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We develop a new algorithm for non-convex stochastic optimization that findsan $\epsilon$-critical point in the optimal $O(\epsilon^{-3})$ stochasticgradient and Hessian-vector product computations. Our algorithm usesHessian-vector products to "correct" a bias term in the momentum of SGD withmomentum. This leads to better gradient estimates in a manner analogous tovariance reduction methods. In contrast to prior work, we do not requireexcessively large batch sizes, and are able to provide an adaptive algorithmwhose convergence rate automatically improves with decreasing variance in thegradient estimates. We validate our results on a variety of large-scale deeplearning architectures and benchmarks tasks.

Better SGD using Second-order Momentum