This paper addresses the problem of distributed learning of average belief with sequential observations, in which a network of $n>1$ agents aim to reach a consensus on the average value of their beliefs, by exchanging information only with their neighbors. Each agent has sequentially arriving samples of its belief in an online manner. The neighbor relationships among the $n$ agents are described by a graph which is possibly time-varying, whose vertices correspond to agents and whose edges depict neighbor relationships. Two distributed online algorithms are introduced for undirected and directed graphs, which are both shown to converge to the average belief almost surely. Moreover, the sequences generated by both algorithms are shown to reach consensus with an $O(1∕t)$ rate with high probability, where $t$ is the number of iterations For undirected graphs, the corresponding algorithm is modified for the case with quantized communication and limited precision of the division operation. It is shown that the modified algorithm causes all $n$ agents to either reach a quantized consensus or enter a small neighborhood around the average of their beliefs. Numerical simulations are then provided to corroborate the theoretical results.

本文通过顺序观察解决了平均信念的分布式学习问题，其中一个由 $ñ>\mathrm{1个}$代理商旨在通过仅与邻居交换信息来就其信仰的平均价值达成共识。每个代理都以在线方式顺序到达其信念样本。之间的邻居关系$ñ$代理由可能随时间变化的图形描述，其顶点对应于代理，并且其边线描绘了邻居关系。针对无向图和有向图引入了两种分布式在线算法，这两种算法都显示几乎可以肯定地收敛于平均信念。此外，两种算法生成的序列均显示出与$Ø（\mathrm{1个}∕Ť）$ 率很高，在哪里 $Ť$是迭代次数，对于无向图，将针对量化通信和除法运算精度有限的情况修改相应的算法。结果表明，改进后的算法导致了所有$ñ$代理商要么达成量化共识，要么在他们的信念平均值附近进入一个小邻居。然后提供数值模拟以证实理论结果。