分布式社会采样算法的渐近性质
Asymptotic Properties of Distributed Social Sampling Algorithm
AI总结：
本文研究随机网络上分布式社会采样算法的理论性质，提出基于随机近似的分析框架，证明了在温和条件下估计的收敛性、强一致性及渐近正态性，并通过数值模拟验证了理论结果。
摘要：
社会采样是一种受社交网络中意见形成的社交通信启发的新型随机消息传递协议。在典型的社会采样算法中，每个智能体在初始时刻持有来自社会意见经验分布的一个样本，并以分布式方式与其他智能体协作，通过从当前分布估计中随机采样一条消息来估计初始经验分布。本文重点分析分布式社会采样算法在随机网络上的理论性质。首先，我们提供了一个基于随机近似的框架来研究该算法的渐近性质。然后，在温和条件下，我们证明所有智能体的估计收敛到一个共同的随机分布，该分布由初始经验分布和量化误差的累积组成。此外，通过调整算法参数，我们证明了强一致性，即智能体的分布估计几乎必然收敛到初始经验分布。进一步，讨论了分布式社会采样算法产生的估计误差的渐近正态性。最后，我们提供了数值模拟以验证本文的理论结果。
英文摘要：
Social sampling is a novel randomized message passing protocol inspired by social communication for opinion formation in social networks. In a typical social sampling algorithm, each agent holds a sample from the empirical distribution of social opinions at initial time, and it collaborates with other agents in a distributed manner to estimate the initial empirical distribution by randomly sampling a message from current distribution estimate. In this paper, we focus on analyzing the theoretical properties of the distributed social sampling algorithm over random networks. Firstly, we provide a framework based on stochastic approximation to study the asymptotic properties of the algorithm. Then, under mild conditions, we prove that the estimates of all agents converge to a common random distribution, which is composed of the initial empirical distribution and the accumulation of quantized error. Besides, by tuning algorithm parameters, we prove the strong consistency, namely, the distribution estimates of agents almost surely converge to the initial empirical distribution. Furthermore, the asymptotic normality of estimation error generated by distributed social sampling algorithm is addressed. Finally, we provide a numerical simulation to validate the theoretical results of this paper.
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