Entropic Independence II: Optimal Sampling and Concentration via Restricted Modified Log-Sobolev Inequalities

We introduce a framework for obtaining tight mixing time bounds for Markov chains based on what we call restricted modified log-Sobolev inequalities. Modified log-Sobolev inequalities quantify the rate of relative entropy contraction for the Markov operator, and are notoriously difficult to establish. However, for distributions infinitesimally close to stationarity, entropy contraction becomes equivalent to variance contraction, a.k.a. a Poincare inequality, which is significantly easier to establish through, for example, spectral analysis. Motivated by this observation, we study restricted modified log-Sobolev inequalities that guarantee entropy contraction not for all starting distributions, but for those in a large neighborhood of the stationary distribution. We use our framework to show that we can sample from the hardcore and Ising models on n-node graphs that have a constant δ relative gap to the tree-uniqueness threshold, in nearly-linear time Õδ(n). Notably, our bound does not depend on the maximum degree ∆ of the graph, and is therefore optimal even for high-degree graphs. Our work improves on prior mixing time bounds of Õδ,∆(n) and Õδ(n), established via (non-restricted) modified log-Sobolev and Poincare inequalities respectively. As an additional corollary of our results we show that optimal concentration inequalities can still be achieved from the restricted form of modified log-Sobolev inequalities. To establish restricted entropy contraction for these distributions, we extend the entropic independence framework of Anari, Jain, Koehler, Pham, and Vuong to distributions that satisfy spectral independence under a restricted set of external fields. We also develop an orthogonal trick that might be of independent interest: utilizing Bernoulli factories we show how to implement Glauber dynamics updates on high-degree graphs in O(1) time, assuming the graph is represented so that one can sample random neighbors of any desired node in O(1) time. Put together, we obtain the perhaps surprising result that we can sample from tree-unique hardcore and Ising models in time Õδ(n), i.e., without even necessarily having enough time to read all edges of the graph.