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On the Geometric Convergence of Byzantine-Resilient Distributed Optimization Algorithms

Kananart Kuwaranancharoen, Shreyas Sundaram

TL;DR

This work introduces REDGRAF, a unifying framework for Byzantine-resilient distributed optimization in peer-to-peer networks, unifying several existing algorithms under a single contraction-and-mixing analysis. Under μ-strong convexity and L-Lipschitz gradients, it proves geometric convergence of regular agents to a ball around the true minimizer with radius R^* and characterizes the convergence rate via γ and the step-size, while maintaining approximate consensus with diameter D^*. The approach handles F-local adversaries and leverages robust graph properties to guarantee information flow despite Byzantine behavior, providing explicit bounds and insights into the trade-offs between convergence region, step-size, and function conditioning. The results extend prior sublinear or stochastic guarantees by delivering linear convergence to a neighborhood for four concrete algorithms (SDMMFD, SDFD, CWTM, RVO) and offer numerical validation on synthetic networks, highlighting practical implications for resilient distributed optimization. Overall, REDGRAF offers a principled, scalable path to designing and analyzing robust multi-agent optimization in the presence of adversaries, with concrete convergence and consensus guarantees under mild structural assumptions.

Abstract

The problem of designing distributed optimization algorithms that are resilient to Byzantine adversaries has received significant attention. For the Byzantine-resilient distributed optimization problem, the goal is to (approximately) minimize the average of the local cost functions held by the regular (non adversarial) agents in the network. In this paper, we provide a general algorithmic framework for Byzantine-resilient distributed optimization which includes some state-of-the-art algorithms as special cases. We analyze the convergence of algorithms within the framework, and derive a geometric rate of convergence of all regular agents to a ball around the optimal solution (whose size we characterize). Furthermore, we show that approximate consensus can be achieved geometrically fast under some minimal conditions. Our analysis provides insights into the relationship among the convergence region, distance between regular agents' values, step-size, and properties of the agents' functions for Byzantine-resilient distributed optimization.

On the Geometric Convergence of Byzantine-Resilient Distributed Optimization Algorithms

TL;DR

This work introduces REDGRAF, a unifying framework for Byzantine-resilient distributed optimization in peer-to-peer networks, unifying several existing algorithms under a single contraction-and-mixing analysis. Under μ-strong convexity and L-Lipschitz gradients, it proves geometric convergence of regular agents to a ball around the true minimizer with radius R^* and characterizes the convergence rate via γ and the step-size, while maintaining approximate consensus with diameter D^*. The approach handles F-local adversaries and leverages robust graph properties to guarantee information flow despite Byzantine behavior, providing explicit bounds and insights into the trade-offs between convergence region, step-size, and function conditioning. The results extend prior sublinear or stochastic guarantees by delivering linear convergence to a neighborhood for four concrete algorithms (SDMMFD, SDFD, CWTM, RVO) and offer numerical validation on synthetic networks, highlighting practical implications for resilient distributed optimization. Overall, REDGRAF offers a principled, scalable path to designing and analyzing robust multi-agent optimization in the presence of adversaries, with concrete convergence and consensus guarantees under mild structural assumptions.

Abstract

The problem of designing distributed optimization algorithms that are resilient to Byzantine adversaries has received significant attention. For the Byzantine-resilient distributed optimization problem, the goal is to (approximately) minimize the average of the local cost functions held by the regular (non adversarial) agents in the network. In this paper, we provide a general algorithmic framework for Byzantine-resilient distributed optimization which includes some state-of-the-art algorithms as special cases. We analyze the convergence of algorithms within the framework, and derive a geometric rate of convergence of all regular agents to a ball around the optimal solution (whose size we characterize). Furthermore, we show that approximate consensus can be achieved geometrically fast under some minimal conditions. Our analysis provides insights into the relationship among the convergence region, distance between regular agents' values, step-size, and properties of the agents' functions for Byzantine-resilient distributed optimization.
Paper Structure (36 sections, 18 theorems, 100 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 36 sections, 18 theorems, 100 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background and Problem Formulation
  4. Background
  5. Linear Algebra
  6. Functions Properties
  7. Graph Theory
  8. Adversarial Behavior
  9. Problem Formulation
  10. Resilient Distributed Optimization Algorithms
  11. Our Framework
  12. Definition of Some Standard Operations for Resilient Distributed Optimization
  13. Mapping Existing Algorithms into REDGRAF
  14. Assumptions and Main Results
  15. Assumptions and Definitions
  16. ...and 21 more sections

Key Result

Proposition 5.7

Suppose Assumption asm: convex holds. If an algorithm $A$ satisfies the reduction property of Type-I or Type-II, then for all $v_i \in \mathcal{V}_\mathcal{R}$, it holds that where $\boldsymbol{x}_{\text{c}}$, $r_{\text{c}}$ and $\beta$ are defined in Definition def: contraction, def: x_c distance, and def: beta, respectively. Furthermore, if $c[k] = \mathcal{O} (\xi^k)$ and $\xi \in (0, 1) \setm

Figures (3)

  • Figure 1: The convergence rate and the normalized convergence radius for different values of the contraction factor $\gamma$ and for legitimate values of the scaled constant step-size $\Tilde{\alpha}$.
  • Figure 2: The (normalized) approximate consensus diameter $D^*_{\text{normalized}}$ for different values of the contraction factor $\gamma$ and for legitimate values of the (scaled) constant step-size $\Tilde{\alpha}$.
  • Figure 3: The plots show the results obtained from SDMMFD (blue), SDFD (orange), CWTM (green), and RVO (red) for given constant step-sizes $\alpha = 0.02$ (left), and $\alpha = 0.04$ (right).

Theorems & Definitions (48)

  • Definition 3.1: strongly convex function
  • Definition 3.2: Lipschitz gradient
  • Definition 3.3
  • Definition 3.4: $r$-reachable set
  • Definition 3.5: $r$-robust graphs
  • Definition 3.6: composition
  • Definition 3.7: jointly rooted
  • Definition 3.8: repeatedly jointly rooted
  • Definition 3.9
  • Definition 3.10: $F$-local model
  • ...and 38 more