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requests, the probability distribution of the prefetchingMis given by P(M= (M1, ...,MK)) = KY i=1 PM(Mi). We define that a decentralized caching scheme, denoted by PM;F , is a distribution parameterized by the file size F, that specifies the prefetching distribution PM for all possible values of F. Similar to the centralized setting, when K users are making requests, we say that a rate R is achievable given a prefetching distribution PM and a demand d if and only if there exists a message X of length RF such that every active user k is able to recover its desired file Wdk with a probability of error of at most . This is rigorously defined as follows: Definition 10.4. When K users are making requests, R is achievable given a prefetching distribution PM and a demand d if and only if for every possible realization of the prefetchingM, we can find a real number M, such that R is Machievable givenMand d, and E[ M] . We denote R ,K(d, PM) as the minimum achievable rate given K, d and PM, and we define the ratememory tradeoff for the average rate based on this notation. For each K 2 N, and each prefetching scheme PM;F , we define the minimum average rate R K (PM;F ) as the minimum expected rate under uniformly random demand that can be achieved with vanishing error probability for sufficiently large file size. Specifically, R K (PM;F ) = sup >0 lim sup F0!+1 Ed[R ,K(d, PM;F (F = F0))], where the demand d is uniformly distributed on {1, . . . ,N}K. Given the fact that a decentralized prefetching scheme is designed without the knowledge of the number of active users K, we characterize the ratememory tradeoff using an infinite dimensional vector, denoted by {RK}K2N, where each term RK corresponds to the needed communication rates when K users are making requests. We aim to find the region in this infinite dimensional vector space that can be achieved by any decentralized prefetching scheme, and we denote this region by R. Rigorously, we aim to find R = [ PM;F {{RK}K2N  8K 2 N,RK R K (PM;F )}, which is a function of N and M. Similarly, we define the ratememory tradeoff for the peak rate as follows: For each K 2 N, and each prefetching scheme PM;F , we define the minimum peak rate R K ,peak(PM;F ) as the minimum communication rate that can be achieved with vanishing error probability for sufficiently large file 131
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  20200324 
Date submitted  20200804 
Date approved  20200805 
Restricted until  20200805 
Date published  20200805 
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 informationtheoretic 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 orderwise improvements over the state of the arts and significantly generalizes classical codingtheoretic 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 gradienttype computation. We prove their optimality by developing informationtheoretic and linearalgebraic converse bounding techniques. ❧ Finally, we consider the problem of coding for communication reduction. In the context of distributed computation, we focus on a MapReducetype 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 informationtheoretic converse bounding approach. The presented lower bound exactly matches the inverseproportional coding gain achieved by coded distributed computing schemes, completely characterizing the optimal computationcommunication tradeoff. The proposed converse bounding approach strictly improves conventional cutset 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 stateoftheart schemes under both centralized and decentralized settings, for both peak and average communication rates. Moreover, by developing informationtheoretic 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  uscthesesm 
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 email 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 900897002, USA 
Repository email  cisadmin@lib.usc.edu 
Filename  etdYuQian8883.pdf 
Archival file  Volume13/etdYuQian8883.pdf 
Description
Title  Page 146 
Full text  requests, the probability distribution of the prefetchingMis given by P(M= (M1, ...,MK)) = KY i=1 PM(Mi). We define that a decentralized caching scheme, denoted by PM;F , is a distribution parameterized by the file size F, that specifies the prefetching distribution PM for all possible values of F. Similar to the centralized setting, when K users are making requests, we say that a rate R is achievable given a prefetching distribution PM and a demand d if and only if there exists a message X of length RF such that every active user k is able to recover its desired file Wdk with a probability of error of at most . This is rigorously defined as follows: Definition 10.4. When K users are making requests, R is achievable given a prefetching distribution PM and a demand d if and only if for every possible realization of the prefetchingM, we can find a real number M, such that R is Machievable givenMand d, and E[ M] . We denote R ,K(d, PM) as the minimum achievable rate given K, d and PM, and we define the ratememory tradeoff for the average rate based on this notation. For each K 2 N, and each prefetching scheme PM;F , we define the minimum average rate R K (PM;F ) as the minimum expected rate under uniformly random demand that can be achieved with vanishing error probability for sufficiently large file size. Specifically, R K (PM;F ) = sup >0 lim sup F0!+1 Ed[R ,K(d, PM;F (F = F0))], where the demand d is uniformly distributed on {1, . . . ,N}K. Given the fact that a decentralized prefetching scheme is designed without the knowledge of the number of active users K, we characterize the ratememory tradeoff using an infinite dimensional vector, denoted by {RK}K2N, where each term RK corresponds to the needed communication rates when K users are making requests. We aim to find the region in this infinite dimensional vector space that can be achieved by any decentralized prefetching scheme, and we denote this region by R. Rigorously, we aim to find R = [ PM;F {{RK}K2N  8K 2 N,RK R K (PM;F )}, which is a function of N and M. Similarly, we define the ratememory tradeoff for the peak rate as follows: For each K 2 N, and each prefetching scheme PM;F , we define the minimum peak rate R K ,peak(PM;F ) as the minimum communication rate that can be achieved with vanishing error probability for sufficiently large file 131 