Dr T presents Evolutionary Computing Computer Science 348
- Slides: 114
Dr. T presents… Evolutionary Computing Computer Science 348
Introduction • The field of Evolutionary Computing studies theory and application of Evolutionary Algorithms. • Evolutionary Algorithms can be described as a class of stochastic, population-based local search algorithms inspired by neo. Darwinian Evolution Theory.
Computational Basis o Trial-and-error (aka Generate-and-test) o Graduated solution quality o Stochastic local search of solution landscape
Biological Metaphors o Darwinian Evolution n n Macroscopic view of evolution Natural selection Survival of the fittest Random variation
Biological Metaphors o (Mendelian) Genetics n Genotype (functional unit of inheritance) n Genotypes vs. phenotypes n Pleitropy: one gene affects multiple phenotypic traits n Polygeny: one phenotypic trait is affected by multiple genes n Chromosomes (haploid vs. diploid) n Loci and alleles
EA Pros o More general purpose than traditional optimization algorithms; i. e. , less problem specific knowledge required o Ability to solve “difficult” problems o Solution availability o Robustness o Inherent parallelism
EA Cons o Fitness function and genetic operators often not obvious o Premature convergence o Computationally intensive o Difficult parameter optimization
EA components o Search spaces: representation & size o Evaluation of trial solutions: fitness function o Exploration versus exploitation o Selective pressure rate o Premature convergence
Nature versus the digital realm Environment Problem (search space) Fitness Population Fitness function Set Individual Datastructure Genes Elements Alleles Datatype
EA Strategy Parameters o o o o Population size Initialization related parameters Selection related parameters Number of offspring Recombination chance Mutation rate Termination related parameters
Problem solving steps o o o o o Collect problem knowledge Choose gene representation Design fitness function Creation of initial population Parent selection Decide on genetic operators Competition / survival Choose termination condition Find good parameter values
Function optimization problem Given the function f(x, y) = x 2 y + 5 xy – 3 xy 2 for what integer values of x and y is f(x, y) minimal?
Function optimization problem Solution space: Z x Z Trial solution: (x, y) Gene representation: integer Gene initialization: random Fitness function: -f(x, y) Population size: 4 Number of offspring: 2 Parent selection: exponential
Function optimization problem Genetic operators: o 1 -point crossover o Mutation (-1, 0, 1) Competition: remove the two individuals with the lowest fitness value
Measuring performance o Case 1: goal unknown or never reached n Solution quality: global average/best population fitness o Case 2: goal known and sometimes reached n Optimal solution reached percentage o Case 3: goal known and always reached n Convergence speed
Initialization o o o Uniform random Heuristic based Knowledge based Genotypes from previous runs Seeding
Representation (§ 2. 3. 1) o o o Genotype space Phenotype space Encoding & Decoding Knapsack Problem (§ 2. 4. 2) Surjective, injective, and bijective decoder functions
Simple Genetic Algorithm (SGA) o o o Representation: Bit-strings Recombination: 1 -Point Crossover Mutation: Bit Flip Parent Selection: Fitness Proportional Survival Selection: Generational
Trace example errata o Page 39, line 5, 729 -> 784 o Table 3. 4, x Value, 26 -> 28, 18 -> 20 o Table 3. 4, Fitness: n n n 676 -> 784 324 -> 400 2354 -> 2538 588. 5 -> 634. 5 729 -> 784
Representations o Bit Strings n Scaling Hamming Cliffs n Binary vs. Gray coding (Appendix A) o Integers n Ordinal vs. cardinal attributes n Permutations o Absolute order vs. adjacency o Real-Valued, etc. o Homogeneous vs. heterogeneous
Permutation Representation o Order based (e. g. , job shop scheduling) o Adjacency based (e. g. , TSP) o o Problem space: [A, B, C, D] Permutation: [3, 1, 2, 4] Mapping 1: [C, A, B, D] Mapping 2: [B, C, A, D]
Mutation vs. Recombination o Mutation = Stochastic unary variation operator o Recombination = Stochastic multi-ary variation operator
Mutation o Bit-String Representation: n Bit-Flip n E[#flips] = L * pm o Integer Representation: n Random Reset (cardinal attributes) n Creep Mutation (ordinal attributes)
