Gene Expression Messy GAs

1
Gene Expression Messy GAs

• Kargupta, et al
• Presented by
• Abhishek Singh

2
Underpinnings

• Extension of mGA
• Black Box Optimization
• Relevance of SEARCH
• Importance of intra-cellular information flow for
  SEARCH
• No explicit modeling
• Precursor to Model Building GAs

3
Overview

• SEARCH (some gory details)!
• Natural Evolution as a SEARCH
• Basic Gene Expression Messy GA
• Updates on the GEMGA
• Interspersed with Results and Achievements
• Summary and Conclusions
• Punctuated with Discussions and Debates
  (hopefully)

4
SEARCH overview

• Black Box Search
• Enumeration
• Induction
• Search Envisioned as Relation and Class
  Hierarchizing
• Framework to formalize sample complexity,
  difficulty, etc
• Relations, Class, and Samples

5
SEARCH contd.

• Enumerative search exponential
• Alternative - stochastic decision making based on
  sampling
• Consequence Premise
• Assumes inductive relationships
• No relations enumeration!
• Relations classify search domain
• Classes contain optima

6
SEARCH Decomposition

• Relation, R Set of ordered pairs
• Class, C Instantiation of a relations
• Sample, S Instance of a class
• Relations imposed implicitly or explicitly
• Representation
• Operators
• Heuristics
• Direct modeling
• Typical example for GAs follows

7
Relations, Classes, and Samples
1100 1001 1110 1011 0100 0011 0000 0111
f f
1 0 1 0
Relation Class Sample space space
space
8
SEARCH components

• Classification based on relations
• Sampling
• Evaluation, ordering, selection of better
  classes
• Evaluation, ordering selection of better
  relations
• Resolution

9
A little bit of theory!

• ri ith Relation
• ?r Set of all Relations
• Ci Set of classes created by ri,, Ci Ni
• P Perturbation operator (dumb or smart)
• T, Tr Class/relation comparison statistic
• Mi Best from Ci (depends on decision error
  probability)
• Sr Best used from ?r
• Ci , ri Sampled ordered class/relation

10
SEARCH challenges

• Based on T some classes are ordered and best Mi
  are selected
• Pruning of classes search (resolution)
• Relation must classify space such that optimal,
  Ci is within T based Mi
• Ci and Ci may not be same
• Ci must be in Misampled
• Based on Tr relations are ordered and selected
• Pruning of relations search based on past

11
SEARCH challenges contd.

• Ordering dependent on sampling
• SEARCH fails if
• Defining relation is such that for chosen T, CI
  ? Mi
• Stochastic error causes CI ? Misampled

12
Quantifying the Challenges

• Relation Selection success
• Pr(CRS ri ) ? Pr( rk ?Tr ri)min?r- Sr
• Class Selection success
• Pr(CCS ri ) ? Pr( Ck ?T Ci)minNi Mi
• Overall success
• ? Pr(CRSri)Pr(CCSri) ?ri ? Sr

13
Specializing the Equation

• Specialize to a specific class comparison
  statistic and representation
• Similar to earlier work on decision making
  (Goldberg, et al, 1992)
• Order statistics used

14
Ordinal Class Selection

• Prob. of correct
• binary decision
• Pr(FT j,i ?a FT k,i)
• ? 1-2nH(a) (a d)a n
• d - zone of indifference
• F CDF of F
• a Quantile of CDF
• n no. of samples
• H Binary entropy func.

15
Class Selection contd.

• For correct class selection
• Pr(CCSri) ? 1-2nH(a) (a d)a n Ni Mi
• d min F(FT ,i) - F(FT j,i) ?j

16
Ordinal Relation Selection

• Analysis same as for class selection except Tr
  used for comparison.
• For one relation ri
• Pr(CRS ri)? 1-2nr H(ar) (ar dr)ar nr
  ?r-?g
• Overall success probability bound

17
Overall Success

• Combine the search for better classes and
  relations
• q Overall success probability
• d Bound for min d over all classes
  compared to optima containing class
• Nmax maxNi ?ri ?Sr
• Mmin minMi ?ri ?Sr

18
Sample Complexity

• Total number of Function Evaluations
• For non-relational enumerative SC becomes the
  size of search space
• To bound SEARCH, Nmax and Sr need to be bound
  by polynomials

19
Order-k delineable Problems

• Order of a relation 0(ri) defined as log of
  number of defined classes
• If o(ri) o(l) then exponential Nmax
• For polynomial search
• Bound O(ri)?k
• Bound Sr O(poly(l))
• This defines a Generalized Order-k delineable
  problem
• Polynomial bound means simple relations can
  capture solutions!

