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Functional Stochastic Localization

Anming Gu, Bobby Shi, Kevin Tian

TL;DR

The paper generalizes Eldan's stochastic localization to a functional, non-Euclidean setting by leveraging log-Laplace transforms (LLTs) to regularize densities. It constructs a discrete-time localization process with exponential-family updates, establishing martingale properties and a provable relationship E[π_τ]=π, while enabling LLT-based proximal sampling that works in geometries beyond the Euclidean space. A key result is a mixing-time bound under a functional Poincaré inequality, with χ²-contraction that generalizes Euclidean proximal samplers and recovers sharp Euclidean rates in the appropriate limit. The authors apply the framework to differential privacy in ℓ_p geometries, achieving improved zeroth-order query complexities for DP-ERM and DP-SCO, and discuss potential entropic strengthening and broader future directions. Overall, this work broadens the stochastic localization toolkit to non-Euclidean settings, enabling faster, structure-exploiting sampling and optimization in privacy-sensitive, high-dimensional contexts.

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.

Functional Stochastic Localization

TL;DR

The paper generalizes Eldan's stochastic localization to a functional, non-Euclidean setting by leveraging log-Laplace transforms (LLTs) to regularize densities. It constructs a discrete-time localization process with exponential-family updates, establishing martingale properties and a provable relationship E[π_τ]=π, while enabling LLT-based proximal sampling that works in geometries beyond the Euclidean space. A key result is a mixing-time bound under a functional Poincaré inequality, with χ²-contraction that generalizes Euclidean proximal samplers and recovers sharp Euclidean rates in the appropriate limit. The authors apply the framework to differential privacy in ℓ_p geometries, achieving improved zeroth-order query complexities for DP-ERM and DP-SCO, and discuss potential entropic strengthening and broader future directions. Overall, this work broadens the stochastic localization toolkit to non-Euclidean settings, enabling faster, structure-exploiting sampling and optimization in privacy-sensitive, high-dimensional contexts.

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 norms for , where we improve state-of-the-art query complexities in a zeroth-order model.
Paper Structure (37 sections, 27 theorems, 116 equations)

This paper contains 37 sections, 27 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Functional stochastic localization.
  3. Applications to sampling.
  4. Our approach
  5. Defining functional stochastic localization.
  6. Non-Euclidean sampling.
  7. Prior work
  8. Proximal sampling.
  9. Localization schemes.
  10. Renormalization.
  11. Preliminaries
  12. Probability.
  13. Optimization.
  14. Log-Laplace transforms.
  15. Localization processes.
  16. ...and 22 more sections

Key Result

Theorem 1

Let $\varphi: \mathbb{R}^d \to \mathbb{R}$ be convex, $\psi := \varphi^\sharp$, and $\pi$ satisfy an $\alpha$-$\psi$-Poincaré inequality (Definition def:vpi). For any $\mu_0\ll \pi$, iterate $\mathbf{x}_k$ of Algorithm alg:alternate has density $\mu_k$ satisfying

Theorems & Definitions (60)

  • Theorem 1
  • Definition 1
  • Lemma 1: LLT convolution
  • proof
  • Definition 2
  • proof
  • Definition 3: $\varphi$-PI
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 50 more