Entropy contraction of the Gibbs sampler under log-concavity

2024/10/01 by Filippo Ascolani, Hugo Lavenant, Ascolani, Filippo +3 · 1 voice · 4 citations
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Theoretical and Computational Physics #math.PR #math.ST #stat.CO #stat.ML

paper · pdf · doi:10.48550/arxiv.2410.00858

openalex publication_date 2024/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Gibbs sampler (a.k.a. Glauber dynamics and heat-bath algorithm) is a popular Markov Chain Monte Carlo algorithm which iteratively samples from the conditional distributions of a probability measure π of interest. Under the assumption that π is strongly log-concave, we show that the random scan Gibbs sampler contracts in relative entropy and provide a sharp characterization of the associated contraction rate. Assuming that evaluating conditionals is cheap compared to evaluating the joint density, our results imply that the number of full evaluations of π needed for the Gibbs sampler to mix grows linearly with the condition number and is independent of the dimension. If π is non-strongly log-concave, the convergence rate in entropy degrades from exponential to polynomial. Our techniques are versatile and extend to Metropolis-within-Gibbs schemes and the Hit-and-Run algorithm. A comparison with gradient-based schemes and the connection with the optimization literature are also discussed.

Cited by

• Wasserstein Contraction of Coordinate Ascent Variational Inference
• A fast non-reversible sampler for Bayesian finite mixture models
• Spectral gap of Metropolis-within-Gibbs under log-concavity
• Zeroth-order parallel sampling

Discussions

• Finally got around to reading the maths of this, after being aware of the paper and results for most of two months - and it was definitely worth it! arxiv.org/abs/2410.00858 'Entropy contraction of t [bsky, 32 points, 2 comments]

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