# Population Dynamics in Genetic Programming for Dynamic Symbolic Regression
## 4 Discussion
In contrast, the ALPS relies on its reseeding mechanism to reintroduce variables that have become extinct. Surprisingly, our results strongly suggest that the ALPS is neither sensitive to mutation rates nor reliant on mutation at all. It is essential to note that our experiments involved local optimization of numeric parameters through gradient descent, which also reduces the reliance on mutation. Without local optimization, mutation typically serves as the sole mechanism for adapting numeric parameters. A noteworthy consideration regarding the findings of the conducted experiments is related to the definition of the benchmarks. All presented benchmarks essentially consist of weighted sums of terms, where the weights change dynamically over the epochs, and variables within the terms are exclusively used therein. This implies that the relevance of terms and their building blocks is independent of other terms. Introducing different benchmarks where terms share common subtrees or variables could alter the population dynamics. However, we suspect that having common building blocks across multiple terms might simplify the problems, as relevant building blocks could be preserved to some extent across epoch changes. Therefore, the benchmarks in this paper represent a worst-case scenario where no information carries over into another epoch for some terms. In the conducted experiments, we exclusively employed generational epoch clocks. Using evaluation-based epoch clocks would require an adapted tuning of the algorithm parameters to achieve faster convergence up until the next epoch change. When analyzing the performance of algorithms in dynamic environments, it showed that we currently lack a standardized approach. For instance, in comparing the overall performance in the mutation rate experiment, we resorted to simply averaging the qualities over time due to a lack of better alternatives. In contrast to static optimization problems, where standardized methods like run-length distributions and expected runtime analysis provide a statistically sound approach for analyzing performance and convergence speed [44], dynamic problems currently lack such a framework. In scenarios where epoch changes are known in advance, one could treat each epoch as an isolated run and observe run-length distributions within that epoch. This approach could yield valuable insights into algorithm performance on individual epochs. However, this method requires predefined epochs, which are often unknown in practical applications. Additionally, for gradual shifts in the data, as seen in the Winkler-based benchmarks, different considerations would be needed for analyzing such shifts. In this study, we exclusively compared a standard genetic algorithm (GA) as a baseline with the ALPS chosen as a representative of open-ended evolutionary algorithms. Despitedifferences in the dependency on mutation, both algorithms yielded comparable results and demonstrated similar population dynamics. A potential direction for future research involves exploring the adaptation of alternative algorithms. For instance, an offspring-selection GA (OSGA), known for its effectiveness in non-dynamic problems, encounters challenges in dynamic scenarios, due to the offspring-selection mechanism relying on the parent qualities, which may become obsolete after an epoch change. Strategies for reevaluation post-epoch changes, akin to the reevaluation of elites in the GA and ALPS, would be essential. Exploring multi-objective algorithms, such as NSGA-II, also presents an interesting line of research. Considering the quality of previous epochs as an additional objective could motivate the algorithm to preserve valuable building blocks from past epochs. In summary, the insights derived from this paper are not limited to symbolic regression problems alone. Various metrics exist for quantifying relevant building blocks in different problem domains, for example, relevant edges in a traveling salesman problem.