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Coded Computing for Resilient Distributed Computing: A Learning-Theoretic Framework

Parsa Moradi, Behrooz Tahmasebi, Mohammad Ali Maddah-Ali

TL;DR

This work proposes a novel foundation for coded computing, integrating the principles of learning theory, and developing a framework that seamlessly adapts with machine learning applications, and demonstrates that the proposed framework outperforms the state-of-the-art in terms of accuracy and rate of convergence.

Abstract

Coded computing has emerged as a promising framework for tackling significant challenges in large-scale distributed computing, including the presence of slow, faulty, or compromised servers. In this approach, each worker node processes a combination of the data, rather than the raw data itself. The final result then is decoded from the collective outputs of the worker nodes. However, there is a significant gap between current coded computing approaches and the broader landscape of general distributed computing, particularly when it comes to machine learning workloads. To bridge this gap, we propose a novel foundation for coded computing, integrating the principles of learning theory, and developing a framework that seamlessly adapts with machine learning applications. In this framework, the objective is to find the encoder and decoder functions that minimize the loss function, defined as the mean squared error between the estimated and true values. Facilitating the search for the optimum decoding and functions, we show that the loss function can be upper-bounded by the summation of two terms: the generalization error of the decoding function and the training error of the encoding function. Focusing on the second-order Sobolev space, we then derive the optimal encoder and decoder. We show that in the proposed solution, the mean squared error of the estimation decays with the rate of $\mathcal{O}(S^3 N^{-3})$ and $\mathcal{O}(S^{\frac{8}{5}}N^{\frac{-3}{5}})$ in noiseless and noisy computation settings, respectively, where $N$ is the number of worker nodes with at most $S$ slow servers (stragglers). Finally, we evaluate the proposed scheme on inference tasks for various machine learning models and demonstrate that the proposed framework outperforms the state-of-the-art in terms of accuracy and rate of convergence.

Coded Computing for Resilient Distributed Computing: A Learning-Theoretic Framework

TL;DR

This work proposes a novel foundation for coded computing, integrating the principles of learning theory, and developing a framework that seamlessly adapts with machine learning applications, and demonstrates that the proposed framework outperforms the state-of-the-art in terms of accuracy and rate of convergence.

Abstract

Coded computing has emerged as a promising framework for tackling significant challenges in large-scale distributed computing, including the presence of slow, faulty, or compromised servers. In this approach, each worker node processes a combination of the data, rather than the raw data itself. The final result then is decoded from the collective outputs of the worker nodes. However, there is a significant gap between current coded computing approaches and the broader landscape of general distributed computing, particularly when it comes to machine learning workloads. To bridge this gap, we propose a novel foundation for coded computing, integrating the principles of learning theory, and developing a framework that seamlessly adapts with machine learning applications. In this framework, the objective is to find the encoder and decoder functions that minimize the loss function, defined as the mean squared error between the estimated and true values. Facilitating the search for the optimum decoding and functions, we show that the loss function can be upper-bounded by the summation of two terms: the generalization error of the decoding function and the training error of the encoding function. Focusing on the second-order Sobolev space, we then derive the optimal encoder and decoder. We show that in the proposed solution, the mean squared error of the estimation decays with the rate of and in noiseless and noisy computation settings, respectively, where is the number of worker nodes with at most slow servers (stragglers). Finally, we evaluate the proposed scheme on inference tasks for various machine learning models and demonstrate that the proposed framework outperforms the state-of-the-art in terms of accuracy and rate of convergence.
Paper Structure (31 sections, 20 theorems, 122 equations, 7 figures, 3 tables)

This paper contains 31 sections, 20 theorems, 122 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Definition
  3. Notations
  4. Problem Setting
  5. Proposed Framework: LeTCC
  6. Objective
  7. Main Results
  8. Experimental Results
  9. Related Work
  10. Conclusions and Future Work
  11. Acknowledgments
  12. Preliminaries
  13. Sobolev spaces and Sobolev norms
  14. Smoothing Splines
  15. Proof of Theorems
  16. ...and 16 more sections

Key Result

Theorem 1

Consider the $\texttt{LeTCC}$ framework with $N$ worker nodes and at most $S$ stragglers. Assume $\{\alpha_k\}^K_{k=1}$ are arbitrary and distinct points in $\Omega=(-1, 1)$ and there exist a constant $J$ that $\Delta_{\textrm{max}} \leqslant \frac{J}{N}$. If $f(\cdot)$ is a $q$-Lipschitz continuous where $C_1$ is a constant.

Figures (7)

  • Figure 1: $\texttt{LeTCC}$ framework.
  • Figure 2: Performance comparison of $\texttt{LeTCC}$ and BACC with a $95\%$ confidence interval across a diverse range of stragglers for different models in a low-redundancy regime (smaller $\frac{N}{K}$).
  • Figure 3: Performance comparison of $\texttt{LeTCC}$ and BACC with a $95\%$ confidence interval across a diverse range of stragglers for different models in a high-redundancy regime (larger $\frac{N}{K}$).
  • Figure 4: Average performance of $\texttt{LeTCC}$ and Lagrange Coded Computing, with a 95% confidence interval. Plots (a) and (d) show the overall performance, while the zoomed-in subplots (b) and (c) highlight the performance for smaller range of stragglers.
  • Figure 5: Sensitivity of $\texttt{LeTCC}$ performance with respect to $\log_{10}(\lambda_\textrm{d})$ and $\log_{10}(\lambda_\textrm{e})$. The yellow line represents the performance when the variable smoothing parameter is set to zero.
  • ...and 2 more figures

Theorems & Definitions (31)

  • Theorem 1: Upper bound for noiseless computation, $\sigma_0 = 0$
  • Theorem 2: Upper bound for noisy computation
  • Theorem 3
  • Proposition 1
  • Theorem 4: Convergence rate
  • Definition 1: Sobolev Space
  • Definition 2
  • Theorem 5: Theorem 7.34, leoni2024first
  • Corollary 1
  • proof
  • ...and 21 more