## Appendix A Theorems on Solving the Score-Matching Equation  
In this appendix, we provide theorems and remarks on solving the score-matching Equation (3), including explicit solutions for the normal/linear model (Theorem A2), the gamma model (Theorem A5), and the inverse Gaussian model (Theorem A6). With the suggested second split based on \(y_{i}>G(\eta_{i})\) and further split for Bernoulli models (Remark A2) and Poisson model (Remark A3), we largely resolve the issue of multiple solutions to (3) in the original SMR algorithm, as explained in Section 2.2. **Theorem A1**. : _Suppose both \(v(\eta)\) and \(G(\eta)\) are continuous on \([\eta_{k}^{\wedge},\eta_{k}^{\vee}]\). If \(S(\eta)\) is strictly monotone on \([\eta_{k}^{\wedge},\eta_{k}^{\vee}]\), then there exists a unique \(\eta_{*}\in[\eta_{k}^{\wedge},\eta_{k}^{\vee}]\) that solves (3)._  
**Theorem A2**. : _For the normal model with an identity link, that is, the usual linear model, there are up to two solutions solving (3):_  
\[\tilde{\eta}_{k,1}=\frac{1}{2}\bigg{(}\tilde{y}_{k}-\sqrt{\tilde{y}_{k}^{2}-4 \tilde{S}}\bigg{)}\,,\ \tilde{\eta}_{k,2}=\frac{1}{2}\bigg{(}\tilde{y}_{k}+\sqrt{\tilde{y}_{k}^{2}-4 \tilde{S}}\bigg{)}\,,\] (A1)  
_where \(\tilde{S}=\eta_{k}\tilde{y}_{k}-n_{k}^{-1}\sum_{i\in I_{k}}\eta_{i}^{2}\) and \(\eta_{k}=n_{k}^{-1}\sum_{i\in I_{k}}\eta_{i}\). Furthermore,_  
_(1) If \(\eta_{i}<0\) and \(y_{i}>G(\eta_{i})\) for all \(i\in I_{k}\), then the only solution is \(\tilde{\eta}_{k,1}\)._  
_(2) If \(\eta_{i}>0\) and \(y_{i}<G(\eta_{i})\) for all \(i\in I_{k}\), then the only solution is \(\tilde{\eta}_{k,2}\)._  
_In general, if \(\tilde{y}_{k}/2\
otin(\eta_{k}^{\wedge},\eta_{k}^{\vee})\), then the solution is unique._  
Note that the essentially same explicit solutions as in (A1) are described in Section 4.2.2.1 of [64]. Along with Theorem A1 and Lemmas S1-S5 in the Supplementary Materials (see Section S6), we conclude the following summarized results for Bernoulli models. **Theorem A3**. : _For Bernoulli models with logit, probit, cloglog, loglog, or caudit links, a data block \(I_{k}\) satisfying one of the following six conditions yields a unique solution solving (3):_  
_(i) \(\eta_{i}<\eta_{i}\) and \(y_{i}<G(\eta_{i})\) for all \(i\in I_{k}\);_  
_(ii) \(\eta_{l}<\eta_{i}<0\) and \(y_{i}<G(\eta_{l})\) for all \(i\in I_{k}\);_  
_(iii) \(\eta_{i}>0\) and \(y_{i}<G(\eta_{i})\) for all \(i\in I_{k}\);_  
_(iv) \(\eta_{i}<0\) and \(y_{i}>G(\eta_{i})\) for all \(i\in I_{k}\);_  
_(v) \(0<\eta_{i}<\eta_{r}\) and \(y_{i}>G(\eta_{l})\) for all \(i\in I_{k}\);_  
_(vi) \(\eta_{i}>\eta_{r}\) and \(y_{i}>G(\eta_{i})\) for all \(i\in I_{k}\),_  
_where \(\eta_{l}<0\) and \(\eta_{r}>0\) are constants listed in Table 1. A general data block under Bernoulli models may yield up to two solutions solving (3)._  
**Remark A1**. : _In practice, the \"\(<\)\" and \"\(>\)\" in Theorem A3 can be relaxed to \"\(\leq\)\" and \"\(\geq\)\", respectively, as long as \"\(=\)\" can only be attained by a small portion of the observations._  
**Remark A2**. : _Theorem A3 suggests further split sub-blocks according to \(\eta_{l}\) or \(\eta_{r}\), which are constants associated with link functions. More specifically, if \(\eta_{i}<0\) and \(y_{i}<C(\eta_{i})\) for all \(i\in I_{k}\), we further split \(I_{k}\) into two sub-blocks at the splitting point \(\eta_{l}\) (see cases (i) and (ii)); if \(\eta_{l}>0\) and \(y_{i}>G(\eta_{i})\) for all \(i\in I_{k}\), we further split \(I_{k}\) at the splitting point \(\eta_{r}\) (see cases (v) and (vi)). Each resulting block yields a unique solution solving (3), which is equivalent to solving \(S(\eta)=\bar{S}\) (see (6)). For example, for Bernoulli models with logit link, that is, logistic regression models, we have the following:_  
\[S(\eta)=\left\{ \begin{array}{ll}-\frac{p\epsilon^{\beta}}{\eta+\sigma^{ \beta}}\;,&\text{for cases (i), (ii) and (iii);}\\ \frac{1}{1+\epsilon^{\beta}}\;,&\text{for cases (iv), (v), and (vi).}\end{array}