Simple parallel estimation of the partition ratio for Gibbs distributions
David G. Harris, Vladimir Kolmogorov
TL;DR
The paper addresses estimating the partition ratio $Q = Z(β_{\text{max}})/Z(β_{\text{min}})$ for Gibbs distributions by leveraging a PPE-based framework guided by carefully designed schedules over the inverse-temperature parameter. It derives rigorous curvature bounds for the partition-function landscape and uses them to control estimator variance, enabling efficient parallelizable estimation. The authors present a non-adaptive algorithm with sample complexity $O\left( \frac{q \log^2 n}{ε^2} \right)$ and a two-round adaptive algorithm matching the sequential complexity with $O\left( \frac{q \log n}{ε^2} \right)$ samples, aided by a simplified PPE that uses a single estimator. They also develop a parallelizable two-round strategy via ${\tt PseudoTPA}(\theta)$, improving practical scalability for large-scale Gibbs models and offering insights into the role of adaptivity in query-based counting problems.
Abstract
We consider the problem of estimating the partition function $Z(β)=\sum_x \exp(β(H(x))$ of a Gibbs distribution with the Hamiltonian $H:Ω\rightarrow\{0\}\cup[1,n]$. As shown in [Harris & Kolmogorov 2024], the log-ratio $q=\ln (Z(β_{\max})/Z(β_{\min}))$ can be estimated with accuracy $ε$ using $O(\frac{q \log n}{ε^2})$ calls to an oracle that produces a sample from the Gibbs distribution for parameter $β\in[β_{\min},β_{\max}]$. That algorithm is inherently sequential, or {\em adaptive}: the queried values of $β$ depend on previous samples. Recently, [Liu, Yin & Zhang 2024] developed a non-adaptive version that needs $O( q (\log^2 n) (\log q + \log \log n + ε^{-2}) )$ samples. We improve the number of samples to $O(\frac{q \log^2 n}{ε^2})$ for a non-adaptive algorithm, and to $O(\frac{q \log n}{ε^2})$ 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.