A mixing time bound for Gibbs sampling from log-smooth log-concave distributions

• URL: http://arxiv.org/abs/2412.17899v1
• Date: Mon, 23 Dec 2024 19:00:02 GMT
• Title: A mixing time bound for Gibbs sampling from log-smooth log-concave distributions
• Authors: Neha S. Wadia,
• Abstract summary: We study the behavior of the Gibbs sampler on the class of log-smooth and strongly log-concave target distributions. We show that the Gibbs sampler requires at most $Ostarleft(kappa2 n7.5left(maxleft1,sqrtfrac1nlog frac2Mgammarightright2right)$ steps to produce a sample with error no more than $gamma$ in total variation distance from a distribution with condition number $kappa$.
• Score: 0.8702432681310401
• License:
• Abstract: 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 $O^{\star}\left(\kappa^2 n^{7.5}\left(\max\left\{1,\sqrt{\frac{1}{n}\log \frac{2M}{\gamma}}\right\}\right)^2\right)$ steps to produce a sample with error no more than $\gamma$ in total variation distance from a distribution with condition number $\kappa$.

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