[Paper Review] Fault-Tolerant Distributed Optimization (Part IV): Constrained Optimization with Arbitrary Directed Networks

Lili Su, Nitin H. Vaidya|arXiv (Cornell University)|Nov 5, 2015

Distributed systems and fault tolerance28 references18 citations

TL;DR

This paper presents a fault-tolerant distributed optimization algorithm for multi-agent systems with arbitrary directed networks, supporting crash and Byzantine faults under convex constraints. It generalizes prior work on fully-connected networks to directed graphs using a matrix representation of iterative consensus, achieving convergence to a weighted average of local objectives with performance bounded by parameters β and γ.

ABSTRACT

We study the problem of constrained distributed optimization in multi-agent networks when some of the computing agents may be faulty. In this problem, the system goal is to have all the non-faulty agents collectively minimize a global objective given by weighted average of local cost functions, each of which is initially known to a non-faulty agent only. In particular, we are interested in the scenario when the computing agents are connected by an arbitrary directed communication network, some of the agents may suffer from crash faults or Byzantine faults, and the estimate of each agent is restricted to lie in a common constraint set. This problem finds its applications in social computing and distributed large-scale machine learning. The fault-tolerant multi-agent optimization problem was first formulated by Su and Vaidya, and is solved when the local functions are defined over the whole real line, and the networks are fully-connected. In this report, we consider arbitrary directed communication networks and focus on the scenario where, local estimates at the non-faulty agents are constrained, and only local communication and minimal memory carried across iterations are allowed. In particular, we generalize our previous results on fully-connected networks and unconstrained optimization to arbitrary directed networks and constrained optimization. As a byproduct, we provide a matrix representation for iterative approximate crash consensus. The matrix representation allows us to characterize the convergence rate for crash iterative consensus.

Motivation & Objective

• Address constrained distributed optimization in multi-agent systems with arbitrary directed communication networks under crash and Byzantine faults.
• Extend prior results on fully-connected networks to general directed graphs, maintaining fault tolerance and constraint handling.
• Characterize system performance using two parameters β (weight threshold) and γ (minimum number of weights exceeding β), which define the quality of the global objective.
• Ensure all non-faulty agents converge to a solution that minimizes a weighted average of local cost functions, even under adversarial behavior.
• Provide a matrix representation for iterative approximate crash consensus to analyze convergence rates and system behavior.

Proposed method

• Use a distributed gradient-based algorithm where agents iteratively update their local estimates using incoming messages from neighbors.
• Apply a projection operator $ P_X $ to enforce constraints, ensuring all estimates remain within a common convex set $ X $.
• Introduce a matrix representation $ \mathbf{\Psi}_{ik}(t) $ to model the evolution of agent states, enabling analysis of consensus convergence.
• Define weights based on in-degree and fault counts: $ a_j[t-1] = \frac{1}{\ell_j + 1} $, where $ \ell_j $ is the number of received messages.
• Use a diminishing step size $ \lambda[t-1] $ to ensure convergence in the presence of faults.
• Analyze convergence using left eigenvectors of the weight matrices, particularly $ \pi_k(t) $, representing long-term influence in the system.

Experimental results

Research questions

• RQ1 Can fault-tolerant distributed optimization be achieved in arbitrary directed networks with both crash and Byzantine faults?
• RQ2 How does the system performance depend on network topology and fault distribution, particularly in terms of β and γ?
• RQ3 Can a matrix representation of iterative consensus be used to characterize convergence rates in crash fault scenarios?
• RQ4 What is the impact of constraint sets on the convergence and quality of the global objective in the presence of faulty agents?
• RQ5 How does the solvable γ parameter depend on the in-degree and fault count of individual agents in directed networks?

Key findings

• The proposed Algorithm 6 achieves convergence to a function in $ \widetilde{\mathcal{A}}(\beta, \gamma) $, where $ \tilde{\beta} = \min\left\{ \frac{1}{\max_i(d_i^{-}+1 - \phi_i)}, \frac{1}{|\mathcal{N}|} \right\} $ and $ \tilde{\gamma} = \min_i(d_i^{-}+1 - \phi_i) $, with $ \phi_i $ being the number of faulty incoming neighbors.
• The matrix representation of iterative consensus allows precise characterization of convergence rates for crash fault scenarios, enabling analysis of long-term behavior.
• The algorithm ensures all non-faulty agents converge to a common limit point $ y[\bar{t}] $, which depends on initial states, gradients, and fault-influenced weights.
• The convergence limit is derived as $ \lim_{s\to\infty} \bar{\mathbf{x}}_i[\bar{t}+s] = \sum_k \pi_k(1)x_k[0] - \sum_r \lambda[r-1]\sum_k \pi_k(r+1)d_k[r-1] + \sum_r \sum_k \pi_k(r+1)e_k[r-1] $, showing dependency on influence weights and fault dynamics.
• For undirected graphs, a modified update rule achieves exact average minimization $ \frac{1}{|\mathcal{N}|}\sum_{i\in\mathcal{N}} h_i(x) $, demonstrating improved performance under symmetry.
• The solvable γ in Algorithm 6 depends only on the in-degree and fault count of each agent, unlike Algorithm 1 which relies on source component structure, suggesting a more scalable and topology-aware bound.