📝 SummarXiv

Abstract

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 introduce the concept of random Tikhonov regularization path and decompose the tracking error of the algorithm's output for the regularization path into random difference equations in RKHS, whose non-homogeneous terms are martingale difference sequences. Investigating the mean square asymptotic stability of the equations, we show that if the regularization path is slowly time-varying, then the algorithm's output achieves mean square consistency with the regularization path. Leveraging operator theory, particularly the monotonicity of the inverses of operators and the spectral decomposition of compact operators, we introduce the RKHS persistence of excitation condition (i.e. there exists a fixed-length time period, such that the conditional expectation of the operators induced by the input data accumulated over every period has a uniformly strictly positive compact lower bound) and develop a dominated convergence method to prove the mean square consistency between the algorithm's output and an unknown function. Finally, for independent and non-identically distributed data streams, the algorithm achieves the mean square consistency if the input data's marginal probability measures are slowly time-varying and the average measure over each fixed-length time period has a uniformly strictly positive lower bound.