# Mean-Square Analysis of Discretized Ito Diffusions for Heavy-tailed Sampling

arXiv: https://arxiv.org/abs/2303.00570 · HTML: https://mlsanity.com/paper/2303.00570/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 88 references that are in the map.
- First posted on arXiv 1 Mar 2023.

## Builds on (most cited first, 30 of 41)

- [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)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Structured Logconcave Sampling with a Restricted Gaussian Oracle](https://mlsanity.com/paper/2010.03106/) (2020)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [Improved analysis for a proximal algorithm for sampling](https://mlsanity.com/paper/2202.06386/) (2022)
- [Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo](https://mlsanity.com/paper/2202.05214/) (2022)
- [On sampling from a log-concave density using kinetic Langevin diffusions](https://mlsanity.com/paper/1807.09382/) (2018)
- [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018)
- [Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm](https://mlsanity.com/paper/2012.12810/) (2020)
- [Wasserstein Control of Mirror Langevin Monte Carlo](https://mlsanity.com/paper/2002.04363/) (2020)
- [On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness](https://mlsanity.com/paper/2005.13097/) (2020)
- [Efficient constrained sampling via the mirror-Langevin algorithm](https://mlsanity.com/paper/2010.16212/) (2020)
- [Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients](https://mlsanity.com/paper/1905.12247/) (2019)
- [Learning Halfspaces with Massart Noise Under Structured Distributions](https://mlsanity.com/paper/2002.05632/) (2020)
- [Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Sampling](https://mlsanity.com/paper/2109.13055/) (2021)
- [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)
- [Coupling and Convergence for Hamiltonian Monte Carlo](https://mlsanity.com/paper/1805.00452/) (2018)
- [Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond](https://mlsanity.com/paper/1906.07868/) (2019)
- [Fractional Underdamped Langevin Dynamics: Retargeting SGD with Momentum under Heavy-Tailed Gradient Noise](https://mlsanity.com/paper/2002.05685/) (2020)
- [Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions](https://mlsanity.com/paper/1905.02313/) (2019)
- [Measuring Sample Quality with Diffusions](https://mlsanity.com/paper/1611.06972/) (2016)
- [On the Ergodicity, Bias and Asymptotic Normality of Randomized Midpoint Sampling Method](https://mlsanity.com/paper/2011.03176/) (2020)
- [Efficient MCMC Sampling with Dimension-Free Convergence Rate using ADMM-type Splitting](https://mlsanity.com/paper/1905.11937/) (2019)
- [Logsmooth Gradient Concentration and Tighter Runtimes for Metropolized Hamiltonian Monte Carlo](https://mlsanity.com/paper/2002.04121/) (2020)
- [Riemannian Langevin Algorithm for Solving Semidefinite Programs](https://mlsanity.com/paper/2010.11176/) (2020)

## Built on by (most cited first, 4 of 4)

- [Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequality](https://mlsanity.com/paper/2303.03589/) (2023)
- [Zeroth-Order Sampling Methods for Non-Log-Concave Distributions: Alleviating Metastability by Denoising Diffusion](https://mlsanity.com/paper/2402.17886/) (2024)
- [Anchored Langevin Algorithms](https://mlsanity.com/paper/2509.19455/) (2025)
- [Sampling by averaging: A multiscale approach to score estimation](https://mlsanity.com/paper/2508.15069/) (2025)

## Abstract (arXiv)

We analyze the complexity of sampling from a class of heavy-tailed distributions by discretizing a natural class of Ito diffusions associated with weighted Poincar'e inequalities. Based on a mean-square analysis, we establish the iteration complexity for obtaining a sample whose distribution is ε close to the target distribution in the Wasserstein-2 metric. In this paper, our results take the mean-square analysis to its limits, i.e., we invariably only require that the target density has finite variance, the minimal requirement for a mean-square analysis. To obtain explicit estimates, we compute upper bounds on certain moments associated with heavy-tailed targets under various assumptions. We also provide similar iteration complexity results for the case where only function evaluations of the unnormalized target density are available by estimating the gradients using a Gaussian smoothing technique. We provide illustrative examples based on the multivariate t-distribution.