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Remark 4.5. The key proof idea of Theorem 4.3 is twofold. We first demonstrate a one-to-one
correspondence between linear computation strategies and upper bound constructions7 for bilinear
complexity, which enables converting a matrix multiplication problem into computing the element-wise
product of two vectors of length R(p, m, n). Then we show that an optimal computation
strategy can be developed for this augmented problem, which achieves the stated recovery threshold.
Similarly to this result, factor-of-2 characterization can also be obtained for non-linear codes, as
discussed in Section 4.5.
Remark 4.6. The coding construction we developed for proving Theorem 4.3 provides an improved
version of the entangled polynomial code. Explicitly, given any upper bound construction for
R(p, m, n) with rank R, the coding scheme achieves a recovery threshold of 2R − 1, while tolerating
arbitrarily many stragglers. This improved version further and orderwise reduces the needed
recovery threshold on top of its basic version. For example, by simply applying the well-know
Strassen’s construction [44], which provides an upper bound R(2k, 2k, 2k) 7k for any k 2 N, the
proposed coding scheme achieves a recovery threshold of 2 · 7k − 1, which orderwise improves upon
Kentangled-poly = 8k + 2k − 1 achieved by the entangled polynomial code. Further improvements can
be achieved by applying constructions with lower ranks, up to 2R(p, m, n) − 1.
Remark 4.7. In parallel to this work, the Generalized PolyDot scheme was proposed in [45] to extend
the PolyDot construction [15] to asymmetric matrix-vector multiplication. Generalized PolyDot
can be applied to achieve the same recovery threshold of the entangled polynomial code for special
case of m = 1 or n = 1. However, entangled polynomial codes achieve order-wise better recovery
thresholds for general values of p, m, and n.
The techniques we developed in this chapter can also be extended to several other problems such
as fault-tolerant computing [36,37], leading to tight characterizations. Unlike the straggler effects we
studied in this paper, fault tolerance considers scenarios where arbitrary errors can be injected into
the computation, and the master has no information about which subset of workers are returning
errors. We show that the techniques we developed for straggler mitigation can also be applied in
this setting to improve robustness against computing failures, and the optimality of any encoding
function in terms of recovery threshold also preserves when applied in the fault-tolerant computing
setting. As an example, we present the following theorem, demonstrating this connection.
Theorem 4.4. For a distributed matrix multiplication problem of computing A|B using N workers,
with parameters p, m, and n, if m = 1 or n = 1, the entangled polynomial code can detect up to
E
detect = N − Kentangled-poly (4.16)
7Formally defined in Section 4.5.
35
Object Description
| Title | Coded computing: a transformative framework for resilient, secure, private, and communication efficient large scale distributed computing |
| Author | Yu, Qian |
| Author email | qyu880@usc.edu;qianyu0929@gmail.com |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Electrical Engineering |
| School | Viterbi School of Engineering |
| Date defended/completed | 2020-03-24 |
| Date submitted | 2020-08-04 |
| Date approved | 2020-08-05 |
| Restricted until | 2020-08-05 |
| Date published | 2020-08-05 |
| Advisor (committee chair) | Avestimehr, Salman |
| Advisor (committee member) |
Luo, HaiPeng Ortega, Antonio Soltanolkotabi, Mahdi |
| Abstract | Modern computing applications often require handling massive amounts of data in a distributed setting, where significant issues on resiliency, security, or privacy could arise. This dissertation presents new computing designs and optimality proofs, that address these issues through coding and information-theoretic approaches. ❧ The first part of this thesis focuses on a standard setup, where the computation is carried out using a set of worker nodes, each can store and process a fraction of the input dataset. The goal is to find computation schemes for providing the optimal resiliency against stragglers given the computation task, the number of workers, and the functions computed by the workers. The resiliency is measured in terms of the recovery threshold, defined as the minimum number of workers to wait for in order to compute the final result. We propose optimal solutions for broad classes of computation tasks, from basic building blocks such as matrix multiplication (entangled polynomial codes), Fourier transform (coded FFT), and convolution (polynomial code), to general functions such as multivariate polynomial evaluation (Lagrange coded computing). We develop optimal computing strategies by introducing a general coding framework called “polynomial coded computing”, to exploit the algebraic structure of the computation task and create computation redundancy in a novel coded form across workers. Polynomial coded computing allows for order-wise improvements over the state of the arts and significantly generalizes classical coding-theoretic results to go beyond linear computations. The encoding and decoding process of polynomial coded computing designs can be mapped to polynomial evaluation and interpolation, which can be computed efficiently. ❧ Then we show that polynomial coded computing can be extended to provide unified frameworks that also enable security and privacy in the computation. We present the optimal designs for three important problems: distributed matrix