High-probability zeroth-order online convex optimisation beyond Euclidean geometry

We study online convex optimisation with $\ell_q$-Lipschitz losses, $\ell_p$-regularised FTRL, and randomised two-point finite-difference gradient estimators based on cone-measure sampling from $\ell_r$-spheres. For random Lipschitz losses whose mean is convex, we prove unified high-probability regret bounds for all $p,q,r \in [1,\infty]$. The analysis is driven by all-moment bounds for the gradient estimator in the dual FTRL norm, yielding time-uniform control of the quadratic variation. The algorithm is anytime and data-driven; in the special cases previously studied, its rates recover the known in-expectation guarantees while strengthening them to time-uniform high probability. Together with constant-probability lower bounds, these results establish optimality for $q\in[1,2]$ under appropriate sampling geometry, and expose a gap for $q>2$ that appears intrinsic to the estimators themselves.