# Spectral gap of Metropolis-within-Gibbs under log-concavity

arXiv: https://arxiv.org/abs/2509.26175 · HTML: https://mlsanity.com/paper/2509.26175/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 36 references that are in the map.
- First posted on arXiv 30 Sep 2025.

## Builds on (most cited first, 13 of 13)

- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients](https://mlsanity.com/paper/1905.12247/) (2019)
- [Entropy contraction of the Gibbs sampler under log-concavity](https://mlsanity.com/paper/2410.00858/) (2024)
- [MALA-within-Gibbs samplers for high-dimensional distributions with sparse conditional structure](https://mlsanity.com/paper/1908.09429/) (2019)
- [Random Coordinate Langevin Monte Carlo](https://mlsanity.com/paper/2010.01405/) (2020)
- [Random Coordinate Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2010.11366/) (2020)
- [Clustering consistency with Dirichlet process mixtures](https://mlsanity.com/paper/2205.12924/) (2022)
- [Scalability of Metropolis-within-Gibbs schemes for high-dimensional Bayesian models](https://mlsanity.com/paper/2403.09416/) (2024)
- [Fast Mixing of Data Augmentation Algorithms: Bayesian Probit, Logit, and Lasso Regression](https://mlsanity.com/paper/2412.07999/) (2024)
- [Spectral Telescope: Convergence Rate Bounds for Random-Scan Gibbs Samplers Based on a Hierarchical Structure](https://mlsanity.com/paper/2208.11299/) (2022)
- [Is Gibbs sampling faster than Hamiltonian Monte Carlo on GLMs?](https://mlsanity.com/paper/2410.03630/) (2024)
- [Mixing times of data-augmentation Gibbs samplers for high-dimensional probit regression](https://mlsanity.com/paper/2505.14343/) (2025)
- [A practical sequential stopping rule for high-dimensional MCMC and its application to spatial-temporal Bayesian models](https://mlsanity.com/paper/1403.5536/) (2014)

## Abstract (arXiv)

The Metropolis-within-Gibbs (MwG) algorithm is a widely used Markov chain Monte Carlo method for sampling from high-dimensional distributions when exact conditional sampling is intractable. We study MwG with Random Walk Metropolis (RWM) updates, whose proposal variances are uniformly comparable to the corresponding conditional variances. Assuming the target π is a d-dimensional log-concave distribution with condition number kappa, we establish a spectral gap lower bound of order Omega((kappa d)-1) for the random-scan version of MwG, improving on the previously available Omega((kappa2 d)-1) bound. This is obtained by developing sharp estimates of the conductance of one-dimensional RWM kernels, which may be of independent interest. The result shows that MwG can mix substantially faster under the stated uniform tuning condition and that its mixing performance is just a constant factor worse than that of the exact Gibbs sampler, thus providing theoretical support to previously observed empirical behavior.