Iterative Refinement of Computational Circuits using Genetic Programming

1. Iterative Refinement of Computational Circuits using Genetic Programming Martin A. Keane Econometrics, Inc. Chicago, Illinois makeane@ix.netcom.com Matthew J. Streeter Genetic Programming Inc. Mountain View, California mjs@tmolp.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-x3/6+x5/120 Error of existing approximation: sin(x) – (x-x3/6+x5/120) Evolve approximation to error function New approximation is: (x-x3/6+x5/120) + [evolved approximation]

4. Iterative Refinement of Rational Polynomial Approximations to Functions Rfinal: 772.65 • Refinement of rational polynomial approximations to sin(x), [0, /2] • Large overall improvement • Overfitting on last iteration (spike between two points)

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. Results for Square Root Circuit Square root output Square root error Rfinal: 3.4830 • First iteration eliminates gap for 0 mV through 200 mV inputs • Second iteration provides miniscule improvement

8. Results for Squaring Circuit Squaring error Squaring output Rfinal: 8.4193

9. Results for Cubing Circuit Cubing error Cubing output Rfinal: 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 output Cubing error Rfinal: 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