Evolutionary Algorithms. Lehrstuhl für Informatik 2 Gabriella Kókai: Maschine Learning Overview Motivation Nature as a Standard Genetic Algorithms Genetic

Evolutionary Algorithms

Lehrstuhl für Informatik 2

Gabriella Kókai: Maschine Learning

Overview

➔ Motivation Nature as a Standard Genetic Algorithms Genetic Programming Evolutionary Strategies Conclusion



Motivation Since millions of years creatures populate Earth By changes in the biosphere there are again and again new environmental

conditions Populations had to learn to adapt to the new conditions; permanent stepwise

development, few stagnancy Organisms are optimally adapted with respect to their needs Nature has its own laws, rules, strategies, and mechanisms

' Evolution: successful, robust mechanism, allows creatures over generations to adapt to environmental conditions

' Goal of evolution is not predefined; optimisation, innovation, creativity

' Selection factors: competition, food supply, enemies, climate, environment, via human beings additionally breed,



Overview

Motivation➔ Nature as a Standard Genetic Algorithms Genetic Programming Evolutionary Strategies Conclusion



Nature as a Standard - Evolution, Genome

Lamarck's thesis (1809): Adaptation, urge to perfection (by specific needs) spontaneous creations, heredity of acquired characteristics (somatic induction) -> no feedback in genome

Darwin's thesis (1859): permanent evolution, common descent, multiplication of species, gradual change, natural selection, descending of characteristics with modificationBasic conditions: too rich production of genetic variations, limitation of resources (competition)Fitness: suitability, result of multiple interactions with selection factors of the environment



Nature as a Standard - Evolution, Genome

Gene: functional unit, relative short segment of DNA, information how to build a protein molecule

Gene-pool: sum of all genotype-variants of a population Genotype: all the genes (genome), generally structures, contain

information, instructions to define individual characteristics Phenotype: interpretation of the genes, expression of the genome as

individual characteristics, competes with other phenotypes for reproductive success in a specific setting (basic conditions of the environment) => selection filter



Overview Motivation Nature as a Standard➔ Genetic Algorithms

Classic Always Algorithm Selection Representation of Hypothesis Genetic Operators Procedure of Evolution Schema Theorem Applications

Genetic Programming Evolutionary Strategies Conclusion



Genetic Algorithms John H. Holland 1975 ; David E. Goldberg 1986 Goal of optimisation, "generate-and-test beam search" Variability (Heterogenity of the characteristics, singleness, variety) Differential fitness (propagation rate depends on the ability to survive

in a specific setting, to reproduce descendants) Heritable fitness (circulate the

genome, incomplete copy, by mixture of different descendants)

Dualism Genotype/Phenotype



Classic Always Algorithm

πη

τσ

η π s

s P 0

τ t,P t , η π s

P' = σ P t ,η P t

η P t

Coding, structures representation of hypothesis and individuals Interpretation function what does the coding represent? Fitness function shall be optimised Termination criteria is the optimum approximately reached? Selection function which individuals determine the next population? Initialise: generate randomly n individuals for the initial population P(0) Evaluate: determine for all t := 0 Generation 0 while not Selection: choose stochastically individuals according to their fitness Crossover: create children via the recombination of parental individuals from P' Mutation: change randomly the representation of child individuals from P' Update: put n, randomly picked child individuals from P' to P(t+1) t := t + 1 increment generation Evaluate return Individual with highest fitness value



Representation of Hypothesis

CodingRepresentation of the parameters (hypothesis, individuals) to be optimised by structures over a discrete alphabet, mostly bit-strings s = (0100111101) s = (atggcaact) with alphabet A = {a, t, g, c}

InterpretationMapping p from the genotypical structure space into the phenotypical characteristics and behaviour space

Production systems = (10 01 1 11 10 0) : IF a1=T & a2=F THEN c=T ; IF a2=T THEN c=F

Triplet : amino acid



Selection Best fitting individuals shall build a descendant

population, one-sidedness shall be avoided by stochastic selection algorithms

Fitness-proportional selection: Roulette algorithm

proportional to their own fitness, indirectly proportional to competitors. Problem: Super individuals may dominate too much

Rank-based selection: Individuals are sorted ascendingly according to their fitness; selection is done by a roulette algorithm based on the position in this ranking



Genetic Operators

MutationMutation probability, uniformly distributed random number

With a discrete alphabet and a maximal mutation distance, to limit variation. Defining the distance measure of the alphabet

