Oracle complexity for the full spectrum of Hölder-smooth optimization settings

Determine the oracle complexity of the entire spectrum of Hölder-smooth convex optimization settings extending beyond the nonsmooth [...].

Background

The paper establishes nearly optimal, polynomial-time oracle rates for nonsmooth Lipschitz convex optimization over [...]-balls when the domain exponent is smaller than both the regularity exponent and 2. The authors note that the broader question cited from prior work also includes Hölder-smooth analogues, which are not treated by the present results. Fully resolving that broader question therefore requires determining oracle complexity across the remaining Hölder-smooth regimes.

References

For fully resolving the open question in \citep{guzman2015open} the oracle complexity of the entire spectrum of H\"older-smooth settings should be considered.

— Stable Movement for Nondual Lipschitz Convex Optimization: Efficiency and Nearly Optimal Oracle Rates  (2609.20701 - Martínez-Rubio et al., 17 Sep 2026) in Section Discussion

A direct $\kappa$-power proximal envelope suggests the same descent profile, but we do not establish the required linear-time proximal-solve bound for that construction.

— Near-Optimal Acceleration for Smooth $\ell_p$ / $\ell_q$ Nondual Convex First-Order Oracle Optimization  (2609.21880 - Martínez-Rubio et al., 18 Sep 2026) in Section 5, “Why constrained feasibility requires a new module”