## 2 RASMR for Big Data Analysis under GLM  
### SMR Approach for GLM  
In this section, we review the key components of the original SMR approach proposed by [43], along with its potential issues. Following the notation of [43], the original data \(\{(\mathbf{x}_{i},y_{i}),i=1,\ldots,N\}\) are provided with an index partition \(\{I_{1},\ldots,I_{K}\}\) of \(I=\{1,\ldots,N\}\), where \(\mathbf{x}_{i}\in\mathbb{R}^{d}\) represents the \(i\)th covariate vector, \(y_{i}\in\mathbb{R}\) is the corresponding response, and \(I=\cup_{j=1}^{K}I_{j}\), \(I_{j}\
eq\emptyset\) for each \(j\), and \(I_{i}\cap I_{j}=\emptyset\) for each \(i\
eq j\). Under a generalized linear model (GLM) [46; 47], there is a link function \(g\), \(p\) known predictor functions \(h_{1},\ldots,h_{p}\), and \(p\) unknown regression parameters \(\beta_{1},\ldots,\beta_{p}\), such that the expectation of the response variable \(Y_{i}\) given \(\mathbf{x}_{i}\) satisfies the following:  
\[E(Y_{i}\mid\mathbf{x}_{i})=\mu_{i}\;\text{and}\;\eta_{i}=g(\mu_{i})=\mathbf{X} _{i}^{T}\boldsymbol{\beta}, \tag{1}\]  
where \(\mathbf{X}_{i}=(h_{1}(\mathbf{x}_{i}),\ldots,h_{p}(\mathbf{x}_{i}))^{T}\), \(i=1,\ldots,N\), and \(\boldsymbol{\beta}=(\beta_{1},\ldots,\beta_{p})^{T}\). Given the GLM (1), we denote the \(k\)th data block by \(\mathcal{D}_{k}=\{(\mathbf{X}_{i},y_{i}),i\in I_{k}\}\) and let \(s_{k}(\boldsymbol{\beta})=\sum_{i\in I_{k}}[y_{i}-G(\eta_{i})]v(\eta_{i}) \mathbf{X}_{i}\) be the contribution made by the \(k\)th data block to the score function \(s(\boldsymbol{\beta})=\sum_{k=1}^{K}s_{k}(\boldsymbol{\beta})\)[43; 46], where \(G(\eta)=g^{-1}(\eta)\), \(v(\eta)=G^{\prime}(\eta)/h(\eta)\), and \(h(\eta_{i})=\text{Var}(Y_{i})\) are functions of \(\eta\) or \(\eta_{i}\). According to Section 2.5 of [46], the maximum likelihood estimate (MLE) of \(\boldsymbol{\beta}\) solves the score equation \(s(\boldsymbol{\beta})=0\). The SMR algorithm [43] was designed to find \(\tilde{y}_{k}\in\mathbb{R}\) and \(\tilde{\mathbf{X}}_{k}\in\mathbb{R}^{p}\) solving \(s_{k}(\boldsymbol{\beta})=n_{k}[\tilde{y}_{k}-G(\tilde{\eta}_{k})]v(\tilde{ \eta}_{k})\tilde{\mathbf{X}}_{k}\stackrel{{\triangle}}{{=}}\tilde {s}_{k}(\boldsymbol{\beta})\), where \(n_{k}=|I_{k}|\) is the size of \(I_{k}\), and \(\tilde{\eta}_{k}=\tilde{\mathbf{X}}_{k}^{T}\boldsymbol{\beta}\). More specifically, the SMR algorithm first chooses the following:  
\[\tilde{y}_{k}=\left[\sum_{i\in I_{k}}v(\eta_{i})\eta_{i}\right]^{-1}\sum_{i\in I _{k}}v(\eta_{i})\eta_{i}y_{i}\;, \tag{2}\]  
then solves the score-matching equation as follows:  
\[n_{k}v(\tilde{\eta}_{k})[\tilde{y}_{k}-G(\tilde{\eta}_{k})]\tilde{\eta}_{k}= \sum_{i\in I_{k}}v(\eta_{i})[y_{i}-G(\eta_{i})]\eta_{i} \tag{3}\]  
for \(\tilde{\eta}_{k}\in\mathbb{R}\), and then constructs the \(k\)th representative \((\tilde{\mathbf{X}}_{k},\tilde{y}_{k})\) by calculating the following:  
\[\tilde{\mathbf{X}}_{k}=\{n_{k}v(\tilde{\eta}_{k})[\tilde{y}_{k}-G(\tilde{\eta} _{k})]\}^{-1}\sum_{i\in I_{k}}\eta_{i}[y_{i}-G(\eta_{i})]\mathbf{X}_{i}\,. \tag{4}\]  
Figure 1: Graphical display for SMR and RASMR. Then the score function \(s(\mathbf{\beta})=\sum_{i=1}^{N}[y_{i}-G(\eta_{i})]v(\eta_{i})\mathbf{X}_{i}=\sum_{k=1 }^{K}s_{k}(\mathbf{\beta})=\sum_{k=1}^{K}s_{k}(\mathbf{\beta})\stackrel{{ \triangle}}{{=}}\tilde{s}(\mathbf{\beta})\). The MLE \(\mathbf{\hat{\beta}}\) of \(\mathbf{\beta}\) based on the full data, which solves the score equation \(s(\mathbf{\beta})=0\), is expected to be the same as the SMR estimate \(\mathbf{\tilde{\beta}}\) based on the weighted representative data \(\{(n_{k},\tilde{\mathbf{X}}_{k},\tilde{y}_{k}),k=1,\dots,K\}\), which solves the matched score equation \(\tilde{s}(\mathbf{\beta})=0\). In practice, since \(\mathbf{\beta}\) is unknown, the SMR algorithm first solves the matched score function \(\tilde{s}(\mathbf{\beta})=0\) with some initial representatives for \(\mathbf{\tilde{\beta}}\), and then solves the score-matching Equation (3) for the representatives.