P = 0.50110011010 --> 0101010001 bit-wise cgeehadcdhh --> chdcgadcdfh Mutation distance 2 (lexicographic)

X



Genetic Operators (2)

Multipoint analogous, e.g. uniformly or odd-even: 0110011010 0011001111 Mask: 1212121212 1011001111 1110011010

Multi-recombination (more than 2 parental chromosomes): random selection of 2 parents, as above several parents

0110011010 \ 1011001111 0100000001 - 0110010001 Mask: 3311144443 1101110000 /

X



Genetic Operators (2) Recombination (Crossover)

Crossing point(s) randomly determined or by a fixed mask Single-point:

0110011010 0111001111 Crossing point: 3 1011001111 1010011010 Mask: 1112222222

Dual-point: bbafdeacca bbabacacca Crossing points: 3, 6 edebacbfbb edefdebfbb Mask: 1112221111

X

X



Genetic Operators (3)

Inversion mirrored (generally: permuted) insertion of the middle part 1010011101 --> 1010100001 inverted fgbbcdadace --> fgbbcdcdaea permuted

Deletionloss of an arbitrary part 1010011101 --> 101001 intercalar fgbbcdadace --> fgbbcda terminal

Duplication duplication of an arbitrary part 1010011101 --> 1010011011101 fgbbcdadace --> fgbbcdadacedace



Procedure of Evolution Example: global maximum of a (multi-modal) function

Bit-vectors: Interpretation: as in the example Evaluation: compute the function at the interpreted location

5 (3) Populations independently of each other Strategies: population size, recombination partner, create descendants,

mutate Plus selection from parents and mutated children Comma selection from mutated children, individuals

survive at most one generation Variants

x y x,1 x,kx y,1 y,kys = s ,s = s , ,s ,s , s ,

η s = f π s

μ / ρ + λ

ρμ

μ / ρ,λ

λ



Schema Theorem

Schema: word over alphabet A* Instance: all words which are equal to at the fixed positions Order: o(H) number of fixed elements Defining length: segment length between the outermost fixed positions e.g. A = {a,b,c}, = (b, *, c, a, *, *) ; o( ) = 3 , = 4 - 1 = 3

Instances: (b, a, c, a, a, a) , (b, b, c, a, b, c) , (b, c, c, a, c, a) Premises: infinite large population, single-point-crossover, punctual mutation Which templates survive (stay instances of the schema)?

exponential propagation, if Selection: more than average fitness Recombination: short defining length Mutation: few fixed positions

As compact as possible conglomeration of gene groups, which are responsible for the increased fitness: building blocks

AH

Aδ H

AI H

AH AH Aδ H

AH



Applications

Can be run easily in parallel In combination with gradient algorithms (hill-climbing): Maximum search for rough

restriction of the search space Simulation of living cells Production system as an extension to expert systems Planning optimisation (storage, production processes, ...) Optimal game strategies Travelling-Salesman-Problem: structure contains indices of the nodes in visiting

order. To visit each node exactly once: modification of the genetic operators Evolution of the structure of neural nets: representation organised in segments

depending on the number of output-neurones; codes the number of layers, hidden neurons and according weights



Overview

Motivation Nature as a Standard Genetic Algorithms➔ Genetic Programming

Representation of the Hypothesis Differences to GA Applications

Evolutionary Strategies Conclusion



Genetic Programming

John R. Koza, 1989 Further development of the idea of genetic algorithms Genetic creation and optimisation of computer programs for special

problem-areas Representation of the Hypothesis Differences to GA Applications



Representation of the Hypothesis Computer program as tree structure (like parse-tree, LISP-Syntax) Combining elements: definition of terms and functions

Arithmetic expressions: {PLUS2, MINUS2, MULT2, DIV2} Functions: {SIN1, COS1, EXP2, LOG2, ...} Relations, conditional statement: {LESS2, EQUAL2, IF-THEN-ELSE3, ...} Problem related: {TURN-LEFT, PICK-UP, MOVE-RANDOM, ...} Tree structure: IF-THEN-ELSE

LESS MULT ADD A B A C B C LISP-Syntax: ( IF-THEN-ELSE ( LESS A B ) ( MULT A C ) ( ADD B C ) )

