  In this paper, we study the generalization performance of regularized
multi-task learning (RMTL) in a vector-valued framework, where MTL is
considered as a learning process for vector-valued functions. We are mainly
concerned with two theoretical questions: 1) under what conditions does RMTL
perform better with a smaller task sample size than STL? 2) under what
conditions is RMTL generalizable and can guarantee the consistency of each task
during simultaneous learning?
  In particular, we investigate two types of task-group relatedness: the
observed discrepancy-dependence measure (ODDM) and the empirical
discrepancy-dependence measure (EDDM), both of which detect the dependence
between two groups of multiple related tasks (MRTs). We then introduce the
Cartesian product-based uniform entropy number (CPUEN) to measure the
complexities of vector-valued function classes. By applying the specific
deviation and the symmetrization inequalities to the vector-valued framework,
we obtain the generalization bound for RMTL, which is the upper bound of the
joint probability of the event that there is at least one task with a large
empirical discrepancy between the expected and empirical risks. Finally, we
present a sufficient condition to guarantee the consistency of each task in the
simultaneous learning process, and we discuss how task relatedness affects the
generalization performance of RMTL. Our theoretical findings answer the
aforementioned two questions.
