[Paper Review] BRIDGE: Byzantine-resilient Decentralized Gradient Descent

Cheng Fang, Zhixiong Yang+1 more|arXiv (Cornell University)|Aug 21, 2019

Sparse and Compressive Sensing Techniques20 citations

TL;DR

This paper proposes BRIDGE, a Byzantine-resilient decentralized gradient descent framework that ensures convergence under arbitrary node failures in decentralized machine learning systems. It achieves provable convergence for both strongly convex and nonconvex problems, with a sublinear rate of O(log t / t), making it scalable and robust against adversarial behavior in real-world networks.

ABSTRACT

Machine learning has begun to play a central role in many applications. A multitude of these applications typically also involve datasets that are distributed across multiple computing devices/machines due to either design constraints (e.g., multiagent systems) or computational/privacy reasons (e.g., learning on smartphone data). Such applications often require the learning tasks to be carried out in a decentralized fashion, in which there is no central server that is directly connected to all nodes. In real-world decentralized settings, nodes are prone to undetected failures due to malfunctioning equipment, cyberattacks, etc., which are likely to crash non-robust learning algorithms. The focus of this paper is on robustification of decentralized learning in the presence of nodes that have undergone Byzantine failures. The Byzantine failure model allows faulty nodes to arbitrarily deviate from their intended behaviors, thereby ensuring designs of the most robust of algorithms. But the study of Byzantine resilience within decentralized learning, in contrast to distributed learning, is still in its infancy. In particular, existing Byzantine-resilient decentralized learning methods either do not scale well to large-scale machine learning models, or they lack statistical convergence guarantees that help characterize their generalization errors. In this paper, a scalable, Byzantine-resilient decentralized machine learning framework termed Byzantine-resilient decentralized gradient descent (BRIDGE) is introduced. Algorithmic and statistical convergence guarantees for one variant of BRIDGE are also provided in the paper for both strongly convex problems and a class of nonconvex problems. In addition, large-scale decentralized learning experiments are used to establish that the BRIDGE framework is scalable and it delivers competitive results for Byzantine-resilient convex and nonconvex learning.

Motivation & Objective

• To address the lack of robust decentralized learning algorithms in the presence of Byzantine failures, where nodes may behave arbitrarily due to compromise or malfunction.
• To develop a scalable decentralized optimization framework that maintains convergence even when a subset of nodes exhibit Byzantine behavior.
• To provide theoretical guarantees on convergence rate and statistical performance for both convex and nonconvex loss functions under Byzantine conditions.
• To close the gap between Byzantine-resilient distributed learning and decentralized learning, where prior work lacks scalability or convergence analysis.

Proposed method

• BRIDGE employs a decentralized gradient descent framework without a central server, enabling peer-to-peer communication among nodes.
• It incorporates a robust aggregation rule that filters out adversarial updates by leveraging local consensus and gradient magnitude constraints.
• The algorithm uses a diminishing step size and adaptive weight updates to ensure convergence despite Byzantine nodes injecting arbitrary gradients.
• A key component is the use of a Lyapunov function to analyze stability and convergence, bounding the deviation of local models from the optimal solution.
• The framework introduces a resilience mechanism based on bounding the influence of corrupted gradients through a parameterized decay function.
• It establishes convergence via a recursive error bound that accounts for both Byzantine noise and stochastic gradient variance.

Experimental results

Research questions

• RQ1 Can a decentralized learning algorithm maintain convergence when a fraction of nodes exhibit Byzantine behavior, i.e., arbitrary deviation from protocol?
• RQ2 What is the convergence rate of a Byzantine-resilient decentralized optimization algorithm for both convex and nonconvex problems?
• RQ3 How can robustness be achieved in decentralized systems without relying on a central coordinator or trusted server?
• RQ4 What theoretical guarantees can be provided on the generalization error and sample complexity in the presence of Byzantine failures?
• RQ5 How does the proposed framework scale to large-scale machine learning models under adversarial conditions?

Key findings

• BRIDGE achieves a convergence rate of O(log t / t) for both strongly convex and a class of nonconvex problems, indicating sublinear but provable convergence under Byzantine failures.
• The framework ensures that all non-faulty nodes converge to the optimal solution with probability at least 1−δ, even when up to a constant fraction of nodes are Byzantine.
• Theoretical analysis confirms that the error bound decreases monotonically over time, with the slowest-decaying term being O(log t / t), which governs the overall convergence rate.
• Empirical evaluations demonstrate that BRIDGE scales well to large-scale decentralized learning tasks and delivers competitive performance in both convex and nonconvex settings.
• The algorithm maintains robustness even when Byzantine nodes inject large-magnitude adversarial gradients, due to the bounded influence mechanism in the aggregation rule.
• The framework is shown to be resilient against single-node Byzantine attacks, which can otherwise break standard decentralized learning algorithms.