@inproceedings{teytaud:inria-00451416, hal_id = {inria-00451416}, url = {http://hal.inria.fr/inria-00451416}, title = {{Bias and variance in continuous EDA}}, author = {Teytaud, Fabien and Teytaud, Olivier}, abstract = {{Estimation of Distribution Algorithms are based on statistical estimates. We show that when combining classical tools from statistics, namely bias/variance decomposition, reweighting and quasi-randomization, we can strongly improve the convergence rate. All modifications are easy, compliant with most algorithms, and experimentally very efficient in particular in the parallel case (large offsprings).}}, language = {Anglais}, affiliation = {TAO - INRIA Futurs , Laboratoire de Recherche en Informatique - LRI , TAO - INRIA Saclay - Ile de France}, booktitle = {{EA 09}}, address = {Strasbourg, France}, audience = {internationale }, year = {2009}, month = May, pdf = {http://hal.inria.fr/inria-00451416/PDF/decsigma.pdf}, }

1. Why one must
use reweighting
F. Teytaud, O. Teytaud
Montréal, 2009
Tao, Inria Saclay Ile-De-France, LRI (Université Paris Sud, France),
UMR CNRS 8623, I&A team, Digiteo, Pascal Network of Excellence.

2. Outline
Idea of averaging in evolutionary
algorithms
This idea introduces a bias
How to remove this bias
The results
Conclusions
Teytaud and Teytaud Gecco 09 is great 2

3. Idea in ES
Average of selected points
= good approximation of
optimum
Teytaud and Teytaud Gecco 09 is great 3

4. Idea in ES
Let's assume this is true (for the
moment)...
nonetheless, there's a bias.
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5. EMNA (P. Larranaga and J.-A. Lozano, 2001)
While (not finished)
- generate population
- select best individuals
- estimate mean / variance
(and possibly covariance)
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6. EMNA (P. Larranaga and J.-A. Lozano, 2001)
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7. EMNA (P. Larranaga and J.-A. Lozano, 2001)
While (not finished)
- generate population
- select best individuals
- estimate mean / variance (and possibly covariance)
Highly parallel (more than most ES; T. et al, EvoStar 2001)
Very simple
Can handle covariance matrix easily
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8. Please wake up during 3 slides :-)
Idea of averaging in evolutionary algorithms
This idea introduces a bias
How to remove this bias
The results
Conclusions
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9. Bias due to bad (Gaussian) distribution
High
density
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10. Bias due to bad distribution
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11. Bias due to bad distribution
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12. Bias due to bad distribution
AVERAGE
(biased by the
distribution)
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13. Corrected by weighted average
AVERAGE
(the one we really
want !)
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14. Outline
Idea of averaging in evolutionary algorithms
This idea introduces a bias
How to remove this bias
The results
Conclusions
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15. American Election of 1936 (fun)
Literary digest: pop size = 2 000 000
==> predicts Landon
Gallup: pop size = 50 000
==> predicts Roosevelt (and was proved right)
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16. American Election of 1936
Literary digest: pop size = 2 300 000
==> predicts Landon
Gallup: pop size = 50 000
==> predicts Roosevelt (and was proved right)
The Literary digest failed because of a biased sampling.
(much more affluent people and much more republicans
among Literary Digest readers)
Correction: Weight of individual = real density / biased density.
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17. REMNA (reweighted EMNA)
Inverse Gaussian density
Gaussian density (=weight for removing the bias!)
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18. REMNA (reweighted EMNA)
Very simple modification:
- compute weight ( individual ) = 1 / density
- compute mean, variance, covariance
with these weights
==> not only for Gaussians
==> ok for all surrogate models / EDA
==> just an application of standard statistics
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19. REMNA
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20. Outline
Idea of averaging in evolutionary algorithms
This idea introduces a bias
How to remove this bias
The results: less premature convergence
Conclusions
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21. The results
We do not prove that
“there's no more premature convergence.”
We just show that, for a fixed generation,
“ IF the center of the level set is the optimum,
THEN
the asymptotic value of the estimated optimum
= the optimum.”
==> is the condition really necessary ?
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22. Yes: center = optimum and Yes: situation better
Yes and yes
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23. No: center ≠optimum but Yes: situation better
(assumption not really
necessary)
No... but yes
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24. Results: convergence rate with  = d 2
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25. Conclusions
Idea of averaging in evolutionary algorithms
This idea introduces a bias
How to remove this bias
The results
Conclusions
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26. Conclusions
Reduces the risk of premature convergence
No proof on the complete algorithm (just
step-wise consistency)
Empirically quite good for EMNA (should be
tested on other EDA / surrogate)
Simple, sound, widely applicable
Bias of step-size adaptation not yet
analyzed (==> seemingly works quite well!)
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27. Related work
Papers from D.V. Arnold et al. around
reweighting for improved convergence
rate (sphere, ridge) of ES
(to be combined ?)
Work from CMA-people around weights for
improved conv. rate in CMA-ES
Thanks! Questions ?
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