Mutation cont. o Floating-Point n Uniform n Nonuniform from fixed distribution o Gaussian, Cauche, Levy, etc. o Permutation n n Swap Insert Scramble Inversion
Permutation Mutation o o Swap Mutation Insert Mutation Scramble Mutation Inversion Mutation (good for adjacency based problems)
Recombination o o o o Recombination rate: asexual vs. sexual N-Point Crossover (positional bias) Uniform Crossover (distributional bias) Discrete recombination (no new alleles) (Uniform) arithmetic recombination Simple recombination Single arithmetic recombination Whole arithmetic recombination
Recombination (cont. ) o Adjacency-based permutation n Partially Mapped Crossover (PMX) n Edge Crossover o Order-based permutation n Order Crossover n Cycle Crossover
Permutation Recombination Adjacency based problems o Partially Mapped Crossover (PMX) o Edge Crossover Order based problems o Order Crossover o Cycle Crossover
PMX o Choose 2 random crossover points & copy midsegment from p 1 to offspring o Look for elements in mid-segment of p 2 that were not copied o For each of these (i), look in offspring to see what copied in its place (j) o Place i into position occupied by j in p 2 o If place occupied by j in p 2 already filled in offspring by k, put i in position occupied by k in p 2 o Rest of offspring filled by copying p 2
Order Crossover o Choose 2 random crossover points & copy mid-segment from p 1 to offspring o Starting from 2 nd crossover point in p 2, copy unused numbers into offspring in the order they appear in p 2, wrapping around at end of list
Population Models o Two historical models n Generational Model n Steady State Model o Generational Gap o General model n Population size n Mating pool size n Offspring pool size
Parent selection o Fitness Proportional Selection (FPS) n n High risk of premature convergence Uneven selective pressure Fitness function not transposition invariant Windowing, Sigma Scaling o Rank-Based Selection n Mapping function (ala SA cooling schedule) n Linear ranking vs. exponential ranking
Sampling methods o Roulette Wheel o Stochastic Universal Sampling (SUS)
Rank based sampling methods o Tournament Selection n Tournament Size
Survivor selection o Age-based o Fitness-based n Truncation o Elitism
Termination o o o CPU time / wall time Number of fitness evaluations Lack of fitness improvement Lack of genetic diversity Solution quality / solution found Combination of the above
Behavioral observables o Selective pressure o Population diversity n n Fitness values Phenotypes Genotypes Alleles
Report writing tips o Use easily readable fonts, including in tables & graphs (11 pnt fonts are typically best, 10 pnt is the absolute smallest) o Number all figures and tables and refer to each and every one in the main text body (hint: use autonumbering) o Capitalize named articles (e. g. , ``see Table 5'', not ``see table 5'') o Keep important figures and tables as close to the referring text as possible, while placing less important ones in an appendix o Always provide standard deviations (typically in between parentheses) when listing averages
Report writing tips o Use descriptive titles, captions on tables and figures so that they are self-explanatory o Always include axis labels in graphs o Write in a formal style (never use first person, instead say, for instance, ``the author'') o Format tabular material in proper tables with grid lines o Provide all the required information, but avoid extraneous data (information is good, data is bad)
Evolution Strategies (ES) o Birth year: 1963 o Birth place: Technical University of Berlin, Germany o Parents: Ingo Rechenberg & Hans. Paul Schwefel
ES history & parameter control o o o Two-membered ES: (1+1) Original multi-membered ES: (µ+1) Multi-membered ES: (µ+λ), (µ, λ) Parameter tuning vs. parameter control Fixed parameter control n Rechenberg’s 1/5 success rule o Self-adaptation n Mutation Step control
Uncorrelated mutation with one o o o Chromosomes: x 1, …, xn, ’ = • exp( • N(0, 1)) x’i = xi + ’ • N(0, 1) Typically the “learning rate” 1/ n½ And we have a boundary rule ’ < 0 ’ = 0
Mutants with equal likelihood Circle: mutants having same chance to be created
Mutation case 2: Uncorrelated mutation with n ’s o o Chromosomes: x 1, …, xn, 1, …, n ’i = i • exp( ’ • N(0, 1) + • Ni (0, 1)) x’i = xi + ’i • Ni (0, 1) Two learning rate parmeters: n ’ overall learning rate n coordinate wise learning rate o ’ 1/(2 n)½ and 1/(2 n½) ½ o ’ and have individual proportionality constants which both have default values of 1 o i ’ < 0 i ’ = 0
