Approximation algorithms for the normalizing constant of Gibbs distributions

Abstract

Consider a family of distributions $\{π_β\}$ where $X\simπ_β$ means that $\mathbb{P}(X=x)=\exp(-βH(x))/Z(β)$. Here $Z(β)$ is the proper normalizing constant, equal to $\sum_x\exp(-βH(x))$. Then $\{π_β\}$ is known as a Gibbs distribution, and $Z(β)$ is the partition function. This work presents a new method for approximating the partition function to a specified level of relative accuracy using only a number of samples, that is, $O(\ln(Z(β))\ln(\ln(Z(β))))$ when $Z(0)\geq1$. This is a sharp improvement over previous, similar approaches that used a much more complicated algorithm, requiring $O(\ln(Z(β))\ln(\ln(Z(β)))^5)$ samples.