Exact GP Schema Theorems. We now turn to Ch. 5 and the claim that, finally, schema theorems can be m

1
Lecture 5

• Exact GP Schema Theorems. We now turn to Ch. 5
  and the claim that, finally, schema theorems can
  be made precise enough to provide a reliable
  predictive (at least theoretical) tool. The
  primary tool is the capability of accounting for
  schema creation (recall the results of Altenberg,
  which were really too complex to be useful), and
  those of Goldberg, use in the study of deception.
• Let H denote a schema over some population, let
  a(H, t) denote the schema transition probability
  the probability that, at generation t, the
  individual produced by selection, cloning,
  crossover and mutation will match schema H.
• The previous chapters show, if nothing else, that
  computing a(H, t) is difficult. Lets start from
  the other end if a(H, t) is known, what becomes
  easy?

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Lecture 5

• The first observation is if each
  selection/crossover is independent (usually so in
  GA and GP contexts) then the chance of each child
  matching H will be the same. But this means that
  m(H, t 1), the number of instances of schema H
  at time t 1, is binomially distributed the
  probability that the population contain exactly k
  instances (in M trials) of the schema at
  generation t 1 is given by
• (For a quick introduction to Probability and
  Queuing Theory you may want to check some of the
  materials in giam/Mikkeli/Lecture5.ppt, and
  ../IntroToProbabilityAndApplications.pdf,
  ../IntroQueTheo.ppt)

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Lecture5

• We can compute the expected number of instances
  
• as well as the variance (see Intropdf for
  details)
• The schema transition probability at generation t
  corresponds to the expected proportion of the
  population sampling the schema H at generation t
  1.
• Another useful idea is that of the
  Signal-to-Noise ratio for a schema

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Lecture 5

• We can also compute the probability of schema
  extinction in one generation Prm(H, t 1)
  0 (1 - a(H, t))M.
• We now turn to another result that relates means
  and variances
• Theorem. (Chebyshev Inequality) Let X be a
  random variable with mean m E(X) and variance
  s2 Var(X). Then, for any t gt 0,
• Proof. Feller, p. 219.
• If we take t ks and use the assumption that
  m(H, t 1) is binomially distributed with mean
  Ma(H, t) and variance Ma (H, t) (1 - a (H,
  t)), we can rewrite Chebyshevs Inequality to
  provide

5
Lecture 5

• Theorem 5.2.1 (Two-Sided Probabilistic Schema
  Theorem). For any given constant k gt 0,
• Proof. Let t ks. The previous theorem - with X
  replaced by m(H, t 1) - gives
• Turning the inequality around, we obtain
  immediately
• Replacing m and s

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Lecture 5

• Theorem 5.2.2 (Probabilistic Schema Theorem). For
  any given constant k gt 0,
• Proof. This follows from the observation
  that since the first
  expression contains the area of the right tail,
  while the second doesnt. The previous theorem
  does the rest.
• Note the results are valid for GA and GP
  contexts, They depend only on the transmission
  probability.

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Lecture 5

• Stephens and Waelbroeck (1997, 1999) For a
  bit-string genetic algorithm with one-point
  crossover applied with probability pxo,
  where L(H, i) the schema
  obtained by replacing all elements of H from i
  1 to N with dont care () symbols R(H, i) the
  schema obtained by replacing all elements of H
  from 1 to i with dont care () symbols p(H, t)
  is the probability of selecting an individual
  matching schema H to be a parent, which, under
  fitness proportionate reproduction is given by
• Note although a(H, t) and p(H, t) are
  probabilities associated with the current
  generation, a(H, t) provides us with information
  about the next generation, while p(H, t) does not
  (except through this formula).

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Lecture 5

• Example of L(H, t) and R(H, t)

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Lecture 5

• How do we extend these results to GP (where
  things are more complicated)? Lets start with
  the simplest case all programs are of exactly
  the same size and shape. The formula becomes,
  allowing for translation into GP
  where N(H) number of nodes in schema
  H, with same size and shape as programs in
  population1/N(H) is the probability of choosing
  any one of them for crossover l(H, i) schema
  obtained by replacing all nodes above crossover
  by dont care symbols lower part of the schema
  u(H, i) schema obtained by replacing all nodes
  below crossover by dont care symbols upper
  part of the schema the summation is over all
  valid crossover points.

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Lecture 5

• Example

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Lecture 5

• Note the detailed internal structure of the
  formula depends on the specific ordering of the
  crossover points the result does not. Thus we
  need not worry about how we numbered the
  crossover points, since it is the shape and size
  of the program tree alone that determines the
  result.
• The formula would be identical to that for GAs if
  all the functions at the interior nodes were
  unary functions See the text for a more
  detailed explanation.
• We now need to extend all of this to populations
  of programs of arbitrary size and shape and
  functions of arbitrary arity. It will take a
  while

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Lecture 5

• Definition 5.4.1 (GP hyperschema). A GP
  hyperschema is a rooted tree composed of internal
  nodes from the set F ? and leaves from T ?
  , . The hyperschema function set is the
  function set used in a GP run, plus the
  hyperschema terminal set is the GP terminal set
  plus and . The operator is a dont care
  symbol which stands for exactly one node, while
  the operator stands for any valid subtree.
• Note the introduction of the second terminal
  operator allows trees to change arbitrarily.