Mutants with equal likelihood Ellipse: mutants having the same chance to be
Mutation case 3: Correlated mutations o Chromosomes: x 1, …, xn, 1, …, n , 1, …, k o where k = n • (n-1)/2 o and the covariance matrix C is defined as: n cii = i 2 n cij = 0 if i and j are not correlated n cij = ½ • ( i 2 - j 2 ) • tan(2 ij) if i and j are correlated o Note the numbering / indices of the
Correlated mutations cont’d The mutation mechanism is then: o ’i = i • exp( ’ • N(0, 1) + • Ni (0, 1)) o ’j = j + • N (0, 1) o x ’ = x + N(0, C’) n x stands for the vector x 1, …, xn n C’ is the covariance matrix C after mutation of the values o 1/(2 n)½ and 1/(2 n½) o i’ < 0 i’ = 0 and ½ and 5° o | ’j | > ’j = ’j - 2 sign( ’j)
Mutants with equal likelihood Ellipse: mutants having the same chance to be
Recombination o Creates one child o Acts per variable / position by either n Averaging parental values, or n Selecting one of the parental values o From two or more parents by either: n Using two selected parents to make a child n Selecting two parents for each position anew
Names of recombinations Two fixed parents Two parents selected for each i Local zi = (xi + yi)/2 intermediary Global intermediary zi is xi or yi chosen randomly Global discrete Local discrete
Evolutionary Programming (EP) o Traditional application domain: machine learning by FSMs o Contemporary application domain: (numerical) optimization o arbitrary representation and mutation operators, no recombination o contemporary EP = traditional EP + ES n self-adaptation of parameters
EP technical summary tableau Representation Real-valued vectors Recombination None Mutation Gaussian perturbation Parent selection Deterministic Survivor selection Probabilistic ( + ) Specialty Self-adaptation of mutation step sizes (in meta-EP)
Historical EP perspective o EP aimed at achieving intelligence o Intelligence viewed as adaptive behaviour o Prediction of the environment was considered a prerequisite to adaptive behaviour o Thus: capability to predict is key to intelligence
Prediction by finite state machines o Finite state machine (FSM): n n n States S Inputs I Outputs O Transition function : S x I S x O Transforms input stream into output stream o Can be used for predictions, e. g. to predict next input symbol in a sequence
FSM example o Consider the FSM with: n n S = {A, B, C} I = {0, 1} O = {a, b, c} given by a diagram
FSM as predictor o o o o Consider the following FSM Task: predict next input Quality: % of in(i+1) = outi Given initial state C Input sequence 011101 Leads to output 110111 Quality: 3 out of 5
Introductory example: evolving FSMs to predict primes o o o P(n) = 1 if n is prime, 0 otherwise I = N = {1, 2, 3, …, n, …} O = {0, 1} Correct prediction: outi= P(in(i+1)) Fitness function: n 1 point for correct prediction of next input n 0 point for incorrect prediction n Penalty for “too many” states
Introductory example: evolving FSMs to predict primes o Parent selection: each FSM is mutated once o Mutation operators (one selected randomly): n n n Change an output symbol Change a state transition (i. e. redirect edge) Add a state Delete a state Change the initial state o Survivor selection: ( + ) o Results: overfitting, after 202 inputs best FSM had one state and both outputs were 0, i. e. , it always predicted “not prime”
Modern EP o No predefined representation in general o Thus: no predefined mutation (must match representation) o Often applies self-adaptation of mutation parameters o In the sequel we present one EP variant, not the canonical EP
Representation o For continuous parameter optimisation o Chromosomes consist of two parts: n Object variables: x 1, …, xn n Mutation step sizes: 1, …, n o Full size: x 1, …, xn, 1, …, n
Mutation o o o Chromosomes: x 1, …, xn, 1, …, n i’ = i • (1 + • N(0, 1)) x’i = xi + i’ • Ni(0, 1) 0. 2 boundary rule: ’ < 0 ’ = 0 Other variants proposed & tried: n n Lognormal scheme as in ES Using variance instead of standard deviation Mutate -last Other distributions, e. g, Cauchy instead of Gaussian
Recombination o None o Rationale: one point in the search space stands for a species, not for an individual and there can be no crossover between species o Much historical debate “mutation vs. crossover” o Pragmatic approach seems to prevail today
