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Approximate Gradient Coding for Distributed Learning with Heterogeneous Stragglers

Heekang Song, Wan Choi

TL;DR

This work tackles stragglers in distributed gradient descent under heterogeneous conditions by formulating an optimization to minimize the residual gradient error while enforcing an unbiased gradient estimator. It derives an optimal gradient-code structure and provides two concrete, low-load schemes (Scheme I and Scheme II) that realize the structure with a data allocation that achieves a computation load $d$ strictly below 2, specifically $d=1+\frac{k-1}{n}$. The paper proves convergence guarantees for $\lambda$-strongly convex and $\mu$-smooth losses, yielding an $O(1/T)$ rate under appropriate step-sizes and ensuring convergence to stationary points in non-convex settings, with bounds that depend on the straggler probabilities through $\delta_i=(p_i/(1-p_i))$. Empirical results on COCO with MobileNetV3 demonstrate faster convergence and stronger resilience to stragglers compared with baselines, highlighting the practical impact for real-world heterogeneous distributed training.

Abstract

In this paper, we propose an optimally structured gradient coding scheme to mitigate the straggler problem in distributed learning. Conventional gradient coding methods often assume homogeneous straggler models or rely on excessive data replication, limiting performance in real-world heterogeneous systems. To address these limitations, we formulate an optimization problem minimizing residual error while ensuring unbiased gradient estimation by explicitly considering individual straggler probabilities. We derive closed-form solutions for optimal encoding and decoding coefficients via Lagrangian duality and convex optimization, and propose data allocation strategies that reduce both redundancy and computation load. We also analyze convergence behavior for $λ$-strongly convex and $μ$-smooth loss functions. Numerical results show that our approach significantly reduces the impact of stragglers and accelerates convergence compared to existing methods.

Approximate Gradient Coding for Distributed Learning with Heterogeneous Stragglers

TL;DR

This work tackles stragglers in distributed gradient descent under heterogeneous conditions by formulating an optimization to minimize the residual gradient error while enforcing an unbiased gradient estimator. It derives an optimal gradient-code structure and provides two concrete, low-load schemes (Scheme I and Scheme II) that realize the structure with a data allocation that achieves a computation load strictly below 2, specifically . The paper proves convergence guarantees for -strongly convex and -smooth losses, yielding an rate under appropriate step-sizes and ensuring convergence to stationary points in non-convex settings, with bounds that depend on the straggler probabilities through . Empirical results on COCO with MobileNetV3 demonstrate faster convergence and stronger resilience to stragglers compared with baselines, highlighting the practical impact for real-world heterogeneous distributed training.

Abstract

In this paper, we propose an optimally structured gradient coding scheme to mitigate the straggler problem in distributed learning. Conventional gradient coding methods often assume homogeneous straggler models or rely on excessive data replication, limiting performance in real-world heterogeneous systems. To address these limitations, we formulate an optimization problem minimizing residual error while ensuring unbiased gradient estimation by explicitly considering individual straggler probabilities. We derive closed-form solutions for optimal encoding and decoding coefficients via Lagrangian duality and convex optimization, and propose data allocation strategies that reduce both redundancy and computation load. We also analyze convergence behavior for -strongly convex and -smooth loss functions. Numerical results show that our approach significantly reduces the impact of stragglers and accelerates convergence compared to existing methods.
Paper Structure (36 sections, 7 theorems, 76 equations, 7 figures, 1 algorithm)

This paper contains 36 sections, 7 theorems, 76 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Distributed Learning with Gradient Coding
  4. Motivations
  5. Optimally Structured Gradient Coding
  6. Optimal Structure of Gradient Coding
  7. Optimally Structured Gradient Code Construction
  8. Scheme I
  9. Scheme II
  10. Code construction and computation load
  11. Theoretical analysis of convergence behavior
  12. Experiments
  13. Conclusions
  14. Acknowledgements
  15. Closed-form Expressions of the Proposed Schemes
  16. ...and 21 more sections

Key Result

Lemma 1

Suppose that Assumption assumption:const is satisfied and gradient estimator $\hat{g}^{(t)}$ is unbiased. Then,

Figures (7)

  • Figure 1: Motivating example of gradient coding.
  • Figure 2: Illustrative example of the proposed schemes: (a) Scheme I and (b) Scheme II.
  • Figure 3: Convergence graph with respect to the training iteration $T$: (a) $\tau_{th}=1.1$ ($k=10$) (b) $\tau_{th}=1.5$ ($k=10$) (c) $\tau_{th}=1.1$ ($k=100$) (d) $\tau_{th}=1.5$ ($k=100$).
  • Figure 4: Convergence graph with respect to the computation load $d$: (a) $\tau_{th}=1.1$ (b) $\tau_{th}=1.5$.
  • Figure 5: Detected objects of sampled image: (a) GD (b) Proposed (c) SGC (d) EHD (e) BGC (f) OD (g) IS-SGD.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 3