What are the effective degrees of freedom/collective modes

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Just what are building blocks?How do (should)
Evolutionary Algorithms work?
Chris Stephens and Jorge Cervantes, Instituto de
Ciencias Nucleares, UNAM FOGA 2007,
9/1/2007 stephens_at_nucleares.unam.mx
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Its mathematically rigorous
Its intuitive
Theory
Its useful for practitioners
Its exact
What should it do?
It unifies phenomena
It predicts well
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Old Schema Theory and the BBH
Statistical Mechanics Approach
Theory
Dynamical Systems Model
Engineering Rules of thumb
Whats the best approach?
Coarse Grained models
Population Biology Models
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The Problem of Theory
Theory
Experiment
The ideal
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The Problem of Theory
?
?
In EC
?
?
New Applications New Algorithms
Theory
Experiment
e.g. Multi-Resource Traveling Gravedigger
Problem with Variable Coffin Size
Most algorithms are NEVER used (except by the
people who created them) - Darrell Whitley,
GECCO 2003 tutorial
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The Problem of Theory
The EC Expectation Gap
What theoreticians think practitioners are and
what practitioners think theoreticians should be
What practitioners think theoreticians are and
what theoreticians think practitioners should be
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EC Theory the Bare Necessities- the choice
of representation
GP
GAs
?
(1,0,0)
Objects Dim X
z
Linear GP Variable-length GAs
(1.321,2.463,3.149)
y

ES
x
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EC Theory the Bare Necessities
Objects have fitness
Objects have interactions
f
?
Selection
Object

Mutation

Recombination
k
m recombination mode
Dynamics
i
j
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In mathematics
Finite population model determined by Markov
chain. In the infinite population limit for
haploids
Thats most of standard population genetics and
evolutionary computation!
Implicit summation over repeated indices
Probability to mutate genotype J to genotype I
Probability to implement recombination
Probability that given recombination takes place
it is implemented with mode m
Probability to select genotype I
Conditional probability for child J given
parents K and L and a mode m
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Select two parents K and L
Dont recombine it with another
Select an object J
Recombine them with respect to a
recombination mode m applied with
probability pcpc(m) to obtain a child J
Mutate it to object I

• O coupled non-linear difference equations
• There are O3 different ?JKL
• Most of them are zero
• In object/string basis for a given m more than
  one K and L can give
• rise to J
• Equation is written covariantly (in terms of
  tensors) and
• therefore is valid in any coordinate system

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Two Questions

• Can we understand anything qualitatively from
  them?
• How does genetic dynamics work? (Why and when
  are recombination and mutation useful?)
• What are the effective degrees of
  freedom/collective modes?
• Can we solve them?
• Put them on the computer. Not very feasible for N
  100!

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Can we make things simpler?- consider only one
operator

• Selection only can get exact solution in terms
  of objects, e.g. strings (microscopic degrees
  of freedom are good coordinates for selection)
• Mutation only can get exact solution by Fourier
  transforming (coordinate transformation to the
  Walsh/Fourier basis) Diagonalizes the mutation
  matrix - solutions are normal modes
  (collective/effective degrees of freedom)

Can answer both 1) and 2) in these cases But
what about recombination?
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• Consider schemata/marginals and neglect the
  construction term

Hollands Schema theorem for schemata of length l
and order Nm
Smaller for longer schemata Tight linkage
beneficial because tightly linked genes are
more likely to crossover together
Smaller for higher order schemata
Bigger for fitter schemata
Dynamic schema fitness is population dependent
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• building block Hypothesis
• A GA works by combining short, low-order,
• highly fit schemata (building blocks) into
• fitter higher order schemata

• But how would we recognise one if we saw one?
• Building what?
• How many of them are there?
• Just how are they combined together?
• When is recombination beneficial?
• How does the effect of recombination depend
• on the fitness landscape (and on other
• operators/parameters)?

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Fitness landscape linkage
Loosely linked epistatic genes
Tightly linked epistatic genes
Understand the linkage (epistatic) patterns of
the fitness landscape (linkage learning)
a
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a
a
a
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Create a representation so that epistatic genes
are tightly linked
Epistatic genes
But
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What is the relationship between landscape
blocks and building blocks?
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Does recombination favour tight linkage?
Perform a coarse graining (i.e. write it in
terms of schemata) of the RHS of the exact
microscopic equations or, equivalently, do a
linear coordinate transformation using
Selection-weighted linkage disequilibrium
coefficient
Depends on population state, fitness landscape
and recombination distribution
Gives a complete description of the utility of
recombination mode by mode and generation by
generation
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Building Block schemata

• Object/string construction is now written in
  terms of schemata/marginals
• - Building Block schemata
• These BBs are not the same as those of the
  building block
• hypothesis they are not necessarily short or
  low-order or even fit!
• For every recombination mode/channel there is a
  corresponding
• unique BB pair
• The number of BB schemata is precisely defined
  (e.g. 2N for
• binary strings)
• They form a coordinate basis (many in fact, one
  for ech object)
• Hierarchical solutions objects have BBs, these
  BBs have their BBs etc.
• Hierarchy can be represented diagramatically

