Parallel implementations of stochastic gradient descent (SGD) have received significant research attention, thanks to its excellent scalability properties. A fundamental barrier when parallelizing SGD is the high bandwidth cost of communicating gradient updates between nodes; consequently, several lossy compresion heuristics have been proposed, by which nodes only communicate quantized gradients. Although effective in practice, these heuristics do not always converge. In this paper, we propose Quantized SGD (QSGD), a family of compression schemes with convergence guarantees and good practical performance. QSGD allows the user to smoothly trade off communication bandwidth and convergence time: nodes can adjust the number of bits sent per iteration, at the cost of possibly higher variance. We show that this trade-off is inherent, in the sense that improving it past some threshold would violate information-theoretic lower bounds. QSGD guarantees convergence for convex and non-convex objectives, under asynchrony, and can be extended to stochastic variance-reduced techniques. When applied to training deep neural networks for image classification and automated speech recognition, QSGD leads to significant reductions in end-to-end training time. For instance, on 16GPUs, we can train the ResNet-152 network to full accuracy on ImageNet 1.8× faster than the full-precision variant. time to the same target accuracy is 2.7×. Further, even computationally-heavy architectures such as Inception and ResNet can benefit from the reduction in communication: on 16GPUs, QSGD reduces the end-to-end convergence time of ResNet152 by approximately 2×. Networks trained with QSGD can converge to virtually the same accuracy as full-precision variants, and that gradient quantization may even slightly improve accuracy in some settings. Related Work. One line of related research studies the communication complexity of convex optimization. In particular, [40] studied two-processor convex minimization in the same model, provided a lower bound of Ω(n(log n + log(1/ ))) bits on the communication cost of n-dimensional convex problems, and proposed a non-stochastic algorithm for strongly convex problems, whose communication cost is within a log factor of the lower bound. By contrast, our focus is on stochastic gradient methods. Recent work [5] focused on round complexity lower bounds on the number of communication rounds necessary for convex learning.Buckwild! [10] was the first to consider the convergence guarantees of low-precision SGD. It gave upper bounds on the error probability of SGD, assuming unbiased stochastic quantization, convexity, and gradient sparsity, and showed significant speedup when solving convex problems on CPUs. QSGD refines these results by focusing on the trade-off between communication and convergence. We view quantization as an independent source of variance for SGD, which allows us to employ standard convergence results [7]. The main differences from Buckw

现代深度学习模型通常在分布式机器集合中并行培训，以减少训练时间。在这种情况下，机器之间模型更新的通信变成了一个重要的性能瓶颈，并且已经提出了各种有损的压缩技术来减轻此问题。在这项工作中，我们介绍了一种新的，简单但理论上和实践上有效的压缩技术：自然压缩（NC）。我们的技术分别应用于要进行压缩的更新向量的所有条目，并通过随机舍入到两个的（负或正）两种功能，可以通过忽略Mantissa来以“自然”方式计算。我们表明，与没有压缩相比，NC将压缩向量的第二刻增加不超过微小因子$ \ frac {9} {8} $，这意味着NC对流行训练算法的收敛速度的影响，例如分布式SGD，可以忽略不计。但是，NC启用的通信节省是可观的，导致$ 3 $ - $ 4 \ times $ $改善整体理论运行时间。对于需要更具侵略性压缩的应用，我们将NC推广到自然抖动，我们证明这比常见的随机抖动技术要好得多。我们的压缩操作员可以自行使用，也可以与现有操作员结合使用，从而产生更具侵略性的结合效果，并在理论和实践中提供新的最先进。

Distributed training of massive machine learning models, in particular deep neural networks, via Stochastic Gradient Descent (SGD) is becoming commonplace. Several families of communication-reduction methods, such as quantization, largebatch methods, and gradient sparsification, have been proposed. To date, gradient sparsification methods-where each node sorts gradients by magnitude, and only communicates a subset of the components, accumulating the rest locally-are known to yield some of the largest practical gains. Such methods can reduce the amount of communication per step by up to three orders of magnitude, while preserving model accuracy. Yet, this family of methods currently has no theoretical justification. This is the question we address in this paper. We prove that, under analytic assumptions, sparsifying gradients by magnitude with local error correction provides convergence guarantees, for both convex and non-convex smooth objectives, for data-parallel SGD. The main insight is that sparsification methods implicitly maintain bounds on the maximum impact of stale updates, thanks to selection by magnitude. Our analysis and empirical validation also reveal that these methods do require analytical conditions to converge well, justifying existing heuristics.

