## 4 Simulation Studies and Numerical Justifications
### Performance of RASMR, Algorithm 1
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Simulation**} & \multicolumn{8}{c}{**Binary Classification, K-Means (K = 1000), True Link Function = Logit**} \\ \hline
**Representatives** & \multicolumn{4}{c}{**MR**} & \multicolumn{4}{c}{**Original SMR**} & \multicolumn{4}{c}{**RASMR**} \\ \hline
**Setup** & **Logit** & **Cloglog** & **Probit** & **Cauchi** & **Logit** & **Cloglog** & **Probit** & **Cauchi** & **Logit** & **Cloglog** & **Probit** & **Cauchi** \\ \hline mzNormal & 17.9 & 9.2 & 9.2 & 33.4 & 1.8 & 1.6 & 1.0 & 2.4 & \(3.7\times 10^{-5}\) & \(2.3\times 10^{-1}\) & \(6.4\times 10^{-5}\) & \(1.0\times 10^{-2}\) \\  (0.3) & (0.3) & (0.2) & (1.2) & (0.6) & (0.5) & (0.3) & (2.4) & \((1.0\times 10^{-5})\) & \((1.8\times 10^{-1})\) & \((2.6\times 10^{-5})\) & \((4.2\times 10^{-3})\) \\ nzNormal & 14.3 & 5.0 & 7.0 & 15.5 & 4.0 & 0.8 & 1.5 & 51.6 & \(1.5\times 10^{-3}\) & \(2.5\times 10^{-2}\) & \(2.5\times 10^{-4}\) & \(69.6\times 10^{-3}\) \\  & (0.7) & (0.3) & (0.3) & (0.3) & (45.8) & (1.1) & (0.4) & (0.5) & (34.6) & \((50\times 10^{-4})\) & \((3.0\times 10^{-2})\) & \((7.4\times 10^{-5})\) & \((20.0)\) \\  ueNormal & 211 & 114 & 110 & 455 & 3.3 & 11 & 2.1 & 19.4 & \(7.2\times 10^{-5}\) & \(4.3\) & \(4.7\times 10^{-4}\) & \(10.4\) \\  & (1.4) & (1.5) & (0.7) & (5.2) & (1.4) & (3.4) & (1.1) & (10.1) & \((3.3\times 10^{-6})\) & \((1.2)\) & \((9.9\times 10^{-5})\) & \((10.2)\) \\ mixNormal & 17.5 & 8.6 & 8.6 & 48.1 & 3.0 & 1.8 & 1.2 & 3.2 & \(3.6\times 10^{-4}\) & \(5.0\times 10^{-1}\) & \((1.6\times 10^{-4})\) & \((5.8\times 10^{-2})\) \\  & (0.4) & (0.6) & (0.2) & (3.1) & (0.9) & (0.7) & (0.3) & (2.5) & \((1.6\times 10^{-4})\) & \((1.8\times 10^{-1})\) & \((1.8\times 10^{-4})\) & \((5.5\times 10^{-2})\) \\ \(T_{3}\) & 12.2 & 10.1 & 7.8 & 10.0 & 10.7 & 15.0 & 6.7 & 8.6 & \(6.3\times 10^{-2}\) & \(1.2\times 10^{-1}\) & \(5.2\times 10^{-2}\) & \(2.8\times 10^{-2}\) \\  & (3.1) & (2.6) & (1.9) & (2.7) & (3.1) & (39.5) & (1.9) & (2.7) & \((3.5\times 10^{-2})\) & \((7.2\times 10^{-2})\) & \((5.2\times 10^{-2})\) & \((2.8\times 10^{-2})\) \\ EXP & 12.4 & 3.9 & 6.2 & 10.4 & 5.8 & 1.4 & 2.8 & 4.5 & \(1.4\times 10^{-6}\) & \(3.3\times 10^{-4}\) & \(5.3\times 10^{-6}\) & \(9.6\times 10^{-3}\) \\  & (0.9) & (0.5) & (0.5) & (1.4) & (1.0) & (0.3) & (0.5) & (1.4) & \((3.3\times 10^{-7})\) & \((8.1\times 10^{-5})\) & \((6.9\times 10^{-7})\) & \((4.4\times 10^{-3})\) \\ BETA & 3.1 & 1.3 & 1.7 & 7.6 & 2.0 & 0.9 & 1.0 & 11.3 & \(9.2\times 10^{-7}\) & \(4.7\times 10^{-5}\) & \(2.7\times 10^{-5}\) & \(5.6\times 10^{-3}\) \\  & (0.8) & (0.4) & (0.5) & (1.7) & (0.7) & (0.3) & (0.3) & (2.3) & \((6.2\times 10^{-7})\) & \((2.4\times 10^{-5})\) & \((2.6\times 10^{-6})\) & \((6.7\times 10^{-3})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average (std) of RMSEs (\(10^{-3}\)) over 100 simulations for binary classification. In [43], extensive comparisons between SMR and other big data approaches have been made in terms of computing speed. According to [43], given a partition of data, SMR is noticeably faster than a divide-and-conquer (DC) approach proposed by [5], but slower than the subsampling approach under A-optimality [56] for logistic regression. Using their results of computing time comparison as a reference, we compare the computing speed of RASMR with SMR for binary classification with four link functions: logit, cloglog, probit, and cauchit. The covariates are generated from the seven distributions described previously and the computation is conducted on a PC running Windows 10 Home (Version 2004) with 2.80 GHz 4-core Intel i7-7700 HQ and 16 GB of memory. The simulation is repeated 100 times and the average CPU time for one iteration of SMR and RASMR is recorded in Table 5.  
According to Table 5, RASMR is slightly slower than SMR. It is because additional splits are made on a proportion of data blocks, which result in a larger number of data blocks in RASMR, given the same partition of data.