Efficient constrained sampling via the mirror-Langevin algorithm
This addresses efficient sampling in constrained domains for applications like Bayesian inference, though it appears incremental as it builds on existing mirror-Langevin methods.
The paper tackles the problem of constrained sampling by proposing a new discretization of the mirror-Langevin algorithm, achieving vanishing bias as step size decreases and significantly better theoretical guarantees for log-concave distributions on compact sets compared to prior works.
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.