20
Milestone 1

• Discussed the basic motivation behind GEMGA
• SEARCH challenges (déjà vu)
• Need for Relational search
• Enumeration bounds any BBS
• Bound on Relation set cardinality for polynomial
  search
• Next step SEARCH implementation as GEMGA

21
Food for thought

• Linkage specific spatial operator-defined
  relation/problem
• Precursors to model builders
• Work relates to earlier and contemporary work on
  problem difficulty and decision making
• Break

22
Recap

• Introduced SEARCH (in some detail)
• SEARCH challenges (intuitive and quantified)
• Order statistics for bounds on SEARCH complexity
• Without hierarchical search BBS is too expensive
• Restrict sampling and use Intelligent Guessing

23
Nature Questions and answers

• Natural evolution evolved fitter (?) organisms
• 3x108 base pairs in humans implies HUGE search
  space
• Without a priori knowledge evolution is a BBS
• Not enough time for enumerative search

24
Some Questions

• Problem of adequate time
• Shapiro Junkyard Tornedo!
• Holland Schema processing
• Goldberg Problem Decomposition
• Kauffman Gene Expression
• Problem of selection space
• Problem of recombination
• Recombination good if we know what to combine
• Natural recombination different from GAs

25
Evolution as Information Flow

• Extracellular storage, exploration and
  transmission within generations
• Intracellular Gene expression
• Most GAs model extracellular flow
• What about intracellular mechanisms (introns,
  diploidy, gene expression etc)

26
Expression Mechanisms

• Transcription DNA mRNA
• Translation mRNA proteins
• Protein folding
• Protein Phenotype
• Lets see these in some detail

27
Transcription

• Initiated and terminated by a specific sequence
  of genes
• RNA polymerase transcribes portion in between
  (AGCT AGCU)
• Regulatory proteins bind to DNA portions and
  control transcription
• Gene activator
• Gene repressor

28
Translation

• mRNA is template for protein formation
• (61) Base triplets correspond to (20) amino acids
• 3 for promotion and termination
• Many to one mapping
• Regulated by control systems of repressors,
  promoters, and operators
• Specialization within cells

29
Protein folding

• 3-D structure determines protein function
  (phenotype)
• This defines fitness space
• Phenotypic Genotypic correspondence

30
Intracellular Information Flow
31
The SEARCH perspective

• Sample space
• DNA population
• Class Space Amino acid sequences
• mRNA correspond to DNA schemas
• mRNA translate to amino acid seq.
• Define equivalence class
• Relation space Regulatory mech.
• Transcription process defines classes
• Transcription controlled by extra and intra DNA
  components (feed back loop)

32
SEARCH answers questions (?)

• Time
• Nature searched for relations too
• Selection
• Feed back loops apportion selection pressure
• Recombination
• Resolution of classes
• Representation (diploidy, introns etc)

33
Doubts, concerns

• Concept of natural optimality
• Evolution as adaptation not SEARCH
• Temporal and Spatial niches

34
Messy GAs

• Separate relation/class space from sample space
• Deterministically processed order k relations and
  classes

35
Messy GAs contd.

• Relation still not separate from class
• Sample space consisted of one template
• No implicit parallelism thus expensive
• fmGA tackled this issue (still problems of cross
  competition between classes from different
  relations)

36
GEMGA overview

• Messy GA continued
• SC is for order-k problem, of length l, and
  alphabet ?
• Separates relation, class, and sample space
• Explicit relation learning
• Many changes and updates

37
Representation

• Messy schemes maintained (locus, value)
• Gene has additional variables
• Weights, linkage lists, capacity
• Start with simple GEMGA with only weights
  (initialized to 1)
• No under/over-specification

38
Population Sizing

• C is signal/noise coefficient
• At least one instance of optimal order-k class in
  population of ?k
• similar structure to previous equations
• mGA was O(( ?k) l )
• Order of relations processed
• 2l relations but only O(k) processed

39
Basic Operators

• Transcription
• Selection
• Class Selection
• String Selection
• Recombination
• Each in detail

40
Transcription

• Detects appropriate order-k relations
• Relations need to be compared
• GEMGA processes relations in distributed manner
• Every chromosome evaluates its genes for instance
  of good class
• Quality of good classes determine quality of
  relation

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Transcription continued

• Flip each gene
• Note change in Fitness function
• Fitness Increases
• Gene not part of good class
• Make weight Zero
• Fitness Decreases
• Gene may be part of good class
• Make weight ?fitness
• repeat for C lt ?

42
Selection

• Class Selection
• Grow better classes
• Gene with higher weight overwrites one with lower
  weight on other string
• String Selection
• Binary Tournament Selection

43
Recombination

• Randomly pick two strings
• Consider all genes for swapping with some
  probability
• If weight of gene is greater than corresponding
  gene then swap it
• What does this do?
• Preserve tight linkage

44
The Algorithm

• Primordial Phase
• l generations (all genes considered)
• Juxtapositional Phase
• Selection and recombination applied
• Every chromosome converges to optimal class when
  
• Substituting n
• Overall SC O( ?k (lk))
• Solution quality?

