## 4 Proposed Solution
### Main Characteristics of the Multi-Agent Gradient Descent Algorithm (MAGD)
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function Name** & **MAGD** & **FMA** & **SBGD** & **Adam** & **GD** \\ \hline Rastrigin & 10 & **50** & **50** & 0 & 0 \\ Ackley & 0 & 0 & **100** & 0 & 0 \\ Goldstein–Price & **100** & **100** & **100** & 30 & 0 \\ Levy-n13 & **100** & **100** & 70 & 10 & 0 \\ Himmelbau & **100** & **100** & **100** & 80 & **100** \\ Three-Hump Camel & 80 & **100** & 50 & 10 & 30 \\ Griewank & **0** & **0** & **0** & **0** & **0** \\ Cross-in-Tray & **0** & **0** & **0** & **0** & **0** \\ EggHolder & **0** & **0** & **0** & **0** & **0** \\ Michalewicz & **100** & **100** & **100** & 10 & 0 \\ Alpine & 30 & **100** & 70 & 0 & 0 \\ Schaffer 2 & **0** & **0** & **0** & **0** & **0** \\ Schaffer 4 & **0** & **0** & **0** & **0** & **0** \\ Easom & **0** & **0** & **0** & **0** & **0** \\ Shubert & 10 & 60 & **100** & 0 & 0 \\ Schwefel & **10** & **10** & 0 & 0 & 0 \\ Zettl & **100** & 80 & 60 & 20 & 0 \\ [PERSON] & 80 & 70 & 80 & **90** & 0 \\ [PERSON] & 10 & 10 & **90** & 0 & 0 \\ [PERSON] & 90 & 90 & **100** & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Hit rate (in %) of various optimization algorithms across mathematical benchmark functions. The hit rate indicates the percentage of seeds for which each algorithm successfully located the global minimum. Bold values represent the best performance. ever, the difference between MAGD and Adam, as well as between MAGD and GD, is significant. It also shows a significant difference between MAGD and all other algorithms. Compared with GD and Adam, this difference indicates a relative drawback due to MAGD's longer duration. However, the significantly shorter execution time of MAGD compared to SBGD suggests a clear advantage, especially considering its similar effectiveness in achieving optimal solutions. #### 5.2.4 Mathematical Benchmarks Experiments: Execution Time

Table 8 presents the median execution time spent by different algorithms to find the optimal value for each mathematical function across 10 randomly generated seeds. Notably, FMA requires an excessively long time, which is not justified given its comparable performance to MAGD and SBGD. SBGD and MAGD require longer execution times than GD and Adam. Although the execution times of MAGD and SBGD exceed those of GD and Adam, both algorithms achieve significantly better results in finding optima. However, MAGD's execution time remains lower than that of SBGD in most experiments, while offering comparable accuracy. Since MAGD does not always consume less time than SBGD and the performance difference is not immediately apparent, statistical analysis is necessary to provide a broader perspective. To that end, we computed the mean, standard deviation (std), median, and 95% \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Comparison** & \(\mathbf{p}\)**-Value** & **Significant Difference** & **MAGD Performance** \\ \hline MAGD vs. FMA & 0.602 & - & - \\ MAGD vs. SBGD & 0.377 & - & - \\ MAGD vs. Adam & 0.019 & Yes & \(\uparrow\) \\ MAGD vs. GD & 0.004 & Yes & \(\uparrow\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: \(p\)-values comparing MAGD hit rates with other algorithms for mathematical benchmark functions. Symbols (\(\uparrow\)/\(\downarrow\)) denote whether the difference signifies outperforming (\(\uparrow\)) or underperforming (\(\downarrow\)) of MAGD, while the absence of a symbol indicates no significant difference.