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 $\kappa$.

翻译：吉布斯采样器，也称为坐标命中-运行算法，是一种广泛应用于任意维度概率分布抽样的马尔可夫链。该算法的每次迭代中，随机选择一个坐标，根据所有其他坐标条件化后的分布进行重采样。我们研究了吉布斯采样器在支撑集为$\mathbb{R}^n$的对数光滑且强对数凹目标分布类上的行为。假设初始分布相对于目标分布是$M$-温的，我们证明对于条件数为$\kappa$的分布，吉布斯采样器至多需要$O^{\star}\left(\kappa^2 n^{7.5}\left(\max\left\{1,\sqrt{\frac{1}{n}\log \frac{2M}{\gamma}}\right\}\right)^2\right)$步，即可生成总变差距离误差不超过$\gamma$的样本。