The aim of Symbolic Regression (SR) is to discover interpretable expressions that accurately describe data. The accuracy of an expression depends on both its structure and coefficients. To keep the structure simple enough to be interpretable, effective coefficient optimisation becomes key. Gradient-based optimisation is clearly effective at training neural networks in Deep Learning (DL), which can essentially be viewed as large, over-parameterised expressions: in this paper, we study how gradient-based optimisation techniques as often used in DL transfer to SR. In particular, we first assess what techniques work well across random SR expressions, independent of any specific SR algorithm. We find that mini-batching and gradient-clipping can be helpful (similar to DL), while second-order optimisers outperform first-order ones (different from DL). Next, we consider whether including gradient-based optimisation in Genetic Programming (GP), a classic SR algorithm, is beneficial. On five real-world datasets, in a generation-based comparison, we find that second-order optimisation outperforms coefficient mutation (or no optimisation). However, in time-based comparisons, performance gaps shrink substantially because the computational expensiveness of second-order optimisation causes GP to perform fewer generations. The interplay of computational costs between the optimisation of structure and coefficients is thus a critical aspect to consider.

Genetic programming (GP) is one of the best approaches today to discover symbolic regression models. To find models that trade off accuracy and complexity, the non-dominated sorting genetic algorithm II (NSGA-II) is widely used. Unfortunately, it has been shown that NSGA-II can be inefficient: in early generations, low-complexity models over-replicate and take over most of the population. Consequently, studies have proposed different approaches to promote diversity. Here, we study the root of this problem, in order to design a superior approach. We find that the over-replication of low complexity-models is due to a lack of evolvability, i.e., the inability to produce offspring with improved accuracy. We therefore extend NSGA-II to track, over time, the evolvability of models of different levels of complexity. With this information, we limit how many models of each complexity level are allowed to survive the generation. We compare this new version of NSGA-II, evoNSGA-II, with the use of seven existing multi-objective GP approaches on ten widely-used data sets, and find that evoNSGA-II is equal or superior to using these approaches in almost all comparisons. Furthermore, our results confirm that evoNSGA-II behaves as intended: models that are more evolvable form the majority of the population. Code: https://github.com/dzhliu/evoNSGA-II