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

arXiv:2606.05438v2 Announce Type: replace 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 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 (Omega(epsilon^{-7/4})) lower bound in the Hessian-Lipschitz case and a matching (Omega(epsilon^{-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.

Source: arXiv cs.LG | 2026-09-23