{"ID":2862748,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.26175","arxiv_id":"2509.26175","title":"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 $π$ is a $d$-dimensional log-concave distribution with condition number $κ$, we establish a spectral gap lower bound of order $\\mathcal{O}(1/κd)$ for the random-scan version of MwG, improving on the previously available $\\mathcal{O}(1/κ^2 d)$ 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.","short_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...","url_abs":"https://arxiv.org/abs/2509.26175","url_pdf":"https://arxiv.org/pdf/2509.26175v1","authors":"[\"Cecilia Secchi\",\"Giacomo Zanella\"]","published":"2025-09-30T12:31:22Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"math.ST\",\"stat.ME\"]","methods":"[]","has_code":false}