张量是分布式计算（例如机器学习）中的一种基础运算，针对大型数据集，张量运算通常会被拆分为多个并行任务执行。掉队节点（straggler）及其他故障会严重影响整体完成时间。编码计算领域的最新研究提出了一种利用编码任务缓解掉队节点问题的新策略，其目标是最小化恢复整体结果所需的任务数量，该数量被称为恢复阈值。但我们证明，这种严格的组合定义并不能直接优化故障概率。\n 本文聚焦于最可能发生的事件，通过解码概率更直接地衡量编码方案的最优性。基于这种概率方法，我们提出了一种实用的矩阵乘随机构造方法，即局部随机合金码（locally random alloy codes），该编码在上述衡量指标下是最优的。此外，借助这种概率方法，我们还发现了一个同时适用于随机和确定性编码分布式张量的出人意料的不可能性定理。
英文摘要：
Tensors are a fundamental operation in distributed computing, \emph{e.g.,} machine learning, that are commonly distributed into multiple parallel tasks for large datasets. Stragglers and other failures can severely impact the overall completion time. Recent works in coded computing provide a novel strategy to mitigate stragglers with coded tasks, with an objective of minimizing the number of tasks needed to recover the overall result, known as the recovery threshold. However, we demonstrate that this strict combinatorial definition does not directly optimize the probability of failure.
In this paper, we focus on the most likely event and measure the optimality of a coding scheme more directly by its probability of decoding. Our probabilistic approach leads us to a practical construction of random codes for matrix multiplication, i.e., locally random alloy codes, which are optimal with respect to the measures. Furthermore, the probabilistic approach allows us to discover a surprising impossibility theorem about both random and deterministic coded distributed tensors.
Imperial College London（帝国理工学院）；University of Maryland（马里兰大学）
Coded computation techniques provide robustness against straggling servers in distributed computing, with the following limitations: First, they increase decoding complexity. Second, they ignore computations carried out by straggling servers; and they are typically designed to recover the full gradient, and thus, cannot provide a balance between the accuracy of the gradient and per-iteration completion time. Here we introduce a hybrid approach, called coded partial gradient computation (CPGC), that benefits from the advantages of both coded and uncoded computation schemes, and reduces both the…
Flexible Coded Distributed Convolution Computing for Enhanced Straggler Resilience and Numerical Stability in Distributed CNNs
College of Computer Science, Chongqing University（重庆大学计算机学院）；School of Electrical Engineering, Korea University（韩国大学电气工程学院）；School of Electronic Information and Communications, Huazhong University of Science and Technology（华中科技大学电子信息与通信学院）；City University of Hong Kong（香港城市大学）
We consider the recently proposed Coded Distributed Computing (CDC) framework that leverages carefully designed redundant computations to enable coding opportunities that substantially reduce the communication load of distributed computing. We generalize this framework to heterogeneous systems where different nodes in the computing cluster can have different storage (or processing) capabilities. We provide the information-theoretically optimal data set placement and coded data shuffling scheme that minimizes the communication load in a cluster with 3 nodes. For clusters with $K>3$ nodes, we pr…
In this paper, we investigate network nest coded modulation schemes for multiple access relay channels. The performance of the distributed systems which are based on distributed convolutional codes with network coded modulation is presented. An analytical upper bound on bit error probability performance for the studied distributed systems with Maximum Likelihood Sequence Detection (MLSD) is derived. The constructured bounds for the investigated systems are shown to be asymptotically tight for increasing channel Signal-to-Noise Ratio (SNR) values
We consider the problem of coded distributed computing where a large linear computational job, such as a matrix multiplication, is divided into $k$ smaller tasks, encoded using an $(n,k)$ linear code, and performed over $n$ distributed nodes. The goal is to reduce the average execution time of the computational job. We provide a connection between the problem of characterizing the average execution time of a coded distributed computing system and the problem of analyzing the error probability of codes of length $n$ used over erasure channels. Accordingly, we present closed-form expressions for…
Imperial College London（帝国理工学院）；University of Maryland（马里兰大学）
Coded computation techniques provide robustness against straggling workers in distributed computing. However, most of the existing schemes require exact provisioning of the straggling behaviour and ignore the computations carried out by straggling workers. Moreover, these schemes are typically designed to recover the desired computation results accurately, while in many machine learning and iterative optimization algorithms, faster approximate solutions are known to result in an improvement in the overall convergence time. In this paper, we first introduce a novel coded matrix-vector multiplic…
This paper considers the MapReduce-like coded distributed computing framework originally proposed by Li et al., which uses coding techniques when distributed computing servers exchange their computed intermediate values, in order to reduce the overall traffic load. Their original model servers are connected via an error-free common communication bus allowing broadcast transmissions. However, this assumption is one of the major limitations in practice since the practical cloud computing network topologies are far more involved than a simple single bus. We formulate a topological coded distribut…
