## 4 Proposed Solution
### Main Characteristics of the Multi-Agent Gradient Descent Algorithm (MAGD)
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Dataset Name** & **MAGD-500** & **Adam-500** & **Adam-750** & **Adam-1000** \\ \hline Iris & 2.01 & **0.78** & 1.19 & 2.05 \\ Wine & 1.82 & **0.86** & 1.24 & 2.16 \\ Breast-Cancer-Wisconsin & 2.25 & **1.10** & 1.71 & 3 \\ Digits & 11.58 & **6.43** & 11.31 & 21.73 \\ Pima-Indians-Diabetes & 1.97 & **1.01** & 2.03 & 3 \\ Connectionist Bench & 2.66 & **2.55** & 3.86 & 9.38 \\ Glass Identification & 1.39 & **0.91** & 1.34 & 3.41 \\ Balance Scale & 1.76 & **0.93** & 1.38 & 2.96 \\ Banknote Authentication & 1.88 & **1.02** & 1.53 & 3.37 \\ Adult & 160.42 & **89.43** & 166.8 & 206.56 \\ Diabetes & 1.95 & **0.81** & 1.22 & 1.64 \\ California Housing & 8.02 & **7.18** & 12.17 & 15.99 \\ Boston Housing & 1.91 & **0.84** & 1.27 & 1.78 \\ Auto MPG & 1.78 & **0.72** & 1.11 & 1.53 \\ Concrete Compressive Strength & 1.23 & **0.80** & 1.27 & 1.71 \\ Energy Efficiency & 1.77 & **0.76** & 1.25 & 1.64 \\ Yacht Hydrodynamics & 1.09 & **0.70** & 1.1 & 1.49 \\ Forest Fires & 2.17 & **0.92** & 1.51 & 1.93 \\ Airfoil Self-Noise & 2.07 & **0.81** & 1.27 & 1.66 \\ Concrete Slump Test & 1.65 & **0.67** & 1.05 & 1.49 \\ \hline \hline \end{tabular}
\end{table} 
Table 27: Mean and standard deviation of execution times for MAGD and Adam with varying maximum iterations for neural network training. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Convergence Epochs** & **Execution Time (Mean \(\pm\) std)** \\ \hline MAGD-500 & \(499\pm 1\) & \(10.57\pm 34.47\) \\ Adam-500 & \(500\pm 0\) & \(5.96\pm 19.23\) \\ Adam-750 & \(750\pm 0\) & \(10.78\pm 35.93\) \\ Adam-1000 & \(997\pm 13\) & \(14.42\pm 44.39\) \\ \hline \hline \end{tabular}
\end{table} 
Table 26: Execution time comparison among MAGD and Adam variants for neural network training across datasets. Bold values indicate the shortest time for each dataset. of observations. Thus, \(T_{A_{1}}=T_{A_{2}}+ATP(A_{1},A_{2})\cdot T_{A_{2}}\). A negative \(ATP\) value indicates that \(A_{1}\) is computationally faster than \(A_{2}\). We computed both the \(P_{super}\) and the corresponding \(ATP\) for MAGD considering all datasets for each types of the tasks, means \(n=10\) for fixed 500 maximum iterations and Adam for three different values of maximum iterations and the results are presented in Table 30 for classification tasks and in Table 31 for regression tasks. \begin{table}
\begin{tabular}{c c c c c} \hline
**Dataset Name** & **MAGD-500** & **Adam-500** & **Adam-750** & **Adam-1000** \\ \hline Iris & **100\%** & **100\%** & **100\%** & **100\%** \\ Wine & **98\%** & **98\%** & **98\%** & **98\%** \\ Breast-Cancer-Wisconsin & **96\%** & **96\%** & **96\%** & **96\%** \\ Digits & **97\%** & **97\%** & **97\%** & **97\%** \\ Pima-Indians-Diabetes & **79\%** & 76\% & 78\% & 76\% \\ Connectionist Bench & **78\%** & 76\% & 76\% & 76\% \\ Glass Identification & **62\%** & 55\% & 54\% & 58\% \\ Balance Scale & **90\%** & 87\% & 88\% & 88\% \\ Banknote Authentication & **99\%** & 98\% & 98\% & 98\% \\ Adult & **85\%** & **85\%** & **85\%** & **85\%** \\ \hline \end{tabular}
\end{table} 
Table 28: Accuracy scores of MAGD with 500 maximum iterations compared to Adam with 500, 750, and 1000 maximum iterations across various classification datasets. Bold values indicate the highest score achieved for each dataset. \begin{table}
\begin{tabular}{c c c c} \hline
**Comparison** & \(ATP\) (\(MAGD\), \(Adam\)) & \(P_{super}\) (\(MAGD\)\(\succ\)\(Adam\)) \\ \hline MAGD-500 vs. Adam-500 & 0.86 & 50\% \\ MAGD-500 vs. Adam-750 & 0.17 & 50\% \\ MAGD-500 vs. Adam-1000 & \(-\)0.36 & 50\% \\ \hline \end{tabular}
\end{table} 
Table 30: Comparison of MAGD for fixed maximum iterations and Adam with varying maximum iterations using \(ATP\) (average time penalty) and \(P_{super}\) (performance superiority) in classification tasks.