Chapter 10 e volutionary c omputation ii g eneral m ethods and t heory
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Slides for Introduction to Stochastic Search and Optimization ( ISSO ) by J. C. Spall. CHAPTER 10 E VOLUTIONARY C OMPUTATION II : G ENERAL M ETHODS AND T HEORY. Organization of chapter in ISSO Introduction Evolution strategy and evolutionary programming; comparisons with GAs

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Chapter 10 e volutionary c omputation ii g eneral m ethods and t heory l.jpg

Slides for Introduction to Stochastic Search and Optimization (ISSO)by J. C. Spall

CHAPTER 10EVOLUTIONARYCOMPUTATIONII: GENERAL METHODS AND THEORY

Organization of chapter in ISSO

Introduction

Evolution strategy and evolutionary programming; comparisons with GAs

Schema theory for GAs

What makes a problem hard?

Convergence theory

No free lunch theorems


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Methods of EC

  • Genetic algorithms (GAs), evolution strategy (ES), and evolutionary programming (EP) are most common EC methods

  • Many modern EC implementations borrow aspects from one or more EC methods

  • Generally: ES generally for function optimization; EP for AI applications such as automatic programming


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ES Algorithm with Noise-Free Loss Measurements

Step 0 (initialization)Randomly or deterministically generate initial population of N values of  and evaluate L for each of the values.

Step 1 (offspring)Generate  offspring from current population of N candidate  values such that all  values satisfy direct or indirect constraints on .

Step 2 (selection)For (N+)-ES, select N best values from combined population of Noriginal values plus  offspring; for (N,)-ES, select N best values from population of  > N offspring only.

Step 3 (repeat or terminate) Repeat steps 1 and 2 or terminate.


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Schema Theory for GAs

  • Key innovation in Holland (1975) is a form of theoretical foundation for GAs based on schemas

    • Represents first attempt at serious theoretical analysis

    • But not entirely successful, as “leap of faith” required to relate schema theory to actual convergence of GA

  • “GAs work by discovering, emphasizing, and recombining good ‘building blocks’ of solutions in a highly parallel fashion.” (Melanie Mitchell, An Introduction to Genetic Algorithms [p. 27], 1996, paraphrasing John Holland)

    • Statement above more intuitive than formal

    • Notion of building block is characterized via schemas

    • Schemas are propagated or destroyed according to the laws of probability


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Schema Theory for GAs

  • Schema is template for chromosomes in GAs

  • Example: [* 1 0 * * * * 1], where the * symbol represents a don’t care (or free) element

    • [11001101] is specific instance of this schema

  • Schemas sometimes called building blocks of GAs

  • Two fundamental results: Schema theorem and implicit parallelism

  • Schema theorem says that better templates dominate the population as generations proceed

  • Implicit parallelism says that GA processes >> N schemas at each iteration

  • Schema theory is controversial

    • Not connected to algorithm performance in same direct way as usual convergence theory for iterates of algorithm


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Convergence Theory via Markov Chains

  • Schema theory inadequate

    • Mathematics behind schema theory not fully rigorous

    • Unjustified claims about implications of schema theory

  • More rigorous convergence theory exists

    • Pertains to noise-free loss (fitness) measurements

    • Pertains to finite representation (e.g., bit coding or floating point representation on digital computer)

  • Convergence theory relies on Markov chains

  • Each state in chain represents possible population

  • Markov transition matrix P contains all information for Markov chain analysis


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GA Markov Chain Model

  • GAs with binary bit coding can be modeled as (discrete state) Markov chains

  • Recall states in chain represent possible populations

  • ith element of probability vector pk represents probability of achieving ith population at iteration k

  • Transition matrix: The i, j element of P represents the probability of population i producing population j through the selection, crossover and mutation operations

    • Depends on loss (fitness) function, selection method, and reproduction and mutation parameters

  • Given transition matrix P, it is known that


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Rudolph (1994) and Markov Chain Analysis for Canonical GA

  • Rudolph (1994, IEEE Trans. Neural Nets.) uses Markov chain analysis to study “canonical GA” (CGA)

  • CGA includes binary bit coding, crossover, mutation, and “roulette wheel” selection

    • CGA is focus of seminal book, Holland (1975)

  • CGA does not include elitismlack of elitism is critical aspect of theoretical analysis

  • CGA assumes mutation probability 0 < Pm < 1 and single-point crossover probability 0  Pc 1

  • Key preliminary result: CGA is ergodic Markov chain:

    • Exists a unique limiting distribution for the states of chain

    • Nonzero probability of being in any state regardless of initial condition


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Rudolph (1994) and Markov Chain Analysis for CGA (cont’d)

  • Ergodicity for CGA provides a negative result on convergence in Rudolph (1994)

  • Let denote lowest of N (= population size) loss values within population at iteration k

