Rapidly Mixing Markov Chains: A Comparison of Techniques (A Survey)

For many fundamental sampling problems, the best, and often the only known, approach to solving them is to take a long enough random walk on a certain Markov chain and then return the current state of the chain. Techniques to prove how long “long enough” is, i.e., the number of steps in the chain one needs to take in order to be sufficiently close to the stationary distribution of the chain, are of great importance in obtaining estimates of running times of such sampling algorithms. In this report, we survey existing techniques to bound the mixing time of Markov chains. The mixing time of a Markov chain is exactly captured by the “spectral gap” of its underlying transition matrix. The spectral gap is closely related to a geometric parameter called “conductance” which is a measure of the “edge-expansion” of the Markov chain. Conductance also captures the mixing time up to square factors. Lower bounds on conductance, which give upper bounds on the mixing time, are typically obtained by a technique called “canonical paths” where the idea is to find a set of paths, one between every unequal source-destination pair, such that no edge is very heavily congested. Unlike conductance, the canonical paths approach cannot always show rapid mixing of a rapidly mixing chain. It is known that this “drawback” disappears if we allow the flow between a pair of states to be spread along multiple paths. We prove that for a large class of Markov chains, including all the ones that we use in the sampling applications we will be interested in, canonical paths does capture rapid mixing, i.e., we show that small mixing time implies the existence of some collection of paths with low edge congestion. Allowing multiple paths to route the flow still does help a great deal in the design of such flows, and this is best illustrated by a recent result of Morris and Sinclair [34] on the rapid mixing of a natural Markov chain for sampling 0-1 knapsack solutions; this result seems to rely critically on fractional flows. An entirely different approach to prove rapid mixing, which in fact historically preceded the conductance/canonical paths based approach, is “Coupling”. Coupling is a very elegant technique and has been used to prove rapid mixing of several chains where designing good canonical paths seems to be a hideous task. “Path Coupling” is a related technique discovered by Bubley and Dyer [5] that often tremendously reduces the complexity of designing good Couplings. We present several applications of Path Coupling in proofs of rapid mixing, and these invariably lead to much better bounds on mixing time than known using conductance, and moreover Coupling based proofs usually turn out to be much simpler. These applications motivate the question of whether Coupling indeed can be made to work whenever the chain is rapidly mixing. This question was answered in the negative in very recent work by Kumar and Ramesh [27], who showed that no Coupling strategy can prove the rapid mixing of the famous Jerrum-Sinclair chain for sampling perfect and near-perfect matchings (the chain is known to be rapidly mixing via a canonical paths argument).