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variables (and possibly a list of independent uniformly random keys when privacy is taken into
account16); while a linear decoding function computes a linear combination of workers’ output.
We essentially need to prove that (a) given any multilinear f, any linear encoding scheme that
achieves any (S, A, T) requires at least N (K + T − 1) deg f + S + 2A + 1 workers when T > 0 or
N Kdeg f − 1, and N K(S + 2A + 1) workers in other cases; (b) for a general polynomial f,
any scheme that uses linear encoding and decoding requires at least the same number of workers, if
the characteristic of F is 0 or greater than deg f.
The proof rely on the following key lemma, which characterizes the recovery threshold of any
encoding scheme, defined as the minimum number of workers that the master needs to wait to
guarantee decodability.
Lemma 6.1. Given any multilinear f, the recovery threshold of any valid linear encoding scheme,
denoted by R, satisfies
R RLCC(N,K, f) ,
min{(K − 1) deg f + 1, N − bN/Kc + 1}. (6.3)
Moreover, if the encoding scheme is T private, we have R RLCC(N,K, f) + T · deg f.
The proof of Lemma 6.1 can be found in Appendix C.4, by constructing instances of the
computation process for any assumed scheme that achieves smaller recovery threshold, and proving
that such scheme fails to achieve decodability in these instances. Intuitively, note that the recovery
threshold is exactly the difference between N and the number of stragglers that can be tolerated,
inequality (6.3) in fact proves that LCC (described in Section 6.3 and Appendix C.6) achieves the
optimum resiliency, as it exactly achieves the stated recovery threshold. Similarly, one can verify
that Lemma 6.1 essentially states that LCC achieves the optimal tradeoff between resiliency and
privacy.
Assuming the correctness of Lemma 6.1, the two parts of Theorem 6.2 can be proved as follows.
To prove part (a) of the converses, we need to extend Lemma 6.1 to also take adversaries into
account. This is achieved by using an extended concept of Hamming distance, defined in [16] for
coded computing. Part (b) requires generalizing Lemma 6.1 to arbitrary polynomial functions,
which is proved by showing that for any f that achieves any (S, T) pair, there exists a multilinear
function with the same degree for which a computation scheme can be found to achieves the same
requirement. The detailed proofs can be found in Appendices C.5 and C.6 respectively.
16This is well defined as we assumed that V is finite when T > 0.
78
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 93 |
| Full text | variables (and possibly a list of independent uniformly random keys when privacy is taken into account16); while a linear decoding function computes a linear combination of workers’ output. We essentially need to prove that (a) given any multilinear f, any linear encoding scheme that achieves any (S, A, T) requires at least N (K + T − 1) deg f + S + 2A + 1 workers when T > 0 or N Kdeg f − 1, and N K(S + 2A + 1) workers in other cases; (b) for a general polynomial f, any scheme that uses linear encoding and decoding requires at least the same number of workers, if the characteristic of F is 0 or greater than deg f. The proof rely on the following key lemma, which characterizes the recovery threshold of any encoding scheme, defined as the minimum number of workers that the master needs to wait to guarantee decodability. Lemma 6.1. Given any multilinear f, the recovery threshold of any valid linear encoding scheme, denoted by R, satisfies R RLCC(N,K, f) , min{(K − 1) deg f + 1, N − bN/Kc + 1}. (6.3) Moreover, if the encoding scheme is T private, we have R RLCC(N,K, f) + T · deg f. The proof of Lemma 6.1 can be found in Appendix C.4, by constructing instances of the computation process for any assumed scheme that achieves smaller recovery threshold, and proving that such scheme fails to achieve decodability in these instances. Intuitively, note that the recovery threshold is exactly the difference between N and the number of stragglers that can be tolerated, inequality (6.3) in fact proves that LCC (described in Section 6.3 and Appendix C.6) achieves the optimum resiliency, as it exactly achieves the stated recovery threshold. Similarly, one can verify that Lemma 6.1 essentially states that LCC achieves the optimal tradeoff between resiliency and privacy. Assuming the correctness of Lemma 6.1, the two parts of Theorem 6.2 can be proved as follows. To prove part (a) of the converses, we need to extend Lemma 6.1 to also take adversaries into account. This is achieved by using an extended concept of Hamming distance, defined in [16] for coded computing. Part (b) requires generalizing Lemma 6.1 to arbitrary polynomial functions, which is proved by showing that for any f that achieves any (S, T) pair, there exists a multilinear function with the same degree for which a computation scheme can be found to achieves the same requirement. The detailed proofs can be found in Appendices C.5 and C.6 respectively. 16This is well defined as we assumed that V is finite when T > 0. 78 |
