Simple parallel estimation of the partition ratio for Gibbs distributions

TL;DR

This paper introduces improved non-adaptive and semi-adaptive algorithms for estimating the partition function ratio in Gibbs distributions, reducing sample complexity and simplifying previous methods.

Contribution

It presents a non-adaptive algorithm with fewer samples and a semi-adaptive algorithm matching the sequential complexity, using a single estimator for simplicity.

Findings

Non-adaptive algorithm requires O(q log^2 n / ε^2) samples.

Semi-adaptive algorithm uses just two rounds of adaptivity with O(q log n / ε^2) samples.

The new methods simplify previous techniques by employing a single estimator.

Abstract

We consider the problem of estimating the partition function $Z(\beta)=\sum_x \exp(\beta(H(x))$ of a Gibbs distribution with the Hamiltonian $H:\Omega\rightarrow\{0\}\cup[1,n]$. As shown in [Harris & Kolmogorov 2024], the log-ratio $q=\ln (Z(\beta_{\max})/Z(\beta_{\min}))$ can be estimated with accuracy $\epsilon$ using $O(\frac{q \log n}{\epsilon^2})$ calls to an oracle that produces a sample from the Gibbs distribution for parameter $\beta\in[\beta_{\min},\beta_{\max}]$. That algorithm is inherently sequential, or {\em adaptive}: the queried values of $\beta$ depend on previous samples. Recently, [Liu, Yin & Zhang 2024] developed a non-adaptive version that needs $O( q (\log^2 n) (\log q + \log \log n + \epsilon^{-2}) )$ samples. We improve the number of samples to $O(\frac{q \log^2 n}{\epsilon^2})$ for a non-adaptive algorithm, and to $O(\frac{q \log n}{\epsilon^2})$ for an algorithm that uses just two rounds of adaptivity (matching the complexity of the sequential version). Furthermore, our algorithm simplifies previous techniques. In particular, we use just a single estimator, whereas methods in [Harris & Kolmogorov 2024, Liu, Yin & Zhang 2024] employ two different estimators for different regimes.