Evolvability in Optimization Problems: Towards more Efficient Evolutionary Algorithms

Evolvability in Optimization Problems: Towards more Efﬁcient Evolutionary Algorithms Jorge Tavares INRIA Lille - Nord Europe Research Centre jorge.tavares@ieee.org http://jorgetavares.wordpress.com

Overview • Fundamental issues • Hows and Whys • Case Studies • Insights for more efﬁcient algorithms

Motivation • Combinatorial Optimization Problems are a class of problems with high practical importance but difﬁcult to solve. • Evolutionary algorithms integrate a set of techniques that proved to be an efﬁcient and robust paradigm for complex problem solving. 3

Motivation • Combinatorial Optimization Problems are a class of problems with high practical importance but difﬁcult to solve. • Evolutionary algorithms integrate a set of techniques that proved to be an efﬁcient and robust paradigm for complex problem solving. • The representation of candidate solutions for a given problem is one of the fundamental questions. 3

Motivation • Combinatorial Optimization Problems are a class of problems with high practical importance but difﬁcult to solve. • Evolutionary algorithms integrate a set of techniques that proved to be an efﬁcient and robust paradigm for complex problem solving. • The representation of candidate solutions for a given problem is one of the fundamental questions. • It is important to understand the role of representation and the interplay with heuristics and local methods. 3

Genetic Representations • It’s important to distinguish between genotypes and phenotypes. 4

Genetic Representations • It’s important to distinguish between genotypes and phenotypes. • Genotype: representation manipulated by the evolutionary algorithm's components. 4

Genetic Representations • It’s important to distinguish between genotypes and phenotypes. • Genotype: representation manipulated by the evolutionary algorithm's components. • Phenotype: a candidate solution to a given problem. 4

Genetic Representations • It’s important to distinguish between genotypes and phenotypes. Solution's Phenotype Fitness Function • Genotype: representation Quality manipulated by the evolutionary algorithm's components. Mapping Function Genotype Genetic Operators • Phenotype: a candidate solution to a given problem. • Important: the mapping between these two entities 4

Genetic Representations • Interdependencies between operators and representations can not be neglected. 5

Genetic Representations • Interdependencies between operators and representations can not be neglected. • In a direct representation the genotype is the same as the Individual Genetic Operators phenotype. Genetic operators are directly applied to the phenotype. 5

Genetic Representations • Interdependencies between operators and representations can not be neglected. • In a direct representation the genotype is the same as the Individual Genetic Operators phenotype. Genetic operators are directly applied to the phenotype. • A representation is indirect if the Phenotype genotype and the phenotype are different. Genetic operators are Mapping Function applied to the genotype. Genotype Genetic Operators 5

Heuristics • Heuristic: any method speciﬁcally developed for application with representations which are simple in nature. 6

Heuristics • Heuristic: any method speciﬁcally developed for application with representations which are simple in nature. • Applied at three different levels: mapping function, genetic operators and on the structure. 6

Heuristics • Heuristic: any method specifically developed for application with representations which are simple in nature. Phenotype Heuristics • Applied at three different levels: mapping function, genetic Mapping Function operators and on the structure. Genotype Genetic Operators • Heuristic bias: the influence of the heuristic on the efficiency of the representation. 6

How to perform Analysis? • In the past years, some tools have been used with this aim in mind, from empirical methods to theory-based analysis. 8

How to perform Analysis? • In the past years, some tools have been used with this aim in mind, from empirical methods to theory-based analysis. • Fitness landscape analysis: ‣ the characteristics of the search space can be helpful to identify a search strategy that is likely to improve the algorithm performance 8

How to perform Analysis? • In the past years, some tools have been used with this aim in mind, from empirical methods to theory-based analysis. • Fitness landscape analysis: ‣ the characteristics of the search space can be helpful to identify a search strategy that is likely to improve the algorithm performance • Empirical Analysis of Representation Properties: ‣ measurement and quantiﬁcation of essential aspects for evolutionary search allowing an inexpensive pre-evaluation of an algorithm 8

Landscape Analysis • Evolutionary search can be represented by three spaces: ‣ the search space ‣ the phenotype space ‣ the ﬁtness space • Fitness landscapes describe the relation between the search and the ﬁtness space. 9

Fitness Distance Correlation • One way to measure problem difficulty is determining how close is the relation between fitness value and distance to the nearest optimum. • The search should be easy, for selection-based algorithms, when fitness increases as the distance to the optimum decreases. This indicates the existence of a path via solutions with increasing fitness values. • Measure: 10

Auto-Correlation • The structure of a fitness landscape can be examined by measuring the degree of correlation between points on the landscape. The degree depends on the difference between the fitness values of the points. • Smoother landscapes are highly correlated, making the search for an evolutionary algorithm easier. If the difference of fitness values is higher, the landscape is less correlated, which implies a rugged landscape, thus being harder to search in it. • Measure: 11

Representations Properties 12

Representations Properties • Locality: ‣ Small steps in the search space cause small phenotypic changes. 12

Representations Properties • Locality: ‣ Small steps in the search space cause small phenotypic changes. • Heritability: ‣ The ability of crossover operators to produce offspring that combine meaningful features of their parents. 12

Representations Properties • Locality: ‣ Small steps in the search space cause small phenotypic changes. • Heritability: ‣ The ability of crossover operators to produce offspring that combine meaningful features of their parents. • Heuristic Bias: ‣ Phenotypes with higher probability to be created without selection pressure, induced by heuristics or problem speciﬁc operators. 12

