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4.5 Factor of 2 characterization of Optimum Linear Recovery Thresh-old
In this section, we provide the proof of Theorem 4.3. Specifically, we need to provide a computation
strategy that achieves a recovery threshold of at most 2R(p, m, n) − 1 for all possible values of p, m,
n, and N, as well as a converse result showing that any linear computation strategy requires at
least N R(p, m, n) workers for any p, m, and n.
The proof is accomplished in 2 steps. In Step 1, we show that any linear code for matrix
multiplication is equivalently an upper bound construction of the bilinear complexity R(p, m, n),
and vice versa. This result indicates the equality between R(p, m, n) and the minimum required
number of workers, which proves the needed converse. It also converts any matrix multiplication
into the computation of element-wise products given two vectors of length R(p, m, n). Then in Step
2, we show that we can find an optimal computation strategy for this augmented computing task.
We develop a variation of the entangled polynomial code, which achieves a recovery threshold of
2R(p, m, n) − 1.
For Step 1, we first formally define upper bound constructions for bilinear complexity.
Definition 4.5. Given parameters p, m, n, an upper bound construction for bilinear complexity
R(p, m, n) with rank R is a tuple of tensors a 2 FR×p×m, b 2 FR×p×n, and c 2 FR×m×n such that
for any matrices A 2 Fp×m, B 2 Fp×n,
X
i
cijk
0
@
X
j0,k0
Aj0k0aij0k0
1
A
0
@
X
j00,k00
Bj00k00bij00k00
1
A
=
X
`
A`jB`k. (4.39)
Recall the definition of linear codes. One can verify that any upper bound construction with
rank R is equivalently a linear computing design using R workers when the sizes of input matrices
are given by A 2 Fp×m, B 2 Fp×n. Note that matrix multiplication follows the same rules for any
block matrices, this equivalence holds true for any input sizes.12 Specifically, given an upper bound
construction (a, b, c) with rank R, and for general inputs A 2 Fs×r, B 2 Fs×t, any block of the final
output C can be computed as
Cj,k =
X
i
cijkA˜|
i,vec ˜B
i,vec, (4.40)
12Rigorously, it also requires the linear independence of the A|
iBj ’s, which can be easily proved.
45
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 60 |
| Full text | 4.5 Factor of 2 characterization of Optimum Linear Recovery Thresh-old In this section, we provide the proof of Theorem 4.3. Specifically, we need to provide a computation strategy that achieves a recovery threshold of at most 2R(p, m, n) − 1 for all possible values of p, m, n, and N, as well as a converse result showing that any linear computation strategy requires at least N R(p, m, n) workers for any p, m, and n. The proof is accomplished in 2 steps. In Step 1, we show that any linear code for matrix multiplication is equivalently an upper bound construction of the bilinear complexity R(p, m, n), and vice versa. This result indicates the equality between R(p, m, n) and the minimum required number of workers, which proves the needed converse. It also converts any matrix multiplication into the computation of element-wise products given two vectors of length R(p, m, n). Then in Step 2, we show that we can find an optimal computation strategy for this augmented computing task. We develop a variation of the entangled polynomial code, which achieves a recovery threshold of 2R(p, m, n) − 1. For Step 1, we first formally define upper bound constructions for bilinear complexity. Definition 4.5. Given parameters p, m, n, an upper bound construction for bilinear complexity R(p, m, n) with rank R is a tuple of tensors a 2 FR×p×m, b 2 FR×p×n, and c 2 FR×m×n such that for any matrices A 2 Fp×m, B 2 Fp×n, X i cijk 0 @ X j0,k0 Aj0k0aij0k0 1 A 0 @ X j00,k00 Bj00k00bij00k00 1 A = X ` A`jB`k. (4.39) Recall the definition of linear codes. One can verify that any upper bound construction with rank R is equivalently a linear computing design using R workers when the sizes of input matrices are given by A 2 Fp×m, B 2 Fp×n. Note that matrix multiplication follows the same rules for any block matrices, this equivalence holds true for any input sizes.12 Specifically, given an upper bound construction (a, b, c) with rank R, and for general inputs A 2 Fs×r, B 2 Fs×t, any block of the final output C can be computed as Cj,k = X i cijkA˜| i,vec ˜B i,vec, (4.40) 12Rigorously, it also requires the linear independence of the A| iBj ’s, which can be easily proved. 45 |
