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22
C.G.: Geometric semantic genetic programming
 PPSN 2012, Part I. LNCS
, 2012
"... Abstract. Traditional Genetic Programming (GP) searches the space of functions/programs by using search operators that manipulate their syntactic representation, regardless of their actual semantics/behaviour. Recently, semantically aware search operators have been shown to outperform purely syntac ..."
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Cited by 14 (4 self)
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Abstract. Traditional Genetic Programming (GP) searches the space of functions/programs by using search operators that manipulate their syntactic representation, regardless of their actual semantics/behaviour. Recently, semantically aware search operators have been shown to outperform purely syntactic operators. In this work, using a formal geometric view on search operators and representations, we bring the semantic approach to its extreme consequences and introduce a novel form of GP – Geometric Semantic GP (GSGP) – that searches directly the space of the underlying semantics of the programs. This perspective provides new insights on the relation between program syntax and semantics, search operators and fitness landscape, and allows for principled formal design of semantic search operators for different classes of problems. We derive specific forms of GSGP for a number of classic GP domains and experimentally demonstrate their superiority to conventional operators. 1
Geometric Crossover for the Permutation Representation
, 2005
"... Abstract. Abstract crossover and abstract mutation are representationindependent operators that are welldefined 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 ..."
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Cited by 9 (6 self)
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Abstract. Abstract crossover and abstract mutation are representationindependent operators that are welldefined 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 permutationlike 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 wellknown problems naturally connected with the permutation representation. 1.
Geometric generalisation of surrogate model based optimisation to combinatorial spaces
 In Proceedings of the 11th European Conference on Evolutionary Computation in Combinatorial Optimisation
, 2011
"... Abstract. In continuous optimisation, Surrogate Models (SMs) are often indispensable components of optimisation algorithms aimed at tackling realworld problems whose candidate solutions are very expensive to evaluate. Because of the inherent spatial intuition behind these models, they are naturally ..."
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Cited by 8 (6 self)
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Abstract. In continuous optimisation, Surrogate Models (SMs) are often indispensable components of optimisation algorithms aimed at tackling realworld problems whose candidate solutions are very expensive to evaluate. Because of the inherent spatial intuition behind these models, they are naturally suited to continuous problems but they do not seem applicable to combinatorial problems except for the special case when solutions are naturally encoded as integer vectors. In this paper, we show that SMs can be naturally generalised to encompass combinatorial spaces based in principle on any arbitrarily complex underlying solution representation by generalising their geometric interpretation from continuous to general metric spaces. As an initial illustrative example, we show how Radial Basis Function Networks (RBFNs) can be used successfully as surrogate models to optimise combinatorial problems defined on the Hamming space associated with binary strings. 1
Geometric generalization of the neldermead algorithm
 In Proceedings of the 10th European Conference on Evolutionary Computation in Combinatorial Optimization
, 2010
"... Abstract. The NelderMead Algorithm (NMA) is an almost halfcentury old method for numerical optimization, and it is a close relative of Particle Swarm Optimization (PSO) and Differential Evolution (DE). Geometric Particle Swarm Optimization (GPSO) and Geometric Differential Evolution (GDE) are rece ..."
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Cited by 4 (3 self)
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Abstract. The NelderMead Algorithm (NMA) is an almost halfcentury old method for numerical optimization, and it is a close relative of Particle Swarm Optimization (PSO) and Differential Evolution (DE). Geometric Particle Swarm Optimization (GPSO) and Geometric Differential Evolution (GDE) are recently introduced formal generalization of traditional PSO and DE that apply naturally to both continuous and combinatorial spaces. In this paper, we generalize NMA to combinatorial search spaces by naturally extending its geometric interpretation to these spaces, analogously as what was done for the traditional PSO and DE algorithms, obtaining the Geometric NelderMead Algorithm (GNMA). 1
Geometric differential evolution on the space of genetic programs
 In Proceedings of the 13th European Conference on Genetic Programming
, 2010
"... Abstract. Geometric Differential Evolution (GDE) is a very recently introduced formal generalization of traditional Differential Evolution (DE) that can be used to derive specific GDE for both continuous and combinatorial spaces retaining the same geometric interpretation of the dynamics of the DE s ..."
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Cited by 3 (2 self)
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Abstract. Geometric Differential Evolution (GDE) is a very recently introduced formal generalization of traditional Differential Evolution (DE) that can be used to derive specific GDE for both continuous and combinatorial spaces retaining the same geometric interpretation of the dynamics of the DE search across representations. In this paper, we derive formally a specific GDE for the space of genetic programs. The result is a Differential Evolution algorithm searching the space of genetic programs by acting directly on their tree representation. We present experimental results for the new algorithm. 1
Runtime analysis of mutationbased geometric semantic genetic programming on boolean functions,” in Foundations of Genetic Algorithms
, 2013
"... 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 tr ..."
