Straggler Mitigation in Distributed Matrix Multiplication: Fundamental Limits and Optimal Coding

We consider the problem of massive matrix multiplication, which underlies many data analytic applications, in a large-scale distributed system comprising a group of worker nodes. We target the stragglers' delay performance bottleneck, which is due to the unpredictable latency in waiting for slo...

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Published in:	IEEE transactions on information theory Vol. 66; no. 3; pp. 1920 - 1933
Main Authors:	Yu, Qian, Maddah-Ali, Mohammad Ali, Avestimehr, A. Salman
Format:	Journal Article
Language:	English
Published:	New York IEEE 01.03.2020 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	coded computing Coding Complexity Complexity theory Computer networks Convolution Data analysis Delays Distributed computing Encoding Fault tolerance Matrices (mathematics) matrix multiplication Multiplication Nodes Optimization Polynomials Recovery Redundancy straggler mitigation Task analysis
ISSN:	0018-9448, 1557-9654
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Abstract	We consider the problem of massive matrix multiplication, which underlies many data analytic applications, in a large-scale distributed system comprising a group of worker nodes. We target the stragglers' delay performance bottleneck, which is due to the unpredictable latency in waiting for slowest nodes (or stragglers) to finish their tasks. We propose a novel coding strategy, named entangled polynomial code, for designing the intermediate computations at the worker nodes in order to minimize the recovery threshold (i.e., the number of workers that we need to wait for in order to compute the final output). We demonstrate the optimality of entangled polynomial code in several cases, and show that it provides orderwise improvement over the conventional schemes for straggler mitigation. Furthermore, we characterize the optimal recovery threshold among all linear coding strategies within a factor of 2 using bilinear complexity, by developing an improved version of the entangled polynomial code. In particular, while evaluating bilinear complexity is a well-known challenging problem, we show that optimal recovery threshold for linear coding strategies can be approximated within a factor of 2 of this fundamental quantity. On the other hand, the improved version of the entangled polynomial code enables further and orderwise reduction in the recovery threshold, compared to its basic version. Finally, we show that the techniques developed in this paper can also be extended to several other problems such as coded convolution and fault-tolerant computing, leading to tight characterizations.
AbstractList	We consider the problem of massive matrix multiplication, which underlies many data analytic applications, in a large-scale distributed system comprising a group of worker nodes. We target the stragglers' delay performance bottleneck, which is due to the unpredictable latency in waiting for slowest nodes (or stragglers) to finish their tasks. We propose a novel coding strategy, named entangled polynomial code, for designing the intermediate computations at the worker nodes in order to minimize the recovery threshold (i.e., the number of workers that we need to wait for in order to compute the final output). We demonstrate the optimality of entangled polynomial code in several cases, and show that it provides orderwise improvement over the conventional schemes for straggler mitigation. Furthermore, we characterize the optimal recovery threshold among all linear coding strategies within a factor of 2 using bilinear complexity, by developing an improved version of the entangled polynomial code. In particular, while evaluating bilinear complexity is a well-known challenging problem, we show that optimal recovery threshold for linear coding strategies can be approximated within a factor of 2 of this fundamental quantity. On the other hand, the improved version of the entangled polynomial code enables further and orderwise reduction in the recovery threshold, compared to its basic version. Finally, we show that the techniques developed in this paper can also be extended to several other problems such as coded convolution and fault-tolerant computing, leading to tight characterizations.
Author	Yu, Qian Maddah-Ali, Mohammad Ali Avestimehr, A. Salman
Author_xml	– sequence: 1 givenname: Qian orcidid: 0000-0002-2034-5941 surname: Yu fullname: Yu, Qian email: qyu880@usc.edu organization: Department of Electrical Engineering, University of Southern California, Los Angeles, CA, USA – sequence: 2 givenname: Mohammad Ali orcidid: 0000-0002-3222-1874 surname: Maddah-Ali fullname: Maddah-Ali, Mohammad Ali email: maddah_ali@sharif.edu organization: Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran – sequence: 3 givenname: A. Salman surname: Avestimehr fullname: Avestimehr, A. Salman email: avestimehr@ee.usc.edu organization: Department of Electrical Engineering, University of Southern California, Los Angeles, CA, USA
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SubjectTerms	coded computing Coding Complexity Complexity theory Computer networks Convolution Data analysis Delays Distributed computing Encoding Fault tolerance Matrices (mathematics) matrix multiplication Multiplication Nodes Optimization Polynomials Recovery Redundancy straggler mitigation Task analysis
Title	Straggler Mitigation in Distributed Matrix Multiplication: Fundamental Limits and Optimal Coding
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