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：我们建立了图结构上具有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去中心化在线学习算法，该算法统一了Hilbert空间中的分布式参数估计与再生核Hilbert空间中的最小均方问题（RKHS-LMS）。我们将算法收敛性问题转化为Hilbert空间中带有L2有界鞅差项的随机时变差分方程的渐近稳定性问题，并发展了L2-渐近稳定性理论。研究表明，若网络图连通且前向算子序列满足无穷维时空持续激励条件，则所有节点的估计值均具有均方和几乎必然强一致性。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，我们提出了基于非平稳非独立在线数据流的RKHS去中心化在线学习算法，并证明当随机输入数据诱导的算子满足无穷维时空持续激励条件时，该算法具有均方和几乎必然强一致性。