Functional Stochastic Localization

Anming Gu, Bobby Shi, Kevin Tian
Proceedings of Thirty Ninth Conference on Learning Theory, PMLR 336:2963-3004, 2026.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v336-gu26b, title = {Functional Stochastic Localization}, author = {Gu, Anming and Shi, Bobby and Tian, Kevin}, booktitle = {Proceedings of Thirty Ninth Conference on Learning Theory}, pages = {2963--3004}, year = {2026}, editor = {Hanneke, Steve and Lattimore, Tor}, volume = {336}, series = {Proceedings of Machine Learning Research}, month = {29 Jun--03 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v336/main/assets/gu26b/gu26b.pdf}, url = {https://proceedings.mlr.press/v336/gu26b.html}, 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.} }
Endnote
%0 Conference Paper %T Functional Stochastic Localization %A Anming Gu %A Bobby Shi %A Kevin Tian %B Proceedings of Thirty Ninth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2026 %E Steve Hanneke %E Tor Lattimore %F pmlr-v336-gu26b %I PMLR %P 2963--3004 %U https://proceedings.mlr.press/v336/gu26b.html %V 336 %X 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.
APA
Gu, A., Shi, B. & Tian, K.. (2026). Functional Stochastic Localization. Proceedings of Thirty Ninth Conference on Learning Theory, in Proceedings of Machine Learning Research 336:2963-3004 Available from https://proceedings.mlr.press/v336/gu26b.html.

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