Structure of search space, complexity of stochastic combinatorial optimization algorithms and applic

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Structure of search space, complexity of
stochastic combinatorial optimization algorithms
and application to biological motifs discovery
Robin Gras INRIA Rennes
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• Black box global combinatorial optimization
• Search of the maximums of a function F (fitness)
  with integer variables.
• Search space C set of all legal
  instantiations.
• Global the search space is the Cartesian
  product of the domains of the variables.
• Black box the analytical form of F is not
  known but its value is computable in each point.

A lot of difficult bioinformatics problems (most
of them are NP-hard) can be represented as black
box combinatorial optimisation problems.
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• A few definitions
• Operator o Move (exploration) from a set of
  points Ec (a sample) to another set of points.
• Neighborhood Vo (given an operator) set of all
  reachable points from a set Ec by one application
  of the operator.
• Landscape triplet (C,F,Vo)
• Local maximum Mo (given an operator) X is a
  local maximum given o iff
• Metaheuristic heuristic exploration algorithm
  for black box combinatorial optimization problems.

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What is complexity? All difficult problems are
not equal
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• Influence of the maximums
• Basin of attraction of Mo the set of points of
  C from which Mo is reachable by a sequence of
  applications of o.
• Study of basins of attraction.
• Study of fitness cloud (variation of fitness
  values in function of fitness values).

Results (empirical) Number of global maximums
compared with the size of C Number of global
maximums compared with number of local
maximums Size and overlap of basins of
attractions Linked to the neighborhood!
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• Problem decomposability epistasie
• A problem of size n which is decomposable in n/k
  independent sub-problems of size k has a
  complexity in 2k.n/k
• Epistasie maximum number of variables of which
  depend each of the n variables level of
  non-linearity ?measures of non-linearity.
• Example spin glasses
• Function with a tunable level of epistasie
  NK-landscape
• With N the size of the problem and K the number
  of dependencies

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• Fitness and instantiation deceptive functions
• Function built to be difficult for hill-climber.
• Function that is almost linear except for a few
  points of the search space.
• Example trap5 function

Trap5(X)
Independent of the neighborhood!
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Efficiency of evolutionary approaches. Two main
strategies - A priori definition of the
operators - Discovery of the operators
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• Classic genetic algorithms (Holland 75)
• Exploration by sampling.
• Population sample
• Evaluation and bias by selection
• Generation of a new population by application of
  operators
• Experimental studies of the efficiency and the
  behavior ?various conclusions.
• Theoretical studies.
• Convergence proofs in simple cases (Goldberg
  87).
• Introduction to the notion of schemes (Bagley
  67)
• Computation of the efficiency on deceptive
  problems (Goldberg 93)

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• Probabilistic model building algorithms
• Principle discovering the dependencies and the
  structure using the sample.
• First model univariate distribution of the
  variables (Muhlenbein et Voigt 96).
• BOA Bayesian network (Pelikan et Golderg 99).
• hBOA Bayesian network decision graph
  ecological niches (Pelikan et Goldberg 00).
• building of the network
• Population generation
• Population size
• Number of generations
• Global complexity
• Validation on spin glasses, MAXSAT and deceptive
  functions.

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• Still limitations
• Convergence proofs do not take into account the
  strong heuristic (greedy) used to build the
  network.
• The quality measure is not computed at each
  step.
• High global computation cost.
• ?What are the consequences on the real
  efficiency of hBOA?

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Efficiency of evolutionary algorithms on high
epistasie level deceptive problems
Problem of size 120 and 100 with sub-functions of
size between 4 and 12
Deceptive function
Non deceptive function
Adjacent Configuration Xi xi, xim, ,
xi(k-1)m with i ?1, ,m
Non-adjacent configuration Xi x(i-1).k1, x
(i-1).k2, , x (i-1).kk with i ?1, ,m
Classical genetic algorithm
Non-adjacent configuration 100 of failure !
Classic genetic algorithm with adjacent
configuration
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hBOA behavior

• Adjacent configuration
• When the epistasie level is above 6 the global
  maximum is never reached in 100 generations.
• Computation time became prohibitive more than
  60 hours with epistasie of 12.
• High dependency with the structure of the
  deceptive function.
• Non-adjacent configuration
• Same results so hBOA is not dependent of the
  configuration .
• ? real capacity to detect and handle the
  dependencies but the heuristics used do not allow
  the obtaining of demonstrated results.

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• Simple PMBGA algorithm dedicated to deceptive
  problems
• Test of several measures on bivariate
  frequencies only frequencies, conditional
  probability, mutual information and statistical
  implication.
• Algorithm taking into account the deceptive
  property
• Random generation of the initial population
• Tournament selection
• Computation of the measures for each couple of
  variables
• Building of a solution

Direct obtaining of the solution in each case
(adjacent or not) in few minutes! With mixed
sub-functions (deceptive and non-deceptive) Imposs
ible to discover the solution. hBOA no
difference with mixed sub-functions
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• Conclusions on black box problems
• Easy if there is no dependency.
• As soon as the epistasie level is 2 or above and
  there is overlap between dependencies the problem
  is NP-hard.
• In general the problem is NP-hard when the
  dependencies are not known.
• Two possible approaches
• Expert Use of expert knowledge about the
  problem to build pertinent neighborhood and
  operators.
• Automatic discovery of the dependencies by
  sampling and model building. Limited by the
  number of dependencies and necessity of new
  heuristics.

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• Further works new algorithms
• Definition of a more structured benchmark than
  NK-landscape.
• New PMBGA algorithm
• New probabilistic model.
• New quality measure for the model.
• More efficient heuristics for the model
  building.
• Learning of the number of dependencies.
• PMBGP algorithm
• Specialization for non-linear regression.
• Use of dependencies discovery.