Beyond Log-concavity: Provable Guarantees for Sampling Multi-modal Distributions using Simulated Tempering Langevin Monte Carlo

A key task in Bayesian machine learning is sampling from distributions that are only specified up to a partition function (i.e., constant of proportionality). One prevalent example of this is sampling posteriors in parametric distributions, such as latent-variable generative models. However sampling (even very approximately) can be #P-hard.

Classical results (going back to Bakry and Emery) on sampling focus on log-concave distributions, and show a natural Markov chain called Langevin diffusion mix in polynomial time. However, all log-concave distributions are uni-modal, while in practice it is very common for the distribution of interest to have multiple modes. In this case, Langevin diffusion suffers from torpid mixing.

We address this problem by combining Langevin diffusion with simulated tempering. The result is a Markov chain that mixes more rapidly by transitioning between different temperatures of the distribution. We analyze this Markov chain for a mixture of (strongly) log-concave distributions of the same shape. In particular, our technique applies to the canonical multi-modal distribution: a mixture of gaussians (of equal variance). Our algorithm efficiently samples from these distributions given only access to the gradient of the log-pdf. To the best of our knowledge, this is the first result that proves fast mixing for multimodal distributions.