Content delivery networks store information distributed across multiple servers, so as to balance the load and avoid unrecoverable losses in case of node or disk failures. Coded caching has been shown to be a useful technique which can reduce peak traffic rates by pre-fetching popular content at the end users and encoding transmissions so that different users can extract different information from the same packet. On one hand, distributed storage limits the capability of combining content from different servers into a single message, causing performance losses in coded caching schemes. But, on…
Several network communication problems are highly related such as coded caching and distributed computation. The centralized coded caching focuses on reducing the network burden in peak times in a wireless network system and the coded distributed computation studies the tradeoff between computation and communication in distributed system. In this paper, motivated by the study of the only rainbow $3$-term arithmetic progressions set, we propose a unified framework for constructing coded caching schemes. This framework builds bridges between coded caching schemes and lots of combinatorial object…
Communication overhead is one of the major performance bottlenecks in large-scale distributed computing systems, in particular for machine learning applications. Conventionally, compression techniques are used to reduce the load of communication by combining intermediate results of the same computation task as much as possible. Recently, via the development of coded distributed computing (CDC), it has been shown that it is possible to enable coding opportunities across intermediate results of different computation tasks to further reduce the communication load. We propose a new scheme, named c…
Building on the previous work of Lee et al. and Ferdinand et al. on coded computation, we propose a sequential approximation framework for solving optimization problems in a distributed manner. In a distributed computation system, latency caused by individual processors ("stragglers") usually causes a significant delay in the overall process. The proposed method is powered by a sequential computation scheme, which is designed specifically for systems with stragglers. This scheme has the desirable property that the user is guaranteed to receive useful (approximate) computation results whenever …
Coded distributed computing introduced by Li et al. in 2015 is an efficient approach to trade computing power to reduce the communication load in general distributed computing frameworks such as MapReduce. In particular, Li et al. show that increasing the computation load in the Map phase by a factor of $r$ can create coded multicasting opportunities to reduce the communication load in the Reduce phase by the same factor. However, there are two major limitations in practice. First, it requires an exponentially large number of input files (data batches) when the number of computing nodes gets l…
In large scale distributed linear transform problems, coded computation plays an important role to effectively deal with "stragglers" (distributed computations that may get delayed due to few slow or faulty processors). We propose a coded computation strategy, referred to as diagonal code, that achieves the optimum recovery threshold and the optimum computation load. This is the first code that simultaneously achieves two-fold optimality in coded distributed linear transforms. Furthermore, by leveraging the idea from random proposal graph theory, we design two random codes that can guarantee o…
In this paper, we consider a distributed reception scenario where a transmitter broadcasts a signal to multiple geographically separated receive nodes over fading channels, and each node forwards a few bits representing a processed version of the received signal to a fusion center. The fusion center then tries to decode the transmitted signal based on the forwarded information from the receive nodes and possible channel state information. We show that there is a strong connection between the problem of minimizing a symbol error probability at the fusion center in distributed reception and chan…
The performance of replication-based distributed databases is affected due to non-uniform storage across storage nodes (also called \textit{data skew}) and reduction in the replication factor during operation, particularly due to node additions or removals. Data rebalancing refers to the communication involved between the nodes in correcting this data skew, while maintaining the replication factor. For carefully designed distributed databases, transmitting coded symbols during the rebalancing phase has been recently shown to reduce the communication load of rebalancing. In this work, we look a…
Matrix factorization is an important representation learning algorithm, e.g., recommender systems, where a large matrix can be factorized into the product of two low dimensional matrices termed as latent representations. This paper investigates the problem of matrix factorization in distributed computing systems with stragglers, those compute nodes that are slow to return computation results. A computation procedure, called coded Alternative Least Square (ALS), is proposed for mitigating the effect of stragglers in such systems. The coded ALS algorithm iteratively computes two low dimensional …
Matrix multiplication is a fundamental building block for large scale computations arising in various applications, including machine learning. There has been significant recent interest in using coding to speed up distributed matrix multiplication, that are robust to stragglers (i.e., machines that may perform slower computations). In many scenarios, instead of exact computation, approximate matrix multiplication, i.e., allowing for a tolerable error is also sufficient. Such approximate schemes make use of randomization techniques to speed up the computation process. In this paper, we initiat…
