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A key task in Bayesian machine learning is sampling from distributions thatare only specified up to a partition function (i.e., constant ofproportionality). One prevalent example of this is sampling posteriors inparametric distributions, such as latent-variable generative models. Howeversampling (even very approximately) can be #P-hard.  Classical results going back to Bakry and \'Emery (1985) on sampling focus onlog-concave distributions, and show a natural Markov chain called Langevindiffusion mixes in polynomial time. However, all log-concave distributions areuni-modal, while in practice it is very common for the distribution of interestto have multiple modes. In this case, Langevin diffusion suffers from torpidmixing.  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 a mixture of (strongly) log-concave distributions of thesame shape. In particular, our technique applies to the canonical multi-modaldistribution: a mixture of gaussians (of equal variance). Our algorithmefficiently samples from these distributions given only access to the gradientof the log-pdf.  For the analysis, we introduce novel techniques for proving spectral gapsbased on decomposing the action of the generator of the diffusion. Previousapproaches rely on decomposing the state space as a partition of sets, whileour approach can be thought of as decomposing the stationary measure as amixture of distributions (a "soft partition").  Additional materials for the paper can be found athttp://holdenlee.github.io/Simulated%20tempering%20Langevin%20Monte%20Carlo.html.The proof and results have been improved and generalized from the precursor atarXiv:1710.02736.

Simulated Tempering Langevin Monte Carlo II: An Improved Proof using Soft Markov Chain Decomposition