Geometry of evolutionary algorithms

• Authors: Alberto Moraglio
• Venue: Proceedings of the 13th annual conference companion on Genetic and evolutionary computation
• Year: 2011

The various flavors of Evolutionary Algorithms look very similar when cleared of algorithmically irrelevant differences such as domain of application and phenotype interpretation. Representation-independent algorithmic characteristics like the selection scheme can be freely exchanged between algorithms. Ultimately, the origin of the differences of the various flavors of Evolutionary Algorithms is rooted in the solution representation and relative genetic operators. Are these differences only superficial? Is there a deeper unity encompassing all Evolutionary Algorithms beyond the specific representation? Is a general mathematical framework unifying search operators for all solution representations at all possible? The aim of the tutorial is to introduce a formal, but intuitive, unified point of view on Evolutionary Algorithms across representations based on geometric ideas, which provides a possible answer to the above questions. It also presents the benefits for both theory and practice brought by this novel perspective. The key idea behind the geometric framework is that search operators have a dual nature. The same search operator can be defined (i) on the underlying solution representations and, equivalently, (ii) on the structure of the search space by means of simple geometric shapes, like balls and segments. These shapes are used to delimit the region of space that includes all possible offspring with respect to the location of their parents. The geometric definition of a search operator is of interest because it can be applied - unchanged - to different search spaces associated with different representations. This, in effect, allows us to define exactly the same search operator across representations in a rigorous way. The geometric view on search operators has a number of interesting consequences of which this tutorial will give a comprehensive overview. These include (i) a straightforward view on the fitness landscape seen by recombination operators, (ii) a formal unification of many pre-existing search operators across representations, (iii) a principled way of designing crossover operators for new representations, (iv) a principled way of generalizing search algorithms from continuous to combinatorial spaces, and (v) the potential for a unified theory of evolutionary algorithms across representations.