我们考虑一个标准的分布式优化设置，其中$ n $ machines，每个持有$ d $ -dimension函数$ f_i $，旨在共同最大限度地减少函数$ \ sum_ {i = 1} ^ n f_i（x）$ 。该问题自然地出现在大规模分布式优化中，其中标准解决方案是施加（随机）梯度下降的变体。我们专注于这个问题的通信复杂性：我们的主要结果在$ N $ Machines中提供了需要发送和接收的比特总数的第一个完全无条件的界限，以便在点对点通信下解决这个问题给定的差错。具体来说，我们显示$ \ omega（ND \ log d / n \ varepsilon）$总比特在机器之间传达，找到一个添加剂$ \ epsilon $-xprupmation到$ \ sum_ {i = 1} ^ n f_i（x）$。结果适用于确定性和随机算法，并且重要的是，不需要对算法结构上的假设。在参数值的某些限制下，下限是紧张的，并且通过量化梯度下降的新变种在恒定因子中匹配，我们描述和分析。我们的结果带来了从通信复杂性到分布式优化的工具，这具有进一步应用的潜力。

联合学习（FL）是一种新兴的范式，可实现对机器学习模型的大规模分布培训，同时仍提供隐私保证。在这项工作中，我们在将联合优化扩展到大节点计数时共同解决了两个主要的实际挑战：中央权威和单个计算节点之间紧密同步的需求以及中央服务器和客户端之间的传输成本较大。具体而言，我们提出了经典联合平均（FedAvg）算法的新变体，该算法支持异步通信和通信压缩。我们提供了一种新的分析技术，该技术表明，尽管有这些系统放松，但在合理的参数设置下，我们的算法基本上与FedAvg的最著名界限相匹配。在实验方面，我们表明我们的算法确保标准联合任务的快速实用收敛。

We initiate a formal study of reproducibility in optimization. We define a quantitative measure of reproducibility of optimization procedures in the face of noisy or error-prone operations such as inexact or stochastic gradient computations or inexact initialization. We then analyze several convex optimization settings of interest such as smooth, non-smooth, and strongly-convex objective functions and establish tight bounds on the limits of reproducibility in each setting. Our analysis reveals a fundamental trade-off between computation and reproducibility: more computation is necessary (and sufficient) for better reproducibility.

使用多个计算节点通常可以加速在大型数据集上的深度神经网络。这种方法称为分布式训练，可以通过专门的消息传递协议，例如环形全部减少。但是，以比例运行这些协议需要可靠的高速网络，其仅在专用集群中可用。相比之下，许多现实世界应用程序，例如联合学习和基于云的分布式训练，在具有不稳定的网络带宽的不可靠的设备上运行。因此，这些应用程序仅限于使用参数服务器或基于Gossip的平均协议。在这项工作中，我们通过提出MOSHPIT全部减少的迭代平均协议来提升该限制，该协议指数地收敛于全局平均值。我们展示了我们对具有强烈理论保证的分布式优化方案的效率。该实验显示了与使用抢占从头开始训练的竞争性八卦的策略和1.5倍的加速，显示了1.3倍的Imagenet培训的加速。

