  In recent years, different types of distributed and parallel learning schemes
have received increasing attention for their strong advantages in handling
large-scale data information. In the information era, to face the big data
challenges {that} stem from functional data analysis very recently, we propose
a novel distributed gradient descent functional learning (DGDFL) algorithm to
tackle functional data across numerous local machines (processors) in the
framework of reproducing kernel Hilbert space. Based on integral operator
approaches, we provide the first theoretical understanding of the DGDFL
algorithm in many different aspects of the literature. On the way of
understanding DGDFL, firstly, a data-based gradient descent functional learning
(GDFL) algorithm associated with a single-machine model is proposed and
comprehensively studied. Under mild conditions, confidence-based optimal
learning rates of DGDFL are obtained without the saturation boundary on the
regularity index suffered in previous works in functional regression. We
further provide a semi-supervised DGDFL approach to weaken the restriction on
the maximal number of local machines to ensure optimal rates. To our best
knowledge, the DGDFL provides the first divide-and-conquer iterative training
approach to functional learning based on data samples of intrinsically
infinite-dimensional random functions (functional covariates) and enriches the
methodologies for functional data analysis.
