# Learning Algorithm of Boltzmann Machine Based on Spatial Monte Carlo Integration Method
## 1 Introduction  
The machine learning techniques for Markov random fields (MRFs) are fundamental in various fields involving pattern recognition [1; 2], image processing [3], sparse modeling [4], and Earth science [5; 6], and a Boltzmann machine [7; 8; 9] is one of the most important models in MRFs. The inference and learning problems in the Boltzmann machine are NP-hard, because they include intractable multiple summations over all the possible configurations of variables. Thus, one of the major challenges of the Boltzmann machine is the design of the efficient inference and learning algorithms that it requires.  
Various effective algorithms for Boltzmann machine learning were proposed by many researchers, a few of which are mean-field learning algorithms [10; 11; 12; 13; 14; 15], maximum pseudo-likelihood estimation (MPLE) [16; 17], contrastive divergence (CD) [18], ratio matching (RM) [19], and minimum probability flow (MPF) [20; 21]. In particular, the CD and MPLE methods are widely used. More recently, the author proposed an effective learning algorithm based on the spatial Monte Carlo integration (SMCI) method [22]. The SMCI method is a Monte Carlo integration method that takes spatial information around the region of focus into account; it was proven that this method is more effective than the standard Monte Carlo integration method. The main target of this study is Boltzmann machine learning based on the first-order SMCI (1-SMCI) method, which is the simplest version of the SMCI method. We refer it to as the 1-SMCI learning method in this paper.  
It was empirically shown through the numerical experiments that Boltzmann machine learning based on the 1-SMCI learning method is more effective than MPLE in the case where no model error exists, i.e., in the case where the learning model includes the generative model [22]. However,the theoretical reason for this was not revealed at all. In this paper, theoretical insights into the effectiveness of the 1-SMCI learning method as compared to that of MPLE are given from an asymptotic point of view. The theoretical results obtained in this paper state that the gradients of the log-likelihood function obtained by the 1-SMCI learning method constitute a quantitatively better approximation of the exact gradients than those obtained by the MPLE method in the case where the generative model and the learning model are the same Boltzmann machine (in Section 4.1). This is one of the contributions of this paper. In the previous paper [22], the 1-SMCI learning method was compared with only the MPLE. In this paper, we compare the 1-SMCI learning method with other effective learning algorithms, RM and MPF, through numerical experiments, and show that the 1-SMCI learning method is superior to them (in Section 5). This is the second contribution of this paper.  
The remainder of this paper is organized as follows. The definition of Boltzmann machine learning and a briefly explanation of the MPLE method are given in Section 2. In Section 3, we explain Boltzmann machine learning based on the 1-SMCI method: reviews of the SMCI and 1-SMCI learning methods are presented in Sections 3.1 and 3.2, respectively. In Section 4, the 1-SMCI learning method and MPLE are compared using two different approaches, the theoretical approach (in Section 4.1) and the numerical approach (in Section 4.2), and the effectiveness of the 1-SMCI learning method as compared to the MPLE is shown. In Section 5, we numerically compare the 1-SMCI method with other effective learning algorithms and observe that the 1-SMCI learning method yields the best approximation. Finally, the conclusion is given in Section 6.