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

Sampling from probability distributions in high dimensional spaces is a fundamental computational primitive; it forms the basis of efficient numerical methods for approximating arbitrary integrals. The problem statement is the following: given a density function π, compute a point x with density proportional to π(x). A general approach to solving this problem is to design a reversible, ergodic Markov chain with a unique stationary distribution that is equal to the target distribution from which samples are needed. It is often possible to design relatively simple chains with low per-iteration computational complexity that are fit for purpose by implementing the Metropolis-Hastings filter [1, 2], a rule by which to either accept the next step in the dynamics or remain put and so tailor the dynamics toward a specific stationary distribution. The resulting Metropolized or Markov Chain Monte Carlo algorithms are known to converge asymptotically to their stationary distributions under mild regularity conditions. Non-asymptotic rates of convergence or mixing times are comparatively few in number and are both algorithm-and target-specific. They are important because downstream estimators computed using samples drawn from a dynamics that has not converged will suffer from bias. The class of log-concave target distributions is of particular interest.