Spatial Monte Carlo Integration with Annealed Importance Sampling

TL;DR

This paper introduces a novel method combining annealed importance sampling with spatial Monte Carlo integration to efficiently evaluate expectations on Ising models across temperature ranges, improving accuracy especially at low temperatures.

Contribution

The paper proposes a new approach that integrates AIS with SMCI, enhancing the accuracy of expectation evaluations on Ising models in challenging low-temperature conditions.

Findings

Method performs well in high-temperature regions.

Method maintains accuracy in low-temperature regions.

Theoretical and numerical validation of effectiveness.

Abstract

Evaluating expectations on an Ising model (or Boltzmann machine) is essential for various applications, including statistical machine learning. However, in general, the evaluation is computationally difficult because it involves intractable multiple summations or integrations; therefore, it requires approximation. Monte Carlo integration (MCI) is a well-known approximation method; a more effective MCI-like approximation method was proposed recently, called spatial Monte Carlo integration (SMCI). However, the estimations obtained using SMCI (and MCI) exhibit a low accuracy in Ising models under a low temperature owing to degradation of the sampling quality. Annealed importance sampling (AIS) is a type of importance sampling based on Markov chain Monte Carlo methods that can suppress performance degradation in low-temperature regions with the force of importance weights. In this study, a new method is proposed to evaluate the expectations on Ising models combining AIS and SMCI. The proposed method performs efficiently in both high- and low-temperature regions, which is demonstrated theoretically and numerically.