General Convex Agreement with Near-Optimal Communication

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.