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Robust Analog Lagrange Coded Computing: Theory and Algorithms via Discrete Fourier Transforms

Rimpi Borah, J. Harshan

TL;DR

The paper addresses the vulnerability of Analog Lagrange Coded Computing (ALCC) to Byzantine workers by coupling ALCC with Discrete Fourier Transform (DFT) codes to enable error localization and correction in the analog, floating-point setting. It develops a secure ALCC framework that uses DFT-based syndrome decoding to detect and correct Byzantine perturbations, introduces joint error localization across multiple codewords, and proposes profile-aware share assignment to improve end-to-end accuracy when worker trust varies. The work also analyzes the impact of precision noise on localization, provides methods for robust reconstruction under colluding Byzantine attacks, and proposes strategies to mitigate such threats through joint localization, optimized base-matrix structures, and probabilistic collusion models. Overall, the approach yields significantly improved accuracy over prior ALCC by enabling resilience to Byzantine and colluding faults while maintaining privacy and straggler resilience, with practical implications for secure distributed analog computing.

Abstract

Analog Lagrange Coded Computing (ALCC) is a recently proposed computational paradigm wherein certain computations over analog datasets are efficiently performed using distributed worker nodes through floating point representation. While the vanilla version of ALCC is known to preserve the privacy of the datasets from the workers and also achieve resilience against stragglers, it is not robust against Byzantine workers that return erroneous results. Highlighting this vulnerability, we propose a secure ALCC framework that is resilient against a wide range of integrity threats from the Byzantine workers. As a foundational step, we use error-correction algorithms for Discrete Fourier Transform (DFT) codes to build novel reconstruction strategies for ALCC thereby improving its computational accuracy in the presence of a bounded number of Byzantine workers. Furthermore, capitalizing on some theoretical results on the performance of the DFT decoders, we propose novel strategies for distributing the ALCC computational tasks to the workers, and show that such methods significantly improve the accuracy when the workers' trust profiles are available at the master server. Finally, we study the robustness of the proposed framework against colluding attacks, and show that interesting attack strategies can be executed by exploiting the inherent precision noise owing to floating point implementation.

Robust Analog Lagrange Coded Computing: Theory and Algorithms via Discrete Fourier Transforms

TL;DR

The paper addresses the vulnerability of Analog Lagrange Coded Computing (ALCC) to Byzantine workers by coupling ALCC with Discrete Fourier Transform (DFT) codes to enable error localization and correction in the analog, floating-point setting. It develops a secure ALCC framework that uses DFT-based syndrome decoding to detect and correct Byzantine perturbations, introduces joint error localization across multiple codewords, and proposes profile-aware share assignment to improve end-to-end accuracy when worker trust varies. The work also analyzes the impact of precision noise on localization, provides methods for robust reconstruction under colluding Byzantine attacks, and proposes strategies to mitigate such threats through joint localization, optimized base-matrix structures, and probabilistic collusion models. Overall, the approach yields significantly improved accuracy over prior ALCC by enabling resilience to Byzantine and colluding faults while maintaining privacy and straggler resilience, with practical implications for secure distributed analog computing.

Abstract

Analog Lagrange Coded Computing (ALCC) is a recently proposed computational paradigm wherein certain computations over analog datasets are efficiently performed using distributed worker nodes through floating point representation. While the vanilla version of ALCC is known to preserve the privacy of the datasets from the workers and also achieve resilience against stragglers, it is not robust against Byzantine workers that return erroneous results. Highlighting this vulnerability, we propose a secure ALCC framework that is resilient against a wide range of integrity threats from the Byzantine workers. As a foundational step, we use error-correction algorithms for Discrete Fourier Transform (DFT) codes to build novel reconstruction strategies for ALCC thereby improving its computational accuracy in the presence of a bounded number of Byzantine workers. Furthermore, capitalizing on some theoretical results on the performance of the DFT decoders, we propose novel strategies for distributing the ALCC computational tasks to the workers, and show that such methods significantly improve the accuracy when the workers' trust profiles are available at the master server. Finally, we study the robustness of the proposed framework against colluding attacks, and show that interesting attack strategies can be executed by exploiting the inherent precision noise owing to floating point implementation.
Paper Structure (27 sections, 9 theorems, 39 equations, 6 figures, 2 tables)

This paper contains 27 sections, 9 theorems, 39 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Contributions
  4. Novelty
  5. ALCC with Byzantine worker nodes
  6. Encoding in ALCC
  7. Distribution of Shares among the Worker Nodes
  8. Computation at the Workers
  9. Function Reconstruction in ALCC
  10. Secure ALCC Framework
  11. DFT Based Error Correction Algorithm for ALCC
  12. Error Localization for Secure ALCC
  13. Independent Error Localization in ALCC
  14. ALCC Based Joint Error Localization
  15. Secure ALCC with Profiled Worker Nodes
  16. ...and 12 more sections

Key Result

Proposition 1

For an ALCC setting with parameters N and $K = (k+t-1)D + 1$, the erroneous computations returned by the $A$ Byzantine worker nodes can be nullified as long as $A \leq v \triangleq \lfloor \frac{N-K}{2}\rfloor$ and the floating-point operations of ALCC have infinite precision.

Figures (6)

  • Figure 1: Secure ALCC framework involving Byzantine worker nodes: (a) captures the phases of encoding at the master sever and distributed computation at the workers where Byzantine workers inject noise into their computations. (b) depicts the reconstruction stage at the master server wherein DFT decoders are used to nullify the noise introduced by the Byzantine workers.
  • Figure 2: Average relative error (in dB scale) of ALCC with parameters $N = 31, K = 15, \beta = 1.5, t = 3, \sigma = 10^6$. Here, the non-zero entries of noise matrices $\{\mathbf{E}_{i_{a}}\}$ introduced by the Byzantine worker nodes are i.i.d. as $\mathcal{CN}(10, 10^3)$.
  • Figure 3: Average relative error (in dB scale) of ALCC with parameters $N = 31, \tau=0, K = 15, \beta = 1.5, t = 3, \sigma = 10^6$ with and without the DFT decoders in the presence of Byzantine worker nodes. Here, the non-zero entries of noise matrices $\{\mathbf{E}_{i_{a}}\}$ are i.i.d. as $\mathcal{CN}(10, 10^3)$.
  • Figure 4: Average relative error (in dB scale) of ALCC with parameters used for Fig. \ref{['Fig: ind localization']}. For this plot, DFT decoders involving joint error localization are used in the presence of Byzantine worker nodes.
  • Figure 5: Average relative error (in dB) when (i) indices are obtained by minimizing the average relative error of the framework, (ii) indices are obtained by solving Problem \ref{['optprob:11']}, and (iii) using contiguous indices.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Proposition 1
  • Theorem 1
  • Corollary 1
  • Proposition 2
  • Corollary 2
  • remark 1
  • Proposition 3
  • Lemma 1
  • Proposition 4
  • Definition 1
  • ...and 2 more