# Fast Conditional Mixing of MCMC Algorithms for Non-log-concave Distributions

arXiv: https://arxiv.org/abs/2306.10506 · HTML: https://mlsanity.com/paper/2306.10506/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 26 references that are in the map.
- First posted on arXiv 18 Jun 2023.

## Builds on (most cited first, 16 of 16)

- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [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)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [Sampling Can Be Faster Than Optimization](https://mlsanity.com/paper/1811.08413/) (2018)
- [Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo](https://mlsanity.com/paper/2202.05214/) (2022)
- [On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness](https://mlsanity.com/paper/2005.13097/) (2020)
- [Convergence of Langevin Monte Carlo in Chi-Squared and Renyi Divergence](https://mlsanity.com/paper/2007.11612/) (2020)
- [Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity](https://mlsanity.com/paper/1907.11331/) (2019)
- [Non-asymptotic bounds for sampling algorithms without log-concavity](https://mlsanity.com/paper/1808.07105/) (2018)
- [Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetry](https://mlsanity.com/paper/1911.01469/) (2019)
- [Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequality](https://mlsanity.com/paper/2303.03589/) (2023)
- [Sampling for Bayesian Mixture Models: MCMC with Polynomial-Time Mixing](https://mlsanity.com/paper/1912.05153/) (2019)

## Built on by (most cited first, 5 of 5)

- [Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling](https://mlsanity.com/paper/2407.16936/) (2024)
- [Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality](https://mlsanity.com/paper/2502.05623/) (2025)
- [Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyond](https://mlsanity.com/paper/2502.04575/) (2025)
- [Complexity of Non-Log-Concave Sampling in Fisher Information](https://mlsanity.com/paper/2605.15859/) (2026)
- [Local and Global Contraction Principles for MCMC Mixing](https://mlsanity.com/paper/2606.03033/) (2026)

## Abstract (arXiv)

MCMC algorithms offer empirically efficient tools for sampling from a target distribution π(x) propto exp(-V(x)). However, on the theory side, MCMC algorithms suffer from slow mixing rate when π(x) is non-log-concave. Our work examines this gap and shows that when Poincar'e-style inequality holds on a subset X of the state space, the conditional distribution of MCMC iterates over X mixes fast to the true conditional distribution. This fast mixing guarantee can hold in cases when global mixing is provably slow. We formalize the statement and quantify the conditional mixing rate. We further show that conditional mixing can have interesting implications for sampling from mixtures of Gaussians, parameter estimation for Gaussian mixture models and Gibbs-sampling with well-connected local minima.