Hierarchical Coding for Distributed Computing

Transcription

1 Hierarchical Coding for Distributed Computing Hyegyeong Park, Kangwook Lee, Jy-yong Sohn, Changho Suh and Jaekyun Moon School of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST) {parkh, kwjjang, jysohn08, Abstract Coding for distributed computing supports lowlatency computation by relieving the burden of straggling workers. hile most existing works assume a simple master-worker model, we consider a hierarchical computational structure consisting of groups of workers, motivated by the need to reflect the architectures of real-world distributed computing systems. In this work, we propose a hierarchical coding scheme for this model, as well as analyze its decoding cost and expected computation time. Specifically, we first provide upper and lower bounds on the expected computing time of the proposed scheme. e also show that our scheme enables efficient parallel decoding, thus reducing decoding costs by orders of magnitude over non-hierarchical schemes. hen considering both decoding cost and computing time, the proposed hierarchical coding is shown to outperform existing schemes in many practical scenarios. I. INTRODUCTION Enabling large-scale computations for big data analytics, distributed computing systems have received significant attention in recent years []. The distributed computing system divides a computational task to a number of subtasks, each of which is allocated to a different worker. This helps reduce computing time by exploiting parallel computing options and thus enables handling of large-scale computing tasks. In a distributed computing system, the stragglers, which refers to the computing nodes that slow down in some random fashion due to a variety of factors, may increase the total runtime of the computing system. To address this problem, the notion of coded computation is introduced in [] where an (n, k) maximum distance separable (MDS) code is employed to speed up distributed matrix multiplications. The authors show that for linear computing tasks, one can design n distributed computing tasks such that any k out of n tasks suffice to complete the assigned task. Since then, coded computation has been applied to a wide variety of task scenarios such as matrix-matrix multiplication [3], [4], distributed gradient computation [5] [8], convolution [9], Fourier transform [0] and matrix sparsification [], []. hile the idea of coded computation has been studied in various settings, existing works have not taken into account the underlying hierarchical nature of practical distributed systems [3] [5]. In modern distributed computing systems, each group of workers is collocated in the same rack, which contains a Top of Rack (ToR) switch, and cross-rack communication is available only via these ToR switches. Surveys on real cloud computing systems show that cross-rack communication through the ToR switches is highly unstable due to the limited bandwidth, whereas intra-rack communication is faster and more reliable [4], [5]. A natural question is whether one -master communication Completion time of the worker... M Fig.. Illustration of the hierarchical computing system... can devise a coded computation scheme that exploits such hierarchical structure. A. Contribution In this work, we first model a distributed computing system with a tree-like hierarchical structure illustrated in Fig., which is inspired by the