Mutation Innovation • The distance between the individuals involved in a mutation is used to predict the effect of the application this operator. • MI illustrates how much innovation the mutation operator introduces, i.e., it aims to determine how much this operator modiﬁes the semantic properties of an individual. • Measure: 13

Crossover Innovation • It measures the ability of this operator to create descendants that are different from their parents, i.e., the phenotypic distance between a child and its phenotypical closer parent. • Similar parents tend to create descendants that are also close to both of them. On the contrary, dissimilar parents tend to originate larger crossover innovations. • Measure: 14

Other Measures 15

Other Measures • Heritability: ‣ Crossover Loss: measures the total size of phenotypic substructures in an offspring that is not inherited from either of the parents but newly introduced. 15

Other Measures • Heritability: ‣ Crossover Loss: measures the total size of phenotypic substructures in an offspring that is not inherited from either of the parents but newly introduced. • Heuristic Bias: ‣ Bias in the encodings: mean and standard deviaton of each individual to the best known solution. ‣ Bias in the operators: mean and standard deviation on a evolutionary run without selection. 15

Landscape Analysis: Golomb Rulers • A Golomb Ruler is defined as a ruler that has marks unevenly spaced at integer locations where the distance between any two marks is unique. • Unlike usual rulers, they have the ability to measure more discrete measures than the number of marks they possess. • Golomb Rulers are not redundant since they do not measure the same distance twice. • An Optimal Golomb Ruler (OGR) is defined as the shortest length ruler for a given number of marks. • There may exist multiple different OGRs for a specific number of marks. 17

Landscape Analysis: Golomb Rulers • An example: • Representations: binary, integer, permutation, random-key • Operators: ﬂip (for all mutation variants), shift, 1-point cx, uniform cx 18

Landscape Analysis: Golomb Rulers 19

Fitness Distance Correlation: Mutation Binary with Flip Binary with Shift Integer Permutation Random-key Incomplete Permutation 20

Fitness Distance Correlation: Crossover Binary with 1Pt CX Integer with 1Pt CX Permutation with 1Pt CX Random-key with 1Pt CX Binary with Uniform CX Integer with Uniform CX Permutation with Uniform CX Random-key with Uniform CX 21

Fitness Distance Correlation: adding Heuristics Binary + Flip + Correction Binary + Flip + Insertion Binary + Flip + Correction + Insertion Binary + Shift + Correction Binary + Shift + Insertion Binary + Shift + Correction + Insertion 22 Inc. Permutation + Heuristic Permutation + Heuristic Random-key + Heuristic

Locality Analysis: Protein-Ligand Docking • Protein-ligand docking is an energy minimization search problem with the aim to find the best ligand conformation and orientation relative to the active site of a target protein. • The docking problem can be very difficult since the relative orientation and conformations of the two molecules must be considered. • Typically, the receptor (usually a protein) is fixed in a three-dimensional coordinate system. By contrast, the ligand can be repositioned and rotated. 23

Locality Analysis: Evolutionary Model • Auto-Dock: this approach is a conformational search method which uses an approximate physical model to evaluate possible protein-ligand conformations. • It incorporates flexibility by allowing the ligand to change its conformation during the docking simulation. • A genotype is encoded by a vector of real-valued numbers which represent the ligand’s translation and orientation. Cartesian coordinates represent the ligand translation, three variables in the vector, whereas four variables defining a quaternion represent the ligand orientation. For each flexible torsion angle one variable is used. 24

Locality Analysis: Evolutionary Model • The phenotype of a candidate solution is composed of the atomic coordinates that represent the three-dimensional structure of the ligand. • The atomic structure of the ligand is built from the translation and orientation coordinates in the ligand crystal structure with the application of the torsion angles. • Since the encoding is a real-valued vector, mutation is performed by using evolutionary strategies based operators: 25

Locality: the effect of Mutation 26

Locality: the impact of Local Search 30

Understanding Algorithms • The analysis of adding heuristics and local improvement techniques; although in general the addition of these type of mechanisms provide a better performance, this is not always true. • Is important for an operator to induce strong locality to obtain good optimization results. • It provides hints on how future genetic operators, representations, heuristics and local search methods can be developed. 34

The Generic Evolutionary Algorithm • Organization of the Evolution engine: 35

Example: Evolving Operators • Evolution by means of Genetic Programing, or similar techniques, of genetic operators for a speciﬁc problem. 36

Example: Evolving Operators • Evolution by means of Genetic Programing, or similar techniques, of genetic operators for a speciﬁc problem. • The operators are evolved according to: ➡ The degree of representation properties induced by the operator, e.g., we want a locally-strong operator. ➡ How an operator can navigate more smoothly on a landscape. 36

Evolution and Self-Organization • With these elements, it will be possible to provide a framework dedicated to the evolution of genetic representations and other evolutionary components. • Understanding the effects of a representation and operators when solving a problem is important so that new and more efﬁcient encodings can be developed to the problem being solved. • Using representation properties and the effects of genetic operators, as well other components, to guide the algorithm in the search for its own conﬁguration. 37

Questions? jorge.tavares@ieee.org http://jorgetavares.wordpress.com Images by Penousal Machado, University of Coimbra http://eden.dei.uc.pt/~machado

Evolvability in Optimization Problems: Towards more Efficient Evolutionary Algorithms

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