1. Estimation of Distribution Algorithm based on Probabilistic Grammar with Latent Annotations
Written by Yoshihiko Hasegawa and Hitoshi Iba
Summarized by Minhyeok Kim

2. Contents Introduction PCFG-LA
Two groups in GP-EDA
PCFG-LA
PCFG and PCFG-LA
probability of the annotated tree
Probability of a observed tree
Log-likelihood and Update formula
Assumptions
Forward-backward probability
P(T;Θ) by forward and backward
Parameter update formula
Initial parameters
PAGE(Programming with Anntated Grammar Estimation)
Experiment
Royal Tree Problem
DMAX Problem
Conclusion

3. Two groups in GP-EDA Proto-type tree based method PCFG based method
It translates variable length tree-structures into fixed length structures
PCFG based method
It is considered to be well suited for expressing functions in GP
It’s production rules do not depend on the ascendant nodes or sibling nodes
It can not take into account the interactions among nodes

4. PCFG-LA(1/10) - PCFG and PCFG-LA
0.7 VP → V NP
0.3 VP → V NP NP
PCFG-LA
PCFG + Latent annotations

5. PCFG-LA (2/10) -Probability of the annotated tree
The probability of the annotated tree
T : derivation tree
xi : annotation of ith non-terminal (all the non-terminals are numbered from the root)
X = {x1, x2, ...}
π(S[x]) : probability of S[x] at the root position
β(r) : probability of annotated production rule r
DT[X] : multi-set of used annotated rules in tree T
Θ : set of parameters Θ = {π, β}.

6. PCFG-LA (3/10) -Probability of a observed tree
The probability of a observed tree
It can be calculated with summing over annotations
The parameters (π and β) have to be estimated by EM algorithm

7. PCFG-LA (4/10) -Log-likelihood and Update formula
The difference of log-likelihood between parameters Θ’ and Θ
the update formula can be obtained by optimizing Q(Θ’|Θ)

8. PCFG-LA (5/10) -Assumptions
Using not CNF but GNF
To reduce the number of parameters, assume that all right-side non-terminal symbols have the same annotation

9. PCFG-LA (6/10) -Forward-backward probability(1/2)
Backward probability biT(x)
The probability that the tree beneath ith non-terminal S[x] is generated
Forward probability fiT(y)
The probability that the tree above ith non-terminal S[y] is generated

10. PCFG-LA (7/10) -Forward-backward probability(2/2)
Forward probability
ch(i,T) :function which returns the set of non-terminal children indices of ith non-terminal in T
pa(i, T) : returns a parent index of ith non-terminal in T
giT :terminal symbol in CFG and is connected to ith non-terminal symbol in T

11. PCFG-LA (8/10) -P(T;Θ) by forward and backward
cover(g,Ti) : function which returns a set of non-terminal indices at which the production rule generating g without annotations is rooted in Ti

12. PCFG-LA (9/10) -Parameter update formula
By using the forward-backward probability and optimizing Q(Θ’|Θ)

13. PCFG-LA (10/10) -Initial parameters
EM algorithm maximizes the log-likelihood monotonically from the initial parameters
Initial parameters
κ : random value which is uniformly distributed over [−log 3, log 3]
γ(S → g S....S) : probability of observed production rule (without annotations)
β(S[x] → g S[x]...S[x]) = 0.

14. PAGE (Programming with Anntated Grammar Estimation)
Flowchart
Initialization of individuals
Evaluation of individuals
Selection of individuals
Estimation of parameters
Generation of new individuals

15. Experiment -Royal Tree Problem
Each function has increasing arity
a has 1 arity, b has 2,…
Perfect tree whose level is smaller by 1 level than the level
P-tree of level c is composed of the P-tree of level b
Level d royal tree problem in this experiments
To compare the performance between PAGE and PCFG-GP

17. Experiment -DMAX Problem
MAX problem + deceptiveness
{＋m,×m}, {λ,0.95}, λr=1
Depth 4, m(arity) 5, r(power) 3
To show the superiority over simple GP

19. Conclusion
In the royal tree problem, we showed that the number of annotations greatly affects the search performance and larger annotation size offered better performance
The result of DMAX problem showed that PAGE is highly effective for the problem with strong deceptiveness
PAGE uses EM algorithm, so it is more computationally expensive
The performance of PAGE is much more superior than none-annotation algorithm
It is important to optimize these two contradicting factors which will be examined in the future work