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Genetic Algorithms for Fast Matrix Multiplication

Genetic Algorithms for Fast Matrix Multiplication. András Joó Anikó Ekárt Juan Neirotti United Kingdom. The Problem: Recursive Matrix Multiplication.

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Genetic Algorithms for Fast Matrix Multiplication

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  1. Genetic Algorithms for Fast Matrix Multiplication András Joó AnikóEkárt Juan Neirotti United Kingdom GECCO 2011 Humies awards

  2. The Problem: Recursive Matrix Multiplication • Standard algorithm for multiplying two square matrices of size requires multiplications and d additions • Strassen’s algorithm reduces the number of required multiplications to if is a power of 2 (1969) GECCO 2011 Humies awards

  3. Known Limits • For matrices of size at least 7 multiplications needed • For matrices of size at least 19 multiplications needed • Best known exact algorithm for size contains 23 multiplications GECCO 2011 Humies awards

  4. Practical Significance • An exact algorithm using 22 multiplications on matrices of size would be an improvement on the best known algorithm for this size • An exact algorithm using 21 multiplications on matrices of size would be an overall improvement on how recursive matrix multiplication is currently performed on large matrices • As the search space has size 2.25e+180 for 21 multiplications and 8.71e+188 for 22 multiplications, respectively, it is highly unlikely that a human or a simple algorithm would discover a solution! GECCO 2011 Humies awards

  5. Our solution: Parallel GA • Parallel island model, with unidirectional ring topology and migration • Steady-state elitist GA • Continuous real-valued representation • Variety of crossover and mutation operators • Periodic explicit enforcing of diversity GECCO 2011 Humies awards

  6. GA Results On matrices of size • reproduced a solution with 23 multiplications • found an approximate solution of fitness 0.9978 for 22 multiplications GECCO 2011 Humies awards

  7. Why Human-Competitive? • In 1976, J. D. Laderman published his article “A noncommutative algorithm for multiplying matrices using 23 multiplications” in the Bulletin of the American Mathematical Society . Others published equivalent algorithms. • The theoretically proven lower bound is 19 multiplications, but no exact algorithm with less than 23 multiplications is known to date. • Our GA approach could reproduce matrix multiplication algorithms using 23 multiplications and also led to an approximate algorithm requiring 22 multiplications. GECCO 2011 Humies awards

  8. Which Criteria? • B: The result is equal to or better than a result that was accepted as a new scientific result at the time when it was published in a peer-reviewed scientific journal. • D: The result is publishable in its own right as a new scientific result independent of the fact that the result was mechanically created. • F: The result is equal to or better than a result that was considered an achievement in its field at the time it was first discovered. • G: The result solves a problem of indisputable difficulty in its field. GECCO 2011 Humies awards

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