We address the use of maximum distance separable (MDS) codes for distributed storage (DS) to enable efficient content delivery in wireless networks. Content is stored in a number of the mobile devices and can be retrieved from them using device-to-device communication or, alternatively, from the base station (BS). We derive an analytical expression for the download delay in the hypothesis that the reliability state of the network is periodically restored. Our analysis shows that MDS-coded DS can dramatically reduce the download time with respect to the reference scenario where content is alway…
A user wants to retrieve a file from a database without revealing the identity of the file retrieved at the database, which is known as the problem of private information retrieval (PIR). If it is further required that the user obtains no information about the database other than the desired file, the concept of symmetric private information retrieval (SPIR) is introduced to guarantee privacy for both parties. In this paper, the problem of SPIR is studied for a database stored among $N$ nodes in a distributed way, by using an $(N,M)$-MDS storage code. The information-theoretic capacity of SPIR…
We propose a unified coded framework for distributed computing with straggling servers, by introducing a tradeoff between "latency of computation" and "load of communication" for some linear computation tasks. We show that the coded scheme of [1]-[3] that repeats the intermediate computations to create coded multicasting opportunities to reduce communication load, and the coded scheme of [4], [5] that generates redundant intermediate computations to combat against straggling servers can be viewed as special instances of the proposed framework, by considering two extremes of this tradeoff: mini…
Coded Distributed Computing (CDC) introduced by Li et al. in 2015 offers an efficient approach to trade computing power to reduce the communication load in general distributed computing frameworks such as MapReduce and Spark. In particular, increasing the computation load in the Map phase by a factor of r can create coded multicasting opportunities to reduce the communication load in the Shuffle phase by the same factor. However, the CDC scheme is designed for the homogeneous settings, where the storage, computation load and communication load on the computing nodes are the same. In addition, …
Imperial College London（帝国理工学院）；University of Maryland（马里兰大学）
Coded computation can be used to speed up distributed learning in the presence of straggling workers. Partial recovery of the gradient vector can further reduce the computation time at each iteration; however, this can result in biased estimators, which may slow down convergence, or even cause divergence. Estimator bias will be particularly prevalent when the straggling behavior is correlated over time, which results in the gradient estimators being dominated by a few fast servers. To mitigate biased estimators, we design a $timely$ dynamic encoding framework for partial recovery that includes…
Data shuffling between distributed cluster of nodes is one of the critical steps in implementing large-scale learning algorithms. Randomly shuffling the data-set among a cluster of workers allows different nodes to obtain fresh data assignments at each learning epoch. This process has been shown to provide improvements in the learning process. However, the statistical benefits of distributed data shuffling come at the cost of extra communication overhead from the master node to worker nodes, and can act as one of the major bottlenecks in the overall time for computation. There has been signifi…
Li {\it et al}. introduced coded distributed computing (CDC) scheme to reduce the communication load in general distributed computing frameworks such as MapReduce. They also proposed cascaded CDC schemes where each output function is computed multiple times, and proved that such schemes achieved the fundamental trade-off between computation load and communication load. However, these schemes require exponentially large numbers of input files and output functions when the number of computing nodes gets large. In this paper, by using the structure of placement delivery arrays (PDAs), we construc…
In this paper, we propose a new rateless coded cooperation scheme for a general multi-user cooperative wireless system. We develop cooperation methods based on Raptor codes with the assumption that the channels face erasure with specific erasure probabilities and transmitters have no channel state information. A fully coded cooperation (FCC) and a partially coded cooperation (PCC) strategy are developed to maximize the average system throughput. Both PCC and FCC schemes have been analyzed through AND-OR tree analysis and a linear programming optimization problem is then formulated to find the …
We consider the problem of coded distributed computing where a large linear computational job, such as a matrix multiplication, is divided into $k$ smaller tasks, encoded using an $(n,k)$ linear code, and performed over $n$ distributed nodes. The goal is to reduce the average execution time of the computational job. We provide a connection between the problem of characterizing the average execution time of a coded distributed computing system and the problem of analyzing the error probability of codes of length $n$ used over erasure channels. Accordingly, we present closed-form expressions for…
