平均算法收敛速度的几何界
Geometric Bounds for Convergence Rates of Averaging Algorithms
摘要：
我们开发了一种通用的方法，用于约束运行在具有时变网络的多智能体系统中的平均算法的收敛速度，其中相关的随机矩阵具有与时间无关的Perron向量。该方法提供了收敛速度的界限，统一和细化了大多数以前已知的界限。它们依赖于动态通信图的几何参数，例如归一化直径或瓶颈度量。作为这些几何界限的推论，我们表明，在$n$代理系统的大都市算法的收敛速度小于$1-1/4n^2$与任何通信图，可能会随时间而变化，但永久连接和双向。我们证明了一个类似的上界的额外的假设下，每个代理的邻居的数量是恒定的，通信图是不是太不规则的邻居算法。此外，我们的边界提供了改进的收敛速度的几个平均算法和特定的家庭的通信图。最后，我们将我们的方法扩展到一个随时间变化的Perron向量，并显示如何收敛时间可能会显着降低甚至有限的变化Perron向量。
英文摘要：
We develop a generic method for bounding the convergence rate of an averaging algorithm running in a multi-agent system with a time-varying network, where the associated stochastic matrices have a time-independent Perron vector. This method provides bounds on convergence rates that unify and refine most of the previously known bounds. They depend on geometric parameters of the dynamic communication graph such as the normalized diameter or the bottleneck measure. As corollaries of these geometric bounds, we show that the convergence rate of the Metropolis algorithm in a system of $n$ agents is less than $1-1/4n^2$ with any communication graph that may vary in time, but is permanently connected and bidirectional. We prove a similar upper bound for the EqualNeighbor algorithm under the additional assumptions that the number of neighbors of each agent is constant and that the communication graph is not too irregular. Moreover our bounds offer improved convergence rates for several averaging algorithms and specific families of communication graphs. Finally we extend our methodology to a time-varying Perron vector and show how convergence times may dramatically degrade with even limited variations of Perron vectors.
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