Page 72 |
Save page Remove page | Previous | 72 of 243 | Next |
|
small (250x250 max)
medium (500x500 max)
Large (1000x1000 max)
Extra Large
large ( > 500x500)
Full Resolution
All (PDF)
|
This page
All
|
Based on this observation, we can naturally view the distributed Fourier transform problem as
a problem of distributedly computing a list of linear transformations, i.e., computing the Fourier
transform of ci’s. We inject the redundancy as follows to provide robustness to the computation:
We first encode the c0, c1, ..., cm−1 using an arbitrary (N,m)-MDS code, where the coded vectors
are denoted a0, ..., aN−1 and are assigned to the workers correspondingly. Then each worker i
computes the Fourier of ai, and return it to the master. Given the linearity of Fourier transform,
the computing results b0, ..., bN−1 are essentially linear combinations of the Fourier transform Ci’s,
which are coded by the same MDS code. Hence, after the master receives any m computing results,
it can decode the message Ci’s, and proceed to recover the final result. This allows achieving the
recovery threshold of m.
Remark 5.4. The recovery threshold K = m achieved by coded FFT can not be achieved using
computation strategies that were developed for generic matrix-by-vector multiplication in the
literature [1, 3]. Specifically, the conventional uncoded repetition strategy requires a recovery
threshold of N − N
m2 + 1, and the short-dot (or short-MDS) strategy provided in [1, 3] requires
N − N
m + m. Hence, by developing a coding strategy for the specific purpose of computing Fourier
transform, we can achieve order-wise improvement in the recovery threshold.
5.2.3 Decoding Complexity of Coded FFT
Now we show that coded FFT allows an efficient decoding algorithm at the master for recovering
the output. After receiving the computing results, the master needs to recover the output in two
steps: decoding the MDS code and then computing X from the intermediate value Ci’s.
For the first step, the master needs of decode an (N,m)-MDS code by s
m times. This can
be computed efficiently, by selecting an MDS code with low decoding complexity for the coded
FFT design. There has been various works on finding efficiently decodable MDS codes (e.g.,
[38, 87]). In general, an upper bound on the decoding complexity of (N,m)-MDS code is given
by O(mlog2mlog logm), which can be attained by the Reed-Solomon codes [40] and using fast
polynomial interpolation [39] as the decoding algorithm. Consequently, the first step of the decoding
algorithm has a complexity of at most O(s log2mlog logm), which scales linearly with respect to s.
For the second step, the master node needs to evaluate equation (5.23) to recover the final result.
Equivalently, the master needs to compute
Xi+j s
m
=
mX−1
k=0
Ck,i!
ik+jk s
m
s (5.24)
57
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 72 |
| Full text | Based on this observation, we can naturally view the distributed Fourier transform problem as a problem of distributedly computing a list of linear transformations, i.e., computing the Fourier transform of ci’s. We inject the redundancy as follows to provide robustness to the computation: We first encode the c0, c1, ..., cm−1 using an arbitrary (N,m)-MDS code, where the coded vectors are denoted a0, ..., aN−1 and are assigned to the workers correspondingly. Then each worker i computes the Fourier of ai, and return it to the master. Given the linearity of Fourier transform, the computing results b0, ..., bN−1 are essentially linear combinations of the Fourier transform Ci’s, which are coded by the same MDS code. Hence, after the master receives any m computing results, it can decode the message Ci’s, and proceed to recover the final result. This allows achieving the recovery threshold of m. Remark 5.4. The recovery threshold K = m achieved by coded FFT can not be achieved using computation strategies that were developed for generic matrix-by-vector multiplication in the literature [1, 3]. Specifically, the conventional uncoded repetition strategy requires a recovery threshold of N − N m2 + 1, and the short-dot (or short-MDS) strategy provided in [1, 3] requires N − N m + m. Hence, by developing a coding strategy for the specific purpose of computing Fourier transform, we can achieve order-wise improvement in the recovery threshold. 5.2.3 Decoding Complexity of Coded FFT Now we show that coded FFT allows an efficient decoding algorithm at the master for recovering the output. After receiving the computing results, the master needs to recover the output in two steps: decoding the MDS code and then computing X from the intermediate value Ci’s. For the first step, the master needs of decode an (N,m)-MDS code by s m times. This can be computed efficiently, by selecting an MDS code with low decoding complexity for the coded FFT design. There has been various works on finding efficiently decodable MDS codes (e.g., [38, 87]). In general, an upper bound on the decoding complexity of (N,m)-MDS code is given by O(mlog2mlog logm), which can be attained by the Reed-Solomon codes [40] and using fast polynomial interpolation [39] as the decoding algorithm. Consequently, the first step of the decoding algorithm has a complexity of at most O(s log2mlog logm), which scales linearly with respect to s. For the second step, the master node needs to evaluate equation (5.23) to recover the final result. Equivalently, the master needs to compute Xi+j s m = mX−1 k=0 Ck,i! ik+jk s m s (5.24) 57 |
