconvex Optimization

Variance reduction techniques like SVRG provide simple and fast algorithms for optimizing a convex finite-sum objective. For nonconvex objectives, these techniques can also find a first-order stationary point (with small gradient). However, in nonconvex optimization it is often crucial to find a second-order stationary point (with small gradient and almost PSD hessian). In this paper, we show that Stabilized SVRG (a simple variant of SVRG) can find an $epsilon$-second-order stationary point using only $widetilde{O}(n^{2/3}/epsilon^2+n/epsilon^{1.5})$ stochastic gradients. To our best knowledge, this is the first second-order guarantee for a simple variant of SVRG. The running time almost matches the known guarantees for finding $epsilon$-first-order stationary points.