# Analysis of Langevin Monte Carlo from Poincar'e to Log-Sobolev

arXiv: https://arxiv.org/abs/2112.12662 · HTML: https://mlsanity.com/paper/2112.12662/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Dec 2021.

## Built on by (most cited first, 2 of 2)

- [Langevin Dynamics: A Unified Perspective on Optimization via Lyapunov Potentials](https://mlsanity.com/paper/2407.04264/) (2024)
- [Near-Linear Accuracy Bounds for Moreau--Yosida Unadjusted Langevin Sampling](https://mlsanity.com/paper/2609.40193/) (2026)

## Abstract (arXiv)

Classically, the continuous-time Langevin diffusion converges exponentially fast to its stationary distribution π under the sole assumption that π satisfies a Poincar'e inequality. Using this fact to provide guarantees for the discrete-time Langevin Monte Carlo (LMC) algorithm, however, is considerably more challenging due to the need for working with chi-squared or R'enyi divergences, and prior works have largely focused on strongly log-concave targets. In this work, we provide the first convergence guarantees for LMC assuming that π satisfies either a Latala--Oleszkiewicz or modified log-Sobolev inequality, which interpolates between the Poincar'e and log-Sobolev settings. Unlike prior works, our results allow for weak smoothness and do not require convexity or dissipativity conditions.