Federated learning is a distributed framework according to which a model is trained over a set of devices, while keeping data localized. This framework faces several systemsoriented challenges which include (i) communication bottleneck since a large number of devices upload their local updates to a parameter server, and (ii) scalability as the federated network consists of millions of devices. Due to these systems challenges as well as issues related to statistical heterogeneity of data and privacy concerns, designing a provably efficient federated learning method is of significant importance yet it remains challenging. In this paper, we present FedPAQ, a communication-efficient Federated Learning method with Periodic Averaging and Quantization. FedPAQ relies on three key features: (1) periodic averaging where models are updated locally at devices and only periodically averaged at the server; (2) partial device participation where only a fraction of devices participate in each round of the training; and (3) quantized messagepassing where the edge nodes quantize their updates before uploading to the parameter server. These features address the communications and scalability challenges in federated learning. We also show that FedPAQ achieves near-optimal theoretical guarantees for strongly convex and non-convex loss functions and empirically demonstrate the communication-computation tradeoff provided by our method.

We show that parametric models trained by a stochastic gradient method (SGM) with few iterations have vanishing generalization error. We prove our results by arguing that SGM is algorithmically stable in the sense of Bousquet and Elisseeff. Our analysis only employs elementary tools from convex and continuous optimization. We derive stability bounds for both convex and non-convex optimization under standard Lipschitz and smoothness assumptions.Applying our results to the convex case, we provide new insights for why multiple epochs of stochastic gradient methods generalize well in practice. In the non-convex case, we give a new interpretation of common practices in neural networks, and formally show that popular techniques for training large deep models are indeed stability-promoting. Our findings conceptually underscore the importance of reducing training time beyond its obvious benefit.

在过去的几年中，各种通信压缩技术已经出现为一个不可或缺的工具，有助于缓解分布式学习中的通信瓶颈。然而，尽管{\ em偏见}压缩机经常在实践中显示出卓越的性能，但与更多的研究和理解的{\ EM无偏见}压缩机相比，非常少见。在这项工作中，我们研究了三类偏置压缩操作员，其中两个是新的，并且它们在施加到（随机）梯度下降和分布（随机）梯度下降时的性能。我们首次展示偏置压缩机可以在单个节点和分布式设置中导致线性收敛速率。我们证明了具有错误反馈机制的分布式压缩SGD方法，享受ergodic速率$ \ mathcal {o} \ left（\ delta l \ exp [ - \ frac {\ mu k} {\ delta l}] + \ frac {（c + \ delta d）} {k \ mu} \右）$，其中$ \ delta \ ge1 $是一个压缩参数，它在应用更多压缩时增长，$ l $和$ \ mu $是平滑性和强凸常数，$ C $捕获随机渐变噪声（如果在每个节点上计算完整渐变，则$ C = 0 $如果在每个节点上计算），则$ D $以最佳（$ d = 0 $ for over参数化模型）捕获渐变的方差）。此外，通过对若干合成和经验的通信梯度分布的理论研究，我们阐明了为什么和通过多少偏置压缩机优于其无偏的变体。最后，我们提出了几种具有有希望理论担保和实际表现的新型偏置压缩机。

我们在限制下研究了一阶优化算法，即使用每个维度的$ r $ bits预算进行量化下降方向，其中$ r \ in（0，\ infty）$。我们提出了具有收敛速率的计算有效优化算法，与信息理论性能匹配：（i）：（i）具有访问精确梯度甲骨文的平稳且强烈的符合目标，以及（ii）一般凸面和非平滑目标访问嘈杂的亚级别甲骨文。这些算法的关键是一种多项式复杂源编码方案，它在量化它之前将矢量嵌入随机子空间中。这些嵌入使得具有很高的概率，它们沿着转换空间的任何规范方向的投影很小。结果，量化这些嵌入，然后对原始空间进行逆变换产生一种源编码方法，具有最佳的覆盖效率，同时仅利用每个维度的$ r $ bits。我们的算法保证了位预算$ r $的任意值的最佳性，其中包括次线性预算制度（$ r <1 $），以及高预算制度（$ r \ geq 1 $），虽然需要$ o \ left（n^2 \右）$乘法，其中$ n $是尺寸。我们还提出了使用Hadamard子空间对这种编码方案的有效放松扩展以显着提高梯度稀疏方案的性能。数值模拟验证我们的理论主张。我们的实现可在https://github.com/rajarshisaha95/distoptconstrocncomm上获得。

