Faster mixing via average conductance

Abstract

The notion of conductance introduced by Jerrum and Sinclair has been widely used to prove rapid mixing of Markov chains. Here we introduce a variant of this - instead of measuring the conductance of the worst subset of states, we show that it is enough to bound a certain weighted average conductance (where the average is taken over subsets of states with different sizes.) In the case of convex bodies, we show that this average conductance is better than the known bounds for the worst case; this helps us save a factor of O(n) which is incurred in all proofs as a `penalty' for a `bad start' (i.e., because the starting distribution may be arbitrary). We show that in a convex body in IRn, with diameter D, random walk with steps in a ball with radius δ mixes in O*(nD2/δ2) time (if idle steps at the boundary are not counted). This gives an O*(n3) sampling algorithm after appropriate preprocessing, improving the previous bound of O*(n4).