On the Robustness of Lexicase Selection to Contradictory Objectives

TL;DR

This work develops theory identifying circumstances under which lexicase selection will succeed or fail to find a Pareto-optimal solution to a theoretical problem with maximally contradictory objectives and proposes theoretically-backed guidelines for parameter choice.

Abstract

Lexicase and epsilon-lexicase selection are state of the art parent selection techniques for problems featuring multiple selection criteria. Originally, lexicase selection was developed for cases where these selection criteria are unlikely to be in conflict with each other, but preliminary work suggests it is also a highly effective many-objective optimization algorithm. However, to predict whether these results generalize, we must understand lexicase selection's performance on contradictory objectives. Prior work has shown mixed results on this question. Here, we develop theory identifying circumstances under which lexicase selection will succeed or fail to find a Pareto-optimal solution. To make this analysis tractable, we restrict our investigation to a theoretical problem with maximally contradictory objectives. Ultimately, we find that lexicase and epsilon-lexicase selection each have a region of parameter space where they are incapable of optimizing contradictory objectives. Outside of this region, however, they perform well despite the presence of contradictory objectives. Based on these findings, we propose theoretically-backed guidelines for parameter choice. Additionally, we identify other properties that may affect whether a many-objective optimization problem is a good fit for lexicase or epsilon-lexicase selection.

Figures (5)

• Figure 1: The full Pareto front vs. the solution set maintained by lexicase selection. Surface shows the set of points that fall along the Pareto front for a hypothetical 3-objective problem. The two points in black represent points that lexicase selection would maintain if the entire theoretical Pareto front were in the population. A third point could be maintained if the points with the highest x and y values were different.
• Figure 2: Visualization of the regions of parameter space where lexicase selection can find Pareto-optimal solutions to a problem with contradictory objectives, as given by Equation \ref{['eq:p_needed_vs_D']}, with $t=.5$. Colors indicate upper bounds on the dimension (i.e. number of objectives) that can be used for each combination of $S$ and $G$. Note that the x, y, and color axes are all on log scales.
• Figure 3: Probability that lexicase selection fails to find a Pareto-optimal solution over various values of $S$, $D$, and $\epsilon$. In figures (A), (B) and (C), the algorithm was run for 10,000 time steps while in figures (D), (E) and (F) it was run for 100,000 time steps (n = 30 per cell, $\mu$=.01, $G$=500).
• Figure 4: Probability that $\epsilon$-lexicase selection fails to find a Pareto-optimal solution over various values of S and D when $\epsilon$ changes according to equation \ref{['eq:medianepsilon']}. The algorithm was run for 100,000 time steps (n = 30 per cell, $\mu$=.01, $G$=500)
• Figure 5: State space explored in a down-scaled reachability analysis graph for $D=5$, $S=30$, $G=50$. A) $\epsilon$=0, B) $\epsilon$=1, C) $\epsilon$=2. Position along the x axis indicates the number of unique genotypes. Position along the y axis correlates with the distance of genotype values from zero. To make the structure visually clear, nodes with similar properties in these regards are plotted on top of each other. Cyan nodes were discovered but not fully explored in the reachability analysis.

Theorems & Definitions (1)

• definition 1