1 Simulated Annealing (Reading – Section 10.9 of NRC) Genetic Algorithms Optimisation Methods

2 Simulated Annealing Optimisation methods to date only find minimum of current basin on hyper-surface SA (and Gas) are optimisation methods that can handle multiple local minima. Analogy with thermodynamics, cooling and annealing of metals, cooling and freezing of liquids Must provide the following elements: A description of possible system states A generator of random changes in the system (options for next system state) An objective function (analogue of system energy) A control parameter (analogue of temperature) and a cooling schedule which describes how the control parameter is lowered from high to low values.

3 Simulated Annealing

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10 Simulated Annealing

11 Genetic Algorithms in a slide  Premise Evolution worked once (it produced us!), it might work again  Basics Pool of solutions Mate existing solutions to produce new solutions Mutate current solutions for long-term diversity Cull population

12 Genetic Algorithms in a slide randomly initialise a pool of solutions Next_Generation Mutation_Loop select a solution from pool using relative fitness mutate solution and save end_Loop Crossover_Loop select pairs of solutions from pool using relative fitness crossover and save both child solutions end_Loop Termination_Check if not finished create new pool from saved solutions goto Next_GenerationGA DemoGA Demo

13 Originator  John Holland  Seminal work Adaptation in Natural and Artificial Systems introduced main GA concepts, 1975

14 Introduction  Computing pioneers (especially in AI) looked to natural systems as guiding metaphors  Evolutionary computation Any biologically-motivated computing activity simulating natural evolution  Genetic Algorithms are one form of this activity  Original goals Formal study of the phenomenon of adaptation  John Holland An optimization tool for engineering problems

15 Main idea  Take a population of candidate solutions to a given problem  Use operators inspired by the mechanisms of natural genetic variation  Apply selective pressure toward certain properties  Evolve a more fit solution

16 Why evolution as a metaphor  Ability to efficiently guide a search through a large solution space  Ability to adapt solutions to changing environments  “Emergent” behavior is the goal  “The hoped-for emergent behavior is the design of high-quality solutions to difficult problems and the ability to adapt these solutions in the face of a changing environment”  Melanie Mitchell, An Introduction to Genetic Algorithms

17 Evolutionary terminology  Abstractions imported from biology Chromosomes, Genes, Alleles Fitness, Selection Crossover, Mutation

18 GA terminology  In the spirit – but not the letter – of biology GA chromosomes are strings of genes  Each gene has a number of alleles; i.e., settings Each chromosome is an encoding of a solution to a problem A population of such chromosomes is operated on by a GA

19 Encoding  A data structure for representing candidate solutions Often takes the form of a bit string Usually has internal structure; i.e., different parts of the string represent different aspects of the solution)

20 Crossover  Mimics biological recombination Some portion of genetic material is swapped between chromosomes Typically the swapping produces an offspring  Mechanism for the dissemination of “building blocks” (schemas)

21 Mutation  Selects a random locus – gene location – with some probability and alters the allele at that locus  The intuitive mechanism for the preservation of variety in the population

22 Fitness  A measure of the goodness of the organism  Expressed as the probability that the organism will live another cycle (generation)  Basis for the natural selection simulation Organisms are selected to mate with probabilities proportional to their fitness  Probabilistically better solutions have a better chance of conferring their building blocks to the next generation (cycle)

23 A Simple GA Generate initial population do Calculate the fitness of each member // simulate another generation do Select parents from current population Perform crossover add offspring to the new population while new population is not full Merge new population into the current population Mutate current population while not converged

24 How do GAs work  The structure of a GA is simple to comprehend, but the dynamic behavior is complex  Holland has done significant work on the theoretical foundations of GAs  “GAs work by discovering, emphasizing, and recombining good ‘building blocks’ of solutions in a highly parallel fashion.”  Melanie Mitchell, paraphrasing John Holland  Using formalism Notion of a building block is formalized as a schema Schemas are propagated or destroyed according to the laws of probability

25 Genetic algorithm (GA), I  GA works by using a large population to explore many options in parallel.  The state of a GA is given by a population, with each member of the population being a complete set of parameters

26 Genetic algorithm - Overview  The whole population is updated in generations by four steps: Fitness: evaluate the function being searched Reproduction: the members of the new population are selected based on their fitness. Members with a low fitness might disappear, and one with a high fitness can be duplicated. Crossover: After two parents are randomly chosen based on their fitness, the offspring gets its parameter values based on some kind of random selection from the parents. Mutation: randomly or in some other way change the parameter values

27 Genetic algorithm - Overview Distribution of Individuals in Generation 0 Distribution of Individuals in Generation N

28 Example - Cumulative Selection Methinks it is like a weasel  28 characters including blank  random cases.  Starting from random sentences, we can find the desired sentence by the following procedure 1.Generate 10 sentences of 27 randomly chosen characters 2.Select the sentence that has the most correct letters 3.Duplicate this best sentence ten times 4.For each duplicate, randomly replace a few letters (mutation rate) 5.Repeat step 2-4 until the target sentence is matched.  The Weasel Applet:

