  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 (1985) on sampling focus on
log-concave distributions, and show a natural Markov chain called Langevin
diffusion mixes 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.
  For the analysis, we introduce novel techniques for proving spectral gaps
based on decomposing the action of the generator of the diffusion. Previous
approaches rely on decomposing the state space as a partition of sets, while
our approach can be thought of as decomposing the stationary measure as a
mixture of distributions (a "soft partition").
  Additional materials for the paper can be found at
http://holdenlee.github.io/Simulated%20tempering%20Langevin%20Monte%20Carlo.html.
The proof and results have been improved and generalized from the precursor at
arXiv:1710.02736.
