Runtime analysis of mutation-based geometric semantic genetic programming for basis functions regression.

Geometric Semantic Genetic Programming (GSGP) is a recently introduced form of Genetic Programming (GP) that searches the semantic space of functions/programs. The fitness landscape seen by GSGP is always -- for any domain and for any problem -- unimodal with a linear slope by construction. This makes the search for the optimum much easier than for traditional GP, and it opens the way to analyse theoretically in a easy manner the optimisation time of GSGP in a general setting. Very recent work proposed a runtime analysis of mutation-based GSGP on the class of all Boolean functions. We present a runtime analysis of mutation-based GSGP on the class of all regression problems with generic basis functions (encompassing e.g., polynomial regression and trigonometric regression).Alberto Moraglio was supported by EPSRC grant EP/I010297/1

CSM429: Abstract Geometric Crossover for the Permutation Representation

Abstract crossover and abstract mutation are representation-independent operators that are well-defined once a notion of distance over the solution space is defined. They were obtained as generalization of genetic operators for binary strings and real vectors. In this paper we explore how the abstract geometric framework applies to the permutation representation. This representation is challenging for various reasons: because of the inherent difference between permutations and the representations that inspired the abstraction; because the whole notion of geometry over permutation spaces radically departs from traditional geometries and it is almost unexplored mathematical territory; because the many notions of distance available and their subtle interconnections make it hard to see the right distance to use, if any; because the various available interpretations of permutations make ambiguous what a permutation represents, hence, how to treat it; because of the existence of various permutation-like representations that are incorrectly confused with permutations; and finally because of the existence of many mutation and recombination operators and their many variations for the same representation. This article shows that the application of our geometric framework naturally clarifies and unifies an important domain,the permutation representation and the related operators, in which there was little or no hope to find order. In addition the abstract geometric framework is used to improve the design of crossover operators for well-known problems naturally connected with the permutation representation

Geometric semantic genetic programming for recursive boolean programs

This is the author accepted manuscript. The final version is available from ACM via the DOI in this record.Geometric Semantic Genetic Programming (GSGP) induces a unimodal fitness landscape for any problem that consists in finding a function fitting given input/output examples. Most of the work around GSGP to date has focused on real-world applications and on improving the originally proposed search operators, rather than on broadening its theoretical framework to new domains. We extend GSGP to recursive programs, a notoriously challenging domain with highly discontinuous fitness landscapes. We focus on programs that map variable-length Boolean lists to Boolean values, and design search operators that are provably efficient in the training phase and attain perfect generalization. Computational experiments complement the theory and demonstrate the superiority of the new operators to the conventional ones. This work provides new insights into the relations between program syntax and semantics, search operators and fitness landscapes, also for more general recursive domains.© 2017 Copyright held by the owner/author(s). Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

CSM-430: Geometric Landscape of Homologous Crossover for Syntactic Trees

Geometric crossover and geometric mutation are representation-independent operators that are welldefined once a notion of distance over the solution space is defined. They were obtained as generalizations of genetic operators for binary strings and real vectors. Our geometric framework has been successfully applied to the permutation representation leading to a clarification and a natural unification of this domain. The relationship between search space, distances and genetic operators for syntactic trees is little understood. In this paper we apply the geometric framework to the syntactic tree representation and show how the wellknown structural distance is naturally associated with homologous crossover and subtree mutation

Runtime analysis of mutation-based geometric semantic genetic programming on boolean functions.

Geometric Semantic Genetic Programming (GSGP) is a recently introduced form of Genetic Programming (GP), rooted in a geometric theory of representations, that searches directly the semantic space of functions/programs, rather than the space of their syntactic representations (e.g., trees) as in traditional GP. Remarkably, the fitness landscape seen by GSGP is always – for any domain and for any problem – unimodal with a linear slope by construction. This has two important consequences: (i) it makes the search for the optimum much easier than for traditional GP; (ii) it opens the way to analyse theoretically in a easy manner the optimisation time of GSGP in a general setting. The runtime analysis of GP has been very hard to tackle, and only simplified forms of GP on specific, unrealistic problems have been studied so far. We present a runtime analysis of GSGP with various types of mutations on the class of all Boolean functionsThe authors are grateful to Dirk Sudholt for helping check the proofs. Alberto Moraglio was supported by EPSRC grant EP/I010297/

Automated Problem Decomposition for the Boolean Domain with Genetic Programming

Researchers have been interested in exploring the regularities and modularity of the problem space in genetic programming (GP) with the aim of decomposing the original problem into several smaller subproblems. The main motivation is to allow GP to deal with more complex problems. Most previous works on modularity in GP emphasise the structure of modules used to encapsulate code and/or promote code reuse, instead of in the decomposition of the original problem. In this paper we propose a problem decomposition strategy that allows the use of a GP search to find solutions for subproblems and combine the individual solutions into the complete solution to the problem

Evolving Recursive Programs using Non-recursive Scaffolding

Genetic programming has proven capable of evolving solutions to a wide variety of problems. However, the successes have largely been with programs without iteration or recursion; evolving recursive programs has turned out to be particularly challenging. The main obstacle to evolving recursive programs seems to be that they are particularly fragile to the application of search operators: a small change in a correct recursive program generally produces a completely wrong program. In this paper, we present a simple and general method that allows us to pass back and forth from a recursive program to an associated non-recursive program. Finding a recursive program can be reduced to evolving non-recursive programs followed by converting the optimum non-recursive program found to the associated optimum recursive program. This avoids the fragility problem above, as evolution does not search the space of recursive programs. We present promising experimental results on a test-bed of recursive problems

A Unifying View on Recombination Spaces and Abstract Convex Evolutionary Search

This is the author accepted manuscript. The final version is available from Springer via the DOI in this record.Proceedings of EvoCOP 2019 - 19th European Conference on Evolutionary Computation, 24-26 April 2019, Leipzig, GermanyPrevious work proposed to unify an algebraic theory of fitness landscapes and a geometric framework of evolutionary algorithms (EAs). One of the main goals behind this unification is to develop an analytical method that verifies if a problem's landscape belongs to certain abstract convex landscapes classes, where certain recombination-based EAs (without mutation) have polynomial runtime performance. This paper advances such unification by showing that: (a) crossovers can be formally classified according to geometric or algebraic axiomatic properties; and (b) the population behaviour induced by certain crossovers in recombination-based EAs can be formalised in the geometric and algebraic theories. These results make a significant contribution to the basis of an integrated geometric-algebraic framework with which analyse recombination spaces and recombination-based EAs

Runtime analysis of convex evolutionary search algorithm with standard crossover

This is the final version. Available on open access from Elsevier via the DOI in this recordEvolutionary Algorithms (EAs) with no mutation can be generalized across representations as Convex Evolu- tionary Search algorithms (CSs). However, the crossover operator used by CSs does not faithfully generalize the standard two-parents crossover: it samples a convex hull instead of a segment. Segmentwise Evolutionary Search algorithms (SESs) are defined as a more faithful generalization, equipped with a crossover operator that samples the metric segment of two parents. In metric spaces where the union of all possible segments of a given set is always a convex set, a SES is a particular CS. Consequently, the representation-free analysis of the CS on quasi- concave landscapes can be extended to the SES in these particular metric spaces. When instantiated to binary strings of the Hamming space (resp. -ary strings of the Manhattan space), a polynomial expected runtime upper bound is obtained for quasi-concave landscapes with at most polynomially many level sets for well-chosen popu- lation sizes. In particular, the SES solves Leading Ones in at most 288 ln [4 (2 + 1)] expected fitness evaluations when the population size is equal to 144 ln [4 (2 + 1)]