29 Genetic algorithm - Overview  Maximization problem in 2D space ( 0<x<1, 0<y<1 )  Encoding: Individual: a point at (x, y); P1=( , ), P2=( , ) Encoding to chromosome-like string: P1=“ ”, P2=“ ”

30 Genetic algorithm - Overview  Breeding: Crossover: P1 =  O1 = P2 =  O2 = Mutation: O2 =  O2 =  Decoding: O1 =  ( , ) O2 =  ( , )

31 Genetic algorithm - Overview  Elitism: Store away the parameters defining the fittest member of the current population. And later copy it intact in the offspring population  Variable mutation rate At any given time, keep track of the fitness value of the fittest population members, and of the median ranked member. The fitness difference f between those two indivituals is a measure of population convergenece. If f becomes too small, increase the mutation rate. If f becomes too large, decrease the mutation rate.

32 Genetic algorithm - Overview  Hamming wall & creep mutation Example: optimal = 21000, current = Choose a digit Instead of replacing the digit, add either 1 or -1 Example: creep mutation hitting the middle “ 9” with 1  20094

33 Schema  A template, much like a regular expression, describing a set of strings  The set of strings represented by a given schema characterizes a set of candidate solutions sharing a property  This property is the encoded equivalent of a building block

34 Example  0 or 1 represents a fixed bit  Asterisk represents a “don’t care”  11****00 is the set of all solutions encoded in 8 bits, beginning with two ones and ending with two zeros Solutions in this set all share the same variants of the properties encoded at these loci

35 Schema qualifiers  Length The inclusive distance between the two bits in a schema which are furthest apart (the defining length of the previous example is 8)  Order  The number of fixed bits in a schema (the order of the previous example is 4)

36 Not just sum of the parts  GAs explicitly evaluate and operate on whole solutions  GAs implicitly evaluate and operate on building blocks Existing schemas may be destroyed or weakened by crossover New schemas may be spliced together from existing schema  Crossover includes no notion of a schema – only of the chromosomes

37 Why do they work  Schemas can be destroyed or conserved  So how are good schemas propagated through generations? Conserved – good – schemas confer higher fitness on the offspring inheriting them Fitter offspring are probabilistically more likely to be chosen to reproduce

38 Approximating schema dynamics  Let H be a schema with at least one instance present in the population at time t  Let m(H, t) be the number of instances of H at time t  Let x be an instance of H and f(x) be its fitness  The expected number of offspring of x is f(x)/f(pop) (by fitness proportionate selection)  To know E(m(H, t +1)) (the expected number of instances of schema H at the next time unit), sum f(x)/f(pop) for all x in H GA never explicitly calculates the average fitness of a schema, but schema proliferation depends on its value

39 Approximating schema dynamics  Approximation can be refined by taking into account the operators Schemas of long defining length are less likely to survive crossover  Offspring are less likely to be instances of such schemas Schemas of higher order are less likely to survive mutation Effects can be used to bound the approximate rates at which schemas proliferate

40 Implications  Instances of short, low-order schemas whose average fitness tends to stay above the mean will increase exponentially  Changing the semantics of the operators can change the selective pressures toward different types of schemas

41 Theoretical Foundations  Empirical observation GAs can work  Goal Learn how to best use the tool  Strategy Understand the dynamics of the model Develop performance metrics in order to quantify success

42 Theoretical Foundations  Issues surrounding the dynamics of the model What laws characterize the macroscopic behavior of GAs? How do microscopic events give rise to this macroscopic behavior?

43 Theoretical Foundation  Holland’s motivation Construct a theoretical framework for adaptive systems as seen in nature Apply this framework to the design of artificial adaptive systems  Issues in performance evaluation According to what criteria should GAs be evaluated? What does it mean for a GA to do well or poorly? Under what conditions is a GA an appropriate solution strategy for a problem?

44 Theoretical Foundation  Holland’s observations An adaptive system must persistently identify, test, and incorporate structural properties hypothesized to give better performance in some environment Adaptation is impossible in a sufficiently random environment

45 Theoretical Foundation  Holland’s intuition A GA is capable of modeling the necessary tasks in an adaptive system It does so through a combination of explicit computation and implicit estimation of state combined with incremental change of state in directions motivated by these calculations

46 Theoretical Foundation  Holland’s assertion The ‘identify and test’ requirement is satisfied by the calculation of the fitnesses of various schemas The ‘incorporate’ requirement is satisfied by implication of the Schema Theorem

47 Theoretical Foundation  How does a GA identify and test properties? A schema is the formalization of a property A GA explicitly calculates fitnesses of individuals and thereby schemas in the population It implicitly estimates fitnesses of hypothetical individuals sharing known schemas In this way it efficiently manages information regarding the entire search space

48 Theoretical Foundation  How does a GA incorporate observed good properties into the population? Implication of the Schema Theorem  Short, low-order, higher than average fitness schemas will receive exponentially increasing numbers of samples over time

49 Theoretical Foundation  Lemmas to the Schema Theorem Selection focuses the search Crossover combines good schemas Mutation is the insurance policy

50 Theoretical Foundation  Holland’s characterization Adaptation in natural systems is framed by a tension between exploration and exploitation Any move toward the testing of previously unseen schemas or of those with instances of low fitness takes away from the wholesale incorporation of known high fitness schemas But without exploration, schemas of even higher fitness can not be discovered

51 Theoretical Foundation  Goal of Holland’s first offering The original GA was proposed as an “adaptive plan” for accomplishing a proper balance between exploration and exploitation

52 Theoretical Foundation  GA does in fact model this Given certain assumptions, the balance is achieved  A key assumption is that the observed and actual fitnesses of schemas are correlated  This assumption creates a stumbling block to which we will return

53 Traveling Salesperson Problem Find the minimum distance tour around a set of cities, visiting each city only once and ending back where you started from.