practical computing systems in [3] [5]. The workers (denoted by ) are divided into groups, each of which has a submaster (denoted by ). Each submaster sends the computational result of its group to the master (denoted by M ). The suggested model can be viewed as a generalization of the existing non-hierarchical coded computation. In this framework, we propose a hierarchical coding scheme which employs an (n (i), k(i) ) MDS code within group i and another (n, k ) outer MDS code across the groups as depicted in Fig.. e also develop a parallel decoding algorithm which exploits the concatenated code structure and allows low complexity. Moreover, we analyze the latency performance of our proposed solution. It turns out that the latency performance of our scheme cannot be analyzed via simple order statistics as in other existing schemes. Here we resort to find lower and upper bounds on the average latency performance: Our upper bound relies on concentration inequalities, and our lower bound is obtained via constructing and analyzing an auxiliary Markov chain. B. Related ork Previous works on coded computation have rarely considered the inherent hierarchical structure of most real-world systems. hereas a very recent work [6] deals with the multirack computing system reflecting imbalance between intraand cross-rack communications, it is based on the settings of the coded MapReduce architecture which do not include general linear computation tasks that we focus on in this work. Another distinction is that the analysis of [6] includes only

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sha_base64="qxr8aegc8e7sylrxfym7vici=">aaab8xicbvbns8naej3urq/qh69lbahqiljl3osepfywbzig8jmoxbjzhdyou0h/hxymixv033vw3btsctpxbwoo9gbmhyngrjutpynza3inulvbdw6pyscnhrnimgbxsjdyhvkljetufg4eoikeahwg44uzn73sdumsfy3kwt9cm6knzigtveqzkofejk6bxgzqrbtdgkwtlycvynekyl/9qczsckvhgmrd89ze+blvhjobsi/zhqnqej7fkqaytazxyxz8ifvqzkgctb0pcf+nsio5h0yi0nrey73qzcx/vf5qhtd+xmsgprsuiycmjimn+fdlhczstuesout7csnqakmmndktkqvnx0mnuffcunfnvpqpi4inme5vmgdkjclbsgdqwkpmmrvdnaexheny9la8hjz07hd5zph+cjw=</latexit> MDS coding across groups w(, ) w(, ) w(,n () <latexit 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sha_base64="imnazmb7cw+yugn5nyknfkt3ua=">aaab7xicbvbntwixej3fl8qvkoxrmkccsg7xpri4sujjrkqwiz0sxcq3xbtdjvkw3/w4kfjvpp/vplvllahbv8yyct7m5mzfyacaeo6305hy3nre6e49rbpzg8kh+f+fqmitakvyqbog5uzqtmgg06iki5dtjvh5gbudx6p0kykeznnabdjkari9hyyx+qnmre5abccevuamidedmpqi7opzvh0qsxlqywrhpc9ntjbhzrjhdfbqp5ommezwipysftimosg87qhvgkjlkljboof6eyhcs9tqobemzvivenpxp6+xmug6yjhiukmfs6kuo6mrppx0zapsgyfokjyvzrmzyyjsqcubgrf68jrxg3xprxt3bqvzy+mowhmcqxu8uiim3eilkdgaz7hfd4c6bw4787hsrxg5don8afo5w/td43z</latexit> w(,n () ) Master ea i =[ e A i,; e A i,] b A,x ba,x (3, ) MDS ba,x ba,x [ b A i,; b A i,; b A i,3 := b A i, + b A i,] <latexit 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sha_base64="eqtstc9kwqtvojbgm03kge/urg=">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</latexit> <latexit sha_base64="i0htm8l8gsrmhev5fgtye5yaza=">aaaclhicdvfnaxsxenvup5k6h3fa6kuxtazqmmb7qgb5jascjvfookyc9gk8/awlpplcmeu/iz+u/6d/izdobr8apx4qpn6b0tzezksdppktxq/epjo8c7uk9btz89f7lx3x44ukbfgusvczd6ckhj5kvhbvubfpuaym50+uuvse4a/rmxjaqfnizs8exukp9d445hyreefhynv5/8x5uy0pmj+kreep9u0t74bhxpmm+3njf/sfxr+xtu7itdzfn0pmbr0chroh+/w7hrlqfabskozdgsylpzskosa4qhyuxmz4baybal6as+tlfj6+d8yy5sagp5eux8nal44tyiy0nk4dztaq/5pgsyh6gfomvquv4pioyt6bjaugggtahzxbkxvtbrnacle75q/0rjlfiigbqzxhz0uizpsh+sc3q4dmmxvchvyafcygdysr6rc9ingtxeb6op0uh8oj6jz+kvq9y4s+8incq/n4laaxncq==</latexit> ba,3x =( b A, + b A,)x ba,3x =( b A, + b A,)x n w(n, ) w(n, ) <latexit sha_base64="pnbe6kotf49j8hzklo/pqejdo=">aaab73icbvbnswmxej3urqrvj6crahqim7veix4mvjc/ydqvk0wbmsusvyps3+d4mdir36d7z5b0w/dtr6yodx3gwz8/xycg0c5xtltrz3dvey+7md/ohrcehktkjrfhpjgivmcnmgkudnwignvoyevmbtf3w799uptgkeyxszizkxkqhkaafeknzvjl9atm96hektsvzag8sd0ktfxzywya9x7hqzeiabiyaaggnddjzzespthvlbprpdofhm6jkptvsskgkvxdw7xzdgeagurakwqv90rkqq0no87qjgetbi/953cqen7kzzwyjulyuzaibci8fx4pugluiiklhcpub8v0rbshxkausyg46y9vkla4jovthtkmmstihcyibc9dqgzuoqxmochibn3hhd+gvfadzsjdvjnn8afo8wffcjcg</latexit> sha_base64="8oilhb4+cinrs8qmyodvwmmu=">aaab73icbvdlsgnbeoynrxijrj0kmhiecchs5qlhgbepczghjeuyncwmqznnzozsljydyixqu8cn6nn39gndwomljqufr093lrzwpbdtfvmptfnzk7dcnu7u3ndg4bkowloxus8lcpkwoz4lndoctijjcebxvsgvo/euulyqg40eoiughuc+yzgrrncf0s0xnfnulm+x7bnqknejf/j3tc+nr5pqt3cz6cxkjigqhoolo7dqtdbevnckettcdnmjkipu0bajaavvumrt3gs6m0kn+ke0jjbq74keb0qna890blgpli3ff/zrhl9eisjvjd5ij/msido+jzqmumj5mndmjhm3iriaetmtikoy0jwll9ejyyybfltskuyq50nampaaby6gatdqhtoq4paaz/bijaxh69v6m7emrmxmefyb9f4dovsyg==</latexit> 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sha_base64="tsdjgtcjmxavejnsmstwcik4yj8=">aaab73icbvdlsgnbeoynrxijrj0kmhiecchs5qlhgbepczghjeuyncwmqznnzozsljydyixqu8cn6nn39gndwomljqufr093lrzwpbdtfvmptfnzk7dcnu7u3ndg4bkowloxus8lcpkwoz4lndoctijjcebxvsgvo/euulyqg40eoiughuc+yzgrrncf0s0xy+fdxn4utogveissl6sva99phfvlu5z04vjhfahsyck9v7ei7czaaeu4nmu6saitjepdpcba6rczhbvbj0zpyf8ujosgs3u3xmjdpqab57pdlaeqgvvkv7ntptx7ojegsqsdzrx7mkq7r9hnuy5iszcegyckzurrazayabnrxotgll+8shrlkmoxnjpjowhzpoeytqeadlxaba6hcnugwoebnuhfglmpqvnm9nyuzi/gd6/0houksyw==</latexit> sha_base64="gotkainqxtyc/u54olzck8jgw=">aaab73icbvdlsgnbeoynrxhfuy9ebomqiytdxpqy8oixgnlasotzywyzhznelvwpkf8ojbea/+jjf/xsnjoikfduvvn9dqskfqdf9dnibmvbo/ndwt7+wefr8fikzejum95ksyxj6cgs6f4ewvk3kk0pegetsy38z89ipxrstqhicj9ym6vciujkkvok9la9vapf9ysmtunoqdeitsqmapslx7bznkikssgtp3at9jgoutpjpozcanlapkpetvtribs/m987jrdgzawryukrn6eykjktgtklcdecrfvm4n9en8xws+esllkii0hakkgjpz8qgngcoj5zqpo9lbarzshjahgq/bx4nrvrvc6venvuqv5zx5oemzqemhlxbh6haugioezxuhnexbenhfny9gac5yzp/ahzucpdcsoq==</latexit> w(n,n (n) ) Submaster 3 ba 3,x =( b A, + b A,)x b A3,x =( b A, + b A,)x Fig.. Illustration of the proposed coding scheme applied to the hierarchical computing system. An (n (i), k(i) ) MDS code is employed within group i, and an (n, k ) MDS code is applied across the groups. w(i, j) denotes worker j in group i. the cross-rack redundancy whereas our analysis considers both intra- and cross-group coding. C. Notations e use boldface uppercase letters for matrices and boldface lowercase letters for vectors. The transpose of a matrix A is denoted by A T. For a matrix A satisfying A T = [A T A T ], we write A = [A ; A ]. For a positive integer n, the set {,,..., n} is denoted by [n]. The j th worker in group i is represented by w(i, j) for i [n ] and j [n (i) ]. The symbol r indicates the largest integer less than or equal to a real number r. II. HIERARCHICAL CODED COMPUTATION A. Proposed Coding Scheme Consider a matrix-vector multiplication task, i.e., computing Ax for a matrix A R m d and a vector x R d. The input matrix A is split into k submatrices as A = [A ; A ;... ; A k ], where A i R m k d for i [k ]. Here we assume that m is divisible by k for simplicity. Then, we apply an (n, k ) MDS code to set {A i } i [k] in obtaining {Ãi} i [n]. Then, each coded matrix Ãi is further