A connection between Tempering and Entropic Mirror Descent

Crucinio Francesca Romana, King's College London

This talk 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. We establish a parallel result to that of \cite{jordan1998variational} which shows that algorithms based on the Langevin diffusion can be seen as numerical approximations of gradient descent (in the appropriate geometry) applied to the KL. 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. Joint work with Nicolas Chopin and Anna Korba (CREST, ENSAE, IP Paris), based on the results in \cite{chopin2023connection}.

Area: CS35 - Monte Carlo methods and Applications I (Francesca R Crucinio, Alessandra Iacobucci, Andrea Bertazzi)

Keywords: monte carlo methods, optimisation

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