# Learning Algorithm of Boltzmann Machine Based on Spatial Monte Carlo Integration Method
## 5 Numerical Comparison with Other Methods  
In this section, we demonstrate a numerical comparison of the 1-SMCI learning method with other approximation methods, RM [19] and MPF [20; 21]. The orders of the computational complexity of these two methods are the same as that of the MPLE and 1-SMCI learning methods. The four methods were implemented by a simple gradient ascent method, \(w_{ij}^{(t+1)}\gets w_{ij}^{(t)}+\eta\Delta_{ij}\), where \(\eta>0\) is the learning rate.  
As described in Section 4.2, we generated \(N\) data points from a generative Boltzmann machine, \(P_{\text{BM}}(\mathbf{x}\mid\mathbf{w}^{\text{gen}})\) and then trained a learning Boltzmann machine of the same size as the generative Boltzmann machine using the generated data points. The coupling parameters in the generative Boltzmann machine, \(w^{\text{gen}}\), were generated from \(U[-0.3,0.3]\). The graph structures of the generative Boltzmann machine and of the learning Boltzmann machine are the same: a \(4\times 4\) square grid. Figure 3 shows the learning curves of the four methods. The horizontal axis represents the number of the step, \(t\), of the gradient ascent method, and the vertical axis represents the mean absolute errors between the solution to the MLE, \(\mathbf{w}^{\text{MLE}}\), and the values of the coupling parameters at the step, \(\mathbf{w}^{(t)}\). In this experiment, we set \(\eta=0.2\), and the values of \(\mathbf{w}\) were initialized as zero.  
Figure 2: The mean absolute errors (MAEs) for various \(N\): (**a**) the case without the model error and (**b**) the case with the model error. Each plot shows the average over 200 trials. MPLE, maximum pseudo-likelihood estimation; 1-SMCI, first-order spatial Monte Carlo integration method.  
Since the vertical axises in Figure 3 represents the the mean absolute error from the solution to the MLE, the lower one is the better approximation of the MLE. We can observe that the MPF shows the fastest convergence and the MPLE, RM, and MPF converge to almost the same values, while the 1-SMCI learning method converges to the lowest values. This concludes that, among the four methods, the 1-SMCI learning method is the best as the approximation of the MLE. However, the 1-SMCI learning method has a drawback. The MPLE, RM, and MPF are convex optimization problems and they have unique solutions, whereas, we do not specify whether the 1-SMCI learning method is a convex problem or not in the present stage. We cannot eliminate the possibility of the existence of local optimal solutions that degrade the accuracy of approximation.  
As mentioned above, the orders of the computational complexity of these four methods, the MPLE, RM, MPF, and 1-SMCI learning methods, are the same, \(O(N|E|)\). However, it is important to check the real computational times of these methods. Table 1 shows the total computational times needed for the one-time learning (until convergence), where the setting of the experiment is the same as that of Figure 3b.  
The MPF method is the fastest, and the 1-SMCI learning method is the slowest which is about 6-7 times slower than the MPF method.