GENETIC ALGORITHMS FOR FAST MATRIX MULTIPLICATION

András Joó

Anikó Ekárt

Juan Neirotti

United Kingdom

14/07/2011GECCO 2011 HUMIES AWARDS

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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)

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GECCO 2011 HUMIES AWARDS

3nnn 12 nn

72logn n

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

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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!

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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

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GA RESULTS

On matrices of size

reproduced a solution with 23 multiplications

found an approximate solution of fitness 0.9978 for 22 multiplications

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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.

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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.

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