# Stochastic Langevin Monte Carlo for (weakly) log-concave posterior distributions

arXiv: https://arxiv.org/abs/2301.03077 · HTML: https://mlsanity.com/paper/2301.03077/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 38 references that are in the map.
- First posted on arXiv 8 Jan 2023.

## Builds on (most cited first, 12 of 12)

- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization](https://mlsanity.com/paper/1707.06618/) (2017)
- [Sampling Can Be Faster Than Optimization](https://mlsanity.com/paper/1811.08413/) (2018)
- [On sampling from a log-concave density using kinetic Langevin diffusions](https://mlsanity.com/paper/1807.09382/) (2018)
- [Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity](https://mlsanity.com/paper/1907.11331/) (2019)
- [Bounding the error of discretized Langevin algorithms for non-strongly log-concave targets](https://mlsanity.com/paper/1906.08530/) (2019)
- [How to compute the barycenter of a weighted graph](https://mlsanity.com/paper/1605.04148/) (2016)
- [Laplacian Smoothing Stochastic Gradient Markov Chain Monte Carlo](https://mlsanity.com/paper/1911.00782/) (2019)
- [On the cost of Bayesian posterior mean strategy for log-concave models](https://mlsanity.com/paper/2010.06420/) (2020)

## Abstract (arXiv)

In this paper, we investigate a continuous time version of the Stochastic Langevin Monte Carlo method, introduced in [WT11], that incorporates a stochastic sampling step inside the traditional over-damped Langevin diffusion. This method is popular in machine learning for sampling posterior distribution. We will pay specific attention in our work to the computational cost in terms of n (the number of observations that produces the posterior distribution), and d (the dimension of the ambient space where the parameter of interest is living). We derive our analysis in the weakly convex framework, which is parameterized with the help of the Kurdyka-L ojasiewicz (KL) inequality, that permits to handle a vanishing curvature settings, which is far less restrictive when compared to the simple strongly convex case. We establish that the final horizon of simulation to obtain an varepsilon approximation (in terms of entropy) is of the order ( d log(n)2 )(1+r)2 [log2(varepsilon-1) + n2 d2(1+r) log4(1+r)(n) ] with a Poissonian subsampling of parameter left(n ( d log2(n))1+rright)-1, where the parameter r is involved in the KL inequality and varies between 0 (strongly convex case) and 1 (limiting Laplace situation).