Ramped Half-n-Half Initialisation Bias in GP

Tree initialisation techniques for genetic programming (GP) are examined in [4,3], highlighting a bias in the standard implementation of the initialisation method Ramped Half-n-Half (RHH) [1]. GP trees typically evolve to random shapes, even when populations were initially full or minimal trees [2]. In canonical GP, unbalanced and sparse trees increase the probability that bigger subtrees are selected for recombination, ensuring code growth occurs faster and that subtree crossover will have more difficultly in producing trees within specified depth limits. The ability to evolve tree shapes which allow more legal crossover operations, by providing more possible crossover points (by being bushier), and control code growth is critical. The GP community often uses RHH [4]. The standard implementation of the RHH method selects either the grow or full method with 0.5 probability to produce a tree. If the tree is already in the initial population it is discarded and another is created by grow or full. As duplicates are typically not allowed, this standard implementation of RHH favours full over grow and possibly biases the evolutionary process. The full and grow methods are similar algorithms for recursively producing GP trees. The full algorithm makes trees with branches extending to the maximum initial depth. The grow algorithm does not require this and allows branches of varying length (up to the maximum initial depth). As many more unique trees exist which are full (as full trees contain more nodes), there is a tendency, especially with particular function and terminal sets, to produce more duplicate trees with the grow method. To estimate the bias of the RHH method with a particular function and terminal set, we use the results from Luke [3] (Lemma 1). The expected number of nodes Ed at depth d is defined as: Ed = {1 if d = 0, else Ed−1pb if d > 0} where pb is the expected number of children of a new node (p is the probability of picking a nonterminal, and b is the expected number of children of a nonterminal). Luke [3] uses this to calculate the expected size of trees in the infinite case. Here we bound the calculation to depth d = 4. For our analysis, Ed correctly predicted the full method would contribute more trees to the initial population whenever the expected size of the two methods was not similar (i.e. grow made smaller trees, causing more duplicates and rejected trees). Canonical GP trees grow in size to maximum depth limits, making the initial trees seeds for the evolutionary process. As the full algorithm is more likely to evolve larger and more bushier trees with more nodes than the grow method, we conduct an experimental study to observe these differences in the evolutionary