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8.2 Secure, Private, Batch Distributed Matrix Multiplication and
Main Results
We show that entangled polynomial codes can be adapted to the settings of secure, private and
batch distributed matrix multiplication, achieving order-wise improvement with subcubic recovery
thresholds while meeting the systems’ requirements. To demonstrate the coding gain, we focus on
applying the second version of the entangled polynomial code, the one that achieves a recovery
thrshold of 2R(p, m, n) − 1 for straggler mitigation.
8.2.1 Secure Distributed Matrix Multiplication
Secure distributed matrix multiplication follows a similar setup discussed in Section 8.1, where the
goal to multiply a single pair of matrices, with additional constraints that either one or both of the
input matrices are information-theoretic private to the workers, even if up to a certain number of
them can collude. In particular, we say an encoding scheme is one-sided T-secure, if
I({A˜i}i2T ;A) = 0 (8.1)
for any subset T with size of at most T, where A is generated uniformly at random. Similarly, we
say an encoding scheme is fully T-secure, if instead
I({ ˜ Ai, ˜B
i}i2T ; A,B) = 0 (8.2)
is satisfied for any |T | T, for uniformly randomly generated A and B.
Secure distributed matrix multiplication has been studied in [61–75]. In particular, [65, 66, 74]
presented coded computing designs for general block-wise partitioning of the input matrices, all
requiring at least pmn workers’ computation.5 Entangled polynomial codes achieves subcubic
recovery thresholds for both one-sided and fully secure settings, formally stated in the following
theorem.
Theorem 8.1. For secure distributed matrix multiplication, there are one-sided T-secure linear
coding schemes that achieves a recovery threshold of 2R(p, m, n) + T − 1, and fully T-secure linear
coding schemes that achieves a recovery threshold of 2R(p, m, n) + 2T − 1.
Remark 8.2. Entangled polynomial codes order-wise improves the state of the arts for general
block-wise partitioning [65, 66, 74], by providing explicit constructions that require subcubic number
of workers. Moreover, entangled polynomial codes simultaneously handles data security and straggler
5In addition, at least T extra workers are needed per each input matrix required to be stored securely.
94
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 109 |
| Full text | 8.2 Secure, Private, Batch Distributed Matrix Multiplication and Main Results We show that entangled polynomial codes can be adapted to the settings of secure, private and batch distributed matrix multiplication, achieving order-wise improvement with subcubic recovery thresholds while meeting the systems’ requirements. To demonstrate the coding gain, we focus on applying the second version of the entangled polynomial code, the one that achieves a recovery thrshold of 2R(p, m, n) − 1 for straggler mitigation. 8.2.1 Secure Distributed Matrix Multiplication Secure distributed matrix multiplication follows a similar setup discussed in Section 8.1, where the goal to multiply a single pair of matrices, with additional constraints that either one or both of the input matrices are information-theoretic private to the workers, even if up to a certain number of them can collude. In particular, we say an encoding scheme is one-sided T-secure, if I({A˜i}i2T ;A) = 0 (8.1) for any subset T with size of at most T, where A is generated uniformly at random. Similarly, we say an encoding scheme is fully T-secure, if instead I({ ˜ Ai, ˜B i}i2T ; A,B) = 0 (8.2) is satisfied for any |T | T, for uniformly randomly generated A and B. Secure distributed matrix multiplication has been studied in [61–75]. In particular, [65, 66, 74] presented coded computing designs for general block-wise partitioning of the input matrices, all requiring at least pmn workers’ computation.5 Entangled polynomial codes achieves subcubic recovery thresholds for both one-sided and fully secure settings, formally stated in the following theorem. Theorem 8.1. For secure distributed matrix multiplication, there are one-sided T-secure linear coding schemes that achieves a recovery threshold of 2R(p, m, n) + T − 1, and fully T-secure linear coding schemes that achieves a recovery threshold of 2R(p, m, n) + 2T − 1. Remark 8.2. Entangled polynomial codes order-wise improves the state of the arts for general block-wise partitioning [65, 66, 74], by providing explicit constructions that require subcubic number of workers. Moreover, entangled polynomial codes simultaneously handles data security and straggler 5In addition, at least T extra workers are needed per each input matrix required to be stored securely. 94 |
