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

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 ℝn. Assuming the initial distribution is M-warm with respect to the target, we show that the Gibbs sampler requires at most Ostarleft(kappa2 n7.5left(maxleft1,√frac1nlog frac2Mγrightright)2right) steps to produce a sample with error no more than γ in total variation distance from a distribution with condition number kappa.