Byzantine-Resilient Distributed Optimization of Multi-Dimensional Functions

Abstract

The problem of distributed optimization requires a group of agents to reach agreement on a parameter that minimizes the average of their local cost functions using information received from their neighbors. While there are a variety of distributed optimization algorithms that can solve this problem, they are typically vulnerable to malicious (or "Byzantine") agents that do not follow the algorithm. Recent attempts to address this issue focus on single dimensional functions, or provide analysis under certain assumptions on the statistical properties of the functions at the agents. In this paper, we propose a resilient distributed optimization algorithm for multidimensional convex functions. Our scheme involves two filtering steps at each iteration of the algorithm: (1) distance-based and (2) component-wise removal of extreme states. We show that this algorithm can mitigate the impact of up to F Byzantine agents in the neighborhood of each regular node, without knowing the identities of the Byzantine agents in advance. In particular, we show that if the network topology satisfies certain conditions, all of the regular states are guaranteed to asymptotically converge to a bounded region that contains the global minimizer.

Resilience for Distributed Consensus with Constraints

“…Here, by using the (γ i , α i )-Resilient Convex Combination developed in the previous section, we aim to introduce an approach that allows multi-agent systems to achieve resilient constrained consensus in the presence of Byzantine attacks. This distinguishes the paper with existing ones that are only applicable to unconstrained consensus problems [22]- [24], [26]- [34], [36].…”

“…Under certain conditions of the network topology, the effectiveness of these algorithms can be theoretically validated. Followed by [22], [23], similar approaches have been further applied to resilient distributed optimization [24], [25] and multi-agent machine learning [26]. Note that in the results of [22], [23], the local states for all agents must be scalars.…”

Byzantine Fault-Tolerance in Decentralized Optimization under Minimal Redundancy

“…There are works on approximate fault-tolerance for multivariate cost functions [20], [21], [22]. However, [21] and [22] consider degenerate cases of the multi-agent optimization problem defined in (1).…”

“…Similar to [6], the algorithms proposed in [21], [22] output a minimum of a non-uniformly weighted aggregate of the non-faulty cost functions. On the other hand, the algorithm in [20] outputs a point in a proximity of a true minimum. Unlike these works, we are interested in exact fault-tolerance under the necessary condition of 2f -redundancy.…”