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A key task in Bayesian statistics is sampling from distributions that areonly specified up to a partition function (i.e., constant of proportionality).However, without any assumptions, sampling (even approximately) can be #P-hard,and few works have provided "beyond worst-case" guarantees for such settings.  For log-concave distributions, classical results going back to Bakry and\'Emery (1985) show that natural continuous-time Markov chains called Langevindiffusions mix in polynomial time. The most salient feature of log-concavityviolated in practice is uni-modality: commonly, the distributions we wish tosample from are multi-modal. In the presence of multiple deep andwell-separated modes, Langevin diffusion suffers from torpid mixing.  We address this problem by combining Langevin diffusion with simulatedtempering. The result is a Markov chain that mixes more rapidly bytransitioning between different temperatures of the distribution. We analyzethis Markov chain for the canonical multi-modal distribution: a mixture ofgaussians (of equal variance). The algorithm based on our Markov chain provablysamples from distributions that are close to mixtures of gaussians, givenaccess to the gradient of the log-pdf. For the analysis, we use a spectraldecomposition theorem for graphs (Gharan and Trevisan, 2014) and a Markov chaindecomposition technique (Madras and Randall, 2002).

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