Page 190 |
Save page Remove page | Previous | 190 of 243 | Next |
|
small (250x250 max)
medium (500x500 max)
Large (1000x1000 max)
Extra Large
large ( > 500x500)
Full Resolution
All (PDF)
|
This page
All
|
we can construct the f0-scheme by letting each of the N0 , N − K workers store the coded version
of (Xi,1,Xi,2, . . . ,Xi,d0) that is stored by a unique respective worker in [N] \ K in f-scheme.3
Now it suffices to prove that the above construction achieves a recovery threshold of R (N,K, f)−
(K−1). Equivalently, we need to prove that given any subset S of [N]\K of size R (N,K, f)−(K−1),
the values of f(Xi,1,Xi,2, ...,Xi,d0 , x) for any i 2 [K] and x 2 V are decodable from the computing
results of workers in S.
We exploit the decodability of the computation design for function f. For any j 2 K, the set
S [K\{j} has size R (N,K, f). Consequently, for any vector (x1,d0+1, ..., xK,d0+1) 2 VKd
0+1, we have
that {f(Xi,1,Xi,2, ...,Xi,d0 , xi,d0+1)}i2[K] is decodable given the results from workers in S [ K\{j}
computed in f-scheme, if each xi,d0+1 is used as the (d0 + 1)th entree for each input.
Because columns of G with indices in K form a basis of FK, we can find values for each input
Xi,d0+1 such that workers in K would store 0 for the Xi,d0+1 entry in the f-scheme. We denote these
values by ¯x1,d0+1, ..., ¯xK,d0+1. Note that if these values are taken as inputs, workers in K would
return constant 0 due to the multilinearity of f. Hence, decoding f(Xi,1,Xi,2, ...,Xi,d0 , ¯xi,d0+1) only
requires results from workers not in K, i.e., it can be decoded given computing results from workers
in S using the f-scheme. Note that these results can be directly computed from corresponding
results in the f0-scheme. We have proved the decodability of f(Xi,1,Xi,2, ...,Xi,d0 , x) for x = ¯xi,d0+1.
Now it remains to prove the decodability of f(Xi,1,Xi,2, ...,Xi,d0 , x) for each i for general x 2 V.
For any j 2 K, let a(j) 2 FK be a non-zero vector that is orthogonal to all columns of G with
indices in K\{j}. If a
(j)
i x + ¯xi,d0+1 is used for each input Xi,d0+1 in the f-scheme, then workers
in K\{j} would store 0 for the Xi,d0+1 entry, and return constant 0 due to the multilinearity of
f. Recall that f(Xi,1,Xi,2, ...,Xi,d0 , a
(j)
i x + ¯xi,d0+1) is assumed to be decodable in the f-scheme
given results from workers in S [ K\{j} . Following the same arguments above, one can prove that
f(Xi,1,Xi,2, ...,Xi,d0 , a
(j)
i x + ¯xi,d0+1) is also decodable using the f0-scheme. Hence, the same applies
for a
(j)
i f(Xi,1,Xi,2, ...,Xi,d0 , x) due to multilinearity of f.
Because columns of G with indices in K form a basis of FK, the vectors a(j) for j 2 K also from a
basis. Consequently, for any i there is a non-zero a
(j)
i , and thus f(Xi,1,Xi,2, ...,Xi,d0 , x) is decodable.
This completes the proof of decodability.
To summarize, we have essentially proved that R (N,K, f) − (K − 1) R (N − K,K, f0). We
can verify that the converse bound R (N,K, f) N − bN/Kc + 1 under the condition N Kd − 1
can be derived given the above result and the induction assumption, for any function f with degree
d0 + 1.
3For breivity, in this proof we instead index these N − K workers also using the set [N] \ K, following the natural
bijection.
175
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 190 |
| Full text | we can construct the f0-scheme by letting each of the N0 , N − K workers store the coded version of (Xi,1,Xi,2, . . . ,Xi,d0) that is stored by a unique respective worker in [N] \ K in f-scheme.3 Now it suffices to prove that the above construction achieves a recovery threshold of R (N,K, f)− (K−1). Equivalently, we need to prove that given any subset S of [N]\K of size R (N,K, f)−(K−1), the values of f(Xi,1,Xi,2, ...,Xi,d0 , x) for any i 2 [K] and x 2 V are decodable from the computing results of workers in S. We exploit the decodability of the computation design for function f. For any j 2 K, the set S [K\{j} has size R (N,K, f). Consequently, for any vector (x1,d0+1, ..., xK,d0+1) 2 VKd 0+1, we have that {f(Xi,1,Xi,2, ...,Xi,d0 , xi,d0+1)}i2[K] is decodable given the results from workers in S [ K\{j} computed in f-scheme, if each xi,d0+1 is used as the (d0 + 1)th entree for each input. Because columns of G with indices in K form a basis of FK, we can find values for each input Xi,d0+1 such that workers in K would store 0 for the Xi,d0+1 entry in the f-scheme. We denote these values by ¯x1,d0+1, ..., ¯xK,d0+1. Note that if these values are taken as inputs, workers in K would return constant 0 due to the multilinearity of f. Hence, decoding f(Xi,1,Xi,2, ...,Xi,d0 , ¯xi,d0+1) only requires results from workers not in K, i.e., it can be decoded given computing results from workers in S using the f-scheme. Note that these results can be directly computed from corresponding results in the f0-scheme. We have proved the decodability of f(Xi,1,Xi,2, ...,Xi,d0 , x) for x = ¯xi,d0+1. Now it remains to prove the decodability of f(Xi,1,Xi,2, ...,Xi,d0 , x) for each i for general x 2 V. For any j 2 K, let a(j) 2 FK be a non-zero vector that is orthogonal to all columns of G with indices in K\{j}. If a (j) i x + ¯xi,d0+1 is used for each input Xi,d0+1 in the f-scheme, then workers in K\{j} would store 0 for the Xi,d0+1 entry, and return constant 0 due to the multilinearity of f. Recall that f(Xi,1,Xi,2, ...,Xi,d0 , a (j) i x + ¯xi,d0+1) is assumed to be decodable in the f-scheme given results from workers in S [ K\{j} . Following the same arguments above, one can prove that f(Xi,1,Xi,2, ...,Xi,d0 , a (j) i x + ¯xi,d0+1) is also decodable using the f0-scheme. Hence, the same applies for a (j) i f(Xi,1,Xi,2, ...,Xi,d0 , x) due to multilinearity of f. Because columns of G with indices in K form a basis of FK, the vectors a(j) for j 2 K also from a basis. Consequently, for any i there is a non-zero a (j) i , and thus f(Xi,1,Xi,2, ...,Xi,d0 , x) is decodable. This completes the proof of decodability. To summarize, we have essentially proved that R (N,K, f) − (K − 1) R (N − K,K, f0). We can verify that the converse bound R (N,K, f) N − bN/Kc + 1 under the condition N Kd − 1 can be derived given the above result and the induction assumption, for any function f with degree d0 + 1. 3For breivity, in this proof we instead index these N − K workers also using the set [N] \ K, following the natural bijection. 175 |
