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Scalable Distributed Optimization of Multi-Dimensional Functions Despite Byzantine Adversaries

Kananart Kuwaranancharoen, Lei Xin, Shreyas Sundaram

TL;DR

This work tackles resilient distributed optimization for multi-dimensional convex functions in the presence of Byzantine agents. It introduces two scalable algorithms based on simultaneous distance-based and min-max filtering, augmented by auxiliary states to guide aggregation toward the true minimizer of the regular nodes' average. Under mild convexity and bounded-gradient assumptions, the algorithms guarantee that regular agents converge to a region—a convergence ball—containing the global minimizer, with Algorithm 1 also achieving asymptotic consensus given a robust network $(2d+1)F+1$-robustness, while Algorithm 2 requires only $(2F+1)$-robustness but lacks consensus guarantees. The convergence radius and performance depend on the proximity of local minimizers, function curvature, and filter design, and the methods are validated on synthetic quadratic problems and a banknote-authentication task, demonstrating near-centralized performance despite adversarial interference. The results illuminate a fundamental trade-off between network redundancy, dimensionality, and convergence guarantees, and provide a practical, low-complexity approach for robust multi-dimensional distributed optimization without relying on statistical assumptions or redundant local objectives.

Abstract

The problem of distributed optimization requires a group of networked agents to compute a parameter that minimizes the average of their local cost functions. While there are a variety of distributed optimization algorithms that can solve this problem, they are typically vulnerable to "Byzantine" agents that do not follow the algorithm. Recent attempts to address this issue focus on single dimensional functions, or assume certain statistical properties of the functions at the agents. In this paper, we provide two resilient, scalable, distributed optimization algorithms for multi-dimensional functions. Our schemes involve two filters, (1) a distance-based filter and (2) a min-max filter, which each remove neighborhood states that are extreme (defined precisely in our algorithms) at each iteration. We show that these algorithms can mitigate the impact of up to $F$ (unknown) Byzantine agents in the neighborhood of each regular agent. In particular, we show that if the network topology satisfies certain conditions, all of the regular agents' states are guaranteed to converge to a bounded region that contains the minimizer of the average of the regular agents' functions.

Scalable Distributed Optimization of Multi-Dimensional Functions Despite Byzantine Adversaries

TL;DR

This work tackles resilient distributed optimization for multi-dimensional convex functions in the presence of Byzantine agents. It introduces two scalable algorithms based on simultaneous distance-based and min-max filtering, augmented by auxiliary states to guide aggregation toward the true minimizer of the regular nodes' average. Under mild convexity and bounded-gradient assumptions, the algorithms guarantee that regular agents converge to a region—a convergence ball—containing the global minimizer, with Algorithm 1 also achieving asymptotic consensus given a robust network -robustness, while Algorithm 2 requires only -robustness but lacks consensus guarantees. The convergence radius and performance depend on the proximity of local minimizers, function curvature, and filter design, and the methods are validated on synthetic quadratic problems and a banknote-authentication task, demonstrating near-centralized performance despite adversarial interference. The results illuminate a fundamental trade-off between network redundancy, dimensionality, and convergence guarantees, and provide a practical, low-complexity approach for robust multi-dimensional distributed optimization without relying on statistical assumptions or redundant local objectives.

Abstract

The problem of distributed optimization requires a group of networked agents to compute a parameter that minimizes the average of their local cost functions. While there are a variety of distributed optimization algorithms that can solve this problem, they are typically vulnerable to "Byzantine" agents that do not follow the algorithm. Recent attempts to address this issue focus on single dimensional functions, or assume certain statistical properties of the functions at the agents. In this paper, we provide two resilient, scalable, distributed optimization algorithms for multi-dimensional functions. Our schemes involve two filters, (1) a distance-based filter and (2) a min-max filter, which each remove neighborhood states that are extreme (defined precisely in our algorithms) at each iteration. We show that these algorithms can mitigate the impact of up to (unknown) Byzantine agents in the neighborhood of each regular agent. In particular, we show that if the network topology satisfies certain conditions, all of the regular agents' states are guaranteed to converge to a bounded region that contains the minimizer of the average of the regular agents' functions.
Paper Structure (37 sections, 16 theorems, 115 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 37 sections, 16 theorems, 115 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical Notation and Problem Formulation
  3. Linear Algebra
  4. Graph Theory
  5. Adversarial Behavior
  6. Problem Formulation
  7. Resilient Distributed Optimization Algorithms
  8. Proposed Algorithms
  9. Example of Algorithm \ref{['alg: DMM_filter']}
  10. Assumptions and Main Results
  11. Assumptions
  12. Analysis of Auxiliary Point Update
  13. Convergence to Consensus of States
  14. The Region To Which The States Converge
  15. Proof Sketch of the Convergence Theorem
  16. ...and 22 more sections

Key Result

Proposition 4.6

Suppose Assumption 4 hold, the graph $\mathcal{G}$ is $(2F + 1)$-robust, and the weights $w^{(\ell)}_{y,ij} [k]$ satisfy Assumption asm: weight matrices. Suppose the estimated auxiliary points of the regular agents $\{ \boldsymbol{y}_i[k] \}_{\mathcal{R}}$ follow the update rule described as Line 11 where $\alpha := \frac{1}{| \mathcal{R} | - 1} \log \frac{1}{\gamma} > 0$, $\beta := \frac{1}{\gamm

Figures (2)

  • Figure 1: The local minimizers $\boldsymbol{x}_i^*$ and the global minimizer $\boldsymbol{x}^*$ are shown in the plot. The estimated auxiliary point $\boldsymbol{y} [\infty]$ is in the rectangle formed by the local minimizers (Proposition \ref{['prop: aux convergence']}) whereas the global minimizer $\boldsymbol{x}^*$ is not necessarily in the rectangle kuwaran2018location. However, the ball centered at $\boldsymbol{y} [\infty]$ with radius $\inf_{\epsilon > 0} s^*(0, \epsilon)$ contains both the supremum limit of the state vectors $\boldsymbol{x}_i[k]$ and the global minimizer $\boldsymbol{x}^*$ (Theorem \ref{['thm: convergence']} and \ref{['thm: true_sol']}).
  • Figure 2: The plots show the results obtained from (left) Algorithm \ref{['alg: DMM_filter']} and (right) Algorithm \ref{['alg: D_filter']}. In the first four plots, the shaded regions represent +1/-1 standard deviation from the mean. In the last two plots, the contour lines show the level sets of the global objective function (in this case, a quadratic function) and the red dots represent the global minimizer.

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2: $r$-reachable set
  • Definition 2.3: $r$-robust graph
  • Definition 2.4
  • Definition 2.5: $F$-local model
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 33 more