  In this paper, an online learning algorithm is proposed as sequential
stochastic approximation of a regularization path converging to the regression
function in reproducing kernel Hilbert spaces (RKHSs). We show that it is
possible to produce the best known strong (RKHS norm) convergence rate of batch
learning, through a careful choice of the gain or step size sequences,
depending on regularity assumptions on the regression function. The
corresponding weak (mean square distance) convergence rate is optimal in the
sense that it reaches the minimax and individual lower rates in the literature.
In both cases we deduce almost sure convergence, using Bernstein-type
inequalities for martingales in Hilbert spaces.
  To achieve this we develop a bias-variance decomposition similar to the batch
learning setting; the bias consists in the approximation and drift errors along
the regularization path, which display the same rates of convergence, and the
variance arises from the sample error analysed as a reverse martingale
difference sequence. The rates above are obtained by an optimal trade-off
between the bias and the variance.
