  Information concentration of probability measures have important implications
in learning theory. Recently, it is discovered that the information content of
a log-concave distribution concentrates around their differential entropy,
albeit with an unpleasant dependence on the ambient dimension. In this work, we
prove that if the potentials of the log-concave distribution are exp-concave,
which is a central notion for fast rates in online and statistical learning,
then the concentration of information can be further improved to depend only on
the exp-concavity parameter, and hence, it can be dimension independent.
Central to our proof is a novel yet simple application of the variance
Brascamp-Lieb inequality. In the context of learning theory, our
concentration-of-information result immediately implies high-probability
results to many of the previous bounds that only hold in expectation.
