# Fast Convergence of Phi-Divergence Along the Unadjusted Langevin Algorithm and Proximal Sampler

arXiv: https://arxiv.org/abs/2410.10699 · HTML: https://mlsanity.com/paper/2410.10699/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 55 references that are in the map.
- First posted on arXiv 14 Oct 2024.

## Builds on (most cited first, 16 of 16)

- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [Structured Logconcave Sampling with a Restricted Gaussian Oracle](https://mlsanity.com/paper/2010.03106/) (2020)
- [Improved analysis for a proximal algorithm for sampling](https://mlsanity.com/paper/2202.06386/) (2022)
- [Faster high-accuracy log-concave sampling via algorithmic warm starts](https://mlsanity.com/paper/2302.10249/) (2023)
- [Improved dimension dependence of a proximal algorithm for sampling](https://mlsanity.com/paper/2302.10081/) (2023)
- [Langevin Monte Carlo without smoothness](https://mlsanity.com/paper/1905.13285/) (2019)
- [A Proximal Algorithm for Sampling from Non-smooth Potentials](https://mlsanity.com/paper/2110.04597/) (2021)
- [Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling](https://mlsanity.com/paper/2210.08448/) (2022)
- [An Efficient Sampling Algorithm for Non-smooth Composite Potentials](https://mlsanity.com/paper/1910.00551/) (2019)
- [In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies](https://mlsanity.com/paper/2405.01425/) (2024)
- [The query complexity of sampling from strongly log-concave distributions in one dimension](https://mlsanity.com/paper/2105.14163/) (2021)
- [The Sample Complexity of Simple Binary Hypothesis Testing](https://mlsanity.com/paper/2403.16981/) (2024)

## Built on by (most cited first, 11 of 11)

- [Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality](https://mlsanity.com/paper/2502.05623/) (2025)
- [Beyond Scores: Proximal Diffusion Models](https://mlsanity.com/paper/2507.08956/) (2025)
- [Oracle-based Uniform Sampling from Convex Bodies](https://mlsanity.com/paper/2510.02983/) (2025)
- [A Non-asymptotic Analysis for Learning and Applying a Preconditioner in MCMC](https://mlsanity.com/paper/2602.10714/) (2026)
- [Mixing Times and Privacy Analysis for the Projected Langevin Algorithm under a Modulus of Continuity](https://mlsanity.com/paper/2501.04134/) (2025)
- [Improved sampling algorithms and functional inequalities for non-log-concave distributions](https://mlsanity.com/paper/2507.11236/) (2025)
- [A transport approach to the cutoff phenomenon](https://mlsanity.com/paper/2509.08560/) (2025)
- [Functional Stochastic Localization](https://mlsanity.com/paper/2602.03999/) (2026)
- [Local and Global Contraction Principles for MCMC Mixing](https://mlsanity.com/paper/2606.03033/) (2026)
- [Local Geometric Mixing via Dobrushin Contraction with Applications to Diffusion Path Monte Carlo and the Proximal Sampler](https://mlsanity.com/paper/2609.28338/) (2026)
- [Reflected Proximal Dynamics for Accelerated High-Accuracy Log-Concave Sampling](https://mlsanity.com/paper/2610.16248/) (2026)

## Abstract (arXiv)

We study the mixing time of two popular discrete-time Markov chains in continuous space, the Unadjusted Langevin Algorithm and the Proximal Sampler, which are discretizations of the Langevin dynamics. We extend mixing time analyses for these Markov chains to hold in Phi-divergence. We show that any Phi-divergence arising from a twice-differentiable strictly convex function Phi converges to 0 exponentially fast along these Markov chains, under the assumption that their stationary distributions satisfy the corresponding Phi-Sobolev inequality, which holds for example when the target distribution of the Langevin dynamics is strongly log-concave. Our setting includes as special cases popular mixing time regimes, namely the mixing in chi-squared divergence under a Poincar'e inequality, and the mixing in relative entropy under a log-Sobolev inequality. Our results follow by viewing the sampling algorithms as noisy channels and bounding the contraction coefficients arising in the appropriate strong data processing inequalities.