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

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 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.