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We present a novel method for reducing the computational complexity ofrigorously estimating the partition functions (normalizing constants) of Gibbs(Boltzmann) distributions, which arise ubiquitously in probabilistic graphicalmodels. A major obstacle to practical applications of Gibbs distributions isthe need to estimate their partition functions. The state of the art inaddressing this problem is multi-stage algorithms, which consist of a coolingschedule, and a mean estimator in each step of the schedule. While the coolingschedule in these algorithms is adaptive, the mean estimation computations useMCMC as a black-box to draw approximate samples. We develop a doubly adaptiveapproach, combining the adaptive cooling schedule with an adaptive MCMC meanestimator, whose number of Markov chain steps adapts dynamically to theunderlying chain. Through rigorous theoretical analysis, we prove that ourmethod outperforms the state of the art algorithms in several factors: (1) Thecomputational complexity of our method is smaller; (2) Our method is lesssensitive to loose bounds on mixing times, an inherent component in thesealgorithms; and (3) The improvement obtained by our method is particularlysignificant in the most challenging regime of high-precision estimation. Wedemonstrate the advantage of our method in experiments run on classic factorgraphs, such as voting models and Ising models.

Fast Doubly-Adaptive MCMC to Estimate the Gibbs Partition Function with Weak Mixing Time Bounds