Price of Precision in Coded Distributed Matrix Multiplication: A Dimensional Analysis

Coded distributed matrix multiplication (CDMM) schemes, such as MatDot codes, seek efficient ways to distribute matrix multiplication task(s) to a set of N distributed servers so that the answers returned from any R servers are sufficient to recover the desired product(s). For example, to compute the product of matrices U, V, MatDot codes partition each matrix into $p\gt1$ sub-matrices to create smaller coded computation tasks that reduce the upload/storage at each server by $1 / p$, such that UV can be recovered from the answers returned by any $R=2 p-1$ servers. An important concern in CDMM is to reduce the recovery threshold R for a given storage/upload constraint. Recently, Jeong et al. introduced Approximate MatDot (AMD) codes that are shown to improve the recovery threshold by a factor of nearly 2, from $2 p-1$ to p. A key observation that motivates our work is that the storage/upload required for approximate computing depends not only on the dimensions of the (coded) sub-matrices that are assigned to each server, but also on their precision levels - a critical aspect that is not explored by Jeong et al. Our main contribution is a rudimentary asymptotic dimensional analysis of AMD codes inspired by the Generalized Degrees of Freedom (GDoF) framework previously developed for wireless networks, which indicates that for the same upload/storage, once the precision levels of the task assignments are accounted for, AMD codes are not better than a replication scheme which assigns the full computation task to every server. The dimensional analysis is supported by simple numerical experiments.