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.

References

It remains unclear whether our bound, particularly the $d{1/3}$ dependence, is asymptotically optimal. ... The development of tight lower bounds in our regime as an independent and interesting open problem.

— Balancing Gradient and Hessian Queries in Non-Convex Optimization  (2510.20786 - Adil et al., 23 Oct 2025) in Subsection 'Our Results', paragraph 'Dimension-Dependent Critical Point Computation'

On the theoretical side, while this paper resolves the convex setting up to logarithmic factors, the complexity for nonconvex settings remains an open, where the best-known upper and lower bounds \citep{chen2026faster} for finding an $\epsilon$-stationary point are $(m+m{1/3} \epsilon{-3/2})$ and $\Omega(\epsilon{-3/2})$, respectively.

— Near-Optimal Convex Optimization with Lazy Second-Order Oracles  (2610.03222 - Zhang et al., 2 Oct 2026) in Section Conclusion and Future Works