This is how recombination works For a given
object/schema it specifies the ONLY ways it can
be built
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Recombination via a particular channel
increases/decreases the proportion (effective
fitness) of a given string or schemata I when
lt 0 gt 0
Favours loose linkage
respectively
Favours tight linkage
If lt 0, channel is
non-deceptive higher probability to
select the Building Blocks of the string/schemata
than the string/schemata itself
If gt 0 , channel is deceptive
lower probability to select the
Building Blocks of the string/schemata than the
string/schemata itself
Standard Two-bit deception f(0) gt f(1)
gt 0
i.e. gt 0
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Example three loci, 1-point crossover
Level 1 BBs BBs of the string (e.g. optimum)
Level 2 BBs BBs of the BBs
Level 3 BBs BBs of the BBs of the BBs there
arent any, hierarchy terminates at O(1) BBs
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Landscape blocks
Modular landscapes m1 NIAH mN counting
ones f_00, Royal Road function Concatenated
traps
Useful metrics
Compares the relative effects of two operator
sets e.g. recombination and selection vs
selection only, or recombination and selection vs
selection and mutation
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What can theory tell us about selecto-recombinati
ve EAs?
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Predictions
First, the obvious if a string or schema does
not exist in the population then
If it does exist then there exists a critical
proportion for any string/schema such that if
and hence
recombination is bad, where
is population, mask/mode and landscape dependent
To see interaction between biases of selection
and recombination consider a random population,
then
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Predictions
For 1-block NIAH, N4 only one landscape block
and
(true for any mask)
Recombination is disadvantageous for
all masks
For 4-block NIAH, N4 maximum number of
landscape blocks
(true for any mask)
Recombination is advantageous for all
masks
For 2-block NIAH, N4 intermediate number of
landscape blocks
the relative advantage of
recombination is mask dependent
0011 is compatible with the landscape blocks but
0001 isnt
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Predictions

• Only in extreme cases can you say whether
  recombination is uniformly good or bad
• The more/less epistatic/unmodular the landscape
  the worse/better the effect of recombination
• Better to ask which recombination distribution is
  good or bad
• Which recombination distribution is best depends
  on the landscape
• The best recombination distributions are those
  whose BBs are compatible with the landscapes
  blocks, i.e. the underlying modularity
• Also depends on the population and therefore
  should be time dependent (first search with very
  mixing recombination to explore for blocks then
  restrict the mixing to exploit them)

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When is recombination bad?
Lower order BBs preferred
Shorter BBs preferred
Recombination leads to LESS production of the
optimal string or ANY optimal BB or schemata than
selection only
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When is recombination good?
Preference for O(1) BBs near the string boundary
Higher order BBs/schemata preferred
Longer BB/schemata preferred
Recombination leads to MORE production of ANY
optimal string or optimal BB or schemata than
selection only
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And what about here?
Recombination favours longer optimal
schemata But these arent BBs!
Preference for O(1) BBs near the string boundary
This level 2 O(2) BB is favoured
These BBs are only favoured after a certain
amount of time.
These level 1 O(2) BBs are suppressed
So, is recombination good or bad?
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So, what do the Deltas tell us?
Recombination is particularly bad in trying to
construct these O(2) BBs/optimal schemata
because of their tight linkage!
masks
Better to construct the needle with these masks
than these asymmetric BBs preferred
Recombination is better constructing these O(2)
optimal schemata because of their loose
linkage! But theyre not BBs!
Recombination is bad for ANY mask but some
masks are worse than others!
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So, what do the Deltas tell us?
Better to construct the optimum with these masks
than these symmetric BBS preferred
Recombination is particularly good in trying to
construct these O(2) BBs because of their tight
linkage!
Recombination is good for ANY mask but some
masks are better than others!
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So, what do the Deltas tell us?
Splitting up landscape blocks that are also BBs
is very BAD
Getting the optimum from recombining BBs that
arent landscape blocks isnt good
Note no sign changes
Getting the optimum from recombining BBs that
are also landscape blocks is good. Preference
for the mask 0011, the only one that respects the
landscape blocks
Recombination is good for SOME masks but BAD for
others, and this depends on the landscape!
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And for finite populations?
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2-point crossover, popsize 13, 1000
repetitions
The more crossover the better it gets!
The hard part here is to find the BBs in the
first place. Lots of crossover helps with that.
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Better to cut at block boundaries
Lots of crossover gives random search (or worse)
2-point crossover, popsize 13, 100
repetitions
Here mutation first finds the blocks then
crossover joins them together
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2-point crossover, popsize 25, 100 reps
Mutation is bad once youve got the BBs
easier to get O(1) BBs!
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Conclusions

• Recombination works by joining together BBs (not
  the BBH ones!) thats the only way it works
• Objects have BBs which have their BBs which
• BB basis is the appropriate mathematical
  description of recombination along with the SWLD
  coefficients
• Can glean qualitative information from the
  infinite population equations that is also valid
  for finite populations
• Recombination is only absolutely good or bad in
  the extreme siutations of maximum and minimum
  epistasis, and even then its good if you dont
  have the string/schema you want
• In other cases it depends on the fitness
  landscape and especially its modularity
• It seems to be particularly beneficial in
  modular landscapes

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Conclusions

• Instead of asking if recombination is good or bad
  better to ask what is a good recombination
  distribution
• If recombination distributions are allowed to
  evolve they will do so to respect landscape
  modularity
• Possible explanation for recombination hotspots
• Coevolution of recombination hotspots and modular
  landscapes
• Remember that a gene is a building block, O(1)
  in terms of loci but O(thousands) in terms of
  nucleotides
• Modularity can be lots of intragene epistasis but
  weak intergene epistasis
• Difference between counting ones (nucelotides)
  versus counting ones (genes)