Abstract: We design an algorithm which finds an $\epsilon$-approximate stationary \n point (with $\|\nabla F(x)\|\le \epsilon$) using $O(\epsilon^{-3})$ \n stochastic gradient and Hessian-vector products, matching guarantees \n that were previously available only under a stronger assumption of access \n to multiple queries with the same random seed. \n We prove a lower bound which establishes that this rate\nis optimal and---surprisingly---that it cannot be improved using stochastic $p$th order methods\nfor any $p\ge2$, even when the first $p$ derivatives of the objective \nare Lipschitz. Together, these results characterize the complexity of\nnon-convex stochastic optimization with second-order methods and beyond.\nExpanding our scope to the oracle complexity of finding \n$(\epsilon,\gamma)$-approximate second-order stationary points, we \nestablish nearly matching upper and lower bounds for stochastic \nsecond-order methods. Our lower bounds here are novel even in the noiseless \ncase.