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Basis-Spline Assisted Coded Computing: Strategies and Error Bounds

Rimpi Borah, J. Harshan, V. Lalitha

TL;DR

This work provides a systematic methodology for integrating B-spline interpolation into coded computing and derive theoretical bounds on approximation error in terms of the number of servers and stragglers.

Abstract

Coded computing has become a key framework for reliable distributed computation over decentralized networks, effectively mitigating the impact of stragglers. Although there exists a wide range of coded computing methods to handle both polynomial and non-polynomial functions, computing methods for the latter class have received traction due its inherent challenges in reconstructing non-polynomial functions using a finite number of evaluations. Among them, the state-of-the-art method is Berrut Approximated coded computing, wherein Berrut interpolants, are used for approximating the non-polynomial function. However, since Berrut interpolants have global support characteristics, such methods are known to offer degraded accuracy when the number of stragglers is large. To address this challenge, we propose a coded computing framework based on cubic B-spline interpolation. In our approach, server-side function evaluations are reconstructed at the master node using B-splines, exploiting their local support and smoothness properties to enhance stability and accuracy. We provide a systematic methodology for integrating B-spline interpolation into coded computing and derive theoretical bounds on approximation error in terms of the number of servers and stragglers. Comparative analysis demonstrates that our framework significantly outperforms Berrut-based methods for various non-polynomial functions.

Basis-Spline Assisted Coded Computing: Strategies and Error Bounds

TL;DR

This work provides a systematic methodology for integrating B-spline interpolation into coded computing and derive theoretical bounds on approximation error in terms of the number of servers and stragglers.

Abstract

Coded computing has become a key framework for reliable distributed computation over decentralized networks, effectively mitigating the impact of stragglers. Although there exists a wide range of coded computing methods to handle both polynomial and non-polynomial functions, computing methods for the latter class have received traction due its inherent challenges in reconstructing non-polynomial functions using a finite number of evaluations. Among them, the state-of-the-art method is Berrut Approximated coded computing, wherein Berrut interpolants, are used for approximating the non-polynomial function. However, since Berrut interpolants have global support characteristics, such methods are known to offer degraded accuracy when the number of stragglers is large. To address this challenge, we propose a coded computing framework based on cubic B-spline interpolation. In our approach, server-side function evaluations are reconstructed at the master node using B-splines, exploiting their local support and smoothness properties to enhance stability and accuracy. We provide a systematic methodology for integrating B-spline interpolation into coded computing and derive theoretical bounds on approximation error in terms of the number of servers and stragglers. Comparative analysis demonstrates that our framework significantly outperforms Berrut-based methods for various non-polynomial functions.
Paper Structure (9 sections, 2 theorems, 13 equations, 1 figure)

This paper contains 9 sections, 2 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on B-Splines
  3. B-Spline Assisted Coded Computing
  4. Encoding
  5. Distribution of Shares among the Worker Nodes
  6. Computation at the Worker Nodes
  7. Function Reconstruction using Spline
  8. Clamped B-Spline Interpolation for BSCC
  9. Experimental Results

Key Result

Proposition 1

For cubic B-spline i.e., $p=3$, the B-spline matrix $\mathbf{B}\in \mathbb{R}^{(N+2)\times (N+2)}$ in eq:square_bspline_matrix is banded and full rank, and hence the associated linear system $\mathbf{B}\mathbf{c} = \bar{\mathbf{f}}$ admits a unique solution.

Figures (1)

  • Figure 1: Average relative error (in dB) of the BSCC and BACC decoding schemes with Lagrange and Berrut encoding for $f(\mathbf{X}) = \mathbf{X}\sin(\mathbf{X})$ and $f(\mathbf{X}) = \frac{1}{1+e^{-\mathbf{X}}}$, with parameters $N = 100$, $K = 8$, and $\mathbf{X} \in \mathbb{R}^{40 \times 5}$.

Theorems & Definitions (8)

  • Definition 1: Clamped (open) knot vectorz1
  • Definition 2: B-spline basis z9
  • Definition 3: Support and smoothness of B-splinesz2
  • Definition 4: Spline space
  • Remark 1: Effect of knot multiplicityz1
  • Definition 5: B-spline representation of spline interpolant z2
  • Proposition 1
  • Proposition 2