Learning Classifier Systems

1
Learning Classifier Systems

• Navigating the fitness landscape?
• Why use evolutionary computation?
• Whats the concept of LCS?
• Early pioneers
• Competitive vs Grouped Classifiers
• Beware the Swampy bits!
• Niching
• Selection for mating and effecting
• Balance exploration with exploitation.
• Balance the pressures
• Zeroth level classifier system
• The X-factor
• Alphabet soup
• New Slants - piecewise linear approximators
• Why don't LCS rule the world?
• Simplification schemes
• Cognitive Classifiers
• Neuroscience Inspirations
• Application Domains

2
Representation

• Genetic information can be any symbol.
• Require dictionary and rules to manipulate the
  symbols.
• Some symbols make the search space easier to
  explore/exploit
• Some symbols are easier to store, manipulate and
  test
• Different problems may be suited to different
  symbol sets

3
States

• Environmental state is passed to the LCS via the
  message
• Often the environmental state is preprocessed to
  create the message
• Values are normalised
• Out of bound data removed
• Known irrelevant conditions removed
• Missing values addressed

4
Multiple Representations

• Binary, ternary, Grey or enumerated.
• Integer, real, floatingpoint or mantisssa
• Rank, order, series, histogram or array
• Bounded, ellipsoidal
• Horn clauses and second-order logic
• S-type expressions, Gene expressions
• Hybrid fuzzy sets, neural networks
• Piecewise linear approximation
• Problem specific

5
Encoding

• Binary (including gray) is a very crude method to
  solve real valued problems.
• Simply, divide the range of interest by the
  number of intervals encoded by the binary
  representation and determine the real number that
  the binary interval represents. e.g. 0000
  represents 0.0, 1111 represents 6.23 and 0001
  represents 6.23/15.
• The more bits, the more accuracy in your
  solution, but the limitations of this encoding
  are obvious.

6
Schema

• 1
• 1

7
Non-optimum Niching

• rule x lt 4 ? 0 00
• rule x lt 3 ? 0 00 0111
• rule x lt 5 ? 0 ???

8
Binary enumeration for Niching

• Can use enumeration
• 0 ? 0000000
• 1 ? 0000001
• 2 ? 0000011
• 3 ? 0000111
• 4 ? 0001111
• 5 ? 0011111
• 6 ? 0111111
• 7 ? 1111111
• rule x lt 4 ? 0 00
• rule x lt 3 ? 0 00
• rule x lt 5 ? 0 00
• Useful in discretised environments
• Trades increased search space less compactness
  for better Niching

9
Integer and Real Encodings

• Match encoding to the environmental message
  using upper and lower bounds
• could use centre and spread, but this assumes a
  gaussian distribution and recombination more
  difficult to implement
• For each allele a, lb x ub, to give match.
• Could use lt instead of but lcs determines
  the correct bound automatically
• 0 x 5 is equivalent to 0 x lt 5.01 or
• 0 x 4.99 is equivalent to 0 x lt 5

10
Mutating at the Limits

• Crossover point can either be between alleles or
  in the middle of an allele.
• Mutation increases/decreases either or both of
  the two bounds.
• Repair is occasionally needed to check that
  lower bound lt upper bound
• Note that most bounds have a limit,
• e.g. WBC 0 a 10
• We probabilistically decide to mutate the general
  allele 0 x 10 lower bound
• We decrease by 10 of range to -1
• we then repair back to 0.
• We increase by 10 of range to 1
• we do not repair it as valid!
• Thus some alphabets have a specificity bias

11
Hyper Partitioning

• We have a sparse search space with only two
  classes to identify 0 1
• Its real numbered so we decide to use bounds
  e.g. 0 x 10
• We form Hypercubes with the number of dimensions
  the number of conditions
• Approximates actual niches, maybe problems

. 1 . 0 N(x) S
. 1 . 0 N(x) S
12
Oblique domains

• We have a search space with only two classes to
  identify 0 1
• Its real numbered so we decide to use bounds
  e.g. 0 x 10
• We form Hypercubes / Hyperrectangles are not
  often suited to oblique domains
• Imagine sine wave domains..

