Paper

Sharp First-Order Lower Bounds for Higher-Order Smooth Nonconvex Optimization

arXiv:2606.05438v1 Announce Type: new Abstract: We study the deterministic first-order oracle complexity of finding \(\epsilon\)-stationary points in smooth nonconvex optimization when the objective satisfies higher-order smoothness assumptions. While the classical \(\epsilon^{-2}\) rate is optimal under only Lipschitz gradients, higher-order smoothness leads to accelerated first-order upper bounds, most notably the \(\epsilon^{-7/4}\) rate under Lipschitz Hessians and the \(\epsilon^{-5/3}\) rate under Lipschitz third derivatives. The matching lower bounds, however, have remained open. We re…

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