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

Stochastic Subgradient Algorithms for Strongly Convex Optimization over Distributed Networks

We study diffusion and consensus based optimization of a sum of unknown convex objective functions over distributed networks. The only access to these functions is through stochastic gradient oracles, each of which is only available at a different node, and a limited number of gradient oracle calls is allowed at each node. In this framework, we introduce a convex optimization algorithm based on the stochastic gradient descent (SGD) updates. Particularly, we use a carefully designed time-dependent weighted averaging of the SGD iterates, which yields a convergence rate of $O\left(\frac{N\sqrt{N}}{T}\right)$ after $T$ gradient updates for each node on a network of $N$ nodes. We then show that after $T$ gradient oracle calls, the average SGD iterate achieves a mean square deviation (MSD) of $O\left(\frac{\sqrt{N}}{T}\right)$. This rate of convergence is optimal as it matches the performance lower bound up to constant terms. Similar to the SGD algorithm, the computational complexity of the proposed algorithm also scales linearly with the dimensionality of the data. Furthermore, the communication load of the proposed method is the same as the communication load of the SGD algorithm. Thus, the proposed algorithm is highly efficient in terms of complexity and communication load. We illustrate the merits of the algorithm with respect to the state-of-art methods over benchmark real life data sets and widely studied network topologies