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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 objectivearXiv:2006.13476v1 fatcat:ayf2gykjpfeldogyxjgmstraam