General Convex Agreement with Near-Optimal Communication

2026-02-24

AI Summary

This work addresses the high communication complexity of existing convex consensus protocols in Byzantine settings over general convex spaces, which significantly exceeds theoretical lower bounds. The paper presents the first deterministic synchronous protocol tailored to arbitrary convex spaces, achieving near-optimal communication complexity through an extractor-graph-based committee assignment mechanism resilient to adaptive adversaries. Specifically, the protocol attains $O(L \cdot n \log n)$ communication complexity for finite convex spaces and $O(L \cdot n^{1+o(1)})$ for Euclidean spaces, where the input length satisfies $L = \Omega(n\kappa)$. It operates within $O(n)$ rounds and achieves fault tolerance approaching the theoretical limit dictated by the Helly number of the underlying space.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Convex Agreement (CA) strengthens Byzantine Agreement (BA) by requiring the output agreed upon to lie in the convex hull of the honest parties' inputs. This validity condition is motivated by practical aggregation tasks (e.g., robust learning or sensor fusion) where honest inputs need not coincide but should still constrain the decision. CA inherits BA lower bounds, and optimal synchronous round complexity is easy to obtain (e.g., via Byzantine Broadcast). The main challenge is \emph{communication}: standard approaches for CA have a communication complexity of $Θ(Ln^2)$ for large $L$-bit inputs, leaving a gap in contrast to BA's lower bound of $Ω(Ln)$ bits. While recent work achieves optimal communication complexity of $O(Ln)$ for sufficiently large $L$ [GLW,PODC'25], translating this result to general convexity spaces remained an open problem. We investigate this gap for abstract convexity spaces, and we present deterministic synchronous CA protocols with near-optimal communication complexity: when $L = Ω(n \cdot κ)$, where $κ$ is a security parameter, we achieve $O(L\cdot n\log n)$ communication for finite convexity spaces and $O(L\cdot n^{1+o(1)})$ communication for Euclidean spaces $\mathbb{R}^d$. Our protocols have asymptotically optimal round complexity $O(n)$ and, when a bound on the inputs' lengths $L$ is fixed a priori, we achieve near-optimal resilience $t < n/(ω+\varepsilon)$ for any constant $\varepsilon>0$, where $ω$ is the Helly number of the convexity space. If $L$ is unknown, we still achieve resilience $t<n/(ω+\varepsilon+1)$ for any constant $\varepsilon > 0$. We further note that our protocols can be leveraged to efficiently solve parallel BA. Our main technical contribution is the use of extractor graphs to obtain a deterministic assignment of parties to committees, which is resilient against adaptive adversaries.

Problem

• Convex Agreement
• communication complexity
• Byzantine Agreement
• convexity spaces
• synchronous protocols

Innovation

• Convex Agreement
• communication complexity
• extractor graphs
• Byzantine resilience
• deterministic protocols