Functional Stochastic Localization

2026-02-03

AI Summary

This work investigates the query complexity of zeroth-order differentially private convex optimization under non-Euclidean geometry, with a focus on efficient sampling. We propose a functional generalization of the Eldan stochastic localization process by replacing Gaussian regularization with the log-Laplace transform scaled by an arbitrary positive integer. This novel Markov chain represents the first extension of the stochastic localization framework to a functional setting. By integrating functional Poincaré inequalities with mirror descent principles, we establish an upper bound on the mixing time under the condition that the functional Poincaré inequality holds. As a result, our method achieves the current best-known query complexity for zeroth-order differentially private optimization in ℓ_p spaces with p ∈ [1, 2).

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Privacy

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

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.

Problem

• stochastic localization
• non-Euclidean geometry
• sampling
• functional Poincaré inequality
• differentially private optimization

Innovation

• functional stochastic localization
• log-Laplace regularization
• functional Poincaré inequality
• differentially private optimization
• zeroth-order query complexity