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Byzantine Distributed Function Computation

Hari Krishnan P. Anilkumar, Neha Sangwan, Varun Narayanan, Vinod M. Prabhakaran

TL;DR

This work addresses robust distributed function computation under Byzantine interference, where up to $s$ of $k$ distributed sources may be malicious. It introduces the notion of $s$-robust recoverability and shows it is equivalent to $s$-viability, a condition that enforces consistency across adversarial perturbations via structured Markov constraints. The main contributions include a complete, optimal achievability scheme for all $s\le k$, a converse establishing the necessity of viability, and a linear-programming-based method to verify viability (with an extension to general adversary structures). An explicit $(k=2,s=1)$ treatment elucidates the core ideas using a single-letter $g$ and a method-of-types based decoder; a $k=3,s=2$ toy example demonstrates applicability to larger adversary sets. Overall, the results delineate which functions can be robustly learned from distributed, potentially corrupted observations and provide a constructive decoding framework for Byzantine resilience in distributed source coding.

Abstract

We study the distributed function computation problem with $k$ users of which at most $s$ may be controlled by an adversary and characterize the set of functions of the sources the decoder can reconstruct robustly in the following sense -- if the users behave honestly, the function is recovered with high probability (w.h.p.); if they behave adversarially, w.h.p, either one of the adversarial users will be identified or the function is recovered with vanishingly small distortion.

Byzantine Distributed Function Computation

TL;DR

This work addresses robust distributed function computation under Byzantine interference, where up to of distributed sources may be malicious. It introduces the notion of -robust recoverability and shows it is equivalent to -viability, a condition that enforces consistency across adversarial perturbations via structured Markov constraints. The main contributions include a complete, optimal achievability scheme for all , a converse establishing the necessity of viability, and a linear-programming-based method to verify viability (with an extension to general adversary structures). An explicit treatment elucidates the core ideas using a single-letter and a method-of-types based decoder; a toy example demonstrates applicability to larger adversary sets. Overall, the results delineate which functions can be robustly learned from distributed, potentially corrupted observations and provide a constructive decoding framework for Byzantine resilience in distributed source coding.

Abstract

We study the distributed function computation problem with users of which at most may be controlled by an adversary and characterize the set of functions of the sources the decoder can reconstruct robustly in the following sense -- if the users behave honestly, the function is recovered with high probability (w.h.p.); if they behave adversarially, w.h.p, either one of the adversarial users will be identified or the function is recovered with vanishingly small distortion.

Paper Structure

This paper contains 27 sections, 23 theorems, 110 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Statement
  4. The $(k=2, s=1)$ case
  5. Main Result
  6. An Example
  7. Proof of Theorem \ref{['thm:2user']}
  8. Converse
  9. Achievability
  10. Method of Types
  11. Proof of Lemma \ref{['lem:decoding-function-2user']}
  12. Proof of Lemma \ref{['lem:2user-technical-single-letter']}
  13. Proof of Lemma \ref{['lem:2user-typicality']}
  14. Achievability proof of Theorem \ref{['thm:2user']}
  15. General Adversary Structure
  16. ...and 12 more sections

Key Result

Theorem 1

A function $f$ is 1-robustly recoverable if and only if it is 1-viable.

Figures (2)

  • Figure 1: At most $s$ out of the $k$ users are malicious (i.e., controlled by an adversary who has access to the observations of the malicious users, but no additional side-information). The decoder, with high probability, either identifies a malicious user or outputs a substantially correct estimate of $Z^n$, where $Z=f(X_1,\ldots,X_k,Y)$. The main result is a characterization of functions $f$ for which this is possible for a given $P_{X_1,\ldots,X_kY}$. If the decoder identifies a malicious user, that user can be removed to get a new instance of the problem with $k-1$ users of which at most $s-1$ are malicious and the process repeated.
  • Figure 2: For any $Q_{\underline{X}_1\widetilde{X}_1\underline{X}_2\widetilde{X}_2\underline{Y}}$ satisfying the conditions (i) and (ii) in Definition \ref{['def:1-viable']}, the observations reported to the decoder and the decoder's side-information can be jointly distributed as $Q_{\underline{X}_1\underline{X}_2\underline{Y}}$ i.i.d. under the two scenarios shown to the left and right. Here, the malicious users are shown in color.

Theorems & Definitions (51)

  • Definition 1
  • Theorem 1: 2 users of which at most 1 is corrupt
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 2: $s$-viable
  • Theorem 5
  • Remark 1
  • Remark 2
  • Claim 6
  • ...and 41 more