Newton-like Polynomial-Coded Distributed Computing for Numerical Stability

Abstract

For coded distributed computing (CDC), polynomial code is one prevalent encoding method for CDC (called Poly-CDC). It suffers from poor numerical stability due to the Vandermonde matrix serving as the coefficient matrix which needs to be inverted, and whose condition number increases exponentially with the size of the matrix or equivalently with the number of parallel worker nodes. To improve the numerical stability, especially for large networks, we propose a Newton-like polynomial code (NLPC)-based CDC (NLPC-CDC), with a design dedicated for both matrix–vector and matrix–matrix multiplications. The associated proof of the constructed code possesses a (n,k)-symmetrical combination property (CP), where symmetrical means the worker nodes have identical computation volume, CP means the k-symmetrical original computing tasks are encoded into n(n≥k)-symmetrically coded computing tasks, and the arbitrary k resulting from the n-coded computing tasks can recover the intended computing results. Extensive numerical studies verify the significant numerical stability improvement of our proposed NLPC-CDC over Poly-CDC.