This document summarizes a simple real-coded estimation of distribution algorithm (RECGA). The algorithm discretizes a real-valued population into intervals, models the discrete population using a greedy maximum probability model search, and generates a new population from the model. The population size scales sub-quadratically and the number of function evaluations scales sub-cubically with problem size for this simple RECGA. Next steps proposed include more experiments on scalability analysis, the relationship between discretization and performance, and developing theory on convergence for real-coded genetic algorithms.

1. A Simple Real-Coded ECGA
Luca Fossati, Pier Luca Lanzi, Kumara Sastry,
David E. Goldberg, Osvaldo Gomez
Politecnico di Milano, Italy
Illinois Genetic Algorithms Laboratory,
University of Illinois at Urbana Champaign, USA
OBUPM GECCO'07, July 7--11, 2007, London, UK.

2. Real-coded EDAs
are complex and difficult to analyze
What is our goal?
The simplest real-coded EDA possible
Elementary discretization + χECGA

3. ECGA
New
Selection
Population Population
MPM Model

4. Simple Real Coded ECGA
Restrict Tournament Replacement (RTR)
Real-Valued New Real-Valued
Population Selection Population
Intervals
Ii,j
χECGA
X-ary New X-ary
…
Population Population

5. Simple Real Coded ECGA
k = # of intervals
rp = real population
dp = discrete population
1: procedure RECGA(k)
Ii,j is the j-th interval for gene i
2: rp ← random();
3: Generate a random population rp
4: Evaluate the fitness in rp
5: while stop criterion not true do
6: Undergo tournament selection at a rate S
7: Discretize rp into dp using k and generate Ii,j
8: Model dp using a greedy MPM search
9: If the model has converged, stop
10: Generate a new dp+1 using the model
11: Generate a new rp+1 from dp+1 using Ii,j
12: rp ← ApplyRTR(rp+1,rp)
13: Evaluate the fitness in rp
14: end while
15: end procedure

6. Number of Evaluations for k=5

7. Population Size for k=5

8. Number of Evaluations for k=10

9. Population Size for k=10

10. Number of Evaluations
as Function of k

11. Population Size
as Function of k

12. Class of additively separable problems
The population size scales
sub-quadratically with problem size
The number of function evaluations scales
sub-cubically with problem size
Simple, amenable for further empirical and
theoretical analysis
First step towards a systematic
analysis of real-coded ECGA

13. What next?
More experiments
Scalability analysis
Relation between discretization
and performance
… virtual alphabets?

14. Virtual Alphabets
(Goldberg, 1991)
Theory of convergence for real-coded GAs
Selection
Dominates early GA performance
Restricts subsequent search to intervals
with above average fitness
It does it, dimension by dimension
Intervals form the characters of a virtual
alphabet, searched during recombination

15. Virtual Alphabets

16. Blocking
x1

17. Simple Real-Coded ECGA
Explicitly builds the alphabet
Virtual alphabets & RECGA?
Blocking & Model Building?

18. Thank you!
Any question?