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Remark 11.1. The above theorem characterizes R and R ave within a constant factor of 2.00884
for all possible values of parameters K, N, and M. To the best of our knowledge, this gives the
best characterization to date. Prior to this work, the best proved constant factors were 4 for peak
rate [6] and 4.7 for average rate (under uniform file popularity) [161]. Furthermore, Theorem 11.1
characterizes R and R ave for large N within a constant factor of 2.
Remark 11.2. The converse bound that we develop for proving Theorem 11.1 also immediately results
in better approximation of rate-memory tradeoff in other scenarios, such as online caching [145],
caching with non-uniform demands [146], and hierarchical caching [151]. For example, in the case
of online caching [145], where the current approximation result is within a multiplicative factor of
24, it can be easily shown that this factor can be reduced to 4.01768 using our proposed bounding
techniques.
Remark 11.3. Ru(N,K, r) and Ru,ave(N,K, r), as defined in Definition 11.1, are the optimum peak
rate and the optimum average rate that can be achieved using uncoded prefetching, as we proved
in [11]. This indicates that for the coded caching problem, using uncoded prefetching schemes is
within a factor of 2.00884 optimal for both peak rate and average rate. More interestingly, we can
show that even for the improved decentralized scheme we proposed in [11], where each user fills their
cache independently without coordination but the delivery scheme was designed to fully exploit
the commonality of user demands, the optimum rate is still achieved within a factor of 2.00884 in
general, and a factor of 2 for large N. 5
Remark 11.4. Based on the proof idea of Theorem 11.1, we can completely characterize the rate-memory
tradeoff for the two-user case, for any possible values of N and M, for both peak rate
and average rate. Prior to this work, the peak rate vs. memory tradeoff for the two-user case was
characterized in [140] for N 2, and is characterized in [170] for N 3 very recently. However the
average rate vs. memory tradeoff has never been completely characterized for any non-trivial case.
In this chapter, we prove that the exact optimal tradeoff for the average rate for two-user case can
be achieved using the caching scheme we provided in [11] (see Appendix G.8).
To prove the Theorem 11.1, we derive new converse bounds of R and R ave for all possible values
of K, N, and M. We highlight the converse bound of R in the following theorem:
Theorem 11.2. For a caching system with K users, a database of N files, and a local cache size
of M files at each user, R is lower bounded by
R s − 1 + −
s(s − 1) − `(` − 1) + 2 s
2(N − ` + 1) M, (11.11)
5This can be proved based on the fact that, in the proof of Theorem 11.1, we showed the communication rates of
the decentralized caching scheme we proposed in [11] (e.g., Rdec(M) for the peak rate) are within constant factor
optimal as intermediate steps.
141
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 156 |
| Full text | Remark 11.1. The above theorem characterizes R and R ave within a constant factor of 2.00884 for all possible values of parameters K, N, and M. To the best of our knowledge, this gives the best characterization to date. Prior to this work, the best proved constant factors were 4 for peak rate [6] and 4.7 for average rate (under uniform file popularity) [161]. Furthermore, Theorem 11.1 characterizes R and R ave for large N within a constant factor of 2. Remark 11.2. The converse bound that we develop for proving Theorem 11.1 also immediately results in better approximation of rate-memory tradeoff in other scenarios, such as online caching [145], caching with non-uniform demands [146], and hierarchical caching [151]. For example, in the case of online caching [145], where the current approximation result is within a multiplicative factor of 24, it can be easily shown that this factor can be reduced to 4.01768 using our proposed bounding techniques. Remark 11.3. Ru(N,K, r) and Ru,ave(N,K, r), as defined in Definition 11.1, are the optimum peak rate and the optimum average rate that can be achieved using uncoded prefetching, as we proved in [11]. This indicates that for the coded caching problem, using uncoded prefetching schemes is within a factor of 2.00884 optimal for both peak rate and average rate. More interestingly, we can show that even for the improved decentralized scheme we proposed in [11], where each user fills their cache independently without coordination but the delivery scheme was designed to fully exploit the commonality of user demands, the optimum rate is still achieved within a factor of 2.00884 in general, and a factor of 2 for large N. 5 Remark 11.4. Based on the proof idea of Theorem 11.1, we can completely characterize the rate-memory tradeoff for the two-user case, for any possible values of N and M, for both peak rate and average rate. Prior to this work, the peak rate vs. memory tradeoff for the two-user case was characterized in [140] for N 2, and is characterized in [170] for N 3 very recently. However the average rate vs. memory tradeoff has never been completely characterized for any non-trivial case. In this chapter, we prove that the exact optimal tradeoff for the average rate for two-user case can be achieved using the caching scheme we provided in [11] (see Appendix G.8). To prove the Theorem 11.1, we derive new converse bounds of R and R ave for all possible values of K, N, and M. We highlight the converse bound of R in the following theorem: Theorem 11.2. For a caching system with K users, a database of N files, and a local cache size of M files at each user, R is lower bounded by R s − 1 + − s(s − 1) − `(` − 1) + 2 s 2(N − ` + 1) M, (11.11) 5This can be proved based on the fact that, in the proof of Theorem 11.1, we showed the communication rates of the decentralized caching scheme we proposed in [11] (e.g., Rdec(M) for the peak rate) are within constant factor optimal as intermediate steps. 141 |