. 1 . 0 S
13
Hyper-ellipsoidal

• The general ellipsoid, also called a triaxial
  ellipsoid, is a quadratic surface which is given
  in Cartesian coordinates by
• where the semi-axes are of lengths a, b, and c.
  Wolfram maths
• N-dimensional ellipsoids can be used to more
  effectively represent oblique domains
• Implementation and analysis becomes harder
• Kernel-based, ellipsoidal conditions in the
  real-valued XCS classifier system Butz, M.V.
  Proceedings of the Genetic and Evolutionary
  Computation Conference (GECCO-2005) pp.
  1835-1842.
• Hyper-ellipsoidal conditions in XCS Rotation,
  linear approximation, and solution structure
  Butz, M. V., Lanzi, Pier-Luca, Wilson, S. W.
  Proceedings of the Genetic and Evolutionary
  Computation Conference (GECCO-2006) pp. 1457-1464

14
Horn clause logic

• A Horn clause is a clause with at most one
  positive literal.
• A rule 1 positive literal, at least 1 negative
  literal. A rule has the form "P1 V P2 V ... V
  Pk V Q". This is logically equivalent to
  "P1P2 ... Pk gt Q" thus, an if-then
  implication with any number of conditions but one
  conclusion. Examples "man(X) V mortal(X)" (All
  men are mortal)
• A fact or unit 1 positive literal, 0 negative
  literals. Examples "man(socrates)", (Everyone is
  an ancestor of themselves (in the trivial
  sense).)
• A negated goal 0 positive literals, at least 1
  negative literal. In virtually all
  implementations of Horn clause logic, the negated
  goal is the negation of the statement to be
  proved the knowledge base consists entirely of
  facts and goals. The statement to be proven,
  therefore, called the goal, is therefore a single
  unit or the conjuction of units an existentially
  quantified variable in the goal turns into a free
  variable in the negated goal. E.g. If the goal to
  be proven is "exists (X) male(X)
  ancestor(elizabeth,X)" (show that there exists a
  male descendent of Elizabeth) the negated goal
  will be "male(X) V ancestor(elizabeth,X)".
• The null clause 0 positive and 0 negative
  literals. Appears only as the end of a resolution
  proof.

15
Horn clause logic

• A Horn clause is a clause with at most one
  positive literal.
• a V b V c V V t V u (only u positive)
  or
• (a b c t) ? u (equivalent
  implication)
• A definite clause is a Horn clause that has
  exactly one positive literal.
• A Horn clause without a positive literal is
  called a goal.
• Horn clauses express a subset of statements of
  first-order logic.
• Prolog is built on top of Horn clauses.
• Prolog programs are comprised of definite clauses
  and any question in Prolog is a goal.
• Strict set of operations and useful scaling
  properties.
• FOIL Produces Horn clauses from data expressed
  as relations
• Quinlan, J.R. (1990), "Learning Logical
  Definitions from Relations", Machine Learning 5,
  239-266.
• http//www.cs.cmu.edu/afs/cs/project/ai-repository
  /ai/areas/learning /systems/0.html

16
Fuzzy

• Use of Fuzzy sets to encode membership of message
  to classifier
• LCS encodes membership function of each allele
• Casillas, J. Carse, B. Bull, L. (2007) Fuzzy
  XCS a Michigan Genetic Fuzzy System. IEEE
  Transactions on Fuzzy Systems 15(4) 536-550.

Membership 1 COLD
WARM HOT Temperature
17
Neural

• Use of Neural Networks to encode membership of
  message to classifier
• LCS encodes one NN of each allele.
• Number of rules and composition of rules learnt.
• Three optimised NNs avoid, seek and orientate
• Jacob Hurst, Matt Studley UWE

18
S-Expressions
Lisp like expressions For people who speak in
brackets
( (- sqrt(- ( b b) ( ( 2 2) ( a c)))) b) (
2 a))
Genetic Programming inspired (see work by P-L
Lanzi) Logical functions AND, OR
,NOT Terminals 0, 1, , NULL Exponential/Logs
e, ln, log, Hyperbolic Sine, Cos,
Tan Mathematical SQRT, POW Tailored Value
at, Address of, ... Need to tailor match and
reproduction NB is ternary alphabet a subset?
http//www.genetic-programming.com/
19
S-Expressions
Lisp like expressions For people who speak in
brackets
( (- sqrt(- ( b b) ( ( 2 2) ( a c)))) b) (
2 a))
Bloat? Is a LCS with S-expressions not just
GP? How to tailor functions without introducing
bias? How to identify building blocks of
Subexpressions? When are two Subexpressions
equivalent? Is trade-off between reduced problem
search space to increased alphabet search space
worth it?
http//www.genetic-programming.com/