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Generalized Fractional Repetition Codes for Binary Coded Computations

Neophytos Charalambides, Hessam Mahdavifar, Alfred O. Hero

TL;DR

This work tackles straggler mitigation in distributed optimization by introducing a numerically stable binary gradient coding (BGC) framework that leverages a binary encoding matrix to avoid unstable real/complex arithmetic. It shows how any gradient coding scheme can be extended to coded matrix multiplication (CMM) and derives two binary CMM schemes with different trade-offs in communication, storage, and computation while guaranteeing exact gradient recovery. The authors prove optimality aspects for load balancing via a near-uniform allocation metric $d_s$, present a close-to-balanced encoding design that works without the divisibility constraint $(s+1) \mid n$, and provide efficient online decoding. They also connect their binary GC/CMM constructions with Reed-Solomon, LDPC, and distributed storage concepts, and demonstrate practical advantages through heterogeneity-aware task allocation and streaming decoding. Overall, the paper delivers numerically stable, scalable codes that tolerate stragglers and enable exact gradient recovery, with actionable trade-offs for distributed gradient descent and matrix multiplication tasks.

Abstract

This paper addresses the gradient coding and coded matrix multiplication problems in distributed optimization and coded computing. We present a numerically stable binary coding method which overcomes the drawbacks of the \textit{Fractional Repetition Coding} gradient coding method proposed by Tandon et al., and can also be leveraged by coded computing networks whose servers are of heterogeneous nature. Specifically, we propose a construction for fractional repetition gradient coding; while ensuring that the generator matrix remains close to perfectly balanced for any set of coded parameters, as well as a low complexity decoding step. The proposed binary encoding avoids operations over the real and complex numbers which are inherently numerically unstable, thereby enabling numerically stable distributed encodings of the partial gradients. We then make connections between gradient coding and coded matrix multiplication. Specifically, we show that any gradient coding scheme can be extended to coded matrix multiplication. Furthermore, we show how the proposed binary gradient coding scheme can be used to construct two different coded matrix multiplication schemes, each achieving different trade-offs.

Generalized Fractional Repetition Codes for Binary Coded Computations

TL;DR

This work tackles straggler mitigation in distributed optimization by introducing a numerically stable binary gradient coding (BGC) framework that leverages a binary encoding matrix to avoid unstable real/complex arithmetic. It shows how any gradient coding scheme can be extended to coded matrix multiplication (CMM) and derives two binary CMM schemes with different trade-offs in communication, storage, and computation while guaranteeing exact gradient recovery. The authors prove optimality aspects for load balancing via a near-uniform allocation metric , present a close-to-balanced encoding design that works without the divisibility constraint , and provide efficient online decoding. They also connect their binary GC/CMM constructions with Reed-Solomon, LDPC, and distributed storage concepts, and demonstrate practical advantages through heterogeneity-aware task allocation and streaming decoding. Overall, the paper delivers numerically stable, scalable codes that tolerate stragglers and enable exact gradient recovery, with actionable trade-offs for distributed gradient descent and matrix multiplication tasks.

Abstract

This paper addresses the gradient coding and coded matrix multiplication problems in distributed optimization and coded computing. We present a numerically stable binary coding method which overcomes the drawbacks of the \textit{Fractional Repetition Coding} gradient coding method proposed by Tandon et al., and can also be leveraged by coded computing networks whose servers are of heterogeneous nature. Specifically, we propose a construction for fractional repetition gradient coding; while ensuring that the generator matrix remains close to perfectly balanced for any set of coded parameters, as well as a low complexity decoding step. The proposed binary encoding avoids operations over the real and complex numbers which are inherently numerically unstable, thereby enabling numerically stable distributed encodings of the partial gradients. We then make connections between gradient coding and coded matrix multiplication. Specifically, we show that any gradient coding scheme can be extended to coded matrix multiplication. Furthermore, we show how the proposed binary gradient coding scheme can be used to construct two different coded matrix multiplication schemes, each achieving different trade-offs.

Paper Structure

This paper contains 42 sections, 10 theorems, 79 equations, 2 figures, 2 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Straggler Problem
  4. Gradient Coding
  5. Notational Conventions
  6. Binary Gradient Coding
  7. Binary GC Condition
  8. Close to Uniform Assignment Distribution
  9. Minimum Maximum Load of Workers in a Binary FRC Scheme
  10. Proposed Binary Gradient Coding Scheme
  11. Close to Balanced Encoding Design
  12. Encoding Matrix
  13. Repetition Assignment for Classes $0$ to $r-1$
  14. Repetition Assignment for Classes $r$ to $s$
  15. Decoding Vector
  16. ...and 27 more sections

Key Result

Proposition 3

Let $\bold{B}\in\{0,1\}^{n\times k}$, and partition its rows into $s+1$ nonempty subsets with index sets $\{\mathcal{K}_i\}_{i=0}^{s}$; i.e., $\bigsqcup_{i=0}^{s}\mathcal{K}_i=\mathbb{N}_n$. If for all $i\in\mathbb{N}_{0,s}$: then, for any $\mathcal{I}\in\mathcal{I}_f^n$, it follows that $\bold{1}_{1\times k}\in\mathop{\mathrm{span}}\nolimits(\bold{B}_\mathcal{I})$. This is a sufficient condition

Figures (2)

  • Figure 1: Propagation of error introduced by the FRC scheme, in log-scale.
  • Figure 2: Error plot of our GCS.

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Proposition 3
  • proof
  • Corollary 4
  • proof
  • Lemma 5
  • proof
  • Definition 6
  • Proposition 7
  • ...and 15 more