arXiv:2602.03999 · v3

Functional Stochastic Localization

首次提交 2026-02-04 04:34

版本更新 2026-11-12 00:11

雷达观测 2026-11-13 14:16

摘要

Eldan's stochastic localization is a probabilistic construction that has proved instrumental to modern breakthroughs in high-dimensional geometry and the design of sampling algorithms. Motivated by sampling under non-Euclidean geometries and the mirror descent algorithm in optimization, we develop a functional generalization of Eldan's process that replaces Gaussian regularization with regularization by any positive integer multiple of a log-Laplace transform. We further give a mixing time bound on the Markov chain induced by our localization process, which holds if our target distribution satisfies a functional Poincaré inequality. Finally, we apply our framework to differentially private convex optimization in $\ell_p$ norms for $p \in [1, 2)$, where we improve state-of-the-art query complexities in a zeroth-order model.

作者备注：Fixed error in previous version