# Theoretical guarantees for approximate sampling from smooth and log-concave densities

arXiv: https://arxiv.org/abs/1412.7392 · HTML: https://mlsanity.com/paper/1412.7392/

## Key facts

- Cited by 208 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Dec 2014.

## Built on by (most cited first, 30 of 208)

- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
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- [Convex Analysis of the Mean Field Langevin Dynamics](https://mlsanity.com/paper/2201.10469/) (2022)
- [Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance](https://mlsanity.com/paper/2212.06359/) (2022)
- [Private Convex Optimization via Exponential Mechanism](https://mlsanity.com/paper/2203.00263/) (2022)
- [Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Sampling](https://mlsanity.com/paper/2109.13055/) (2021)
- [Differentiable Annealed Importance Sampling and the Perils of Gradient Noise](https://mlsanity.com/paper/2107.10211/) (2021)
- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [FedPop: A Bayesian Approach for Personalised Federated Learning](https://mlsanity.com/paper/2206.03611/) (2022)
- [Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequality](https://mlsanity.com/paper/2303.03589/) (2023)
- [Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics](https://mlsanity.com/paper/2211.00450/) (2022)
- [Improved dimension dependence of a proximal algorithm for sampling](https://mlsanity.com/paper/2302.10081/) (2023)
- [Provable and Practical: Efficient Exploration in Reinforcement Learning via Langevin Monte Carlo](https://mlsanity.com/paper/2305.18246/) (2023)
- [Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance](https://mlsanity.com/paper/2501.02298/) (2025)
- [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023)
- [Stochastic Localization via Iterative Posterior Sampling](https://mlsanity.com/paper/2402.10758/) (2024)
- [QLSD: Quantised Langevin stochastic dynamics for Bayesian federated learning](https://mlsanity.com/paper/2106.00797/) (2021)
- [Particle algorithms for maximum likelihood training of latent variable models](https://mlsanity.com/paper/2204.12965/) (2022)
- [Towards Predicting Equilibrium Distributions for Molecular Systems with Deep Learning](https://mlsanity.com/paper/2306.05445/) (2023)
- [Sqrt(d) Dimension Dependence of Langevin Monte Carlo](https://mlsanity.com/paper/2109.03839/) (2021)
- [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)
- [Implicit Diffusion: Efficient Optimization through Stochastic Sampling](https://mlsanity.com/paper/2402.05468/) (2024)
- [A Proximal Algorithm for Sampling](https://mlsanity.com/paper/2202.13975/) (2022)

## Abstract (arXiv)

Sampling from various kinds of distributions is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators, test procedures or confidence intervals. In many situations, the exact sampling from a given distribution is impossible or computationally expensive and, therefore, one needs to resort to approximate sampling strategies. However, there is no well-developed theory providing meaningful nonasymptotic guarantees for the approximate sampling procedures, especially in the high-dimensional problems. This paper makes some progress in this direction by considering the problem of sampling from a distribution having a smooth and log-concave density defined on (RRp), for some integer (p>0). We establish nonasymptotic bounds for the error of approximating the target distribution by the one obtained by the Langevin Monte Carlo method and its variants. We illustrate the effectiveness of the established guarantees with various experiments. Underlying our analysis are insights from the theory of continuous-time diffusion processes, which may be of interest beyond the framework of log-concave densities considered in the present work.