Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flows

We consider the problem of sampling from a probability distribution $\pi$. It is well known that this can be written as an optimisation problem over the space of probability distribution in which we aim to minimise the Kullback--Leibler divergence from $\pi$. We consider several partial differential equations (PDEs) whose solution is a minimiser of the Kullback--Leibler divergence from $\pi$ and connect them to well-known Monte Carlo algorithms. We focus in particular on PDEs obtained by considering the Wasserstein--Fisher--Rao geometry over the space of probabilities and show that these lead to a natural implementation using importance sampling and sequential Monte Carlo. We propose a novel algorithm to approximate the Wasserstein--Fisher--Rao flow of the Kullback--Leibler divergence which empirically outperforms the current state-of-the-art. We study tempered versions of these PDEs obtained by replacing the target distribution with a geometric mixture of initial and target distribution and show that these do not lead to a convergence speed up.

Proximal Interacting Particle Langevin Algorithms

We introduce a class of algorithms, termed Proximal Interacting Particle Langevin Algorithms (PIPLA), for inference and learning in latent variable models whose joint probability density is non-differentiable. Leveraging proximal Markov chain Monte Carlo (MCMC) techniques and the recently introduced interacting particle Langevin algorithm (IPLA), we propose several variants within the novel proximal IPLA family, tailored to the problem of estimating parameters in a non-differentiable statistical model. We prove nonasymptotic bounds for the parameter estimates produced by multiple algorithms in the strongly log-concave setting and provide comprehensive numerical experiments on various models to demonstrate the effectiveness of the proposed methods. In particular, we demonstrate the utility of the proposed family of algorithms on a toy hierarchical example where our assumptions can be checked, as well as on the problems of sparse Bayesian logistic regression, sparse Bayesian neural network, and sparse matrix completion. Our theory and experiments together show that PIPLA family can be the de facto choice for parameter estimation problems in latent variable models for non-differentiable models.

A connection between Tempering and Entropic Mirror Descent

This paper explores the connections between tempering (for Sequential Monte Carlo; SMC) and entropic mirror descent to sample from a target probability distribution whose unnormalized density is known. We establish that tempering SMC is a numerical approximation of entropic mirror descent applied to the Kullback-Leibler (KL) divergence and obtain convergence rates for the tempering iterates. Our result motivates the tempering iterates from an optimization point of view, showing that tempering can be used as an alternative to Langevin-based algorithms to minimize the KL divergence. We exploit the connection between tempering and mirror descent iterates to justify common practices in SMC and propose improvements to algorithms in literature.