Deterministic-scan Gibbs sampler worst-case performance under log-concavity

Establish rigorous quantitative worst-case performance bounds for the deterministic-scan Gibbs sampler targeting log-concave probability distributions on R^d, and determine whether its convergence guarantees are strictly worse than those of the random-scan Gibbs sampler (which contracts relative entropy at rate 1/(κ·M) under strong log-concavity).

Background

The paper analyzes the random-scan Gibbs sampler (GS) for log-concave targets and proves sharp, dimension-free entropy contraction rates depending on a coordinate-wise condition number. A commonly used variant is the deterministic-scan GS, where coordinates are updated cyclically rather than randomly. Drawing analogy from convex optimization, random-scan coordinate descent often enjoys better worst-case complexity than deterministic scan.

The authors state an expectation that deterministic-scan GS behaves worse in worst-case performance for log-concave targets but acknowledge the absence of rigorous results. This leaves open the problem of deriving explicit convergence bounds for deterministic-scan GS comparable to those established for random-scan GS.