A mixing time bound for Gibbs sampling from log-smooth log-concave distributions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916541301260288 |
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| author | Wadia, Neha S. |
| author_facet | Wadia, Neha S. |
| contents | The Gibbs sampler, also known as the coordinate hit-and-run algorithm, is a Markov chain that is widely used to draw samples from probability distributions in arbitrary dimensions. At each iteration of the algorithm, a randomly selected coordinate is resampled from the distribution that results from conditioning on all the other coordinates. We study the behavior of the Gibbs sampler on the class of log-smooth and strongly log-concave target distributions supported on $\mathbb{R}^n$. Assuming the initial distribution is $M$-warm with respect to the target, we show that the Gibbs sampler requires at most $O^{\star}\left(κ^2 n^{7.5}\left(\max\left\{1,\sqrt{\frac{1}{n}\log \frac{2M}γ}\right\}\right)^2\right)$ steps to produce a sample with error no more than $γ$ in total variation distance from a distribution with condition number $κ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17899 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A mixing time bound for Gibbs sampling from log-smooth log-concave distributions Wadia, Neha S. Statistics Theory Data Structures and Algorithms Machine Learning The Gibbs sampler, also known as the coordinate hit-and-run algorithm, is a Markov chain that is widely used to draw samples from probability distributions in arbitrary dimensions. At each iteration of the algorithm, a randomly selected coordinate is resampled from the distribution that results from conditioning on all the other coordinates. We study the behavior of the Gibbs sampler on the class of log-smooth and strongly log-concave target distributions supported on $\mathbb{R}^n$. Assuming the initial distribution is $M$-warm with respect to the target, we show that the Gibbs sampler requires at most $O^{\star}\left(κ^2 n^{7.5}\left(\max\left\{1,\sqrt{\frac{1}{n}\log \frac{2M}γ}\right\}\right)^2\right)$ steps to produce a sample with error no more than $γ$ in total variation distance from a distribution with condition number $κ$. |
| title | A mixing time bound for Gibbs sampling from log-smooth log-concave distributions |
| topic | Statistics Theory Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2412.17899 |