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Modeling Selection Pressure in XCS for Proportionate and
Tournament SelectionAlbert Orriols-Puig1,2 Kumara Sastry2
Pier Luca Lanzi1,3 David E. Goldberg2
Ester Bernadó-Mansilla1
1Research Group in Intelligent SystemsEnginyeria i Arquitectura La Salle, Ramon Llull University
2Illinois Genetic Algorithms LaboratoryDepartment of Industrial and Enterprise Systems Engineering
University of Illinois at Urbana Champaign
3Dipartamento di Elettronica e InformazionePolitecnico di Milano
Slide 2GRSI Enginyeria i Arquitectura la Salle
Motivation
Facetwise modeling to permit a successful understanding of complex systems (Goldberg, 2002)
– Model of generalization pressures of XCS (Butz, Kovacs, Lanzi, Wilson,04)
– Learning time bound (Butz, Goldberg & Lanzi, 04)
– Population size bound to guarantee niche support (Butz, Goldberg, Lanzi & Sastry, 07)
Slide 3GRSI Enginyeria i Arquitectura la Salle
Motivation
Analysis of selection schemes in XCS: Proportionate vs. Tournament
– Tournament selection is more robust to parameter settings and noise than proportionate selection (Butz, Goldberg & Tharakunnel, 03) and (Butz, Sastry, Goldberg, 05)
– Proportionate selection is, at least, as robust as tournament selection if the appropriate fitness separation is used (Karbat, Bull & Odeh, 05)
In GA, these schemes were studied through the analysis of takeover time (Goldberg & Deb, 90; Goldberg, 02)
Slide 4GRSI Enginyeria i Arquitectura la Salle
Aim
Model selection pressure in XCS through the analysis of the takeover time
– Consider that XCS has converged to an optimal solution
– Write differential equations that describe the change in proportion of the best individual
– Solve the equations and derive a closed form solution
– Validate the model empirically
Slide 5GRSI Enginyeria i Arquitectura la Salle
Outline
1. Description of XCS
2. Modeling takeover time
3. Comparing the two models
4. Experimental validation
5. Modeling generality
6. Conclusions
Slide 6GRSI Enginyeria i Arquitectura la Salle
Description of XCS
Representation:
– fixed-size rule-based representation
– Rule Parameters: Pk, εk, Fk, nk
Learning interaction:
– At each learning iteration Sample a new example
– Create the match set [M]
– Each classifier in [M] votes in the prediction array
– Select randomly an action and create the action set [A]
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 7GRSI Enginyeria i Arquitectura la Salle
Description of XCS
Rule evaluation: reinforcement learning techniques.
– Prediction:
– Prediction error:
– Accuracy of the prediction:
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 8GRSI Enginyeria i Arquitectura la Salle
Description of XCS
Rule evaluation: reinforcement learning techniques.
– Relative accuracy:
– Fitness computed as a windowed average of the accuracy:
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 9GRSI Enginyeria i Arquitectura la Salle
Description of XCS
Rule discovering:
– Steady-state, niched GA
– Population-wide deletion
Proportionate selection
• Probability proportionate to rule’s fitness.
Tournament selection
• Selects τ percent of classifiers from [A]
• Selects the classifier with higher microclassifier fitness
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 10GRSI Enginyeria i Arquitectura la Salle
Outline
1. Description of XCS
2. Modeling takeover time
3. Comparing the two models
4. Experimental validation
5. Modeling generality
6. Conclusions
Slide 11GRSI Enginyeria i Arquitectura la Salle
Modeling Takeover Time
In GA:
– Usually, the fitness of an individual is constant
– Selection and replacement are performed over the whole population
In XCS:
– Fitness depends on the other rule’s fitness in the same niche
– Selection is niched-based, whilst deletion is population-wide
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 12GRSI Enginyeria i Arquitectura la Salle
Modeling Takeover Time
Assumptions in our model– XCS has evolved a set of non-overlapping niches
– Simplified scenario: niche with two classifiers cl1 and cl2.