Closed under composition Complete according to the problem to be solved



Differences to GA Recombination

Exchange arbitrarily chosen sub-trees

Random determination of the crossing points

Even with identical terms mostly new structure pairs

Both children survive

Mutation Substitution of a sub-tree by a newly

generated sub-tree Random selection of a node Substitution by a randomly new

term which is correctly generated out of building blocks



Differences to GA

Recombination, Mutation: context-sensitive variation Selection: matches the algorithmic solution of the given problem Formulation as fitness value, e.g. Distance measure for numeric problems Successfully solved / identified cases Copy operator: copies a GP-chromosome unchanged into the next

generation Each genome is only modified by a single operator: selection

between operators Extension of terms to symbolic expressions



Genetic Programming (Example) Minimise a Boolean function

24

00000 1 01000 1 10000 0 11000 000001 0 01001 0 10001 0 11001 100010 0 01010 1 10010 1 11010 100011 0 01011 1 10011 0 11011 100100 0 01100 1 10100 0 11100 000101 0 01101 1 10101 0 11101 100110 1 01110 0 10110 0 11110 0



Genetic Programming(Example Results) 1000 individuals, 1000 steps (8 minutes) starting length: 127, results 71, 57



Applications

Ant searching for food on a minimal route Classification of groups belonging together for complex areas, e.g.

swallowed spirals Robots searching for objects and doing precisely oriented moves Robots following walls Random number generator with a distribution as uniformly as possible Backwards docking of a truck with its hanger Steering a robot arm with two joints to points in a field Design of electronic circuits for analogous filters



Overview

Motivation Nature as a Standard Genetic Algorithms Genetic Programming➔ Evolutionary Strategies

Idea, basic Principles Differences to GA Applications

Conclusion



Evolutionary Strategies

Ingo Rechenberg, 1964 / 1994 Adaptation of the basic mechanisms of natural evolution to

technical optimisation problems by engineering sciences Root: evolutionary experimental methods, focussed on the

physical experiment Results of the (at that time) unorthodox methods could not be

analytically founded or reproduced Idea, basic principles Differences to GA Applications



Development, Idea

Given: experimental equipment with variable parameters Mechanic: changing position by pitch and angle Elastic: outline by bending Combination of segments of different sizes Random change of the parameters in a certain area (mostly

binomially distributed: little mutation prefered) Measuring the experimental result: if getting worse then back

propagation of the changes Repeat until optimum is found Representation: Parameter as real-valued vector Original experiment: orthogonal pipe redirection with smallest loss

1, ng = p ,p



Development, Idea (2)



Differences to GA Algorithmic representable, expandable to populations / several descendants Representation expanded by strategy parameters: Describe variance for controlling the mutation spreading of the appropriate

parameter, can be integrated in the optimum search (adaptation of the increment)

Real-valued structures: adaptation of the genetic operators Mutation: numeric deviation ; Gauss distributed

random number, average 0, variance Recombination: discrete (randomly copied from the one or the other parent

chromosome), intermediary (average building), local (single individuals), global (whole population)

Random selection, no proportionality of the fitness Surplus of descendants, selection of the best for succeeding population

1, n 1, ng = p ,p , s ,s

i i 0 ip' = p + N s 0N

si

is



Evolutionary Strategies (Example)

Fluid storage Changeable shape

● Fixed volume● Minimal surface



Evolutionary Strategies(Example Results)

100 individuals 100 generations



Applications Pressure output of a two phases supersonic cone, segments with variable

diameter

Flow resistance of a joint plate, 5 joints with 51 engaging levels (0, +, -) each Rotation body form with little flow resistance, air plane ... dolphin spindle Minimal weight construction of a bow bridge Flexion of a lens for concentration on focus Magic square: 30x30 with magic sum 13525 Networking with minimal lengths and a given branching degree



Conclusion Evolutionary algorithms solve optimisation problems Standard is the natural evolution, which produces permanently new and partly

improved organisms, which must assert themselves in their environment Basis is the biological adaptation as a learning procedure of populations of natural

organisms Hypotheses are interpreted and evaluated by a fitness function The hypothesis room is explored by a stochastic search: Selection as fitness

proportional procedure New hypotheses come up by recombination and mutation, similar to the chromosomes

of organisms The representation can be done by bit-strings/character-strings (GA), programs as term

and function trees (GP) or real-valued parameter vectors (ES) The convergence of the algorithms is mostly very good, but not guaranteed They work also with complex problems, where other algorithms have failed on or are

not (yet) known

Documents

Evolutionary Algorithms. Lehrstuhl für Informatik 2 Gabriella Kókai: Maschine Learning Overview Motivation Nature as a Standard Genetic Algorithms Genetic