In this paper, we study an online algorithm in a reproducing kernel Hilbert spaces (RKHS) based on a class of dependent processes, called the mixing process. For such a process, the degree of dependence is measured by various mixing coefficients. As a representative example, we analyze a strictly stationary Markov chain, where the dependence structure is characterized by the \(\beta-\) and \(\phi-\)mixing coefficients. For these dependent samples, we derive nearly optimal convergence rates. Our findings extend existing error bounds for i.i.d. observations, demonstrating that the i.i.d. case is a special instance of our framework. Moreover, we explicitly account for an additional factor introduced by the dependence structure in the Markov chain.

In this paper, we study a Markov chain-based stochastic gradient algorithm in general Hilbert spaces, aiming to approximate the optimal solution of a quadratic loss function. We establish probabilistic upper bounds on its convergence. We further extend these results to an online regularized learning algorithm in reproducing kernel Hilbert spaces, where the samples are drawn along a Markov chain trajectory hence the samples are of the non i.i.d.