Huge scale machine learning problems are nowadays tackled by distributed optimization algorithms, i.e. algorithms that leverage the compute power of many devices for training. The communication overhead is a key bottleneck that hinders perfect scalability. Various recent works proposed to use quantization or sparsification techniques to reduce the amount of data that needs to be communicated, for instance by only sending the most significant entries of the stochastic gradient (top-k sparsification). Whilst such schemes showed very promising performance in practice, they have eluded theoretical analysis so far. In this work we analyze Stochastic Gradient Descent (SGD) with k-sparsification or compression (for instance top-k or random-k) and show that this scheme converges at the same rate as vanilla SGD when equipped with error compensation (keeping track of accumulated errors in memory). That is, communication can be reduced by a factor of the dimension of the problem (sometimes even more) whilst still converging at the same rate. We present numerical experiments to illustrate the theoretical findings and the good scalability for distributed applications.

Mini-batch stochastic gradient descent (SGD) is state of the art in large scale distributed training. The scheme can reach a linear speedup with respect to the number of workers, but this is rarely seen in practice as the scheme often suffers from large network delays and bandwidth limits. To overcome this communication bottleneck recent works propose to reduce the communication frequency. An algorithm of this type is local SGD that runs SGD independently in parallel on different workers and averages the sequences only once in a while. This scheme shows promising results in practice, but eluded thorough theoretical analysis.We prove concise convergence rates for local SGD on convex problems and show that it converges at the same rate as mini-batch SGD in terms of number of evaluated gradients, that is, the scheme achieves linear speedup in the number of workers and mini-batch size. The number of communication rounds can be reduced up to a factor of T 1/2 -where T denotes the number of total steps-compared to mini-batch SGD. This also holds for asynchronous implementations.Local SGD can also be used for large scale training of deep learning models. The results shown here aim serving as a guideline to further explore the theoretical and practical aspects of local SGD in these applications.

Training large neural networks requires distributing learning across multiple workers, where the cost of communicating gradients can be a significant bottleneck. SIGNSGD alleviates this problem by transmitting just the sign of each minibatch stochastic gradient. We prove that it can get the best of both worlds: compressed gradients and SGD-level convergence rate. The relative 1 / 2 geometry of gradients, noise and curvature informs whether SIGNSGD or SGD is theoretically better suited to a particular problem. On the practical side we find that the momentum counterpart of SIGNSGD is able to match the accuracy and convergence speed of ADAM on deep Imagenet models. We extend our theory to the distributed setting, where the parameter server uses majority vote to aggregate gradient signs from each worker enabling 1-bit compression of worker-server communication in both directions. Using a theorem by Gauss (1823) we prove that majority vote can achieve the same reduction in variance as full precision distributed SGD. Thus, there is great promise for sign-based optimisation schemes to achieve fast communication and fast convergence. Code to reproduce experiments is to be found at https://github.com/jxbz/signSGD.

我们开发了一种新方法来解决中央服务器中分布式学习问题中的通信约束。我们提出和分析了一种执行双向压缩的新算法，并仅使用uplink（从本地工人到中央服务器）压缩达到与算法相同的收敛速率。为了获得此改进，我们设计了MCM，一种算法，使下行链路压缩仅影响本地模型，而整体模型则保留。结果，与以前的工作相反，本地服务器上的梯度是在干扰模型上计算的。因此，融合证明更具挑战性，需要精确控制这种扰动。为了确保它，MCM还将模型压缩与存储机制相结合。该分析打开了新的门，例如纳入依赖工人的随机模型和部分参与。

