Gradient-Hessian Query Tradeoffs: Matching Lower Bounds in Logarithmic Dimension

Abstract

We establish matching lower bounds for the tradeoff between gradient and Hessian queries in nonconvex optimization with Lipschitz Hessian. In normalized units, the initial objective gap is at most $T$, the Hessian Lipschitz constant is at most one, and the target gradient norm is one. For sufficiently large $T$, dimension $d \ge C\log(2+T)$, and a prescribed Hessian budget $h \ge 1$ with $h+1 \le aT$, every randomized algorithm requires $Ω(\min\{Td^{1/3}, T^{3/2}/\sqrt{h+1}\})$ ordinary queries. An ordinary query returns the value and gradient; a Hessian query returns the complete second jet and is charged only to its own budget. Both budgets are worst-case caps. Queries and outputs may be arbitrary real points, and no Hessian-norm bound is imposed. The hard input law consists of fixed, globally $C^3$, coercive functions. The construction combines sequential readouts of a continuous random matrix with localized completion blocks and reliability dimensions that vary across hierarchy levels. Premature disclosures are charged by their expected task cost, while every oracle reply remains the exact jet of the presampled objective. This yields a logarithmic dimension threshold and a depth-independent third-derivative bound. Together with a translation of the NALEN upper bound of Chen, Liu, Luo, and Zhang, the result determines the query tradeoff up to absolute constants throughout the stated range.