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Local SGD Converges Fast and Communicates Little

Sebastian U. Stich

TL;DR

<3-5 sentence high-level summary>

Abstract

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.

Local SGD Converges Fast and Communicates Little

TL;DR

<3-5 sentence high-level summary>

Abstract

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.

Paper Structure

This paper contains 28 sections, 8 theorems, 35 equations, 6 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Outline
  5. Local SGD
  6. Variance reduction in local SGD
  7. Parallel (mini-batch) SGD.
  8. Towards local SGD.
  9. Convergence Result and Discussion
  10. Proof Outline
  11. Perturbed iterate analysis.
  12. Bounding the variance.
  13. Bounding the deviation.
  14. Optimal Averaging.
  15. Proof of Theorem \ref{['thm:main']}.
  16. ...and 13 more sections

Key Result

Theorem 2.2

Let $f$ be $L$-smooth and $\mu$-strongly convex, $\mathop{\mathrm{\mathbb{E}}}\nolimits_i \left\lVert \nabla f_i(\mathbf{x}_t^k)-\nabla f(\mathbf{x}_t^k)\right\rVert^2 \leq \sigma^2$, $\mathop{\mathrm{\mathbb{E}}}\nolimits_i \left\lVert \nabla f_i(\mathbf{x}_t^k)\right\rVert^2 \leq G^2$, for $t=0,\d where $\hat{\mathbf{x}}_T = \frac{1}{K S_T} \sum_{k=1}^K \sum_{t=0}^{T-1} w_t \mathbf{x}_t^k$, for

Figures (6)

  • Figure 1: Illustration of the speedup \ref{['eq:result']} for time-to-accuracy when either increasing mini-batch size $b$$(1\to 2)$ or communication inverval $H$$(1\to 2)$, for compute to communication ratio $\rho =25$.
  • Figure 2: Theoretical speedup of local SGD for different numbers of workers $K$ and $H$.
  • Figure 4: Measured speedup of local SGD with mini-batch $b=1$ for different numbers of workers $K$ and parameters $H$.
  • Figure 5: Measured speedup of local SGD with mini-batch $b=16$ for different numbers of workers $K$ and parameters $H$.
  • Figure : Local SGD
  • ...and 1 more figures

Theorems & Definitions (16)

  • Definition 2.1: gap
  • Theorem 2.2
  • Corollary 2.3
  • Remark 2.4: Mini-batch local SGD
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4: stich2018sparse
  • proof
  • Theorem 5.1
  • ...and 6 more