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Exponential ergodicity of mirror-Langevin diffusions
Sinho Chewi · Thibaut Le Gouic · Chen Lu · Tyler Maunu · Philippe Rigollet · Austin Stromme

Thu Dec 10 09:00 AM -- 11:00 AM (PST) @ Poster Session 5 #1391

Motivated by the problem of sampling from ill-conditioned log-concave distributions, we give a clean non-asymptotic convergence analysis of mirror-Langevin diffusions as introduced in Zhang et al. (2020). As a special case of this framework, we propose a class of diffusions called Newton-Langevin diffusions and prove that they converge to stationarity exponentially fast with a rate which not only is dimension-free, but also has no dependence on the target distribution. We give an application of this result to the problem of sampling from the uniform distribution on a convex body using a strategy inspired by interior-point methods. Our general approach follows the recent trend of linking sampling and optimization and highlights the role of the chi-squared divergence. In particular, it yields new results on the convergence of the vanilla Langevin diffusion in Wasserstein distance.

Author Information

Sinho Chewi (Massachusetts Institute of Technology)
Thibaut Le Gouic (Massachusetts Institute of Technology)
Chen Lu (Massachusetts Institute of Technology)
Tyler Maunu (Massachusetts Institute of Technology)
Philippe Rigollet (MIT)
Austin Stromme (MIT)

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