The document discusses Adaptable Constrained Genetic Programming (ACGP), which aims to automate the discovery of heuristics to guide the genetic programming search. It describes how ACGP develops first-order and second-order heuristics based on patterns observed in high-performing individuals, and uses these heuristics to bias mutation, crossover and regrowth. Experimental results on a target equation with explicit second-order structure show that ACGP with second-order heuristics outperforms both standard GP and ACGP with only first-order heuristics. The document concludes that ACGP is effective at discovering and exploiting problem structure through its adaptive heuristic approach.

1. Second Order Heuristics in ACGP John Aleshunas University of Missouri – Saint Louis jja7w2@umsl.edu Cezary Janikow University of Missouri – Saint Louis janikow@umsl.edu Mark W Hauschild University of Missouri – Saint Louis mwh308@umsl.edu

2. Outline The Genetic Programming (GP) Model Issues with the GP Model Constrained Genetic Programming (CGP) Adaptable Constrained GP (ACGP) Adaptable Constrained GP (ACGP) Heuristics ACGP Learning Results Conclusions 2

3. The Genetic Programming (GP) Model Evolve a solution program from a population of candidates Represents individual solutions as variable size trees Employs standard genetic operators Selection Crossover Mutation 3

4. Issues with the GP Model The Function set and Terminal set (labels) define the representation space of a GP implementation Often the Function set and Terminal set are over specified to ensure that they cover the solution space Closure Sufficiency This over specification increases the representation space and increases search time 4

5. Issues with the GP Model Search is not completely biased Mutation is always random Crossover is random given the current population (driven by the current population but random there) Selection is the only bias Selection drives the population and thus affects crossover How to improve the bias? Building “models” and using them to drive the search 5

6. Constrained Genetic Programming (CGP)An Option to Reduce Complexity Developed by Dr. Cezary Janikow in 1996 It uses user-specified “model” 1st order heuristics parent-child Global at the root Local anywhere Demonstrated to dramatically improve search time and complexity for problems such as SantaFe and 11-multiplexer But the user has to give the “right” heuristics – discovery of the right heuristics can be a long process 6

7. Adaptable Constrained GP (ACGP)Automating the CGP Methodology Developed by Dr. Cezary Janikow in 2003 ACGP discovers 1st order heuristics Assumes that best trees are the “correct” heuristics Adjusts the heuristics at generational intervals Regrows the population using the new heuristics Mutation and crossover are now biased using the heuristics Uses gradual adjustments of the heuristics to control convergence 7

8. GP Building BlocksA Digression 8 The building block hypothesis states: Genetic processes work by combining relatively fit, short schema to form complete solutions. a b

9. Adaptable Constrained GP (ACGP)1st Order Heuristics Controls one argument to a function Develops a set of heuristics Looks at parent node (function) and one child node (function or terminal) Treats those schema that are frequent in the fittest population members as desirable Gradual heuristic adjustment prevents premature convergence 9

10. Adaptable Constrained GP (ACGP)2nd Order Heuristics Controls all arguments to a function Develops a set of heuristics Looks at parent node (function) and all child nodes (function or terminal) Treats those schema that are frequent in the fittest population members as desirable Gradual heuristic adjustment prevents premature convergence 10

11. Adaptable Constrained GP (ACGP)2nd Order Heuristics Consider x*x 1st order can only speak of * and one argument at a time Suppose that x* is 10% and that *x is 10% In the absence of heuristics about the entire subtree x*x, the probability of x*x will be joint as 1% But the true probability of x*x in the problem can be up to 10% Processing 1st order will always process it as 1% We need 2nd order (subtree) to process this as 10% We need Mechanism to discover 1st and 2nd order structures Validate performance gains of 2nd order vs. 1st order 11

12. Where is the Advantage in ACGP? Discovery of fit building blocks Promotes the use of fit building blocks in crossover, mutation and regrow Creates a population of highly fit compact solutions ACGP discovers heuristics that improve fitness primarily but also reduce tree sizes 12

13. How ACGP Develops Its Heuristics Tabulate the frequency of building blocks in the population Tabulate the frequency of building blocks in the fittest individuals in the population Increase the probability of fit heuristics Suppress the probability of less fit heuristics Access to heuristics is constant because of the tabulation Disregarding paging 13

14. Representation of Heuristics in ACGP1st Order Heuristics The general form of the 1st order combinations is Using 4 binary functions, 14 terminals, this creates 162 combinations 14