Parent selection o Each individual creates one child by mutation o Thus: n Deterministic n Not biased by fitness
Survivor selection o P(t): parents, P’(t): offspring o Pairwise competitions, round-robin format: n Each solution x from P(t) P’(t) is evaluated against q other randomly chosen solutions n For each comparison, a "win" is assigned if x is better than its opponent n The solutions with greatest number of wins are retained to be parents of next generation o Parameter q allows tuning selection pressure (typically q = 10)
Example application: the Ackley function (Bäck et al ’ 93) o The Ackley function (with n =30): o Representation: n -30 < xi < 30 (coincidence of 30’s!) n 30 variances as step sizes o o Mutation with changing object variables first! Population size = 200, selection q = 10 Termination after 200, 000 fitness evals Results: average best solution is 1. 4 • 10 – 2
Example application: evolving checkers players (Fogel’ 02) o Neural nets for evaluating future values of moves are evolved o NNs have fixed structure with 5046 weights, these are evolved + one weight for “kings” o Representation: n vector of 5046 real numbers for object variables (weights) n vector of 5046 real numbers for ‘s o Mutation: n Gaussian, lognormal scheme with -first n Plus special mechanism for the kings’ weight o Population size 15
Example application: evolving checkers players (Fogel’ 02) o Tournament size q = 5 o Programs (with NN inside) play against other programs, no human trainer or hard-wired intelligence o After 840 generation (6 months!) best strategy was tested against humans via Internet o Program earned “expert class” ranking outperforming 99. 61% of all rated players
Genetic Programming (GP) o Characteristic property: variable-size hierarchical representation vs. fixedsize linear in traditional EAs o Application domain: model optimization vs. input values in traditional EAs o Unifying Paradigm: Program Induction
Program induction examples o o o o o Optimal control Planning Symbolic regression Automatic programming Discovering game playing strategies Forecasting Inverse problem solving Decision Tree induction Evolution of emergent behavior Evolution of cellular automata
GP specification o o o o S-expressions Function set Terminal set Arity Correct expressions Closure property Strongly typed GP
GP notes o Mutation or recombination (not both) o Bloat (survival of the fattest) o Parsimony pressure
Learning Classifier Systems (LCS) o Note: LCS is technically not a type of EA, but can utilize an EA o Condition-Action Rule Based Systems n rule format: <condition: action> o Reinforcement Learning o LCS rule format: n <condition: action> → predicted payoff n don’t care symbols
LCS specifics o Multi-step credit allocation – Bucket Brigade algorithm o Rule Discovery Cycle – EA o Pitt approach: each individual represents a complete rule set o Michigan approach: each individual represents a single rule, a population represents the complete rule set
Parameter Tuning vs Control o Parameter Tuning: A priori optimization of fixed strategy parameters o Parameter Control: On-the-fly optimization of dynamic strategy parameters
Parameter Tuning methods o Start with stock parameter values o Manually adjust based on user intuition o Monte Carlo sampling of parameter values on a few (short) runs o Meta-tuning algorithm (e. g. , meta-EA)
Parameter Tuning drawbacks o Exhaustive search for optimal values of parameters, even assuming independency, is infeasible o Parameter dependencies o Extremely time consuming o Optimal values are very problem specific o Different values may be optimal at different evolutionary stages
Parameter Control methods o Deterministic n Example: replace pi with pi(t) o akin to cooling schedule in Simulated Annealing o Adaptive n Example: Rechenberg’s 1/5 success rule o Self-adaptive n Example: Mutation-step size control in ES
Evaluation Function Control o Example 1: Parsimony Pressure in GP o Example 2: Penalty Functions in Constraint Satisfaction Problems (aka Constrained Optimization Problems)
Penalty Function Control eval(x)=f(x)+W ·penalty(x) Deterministic ex: W=W(t)=(C ·t)α with C, α≥ 1 Adaptive ex (page 135 of textbook) Self-adaptive ex (pages 135 -136 of textbook) Note: this allows evolution to cheat!
