Geometric Bounds on the Fastest Mixing Markov Chain
(2111.05816)Abstract
In the Fastest Mixing Markov Chain problem, we are given a graph $G = (V, E)$ and desire the discrete-time Markov chain with smallest mixing time $\tau$ subject to having equilibrium distribution uniform on $V$ and non-zero transition probabilities only across edges of the graph. It is well-known that the mixing time $\tau\textsf{RW}$ of the lazy random walk on $G$ is characterised by the edge conductance $\Phi$ of $G$ via Cheeger's inequality: $\Phi{-1} \lesssim \tau\textsf{RW} \lesssim \Phi{-2} \log |V|$. Analogously, we characterise the fastest mixing time $\tau\star$ via a Cheeger-type inequality but for a different geometric quantity, namely the vertex conductance $\Psi$ of $G$: $\Psi{-1} \lesssim \tau\star \lesssim \Psi{-2} (\log |V|)2$. This characterisation forbids fast mixing for graphs with small vertex conductance. To bypass this fundamental barrier, we consider Markov chains on $G$ with equilibrium distribution which need not be uniform, but rather only $\varepsilon$-close to uniform in total variation. We show that it is always possible to construct such a chain with mixing time $\tau \lesssim \varepsilon{-1} (\operatorname{diam} G)2 \log |V|$. Finally, we discuss analogous questions for continuous-time and time-inhomogeneous chains.
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