Asymptotic optimality of the gradient–Hessian query trade-offs

Ascertain whether the query complexities achieved in Theorem 1—namely computing an ε-critical point of a twice-differentiable function with L2-Lipschitz Hessian (and optionally L1-Lipschitz gradient) using at most n_H queries to a δ-approximate Hessian oracle and O(Δ L_2^{1/4} c_δ^{1/2} ε^{-7/4} · polylog(c_ℓ / c_δ)) gradient queries—are asymptotically optimal in the stated oracle model. Establish matching lower bounds across values of n_H or provide faster algorithms to resolve the optimality question.

Background

The main theorem provides a trade-off between the number of gradient queries and (approximate) Hessian queries required to find ε-critical points under smoothness assumptions, recovering and extending several prior results.

In the conclusion, the authors explicitly state that the asymptotic optimality of their query complexities is unknown, pointing to the need for lower bounds or improved algorithms to settle optimality.