# An Improved Analysis of Langevin Algorithms with Prior Diffusion for Non-Log-Concave Sampling

arXiv: https://arxiv.org/abs/2403.06183 · HTML: https://mlsanity.com/paper/2403.06183/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 43 references that are in the map.
- First posted on arXiv 10 Mar 2024.

## Builds on (most cited first, 22 of 22)

- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [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)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (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)
- [Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization](https://mlsanity.com/paper/1707.06618/) (2017)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [Sampling Can Be Faster Than Optimization](https://mlsanity.com/paper/1811.08413/) (2018)
- [Consistency and fluctuations for stochastic gradient Langevin dynamics](https://mlsanity.com/paper/1409.0578/) (2014)
- [Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients](https://mlsanity.com/paper/1905.12247/) (2019)
- [Faster high-accuracy log-concave sampling via algorithmic warm starts](https://mlsanity.com/paper/2302.10249/) (2023)
- [Reverse Diffusion Monte Carlo](https://mlsanity.com/paper/2307.02037/) (2023)
- [Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetry](https://mlsanity.com/paper/1911.01469/) (2019)
- [Faster Convergence of Stochastic Gradient Langevin Dynamics for Non-Log-Concave Sampling](https://mlsanity.com/paper/2010.09597/) (2020)
- [Langevin diffusions and the Metropolis-adjusted Langevin algorithm](https://mlsanity.com/paper/1309.2983/) (2013)
- [Is There an Analog of Nesterov Acceleration for MCMC?](https://mlsanity.com/paper/1902.00996/) (2019)
- [Faster Sampling without Isoperimetry via Diffusion-based Monte Carlo](https://mlsanity.com/paper/2401.06325/) (2024)
- [When is the Convergence Time of Langevin Algorithms Dimension Independent? A Composite Optimization Viewpoint](https://mlsanity.com/paper/2110.01827/) (2021)

## Abstract (arXiv)

Understanding the dimension dependency of computational complexity in high-dimensional sampling problem is a fundamental problem, both from a practical and theoretical perspective. Compared with samplers with unbiased stationary distribution, e.g., Metropolis-adjusted Langevin algorithm (MALA), biased samplers, e.g., Underdamped Langevin Dynamics (ULD), perform better in low-accuracy cases just because a lower dimension dependency in their complexities. Along this line, Freund et al. (2022) suggest that the modified Langevin algorithm with prior diffusion is able to converge dimension independently for strongly log-concave target distributions. Nonetheless, it remains open whether such property establishes for more general cases. In this paper, we investigate the prior diffusion technique for the target distributions satisfying log-Sobolev inequality (LSI), which covers a much broader class of distributions compared to the strongly log-concave ones. In particular, we prove that the modified Langevin algorithm can also obtain the dimension-independent convergence of KL divergence with different step size schedules. The core of our proof technique is a novel construction of an interpolating SDE, which significantly helps to conduct a more accurate characterization of the discrete updates of the overdamped Langevin dynamics. Our theoretical analysis demonstrates the benefits of prior diffusion for a broader class of target distributions and provides new insights into developing faster sampling algorithms.