Parallelising Glauber dynamics

07/14/2023

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by   Holden Lee, et al.

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For distributions over discrete product spaces ∏_i=1^n Ω_i', Glauber dynamics is a Markov chain that at each step, resamples a random coordinate conditioned on the other coordinates. We show that k-Glauber dynamics, which resamples a random subset of k coordinates, mixes k times faster in χ^2-divergence, and assuming approximate tensorization of entropy, mixes k times faster in KL-divergence. We apply this to Ising models μ_J,h(x)∝exp(1/2⟨ x,Jx ⟩ + ⟨ h,x⟩) with J<1-c (the regime where fast mixing is known), where we show that we can implement each step of O(n/J_F)-Glauber dynamics efficiently with a parallel algorithm, resulting in a parallel algorithm with running time O(J_F) = O(√(n)).