We establish a framework of random inverse problems with real-time observations over graphs, and present a decentralized online learning algorithm based on online data streams, which unifies the distributed parameter estimation in Hilbert space and the least mean square problem in reproducing kernel Hilbert space (RKHS-LMS). We transform the algorithm convergence into the asymptotic stability of randomly time-varying difference equations in Hilbert space with L2-bounded martingale difference terms and develop the L2 -asymptotic stability theory. It is shown that if the network graph is connected and the sequence of forward operators satisfies the infinitedimensional spatio-temporal persistence of excitation condition, then the estimates of all nodes are mean square and almost surely strongly consistent. By equivalently transferring the distributed learning problem in RKHS to the random inverse problem over graphs, we propose a decentralized online learning algorithm in RKHS based on non-stationary and non-independent online data streams, and prove that the algorithm is mean square and almost surely strongly consistent if the operators induced by the random input data satisfy the infinite-dimensional spatio-temporal persistence of excitation condition.

翻译：我们建立了一个基于图上实时观测的随机逆问题框架，并提出了一种基于在线数据流的分布式在线学习算法。该算法统一了希尔伯特空间中的分布式参数估计与再生核希尔伯特空间中的最小均方问题（RKHS-LMS）。我们将算法收敛性转化为希尔伯特空间中具有L2有界鞅差分项的随机时变差分方程的渐近稳定性，并发展了L2渐近稳定性理论。研究表明，若网络图是连通的且前向算子序列满足无限维时空激励持续条件，则所有节点的估计在均方意义和几乎必然意义下均为强一致的。通过将RKHS中的分布式学习问题等价转化为图上的随机逆问题，我们提出了一种基于非平稳且非独立的在线数据流的RKHS分布式在线学习算法，并证明若随机输入数据诱导的算子满足无限维时空激励持续条件，则该算法在均方和几乎必然意义下是强一致的。