    • represents loss value for  in population k that has maximum fitness value

  • Main theorem: CGA satisfies

    (above limit on left-hand side exists by ergodicity)

  • Implies CGA does not converge to the global optimum


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Rudolph (1994) and Markov Chain Analysis for CGA (cont’d)

  • Fundamental problem with CGA is that optimal solutions are found but then lost

  • CGA has no mechanism for retaining optimal solution

  • Rudolph discusses modification to CGA yielding positive convergence results

  • Appends “super individual” to each population

    • Super individual represents best chromosome so far

    • Not eligible for GA operations (selection, crossover, mutation)

    • Not same as elitism

  • CGA with added super individual converges in probability


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Contrast of Suzuki (1995) and Rudolph (1994) in Markov Chain Analysis for GA

  • Suzuki (1995, IEEE Trans. Systems, Man, and Cyber.) uses Markov chain analysis to study GA with elitism

    • Same as CGA of Rudolph (1994) except for elitism

  • Suzuki (1995) only considers unique states (populations)

    • Rudolph (1994) includes redundant states

  • With N = population size and B = no. of bits/chromosome:

    unique states in Suzuki (1995),

    2NB states in Rudolph (1994) (much larger than number of unique states above)

  • Above affects bookkeeping; does not fundamentally change relative results of Suzuki (1995) and Rudolph (1994)


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Convergence Under Elitism Analysis for GA

  • In both CGA case (Rudolph, 1994) and case with elitism (Suzuki, 1995) the limit exists:

    (dimension of differs according to definition of states, unique or nonunique as on previous slide)

  • Suzuki (1995) assumes each population includes one elite element and that crossover probability Pc = 1

  • Let represent jth element of , and J represent indices j where population j includes chromosome achieving L()

  • Then from Suzuki (1995):

  • Implies GA with elitism converges in probability to set of optima


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Calculation of Stationary Distribution Analysis for GA

  • Markov chain theory provides useful conceptual device

  • Practical calculation difficult due to explosive growth of number of possible populations (states)

  • Growth is in terms of factorials of N and bit string length (B)

  • Practical calculation of pk usually impossible due to difficulty in getting P

  • Transition matrix can be very large in practice

    • E.g., if N = B = 6, P is 108108 matrix!

    • Real problems have N and Bmuch larger than 6

  • Ongoing work attempts to severelyreduce dimension by limiting states to only most important (e.g., Spears, 1999; Moey and Rowe, 2004)


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Example 10.2 from Analysis for GAISSO: Markov Chain Calculations for Small-Scale Implementation

  • Consider L() =  = [0,15]

  • Function has local and global minimum; plot on next slide

  • Several GA implementations with very small population sizes (N) and numbers of bits (B)

  • Small scale implementations imply Markov transition matrices are computable

    • But still not trivial, as matrix dimensions range from approximately 20002000 to 40004000


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Loss Function for Example 10.2 in Analysis for GAISSOMarkov chain theory provides probability of finding solution ( = 15) in given number of iterations



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Summary of GA Convergence Theory Small-Scale GAs

  • Schema theory (Holland, 1975) was most popular method for theoretical analysis until approximately mid-1990s

    • Schema theory not fully rigorous and not fully connected to actual algorithm performance

  • Markov chain theory provides more formal means of convergence—and convergence rate—analysis

  • Rudolph (1994) used Markov chains to provide largely negative result on convergence for canonical GAs

    • Canonical GA does not converge to optimum

  • Suzuki (1995) considered GAs with elitism; unlike Rudolph (1994), GA is now convergent

  • Challenges exist in practical calculation of Markov transition matrix


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No Free Lunch Theorems (Reprise, Chap. 1) Small-Scale GAs

  • No free lunch (NFL) Theorems apply to EC algorithms

    • Theorems imply there can be no universally efficient EC algorithm

    • Performance of one algorithm when averaged over all problems is identical to that of any other algorithm

  • Suppose EC algorithm A applied to loss L

    • Let denote lowest loss value from most recent N population elements after nN unique function evaluations

  • Consider the probability that after n unique evaluations of the loss:

NFL theorems state that the sum of above probabilities over all loss functions is independent of A


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Comparison of Algorithms for Stochastic Optimization in Chaps. 2 – 10 of ISSO

  • Table next slide is rough summary of relative merits of several algorithms for stochastic optimization

    • Comparisons based on semi-subjective impressions from numerical experience (author and others) and theoretical or analytical evidence

    • NFL theorems not generally relevant as only considering “typical” problems of interest, not all possible problems

  • Table does not consider root-finding per se

  • Table is for “basic” implementation forms of algorithms

  • Ratings range fromL(low),ML(medium-low), M(medium), MH(mediumhigh), andH(high)

    • These scales are for stochastic optimization setting and have no meaning relative to classical deterministic methods


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Comparison of Algorithms Chaps. 2 – 10 of