发表机构
TU Berlin（柏林工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出BallMidpoint协议，在同步与异步通信模型中实现多维近似一致，收缩率达1/√2≈0.707，是首个收敛率更接近下界而非1的上界的多维近似一致协议。
AI 中文摘要
本研究探讨多维近似一致问题：分布式系统中的n个节点，最多有t个可能被拜占庭敌手篡改，需输出彼此接近且位于所有未被篡改输入向量凸包内的向量。假设节点在全连接的认证网络中通信，分析同步与异步通信模型。研究聚焦近似一致协议的收缩因子：早期多维近似一致协议的收缩率为1-1/n（VG，PODC'13）和√[d]{1/2}（MH，STOC'13）；虽收缩率低于1足以实现收敛，但无法满足实际应用需求，目前近似一致算法的最优收敛率为√(7/8)≈0.935（FN，DISC'18），由MidExtremes协议实现，这与一维场景下1/2的收敛率下界形成对比。本研究提出BallMidpoint协议，这是一种新型近似一致协议，在同步与异步通信模型中收缩率达1/√2≈0.707，满足凸有效性下的最优弹性；该收缩率通过选取所谓局部安全区域的最小包围球中点实现，且对所提协议而言是紧的；与（FN，DISC'18）类似，该协议与坐标无关，安全区域内的点可高效计算，是首个收敛率更接近下界而非1的上界的多维近似一致协议。
英文摘要
This work considers the multidimensional approximate agreement problem. In this problem, $n$ parties in a distributed system, up to $t$ of which may be corrupted by a Byzantine adversary, need to output vectors that are close to each other and that lie inside the convex hull of all non-corrupted input vectors. We assume that nodes communicate in a fully-connected authenticated network and analyze synchronous and asynchronous communication models. The focus of this work is on the contraction factor of approximate agreement protocols. The first multidimensional approximate agreement protocols had a contraction rate of $1-1/n$(VG, PODC'13) and $\sqrt[d]{1/2}$(MH, STOC'13). While a rate below $1$ is sufficient for convergence, it is not sufficient for practical applications. To date, the best known convergence rate of approximate agreement algorithms is $\sqrt{7/8}\approx0.935$ (FN, DISC'18), which is achieved through the MidExtremes protocol. This stands in contrast to the lower bound on the convergence rate in the $1$-dimensional setting, which is $1/2$. In this work, we propose BallMidpoint - a novel approximate agreement protocol with a contraction rate of $1/\sqrt{2}\approx 0.707$ in the synchronous and the asynchronous communication models. This algorithm satisfies the optimal resilience under convex validity. The presented contraction rate is achieved by choosing the midpoint of the smallest enclosing ball of the so-called local safe areas, and it is tight for the presented algorithms. Similar to (FN, DISC'18), our algorithm is coordinate-free, and the point inside the safe area can be computed efficiently. This work presents the first multidimensional approximate agreement protocol where the convergence rate is closer to the best known lower bound rather than the upper bound of $1$.