15. Representation of Heuristics in ACGP2nd Order Heuristics The general form of the 2nd order combinations is Using 4 binary functions, 14 terminals, this creates 2592 combinations 15

16. Representation of Heuristics in ACGPHigher Order Heuristics The general form of the 3rd order combinations is Using 4 binary functions, 14 terminals, this creates 13,728,800 combinations 4th OH 3.76 * 1014 combinations 5th OH 2.84 * 1029 combinations 6th OH 1.62 * 1059 combinations 16

17. ACGP 2nd order Implementation was painfully validated But experiments with Santafe and 11-multiplexer using 1st order and 2nd order gave the same results Means that these problems do not have 2nd order structure not implied by joint probabilities For validation, needed a function with obvious difference between 1st and 2nd order probabilities 17

18. Experimental Setup Target Equation: x*x + y*y + z*z Has very explicit 2nd order structure different from that implied by 1st order Function set: {+, -, *, /} (protected divide) Terminal set: {x, y, z, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5} Population size: 500 Generations: 500 Operators: crossover 85%, mutation 10%, selection 5%, regrow 100% at each iteration Number of independent runs: 30 Fitness: sum of square errors on 100 random data points in the range -10 to 10 Iteration length: 20 generations 18

19. ACGP Learning Results 19 Find the equation Comparison of GP-Base, ACGP with 1st order and 2nd order heuristics. Training slope-off with 50% update

20. ACGP Learning Results 20 Find the equation Comparison of GP-Base, ACGP with 1st order and 2nd order heuristics. Training slope-on

21. ACGP Learning Results 21 Find the equation Comparison of GP-Base, ACGP with 1st order and 2nd order heuristics. Training slope-on, first five iterations

22. ACGP Learning Results 22 Find the equation Comparison of GP-Base, Strong 1st Order and Strong 2nd Order Heuristics on 200 generation, no iterations

23. ACGP Learning Results 23 Find the equation Comparison of Base, 1st Order and 2nd Order Heuristics on a Time Scale

24. ACGP Learning Results 24 Find the equation Average tree structure for Base, 1st Order and 2nd Order Heuristics

25. ACGP Learning Limits 25 Find the equation Comparison of Base, 1st Order and 2nd Order Heuristics with Training Slope on for an extended Bowl3 equation

26. ACGP Learning Results 26 Desirable 1st order heuristics Implied 2nd order heuristics Desirable 2nd order heuristics {*1 , x} {*2 , x} {*1 , z} {*2 , z} {*2 , y} {*1 , y} {+1 , *} {+1 , +} {+2 , +} {+2 , *} {*, x, x} {*, x, z} {*, y, x} {*, y, z} {*, x, y} {*, y, y} {*, z, x} {*, z, z} {*, z, y} {+, *, +} {+, *, *} {+, +, *} {*, x, x} {*, y, y} {*, z, z} {+, +, +} {+, *, +} {+, *, *} {+, +, *}

27. ACGP Learning Results 27 1st order heuristics {*1 , x} {*2 , x} {*1 , z} {*2 , z} {*2 , y} {*1 , y} 0.2289 0.2426 0.2403 0.2323 0.2489 0.2087 ACGP 1 2nd order heuristics Implied ACGP 1 0.0532 0.0570 0.0478 0.0604 0.0506 0.0502 0.2545 0.0001 0.0001 0.2368 0.0001 0.2436 {*, x, x} {*, x, z} {*, y, y} {*, z, z} {*, x, y} {*, y, z} Actual ACGP 2

28. ACGP Learning Results 28 First-order heuristics discovered. Root’s heuristics are zero-order for Bowl3 equation

29. ACGP Learning Results 29 Second-order heuristics summary for ‘*’ for Bowl3 equation

30. ACGP Learning Results 30 Second-order heuristics summary for ‘+’ for Bowl3 equation

31. Conclusions and Further Work 31 The 1st and 2nd order heuristics of ACGP provide a significant performance improvement over the base GP If very strong second-order heuristics are present, ACGP is able to process them and also to discover them Running 2nd order doesn’t hurt if there is no explicit 2nd order not implied by 1st Overhead is minimal If a problem does not have explicit second-order structure, ACGP running in first or second-order mode is the same Going higher than 2nd order has drawbacks. The next step will be to move on to standard test or real world problems

32. Summary The Genetic Programming (GP) Model Issues with the GP Model Constrained Genetic Programming (CGP) Adaptable Constrained GP (ACGP) Adaptable Constrained GP (ACGP) Heuristics ACGP Learning Results Conclusions 32

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