Approximating Partition Functions in Constant Time

We study approximations of the partition function of dense graphical models. Partition functions of graphical models play a fundamental role is statistical physics, in statistics and in machine learning. Two of the main methods for approximating the partition function are Markov Chain Monte Carlo and Variational Methods. An impressive body of work in mathematics, physics and theoretical computer science provides conditions under which Markov Chain Monte Carlo methods converge in polynomial time. These methods often lead to polynomial time approximation algorithms for the partition function in cases where the underlying model exhibits correlation decay. There are very few theoretical guarantees for the performance of variational methods. One exception is recent results by Risteski (2016) who considered dense graphical models and showed that using variational methods, it is possible to find an $O(\epsilon n)$ additive approximation to the log partition function in time $n^{O(1/\epsilon^2)}$ even in a regime where correlation decay does not hold. We show that under essentially the same conditions, an $O(\epsilon n)$ additive approximation of the log partition function can be found in constant time, independent of $n$. In particular, our results cover dense Ising and Potts models as well as dense graphical models with $k$-wise interaction. They also apply for low threshold rank models.