  In recent years, functional linear models have attracted growing attention in
statistics and machine learning, with the aim of recovering the slope function
or its functional predictor. This paper considers online regularized learning
algorithm for functional linear models in reproducing kernel Hilbert spaces.
Convergence analysis of excess prediction error and estimation error are
provided with polynomially decaying step-size and constant step-size,
respectively. Fast convergence rates can be derived via a capacity dependent
analysis. By introducing an explicit regularization term, we uplift the
saturation boundary of unregularized online learning algorithms when the
step-size decays polynomially, and establish fast convergence rates of
estimation error without capacity assumption. However, it remains an open
problem to obtain capacity independent convergence rates for the estimation
error of the unregularized online learning algorithm with decaying step-size.
It also shows that convergence rates of both prediction error and estimation
error with constant step-size are competitive with those in the literature.