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Cited by 2 (1 self)
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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 functions. 1.
Quotient Geometric Crossovers and Redundant Encodings
, 2011
"... We extend a geometric framework for the interpretation of search operators to encompass the genotypephenotype mapping derived from an equivalence relation defined by an isometry group. We show that this mapping can be naturally interpreted using the concept of quotient space, in which the original ..."
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Cited by 1 (1 self)
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We extend a geometric framework for the interpretation of search operators to encompass the genotypephenotype mapping derived from an equivalence relation defined by an isometry group. We show that this mapping can be naturally interpreted using the concept of quotient space, in which the original space corresponds to the genotype space and the quotient space corresponds to the phenotype space. Using this characterization, it is possible to define induced geometric crossovers on the phenotype space (called quotient geometric crossovers). These crossovers have very appealing properties for nonsynonymously redundant encodings, such as reducing the size of the search space actually searched, removing the low locality from the encodings, and allowing a more informed search by utilizing distances better tailored to the specific solution interpretation. Interestingly, quotient geometric crossovers act on genotypes but have an effect equivalent to geometric crossovers acting directly on the phenotype space. This property allows us to actually implement them even when phenotypes cannot be represented directly. We give four example applications of quotient geometric crossovers for nonsynonymously redundant encodings and demonstrate their superiority experimentally.
Inertial Geometric Particle Swarm Optimization
"... Abstract — Geometric particle swarm optimization (GPSO) is a recently introduced formal generalization of a simplified form of traditional particle swarm optimization (PSO) without the inertia term that applies naturally to both continuous and combinatorial spaces. In this paper, we propose an exten ..."
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Cited by 1 (0 self)
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Abstract — Geometric particle swarm optimization (GPSO) is a recently introduced formal generalization of a simplified form of traditional particle swarm optimization (PSO) without the inertia term that applies naturally to both continuous and combinatorial spaces. In this paper, we propose an extension of GPSO, the inertial GPSO (IGPSO), that generalizes the traditional PSO endowed with the full equation of motion of particles to generic search spaces. We then formally derive the specific IGPSO for the Hamming space associated with binary strings and present experimental results for this new algorithm. I.
Geometric Semantic Genetic Programming for Financial Data
"... Abstract. We cast financial trading as a symbolic regression problem on the lagged time series, and test a state of the art symbolic regression method on it. The system is geometric semantic genetic programming, which achieves good performance by converting the fitness landscape to a cone landscape ..."
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Abstract. We cast financial trading as a symbolic regression problem on the lagged time series, and test a state of the art symbolic regression method on it. The system is geometric semantic genetic programming, which achieves good performance by converting the fitness landscape to a cone landscape which can be searched by hillclimbing. Two novel variants are introduced and tested also, as well as a standard hillclimbing genetic programming method. Baselines are provided by buyandhold and ARIMA. Results are promising for the novel methods, which produce smaller trees than the existing geometric semantic method. Results are also surprisingly good for standard genetic programming. New insights into the behaviour of geometric semantic genetic programming are also generated.
Geometric NelderMead Algorithm on the Space of Genetic Programs
 GECCO'11
, 2011
"... The NelderMead Algorithm (NMA) is a close relative of Particle Swarm Optimization (PSO) and Differential Evolution (DE). In recent work, PSO, DE and NMA have been generalized using a formal geometric framework that treats solution representations in a uniform way. These formal algorithms can be use ..."
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The NelderMead Algorithm (NMA) is a close relative of Particle Swarm Optimization (PSO) and Differential Evolution (DE). In recent work, PSO, DE and NMA have been generalized using a formal geometric framework that treats solution representations in a uniform way. These formal algorithms can be used as templates to derive rigorously specific PSO, DE and NMA for both continuous and combinatorial spaces retaining the same geometric interpretation of the search dynamics of the original algorithms across representations. In previous work, a geometric NMA has been derived for the binary string representation and permutation representation. Furthermore, PSO and DE have already been derived for the space of genetic programs. In this paper, we continue this line of research and derive formally a specific NMA for the space of genetic programs. The result is a NelderMead Algorithm searching the space of genetic programs by acting directly on their tree representation. We present initial experimental results for the new algorithm. The challenge tackled in the present work compared with earlier work is that the pair NMA and genetic programs is the most complex considered so far. This combination raises a number of issues and casts light on how algorithmic features can interact with representation features to give rise to a highly peculiar search behaviour.