AI 中文总结
本文提出基于最小包围球有效性的多维近似一致算法，通过无维度膨胀定理实现最优弹性与快速收缩，显著优于现有方法。
AI 中文摘要
多维近似一致要求$n$个进程在输入位于$\mathbb{R}^d$的情况下，尽管存在多达$t$个拜占庭故障，仍能输出彼此接近的向量。在凸有效性下，输出必须位于正确输入的凸包内，这导致弹性阈值随维度增长。我们转而研究最小包围球（MEB）有效性及其乘法松弛$c$-MEB有效性。我们的第一个贡献是自适应MEB收缩，一种无坐标算法，在同步模型中，当$n>(d+1)t$时，对于$\alpha=1$，每轮将正确MEB半径收缩$1/\sqrt{2}$，同时满足$\sqrt{2}$-MEB有效性。我们还给出了一个例子，表明该界限对我们的分析是紧的。我们的主要技术贡献是一个无维度的膨胀定理：如果有限族欧几里得球中的每$\beta$个球都有一个公共点，那么将每个半径膨胀$\sqrt{\beta/(\beta-1)}$保证存在公共交集。将该定理应用于定义局部MEB安全区域的候选球，得到一个同步算法，具有最优弹性$n>3t$，收缩因子$\sqrt{3}/2$和$\sqrt{6}$-MEB有效性。据我们所知，这是第一个具有最优弹性、常数$c$-MEB有效性和无坐标收缩的多维近似一致算法。我们进一步将该方法扩展到异步设置。在没有膨胀的情况下，我们获得弹性$n>(d+2)t$，收缩因子$\sqrt{2/3}$和$\sqrt{6}$-MEB有效性，而使用膨胀时，对于$n>4t$，我们得到收缩因子$\sqrt{15}/4$和$2\sqrt{10}$-MEB有效性。最后，我们将我们的保证与现有算法进行比较，包括最小直径平均（MDA），并为其推导出MEB有效性保证。我们的算法实现了严格更好的弹性，同时提供了比MDA更强的MEB有效性保证。
英文摘要
Multidimensional approximate agreement requires $n$ processes with inputs in $\mathbb{R}^d$ to output vectors close to each other, despite up to $t$ Byzantine faults. Under convex validity, outputs must lie in the convex hull of the correct inputs, which leads to resilience thresholds that grow with the dimension. We instead study Minimum Enclosing Ball (MEB) validity and its multiplicative relaxation $c$-MEB validity. Our first contribution is Adaptive MEB Contraction, a coordinate-free algorithm that, in the synchronous model with $n>(d+1)t$ contracts the correct MEB radius by $1/\sqrt2$ per round for $α=1$ while satisfying $\sqrt2$-MEB validity. We also give an example showing that this bound is tight for our analysis. Our main technical contribution is a dimension-free inflation theorem: if every $β$ balls in a finite family of Euclidean balls have a common point, then inflating each radius by $\sqrt{β/(β-1)}$ guarantees a common intersection. Applying the theorem to the candidate balls defining the local MEB-safe areas results in a synchronous algorithm with optimal resilience $n>3t$, contraction factor $\sqrt3/2$ and $\sqrt6$-MEB validity. To the best of our knowledge, this is the first multidimensional approximate agreement algorithm with optimal resilience, constant $c$-MEB validity and coordinate-free contraction. We further extend the approach to the asynchronous setting. Without inflation we obtain resilience $n>(d+2)t$, contraction factor $\sqrt{2/3}$ and $\sqrt6$-MEB validity, whereas with inflation for $n>4t$ we get the contraction factor $\sqrt{15}/4$ and $2\sqrt{10}$-MEB validity. Finally, we compare our guarantees with existing algorithms, including Minimum-Diameter Averaging (MDA), for which we derive MEB-validity guarantees. Our algorithms achieve strictly better resilience while providing substantially stronger MEB-validity guarantees than MDA.
