6 Crossover The Center of the Storm

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6 Crossover -The Center of the Storm
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• Crossover and Building Blocks
• GP crossover mimic the process of sexual
  reproduction.
• GP search is more effective than systems based on
  random transformations (mutations) of the
  candidate solutions.
• GP works faster than systems just based on
  mutations, according to building block
  hypothesis, because good building blocks get
  combined into ever larger and better building
  blocks to form better individuals.
• Crossover - The Controversy
• Does the GP crossover operator outperform
  mutation-based systems by locating and combining
  good building blocks or is GP crossover, itself,
  a form of macromutation?
• What sorts of improvements may be made to the
  crossover operator to improve its performance?

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• A Caveat
• This chapter will focus at length on the
  undoubted shortcomings of the GP crossover
  operator.
• It is important, nevertheless, to remember that
  something is going on with GP crossover.
• GP crossover already has a substantial record of
  accomplishment.
• Chapter Overview
• The theoretical bases for both the building block
  hypothesis and the notion that GP crossover is
  really a macromutation operator
• The empirical evidence about the effect of
  crossover
• Several promising directions for improving GP
  crossover

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The Theoretical Basis for the Building Block
Hypothesis in GP

• The schema theorem of Holland is one of the most
  influential and debated theorems in evolutionary
  algorithms in general and genetic algorithms in
  particular.
• The schema theorem for fixed length genetic
  algorithms states that good schemata will tend to
  multiply exponentially in the population as the
  genetic search progresses and will thereby be
  combined into good overall solutions with other
  such schemata.
• However, the GP case is much more complex because
  GP uses representations of varying length and
  allows genetic material to move from one place to
  another in the genome.
• The crucial issue in the schema theorem is the
  extent to which crossover tends to disrupt or to
  preserve good schemata.

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• Kozas Schema Theorem Analysis
• A schema is a set of subtrees that contains
  (somewhere) one or many subtrees from a special
  schema defining set.
• Kozas argument is informal and he does not
  suggest an ordering or length definition for his
  schemata.
• Kozas statement that GP crossover tends to
  preserve, rather than disrupt, good schemata
  depends crucially on the GP reproduction
  operator.
• Good schemata will be tested and combined by
  crossover operator more often than poorer
  schemata.
• This process results in the combination of
  smaller but good schemata into bigger schemata
  and, ultimately, good overall solutions.

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• OReillys Schema Theorem Analysis
• OReilly defines her schemata similarly to Koza
  but with the presence of a dont care symbol ()
  in one or more subtrees.
• The order of a schema is the number of nodes
  which are not symbols.
• The length is the number of links in the tree
  fragments plus the number of links connecting
  them.

f(H,t) mean fitness of all instances of a
certain schema H (t) average fitness in
generation t Em(H,t) the expected value of the
number of instances of H Pd(H,t) the maximum
probability of disruption pc crossover
probability
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• Whighams Schema Theorem Analysis
• Whigham has formulated a definition of schemata
  in his grammar-based GP system.
• This approach leads to a simpler equation for the
  probability of disruption than OReillys
  approach.
• Newer Schema Theorems
• Poli and Langdon have formulated a new schema
  theorem that asymptotically converges to the GA
  schema theorem.
• The result of their study suggests that there
  might be two different phases in a GP run a
  first phase completely depending on fitness, and
  a second phase depending on fitness and structure
  of the individual (e.g., schema defining length).
• Roscas schema theorem for rooted-tree schemata

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• Inconclusive Schema Theorem Results for GP
• None of the existing formulations of a GP schema
  theorem predicts with any certainty that good
  schemata will propagate during a GP run.
• The principal problem is the variable length of
  the GP representation.
• In the absence of a strong theoretical basis for
  the claim that GP crossover is more than a
  macromutation operator, it is necessary to turn
  to other approaches.

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Preservation and Disruption of Building Blocks A
Gedanken Experiment

• Crossover as a Disruptive Force
• As GP becomes more and more successful in
  assembling small building blocks into larger and
  larger blocks, the whole structure becomes more
  and more fragile because it is more prone to
  being broken up by subsequent crossover.

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• Assume that our building block is almost a
  perfect program.
• But in this case, just before success, the
  probability that the perfect solution will be
  disrupted by crossover is 10/11 or 90.9.