许多深度学习领域都受益于使用越来越大的神经网络接受公共数据训练的培训，就像预先训练的NLP和计算机视觉模型一样。培训此类模型需要大量的计算资源（例如，HPC群集），而小型研究小组和独立研究人员则无法使用。解决问题的一种方法是，几个较小的小组将其计算资源汇总在一起并训练一种使所有参与者受益的模型。不幸的是，在这种情况下，任何参与者都可以通过故意或错误地发送错误的更新来危害整个培训。在此类同龄人的情况下进行培训需要具有拜占庭公差的专门分布式培训算法。这些算法通常通过引入冗余通信或通过受信任的服务器传递所有更新来牺牲效率，从而使它们无法应用于大规模深度学习，在该大规模深度学习中，模型可以具有数十亿个参数。在这项工作中，我们提出了一种新的协议，用于强调沟通效率的安全（容忍）分散培训。

我们介绍了一个框架 - Artemis-，以解决分布式或联合设置中的学习问题，并具有通信约束和设备部分参与。几位工人（随机抽样）使用中央服务器执行优化过程来汇总其计算。为了减轻通信成本，Artemis允许在两个方向上（从工人到服务器，相反）将发送的信息与内存机制相结合。它改进了仅考虑单向压缩（对服务器）的现有算法，或在压缩操作员上使用非常强大的假设，并且通常不考虑设备的部分参与。我们在非I.I.D中的随机梯度（仅在最佳点界定的噪声方差）提供了快速的收敛速率（线性最高到阈值）。设置，突出显示内存对单向和双向压缩的影响，分析Polyak-Ruppert平均。我们在分布中使用收敛性，以获得渐近方差的下限，该方差突出了实际的压缩极限。我们提出了两种方法，以解决设备部分参与的具有挑战性的案例，并提供实验结果以证明我们的分析有效性。

分布式优化和学习的最新进展表明，沟通压缩是减少交流的最有效手段之一。尽管在通信压缩下的收敛速率有很多结果，但理论下限仍然缺失。通过通信压缩的算法的分析将收敛归因于两个抽象属性：无偏见的属性或承包属性。它们可以通过单向压缩（仅从工人到服务器的消息被压缩）或双向压缩来应用它们。在本文中，我们考虑了分布式随机算法，以最大程度地减少通信压缩下的平滑和非凸目标函数。我们为算法建立了收敛的下限，无论是在单向或双向中使用无偏压缩机还是使用承包压缩机。为了缩小下限和现有上限之间的差距，我们进一步提出了一种新石器时代的算法，该算法在轻度条件下几乎达到了我们的下限（达到对数因素）。我们的结果还表明，使用承包双向压缩可以产生迭代方法，该方法的收敛速度与使用无偏见的单向压缩的方法一样快。实验结果验证了我们的发现。

分布式平均值估计（DME）是联邦学习中的一个中央构建块，客户将本地梯度发送到参数服务器，以平均和更新模型。由于通信限制，客户经常使用有损压缩技术来压缩梯度，从而导致估计不准确。当客户拥有多种网络条件（例如限制的通信预算和数据包损失）时，DME更具挑战性。在这种情况下，DME技术通常会导致估计误差显着增加，从而导致学习绩效退化。在这项工作中，我们提出了一种名为Eden的强大DME技术，该技术自然会处理异质通信预算和数据包损失。我们为伊甸园提供了有吸引力的理论保证，并通过经验进行评估。我们的结果表明，伊甸园对最先进的DME技术持续改进。

最近对基于置换的SGD的接地结果进行了证实了广泛观察到的现象：随机排列提供更快的收敛性，而不是更换采样。但是，是随机的最佳状态吗？我们表明这一点在很大程度上取决于我们正在优化的功能，并且最佳和随机排放之间的收敛差距可能因指数而异。我们首先表明，对于具有光滑的第二衍生物的1维强凸功能，与随机相比，存在令人指导的收敛性的排列。但是，对于一般强凸的功能，随机排列是最佳的。最后，我们表明，对于二次，强凸的功能，与随机相比，存在易于构建的置换，从而导致加速会聚。我们的研究结果表明，最佳排列的一般收敛性表征不能捕获各个函数类的细微差别，并且可能错误地表明一个人不能比随机更好。