Online Learning Algorithms in Hilbert Spaces with $\beta-$ and $\phi-$Mixing Sequences

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.

@article{roy2025_2502.03551,
  title={ Online Learning Algorithms in Hilbert Spaces with $\beta-$ and $\phi-$Mixing Sequences },
  author={ Priyanka Roy and Susanne Saminger-Platz },
  journal={arXiv preprint arXiv:2502.03551},
  year={ 2025 }
}