Fully online functional learning with streaming data

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.