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where each Ck,k0 is exactly the coefficient of the (p − 1 + kp + k0pm)-th degree term. Since all xi’s
are selected to be distinct, recovering C given results from any pmn + p − 1 workers is essentially
interpolating h(x) using pmn + p − 1 distinct points. Because the degree of h(x) is pmn + p − 2,
the output C can always be uniquely decoded.
4.3.3 Computational complexities
In terms of complexity, the decoding process of entangled polynomial code can be viewed as
interpolating a degree pmn + p − 2 polynomial for rt
mn times. It is well known that polynomial
interpolation of degree k has a complexity of O(k log2 k log log k) [42].9 Therefore, decoding entangled
polynomial code only requires at most a complexity of O(prt log2(pmn) log log(pmn)), which is
almost linear to the input size of the decoder ( (prt) elements). This complexity can be reduced
by simply swapping in any faster polynomial interpolation algorithm or Reed-Solomon decoding
algorithm. In addition, this decoding complexity can also be further improved by exploiting the
fact that only a subset of the coefficients are needed for recovering the output matrix.
Note that given the presented computation framework, each worker is assigned to multiply two
coded matrices with sizes of r
m × s
p and s
p × t
n, which requires a complexity of O( srt
pmn).10 This
complexity is independent of the coding design, indicating that the entangled polynomial code
strictly improves other designs without requiring extra computation at the workers. Recall that
the decoding complexity of entangled polynomial code grows linearly with respect to the size of
the output matrix. The decoding overhead becomes negligible compared to workers’ computational
load in practical scenarios where the sizes of coded matrices assigned to the workers are sufficiently
large. Moreover, the fast decoding algorithms enabled by the Polynomial coding approach further
reduces this overhead, compared to general linear coding designs.
Entangled polynomial code also enables improved performances for systems where the data has
to encoded online. For instance, if the input matrices are broadcast to the workers and are encoded
distributedly, the linearity of entangled polynomial code allows for an in-place algorithm, which does
not require addition storage or time complexity. Alternatively, if centralized encoding is required,
almost-linear-time algorithms can also be developed similar to decoding: at most a complexity of
O(( sr
pm log2(pm) log log(pm)+ st
pn log2(pn) log log(pn))N) is required using fast polynomial evaluation,
which is almost linear with respect to the output size of the encoder ( (( sr
pm + st
pn )N) elements).
9When the base field supports FFT, this complexity bound can be improved to O(k log2 k).
10More precisely, the commonly used cubic algorithm achieves a complexity of ( srt
pmn) for the general case. Improved
algorithms has been found in certain cases (e.g., [44,46–54]), however, all known approaches requires a super-quadratic
complexity.
40
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 55 |
| Full text | where each Ck,k0 is exactly the coefficient of the (p − 1 + kp + k0pm)-th degree term. Since all xi’s are selected to be distinct, recovering C given results from any pmn + p − 1 workers is essentially interpolating h(x) using pmn + p − 1 distinct points. Because the degree of h(x) is pmn + p − 2, the output C can always be uniquely decoded. 4.3.3 Computational complexities In terms of complexity, the decoding process of entangled polynomial code can be viewed as interpolating a degree pmn + p − 2 polynomial for rt mn times. It is well known that polynomial interpolation of degree k has a complexity of O(k log2 k log log k) [42].9 Therefore, decoding entangled polynomial code only requires at most a complexity of O(prt log2(pmn) log log(pmn)), which is almost linear to the input size of the decoder ( (prt) elements). This complexity can be reduced by simply swapping in any faster polynomial interpolation algorithm or Reed-Solomon decoding algorithm. In addition, this decoding complexity can also be further improved by exploiting the fact that only a subset of the coefficients are needed for recovering the output matrix. Note that given the presented computation framework, each worker is assigned to multiply two coded matrices with sizes of r m × s p and s p × t n, which requires a complexity of O( srt pmn).10 This complexity is independent of the coding design, indicating that the entangled polynomial code strictly improves other designs without requiring extra computation at the workers. Recall that the decoding complexity of entangled polynomial code grows linearly with respect to the size of the output matrix. The decoding overhead becomes negligible compared to workers’ computational load in practical scenarios where the sizes of coded matrices assigned to the workers are sufficiently large. Moreover, the fast decoding algorithms enabled by the Polynomial coding approach further reduces this overhead, compared to general linear coding designs. Entangled polynomial code also enables improved performances for systems where the data has to encoded online. For instance, if the input matrices are broadcast to the workers and are encoded distributedly, the linearity of entangled polynomial code allows for an in-place algorithm, which does not require addition storage or time complexity. Alternatively, if centralized encoding is required, almost-linear-time algorithms can also be developed similar to decoding: at most a complexity of O(( sr pm log2(pm) log log(pm)+ st pn log2(pn) log log(pn))N) is required using fast polynomial evaluation, which is almost linear with respect to the output size of the encoder ( (( sr pm + st pn )N) elements). 9When the base field supports FFT, this complexity bound can be improved to O(k log2 k). 10More precisely, the commonly used cubic algorithm achieves a complexity of ( srt pmn) for the general case. Improved algorithms has been found in certain cases (e.g., [44,46–54]), however, all known approaches requires a super-quadratic complexity. 40 |
