# Learning Algorithm of Boltzmann Machine Based on Spatial Monte Carlo Integration Method
## 6 Conclusions  
In this paper, we examined the effectiveness of Boltzmann machine learning based on the 1-SMCI method proposed in [22] where, by numerical experiments, it was shown that the 1-SMCI learning method is more effective than the MPLE in the case where no model error exists. In Section 4.1, we gave the theoretical results for the empirical observation from the asymptotic point of view. The theoretical results improved our understanding of the advantage of the 1-SMCI learning method as compared to the MPLE. The numerical experiments in Section 4.2 showed that the 1-SMCI learning method is a better approximation of the MPLE in the case with and without model error. Furthermore, we compared the 1-SMCI learning method with the other effective methods, RM and MPF, using the numerical experiments in Section 5. The numerical results showed that the 1-SMCI learning method is the best method.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline  & **MPLE** & **RM** & **MPF** & **1-SMCI** \\ \hline time (s) & 0.08 & 0.1 & 0.04 & 0.26 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Real computational times of the four learning methods. The setting of the experiment is the same as that of Figure 3b.  
Figure 3: Mean absolute errors (MAEs) versus the number of updates of the gradient ascent method: (a) \(N=200\) and (b) \(N=2000\). Each plot shows the average over 200 trials. RM, ratio matching.  
However, issues related to the 1-SMCI learning method still remain. Since the objective function of the 1-SMCI learning method, e.g., Equation (7) for the MPLE, is not revealed, it is not straightforward to specify whether the problem of solving Equation (27), i.e., a gradient ascent method with the gradients in Equation (26), is a convex problem or not. This is one of the most challenging issues of the method. As shown in Section 4.2, the performance of the 1-SMCI learning method decreases when model error exists, i.e., when the learning Boltzmann machine does not include the generative model. The decrease may be caused by the replacement of the sample points, \(\mathcal{S}\), by the data points, \(\mathcal{D}\), as discussed in the same section. It is expected that combining the 1-SMCI learning method with an effective sampling method, e.g., the persistent contrastive divergence [28], relaxes the problem of the performance degradation.  
The presented the 1-SMCI learning method can be applied to other types of Boltzmann machines, e.g., restricted Boltzmann machine [1], deep Boltzmann machine [2; 29]. Although we focused on the Boltzmann machine learning in this paper, the SMCI method can be applied to various MRFs [22]. Hence, there are many future directions of application of the SMCI: for example, graphical LASSO problem [4], Bayesian image processing [3], Earth science [5; 6] and brain-computer interface [30; 31; 32; 33].  
**Acknowledgments:** This work was partially supported by JST CREST, Grant Number JPMJCR1402, ISPS KAKENHI, Grant Numbers, 15K00330 and 15H03699, and MIC SCOPE (Strategic Information and Communications R&D Promotion Programme), Grant Number 172302009.