# Deep Learning-Based Step Size Determination for Hill Climbing Metaheuristics
## 5 Discussion  
In developing the Hybrid ML-HC algorithm, the first question was how many steps to allow the ML model before switching back to the BF method. The first experiments were aimed at determining this. The results show that the number of iterations is significantly affected by this parameter, but this has minimal influence on the best fitness values achieved (Figure 2). The best results were obtained with the algorithm using five ML steps; therefore, this variant was further investigated.  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Method** & **Fitness Value** & \(f\)**cnt. avg.** & \(f\)**cnt. med** & \(f\)**cnt. std.dev.** \\ \hline Fix step size (1.0) & \(1.94\times 10^{+01}\) & \(2.91\times 10^{+06}\) & \(2.91\times 10^{+06}\) & \(2.83\times 10^{+12}\) \\ Fix step size (0.1) & \(1.36\times 10^{+01}\) & \(2.80\times 10^{+07}\) & \(2.79\times 10^{+07}\) & \(2.62\times 10^{+14}\) \\ Fix step size (0.001) & \(1.91\times 10^{+00}\) & \(3.71\times 10^{+09}\) & \(3.71\times 10^{+09}\) & \(4.59\times 10^{+18}\) \\ Fix step size (0.00001) & \(5.89\times 10^{-07}\) & \(3.95\times 10^{+11}\) & \(3.95\times 10^{+11}\) & \(5.19\times 10^{+22}\) \\ BF & \(5.82\times 10^{-15}\) & \(1.73\times 10^{+10}\) & \(1.73\times 10^{+10}\) & \(9.97\times 10^{+19}\) \\ Hybrid ML+BF & \(5.76\times 10^{-15}\) & \(8.60\times 10^{+09}\) & \(8.61\times 10^{+09}\) & \(2.46\times 10^{+19}\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Fitness and step count results for different step size determination methods.  
Figure 2: Average best fitness values for different numbers of ML steps (\(l\) value).  
When comparing the different methods for determining \(\epsilon\), it was immediately obvious that solutions using fixed step sizes are not efficient (Figure 3). It is more interesting to compare the standard BF and the novel Hybrid ML-BF methods (Figure 4). As is visible, the new method has achieved basically the same results, sometimes even outperforming the original (although this advantage is not significant). The main strategy of the BF method is that it always chooses the best step size; therefore, it was surprising that the Hybrid method was able to find better results. The reason for this may be that the BF method uses a set of predefined step sizes, and one of these has to be chosen. The ML model was trained on these data, but the model was able to interpolate and extrapolate; therefore, it can suggest better step sizes than in the given predefined set.  
In the next analysis, we examined the number of iterations performed by the different methods. Decreasing the fixed step size yields better results, but the iteration count can become unmanageable. Comparing the BF and Hybrid methods (Figure 5), it is visible that the Hybrid method slightly outperforms the BF method; however, the difference is also not significant. In summary, the novel method can achieve similar results with a similar number of steps.  
Figure 4: Best fitness values for the best two methods.  
Figure 3: Average best fitness values for different methods.  
The BF and Hybrid methods outperformed the fixed step size-based methods, both in speed (iteration count) and accuracy (best fitness achieved). These two look very similar in both aspects; however, there is a considerable difference between the number of fitness calculations. The number of iterations performed by the methods is the same, but it is worth taking into account that the computation demand of one iteration is very different. As is visible from Figure 6, the number of fitness calculations is significantly lower for the ML-assisted method.