skip to main content


Title: Coded Distributed Computing: Performance Limits and Code Designs
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 the execution time using binary random linear codes and the best execution time any linear-coded distributed computing system can achieve. It is also shown that there exist good binary linear codes that attain, asymptotically, the best performance any linear code, not necessarily binary, can achieve. We also investigate the performance of coded distributed computing systems using polar and Reed-Muller (RM) codes that can benefit from low-complexity decoding, and superior performance, respectively, as well as explicit constructions. The proposed framework in this paper can enable efficient designs of distributed computing systems given the rich literature in the channel coding theory.  more » « less
Award ID(s):
1763348
NSF-PAR ID:
10177956
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
2019 IEEE Information Theory Workshop (ITW)
Page Range / eLocation ID:
1 to 5
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Obeid, I. ; Selesnik, I. ; Picone, J. (Ed.)
    The Neuronix high-performance computing cluster allows us to conduct extensive machine learning experiments on big data [1]. This heterogeneous cluster uses innovative scheduling technology, Slurm [2], that manages a network of CPUs and graphics processing units (GPUs). The GPU farm consists of a variety of processors ranging from low-end consumer grade devices such as the Nvidia GTX 970 to higher-end devices such as the GeForce RTX 2080. These GPUs are essential to our research since they allow extremely compute-intensive deep learning tasks to be executed on massive data resources such as the TUH EEG Corpus [2]. We use TensorFlow [3] as the core machine learning library for our deep learning systems, and routinely employ multiple GPUs to accelerate the training process. Reproducible results are essential to machine learning research. Reproducibility in this context means the ability to replicate an existing experiment – performance metrics such as error rates should be identical and floating-point calculations should match closely. Three examples of ways we typically expect an experiment to be replicable are: (1) The same job run on the same processor should produce the same results each time it is run. (2) A job run on a CPU and GPU should produce identical results. (3) A job should produce comparable results if the data is presented in a different order. System optimization requires an ability to directly compare error rates for algorithms evaluated under comparable operating conditions. However, it is a difficult task to exactly reproduce the results for large, complex deep learning systems that often require more than a trillion calculations per experiment [5]. This is a fairly well-known issue and one we will explore in this poster. Researchers must be able to replicate results on a specific data set to establish the integrity of an implementation. They can then use that implementation as a baseline for comparison purposes. A lack of reproducibility makes it very difficult to debug algorithms and validate changes to the system. Equally important, since many results in deep learning research are dependent on the order in which the system is exposed to the data, the specific processors used, and even the order in which those processors are accessed, it becomes a challenging problem to compare two algorithms since each system must be individually optimized for a specific data set or processor. This is extremely time-consuming for algorithm research in which a single run often taxes a computing environment to its limits. Well-known techniques such as cross-validation [5,6] can be used to mitigate these effects, but this is also computationally expensive. These issues are further compounded by the fact that most deep learning algorithms are susceptible to the way computational noise propagates through the system. GPUs are particularly notorious for this because, in a clustered environment, it becomes more difficult to control which processors are used at various points in time. Another equally frustrating issue is that upgrades to the deep learning package, such as the transition from TensorFlow v1.9 to v1.13, can also result in large fluctuations in error rates when re-running the same experiment. Since TensorFlow is constantly updating functions to support GPU use, maintaining an historical archive of experimental results that can be used to calibrate algorithm research is quite a challenge. This makes it very difficult to optimize the system or select the best configurations. The overall impact of all of these issues described above is significant as error rates can fluctuate by as much as 25% due to these types of computational issues. Cross-validation is one technique used to mitigate this, but that is expensive since you need to do multiple runs over the data, which further taxes a computing infrastructure already running at max capacity. GPUs are preferred when training a large network since these systems train at least two orders of magnitude faster than CPUs [7]. Large-scale experiments are simply not feasible without using GPUs. However, there is a tradeoff to gain this performance. Since all our GPUs use the NVIDIA CUDA® Deep Neural Network library (cuDNN) [8], a GPU-accelerated library of primitives for deep neural networks, it adds an element of randomness into the experiment. When a GPU is used to train a network in TensorFlow, it automatically searches for a cuDNN implementation. NVIDIA’s cuDNN implementation provides algorithms that increase the performance and help the model train quicker, but they are non-deterministic algorithms [9,10]. Since our networks have many complex layers, there is no easy way to avoid this randomness. Instead of comparing each epoch, we compare the average performance of the experiment because it gives us a hint of how our model is performing per experiment, and if the changes we make are efficient. In this poster, we will discuss a variety of issues related