## 4 Proposed Solution
### Main Characteristics of the Multi-Agent Gradient Descent Algorithm (MAGD)
* \(m_{f,s_{1},a}\): The minimum value found by algorithm \(a\) when applied to function \(f\) using the \(i\)-th random seed (\(s_{i}\) where \(i\in\{1,2,\ldots,10\}\)). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function Name** & **MAGD** & **FMA** & **SBGD** & **Adam** & **GD** \\ \hline Rastrigin & 1.3930 & **0.4975** & **0.4975** & 22.9834 & 22.9877 \\ Ackley & 7.5376 & 7.5376 & **0.0001** & 19.3172 & 19.3171 \\ Goldstein–Price & **0.0000** & **0.0000** & **0.0000** & 186.3059 & 113.5452 \\ Levy-n13 & **0.0000** & **0.0000** & 0.0995 & 4.7850 & 4.8066 \\ Himmelblau & 0.0001 & **0.0000** & **0.0000** & 0.0023 & **0.0000** \\ Three-Hump Camel & 0.0597 & **0.0000** & 0.0017 & 0.2700 & 0.2091 \\ Griewank & **1.6096** & **1.6096** & 1.6219 & 75.7318 & 75.7329 \\ Cross-in-Tray & **2.0604** & **2.0604** & **2.0604** & 2.0608 & 2.0611 \\ EggHolder & 87.6805 & **79.2623** & 322.1294 & 844.0750 & 873.2270 \\ Michalewicz & 0.0001 & **0.0000** & 0.0001 & 0.8996 & 1.3997 \\ Alpine & 0.0046 & **0.0001** & 0.0012 & 0.0127 & 0.0559 \\ Schaffer 2 & 0.1183 & 0.1182 & **0.1169** & 0.4865 & 0.4989 \\ Schaffer 4 & **0.1742** & 0.1743 & 0.1756 & 0.1939 & 0.2063 \\ Easom & **1.0000** & **1.0000** & **1.0000** & **1.0000** & **1.0000** \\ [PERSON] & 6.3178 & 0.0015 & **0.0000** & 157.7766 & 157.7990 \\ [PERSON] & 81.5770 & **77.6650** & 193.4469 & 666.3456 & 692.4238 \\ [PERSON] & **0.0000** & 0.0008 & 0.0470 & 0.0912 & 0.8460 \\ [PERSON] & 0.0004 & 0.0012 & **0.0000** & 0.0003 & 0.0353 \\ Drop-Wave & 0.1026 & 0.1026 & **0.0065** & 0.8057 & 0.8054 \\ [PERSON] & 0.2211 & 0.0258 & **0.0003** & 4.7501 & 4.7487 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Median of the absolute difference between the true global minima and the found minima across various optimization algorithms for standard mathematical benchmark functions over 10 different random seeds. Bold values indicate the best-performing algorithms. #### 5.2.3 Mathematical Benchmarks Experiments: Global Optimization Capability

To evaluate the new algorithm's ability to locate global minima in the mathematical function experiments, we propose a measure called the hit rate. The hit rate counts the number of seeds where an algorithm successfully finds the global minimum, determined by whether the absolute difference between the true minimum and the minimum found by the algorithm is within a tolerance of \(10^{-3}\). The equations defining this measure are provided in Equations (22) and (23). \[\text{hit}= \begin{cases}\text{True}&\text{if $\left|\text{opt\_cost}-f^{*} \right|\leq 10^{-3}$,}\\ \text{False}&\text{otherwise.}\end{cases}  \tag{22}\]

\[\text{hit rate}\%=\frac{\sum\text{hit}}{\text{number of seeds}}\times 100 \tag{23}\]

Table 6 presents the hit rate of different algorithms across various benchmark mathematical functions. Table 6 demonstrates that MAGD, FMA, and SBGD outperform GD and Adam in hit-rate performance, with similar performance observed between MAGD, FMA, and SBGD (remember that FMA is not an actual algorithm but rather multiple simultaneous executions of Adam). To substantiate these observations, we conducted a \(t\)-test comparing the hit ratios of MAGD with those of other algorithms. The \(t\)-test, also known as Student's \(t\)-test, is widely used to determine if there is a statistically significant difference between two data samples. This hypothesis test uses the t-distribution to compute the t-value, which is the ratio of the observed difference in means to the variability in the data, scaled by sample sizes. Using the t-distribution, the \(p\)-value is the probability distribution function (PDF) value corresponding to the computed t-value. A \(p\)-value less than 0.05 typically indicates a significant difference between two groups [46] (p. 64). As shown in Table 7, The \(p\)-values indicate no significant difference between MAGD and FMA, or between MAGD and SBGD, in terms of global optimization capability. How