  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.
