E-Approximate Coded Matrix Multiplication is Nearly Twice as Efficient as Exact Multiplication

Viveck R. Cadambe, Flávio P. Calmon, Ateet Devulapalli, Haewon Jeong

Abstract: We study coded distributed matrix multiplication from an approximate recovery viewpoint. We consider a system of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P$</tex> computation nodes where each node stores 1/m of each multiplicand via linear encoding. Our main result shows that the matrix product can be recovered with ∊ relative error from any <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m$</tex> of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P$</tex> nodes for any ∊ >0. We obtain this result through a careful specialization of MatDot codes-a class of matrix multiplication code previously developed in the context of exact recovery (∊ = 0). Since previous results showed that the MatDot code is tight for a class of linear coding schemes for exact recovery, our result shows that allowing for mild approximations leads to a system that is nearly twice as efficient as exact reconstruction. Moreover, we develop an optimization framework based on alternating minimization that enables the discovery of new codes for approximate matrix multiplication.