We propose a decentralized online learning algorithm for distributed random inverse problems over network graphs with online measurements, and 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 $L_{2}$-bounded martingale difference terms and develop the $L_2$-asymptotic stability theory in Hilbert spaces. We show 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 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：我们针对网络图上基于在线测量的分布式随机逆问题，提出了一种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）。我们将算法的收敛性转化为希尔伯特空间中一类带有 $L_{2}$ 有界鞅差项的非齐次随机差分方程的渐近稳定性问题，并建立了希尔伯特空间中的 $L_2$ 渐近稳定性理论。我们证明：若网络图是连通的且前向算子序列满足无限维时空持续激励条件，则所有节点的估计在均方意义下以及几乎必然地具有强一致性。此外，我们提出了一种基于非平稳在线数据流的再生核希尔伯特空间分散式在线学习算法，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，该算法在均方意义下以及几乎必然地具有强一致性。