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arXiv · 2005.09669

Exponential ergodicity of mirror-Langevin diffusions

Sinho Chewi·Thibaut Le Gouic·Chen Lu·Tyler Maunu·Philippe Rigollet·Austin J. Stromme
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Abstract

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.

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Sinho Chewi, Thibaut Le Gouic, Chen Lu, Tyler Maunu, Philippe Rigollet, Austin J. Stromme. 2020-06-02. Exponential ergodicity of mirror-Langevin diffusions. https://arxiv.org/abs/2005.09669

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