## 3 The SMO-like Algorithm for the Nonsmooth Dual Problem of \(\varepsilon\)-In SVR
### Working Set Selection
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Dataset** & **Method** & **(C, \(\sigma\), \(\varepsilon\), \(\eta_{1}\))** & **\# of SVs** & **Test RMSE** & **Training Cpu Time** & **\# of Iterations** \\ \hline Servo (167x4) & \(\varepsilon\)-\(l_{2}\) SVR & \((10^{1},2^{0},0)\) & \(133.5\pm 0.53\) & 0.886 \(\pm\) 0.18 & 0.033 \(\pm\) 0.04 & 459 \(\pm\) 13.6 \\  & \(\varepsilon\)-SVR & \((10^{3},2^{0},0,02)\) & \(130.9\pm 1.79\) & 0.785 \(\pm\) 0.27 & 0.138 \(\pm\) 0.03 & 57,100.2 \(\pm\) 16,430.5 \\  & \(\varepsilon\)-In SVR & \((10^{1},2^{-1},0,2^{1})\) & \(133.6\pm 0.52\) & **0.726 \(\pm\) 0.28** & 0.052 \(\pm\) 0.04 & 1253 \(\pm\) 53.5 \\ \hline Auto-mpg (392x7) & \(\varepsilon\)-\(l_{2}\) SVR & \((10^{1},2^{-1},0.5)\) & \(257.9\pm 7.23\) & 2.776 \(\pm\) 0.33 & 0.064 \(\pm\) 0.03 & 1049.1 \(\pm\) 37.7 \\  & \(\varepsilon\)-SVR & \((10^{2},2^{-1},0.5)\) & \(245.5\pm 4.06\) & 2.674 \(\pm\) 0.36 & 0.022 \(\pm\) 0.03 & 5236.1 \(\pm\) 938.0 \\  & \(\varepsilon\)-In SVR & \((10^{1},2^{-1},1.5,2^{-1})\) & \(160.2\pm 6.11\) & **2.595 \(\pm\) 0.27** & 0.039 \(\pm\) 0.02 & 967.9 \(\pm\) 88.9 \\ \hline Boston (560x13) & \(\varepsilon\)-\(l_{2}\) SVR & \((10^{2},2^{0},0)\) & \(404.8\pm 0.42\) & 4.626 \(\pm\) 1.72 & 0.059 \(\pm\) 0.03 & 4827.3 \(\pm\) 106.9 \\  & \(\varepsilon\)-SVR & \((10^{2},2^{0},1)\) & \(265.8\pm 8.32\) & 3.461 \(\pm\) 0.78 & 0.058 \(\pm\) 0.05 & 4062.1 \(\pm\) 820.9 \\  & \(\varepsilon\)-In SVR & \((10^{2},2^{-1},1,2^{1})\) & \(269.3\pm 5.12\) & **3.151 \(\pm\) 0.38** & 0.061 \(\pm\) 0.03 & 5261.9 \(\pm\) 252.4 \\ \hline Cooling (768x8) & \(\varepsilon\)-\(l_{2}\) SVR & \((10^{1},2^{0},0)\) & \(614.4\pm 0.52\) & 3.031 \(\pm\) 0.33 & 0.044 \(\pm\) 0.03 & 2339.4 \(\pm\) 59.1 \\  & \(\varepsilon\)-SVR & \((10^{2},2^{-1},0)\) & \(614.4\pm 0.52\) & 1.981 \(\pm\) 0.20 & 0.222 \(\pm\) 0.06 & 39,150.1 \(\pm\) 4705.9 \\  & \(\varepsilon\)-In SVR & \((10^{3},2^{0},0,2^{-3})\) & \(614.4\pm 0.52\) & **1.772 \(\pm\) 0.15** & 0.153 \(\pm\) 0.08 & 65,277.4 \(\pm\) 1747.3 \\ \hline Heating (768x8) & \(\varepsilon\)-\(l_{2}\) SVR & \((10^{2},2^{0},0)\) & \(614.3\pm 0.67\) & 1.980 \(\pm\) 0.21 & 0.098 \(\pm\) 0.02 & 9212.6 \(\pm\) 213.4 \\  & \(\varepsilon\)-SVR & \((10^{3},2^{0},0,5)\) & \(421.9\pm 8.28\) & 1.124 \(\pm\) 0.10 & 0.621 \(\pm\) 0.19 & 143,945.9 \(\pm\) 27,282.0 \\  & \(\varepsilon\)-In SVR & \((10^{3},2^{0},0,2^{0})\) & \(614.4\pm 0.52\) & **0.939 \(\pm\) 0.06** & 0.267 \(\pm\) 0.07 & 91,786.0 \(\pm\) 2373.9 \\ \hline Airfoil (1503x5) & \(\varepsilon\)-\(l_{2}\) SVR & \((10^{1},2^{-2},0)\) & \(1202.4\pm 0.52\) & 3.849 \(\pm\) 1.32 & 0.083 \(\pm\) 0.05 & 4840.6 \(\pm\) 130.3 \\  & \(\varepsilon\)-SVR & \((10^{2},2^{-3},0,2)\) & \(1103.5\pm 5.97\) & **2.776 \(\pm\) 0.26** & 0.147 \(\pm\) 0.08 & 28,317.0 \(\pm\) 3876.7 \\  & \(\varepsilon\)-In SVR & \((10^{2},2^{-2},1,2^{-1})\) & \(789.7\pm 9.52\) & 2.778 \(\pm\) 0.27 & 0.120 \(\pm\) 0.03 & 15,367.1 \(\pm\) 451.2 \\ \hline Space ga (3107x6) & \(\varepsilon\)-\(l_{2}\) SVR & \((10^{2},2^{1},0)\) & \(2485.4\pm 0.70\) & 1.232 \(\pm\) 0.40 & 0.243 \(\pm\) 0.03 & 8361.1 \(\pm\) 655.1 \\ (3107x6) & \(\varepsilon\)-SVR & \((10^{1},2^{-2},0,05)\) & \(2184.9\pm 11.01\) & 0.134 \(\pm\) 0.01 & 0.508 \(\pm\) 0.11 & 21,577.9 \(\pm\) 2162.1 \\  & \(\varepsilon\)-In SVR & \((10^{2},2^{1},0,2^{0})\) & \(2485.1\pm 0.74\) & **0.116 \(\pm\) 0.01** & 0.319 \(\pm\) 0.06 & 20,649.9 \(\pm\) 648.4 \\ \hline Abalone & \(\varepsilon\)-\(l_{2}\) SVR & \((10^{0},2^{-2},0)\) & \(3341.2\pm 0.63\) & 2.406 \(\pm\) 0.22 & 0.308 \(\pm\) 0.07 & 6336.3 \(\pm\) 119.7 \\ (4177x8) & \(\varepsilon\)-SVR & \((10^{0},2^{0},1.5)\) & \(1367.8\pm 20.42\) & 2 While \(\varepsilon\)-SVR and \(\varepsilon\)-\(l_{2}\) SVR employ loss functions optimized for specific noise distributions such as Laplace and Gaussian, respectively, the \(\varepsilon\)-ln SVR offers greater flexibility. Its modified Incosh loss is optimal for the broader family of power-raised hyperbolic secant distributions, encompassing Laplace, Gaussian and hyperbolic secants as special cases.