{"ID":3084686,"CreatedAt":"2026-06-05T06:46:15.197025399Z","UpdatedAt":"2026-06-06T20:54:36.964885582Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.05438","arxiv_id":"2606.05438","title":"Sharp First-Order Lower Bounds for Higher-Order Smooth Nonconvex Optimization","abstract":"We study the deterministic first-order oracle complexity of finding \\(ε\\)-stationary points in smooth nonconvex optimization when the objective satisfies higher-order smoothness assumptions. While the classical \\(ε^{-2}\\) rate is optimal under only Lipschitz gradients, higher-order smoothness leads to accelerated first-order upper bounds, most notably the \\(ε^{-7/4}\\) rate under Lipschitz Hessians and the \\(ε^{-5/3}\\) rate under Lipschitz third derivatives. The matching lower bounds, however, have remained open. We resolve this gap by proving a new dimension-free first-order lower bound for higher-order smooth nonconvex functions, valid for every finite smoothness order. In particular, our construction gives a matching \\(Ω(ε^{-7/4})\\) lower bound in the Hessian-Lipschitz case and a matching \\(Ω(ε^{-5/3})\\) lower bound in the third-order-smooth regime. The hard instance is based on a \\emph{block-chain} mechanism that enforces blockwise oracle revelation while preserving the smoothness structure needed for the scalar hard instance. The lower-bound construction was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors.","short_abstract":"We study the deterministic first-order oracle complexity of finding \\(ε\\)-stationary points in smooth nonconvex optimization when the objective satisfies higher-order smoothness assumptions. While the classical \\(ε^{-2}\\) rate is optimal under only Lipschitz gradients, higher-order smoothness leads to accelerated first...","url_abs":"https://arxiv.org/abs/2606.05438","url_pdf":"https://arxiv.org/pdf/2606.05438v1","authors":"[\"Dongruo Zhou\"]","published":"2026-06-03T21:01:21Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.OC\"]","methods":"[]","has_code":false}