Polar Coded Computing: The Role of the Scaling Exponent

TL;DR

This paper explores how polar codes influence the average execution time in distributed computing, linking it to the scaling exponent, and shows potential for improved performance bounds with large kernels.

Contribution

It establishes a connection between the average execution time of polar codes and their scaling exponent, and suggests methods to improve bounds using large kernels.

Findings

The gap between polar codes and MDS codes is O(n^{-1/μ}).

Large kernels can improve the bound to roughly O(n^{-1/2}).

Numerical evidence supports potential further improvements to O(n^{-2/μ}) and O(n^{-1}).

Abstract

We consider the problem of coded distributed computing using polar codes. The average execution time of a coded computing system is related to the error probability for transmission over the binary erasure channel in recent work by Soleymani, Jamali and Mahdavifar, where the performance of binary linear codes is investigated. In this paper, we focus on polar codes and unveil a connection between the average execution time and the scaling exponent $\mu$ of the family of codes. The scaling exponent has emerged as a central object in the finite-length characterization of polar codes, and it captures the speed of convergence to capacity. In particular, we show that (i) the gap between the normalized average execution time of polar codes and that of optimal MDS codes is $O(n^{-1/\mu})$, and (ii) this upper bound can be improved to roughly $O(n^{-1/2})$ by considering polar codes with large kernels. We conjecture that these bounds could be improved to $O(n^{-2/\mu})$ and $O(n^{-1})$, respectively, and provide a heuristic argument as well as numerical evidence supporting this view.