$\epsilon$-Approximate Coded Matrix Multiplication is Nearly Twice as Efficient as Exact Multiplication

We study coded distributed matrix multiplication from an approximate recovery viewpoint. We consider a system of $P$ 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 $\epsilon$ relative error from any $m$ of the $P$ nodes for any $\epsilon > 0$. We obtain this result through a careful specialization of MatDot codes -- a class of matrix multiplication codes previously developed in the context of exact recovery ($\epsilon=0$). Since prior results showed that MatDot codes achieve the best exact recovery threshold for a class of linear coding schemes, our result shows that allowing for mild approximations leads to a system that is nearly twice as efficient as exact reconstruction. As an additional contribution, we develop an optimization framework based on alternating minimization that enables the discovery of new codes for approximate matrix multiplication.