multiplication, multivariate polynomial evaluation, and gradient-type computation. We prove their optimality by developing information-theoretic and linear-algebraic converse bounding techniques. ❧ Finally, we consider the problem of coding for communication reduction. In the context of distributed computation, we focus on a MapReduce-type framework, where the workers need to shuffle their intermediate results to finish the computation. We aim to understand how to optimally exploit extra computing power to reduce communication, i.e., to establish a fundamental tradeoff between computation and communication. We prove a lower bound on the needed communication load for general allocation of the task assignments, by introducing a novel information-theoretic converse bounding approach. The presented lower bound exactly matches the inverse-proportional coding gain achieved by coded distributed computing schemes, completely characterizing the optimal computation-communication tradeoff. The proposed converse bounding approach strictly improves conventional cut-set bounds and can be widely applied to prove exact optimally results for more general settings, as well as more classical communication problems. We also investigate a problem called coded caching, where a single server is connected to multiple users in a cache network through a shared bottleneck link. Each user has an isolated memory that can be used to prefetch content. Then the server needs to deliver users’ demands efficiently in a following delivery phase. We propose caching and delivery designs that improve the state-of-the-art schemes under both centralized and decentralized settings, for both peak and average communication rates. Moreover, by developing information-theoretic bounds, we prove the proposed designs are exactly optimal among all schemes that use uncoded prefetching, and optimal within a factor of 2.00884 among schemes with coded prefetching. |
| Keyword | coding theory; information theory; distributed computing; security; privacy; matrix multiplication; caching |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m |
| Contributing entity | University of Southern California |
| Rights | Yu, Qian |
| Physical access | The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository e-mail address given. |
| Repository name | University of Southern California Digital Library |
| Repository address | USC Digital Library, University of Southern California, University Park Campus MC 7002, 106 University Village, Los Angeles, California 90089-7002, USA |
| Repository email | cisadmin@lib.usc.edu |
| Filename | etd-YuQian-8883.pdf |
| Archival file | Volume13/etd-YuQian-8883.pdf |
Description
| Title | Page 50 |
| Full text | Remark 4.5. The key proof idea of Theorem 4.3 is twofold. We first demonstrate a one-to-one correspondence between linear computation strategies and upper bound constructions7 for bilinear complexity, which enables converting a matrix multiplication problem into computing the element-wise product of two vectors of length R(p, m, n). Then we show that an optimal computation strategy can be developed for this augmented problem, which achieves the stated recovery threshold. Similarly to this result, factor-of-2 characterization can also be obtained for non-linear codes, as discussed in Section 4.5. Remark 4.6. The coding construction we developed for proving Theorem 4.3 provides an improved version of the entangled polynomial code. Explicitly, given any upper bound construction for R(p, m, n) with rank R, the coding scheme achieves a recovery threshold of 2R − 1, while tolerating arbitrarily many stragglers. This improved version further and orderwise reduces the needed recovery threshold on top of its basic version. For example, by simply applying the well-know Strassen’s construction [44], which provides an upper bound R(2k, 2k, 2k) 7k for any k 2 N, the proposed coding scheme achieves a recovery threshold of 2 · 7k − 1, which orderwise improves upon Kentangled-poly = 8k + 2k − 1 achieved by the entangled polynomial code. Further improvements can be achieved by applying constructions with lower ranks, up to 2R(p, m, n) − 1. Remark 4.7. In parallel to this work, the Generalized PolyDot scheme was proposed in [45] to extend the PolyDot construction [15] to asymmetric matrix-vector multiplication. Generalized PolyDot can be applied to achieve the same recovery threshold of the entangled polynomial code for special case of m = 1 or n = 1. However, entangled polynomial codes achieve order-wise better recovery thresholds for general values of p, m, and n. The techniques we developed in this chapter can also be extended to several other problems such as fault-tolerant computing [36,37], leading to tight characterizations. Unlike the straggler effects we studied in this paper, fault tolerance considers scenarios where arbitrary errors can be injected into the computation, and the master has no information about which subset of workers are returning errors. We show that the techniques we developed for straggler mitigation can also be applied in this setting to improve robustness against computing failures, and the optimality of any encoding function in terms of recovery threshold also preserves when applied in the fault-tolerant computing setting. As an example, we present the following theorem, demonstrating this connection. Theorem 4.4. For a distributed matrix multiplication problem of computing A|B using N workers, with parameters p, m, and n, if m = 1 or n = 1, the entangled polynomial code can detect up to E detect = N − Kentangled-poly (4.16) 7Formally defined in Section 4.5. 35 |