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• The conclusion is inevitable crossover operator
  is a disruptive force as well as a constructive
  force - putting building blocks together and then
  tearing them apart.
• The balance is impossible to measure with todays
  techniques.
• It is undoubtedly a dynamic equilibrium that
  changes during the course of evolution.
• We note, however, that for most runs, measured
  destructive crossover rates stay high until the
  very end.

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• Reproduction and Crossover
• The good building blocks in individuals
  duplicated by the reproduction operator will have
  many chances to try to find crossovers that are
  not disruptive.
• This argument depends on the assumption that the
  high quality of the building block will somehow
  be reflected in the quality of the individual in
  which it appears.
• It also depends on the balance between the
  reproduction operator and the destructive effects
  of crossover at any given time in a run.
• Schema Theorem Analysis Is Still Inconclusive.
• It is impossible to predict with any certainty
  yet whether GP crossover is only a macromutation
  operator or something more.

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Empirical Evidence of Crossover Effects

• The Effect of Crossover on the Fitness of
  Offspring
• The effect of crossover on the relative fitness
  of parents and their offspring

How can we measure the effect of crossover? It is
not entirely clear what should be measured
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• Two basic approaches to measuring the effect of
  crossover
• The Result of Measuring the Effect of Crossover
• In all three cases (tree-based GP, linear GP, and
  graph GP), crossover has an overwhelmingly
  negative effect on the fitness of the offspring
  of the crossover.
• The conclusion is compelling crossover routinely
  reduces the fitness of offspring substantially
  relative to their parents in almost every GP
  system.

The average fitness of all parents has been
compared with the average fitness of all
offspring The fitness of children and parents is
compared on an individual basis.
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• The Relative Merits of Program Induction via
  Crossover versus Hill Climbing or Annealing
• Headless Chicken Crossover
• Only one parent is selected and an entirely new
  individual is created randomly. The selected
  parent is then crossed over with the new and
  randomly created individual.
• The offspring is kept if it is better than or
  equal to the parent in fitness. Otherwise, it is
  discarded. Thus, headless chicken crossover is a
  form of hill climbing.
• Mutation techniques may perform as well as and
  sometimes slightly better than traditional GP
  crossover.

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• Crossover vs. Non-Population-Based Operators
• Mutate-simulated annealing and crossover-hill
  climbing
• If the new solution has higher fitness, it
  replaced the original solution. Otherwise, it is
  discarded in crossover-hill climbing but kept
  probabilistically in mutate-SA.
• The mutate-SA and crossover-hill climbing
  algorithms performed as well as or slightly
  better than standard GP on a test suite of six
  different problems.
• Crossover seems to create children with large
  syntactic differences between parents and
  offspring.

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• Conclusions about Crossover as Macromutation
• The empirical evidence lends little credence to
  the notion that traditional GP crossover is,
  somehow, a more efficient or better search
  operator than mutation-based techniques.
• There is no serious support the conclusion that
  hill climbing outperforms GP.
• On the state of the evidence as it exists today,
  one must conclude that traditional GP crossover
  acts primarily as a macromutation operator.
• The failure of the standard GP crossover operator
  may be due to the stagnation of GP runs (bloat
  - in other words, the exponential growth of GP
  introns).

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Improving Crossover - The Argument from Biology

• Biological crossover works in a highly
  constrained and highly controlled context that
  has evolved over billions of years.
• Crossover may be seen as the result of the
  evolution of evolvability.
• Three principal constraints on biological
  crossover
• In nature, most crossover events are successful -
  that is, they results in viable offspring (in
  standard GP, 25).

Biological crossover takes place only between
members of the same species. Biological crossover
occurs with remarkable attention to preservation
of semantics. Biological crossover is
homologous.
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• In the basic GP system, any subtree may be
  crossed over with any other subtree. There is no
  requirement that the two subtrees fulfill similar
  functions.
• There is no requirement that a subtree, after
  being swapped, is in a context in the new
  individual that has any relation to the context
  in the old individual.
• Were GP to develop a good subtree building block,
  it would be very likely to be disrupted by
  crossover rather than preserved and spread.
• There is no reason to suppose that randomly
  initialized individuals in a GP population are
  members of the same species.

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Improving Crossover - New Directions

• Brood Recombination
• pick two parents from the population
• Perform random crossover on the parents N times,
  each time creating a pair of children as a result
  of crossover.
• Evaluate each of the children for fitness. Sort
  them by fitness. Select the best two.
• Time-Saving Evaluation
• Is Brood Recombination Effective?

