Reviews: Mirrored Langevin Dynamics

This is a very well-written paper and excellently presented with some interesting supporting theoretical results. The paper introduces a method (mirror map) from the optimization literature, mirrored descent, to perform scalable Monte Carlo sampling in a constrained state space. The mirror map works by transforming the sampling problem onto an unconstrained space, where stochastic gradient Markov chain Monte Carlo (MCMC) algorithms, in particular, stochastic gradient Langevin dynamics, can be readily applied. The Fenchal dual of the transformation function is used to transform the samples from the unconstrained space back onto the constrained space. In the paper, the authors state that a "good" mirror map is required.