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Lecture 5

• We have now restricted Altenbergs ideas and
  generalized the ideas of Rosca and OReilly it
  may be enough (if done correctly).
• Theorem 5.4.1 (Microscopic Exact GP Schema
  Theorem). The total transmission probability for
  a fixed-size-and-shape GP schema H under
  one-point crossover and no mutation
  is where the first
  two summations are over all individuals in the
  population
• NC(h1, h2) is the number of nodes in the tree
  fragment representing the common region between
  program h1 and program h2
• C(h1, h2) is the set of indices of the crossover
  points in the common region

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Lecture 5

• d(x) is a function that returns 1 if x is true
  and 0 if not
• L(H, i) is the hyperschema obtained by replacing
  all nodes on the path between crossover point i
  and the root node with nodes, and all the
  subtrees connected to those nodes with nodes
• U(H, i) is the hyperschema obtained by replacing
  the subtree below the crossover point i with a
  node.
• Example start with a schema H ( ( x ))
  and examine the figure on the next slide.

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Lecture 5

• Ex. H ( ( x ))

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Lecture 5

• Proof. Let p(h1, h2, i, t) be the probability
  that, at generation t, the selection/crossover
  process will select h1 and h2 as parents and i as
  the crossover point. Consider
• g(h1, h2, i, H) ? d(h1 Î L(H, i))d(h2 Î U(H,
  i)).
• Given two parents, a schema and a crossover
  point, g returns 1 if crossing over h1 and h2 at
  position i yields an offspring which is an
  instance of H, and 0 otherwise. If h1, h2 and i
  are random variables with joint probability
  distribution (dependent on the generation) given
  by p(h1, h2, i, t), the expected value of g is
  just the sum
• Now we need to find a usable expression for p(h1,
  h2, i, t).

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Lecture 5

• We add a reasonable assumption that the choice
  of crossover point is independent of the
  generation, and just depends on the two parents.
  Using conditional probabilities, we can
  obtain p(h1, h2, i, t) p(i h1, h2)p(h1,
  h2, t), where p(h1, h2, t) is the probability
  of choosing h1 and h2 at generation t (and must
  be generation dependent), while p(i h1, h2) is
  just the conditional probability of choosing
  crossover point i, given that we have h1 and h2
  as parents for crossover. Since the parents are
  chosen independently (selection with
  replacement), we can rewrite the formula
  as p(h1, h2, i, t) p(i h1, h2)p(h1,
  t)p(h2, t), where p(h1, t) and p(h2, t) are
  the selection probabilities for the parents.

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Lecture 5

• Also, in one-point crossover, p(i h1,
  h2) 1/NC(h1, h2), so that
• When we substitute this in the summation, and
  adjust for the cloning/crossover possibilities,
  we obtain the formula of the theorem
• We now have an expression for the total
  transmission probability a(H, t) of a GP schema.
  All the derivative results (expectation,
  variance, S/N ratio, extinction probability) can
  be also obtained.

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Lecture 5

• Our final schema theorem will use the set of all
  possible hyperschemata. We can certainly number
  all possible program shapes (schemata of order
  0), G0, G1, G2, There is no canonical
  ordering, but the number of program shapes (over
  finite function and terminal alphabets) of each
  depth is finite, so the numbering up to the
  shapes of any finite depth is, in principle,
  possible. The hyperschemata represent disjoint
  sets of programs, and their union represents the
  whole search space. In particular, for any
  program h1, We start by modifying the second
  term in the previous formula (recall all sums are
  finite)

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Lecture 5

• Rearranging the summations

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Lecture 5

• The expression
  is simply the probability that we extract
  a program h1 which both belongs to the
  hyperschema Gj and matches a specific
  generalization of the schema H. It can be
  restated as Using
  this simplification, we have
• Theorem 5.4.3 (Macroscopic Exact GP Schema
  Theorem for One-point Crossover). The total
  transmission probability for a fixed-size-and-shap
  e GP schema H under one-point crossover and no
  mutation is where the
  sets L(H, i) Ç Gj and U(H, i) Ç Gk either are (or
  can be represented by) fixed-size-and-shape
  schemata or are Æ.

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Lecture 5

• The textbook has several examples where the
  computations are carried out, showing that it
  would be feasible to compare theoretical
  predictions to actual runs. We end this series
  of theoretical lectures with a final idea that
  of Effective Fitness.
• Effective Fitness is a measure of the
  reproductive success of a schema that
  incorporates the creative and disruptive effects
  of the genetic operators. Without any mutation
  or crossover, fitness proportionate reproduction
  leads to the equation where
  all the terms have their usual meaning in
  particular, f(H, t) is the fitness of the schema
  H in generation t. We want an equation of the
  form

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Lecture 5

• Recalling from previous computations that Em(H,
  t 1) Ma(H, t), and that
  where p(H, t)
  is the probability of selecting schema H for
  reproduction under fitness proportionate
  selection,we have feff(H, t) (a(H, t)/p(H,
  t))f(H, t). Since a(H, t) is computable via
  formulae just derived, we have (at least for
  fixed-size-and-shape schemata under one-point
  crossover and no mutation)

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Lecture 5

• A similar formula can be obtained for Genetic
  Algorithm schemata - modifying the earlier result
  of Stephens and Woelbroeck.
• Our text has a discussion comparing the meaning
  of fitness and effective fitness it appears that
  effective fitness can provide at least some
  plausible explanations for certain aspects of
  computational evolution (in particular, program
  bloat in GP).
• It is plausible to conclude that the effective
  fitness is what leads to propagation down the
  generations (more than fitness itself) and that
  the search space is determined as least as much
  by effective fitness as by the fitness formula
  used statically on the individuals of each
  generation.