Functional Stochastic Localization

TL;DR

This paper generalizes Eldan's stochastic localization to non-Gaussian regularizations, providing new mixing time bounds and applications to private convex optimization in non-Euclidean geometries.

Contribution

It introduces a functional generalization of Eldan's process using log-Laplace transforms and establishes mixing bounds under Poincaré inequalities, with applications to private convex optimization.

Findings

Provides a mixing time bound for the localization process.

Improves query complexities in private convex optimization for p in [1,2).

Extends stochastic localization to broader regularizations.

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 Poincar\'e 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.