Chain of Log-Concave Markov Chains

• URL: http://arxiv.org/abs/2305.19473v2
• Date: Fri, 29 Sep 2023 01:58:32 GMT
• Title: Chain of Log-Concave Markov Chains
• Authors: Saeed Saremi, Ji Won Park, Francis Bach
• Abstract summary: We introduce a theoretical framework for sampling from unnormalized densities. We prove one can decompose sampling from a density into a sequence of sampling from log-concave conditional densities. We report a remarkable capacity of our algorithm to "tunnel" between modes of a distribution.
• Score: 2.9465623430708905
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce a theoretical framework for sampling from unnormalized densities based on a smoothing scheme that uses an isotropic Gaussian kernel with a single fixed noise scale. We prove one can 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. Our construction is unique in that it keeps track of a history of samples, making it non-Markovian as a whole, but it is lightweight algorithmically as the history only shows up in the form of a running empirical mean of samples. Our sampling algorithm generalizes walk-jump sampling (Saremi & Hyv\"arinen, 2019). The "walk" phase becomes a (non-Markovian) chain of (log-concave) Markov 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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