to reproducibility and introduce ways we mitigate these effects. For example, TensorFlow uses a random number generator (RNG) which is not seeded by default. TensorFlow determines the initialization point and how certain functions execute using the RNG. The solution for this is seeding all the necessary components before training the model. This forces TensorFlow to use the same initialization point and sets how certain layers work (e.g., dropout layers). However, seeding all the RNGs will not guarantee a controlled experiment. Other variables can affect the outcome of the experiment such as training using GPUs, allowing multi-threading on CPUs, using certain layers, etc. To mitigate our problems with reproducibility, we first make sure that the data is processed in the same order during training. Therefore, we save the data from the last experiment and to make sure the newer experiment follows the same order. If we allow the data to be shuffled, it can affect the performance due to how the model was exposed to the data. We also specify the float data type to be 32-bit since Python defaults to 64-bit. We try to avoid using 64-bit precision because the numbers produced by a GPU can vary significantly depending on the GPU architecture [11-13]. Controlling precision somewhat reduces differences due to computational noise even though technically it increases the amount of computational noise. We are currently developing more advanced techniques for preserving the efficiency of our training process while also maintaining the ability to reproduce models. In our poster presentation we will demonstrate these issues using some novel visualization tools, present several examples of the extent to which these issues influence research results on electroencephalography (EEG) and digital pathology experiments and introduce new ways to manage such computational issues. 
    more » « less
  2. Obeid, Iyad ; Selesnick, Ivan ; Picone, Joseph (Ed.)
    The goal of this work was to design a low-cost computing facility that can support the development of an open source digital pathology corpus containing 1M images [1]. A single image from a clinical-grade digital pathology scanner can range in size from hundreds of megabytes to five gigabytes. A 1M image database requires over a petabyte (PB) of disk space. To do meaningful work in this problem space requires a significant allocation of computing resources. The improvements and expansions to our HPC (highperformance computing) cluster, known as Neuronix [2], required to support working with digital pathology fall into two broad categories: computation and storage. To handle the increased computational burden and increase job throughput, we are using Slurm [3] as our scheduler and resource manager. For storage, we have designed and implemented a multi-layer filesystem architecture to distribute a filesystem across multiple machines. These enhancements, which are entirely based on open source software, have extended the capabilities of our cluster and increased its cost-effectiveness. Slurm has numerous features that allow it to generalize to a number of different scenarios. Among the most notable is its support for GPU (graphics processing unit) scheduling. GPUs can offer a tremendous performance increase in machine learning applications [4] and Slurm’s built-in mechanisms for handling them was a key factor in making this choice. Slurm has a general resource (GRES) mechanism that can be used to configure and enable support for resources beyond the ones provided by the traditional HPC scheduler (e.g. memory, wall-clock time), and GPUs are among the GRES types that can be supported by Slurm [5]. In addition to being able to track resources, Slurm does strict enforcement of resource allocation. This becomes very important as the computational demands of the jobs increase, so that they have all the resources they need, and that they don’t take resources from other jobs. It is a common practice among GPU-enabled frameworks to query the CUDA runtime library/drivers and iterate over the list of GPUs, attempting to establish a context on all of them. Slurm is able to affect the hardware discovery process of these jobs, which enables a number of these jobs to run alongside each other, even if the GPUs are in exclusive-process mode. To store large quantities of digital pathology slides, we developed a robust, extensible distributed storage solution. We utilized a number of open source tools to create a single filesystem, which can be mounted by any machine on the network. At the lowest layer of abstraction are the hard drives, which were split into 4 60-disk chassis, using 8TB drives. To support these disks, we have two server units, each equipped with Intel Xeon CPUs and 128GB of RAM. At the filesystem level, we have implemented a multi-layer solution that: (1) connects the disks together into a single filesystem/mountpoint using the ZFS (Zettabyte File System) [6], and (2) connects filesystems on multiple machines together to form a single mountpoint using Gluster [7]. ZFS, initially developed by Sun Microsystems, provides disk-level awareness and a filesystem which takes advantage of that awareness to provide fault tolerance. At the filesystem level, ZFS protects against data corruption and the infamous RAID write-hole bug by implementing a journaling scheme (the ZFS intent log, or ZIL) and copy-on-write functionality. Each machine (1 controller + 2 disk chassis) has its own separate ZFS filesystem. Gluster, essentially a meta-filesystem, takes each of these, and provides the means to connect them together over the network and using distributed (similar to RAID 0 but without striping individual files), and mirrored (similar to RAID 1) configurations [8]. By implementing these improvements, it has been possible to expand the storage and computational power of the Neuronix cluster arbitrarily to support the most computationally-intensive endeavors by scaling horizontally. We have greatly improved the scalability of the cluster while maintaining its excellent price/performance ratio [1]. 
