Paper 2026/1797

High-Precision Lewis Weights via Fourth-Moment Control and Local Bregman Acceleration

Abstract

We study the high-precision computation of $\ell_p$-Lewis weights for $p\ge4$ in the black-box exact-real full-vector leverage-score oracle model, measuring complexity by the number of adaptive oracle rounds. In this model, Gribling, Sidford, and Zhang [GSZ26] obtained an $O(p^2\log(m/\epsilon))$ bound for computing an $\epsilon$-estimate. We improve this bound to $O(p\log(mp)+\sqrt p\log(1/\epsilon))$. To obtain this result, we isolate the normalized fourth-moment operator governing the nonlinear Hessian of their log-determinant matrix potential and prove that each relative-gradient step with denominator $p$ resets the operator norm to a universal constant. This reset controls the entire update segment and yields an $O(p\log(mp))$ global entrance phase. After entering an $O(1/p)$ spectral neighborhood of the optimum, we switch to a restarted accelerated Bregman-gradient method for the vector potential.