2020/10/30 by Kwangjun Ahn, Ahn, Kwangjun, Sinho Chewi +1 · 5 citations
Biochemistry, Genetics and Molecular Biology · Engineering · Mathematics · #Diffusion and Search Dynamics #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Markov Chains and Monte Carlo Methods #Sparse and Compressive Sensing Techniques #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2010.16212
openalex publication_date 2020/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose a new discretization of the mirror-Langevin diffusion and give a crisp proof of its convergence. Our analysis uses relative convexity/smoothness and self-concordance, ideas which originated in convex optimization, together with a new result in optimal transport that generalizes the displacement convexity of the entropy. Unlike prior works, our result both (1) requires much weaker assumptions on the mirror map and the target distribution, and (2) has vanishing bias as the step size tends to zero. In particular, for the task of sampling from a log-concave distribution supported on a compact set, our theoretical results are significantly better than the existing guarantees.