# Nonsmooth Optimization-Based Hyperparameter-Free Neural Networks for Large-Scale Regression
## 5 LMBNNR Algorithm
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**LMSENNR**} & \multicolumn{2}{c}{**MGDGD**} & \multicolumn{2}{c}{**SGD**} & \multicolumn{2}{c}{**EMD**} & \multicolumn{2}{c}{**SGD**} & \multicolumn{2}{c}{**EMD**} & \multicolumn{2}{c}{**MOMMLP**} \\ \cline{2-13} \cline{11-11}
**H** & **RMSE** & **CPU** & **RMSE** & **CPU** & **RMSE** & **CPU** & **RMSE** & **CPU** & **RMSE** & **CPU** & **RMSE** & **CPU** \\ \hline \multicolumn{13}{c}{**Combined cycle power plant**} \\ \hline
2 & 4.433 & 0.14 & 8.472 & 1.45 & 19.081 & 0.51 & 4.582 & 1036.42 & 18.907 & 0.00 & 6.811 & 17.10 \\
5 & 4.267 & 0.61 & 6.091 & 0.97 & 23.567 & 0.88 & 4.415 & 790.22 & 5.192 & 0.00 & 6.692 & 40.72 \\
10 & 4.216 & 2.202 & 5.784 & 0.78 & 17.809 & 0.47 & 4.278 & 603.95 & 5.192 & 0.02 & 6.647 & 97.91 \\
50 & 4.140 & 12.27 & 4.712 & 1.04 & 18.750 & 0.66 & 4.215 & 679.02 & 5.192 & 0.17 & 9.424 & 1359.97 \\
100 & 4.139 & 17.70 & 4.612 & 0.68 & 17.213 & 0.38 & **4.167** & **69.05** & 5.192 & 0.49 & 17.207 & 1514.94 \\
200 & 4.139 & 31.92 & 4.566 & 0.66 & 17.469 & 0.69 & 4.190 & 691.76 & 5.192 & 2.34 & 1-im \\
500 & & & 4.534 & 1.00 & 16.514 & 0.63 & 4.168 & 761.36 & 5.192 & 8.03 & 1-im & – \\
5000 & & & 4.533 & 1.81 & 16.710 & 1.53 & 4.205 & 897.52 & 5.192 & 1359.99 & +im & – \\ \hline \multicolumn{13}{c}{**Sket**} & 4.138 & 8.14 & \(H=29\) & & & & & & & & & \\ \multicolumn{13}{c}{**ASF**} & **4.139** & **13.73** & \(H=61\) & & & & & & & & & \\ \hline \multicolumn{13}{c}{**Airloff self-noise**} \\ \hline
2 & 4.407 & 0.02 & 6.756 & 0.36 & 10.129 & 0.29 & 4.387 & 102.65 & 87.622 & 0.00 & 5.709 & 4.98 \\
5 & 4.407 & 0.10 & 6.580 & 0.38 & 9.465 & 0.30 & 3.688 & 102.13 & 30.274 & 0.00 & 5.634 & 9.05 \\
10 & 4.410 & 0.30 & 5.888 & 0.35 &

\begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline  & \multicolumn{2}{c}{**LMBNNR**} & \multicolumn{2}{c}{**MBGD**} & \multicolumn{2}{c}{**BCD**} & \multicolumn{2}{c}{**SGD**} & \multicolumn{2}{c}{**EML**} & \multicolumn{2}{c}{**MONMLP**} \\ \cline{2-13} \(H\) & **RMSE** & **CPU** & **RMSE** & **CPU** & **RMSE** & **CPU** & **RMSE** & **CPU** & **RMSE** & **CPU** & **RMSE** & **CPU** \\ \hline \multicolumn{13}{c}{**Physicochemical properties of protein**} \\ \hline
2 & 5.304 & 1.06 & 5.223 & 2.29 & 7.888 & 0.71 & 5.327 (1) & 3368.54



\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**LMENNR**} & \multicolumn{2}{c}{**MICD**} & \multicolumn{2}{c}{**SGD**} & \multicolumn{2}{c}{**SGD**} & \multicolumn{2}{c}{**EMM**} & \multicolumn{2}{c}{**MONMLP**} \\ \cline{2-13}
**H** & **RMSE** & **CPU** & **RMSE** & **CPU** & **RMSE** & **CPU** & **RMSE** & **CPU** & **RMSE** & **CPU** & **RMSE** & **CPU** & **RMSE** & **CPU** \\ \hline \multicolumn{13}{c}{**Online news popularity**} \\ \hline
2 & 123,387.48 & 0.75 & 123,201.20 & 1.68 & 13,5014.63 & 0.63 & 12,457,39 (1) & 2664.11 & 11,228.42 & 0.03 & 10,603.67 & 692.14 \\
5 & 123,377.88 & 13,272.34 & 123,407.9 & 1.68 & 13,502.67 & 0.69 & NAN & - & 11,098.23 & 0.09 & 5986.04 & 2233.04 \\
60 & 123,363.9 & 32,58 & 123,407.9 & 1.68 & 13,502.67 & 0.69 & NAN & - & 11,098.23 & 0.09 & 5986.04 & 2233.04 \\
80 & 123,363.9 & 32,58 & 124,058.1 & 1.77 & 16,463.91 & 0.67 & NAN & - & 11,098.49 & 0.09 & 598.41 & 204.55 \\
100 & 123,365.9 & 320.50 & 127,744.9 & 1.78 & 14,731.9 & 0.68 & NAN & - & 11,043.99 & 3.07 & 1\(\rm\AA\) & - \\
200 & 123,363.9 & 706.29 & 125,076.68 & 1.80 & 13,097.35 & 0.74 & NAN & - & 11,043.69 & 10.28 & \(\rm\AA\) & - \\
500 & & 123,583.3 & 2.10 & 13,091.07 & 0.91 & NAN & - & 11,043.69 & 23.44 & \(\rm\AA\) & - & \(\rm\AA\) \\
5000 & & & 123,232.53 & 6.27 & 12,944.06 & 3.32 & NAN & - & 14,088 & \(\rm\AA\) & - & \(\rm\AA\) & - \\ \hline Best & 123,363.9 & 16.49 & \(H=19\) & & & & & & & & & \\
857 & 123,363.9 & 16.49 & \(H=19\) & & & & & & & & & \\ \hline \multicolumn{13}{c}{\({}^{\dagger}\) 7/100 km km s s/ km s



\begin{table}
\begin{tabular}{l c c c c c c c c c c c c c} \hline  & \multicolumn{2}{c}{**LAMBNN**} & \multicolumn{2}{c}{**MIGD**} & \multicolumn{2}{c}{**BCD**} & \multicolumn{2}{c}{**SGD**} & \multicolumn{2}{c}{**ELM**} & \multicolumn{2}{c}{**MOONLP**} \\ \cline{2-13} \cline{13-1



The predictions with the LMBNNR algorithm in termination are better than those of any other tested NNR algorithm in 5 data sets out of 12.