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

Abstract: 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, using proposal variances tuned to match the target's conditional variances. Assuming the target $\pi$ is a $d$-dimensional log-concave distribution with condition number $\kappa$, we establish a spectral gap lower bound of order $\mathcal{O}(1/\kappa d)$ for the random-scan version of MwG, improving on the previously available $\mathcal{O}(1/\kappa² d)$ bound. This is obtained by developing sharp estimates of the conductance of one-dimensional RWM kernels, which can be of independent interest. The result shows that MwG can mix substantially faster with variance-adaptive proposals 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.