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. . .
Figure 4.1: Overview of the distributed matrix multiplication problem. Each worker computes the product of
the two stored encoded submatrices (A˜i and ˜B
i) and returns the result to the master. By carefully designing
the coding strategy, the master can decode the multiplication result of the input matrices from a subset of
workers, without having to wait for stragglers (worker 1 in this example).
can choose p, m, and n accordingly. We aim to find optimal coding and computation designs for
any choice of parameters p, m and n, to provide optimum straggler effect mitigation for various
situations.
With a careful design of the coded submatrices A˜i and ˜B
i at each worker, the master only needs
results from the fastest workers before it can recover the final output, which effectively mitigates
straggler issues. To measure the robustness against straggler effects of a given coding strategy, we
use the metric recovery threshold, defined earlier in Section 1.1, which is equal to the minimum
number of workers that the master needs to wait for in order to compute the output C. Given this
terminology, our main problem is as follows: What is the minimum possible recovery threshold and
the corresponding coding scheme, for any choice of parameters p, m, n, and N?
We propose a novel coding technique, referred to as entangled polynomial code, which achieves
the recovery threshold of pmn + p − 1 for all possible parameter values. The construction of the
entangled polynomial code is based on the observation that when multiplying an m-by-p matrix and
a p-by-n matrix, we essentially evaluate a subspace of bilinear functions, spanned by the pairwise
product of the elements from the two matrices. Although potentially there are a total of p2mn
pairs of elements, at most pmn pairs are directly related to the matrix product, which is an order of
p less. The particular structure of the proposed code entangles the input matrices to the output
such that the system almost avoids unnecessary multiplications and achieves a recovery threshold
in the order of pmn, while allowing robust straggler mitigation for arbitrarily large systems. This
allows orderwise improvement upon conventional uncoded approaches, random linear codes, and
MDS-coding type approaches for straggler mitigation [1, 3].
27
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 42 |
| Full text | . . . Figure 4.1: Overview of the distributed matrix multiplication problem. Each worker computes the product of the two stored encoded submatrices (A˜i and ˜B i) and returns the result to the master. By carefully designing the coding strategy, the master can decode the multiplication result of the input matrices from a subset of workers, without having to wait for stragglers (worker 1 in this example). can choose p, m, and n accordingly. We aim to find optimal coding and computation designs for any choice of parameters p, m and n, to provide optimum straggler effect mitigation for various situations. With a careful design of the coded submatrices A˜i and ˜B i at each worker, the master only needs results from the fastest workers before it can recover the final output, which effectively mitigates straggler issues. To measure the robustness against straggler effects of a given coding strategy, we use the metric recovery threshold, defined earlier in Section 1.1, which is equal to the minimum number of workers that the master needs to wait for in order to compute the output C. Given this terminology, our main problem is as follows: What is the minimum possible recovery threshold and the corresponding coding scheme, for any choice of parameters p, m, n, and N? We propose a novel coding technique, referred to as entangled polynomial code, which achieves the recovery threshold of pmn + p − 1 for all possible parameter values. The construction of the entangled polynomial code is based on the observation that when multiplying an m-by-p matrix and a p-by-n matrix, we essentially evaluate a subspace of bilinear functions, spanned by the pairwise product of the elements from the two matrices. Although potentially there are a total of p2mn pairs of elements, at most pmn pairs are directly related to the matrix product, which is an order of p less. The particular structure of the proposed code entangles the input matrices to the output such that the system almost avoids unnecessary multiplications and achieves a recovery threshold in the order of pmn, while allowing robust straggler mitigation for arbitrarily large systems. This allows orderwise improvement upon conventional uncoded approaches, random linear codes, and MDS-coding type approaches for straggler mitigation [1, 3]. 27 |
