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Distributed matrix multiplication with straggler tolerance over very small field

Adrián Fidalgo-Díaz, Umberto Martínez-Peñas

TL;DR

This work addresses distributed matrix multiplication with straggler tolerance over finite fields, including very small fields like $\mathbb{F}_2$, where prior approaches fail due to field-size constraints. It introduces multivariate polynomial codes and multivariate matdot codes, leveraging the footprint bound to analyze recovery thresholds via hyperbolic set sizes, and develops regime-specific constructions (box_poly, better_box, separation_of_variables, halfhyperbolic) that allow a large number of workers $N$ independent of $q$. The main contributions include explicit recovery-threshold bounds, scalable encodings for different $(q,\ell)$ regimes, and optimality results within certain families (notably when $D_A=D_B$ via halfhyperbolic). The results enable practical DMM over binary and other small fields with straggler tolerance, mitigating the field-size bottleneck in coded computation and broadening the applicability of distributed linear algebra.

Abstract

The problem of distributed matrix multiplication with straggler tolerance over finite fields is considered, focusing on field sizes for which previous solutions were not applicable (for instance, the field of two elements). We employ Reed-Muller-type codes for explicitly constructing the desired algorithms and study their parameters by translating the problem into a combinatorial problem involving sums of discrete convex sets. We generalize polynomial codes and matdot codes, discussing the impossibility of the latter being applicable for very small field sizes, while providing optimal solutions for some regimes of parameters in both cases.

Distributed matrix multiplication with straggler tolerance over very small field

TL;DR

This work addresses distributed matrix multiplication with straggler tolerance over finite fields, including very small fields like , where prior approaches fail due to field-size constraints. It introduces multivariate polynomial codes and multivariate matdot codes, leveraging the footprint bound to analyze recovery thresholds via hyperbolic set sizes, and develops regime-specific constructions (box_poly, better_box, separation_of_variables, halfhyperbolic) that allow a large number of workers independent of . The main contributions include explicit recovery-threshold bounds, scalable encodings for different regimes, and optimality results within certain families (notably when via halfhyperbolic). The results enable practical DMM over binary and other small fields with straggler tolerance, mitigating the field-size bottleneck in coded computation and broadening the applicability of distributed linear algebra.

Abstract

The problem of distributed matrix multiplication with straggler tolerance over finite fields is considered, focusing on field sizes for which previous solutions were not applicable (for instance, the field of two elements). We employ Reed-Muller-type codes for explicitly constructing the desired algorithms and study their parameters by translating the problem into a combinatorial problem involving sums of discrete convex sets. We generalize polynomial codes and matdot codes, discussing the impossibility of the latter being applicable for very small field sizes, while providing optimal solutions for some regimes of parameters in both cases.

Paper Structure

This paper contains 15 sections, 13 theorems, 38 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. State of the art and our solution
  4. Preliminaries. The Footprint Bound
  5. Multivariate polynomial codes
  6. A bound on the recovery threshold
  7. Few variables and moderately small field size
  8. Many variables and small field size
  9. Multivariate matdot codes
  10. The box and the case $q=2$
  11. Optimal solution when $D_A = D_B$
  12. Complexity
  13. Tables
  14. Multivariate polynomial codes
  15. Multivariate matdot codes

Key Result

Lemma 1

Let $f \in \mathcal{R} \setminus \{0\}$ and $\delta(f) := |\Delta_{q,\prec}(f)|$. If $Z(f)$ denotes the set of zeros of $f$ in $\mathbb{F}_q^l$, then $|Z(f)| \leq \delta(f)$. In particular, if we define $\delta(\mathcal{V}) := \max \{\delta(f) : \,\, f \in \mathcal{V} \setminus \{0\} \}$, then $|Z

Figures (3)

  • Figure 1: General scheme of DMM with straggler tolerance, where each $\tilde{A}_i$ and $\tilde{B}_i$ denotes a matrix of lower size than $A$ and $B$.
  • Figure 2: Hyperbolic sets $\mathop{\mathrm{Hyp}}\nolimits_{11}(53,2) \subseteq \mathop{\mathrm{Hyp}}\nolimits_{11}(24,2) \subseteq \mathop{\mathrm{Hyp}}\nolimits_{11}(8,2)$ represented in yellow, red and blue, respectively.
  • Figure 3:

Theorems & Definitions (37)

  • Lemma 1: geil2000footprints
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Definition 1: garcia2020high
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 27 more