– cl1 is the best rule in the niche: k1 > k2
– Classifier clk has:
• prediction error εk
• fitness Fk
• numerosity nk
• microclassifier fitness fk
– cl1 and cl2 are equally general Same reproduction opportunities
– We assume niched deletion
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 13GRSI Enginyeria i Arquitectura la Salle
Proportionate Selection
Fitness is an average of cl1 and cl2 respective accuracies
Then, the probability of selecting the best classifier cl1 is:
Probability of deletion: Pdel(clj) = nj/n where n=n1+n2
Num. ratio: nr = n2/n1
Accuracy ratio: ρ=k2/k1
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 14GRSI Enginyeria i Arquitectura la Salle
Proportionate Selection
Evolution of cl1 numerosity
1. The numerosity of cl1 increases if cl1 is selected by the GA and another classifier is selected to be deleted
2. The numerosity of cl1 decreases if cl1 is not selected by the GA but it is selected by the deletion operator
3. The numerosity of c1 remain de same otherwise.
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 15GRSI Enginyeria i Arquitectura la Salle
Proportionate Selection
Grouping the above equations we obtain
Rewritten in terms of proportion of classifiers cl1 in the niche:
Considering Pt+1 – Pt ≈ dp/dt
Pt = n1/n
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Integrate: Initial proportion: P0 Final proportion: P
Slide 16GRSI Enginyeria i Arquitectura la Salle
Proportionate Selection
This gives us that the takeover time for proportionate selection is guided by the following expression:
P0: initial proportion of cl1 in the niche
P: final proportion of cl1 in the niche
ρ: accuracy ratio between cl2 and cl1
n: niche size
If ρ 1: trws ≈ ∞
If ρ 0:
A higher separation between fitness enables a higher ability in identifying accuraterules, as announced by Karbat, Bull & Odeh, 2005.
n
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 17GRSI Enginyeria i Arquitectura la Salle
Tournament Selection
Assumptions
– Fixed tournament size s
– cl1 is the best classifier in the niche: f1 > f2 , that is, F1/n1 > F2 /n2
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 18GRSI Enginyeria i Arquitectura la Salle
Tournament Selection
Evolution of cl1 numerosity
1. The numerosity of cl1 increases if cl1 participates in the tournament and another classifier is selected to be deleted
2. The numerosity of cl1 decreases if cl1 does not participate in the tournament but it is selected by the deletion operator
3. The numerosity of c1 remain de same otherwise.
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 19GRSI Enginyeria i Arquitectura la Salle
Tournament Selection
This gives us that the takeover time for tournament selection is guided by the following expression:
P0: initial proportion of cl1 in the niche
P: final proportion of cl1 in the niche
s: tournament size
n: niche size
Tournament selection does not depend on the accuracy ratio between the best classifier and the others in the same [A], as pointed by Butz, Sastry & Goldberg, 2005
As s increases, this expression decreases
It does not depend on the individual fitness
For large s:
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 20GRSI Enginyeria i Arquitectura la Salle
Outline
1. Description of XCS
2. Modeling takeover time
3. Comparing the two models
4. Experimental validation
5. Modeling generality
6. Conclusions
Slide 21GRSI Enginyeria i Arquitectura la Salle
Proportionate vs. Tournament
Values of s and ρ for which both schemes result in the same takeover time. Require: t*RWS = t*TS
– We obtain
For P0 = 0.01 and P = 0.99
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 22GRSI Enginyeria i Arquitectura la Salle
Outline
1. Description of XCS
2. Modeling takeover time
3. Comparing the two models
4. Experimental validation
5. Modeling generality
6. Conclusions
Slide 23GRSI Enginyeria i Arquitectura la Salle
Design of Test Problems
Single-niche problem
– One niche with 2 classifiers:
• Highly accurate classifier cl1:
• Less accurate classifier cl2:
• Varying ρ we are changing the fitness separation between cl1 and cl2
– Population initialized with
• N · P0 copies of cl1
• N · (1 – P0) copies of cl2
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 24GRSI Enginyeria i Arquitectura la Salle
Results on the Single-Niche Problem
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Accuracy ratio: ρ= 0.01
RWS
Tournament s=9
Tournament s=3
Tournament s=2
Slide 25GRSI Enginyeria i Arquitectura la Salle
Results on the Single-Niche Problem
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Accuracy ratio: ρ= 0.50
RWSTournament s=9
Tournament s=3
Tournament s=2
Slide 26GRSI Enginyeria i Arquitectura la Salle
Results on the Single-Niche Problem
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Accuracy ratio: ρ= 0.90
RWS
Tournament s=9
Tournament s=3
Tournament s=2
Slide 27GRSI Enginyeria i Arquitectura la Salle
Design of Test Problems
Multiple-niche problem
– Several niches with 1 maximally accurate classifier each niche.
