Second-Order Information in Non-Convex Stochastic Optimization: Power and Limitations

We design an algorithm which finds an ε-approximate stationary point (with k∇F(x)k ≤ ε) using O(ε⁻³) stochastic gradient and Hessian-vector products, matching guarantees that were previously available only under a stronger assumption of access to multiple queries with the same random seed. We prove a lower bound which establishes that this rate is optimal and—surprisingly—that it cannot be improved using stochastic pth order methods for any p ≥ 2, even when the first p derivatives of the objective are Lipschitz. Together, these results characterize the complexity of non-convex stochastic optimization with second-order methods and beyond. Expanding our scope to the oracle complexity of finding (ε, γ)-approximate second-order stationary points, we establish nearly matching upper and lower bounds for stochastic second-order methods. Our lower bounds here are novel even in the noiseless case.