Parameter estimation for Gibbs distributions

TL;DR

This paper develops near-optimal algorithms for estimating parameters like counts and partition functions of Gibbs distributions, which are crucial in statistical physics and combinatorics, using a number of samples that is nearly the best possible.

Contribution

It introduces sample-efficient algorithms for estimating counts and partition functions of Gibbs distributions, improving upon previous methods and applicable to various combinatorial problems.

Findings

Estimates counts with roughly (q/^2) samples, optimal up to logs.

Provides algorithms to compute the partition function with similar sample complexity.

Applies methods to counting subgraphs, independent sets, and perfect matchings.

Abstract

We consider Gibbs distributions, which are families of probability distributions over a discrete space $\Omega$ with probability mass function of the form $\mu^\Omega_\beta(\omega) \propto e^{\beta H(\omega)}$ for $\beta$ in an interval $[\beta_{\min}, \beta_{\max}]$ and $H( \omega ) \in \{0 \} \cup [1, n]$. The partition function is the normalization factor $Z(\beta)=\sum_{\omega \in\Omega}e^{\beta H(\omega)}$. Two important parameters of these distributions are the log partition ratio $q = \log \tfrac{Z(\beta_{\max})}{Z(\beta_{\min})}$ and the counts $c_x = |H^{-1}(x)|$. These are correlated with system parameters in a number of physical applications and sampling algorithms. Our first main result is to estimate the counts $c_x$ using roughly $\tilde O( \frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and $\tilde O( \frac{n^2}{\varepsilon^2} )$ samples for integer-valued distributions (ignoring some second-order terms and parameters), and we show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs, independent sets, and perfect matchings. As a key subroutine, we also develop algorithms to compute the partition function $Z$ using $\tilde O(\frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and using $\tilde O(\frac{n^2}{\varepsilon^2})$ samples for integer-valued distributions.