– One over-general classifier that participates in all niches
– The population contains:
• N · P0 copies of maximally accurate classifiers
• N · (1 – P0) copies of the overgeneral classifier
– The problem violates two assumptions of the model
• Overlapping niches
• The size of the different niches differ from the population size
– Deletion can select any classifier in [P]
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 28GRSI Enginyeria i Arquitectura la Salle
Results on the Multiple-Niche Problem
For small ρ the theory slightly underestimates the empirical takeover time The model of proportionate selection is accurate in general scenarios if the ratio of accuracies is small
In situations where there is a small proportion of the best classifier in one niche competing with other slightly inaccurate and overgeneral, the overgeneral may take over the population.
Further experiments, show that for ρ > 0.5, the best classifiers is removed from the population
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
RWS ρ = 0.01
RWS ρ = 0.20
RWS ρ = 0.30
RWS ρ = 0.40
RWS ρ = 0.50
Slide 29GRSI Enginyeria i Arquitectura la Salle
Results on the Multiple-Niche Problem
For high s the theory slightly underestimates the empirical takeover time
The model of tournament selection is accurate in general scenarios if the tournament size is high enough
Only in the extreme case (s=2), the experiments strongly disagree with the theory
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
TS s = 2
TS s = 3
TS s = 9
Slide 30GRSI Enginyeria i Arquitectura la Salle
Outline
1. Description of XCS
2. Modeling takeover time
3. Comparing the two models
4. Experimental validation
5. Modeling generality
6. Conclusions
Slide 31GRSI Enginyeria i Arquitectura la Salle
Modeling Generality
Scenario
– The best classifier cl1 appears in the niche with probability 1
– cl2 appears in the niche with probability ρm
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 32GRSI Enginyeria i Arquitectura la Salle
Proportionate Selection
The takeover time for proportionate selection is guided by the following expression:
P0: initial proportion of cl1 in the niche
P: final proportion of cl1 in the niche
ρ: accuracy ratio between cl2 and cl1
ρm: occurrence probability of cl2
N: niche size
If cl1 is either more accurate or more general than cl2, cl1 will take over the population.
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 33GRSI Enginyeria i Arquitectura la Salle
Tournament Selection
The takeover time for tournament selection is guided by the following expression:
P0: initial proportion of cl1 in the niche
P: final proportion of cl1 in the niche
s: tournament size
ρm: occurrence probability of cl2
n: niche size
For low ρm or high s the right-hand logarithm goes to zero, so that the takeover timemainly depends on P0 and P
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Slide 34GRSI Enginyeria i Arquitectura la Salle
Results of the Extended Model onthe one-niched Problem
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
RW
S
To
urn
amen
t
Slide 35GRSI Enginyeria i Arquitectura la Salle
Outline
1. Description of XCS
2. Modeling takeover time
3. Comparing the two models
4. Experimental validation
5. Modeling generality
6. Conclusions
Slide 36GRSI Enginyeria i Arquitectura la Salle
Conclusions
We derived theoretical models for proportionate and tournament under some assumptions
– Models are exact in very simple scenarios
– Models can qualitatively explain both selection schemes in more complicated scenarios
Models support that tournament is more robust (Butz, Sastry & Goldberg, 2005)
Fitness separation is essential to guarantee that the best classifier will take over the population in proportionate selection (Karbhat, Bull & Oates, 2005)
Models show that proportionate selection depends on fitness scaling. It may fail in domains where there are slightly inaccurate classifiers (real-world domains)
1. Description of XCS
2. Modeling Takeover Time
3. Comparing the two Models
4. Experimental Validation
5. Modeling Generality
6. Conclusions
Modeling Selection Pressure in XCS for Proportionate and
Tournament Selection
Albert Orriols-Puig1,2 Kumara Sastry2
Pier Luca Lanzi2 David E. Goldberg2
Ester Bernadó-Mansilla1
1Research Group in Intelligent Systems
Enginyeria i Arquitectura La Salle, Ramon Llull University
2Illinois Genetic Algorithms LaboratoryDepartment of Industrial and Enterprise Systems Engineering
University of Illinois at Urbana Champaign