TY - JOUR
T1 - Numerically Stable Polynomially Coded Computing
AU - Fahim, Mohammad
AU - Cadambe, Viveck R.
N1 - Funding Information:
Manuscript received May 22, 2019; revised September 7, 2020; accepted December 26, 2020. Date of publication January 11, 2021; date of current version April 21, 2021. This work was supported by NSF under Grant CCF 1763657. This article was presented in part at the 2019 IEEE International Symposium on Information Theory (ISIT). (Corresponding author: Mohammad Fahim.) Mohammad Fahim was with the Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802 USA. He is now with Qualcomm Technologies, Inc., San Diego, CA 92121 USA (e-mail: mfahim@qti.qualcomm.com).
Publisher Copyright:
© 1963-2012 IEEE.
PY - 2021/5
Y1 - 2021/5
N2 - We study the numerical stability of polynomial based encoding methods, which has emerged to be a powerful class of techniques for providing straggler and fault tolerance in the area of coded computing. Our contributions are as follows: 1)We construct new codes for matrix multiplication that achieve the same fault/straggler tolerance as the previously constructed MatDot Codes and Polynomial Codes.2)We show that the condition number of every m times m sub-matrix of an m times n, n geq m Chebyshev-Vandermonde matrix, evaluated on the n -point Chebyshev grid, grows as O(n{2(n-m)}) for n > m.3)By specializing our orthogonal polynomial based constructions to Chebyshev polynomials, and using our condition number bound for Chebyshev-Vandermonde matrices, we construct new numerically stable techniques for coded matrix multiplication. We empirically demonstrate that our constructions have significantly lower numerical errors compared to previous approaches which involve inversion of Vandermonde matrices. We generalize our constructions to explore the trade-off between computation/communication and fault-tolerance.4)We propose a numerically stable specialization of Lagrange coded computing. Our approach involves the choice of evaluation points and a suitable decoding procedure. Our approach is demonstrated empirically to have lower numerical errors as compared to standard methods.
AB - We study the numerical stability of polynomial based encoding methods, which has emerged to be a powerful class of techniques for providing straggler and fault tolerance in the area of coded computing. Our contributions are as follows: 1)We construct new codes for matrix multiplication that achieve the same fault/straggler tolerance as the previously constructed MatDot Codes and Polynomial Codes.2)We show that the condition number of every m times m sub-matrix of an m times n, n geq m Chebyshev-Vandermonde matrix, evaluated on the n -point Chebyshev grid, grows as O(n{2(n-m)}) for n > m.3)By specializing our orthogonal polynomial based constructions to Chebyshev polynomials, and using our condition number bound for Chebyshev-Vandermonde matrices, we construct new numerically stable techniques for coded matrix multiplication. We empirically demonstrate that our constructions have significantly lower numerical errors compared to previous approaches which involve inversion of Vandermonde matrices. We generalize our constructions to explore the trade-off between computation/communication and fault-tolerance.4)We propose a numerically stable specialization of Lagrange coded computing. Our approach involves the choice of evaluation points and a suitable decoding procedure. Our approach is demonstrated empirically to have lower numerical errors as compared to standard methods.
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U2 - 10.1109/TIT.2021.3050526
DO - 10.1109/TIT.2021.3050526
M3 - Article
AN - SCOPUS:85099594024
VL - 67
SP - 2758
EP - 2785
JO - IRE Professional Group on Information Theory
JF - IRE Professional Group on Information Theory
SN - 0018-9448
IS - 5
M1 - 9319171
ER -