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It was shown in [16] that this hamming distance behaves similar to its classical counter part: an
encoding scheme is S-resilient and A-secure whenever S+2A d−1. Hence, for any encoding scheme
that is A-secure and S-reselient, it has a hamming distance of at least S + 2A + 1. Consequently
it can tolerate S + 2A stragglers. Combining the above and Lemma 6.1, we have completed the
proof.
C.6 Optimality on the Resiliency-Privacy Tradeoff for General
Multivariate Polynomials
In this appendix, we prove the second part of Theorem 6.2 using Lemma 6.1. Specifically, we
aim to prove that LCC achieves the optimal trade-off between resiliency and privacy, for general
multivariate polynomial f. The proof is carried out by showing that for any function f that allows
S-resilient T-private designs, there exists a multilinear function with the same degree for which a
computation scheme can be found that achieves the same requirement.
Specifically, given any function f with degree d, we aim to provide an explicit construction of
an multilinear function, denoted by f0, which achieves the same requirements. The construction
satisfies certain properties to ensure this fact. Both the construction and the properties are formally
stated in the following lemma (which is proved in Appendix C.7):
Lemma C.2. Given any function f of degree d, let f0 be a map from Vd ! U such that
f0(Z1, ...,Zd) = P
S [d] (−1)|S|f(P
j2S Zj) for any {Zj}j2[d] 2 Vd. Then f0 is multilinear with
respect to the d inputs. Moreover, if the characteristic of the base field F is 0 or greater than d, then
f0 is non-zero.
Assuming the correctness of Lemma C.2, it suffices to prove that f0 enables computation designs
that tolerates at least the same number of stragglers, and provides at least the same level of data
privacy, compared to that of f. We prove this fact by constructing such computing schemes for f0
given any design for f.
Note that f0 is defined as a linear combination of functions f(P
j2S Zj ), each of which is a
composition of a linear map and f. Given the linearity of the encoding design, any computation
scheme of f can be directly applied to any of these functions, achieving the same resiliency and
privacy requirements. Since the decoding functions are linear, the same scheme also applies to linear
combinations of them, which includes f0. Hence, the resiliency-privacy tradeoff achievable for f can
also be achieved by f0. This concludes the proof.
180
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 195 |
| Full text | It was shown in [16] that this hamming distance behaves similar to its classical counter part: an encoding scheme is S-resilient and A-secure whenever S+2A d−1. Hence, for any encoding scheme that is A-secure and S-reselient, it has a hamming distance of at least S + 2A + 1. Consequently it can tolerate S + 2A stragglers. Combining the above and Lemma 6.1, we have completed the proof. C.6 Optimality on the Resiliency-Privacy Tradeoff for General Multivariate Polynomials In this appendix, we prove the second part of Theorem 6.2 using Lemma 6.1. Specifically, we aim to prove that LCC achieves the optimal trade-off between resiliency and privacy, for general multivariate polynomial f. The proof is carried out by showing that for any function f that allows S-resilient T-private designs, there exists a multilinear function with the same degree for which a computation scheme can be found that achieves the same requirement. Specifically, given any function f with degree d, we aim to provide an explicit construction of an multilinear function, denoted by f0, which achieves the same requirements. The construction satisfies certain properties to ensure this fact. Both the construction and the properties are formally stated in the following lemma (which is proved in Appendix C.7): Lemma C.2. Given any function f of degree d, let f0 be a map from Vd ! U such that f0(Z1, ...,Zd) = P S [d] (−1)|S|f(P j2S Zj) for any {Zj}j2[d] 2 Vd. Then f0 is multilinear with respect to the d inputs. Moreover, if the characteristic of the base field F is 0 or greater than d, then f0 is non-zero. Assuming the correctness of Lemma C.2, it suffices to prove that f0 enables computation designs that tolerates at least the same number of stragglers, and provides at least the same level of data privacy, compared to that of f. We prove this fact by constructing such computing schemes for f0 given any design for f. Note that f0 is defined as a linear combination of functions f(P j2S Zj ), each of which is a composition of a linear map and f. Given the linearity of the encoding design, any computation scheme of f can be directly applied to any of these functions, achieving the same resiliency and privacy requirements. Since the decoding functions are linear, the same scheme also applies to linear combinations of them, which includes f0. Hence, the resiliency-privacy tradeoff achievable for f can also be achieved by f0. This concludes the proof. 180 |
