Analyzing large-scale functional data poses significant computational challenges due to high costs and substantial data storage needs. Additionally, traditional batch learning algorithms are not well-equipped to manage streaming data effectively. To address these issues, we propose a fully online learning algorithm designed for functional linear regression, which models the linear relationship between a scalar response and a functional predictor. Our approach employs Tikhonov regularization schemes within the framework of reproducing kernel Hilbert spaces (RKHS). A key feature of this fully online algorithm is its polynomially decaying regularization parameter, which adapts dynamically at each learning step, distinguishing it from the partially online algorithm that uses a fixed parameter. Within the functional linear model framework, we establish sufficient conditions for the convergence of the fully online algorithm in the RKHS norm. Additionally, we employ a capacity-independent approach to derive error bounds and almost sure convergence rates for both prediction and estimation, achieved through careful selection of step sizes and regularization parameters.
Chen et al. (Fri,) studied this question.