54 Initial Population for TSP (5,3,4,6,2)(2,4,6,3,5)(4,3,6,5,2) (2,3,4,6,5)(4,3,6,2,5)(3,4,5,2,6) (3,5,4,6,2)(4,5,3,6,2)(5,4,2,3,6) (4,6,3,2,5)(3,4,2,6,5)(3,6,5,1,4)

55 Select Parents (5,3,4,6,2)(2,4,6,3,5)(4,3,6,5,2) (2,3,4,6,5)(4,3,6,2,5)(3,4,5,2,6) (3,5,4,6,2)(4,5,3,6,2)(5,4,2,3,6) (4,6,3,2,5)(3,4,2,6,5)(3,6,5,1,4) Try to pick the better ones.

56 Create Off-Spring – 1 point (5,3,4,6,2)(2,4,6,3,5)(4,3,6,5,2) (2,3,4,6,5)(4,3,6,2,5)(3,4,5,2,6) (3,5,4,6,2)(4,5,3,6,2)(5,4,2,3,6) (4,6,3,2,5)(3,4,2,6,5)(3,6,5,1,4) (3,4,5,6,2)(3,4,5,6,2)

57 (3,4,5,6,2)(3,4,5,6,2) Create More Offspring (5,3,4,6,2)(2,4,6,3,5)(4,3,6,5,2) (2,3,4,6,5)(4,3,6,2,5)(3,4,5,2,6) (3,5,4,6,2)(4,5,3,6,2)(5,4,2,3,6) (4,6,3,2,5)(3,4,2,6,5)(3,6,5,1,4) (5,4,2,6,3)(5,4,2,6,3)

58 (3,4,5,6,2)(3,4,5,6,2)(5,4,2,6,3)(5,4,2,6,3) Mutate (5,3,4,6,2)(2,4,6,3,5)(4,3,6,5,2) (2,3,4,6,5)(4,3,6,2,5)(3,4,5,2,6) (3,5,4,6,2)(4,5,3,6,2)(5,4,2,3,6) (4,6,3,2,5)(3,4,2,6,5)(3,6,5,1,4)

59 Mutate (5,3,4,6,2)(2,4,6,3,5)(4,3,6,5,2) (2,3,4,6,5)(2,3,6,4,5)(3,4,5,2,6) (3,5,4,6,2)(4,5,3,6,2)(5,4,2,3,6) (4,6,3,2,5)(3,4,2,6,5)(3,6,5,1,4) (3,4,5,6,2)(3,4,5,6,2)(5,4,2,6,3)(5,4,2,6,3)

60 Eliminate (5,3,4,6,2)(2,4,6,3,5)(4,3,6,5,2) (2,3,4,6,5)(2,3,6,4,5)(3,4,5,2,6) (3,5,4,6,2)(4,5,3,6,2)(5,4,2,3,6) (4,6,3,2,5)(3,4,2,6,5)(3,6,5,1,4) Tend to kill off the worst ones. (3,4,5,6,2)(3,4,5,6,2) (5,4,2,6,3)(5,4,2,6,3)

61 Integrate (5,3,4,6,2)(2,4,6,3,5) (2,3,6,4,5)(3,4,5,2,6) (3,5,4,6,2)(4,5,3,6,2)(5,4,2,3,6) (4,6,3,2,5)(3,4,2,6,5)(3,6,5,1,4) (3,4,5,6,2)(3,4,5,6,2) (5,4,2,6,3)(5,4,2,6,3)

62 Restart (5,3,4,6,2)(2,4,6,3,5) (2,3,6,4,5)(3,4,5,2,6) (3,5,4,6,2)(4,5,3,6,2)(5,4,2,3,6) (4,6,3,2,5)(3,4,2,6,5)(3,6,5,1,4) (3,4,5,6,2) (5,4,2,6,3)

63 Genetic Algorithms  Facts Very robust but slow  Can make simulated annealing seem fast In the limit, optimal

64 Other GA-TSP Possibilities  Ordinal Representation  Partially-Mapped Crossover  Edge Recombination Crossover  Problem Operators are not sufficiently exploiting the proper “building blocks” used to create new solutions.

65 Genetic Algorithms  Some ideas Parallelism Punctuated equilibria Jump starting Problem-specific information Synthesize with simulated annealing Perturbation operator

66 Heuristic H Length(MST) < Length(T)Let T be the optimal tour.

67 Heuristic H Tour T’Tour T’’

68 Perturbation of points