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

Yossi Arjevani, Yair Carmon, John C. Duchi, Dylan J. Foster, Ayush Sekhari, Karthik Sridharan
2020 arXiv   pre-print
We design an algorithm which finds an ϵ-approximate stationary point (with ∇ F(x)<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
more » ... 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.
arXiv:2006.13476v1 fatcat:ayf2gykjpfeldogyxjgmstraam