A connection between Tempering and Entropic Mirror Descent

Sampling from a target probability distribution whose density is known up to a normalization constant is a fundamental task in computational statistics and machine learning. It can be naturally formulated as optimizing a functional measuring the dissimilarity to the target probability distribution, typically the Kullback-Leibler (KL) divergence. From there, it is natural to consider optimization schemes over the space of probability distributions, to design a sequence of distributions approximating the target one. Depending on the chosen geometry over the search space and the time discretization, one may obtain different schemes. For instance, one possible framework is to restrict the search space to the Wasserstein space, i.e. probability distributions with bounded second moments equipped with the Wasserstein-2 distance (Ambrosio et al., 2008). The latter is equipped with a rich Riemannian structure (Otto and Villani, 2000) that enables to define Wasserstein-2 gradient flows, i.e. paths of distributions decreasing the objective functional of steepest descent according to this metric. It is well-known that the Wasserstein gradient flow of the KL can be implemented by a Langevin diffusion on the ambient space (Jordan et al., 1998) and easily discretized in time, resulting in the Langevin Monte Carlo (or Unadjusted Langevin) algorithm (Roberts and Tweedie, 1996). The latter is one of the most famous Markov Chain Monte Carlo (MCMC) algorithms - maybe the most canonical - that generate Markov chains in the ambient space, whose law approximates the target distribution for a large time horizon. Many other time discretizations of the KL Wasserstein gradient flow (Salim et al., 2020; Mou et al., 2021) or its gradient flow with respect to similar optimal transport geometries have been considered in the literature (Liu, 2017; Garbuno-Inigo et al., 2020).