Polar Coded Computing: The Role of the Scaling Exponent

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 µ of the family of codes. The scaling exponent has emerged as a central object in the finitelength 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/µ ), 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/µ ) and O(n −1 ), respectively, and provide a heuristic argument as well as numerical evidence supporting this view.

Sub-4.7 Scaling Exponent of Polar Codes

“…Seven: Over wiretap channels, polar code achieves the secrecy capacity but consumes secrete keys shared between Alice and Bob; the scaling exponent gives prediction on the length of the secrete key [WU16;GB17]. Eight: For coded computation, scaling behavior is related to not only the code rate but also the waiting time [FM22].…”