We establish a framework of distributed random inverse problems over network graphs with online measurements, and propose a decentralized online learning algorithm. This unifies the distributed parameter estimation in Hilbert spaces and the least mean square problem in reproducing kernel Hilbert spaces (RKHS-LMS). We transform the convergence of the algorithm into the asymptotic stability of a class of inhomogeneous random difference equations in Hilbert spaces with L2-bounded martingale difference terms and develop the L2 -asymptotic stability theory in Hilbert spaces. It is shown that if the network graph is connected and the sequence of forward operators satisfies the infinite-dimensional spatio-temporal persistence of excitation condition, then the estimates of all nodes are mean square and almost surely strongly consistent. Moreover, 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：我们建立了基于在线测量的网络图上分布式随机逆问题框架，并提出了一种分散在线学习算法。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）。我们将算法的收敛性转化为希尔伯特空间中一类带有L2有界鞅差分项的非齐次随机差分方程的渐近稳定性，并发展了希尔伯特空间中的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限维时空持续激励条件，则所有节点的估计值在均方和几乎必然意义下强一致收敛。此外，我们基于非平稳、非独立的在线数据流，提出了一种再生核希尔伯特空间中的分散在线学习算法，并证明当随机输入数据诱导的算子满足无限维时空持续激励条件时，该算法在均方和几乎必然意义下强一致收敛。