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• Intelligent Crossover
• A Crossover Operator That Learns
• Improving the rate of constructive crossover in
  PADO (a graph-based GP) by letting an intelligent
  crossover operator learn how to select good
  crossover points
• A Crossover Operator Guided by Heuristics
• The performance value for subtrees decides which
  subtrees are potential building blocks to be
  inserted into other trees, and which subtrees are
  to be replaced.
• The intelligent operators found regularities in
  the program structures of very different GP
  systems

There are blocks of code that are best left
together - perhaps these are building
blocks. These blocks of code have characteristics
that can be identified by heuristics or a
learning algorithm. GP produces higher
constructive crossover rates and better results
when these blocks of code are probabilistically
kept together.
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• Context-Sensitive Crossover
• Most crossover does not preserve the context of
  the code - yet context is crucial to the meaning
  of computer code.
• Strong context preserving crossover (SCPC) that
  only permitted crossover between nodes that
  occupied exactly the same position in the two
  parents.
• Modest improvements in results by mixing regular
  crossover and SCPC
• This approach introduced an element of homology
  into the crossover operator.
• Requiring crossover to swap between trees at
  identical locations is somewhat homologous.

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• Explicit Multiple Gene Systems
• Fitness components are affected by all or some of
  the genes.
• This system highly theoretical because the
  fitness of the individual is just the sum of the
  fitness components.
• During evolution, a gene is periodically added.
  If it improves the fitness individual, it is
  kept otherwise, it is discarded.
• Between gene additions, the population evolved by
  intergene crossover.
• Having mutiple fitness fuctions allows the genes
  to be more independent or, in biological terms,
  to be less epistatic.

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• Explicitly Defined Introns
• An integer value (explicitly defined introns
  value - EDIV) is stored between every two nodes
  in the GP individual.
• The probability that crossover occurs between any
  two nodes is the GP program is proportional to
  the integer value between the nodes.
• The EDIV vector evolves during the GP run to
  identify the building blocks in the individual as
  an emergent phenomenon.
• The EDIV values within a good building block
  should become low and, outside the good block,
  high.
• Using real-valued EDIVs and constraining changes
  in the EDIVs by Gaussian distribution of
  permissible mutation to the EDIVs

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• Modeling Other Forms of Genetic Exchange
• There are several ways in which individuals
  exchange genetic material in nature (conjugation,
  transduction, and transformation)
• Conjugation
• Simple conjugation in GAs - donor, recipient
• To foster the spread of potentially advantageous
  genetic information, conjugation might be
  combined with tournament selection.
• Multiple conjugation involving n donors could be
  combined preferentially with n1-tournament
  selection.

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Improving Crossover - A Proposal

• Homologous crossover in GP
• What result does homologous crossover have?

The mechanism by which biology cause homology,
i.e. speciation, almost identical length or
structure of DNA between parents, and strict base
pairing during crossover. The reason the
mechanism has evolved makes the actual mechanism
somewhat irrelevant when changing the medium.
Two parents have a child that combines some of
the genome of each parent. The exchange is
strongly biased toward experimenting with
exchanging very similar chunks of the genome -
specific genes performing specific functions -
that have small variations among them.
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• Mating selection Two trees are selected
  randomly.
• Measurement of structural similarity for each
  edge k in the larger tree, a subtree with
  smallest distance - imin(k) - in the other tree ?
  DS(k,imin(k))
• Measurement of structural similarity
• Selection of crossover points

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Improving Crossover - The Tradeoffs

• Tradeoffs
• Standard GP crossover acts mainly as a
  macromutation operator.
• Much of our discussion has focused on how to
  improve crossover - how to make it more than a
  simple macromutation operator?
• It is important not to under estimate the power
  of a simple mutation operator.
• Digital Overhead and Homology
• There is a cost associated with improving
  crossover in GP.
• This digital overhead may be likened to the large
  amount of biological energy expended to maintain
  homologous crossover in nature.
• Locating the Threshold

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Conclusion

• It certainly stands to benefit from improvements
  through smart mutation or other typed of added
  mechanisms.
• The crossover operator will be much more powerful
  and robust over the next few years.
• One of the strongest arguments for the building
  block hypothesis is the manner in which a GP
  population adapts to the destructive effects of
  crossover.
• GP individuals tend to accumulate code that does
  nothing during a run - we refer to such code as
  introns.
• The important point is that the presence of
  introns underlines how important preventation of
  destructive crossover is in the GP system.
• The challenge in GP for the next few years is to
  tame the crossover operator and to find the
  building blocks.