    more » « less
  3. Our extensive real measurements over Amazon EC2 show that the virtual instances often have different computing speeds even if they share the same configurations. This motivates us to study heterogeneous Coded Storage Elastic Computing (CSEC) systems where machines, with different computing speeds, join and leave the network arbitrarily over different computing steps. In CSEC systems, a Maximum Distance Separable (MDS) code is used for coded storage such that the file placement does not have to be re-defined with each elastic event. Computation assignment algorithms are used to minimize the computation time given computation speeds of different machines. While previous studies of heterogeneous CSEC do not include stragglers - the slow machines during the computation, we develop a new framework in heterogeneous CSEC that introduces straggler tolerance. Based on this framework, we design a novel algorithm using our previously proposed approach for heterogeneous CSEC such that the system can handle any subset of stragglers of a specified size while minimizing the computation time. Furthermore, we establish a trade-off in computation time and straggler tolerance. Another major limitation of existing CSEC designs is the lack of practical evaluations using real applications. In this paper, we evaluate the performance of our designs on Amazon EC2 for applications of the power iteration and linear regression. Evaluation results show that the proposed heterogeneous CSEC algorithms outperform the state-of-the-art designs by more than 30%. 
    more » « less
  4. Braverman, Mark (Ed.)
    A longstanding open problem in coding theory is to determine the best (asymptotic) rate R₂(δ) of binary codes with minimum constant (relative) distance δ. An existential lower bound was given by Gilbert and Varshamov in the 1950s. On the impossibility side, in the 1970s McEliece, Rodemich, Rumsey and Welch (MRRW) proved an upper bound by analyzing Delsarte’s linear programs. To date these results remain the best known lower and upper bounds on R₂(δ) with no improvement even for the important class of linear codes. Asymptotically, these bounds differ by an exponential factor in the blocklength. In this work, we introduce a new hierarchy of linear programs (LPs) that converges to the true size A^{Lin}₂(n,d) of an optimum linear binary code (in fact, over any finite field) of a given blocklength n and distance d. This hierarchy has several notable features: 1) It is a natural generalization of the Delsarte LPs used in the first MRRW bound. 2) It is a hierarchy of linear programs rather than semi-definite programs potentially making it more amenable to theoretical analysis. 3) It is complete in the sense that the optimum code size can be retrieved from level O(n²). 4) It provides an answer in the form of a hierarchy (in larger dimensional spaces) to the question of how to cut Delsarte’s LP polytopes to approximate the true size of linear codes. We obtain our hierarchy by generalizing the Krawtchouk polynomials and MacWilliams inequalities to a suitable "higher-order" version taking into account interactions of 𝓁 words. Our method also generalizes to translation schemes under mild assumptions. 
    more » « less
  5. This paper formulates the cache-aided multi-user Private Information Retrieval (MuPIR) problem, including K u cache-equipped users, each of which wishes to retrieve a desired message efficiently from N distributed databases with access to K independent messages. Privacy of the users’ demands requires that any individual database can not learn anything about the demands of the users. The load of this problem is defined as the average number of downloaded bits per desired message bit. The goal is to find the optimal memory-load trade-off while preserving the demand privacy. Besides the formulation of the MuPIR problem, the contribution of this paper is two-fold. First, we characterize the optimal memory-load trade-off for a system with N = 2 databases, K = 2 messages and K u = 2 users demanding distinct messages; Second, a product design with order optimality guarantee is proposed. In addition, the product design can achieve the optimal load when the cache memory is large enough. The product design embeds the well-known Sun-Jafar PIR scheme into coded caching, in order to benefit from the coded caching gain while preserving the privacy of the users’ demands. 
    more » « less