Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Gu, Anming, Shi, Bobby, Tian, Kevin
Institution: The University of Texas at Austin
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7128083894
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