Page 206 |
Save page Remove page | Previous | 206 of 243 | Next |
|
small (250x250 max)
medium (500x500 max)
Large (1000x1000 max)
Extra Large
large ( > 500x500)
Full Resolution
All (PDF)
|
This page
All
|
Appendix F
Supplement to Chapter 10
F.1 Proof of Lemma 10.2
Proof. The Proof of Lemma 10.2 is organized as follows: We start by proving a lower bound of
the communication rate required for a single demand, i.e., R
(d,M). By averaging this lower
bound over a single demand type Ds, we automatically obtain a lower bound for the rate R
(s,M).
Finally we bound the minimum possible R
(s,M) over all prefetching schemes by solving for the
minimum value of our derived lower bound.
We first use a genie-aided approach to derive a lower bound of R
(d,M) for any demand d and
for any prefetching M: Given a demand d, let U = {u1, ..., uNe(d)} be an arbitrary subset with
Ne(d) users that request distinct files. We construct a virtual user whose cache is initially empty.
Suppose for each ` 2 {1, ...,Ne(d)}, a genie fills the cache with the value of bits that are cached
by u`, but not from files requested by users in {u1, ..., u`−1}. Then with all the cached information
provided by the genie, the virtual user should be able to inductively decode all files requested by
users in U upon receiving the message X. Consequently, a lower bound on the communication rate
R
(d,M) can be obtained by applying a cut-set bound on the virtual user.
Specifically, we prove that the virtual user can decode all Ne(d) requested files with high probability,
by inductively decoding each file du` using the decoding function of user u`, from ` = 1 to ` = Ne(d):
Recall that any communication rate is -achievable if the error probability of each decoding function
is at most . Consequently, the probability that all Ne(d) decoding functions can correctly decode
the requested files is at least 1 − Ne(d) . In this scenario, the virtual user can correctly decode all
the files, given that at every single step of induction, all bits necessary for the decoding function
have either been provided by the genie, or decoded in previous inductive steps.
191
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 206 |
| Full text | Appendix F Supplement to Chapter 10 F.1 Proof of Lemma 10.2 Proof. The Proof of Lemma 10.2 is organized as follows: We start by proving a lower bound of the communication rate required for a single demand, i.e., R (d,M). By averaging this lower bound over a single demand type Ds, we automatically obtain a lower bound for the rate R (s,M). Finally we bound the minimum possible R (s,M) over all prefetching schemes by solving for the minimum value of our derived lower bound. We first use a genie-aided approach to derive a lower bound of R (d,M) for any demand d and for any prefetching M: Given a demand d, let U = {u1, ..., uNe(d)} be an arbitrary subset with Ne(d) users that request distinct files. We construct a virtual user whose cache is initially empty. Suppose for each ` 2 {1, ...,Ne(d)}, a genie fills the cache with the value of bits that are cached by u`, but not from files requested by users in {u1, ..., u`−1}. Then with all the cached information provided by the genie, the virtual user should be able to inductively decode all files requested by users in U upon receiving the message X. Consequently, a lower bound on the communication rate R (d,M) can be obtained by applying a cut-set bound on the virtual user. Specifically, we prove that the virtual user can decode all Ne(d) requested files with high probability, by inductively decoding each file du` using the decoding function of user u`, from ` = 1 to ` = Ne(d): Recall that any communication rate is -achievable if the error probability of each decoding function is at most . Consequently, the probability that all Ne(d) decoding functions can correctly decode the requested files is at least 1 − Ne(d) . In this scenario, the virtual user can correctly decode all the files, given that at every single step of induction, all bits necessary for the decoding function have either been provided by the genie, or decoded in previous inductive steps. 191 |
