Spectral gap of Metropolis-within-Gibbs under log-concavity
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918151629832192 |
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| author | Secchi, Cecilia Zanella, Giacomo |
| author_facet | Secchi, Cecilia Zanella, Giacomo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26175 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral gap of Metropolis-within-Gibbs under log-concavity Secchi, Cecilia Zanella, Giacomo Machine Learning Statistics Theory Methodology 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. |
| title | Spectral gap of Metropolis-within-Gibbs under log-concavity |
| topic | Machine Learning Statistics Theory Methodology |
| url | https://arxiv.org/abs/2509.26175 |