divided into k (i) submatrices as Ãi = [Ãi,; Ãi,;... ; m k à i,j R (i) d k (i) for j [k Ãi,k (i)] where ] and m divisible by k(i) k. Afterwards, for each i [n ], we apply an (n (i), k(i) ) MDS code to set {Ãi,j} (i) j [k ] to obtain { i,j } (i) j [n ]. Then, for each i [n ] and j [n (i) ], worker w(i, j) computes  i,j x. Fig. illustrates the proposed coding scheme for the hierarchical computing system with a different number of workers in each group. In the case of n (i) = n and k (i) = k for all i [n ], we will refer this coding scheme as (n, k ) (n, k ) coded computation. e present our code in Fig. 3 via a toy example. In this example, (n, k ) = (n, k ) = (3, ). That is, the input matrix A = [A ; A ] is encoded via (n, k ) = (3, ) MDS code, yielding [Ã; Ã; à + Ã]. Afterwards, the matrix à i = [Ãi,; Ãi,] is encoded via an (n, k ) = (3, ) MDS code, producing [ i, ;  i, ;  i, + i, ]. For notational simplicity, we define Ã3 = à + à and  i,3 =  i, +  i,. Fig. 3. Allocation of the computational task to workers in a (3, ) (3, ) coded computation For i, j {,, 3}, worker w(i, j) computes  i,j x. Note that group i is assigned a subtask with respect to Ãi. e now describe the decoding algorithm for our proposed coding scheme. hen a worker completes its task, it sends the result to its submaster. ith the aid of the (n, k ) MDS code, submaster i (in group i) can compute Ãix as soon as the task results from any k workers within group i are collected. Once Ãix is computed, it is sent to the master. The master can obtain Ax by retrieving Ãix from any k submasters. For each worker, we define completion time as the sum of the runtime of the worker and the time required for delivering its computation result to the submaster. For each group, we further define intragroup latency as the time for completing its assigned subtask. The total computation time is defined as the time from when the workers start to run until the master completes computing Ax. The proposed computation framework can be applied to practical multi-rack systems where the input data A is coded and distributed into n racks; the i th rack contains Ãi. For instance, in the Facebook s warehouse cluster, data is encoded with a (4, 0) MDS code, and then the 4 encoded chunks are stored across different racks [7]. Once x is given from the master, the i th rack can compute Ãix using the coded data à i that it contains. B. Application: Matrix-Matrix Multiplications Our scheme can be also applied to matrix-matrix multiplications. More specifically, consider computing A T B for given matrices A and B = [b b b k ]. After applying an (n, k ) MDS code to B, we have ˇB = [ˇb ˇb ˇb n ]. Moreover, group i divides A into k (i) equal-sized submatrices as A = [A i, A i, A (i) i,k MDS code, resulting in Ǎi ], and we apply an (n (i), k(i) ) = [Ǎi, Ǎi, ]. The Ǎi,n (i) computation ǍT ˇb i,j i is assigned to worker w(i, j). Using an (n (i), k(i) ) MDS code, submaster i can compute AT ˇb i when any k (i) workers within its group delivered their computation results. The master can calculate A T B by gathering A T ˇb i results from any k submasters, using the (n, k ) MDS code. Under the homogeneous setting of n (i) = n and k (i) = k for all i [n ], the encoding algorithm of the proposed scheme reduces to that of the product coded scheme [3]. However, the suggested scheme with the homogeneous setting is shown to reduce the decoding cost compared to the product coded