The First-Order Oracle Complexity of Lipschitz Convex Optimization in Nondual Settings
We study first-order black-box convex optimization over an $\ell_p$-ball for objectives Lipschitz in the $\ell_q$-norm, solving in the affirmative the nonsmooth version of the COLT open question (Guz15b) on whether the geometry of a smaller feasible set ($p < q$) can improve convergence rates in convex optimization, and matching prior lower bounds up to logarithmic factors. Our rates include \(\widetilde O(1/T)\) for convex Euclidean-Lipschitz optimization over the $\ell_1$-ball, improving on the $O(1/\sqrt{T})$ classical rate under general assumptions. The key technical device is a new online learning game, where the comparator is evaluated using the maximum of affine losses observed so ...
| Authors | David Martínez-Rubio, Brian Bullins, Cristóbal Guzmán et al. |
|---|---|
| Primary category | math.OC |
| Published | 17 September 2026 |
| Last updated at source | 17 September 2026 |
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