Parameter Control aspects o What is changed? n Parameters vs. operators o What evidence informs the change? n Absolute vs. relative o What is the scope of the change? n Gene vs. individual vs. population n Ex: one-bit allele for recombination operator selection (pairwise vs. vote)
Parameter control examples Representation (GP: ADFs, delta coding) Evaluation function (objective function/…) Mutation (ES) Recombination (Davis’ adaptive operator fitness: implicit bucket brigade) o Selection (Boltzmann) o Population o Multiple o o
Parameterless EAs o Previous work o Dr. T’s Evo. Free project
Multimodal Problems o Multimodal def. : multiple local optima and at least one local optimum is not globally optimal o Basins of attraction & Niches o Motivation for identifying a diverse set of high quality solutions: n Allow for human judgement n Sharp peak niches may be overfitted
Restricted Mating o Panmictic vs. restricted mating o Finite pop size + panmictic mating -> genetic drift o Local Adaptation (environmental niche) o Punctuated Equilibria n Evolutionary Stasis n Demes o Speciation (end result of increasingly specialized adaptation to particular environmental niches)
EA spaces Biology EA Geographical Algorithmic Genotype Representation Phenotype Solution
Implicit diverse solution identification (1) o Multiple runs of standard EA n Non-uniform basins of attraction problematic o Island Model (coarse-grain parallel) n n Punctuated Equilibria Epoch, migration Communication characteristics Initialization: number of islands and respective population sizes
Implicit diverse solution identification (2) o Diffusion Model EAs n Single Population, Single Species n Overlapping demes distributed within Algorithmic Space (e. g. , grid) n Equivalent to cellular automata o Automatic Speciation n Genotype/phenotype mating restrictions
Explicit diverse solution identification o Fitness Sharing: individuals share fitness within their niche o Crowding: replace similar parents
Game-Theoretic Problems Adversarial search: multi-agent problem with conflicting utility functions Ultimatum Game o Select two subjects, A and B o Subject A gets 10 units of currency o A has to make an offer (ultimatum) to B, anywhere from 0 to 10 of his units o B has the option to accept or reject (no negotiation) o If B accepts, A keeps the remaining units and B the offered units; otherwise they both loose all units
Real-World Game-Theoretic Problems o Real-world examples: n n economic & military strategy arms control cyber security bargaining o Common problem: real-world games are typically incomputable
Armsraces o Military armsraces o Prisoner’s Dilemma o Biological armsraces
Approximating incomputable games o Consider the space of each user’s actions o Perform local search in these spaces o Solution quality in one space is dependent on the search in the other spaces o The simultaneous search of codependent spaces is naturally modeled as an armsrace
Evolutionary armsraces o Iterated evolutionary armsraces o Biological armsraces revisited o Iterated armsrace optimization is doomed!
