We study the convergence of recursive regularized learning algorithms in the reproducing kernel Hilbert space (RKHS) with dependent and non-stationary online data streams. Firstly, we study the mean square asymptotic stability of a class of random difference equations in RKHS, whose non-homogeneous terms are martingale difference sequences dependent on the homogeneous ones. Secondly, we introduce the concept of random Tikhonov regularization path, and show that if the regularization path is slowly time-varying in some sense, then the output of the algorithm is consistent with the regularization path in mean square. Furthermore, if the data streams also satisfy the RKHS persistence of excitation condition, i.e. there exists a fixed length of time period, such that the conditional expectation of the operators induced by the input data accumulated over every time period has a uniformly strictly positive compact lower bound in the sense of the operator order with respect to time, then the output of the algorithm is consistent with the unknown function in mean square. Finally, for the case with independent and non-identically distributed data streams, the algorithm achieves the mean square consistency provided the marginal probability measures induced by the input data are slowly time-varying and the average measure over each fixed-length time period has a uniformly strictly positive lower bound.

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.

We study the decentralized online regularized linear regression algorithm over random time-varying graphs. At each time step, every node runs an online estimation algorithm consisting of an innovation term processing its own new measurement, a consensus term taking a weighted sum of estimations of its own and its neighbors with additive and multiplicative communication noises and a regularization term preventing over-fitting. It is not required that the regression matrices and graphs satisfy special statistical assumptions such as mutual independence, spatio-temporal independence or stationarity. We develop the nonnegative supermartingale inequality of the estimation error, and prove that the estimations of all nodes converge to the unknown true parameter vector almost surely if the algorithm gains, graphs and regression matrices jointly satisfy the sample path spatio-temporal persistence of excitation condition. Especially, this condition holds by choosing appropriate algorithm gains if the graphs are uniformly conditionally jointly connected and conditionally balanced, and the regression models of all nodes are uniformly conditionally spatio-temporally jointly observable, under which the algorithm converges in mean square and almost surely. In addition, we prove that the regret upper bound is $O(T^{1-\tau}\ln T)$, where $\tau\in (0.5,1)$ is a constant depending on the algorithm gains.