Convergence Conditions of Online Regularized Statistical Learning in
  Reproducing Kernel Hilbert Space With Non-Stationary Data
Xiwei Zhang Tao Li
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 each eigenvalue of the conditional expectation of the operators induced by
the input data accumulated over every time period has a uniformly positive
lower bound 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.