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结构、具有实时观测数据的随机逆问题框架，并提出了一种基于在线数据流的分散式在线学习算法，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穷维时空持续激励条件，则所有节点的估计量在均方意义下和几乎必然意义下均具有强相合性。通过将再生核希尔伯特空间中的分布式学习问题等价转化为基于图的随机逆问题，我们提出了一种基于非平稳、非独立在线数据流的RKHS分散式在线学习算法，并证明了若随机输入数据所诱导的算子满足无穷维时空持续激励条件，则该算法在均方意义下和几乎必然意义下均具有强相合性。