1. Iterative Refinement of Computational Circuits using Genetic Programming Matthew J. Streeter Genetic Programming Inc. Mountain View, California mjs@tmolp.com Martin A. Keane Econometrics, Inc. Chicago, Illinois makeane@ix.netcom.com John R. Koza Stanford University Stanford, California koza@stanford.edu GECCO 2002, New York City, July 9-13

2. Overview Technique to iteratively refine solutions to problems over multiple runs Demonstrated on rational polynomial approximations Iterative refinement of computational circuits for squaring, square root, and cubing Refinement of recently-patented cubic signal generator

3. Basic technique Existing approximation to sin(x): x-x 3 /6+x 5 /120 Error of existing approximation: sin(x) – (x-x 3 /6+x 5 /120) Evolve approximation to error function New approximation is: (x-x 3 /6+x 5 /120) + [evolved approximation]

4. Iterative Refinement of Rational Polynomial Approximations to Functions IterationErrorE/E prev 04.554574e-6  13.483825e-8130.73 22.093228e-81.6643 31.039170e-82.0143 46.302312e-91.6489 52.200276e-63.4912e-4 Refinement of rational polynomial approximations to sin(x), [0,  /2] Large overall improvement Overfitting on last iteration (spike between two points) R final: 772.65

5. Iterative Refinement of Computational Circuits Output of multiple solutions combined through voltage adder Refinement (from scratch) of square root, squaring, cubing circuits Refinement of patented cubic signal generator

6. Control Parameters 20 node Beowulf cluster with 350 MHz Pentium II processors Total population size of 20,000 70% crossover, 20% constant mutation, 9% cloning, 1% subtree mutation 100 generations per iteration

7. Square root outputSquare root error First iteration eliminates gap for 0 mV through 200 mV inputs Second iteration provides miniscule improvement Results for Square Root Circuit IterationErrorE/E prev 011.835  13.44453.4359 23.3981.0137 R final: 3.4830

8. Squaring output Squaring error Results for Squaring Circuit IterationErrorE/E prev 046.84  15.8368.0260 25.56351.0490 R final: 8.4193

9. Cubing output Cubing error Results for Cubing Circuit IterationErrorE/E prev 022.312  120.1981.1047 217.5101.1535 317.0611.0263 R final: 1.3078

10. Efficiency of Iterative Refinement Process Don’t know when to stop an iteration and go to the next one (if ever) Not clear how to manage tradeoff between population size / number of iterations / iteration length Determining this empirically runs into usual difficulties

11. Refinement of Patented Cubic Signal Generator Cubic signal generator patent issued to Cipriani and Takeshian of Conexant Systems on December 12, 2000 Low voltage cubic signal generator works on 2V power supply Has about 7 mV average error over inputs between 0 and 1.26 V Evolved refinement maintains low voltage restriction

12. Results for Patented Cubic Signal Generator Cubing outputCubing error IterationErrorE/E prev 07.128  10.98737.2197 20.92361.0690 R final: 7.2197 Evolved refinement evens out error across range Second iteration revealed to produce large spikes using finer-grained simulation

13. Original and Refined Circuits Top box is original; bottom is refined

14. Conclusions Iterative refinement technique can be successfully applied to rational polynomial approximations and to computational circuits Technique can provide significant improvements to state-of-the-art computational circuits Possible applicability to other areas