

![0_image_1.png](0_image_1.png) 

Figure 6: Evolution of average of best fitness for the 3-bands problem. The lowest curve is that of the 1-recurr model**, and the 3 indistinguishable curves** above the other 2 are those of the other 3 artificial embryogeny models.

![0_image_2.png](0_image_2.png)

Figure 7: Development stages on the three-bands problem for the recurrent NN with 2 chemicals at iterations 16, 32 and 44 (columns) for the phenotype (top row), and both chemicals.

act solution, as shown in figure 3, while the 3 embryogenic models give nearly optimal individuals.

As expected, the disc **target is difficult for the embryogenic approaches: as can be seen on the box-plots (Figure**
4, all 4 are clearly outperformed by the reference model, that was not trapped in the same local optimum. The online results did not reveal any other conclusion, and are not shown here. It is worth noting that here, experiments using 2 chemicals outperform the same model with a single chemical (with statistically significant differences according **to a**
95% confidence T-test).

Finally, the situation is slightly different for the half-discs, the most difficult target (Figure 5): all embryogenic models are, again, clearly outperformed by the reference model, even though this model doesn't reach such a good fitness than for the disc problem. However, the best results among embryogenic approaches are obtained by the recurrent networks, that exhibit a much larger variance, and thus sometimes reaches much better fitnesses - with a slight advantage for the 2-chemicals recurrent model in this respect.

## 3.4.2 Halting Criterion And Robustness

The evolved halting criterion is one of the main original feature of the proposed approach. It thus needs to be studied in detail, especially as it is closely related to the self-healing properties, i.e. the robustness with respect to noise during

![0_image_0.png](0_image_0.png)

Figure 8: Development stages on the half-discs problem for the recurrent NN with 2 chemicals at iterations 28, 64 and 122 (columns) for the phenotype
(top row), and both chemicals. Figure 9: Self-healing on the three-bands problem for the recurrent NN and 2 chemicals: Snapshots of the phenotype at iterations 0 (beginning of the perturbation), 4, 11, 17 and 22.

## The Growth Iterations.

Because all organisms are allowed 1024 iterations in their growth process, it could be feared that several hundreds iterations would be needed before stabilization even for the best solutions found by the algorithm. The total computational costs would henceforth have been tremendously higher that it already is. The good news is that in all cases, and for all embryogenic models, the whole population rapidly contains a large majority of organisms that did stabilize within a few dozens iterations. Illustrations of the growth process are given in Figures 7 and 8. For the easy 3-bands problem, only 44 iterations are needed (and chemical 1 doesn't change after the 16th iteration). For the more difficult halfdiscs problem, 122 iterations are needed.

But another important issue is that of robustness: earlier works [12, 6] have demonstrated that developmental approaches lead to robust solutions as far as development is concerned Here, the robustness of the fixed points was checked by applying a centered Gaussian perturbation with unit standard deviation to the states of all neurons. The good news is that for any perturbation, 100% of the feedfoward controllers and 75% of the recurrent controllers return to the very same state they had before the perturbation. The other 25% reccurent controllers returns to a state very close to the one they had before perturbation. An example of perfect and fast self-healing for the three-bands **problem is shown in** Figure 3.4.2.

To sum up, the embryogenic approach perform often nearly as good as a simple regression (f(x, y) = z **), if using the same**
optimizer. The feedforward and the recurrent networks seem hardly distinguishable across the 4 experiments, and a slight advantage of the 2-chemical over the 1-chemical could be hypothesized. The most interesting result concerns the almost perfect self-healing property of the resulting organisms.