Coded distributed computing (CDC) introduced by Li et al. in 2015 offers an efficient approach to trade computing power to reduce the communication load in general distributed computing frameworks such as MapReduce. For the more general cascaded CDC, Map computations are repeated at $r$ nodes to significantly reduce the communication load among nodes tasked with computing $Q$ Reduce functions $s$ times. While an achievable cascaded CDC scheme was proposed, it only operates on homogeneous networks, where the storage, computation load and communication load of each computing node is the same. In…
Distributed storage systems are known to be susceptible to long tails in response time. In modern online storage systems such as Bing, Facebook, and Amazon, the long tails of the service latency are of particular concern. with 99.9th percentile response times being orders of magnitude worse than the mean. As erasure codes emerge as a popular technique to achieve high data reliability in distributed storage while attaining space efficiency, taming tail latency still remains an open problem due to the lack of mathematical models for analyzing such systems. To this end, we propose a framework for…
Cache-aided coded multicast leverages side information at wireless edge caches to efficiently serve multiple groupcast demands via common multicast transmissions, leading to load reductions that are proportional to the aggregate cache size. However, the increasingly unpredictable and personalized nature of the content that users consume challenges the efficiency of existing caching-based solutions in which only exact content reuse is explored. This paper generalizes the cache-aided coded multicast problem to a source compression with distributed side information problem that specifically accou…
Private Information Retrieval (PIR) was first proposed by B. Chor, O. Goldreich, E. Kushilevitz and M. Sudan in their 1995 FOCS paper. For MDS coded distributed storage system private information retrieval was proposed and the capacity of PIR schemes for MDS coded distributed storage was studied. Star product PIR schemes from general coded distributed storage system with colluding servers were constructed over general finite fields. These star product schemes has no restriction on the sizes of fields and can be constructed for coded distributed storage across large number of servers. In this p…
Training a machine learning model is both compute and data-intensive. Most of the model training is performed on high performance compute nodes and the training data is stored near these nodes for faster training. But there is a growing interest in enabling training near the data. For instance, mobile devices are rich sources of training data. It may not be feasible to consolidate the data from mobile devices into a cloud service, due to bandwidth and data privacy reasons. Training at mobile devices is however fraught with challenges. First mobile devices may join or leave the distributed sett…
The growing size of modern datasets necessitates splitting a large scale computation into smaller computations and operate in a distributed manner. Adversaries in a distributed system deliberately send erroneous data in order to affect the computation for their benefit. Boolean functions are the key components of many applications, e.g., verification functions in blockchain systems and design of cryptographic algorithms. We consider the problem of computing a Boolean function in a distributed computing system with particular focus on \emph{security against Byzantine workers}. Any Boolean funct…
Coded computation is a framework which provides redundancy in distributed computing systems to speed up largescale tasks. Although most existing works assume an error-free scenarios in a master-worker setup, the link failures are common in current wired/wireless networks. In this paper, we consider the straggler problem in coded distributed computing with link failures, by modeling the links between the master node and worker nodes as packet erasure channels. When the master fails to detect the received signal, retransmission is required for each worker which increases the overall run-time to …
We propose a coded computing strategy and examine communication costs of coded computing algorithms to make distributed Fast Fourier Transform (FFT) resilient to errors during the computation. We apply maximum distance separable (MDS) codes to a widely used "Transpose" algorithm for parallel FFT. In the uncoded distributed FFT algorithm, the most expensive step is a single "all-to-all" communication. We compare this with communication overhead of coding in our coded FFT algorithm. We show that by using a systematic MDS code, the communication overhead of coding is negligible in comparison with…
We consider the data shuffling problem in a distributed learning system, in which a master node is connected to a set of worker nodes, via a shared link, in order to communicate a set of files to the worker nodes. The master node has access to a database of files. In every shuffling iteration, each worker node processes a new subset of files, and has excess storage to partially cache the remaining files, assuming the cached files are uncoded. The caches of the worker nodes are updated every iteration, and they should be designed to satisfy any possible unknown permutation of the files in subse…
Distributed computation is a framework used to break down a complex computational task into smaller tasks and distributing them among computational nodes. Erasure correction codes have recently been introduced and have become a popular workaround to the well known ``straggling nodes'' problem, in particular, by matching linear coding for linear computation tasks. It was observed that decoding tends to amplify the computation ``noise'', i.e., the numerical errors at the computation nodes. We propose taking advantage of the case that more nodes return than minimally required. We show how a cleve…
