# Learning Algorithm of Boltzmann Machine Based on Spatial Monte Carlo Integration Method
## 4 Comparison of 1-SMCI Learning Method and MPLE
### Comparison from Asymptotic Point of View
This means that,when \(N\gg 1\), the distribution of \(m^{\text{ISMCI}}_{ij}(\mathbf{w};\mathcal{D})\) converges to a Gaussian around the mean value (i.e., the exact expectation) that is sharper than a Gaussian to which the distribution of \(m^{\text{MPLE}}_{ij}(\mathbf{w};\mathcal{D})\) converges, and therefore, it is likely that \(m^{\text{ISMCI}}_{ij}(\mathbf{w};\mathcal{D})\) is closer to the exact expectation than \(m^{\text{MPLE}}_{ij}(\mathbf{w};\mathcal{D})\) when \(N\gg 1\); that is, \(m^{\text{ISMCI}}_{ij}(\mathbf{w};\mathcal{D})\) is a better approximation of \(\text{EBM}[x_{i}x_{j}\mid\mathbf{w}]\) than \(m^{\text{MPLE}}_{ij}(\mathbf{w};\mathcal{D})\). Next, we consider the differences between the true gradient in Equation (5) and the approximate gradients in Equations (13) and (26) for \(w_{ij}\):  
\[e^{\text{MPLE}}_{ij}(\mathbf{w};\mathcal{D}) :=\Delta^{\text{MPLE}}_{ij}(\mathbf{w};\mathcal{D})-\Delta^{\text{ MLE}}_{ij}(\mathbf{w};\mathcal{D})=m^{\text{MPLE}}_{ij}(\mathbf{w};\mathcal{D})-\text{EBM}[x_{i}x_{j} \mid\mathbf{w}], \tag{32}\] \[e^{\text{ISMCI}}_{ij}(\mathbf{w};\mathcal{D}) :=\Delta^{\text{ISMCI}}_{ij}(\mathbf{w};\mathcal{D})-\Delta^{\text{ MLE}}_{ij}(\mathbf{w};\mathcal{D})=m^{\text{ISMCI}}_{ij}(\mathbf{w};\mathcal{D})-\text{EBM}[x_{i}x_{j} \mid\mathbf{w}]. \tag{33}\]  
For the gradient differences in Equations (32) and (33), we obtain the following theorem. **Theorem 2**. : _For a Boltzmann machine, \(P_{\text{BM}}(\mathbf{x}\mid\mathbf{w})\), defined in Equation (1), the inequality_  
\[P\big{(}\big{|}e^{\text{MPLE}}_{ij}(\mathbf{w};\mathcal{D})\big{|}\leq\varepsilon \big{)}\leq P\big{(}\big{|}e^{\text{ISMCI}}_{ij}(\mathbf{w};\mathcal{D})\big{|} \leq\varepsilon\big{)},\quad\forall\varepsilon>0,\]  
_is satisfied for all \((i,j)\in E\) when \(N\to\infty\), where \(\mathcal{D}\) is the set of \(N\) data points generated independently from \(P_{\text{BM}}(\mathbf{x}\mid\mathbf{w})\)._  
The proof of this theorem is given in Appendix B. Theorem 2 states that it is likely that the magnitude of \(e^{\text{ISMCI}}_{ij}(\mathbf{w};\mathcal{D})\) is smaller than (or equivalent to) that of \(e^{\text{MPLE}}_{ij}(\mathbf{w};\mathcal{D})\) when the data points are generated independently from the same Boltzmann machine and when \(N\gg 1\). Suppose that \(N\) data points are generated independently from a generative Boltzmann machine, \(P_{\text{BM}}(\mathbf{x}\mid\mathbf{w}^{\text{gen}})\), defined on \(G^{\text{gen}}\), and that a learning Boltzmann machine, defined on the same graph as the generative Boltzmann machine, is trained using the generative data points. In this case, since there is no model error, the solutions of the MLE, the MPLE and the 1-SMCI learning methods are expected to approach \(\mathbf{w}^{\text{gen}}\) as \(N\to\infty\), that is \(\Delta^{\text{MLE}}_{ij}(\mathbf{w}^{\text{gen}};\mathcal{D})\), \(\Delta^{\text{MPLE}}_{ij}(\mathbf{w}^{\text{gen}};\mathcal{D})\), and \(\Delta^{\text{ISMCI}}_{ij}(\mathbf{w}^{\text{gen}};\mathcal{D})\) are expected to approach zero as \(N\to\infty\). Consider the case in which \(N\) is very large and \(\Delta^{\text{MLE}}_{ij}(\mathbf{w}^{\text{gen}};\mathcal{D})=0\). From the statement in Theorem 2, \(\Delta^{\text{ISMCI}}_{ij}(\mathbf{w}^{\text{gen}};\mathcal{D})\) is statistically closer to zero than \(\Delta^{\text{MPLE}}_{ij}(\mathbf{w}^{\text{gen}};\mathcal{D})\). This implies that the solution of the 1-SMCI learning method converges to that of the MLE faster than the MPLE.