Spatial Monte Carlo Integration with Annealed Importance Sampling
This work provides an improved method for accurately evaluating expectations on Ising models, which is crucial for researchers and practitioners in statistical machine learning, especially when dealing with low-temperature regimes.
This paper addresses the challenge of evaluating expectations on Ising models, particularly at low temperatures where existing methods like Monte Carlo integration (MCI) and spatial Monte Carlo integration (SMCI) suffer from low accuracy due to sampling degradation. The authors propose a new method that combines Annealed Importance Sampling (AIS) with SMCI, demonstrating its efficiency in both high- and low-temperature regions.
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.