Distributed computing platforms typically assume the availability of reliable and dedicated connections among the processors. This work considers an alternative scenario, relevant for wireless data centers and federated learning, in which the distributed processors, operating on generally distinct coded data, are connected via shared wireless channels accessed via full-duplex transmission. The study accounts for both wireless and computing impairments, including interference, imperfect Channel State Information, and straggling processors, and it assumes a Map-Shuffle-Reduce coded computing par…
Today's data centers have an abundance of computing resources, hosting server clusters consisting of as many as tens or hundreds of thousands of machines. To execute a complex computing task over a data center, it is natural to distribute computations across many nodes to take advantage of parallel processing. However, as we allocate more and more computing resources to a computation task and further distribute the computations, large amounts of (partially) computed data must be moved between consecutive stages of computation tasks among the nodes, hence the communication load can become the b…
Coded distributed computing framework enables large-scale machine learning (ML) models to be trained efficiently in a distributed manner, while mitigating the straggler effect. In this work, we consider a multi-task assignment problem in a coded distributed computing system, where multiple masters, each with a different matrix multiplication task, assign computation tasks to workers with heterogeneous computing capabilities. Both dedicated and probabilistic worker assignment models are considered, with the objective of minimizing the average completion time of all computations. For dedicated w…
We consider private information retrieval (PIR) for distributed storage systems (DSSs) with noncolluding nodes where data is stored using a non maximum distance separable (MDS) linear code. It was recently shown that if data is stored using a particular class of non-MDS linear codes, the MDS-PIR capacity, i.e., the maximum possible PIR rate for MDS-coded DSSs, can be achieved. For this class of codes, we prove that the PIR capacity is indeed equal to the MDS-PIR capacity, giving the first family of non-MDS codes for which the PIR capacity is known. For other codes, we provide asymmetric PIR pr…
An adaptive joint power allocation (JPA) and linear receiver design algorithm using the minimum bit error rate (MBER) criterion for a cooperative Multiple-Input Multiple-Output (MIMO) network is proposed. The system employs multiple relays with Distributed Space-Time Coding (DSTC) schemes and an Amplify-and-Forward (AF) strategy. It is designed according to a joint constrained optimization algorithm to determine the MBER power allocation parameters and the receive filter parameters for each transmitted symbol. The simulation results indicate that the proposed algorithm obtains performance gain…
We investigate the use of maximum distance separable (MDS) codes to cache popular content to reduce the download delay of wireless content delivery. In particular, we consider a cellular system where devices roam in an out of a cell according to a Poisson random process. Popular content is cached in a limited number of the mobile devices using an MDS code and can be downloaded from the mobile devices using device-to-device communication. We derive an analytical expression for the delay incurred in downloading content from the wireless network and show that distributed caching using MDS codes c…
Privacy-aware Berrut Approximated Coded Computing applied to general distributed learning
Galician Research and Development Center in Advanced Telecommunications (GRADIANT)（加利西亚先进电信研究与发展中心（GRADIANT））；atlanTTic, Information & Computing Lab, Telecommunication Engineering School, Universidade de Vigo（atlanTTic信息与计算实验室，电信工程学院，维戈大学）
Erasure codes, such as Reed-Solomon (RS) codes, are being increasingly employed in data centers to combat the cost of reliably storing large amounts of data. Although these codes provide optimal storage efficiency, they require significantly high network and disk usage during recovery of missing data. In this paper, we first present a study on the impact of recovery operations of erasure-coded data on the data-center network, based on measurements from Facebook's warehouse cluster in production. To the best of our knowledge, this is the first study of its kind available in the literature. Our …
This work explores a distributed computing setting where $K$ nodes are assigned fractions (subtasks) of a computational task in order to perform the computation in parallel. In this setting, a well-known main bottleneck has been the inter-node communication cost required to parallelize the task, because unlike the computational cost which could keep decreasing as $K$ increases, the communication cost remains approximately constant, thus bounding the total speedup gains associated to having more computing nodes. This bottleneck was substantially ameliorated by the recent introduction of coded M…
In this paper, we consider a hierarchical distributed multi-task learning (MTL) system where distributed users wish to jointly learn different models orchestrated by a central server with the help of a layer of multiple relays. Since the users need to download different learning models in the downlink transmission, the distributed MTL suffers more severely from the communication bottleneck compared to the single-task learning system. To address this issue, we propose a coded hierarchical MTL scheme that exploits the connection topology and introduces coding techniques to reduce communication l…
