Asymptotic optimality of the d^{1/3} dimension-dependent bound

Determine whether the d^{1/3} dependence in the gradient query complexity bound \~O(d^{1/3} L_2^{1/2} Δ ε^{-3/2} + d) stated in Corollary 1 for computing an ε-critical point of a twice-differentiable function with L1-Lipschitz gradient and L2-Lipschitz Hessian is asymptotically optimal. Develop tight dimension-dependent lower bounds for the number of gradient queries required in this regime to confirm or refute the optimality of the d^{1/3} factor.

Background

The paper proves a dimension-dependent gradient-only complexity via a finite-difference implementation of approximate Hessian queries, yielding ~O(d^{1/3} L_2^{1/2} Δ ε^{-3/2} + d) queries under L1-Lipschitz gradient and L2-Lipschitz Hessian assumptions. This improves prior state-of-the-art bounds for sufficiently large dimensions.

The authors note that while lower bounds are known in dimension-independent settings, tight dimension-dependent lower bounds—especially under Hessian Lipschitzness—are lacking. They flag the asymptotic optimality of the d^{1/3} dependence as unclear and identify developing tight lower bounds in this regime as an open problem.