  We show a Talagrand-type concentration inequality for Multi-Task Learning
(MTL), using which we establish sharp excess risk bounds for MTL in terms of
distribution- and data-dependent versions of the Local Rademacher Complexity
(LRC). We also give a new bound on the LRC for norm regularized as well as
strongly convex hypothesis classes, which applies not only to MTL but also to
the standard i.i.d. setting. Combining both results, one can now easily derive
fast-rate bounds on the excess risk for many prominent MTL methods,
including---as we demonstrate---Schatten-norm, group-norm, and
graph-regularized MTL. The derived bounds reflect a relationship akeen to a
conservation law of asymptotic convergence rates. This very relationship allows
for trading off slower rates w.r.t. the number of tasks for faster rates with
respect to the number of available samples per task, when compared to the rates
obtained via a traditional, global Rademacher analysis.
