Coded Computing via Binary Linear Codes: Designs and Performance Limits

TL;DR

This paper explores the use of binary linear codes in distributed computing to minimize execution time, establishing connections with error probability analysis and demonstrating the effectiveness of Reed-Muller codes with low-complexity decoding.

Contribution

It introduces a novel framework linking coded distributed computing with error probability analysis of codes, and proposes efficient Reed-Muller decoding algorithms for practical implementation.

Findings

Binary random linear codes achieve optimal execution time bounds.

Good binary linear codes can match the performance of non-binary codes.

Reed-Muller codes with low-complexity decoding perform close to optimal.

Abstract

We consider the problem of coded distributed computing where a large linear computational job, such as a matrix multiplication, is divided into $k$ smaller tasks, encoded using an $(n,k)$ linear code, and performed over $n$ distributed nodes. The goal is to reduce the average execution time of the computational job. We provide a connection between the problem of characterizing the average execution time of a coded distributed computing system and the problem of analyzing the error probability of codes of length $n$ used over erasure channels. Accordingly, we present closed-form expressions for the execution time using binary random linear codes and the best execution time any linear-coded distributed computing system can achieve. It is also shown that there exist \textit{good} binary linear codes that not only attain (asymptotically) the best performance that any linear code (not necessarily binary) can achieve but also are numerically stable against the inevitable rounding errors in practice. We then develop a low-complexity algorithm for decoding Reed-Muller (RM) codes over erasure channels. Our decoder only involves additions, subtractions, {and inversion of relatively small matrices of dimensions at most $\log n+1$}, and enables coded computation over real-valued data. Extensive numerical analysis of the fundamental results as well as RM- and polar-coded computing schemes demonstrate the excellence of the RM-coded computation in achieving close-to-optimal performance while having a low-complexity decoding and explicit construction. The proposed framework in this paper enables efficient designs of distributed computing systems given the rich literature in the channel coding theory.