Polynomial Codes: an Optimal Design for High-Dimensional Coded Matrix Multiplication

TL;DR

This paper introduces polynomial codes, a novel distributed computation strategy for matrix multiplication that minimizes the number of worker nodes needed for recovery, improving efficiency and optimality over existing methods.

Contribution

The paper presents polynomial codes, a new coding-based approach that achieves the optimal recovery threshold and extends to distributed convolution, outperforming prior techniques.

Findings

Achieves the minimum possible recovery threshold for distributed matrix multiplication.

Provides an efficient polynomial interpolation-based reconstruction method.

Extends the coding strategy to distributed convolution with similar optimality.

Abstract

We consider a large-scale matrix multiplication problem where the computation is carried out using a distributed system with a master node and multiple worker nodes, where each worker can store parts of the input matrices. We propose a computation strategy that leverages ideas from coding theory to design intermediate computations at the worker nodes, in order to efficiently deal with straggling workers. The proposed strategy, named as \emph{polynomial codes}, achieves the optimum recovery threshold, defined as the minimum number of workers that the master needs to wait for in order to compute the output. Furthermore, by leveraging the algebraic structure of polynomial codes, we can map the reconstruction problem of the final output to a polynomial interpolation problem, which can be solved efficiently. Polynomial codes provide order-wise improvement over the state of the art in terms of recovery threshold, and are also optimal in terms of several other metrics. Furthermore, we extend this code to distributed convolution and show its order-wise optimality.