Local SGD Converges Fast and Communicates Little

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.Comment: to appear at ICLR 2019, 19 page

Compressed Distributed Gradient Descent: Communication-Efficient Consensus over Networks

Network consensus optimization has received increasing attention in recent years and has found important applications in many scientific and engineering fields. To solve network consensus optimization problems, one of the most well-known approaches is the distributed gradient descent method (DGD). However, in networks with slow communication rates, DGD's performance is unsatisfactory for solving high-dimensional network consensus problems due to the communication bottleneck. This motivates us to design a communication-efficient DGD-type algorithm based on compressed information exchanges. Our contributions in this paper are three-fold: i) We develop a communication-efficient algorithm called amplified-differential compression DGD (ADC-DGD) and show that it converges under {\em any} unbiased compression operator; ii) We rigorously prove the convergence performances of ADC-DGD and show that they match with those of DGD without compression; iii) We reveal an interesting phase transition phenomenon in the convergence speed of ADC-DGD. Collectively, our findings advance the state-of-the-art of network consensus optimization theory.Comment: 11 pages, 11 figures, IEEE INFOCOM 201

MindTheStep-AsyncPSGD: Adaptive Asynchronous Parallel Stochastic Gradient Descent

Stochastic Gradient Descent (SGD) is very useful in optimization problems with high-dimensional non-convex target functions, and hence constitutes an important component of several Machine Learning and Data Analytics methods. Recently there have been significant works on understanding the parallelism inherent to SGD, and its convergence properties. Asynchronous, parallel SGD (AsyncPSGD) has received particular attention, due to observed performance benefits. On the other hand, asynchrony implies inherent challenges in understanding the execution of the algorithm and its convergence, stemming from the fact that the contribution of a thread might be based on an old (stale) view of the state. In this work we aim to deepen the understanding of AsyncPSGD in order to increase the statistical efficiency in the presence of stale gradients. We propose new models for capturing the nature of the staleness distribution in a practical setting. Using the proposed models, we derive a staleness-adaptive SGD framework, MindTheStep-AsyncPSGD, for adapting the step size in an online-fashion, which provably reduces the negative impact of asynchrony. Moreover, we provide general convergence time bounds for a wide class of staleness-adaptive step size strategies for convex target functions. We also provide a detailed empirical study, showing how our approach implies faster convergence for deep learning applications.Comment: 12 pages, 3 figures, accepted in IEEE BigData 201

Hogwild! over Distributed Local Data Sets with Linearly Increasing Mini-Batch Sizes

Hogwild! implements asynchronous Stochastic Gradient Descent (SGD) where multiple threads in parallel access a common repository containing training data, perform SGD iterations and update shared state that represents a jointly learned (global) model. We consider big data analysis where training data is distributed among local data sets in a heterogeneous way -- and we wish to move SGD computations to local compute nodes where local data resides. The results of these local SGD computations are aggregated by a central "aggregator" which mimics Hogwild!. We show how local compute nodes can start choosing small mini-batch sizes which increase to larger ones in order to reduce communication cost (round interaction with the aggregator). We improve state-of-the-art literature and show $O(\sqrt{K}$) communication rounds for heterogeneous data for strongly convex problems, where $K$ is the total number of gradient computations across all local compute nodes. For our scheme, we prove a \textit{tight} and novel non-trivial convergence analysis for strongly convex problems for {\em heterogeneous} data which does not use the bounded gradient assumption as seen in many existing publications. The tightness is a consequence of our proofs for lower and upper bounds of the convergence rate, which show a constant factor difference. We show experimental results for plain convex and non-convex problems for biased (i.e., heterogeneous) and unbiased local data sets.Comment: arXiv admin note: substantial text overlap with arXiv:2007.09208 AISTATS 202

Adaptiveness and Lock-free Synchronization in Parallel Stochastic Gradient Descent

The emergence of big data in recent years due to the vast societal digitalization and large-scale sensor deployment has entailed significant interest in machine learning methods to enable automatic data analytics. In a majority of the learning algorithms used in industrial as well as academic settings, the first-order iterative optimization procedure Stochastic gradient descent (SGD), is the backbone. However, SGD is often time-consuming, as it typically requires several passes through the entire dataset in order to converge to a solution of sufficient quality.In order to cope with increasing data volumes, and to facilitate accelerated processing utilizing contemporary hardware, various parallel SGD variants have been proposed. In addition to traditional synchronous parallelization schemes, asynchronous ones have received particular interest in recent literature due to their improved ability to scale due to less coordination, and subsequently waiting time. However, asynchrony implies inherent challenges in understanding the execution of the algorithm and its convergence properties, due the presence of both stale and inconsistent views of the shared state.In this work, we aim to increase the understanding of the convergence properties of SGD for practical applications under asynchronous parallelism and develop tools and frameworks that facilitate improved convergence properties as well as further research and development. First, we focus on understanding the impact of staleness, and introduce models for capturing the dynamics of parallel execution of SGD. This enables (i) quantifying the statistical penalty on the convergence due to staleness and (ii) deriving an adaptation scheme, introducing a staleness-adaptive SGD variant MindTheStep-AsyncSGD, which provably reduces this penalty. Second, we aim at exploring the impact of synchronization mechanisms, in particular consistency-preserving ones, and the overall effect on the convergence properties. To this end, we propose LeashedSGD, an extensible algorithmic framework supporting various synchronization mechanisms for different degrees of consistency, enabling in particular a lock-free and consistency-preserving implementation. In addition, the algorithmic construction of Leashed-SGD enables dynamic memory allocation, claiming memory only when necessary, which reduces the overall memory footprint. We perform an extensive empirical study, benchmarking the proposed methods, together with established baselines, focusing on the prominent application of Deep Learning for image classification on the benchmark datasets MNIST and CIFAR, showing significant improvements in converge time for Leashed-SGD and MindTheStep-AsyncSGD

Breaking (Global) Barriers in Parallel Stochastic Optimization with Wait-Avoiding Group Averaging

Deep learning at scale is dominated by communication time. Distributing samples across nodes usually yields the best performance, but poses scaling challenges due to global information dissemination and load imbalance across uneven sample lengths. State-of-the-art decentralized optimizers mitigate the problem, but require more iterations to achieve the same accuracy as their globally-communicating counterparts. We present Wait-Avoiding Group Model Averaging (WAGMA) SGD, a wait-avoiding stochastic optimizer that reduces global communication via subgroup weight exchange. The key insight is a combination of algorithmic changes to the averaging scheme and the use of a group allreduce operation. We prove the convergence of WAGMA-SGD, and empirically show that it retains convergence rates similar to Allreduce-SGD. For evaluation, we train ResNet-50 on ImageNet; Transformer for machine translation; and deep reinforcement learning for navigation at scale. Compared with state-of-the-art decentralized SGD variants, WAGMA-SGD significantly improves training throughput (e.g., 2.1x on 1,024 GPUs for reinforcement learning), and achieves the fastest time-to-solution (e.g., the highest score using the shortest training time for Transformer).Comment: Published in IEEE Transactions on Parallel and Distributed Systems (IEEE TPDS), vol. 32, no. 7, pp. 1725-1739, 1 July 202