Chain of Log-Concave Markov Chains
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Markov chain Monte Carlo (MCMC) is a class of general-purpose algorithms for sampling from unnormalized densities. There are two well-known problems facing MCMC in high dimensions: (i) The distributions of interest are concentrated in pockets separated by large regions with small probability mass, and (ii) The log-concave pockets themselves are typically ill-conditioned. We introduce a framework to tackle these problems using isotropic Gaussian smoothing. We prove one can always decompose sampling from a density (minimal assumptions made on the density) into a sequence of sampling from log-concave conditional densities via accumulation of noisy measurements with equal noise levels. This construction keeps track of a history of samples, making it non-Markovian as a whole, but the history only shows up in the form of an empirical mean, making the memory footprint minimal. Our sampling algorithm generalizes walk-jump sampling [1]. The "walk" phase becomes a (non-Markovian) chain of log-concave Langevin chains. The "jump" from the accumulated measurements is obtained by empirical Bayes. We study our sampling algorithm quantitatively using the 2-Wasserstein metric and compare it with various Langevin MCMC algorithms. We also report a remarkable capacity of our algorithm to "tunnel" between modes of a distribution.
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