45
Results

• Tested over uniform l bit trap functions of
  length l
• Order-l delineability needed
• Function evaluations grow linearly with l
• Population size is constant for constant l
• Scaling and Noise added

46
Results contd.
47
Milestone 2

• Natural BBS discussed
• SEARCH implemented as GEMGA to solve order-k
  delineable problems
• Polynomial time achieved
• Issues solved
• Relation (linkage) space searched
• Simplistic relation search mechanism
• Scope for improvements
• Similarity to hybrid GAs (local search)

48
GEMGA revisions

• Need for more explicit relation learning
• Linkage set added to gene representation
• Transcription extended
• And class selection linked with linkage

49
Linkage Set

• For each gene the set stores related genes in
  chromosome
• If genes 1,5,10,15 are related then linkage set
  for gene 1 is 5,10,15 and so on
• Linkage space over all genes defines relation
  space

50
Transcription II

• In addition to previous transcription operator
• Tries to identify the exact relations (construct
  the linkage set)
• Not very clear how!
• Transcription II applied for l2-l generations

51
Transcription II contd.

• Pick two points (with weight gt 0) on chromosome
• Keep original fitness value
• Perturb both genes
• If change of fitness ! change due to
  perturbation of single gene then genes are
  related
• Put them in the linkage set
• Change weights to 1

52
Class Selection II

• Here cardinality of linkage set decides gene
  growth
• Same as previous except genes with high
  cardinality overwrite lesser genes
• Linkage sets of genes with greater cardinality
  are copied with genes
• Weight becomes a criterion for gene consideration

53
Recombination

• Same as before except cardinality of linkage set
  is used
• Genes with larger linkage sets are chosen and
  exchanged
• This preserves linkage better

54
Algorithm Complexity

• Transcription I
• l generations
• Transcription II
• l2-l generations in worst case
• Juxtapositional phase
• Same as before O(k)
• Population O(?k )
• Total SC O(?k (l2k))
• Worse than before, but better linkage

55
Final GEMGA

• Linear Sample Complexity achieved
• Change in representation (again)
• Transcription II dropped but relation learning
  maintained
• Recombination and Expression combined

56
Representation

• Chromosome genesi , linki ?i?l
• Gene has locus, value, capacity (for improvement)
• Link has
• Linkage set Set of related genes
• Weights Number of particular linkage in
  population
• Goodness (0,1) How good the linkage
  is w.r.t. fitness contribution
• Trials Number of time linkage has been
  tried

57
Transcription

• Same as before except that capacity is set to 1
  if fitness after perturbation increases, else 0
• All genes with capacity 0 are put in the initial
  linkage set
• Continued for l generations

58
Recombination Expression

• Two phases
• Pre-recombination Expression
• GEMGA Recombination
• Pre-recombination determines related gene
  clusters
• GEMGA recombination ensures growth of proper
  classes and relations

59
Pre-recombination

• Applied several time during first generation
• Pair of chromosomes selected
• Of those in initial linkage set (ILS) genes with
  same values and capacities are extracted
• If this set is present in ILS then weight of
  linkage is increased by one
• If not then this set is added to ILS

60
Pre-recombination contd.

• Gives an lxl conditional probability matrix
• Mi,j prob(genes i and j together)
• Final Linkage Set constructed
• Max(Mi,j) ? j, calculated for i
• All genes with Mi,j within e of Max are included
  in linkage set for I

61
GEMGA recombination

• Element from linkage set of one chromosome chosen
  based on weight and goodness for swapping
• If goodness value of disrupted linkage set of
  other chromosome are less than this one then SWAP
  and adjust linkage set
• Goodness is set by change in fitness due
  recombination

62
GEMGA recombination contd.

• Of the two original chromosome and two recombined
  chromosomes two are selected based on goodness
  and fitness
• Apply iteratively over all pairs
• No fitness evaluations if fitness and disrupted
  fitness is stored

63
GEMGA analysis

• Transcription applied for l generations
• In Pre-recombination no fitness evaluations
• Population O(?k )
• SC O(?k ( l ))
• Linear growth of fitness evaluations and
  relational learning

64
Milestone 3

• GEMGA designed with strong linkage learning and
  linear time
• More complex relation building
• Tested on deceptive multi-modal functions to
  validate conclusions
• Could it be tested for tougher relations

65
Musings

• LLGA solved linkages sequentially, GEMGA solves
  parallely
• Multi-Objective optimization
• Niche specific SEARCH?
• Linkage is a specific relations
• Test on problems with relational complexity
  beyond deception
• Similarity to natural gene expression?
• Memory Drawbacks

66
Summary and Conclusion

• Need for relation based search
• GEMGA spans a class of methods that model
  relationships within individuals
• GEMGA shown to solve difficult problems
  efficiently
• Walsh analysis on GEMGA (Kargupta Park, 1999)
• Used on G.P. (Neill Ryan, 2000)