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Tightness of κ-dependence in spectral gap bounds for Metropolis-within-Gibbs with Random Walk Metropolis updates

Ascertain whether the dependence on the condition number κ (or κ*) in the lower bound Gap(P^MwG) ≥ C/((κ*)^2·M·d_max) for Metropolis-within-Gibbs with Random Walk Metropolis proposals is tight, and determine whether alternative choices of proposal step-size can improve this κ dependence.

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Background

The authors extend their entropy contraction analysis to Metropolis-within-Gibbs (MwG) schemes and provide explicit spectral gap bounds when each coordinate is updated via a Random Walk Metropolis kernel with a specific step-size. The resulting bound scales linearly with dimension and quadratically with κ*, but the optimality of κ dependence is not established.

They note uncertainty about whether the κ dependence can be improved, for example through different step-size selections, leaving open the question of tightness and potential refinement of κ dependence in MwG spectral gap guarantees.

References

By arguments similar to the ones of Section \ref{subsec:tightness}, it is easy to see that the linear dependence with respect to $d$ is tight, while it is unclear to us if the dependence on $\kappa$ is tight, and in particular whether it can be improved with a different choice of step-size in the RWM proposal $Q_m{x_{-m}}$.

— Entropy contraction of the Gibbs sampler under log-concavity (2410.00858 - Ascolani et al., 1 Oct 2024) in Section 6 (Metropolis-within-Gibbs and general coordinate-wise kernels), discussion after Corollary thm:gap_MwG_RWM