Informed Down-Sampled Lexicase Selection: Identifying productive training cases for efficient problem solving.

Genetic Programming (GP) often uses large training sets and requires all individuals to be evaluated on all training cases during selection. Random down-sampled lexicase selection evaluates individuals on only a random subset of the training cases allowing for more individuals to be explored with the same amount of program executions. However, sampling randomly can exclude important cases from the down-sample for a number of generations, while cases that measure the same behavior (synonymous cases) may be overused. In this work, we introduce Informed Down-Sampled Lexicase Selection. This method leverages population statistics to build down-samples that contain more distinct and therefore informative training cases. Through an empirical investigation across two different GP systems (PushGP and Grammar-Guided GP), we find that informed down-sampling significantly outperforms random down-sampling on a set of contemporary program synthesis benchmark problems. Through an analysis of the created down-samples, we find that important training cases are included in the down-sample consistently across independent evolutionary runs and systems. We hypothesize that this improvement can be attributed to the ability of Informed Down-Sampled Lexicase Selection to maintain more specialist individuals over the course of evolution, while still benefiting from reduced per-evaluation costs.

Problem-Solving Benefits of Down-Sampled Lexicase Selection.

In genetic programming, an evolutionary method for producing computer programs that solve specified computational problems, parent selection is ordinarily based on aggregate measures of performance across an entire training set. Lexicase selection, by contrast, selects on the basis of performance on random sequences of training cases; this has been shown to enhance problem-solving power in many circumstances. Lexicase selection can also be seen as better reflecting biological evolution, by modeling sequences of challenges that organisms face over their lifetimes. Recent work has demonstrated that the advantages of lexicase selection can be amplified by down-sampling, meaning that only a random subsample of the training cases is used each generation. This can be seen as modeling the fact that individual organisms encounter only subsets of the possible environments and that environments change over time. Here we provide the most extensive benchmarking of down-sampled lexicase selection to date, showing that its benefits hold up to increased scrutiny. The reasons that down-sampling helps, however, are not yet fully understood. Hypotheses include that down-sampling allows for more generations to be processed with the same budget of program evaluations; that the variation of training data across generations acts as a changing environment, encouraging adaptation; or that it reduces overfitting, leading to more general solutions. We systematically evaluate these hypotheses, finding evidence against all three, and instead draw the conclusion that down-sampled lexicase selection's main benefit stems from the fact that it allows the evolutionary process to examine more individuals within the same computational budget, even though each individual is examined less completely.

A Probabilistic and Multi-Objective Analysis of Lexicase Selection and ε-Lexicase Selection.

Lexicase selection is a parent selection method that considers training cases individually, rather than in aggregate, when performing parent selection. Whereas previous work has demonstrated the ability of lexicase selection to solve difficult problems in program synthesis and symbolic regression, the central goal of this article is to develop the theoretical underpinnings that explain its performance. To this end, we derive an analytical formula that gives the expected probabilities of selection under lexicase selection, given a population and its behavior. In addition, we expand upon the relation of lexicase selection to many-objective optimization methods to describe the behavior of lexicase selection, which is to select individuals on the boundaries of Pareto fronts in high-dimensional space. We show analytically why lexicase selection performs more poorly for certain sizes of population and training cases, and show why it has been shown to perform more poorly in continuous error spaces. To address this last concern, we propose new variants of ε-lexicase selection, a method that modifies the pass condition in lexicase selection to allow near-elite individuals to pass cases, thereby improving selection performance with continuous errors. We show that ε-lexicase outperforms several diversity-maintenance strategies on a number of real-world and synthetic regression problems.