# Functional Stochastic Localization

arXiv: https://arxiv.org/abs/2602.03999 · HTML: https://mlsanity.com/paper/2602.03999/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 40 of its 90 references that are in the map.
- First posted on arXiv 3 Feb 2026.

## Builds on (most cited first, 30 of 40)

- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Structured Logconcave Sampling with a Restricted Gaussian Oracle](https://mlsanity.com/paper/2010.03106/) (2020)
- [Private Stochastic Convex Optimization: Optimal Rates in ℓ1 Geometry](https://mlsanity.com/paper/2103.01516/) (2021)
- [Improved analysis for a proximal algorithm for sampling](https://mlsanity.com/paper/2202.06386/) (2022)
- [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018)
- [Convex Analysis of the Mean Field Langevin Dynamics](https://mlsanity.com/paper/2201.10469/) (2022)
- [Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients](https://mlsanity.com/paper/1905.12247/) (2019)
- [Non-Euclidean Differentially Private Stochastic Convex Optimization: Optimal Rates in Linear Time](https://mlsanity.com/paper/2103.01278/) (2021)
- [Private Convex Optimization via Exponential Mechanism](https://mlsanity.com/paper/2203.00263/) (2022)
- [Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization](https://mlsanity.com/paper/2203.05093/) (2022)
- [Faster high-accuracy log-concave sampling via algorithmic warm starts](https://mlsanity.com/paper/2302.10249/) (2023)
- [Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo](https://mlsanity.com/paper/1705.08964/) (2017)
- [Exponential ergodicity of mirror-Langevin diffusions](https://mlsanity.com/paper/2005.09669/) (2020)
- [The Mirror Langevin Algorithm Converges with Vanishing Bias](https://mlsanity.com/paper/2109.12077/) (2021)
- [Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetry](https://mlsanity.com/paper/1911.01469/) (2019)
- [Improved dimension dependence of a proximal algorithm for sampling](https://mlsanity.com/paper/2302.10081/) (2023)
- [Convergence of the Riemannian Langevin Algorithm](https://mlsanity.com/paper/2204.10818/) (2022)
- [A Proximal Algorithm for Sampling from Non-smooth Potentials](https://mlsanity.com/paper/2110.04597/) (2021)
- [Riemannian Langevin Algorithm for Solving Semidefinite Programs](https://mlsanity.com/paper/2010.11176/) (2020)
- [An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations](https://mlsanity.com/paper/1304.2338/) (2013)
- [Localization Schemes: A Framework for Proving Mixing Bounds for Markov Chains](https://mlsanity.com/paper/2203.04163/) (2022)
- [Private Convex Optimization in General Norms](https://mlsanity.com/paper/2207.08347/) (2022)
- [Sampling and Integration of Logconcave Functions by Algorithmic Diffusion](https://mlsanity.com/paper/2411.13462/) (2024)
- [Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality](https://mlsanity.com/paper/2502.05623/) (2025)
- [Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithm](https://mlsanity.com/paper/2312.08823/) (2023)
- [Fast Convergence of Phi-Divergence Along the Unadjusted Langevin Algorithm and Proximal Sampler](https://mlsanity.com/paper/2410.10699/) (2024)

## Abstract (arXiv)

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 ℓp norms for p ∈ [1, 2), where we improve state-of-the-art query complexities in a zeroth-order model.