G ENETIC A LGORITHMS FOR F AST M ATRIX M ULTIPLICATION András Joó Anikó Ekárt Juan Neirotti United Kingdom 14/07/2011 GECCO 2011 H UMIES AWARDS 2

T HE P ROBLEM : R ECURSIVE M ATRIX M ULTIPLICATION  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) 14/07/ GECCO 2011 H UMIES AWARDS

K NOWN L IMITS  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 14/07/2011 GECCO 2011 H UMIES AWARDS 4

P RACTICAL S IGNIFICANCE  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! 14/07/2011 GECCO 2011 H UMIES AWARDS 5

O UR SOLUTION : P ARALLEL 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 14/07/2011 GECCO 2011 H UMIES AWARDS 6

GA R ESULTS On matrices of size  reproduced a solution with 23 multiplications  found an approximate solution of fitness for 22 multiplications 14/07/2011 GECCO 2011 H UMIES AWARDS 7

W HY H UMAN -C OMPETITIVE ?  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. 14/07/2011 GECCO 2011 H UMIES AWARDS 8

W HICH C RITERIA ?  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. 14/07/2011 GECCO 2011 H UMIES AWARDS 9