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10.6 Concluding Remarks
0 0.5 1 1.5 2 2.5 3 3.5 4
Local Cache Size M
0
5
10
15
20
25
Peak Communication Rates
Optimal Uncoded Prefetching (Proposed)
Tian-Chen Scheme [4]
Amiri-Gunduz Scheme [5]
Gomez-Vilardebo Scheme [10]
Ghasemi-Ramamoorthy Converse [6]
Sengupta et al. Converse [7]
N. et al. Converse [8]
Yu et al. Converse [9]
Figure 10.6: Achievable peak communication rates for centralized schemes that allow coded prefetching. For
N = 20, K = 40, we compare our proposed achievability scheme with prior-art coded-prefetching schemes [4,5],
prior-art converse bounds [6–8], and two recent results [9, 10]. The achievability scheme presented in this
chapter achieves the best performance to date in most cases, and is within a factor of 2 optimal as shown
in [9], even compared with schemes that allow coded prefetching.
In this chapter, we characterized the rate-memory tradeoff for the coded caching problem with
uncoded prefetching. To that end, we proposed the optimal caching schemes for both centralized
setting and decentralized setting, and proved their exact optimality for both average rate and
peak rate. The techniques we presented in this chapter can be directly applied to many other
problems, immediately improving their state of the arts. For instance, the achievability scheme
presented in this chapter has already been applied in various different settings, achieving improved
results [9, 172, 173]. Beyond these works, the techniques can also be applied in directions such as
online caching [145], caching with non-uniform demands [146], and hierarchical caching [151], where
improvements can be immediately achieved by directly plugging in our results.
One interesting follow-up direction is to consider coded caching problem with coded placement.
In this scenario, it has been shown that in the centralized setting, coded prefetching schemes can
achieve better peak communication rates. For example, Figure 10.6 shows that one can improve
the peak communication rate by coded placement when the cache size is small. As we will show
in the following Chapter, through a new converse bounding technique, the achievability scheme
we presented in this chapter is optimal within a factor of 2. However, finding the exact optimal
solution in this regime remains an open problem.
135
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 150 |
| Full text | 10.6 Concluding Remarks 0 0.5 1 1.5 2 2.5 3 3.5 4 Local Cache Size M 0 5 10 15 20 25 Peak Communication Rates Optimal Uncoded Prefetching (Proposed) Tian-Chen Scheme [4] Amiri-Gunduz Scheme [5] Gomez-Vilardebo Scheme [10] Ghasemi-Ramamoorthy Converse [6] Sengupta et al. Converse [7] N. et al. Converse [8] Yu et al. Converse [9] Figure 10.6: Achievable peak communication rates for centralized schemes that allow coded prefetching. For N = 20, K = 40, we compare our proposed achievability scheme with prior-art coded-prefetching schemes [4,5], prior-art converse bounds [6–8], and two recent results [9, 10]. The achievability scheme presented in this chapter achieves the best performance to date in most cases, and is within a factor of 2 optimal as shown in [9], even compared with schemes that allow coded prefetching. In this chapter, we characterized the rate-memory tradeoff for the coded caching problem with uncoded prefetching. To that end, we proposed the optimal caching schemes for both centralized setting and decentralized setting, and proved their exact optimality for both average rate and peak rate. The techniques we presented in this chapter can be directly applied to many other problems, immediately improving their state of the arts. For instance, the achievability scheme presented in this chapter has already been applied in various different settings, achieving improved results [9, 172, 173]. Beyond these works, the techniques can also be applied in directions such as online caching [145], caching with non-uniform demands [146], and hierarchical caching [151], where improvements can be immediately achieved by directly plugging in our results. One interesting follow-up direction is to consider coded caching problem with coded placement. In this scenario, it has been shown that in the centralized setting, coded prefetching schemes can achieve better peak communication rates. For example, Figure 10.6 shows that one can improve the peak communication rate by coded placement when the cache size is small. As we will show in the following Chapter, through a new converse bounding technique, the achievability scheme we presented in this chapter is optimal within a factor of 2. However, finding the exact optimal solution in this regime remains an open problem. 135 |