Coevolutionary Algorithm (Co. EA) A special type of EAs where the fitness of an individual is dependent on other individuals. (i. e. , individuals are explicitely part of the environment) o Single species vs. multiple species o Cooperative vs. competitive coevolution
Co. EA difficulties (1) Disengagement o Occurs when one population evolves so much faster than the other that all individuals of the other are utterly defeated, making it impossible to differentiate between better and worse individuals without which there can be no evolution
Co. EA difficulties (2) Cycling o Occurs when populations have lost the genetic knowledge of how to defeat an earlier generation adversary and that adversary re-evolves o Potentially this can cause an infinite loop in which the populations continue to evolve but do not improve
Co. EA difficulties (3) Suboptimal Equilibrium (aka Mediocre Stability) o Occurs when the system stabilizes in a suboptimal equilibrium
Case Study from Critical Infrastructure Protection Infrastructure Hardening o Hardenings (defenders) versus contingencies (attackers) o Hardenings need to balance spare flow capacity with flow control
Case Study from Automated Software Engineering Automated Software Correction o Programs (defenders) versus test cases (attackers) o Programs encoded with Genetic Programming o Program specification encoded in fitness function (correctness critical!)
Multi-Objective EAs (MOEAs) o Extension of regular EA which maps multiple objective values to single fitness value o Objectives typically conflict o In a standard EA, an individual A is said to be better than an individual B if A has a higher fitness value than B o In a MOEA, an individual A is said to be better than an individual B if A dominates B
Domination in MOEAs o An individual A is said to dominate individual B iff: n A is no worse than B in all objectives n A is strictly better than B in at least one objective
Pareto Optimality (Vilfredo Pareto) o Given a set of alternative allocations of, say, goods or income for a set of individuals, a movement from one allocation to another that can make at least one individual better off without making any other individual worse off is called a Pareto Improvement. An allocation is Pareto Optimal when no further Pareto Improvements can be made. This is often called a Strong Pareto Optimum (SPO).
Pareto Optimality in MOEAs o Among a set of solutions P, the nondominated subset of solutions P’ are those that are not dominated by any member of the set P o The non-dominated subset of the entire feasible search space S is the globally Pareto-optimal set
Goals of MOEAs o Identify the Global Pareto-Optimal set of solutions (aka the Pareto Optimal Front) o Find a sufficient coverage of that set o Find an even distribution of solutions
MOEA metrics o Convergence: How close is a generated solution set to the true Pareto-optimal front o Diversity: Are the generated solutions evenly distributed, or are they in clusters
Deterioration in MOEAs o Competition can result in the loss of a non-dominated solution which dominated a previously generated solution o This loss in its turn can result in the previously generated solution being regenerated and surviving
NSGA-II o Initialization – before primary loop Create initial population P 0 Sort P 0 on the basis of non-domination Best level is level 1 Fitness is set to level number; lower number, higher fitness n Binary Tournament Selection n Mutation and Recombination create Q 0 n n
NSGA-II (cont. ) o Primary Loop n Rt = P t + Q t n Sort Rt on the basis of non-domination n Create Pt + 1 by adding the best individuals from Rt n Create Qt + 1 by performing Binary Tournament Selection, Mutation, and Recombination on Pt + 1
Epsilon-MOEA o Steady State o Elitist o No deterioration
Epsilon-MOEA (cont. ) o Create an initial population P(0) o Epsilon non-dominated solutions from P(0) are put into an archive population E(0) o Choose one individual from E, and one from P o These individuals mate and produce an offspring, c o A special array B is created for c, which consists of abbreviated versions of the objective values from c
Epsilon-MOEA (cont. ) o An attempt to insert c into the archive population E o The domination check is conducted using the B array instead of the actual objective values o If c dominates a member of the archive, that member will be replaced with c o The individual c can also be inserted into P in a similar manner using a standard domination check
SNDL-MOEA o Desired Features n n n n Deterioration Prevention Stored non-domination levels (NSGA-II) Number and size of levels user configurable Selection methods utilizing levels in different ways Problem specific representation Problem specific “compartments” (E-MOEA) Problem specific mutation and crossover
- Introduction to evolutionary computing
- Evolutionary computing ppt
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