Convergence analysis of online algorithms for vector-valued kernel regression

We consider the problem of approximating the regression function from noisy vector-valued data by an online learning algorithm using an appropriate reproducing kernel Hilbert space (RKHS) as prior. In an online algorithm, i.i.d. samples become available one by one by a random process and are successively processed to build approximations to the regression function. We are interested in the asymptotic performance of such online approximation algorithms and show that the expected squared error in the RKHS norm can be bounded by $C^2 (m+1)^{-s/(2+s)}$, where $m$ is the current number of processed data, the parameter $0<s\leq 1$ expresses an additional smoothness assumption on the regression function and the constant $C$ depends on the variance of the input noise, the smoothness of the regression function and further parameters of the algorithm.