  Multi Task Learning (MTL) efficiently leverages useful information contained
in multiple related tasks to help improve the generalization performance of all
tasks. This article conducts a large dimensional analysis of a simple but, as
we shall see, extremely powerful when carefully tuned, Least Square Support
Vector Machine (LSSVM) version of MTL, in the regime where the dimension $p$ of
the data and their number $n$ grow large at the same rate.
  Under mild assumptions on the input data, the theoretical analysis of the
MTL-LSSVM algorithm first reveals the "sufficient statistics" exploited by the
algorithm and their interaction at work. These results demonstrate, as a
striking consequence, that the standard approach to MTL-LSSVM is largely
suboptimal, can lead to severe effects of negative transfer but that these
impairments are easily corrected. These corrections are turned into an improved
MTL-LSSVM algorithm which can only benefit from additional data, and the
theoretical performance of which is also analyzed.
  As evidenced and theoretically sustained in numerous recent works, these
large dimensional results are robust to broad ranges of data distributions,
which our present experiments corroborate. Specifically, the article reports a
systematically close behavior between theoretical and empirical performances on
popular datasets, which is strongly suggestive of the applicability of the
proposed carefully tuned MTL-LSSVM method to real data. This fine-tuning is
fully based on the theoretical analysis and does not in particular require any
cross validation procedure. Besides, the reported performances on real datasets
almost systematically outperform much more elaborate and less intuitive
state-of-the-art multi-task and transfer learning methods.
