1. Stochastic Grammar-Based Genetic Programming
SC-LAB Lee YunGeun

2. What is SG-GP Stochastic Grammar-Based Genetic Programming =
distribution-based evolution +
grammar-based genetic programming

3. Why is SG-GP used SG-GP remedies two main limitations of GP.
It allows for exploiting powerful background knowledge, such as dimensional consistency.
It successfully resists the bloat phenomenon, avoiding the intron growth.

4. Grammar-based GP

5. Example S = a*t*t/2+V0*t N = {< E >, < Op >, < V >}
T = {+, −, ×, /, a, t, V0 }
P =
S := < E > ;
< E > := < Op > < E > < E > | < V > ;
< Op > := + | − | ×| / ;
< V > := a | t | V0 ;

6. Example S = a*t*t/2+V0*t

7. Dimensionally-Aware GP
Physical Units
Quantity
Mass
Length
Time
Variables a 1 -2 V0 -1 t
Solution S

8. Dimensional constraints
The solution is expressed in displacement,
so the start symbol is defined as:
S := <E+0+l+0>;
< E > := < Op > < E > < E > | < V > ;
: : :
< E > := < Op > < E > < E > | < V > ;
:total 5^2 in this problem
< Op > := + | − | ×| / ;
< V > := a | t | V ;

9. Stochastic Grammar-Based GP
1. Representation of the Distribution
2. Generation of the Population
3. Updating the Distribution

10. 1. Representation of the Distribution
Each derivation di in a production rule is attached a weight wi
All Wi are initialized to 1.

11. 2. Generation of the Population
For each occurrence of a non-terminal symbol, all admissible derivations are determined from the maximum tree size allowed and the position of the current non-terminal symbol.
the selection of the derivation di is done with probability pi, where

12. Example S = a*t*t/2+V0*t
< E > := < Op > < E > < E >,0.8 | < V >,0.2 ;
< Op > := +,1.2 | −,1.4 | ×,0.6| / ,0.4;
< V > := a,1 | t,1 | V0,1 ;

13. 3. Updating the Distribution
All individuals in the current population have been evaluated.
The probability distribution is updated from the Nb best and Nw worst individuals according to the following rules:
– Let b denotes the number of individuals among the Nb best individuals that carry derivation di;
weight wi is multiplied by (1 + ε)^b
– Let w denotes the number of individuals among the Nw worst individuals that carry derivation di;
weight wi is divided by (1 + ε)^w;
– Last, weight wi is mutated with probability pm; the mutation either multiplies or divides wi by factor (1 + εm).

14. Parameters of SG-GP

15. Example S = a*t*t/2+V0*t Initial Wi
< E > := < Op > < E > < E >,1 | < V >,1 ;
< Op > := +,1 | −,1 | ×,1| / ,1;
< V > := a+0+1-2,1 | t+0+0+1,1 | V ,1 ;
If ε=0.001, Wb=3, Ww=3 and b=2,w=1 of operator +
wi <- 1* ( )^2/( )^1

16. Vectorial SG-GP vs Scalar SG-GP
Distribution vector Wi is attached to the i-th level of the GP trees (i ranging from 1 to Dmax).
This scheme is referred to as Vectorial SG-GP, as opposed to the previous scheme referred to as Scalar SG-GP.
The distribution update in Vectorial SG-GP is modified in a straightforward manner;
the update of distribution Wi is only based on the derivations actually occurring at the i-th level among the best and worst individuals in the current population.

17. Result
GP vs SG-GP

18. Result
Better results are obtained with a low learning rate and a sufficiently large mutation amplitude.
This can be interpreted as a pressure toward the preservation of diversity in the population.

19. Result The maximum derivation depth, Dmax
Too short, the solution will be missed.
Too large, the search will take a prohibitively long time.

20. Result
Vectorial SG-GP vs Scalar SG-GP

21. Result
Resisting the Bloat