Byzantine Multi-Agent Optimization: Part II

TL;DR

This paper extends Byzantine fault-tolerant distributed optimization to a condition-based setting over directed graphs, proposing algorithms that ensure optimal solutions despite malicious agents and limited side information.

Contribution

It introduces a condition-based variant of Byzantine fault-tolerant optimization with new algorithms suitable for arbitrary directed graphs and limited agent side information.

Findings

The proposed algorithms achieve optimization despite Byzantine faults.

Redundancy in input functions enables fault-tolerant solutions.

Minimal state is required at each agent for convergence.

Abstract

In Part I of this report, we introduced a Byzantine fault-tolerant distributed optimization problem whose goal is to optimize a sum of convex (cost) functions with real-valued scalar input/ouput. In this second part, we introduce a condition-based variant of the original problem over arbitrary directed graphs. Specifically, for a given collection of $k$ input functions $h_1(x), \ldots, h_k(x)$, we consider the scenario when the local cost function stored at agent $j$, denoted by $g_j(x)$, is formed as a convex combination of the $k$ input functions $h_1(x), \ldots, h_k(x)$. The goal of this condition-based problem is to generate an output that is an optimum of $\frac{1}{k}\sum_{i=1}^k h_i(x)$. Depending on the availability of side information at each agent, two slightly different variants are considered. We show that for a given graph, the problem can indeed be solved despite the presence of faulty agents. In particular, even in the absence of side information at each agent, when adequate redundancy is available in the optima of input functions, a distributed algorithm is proposed in which each agent carries minimal state across iterations.