Dimension-free Information Concentration via Exp-Concavity
This provides a dimension-free result for information concentration, which is incremental but improves learning theory bounds from expectation to high probability.
The paper tackles the problem of information concentration in log-concave distributions, which previously had a dimension-dependent bound, and shows that under exp-concavity conditions, the concentration becomes dimension-independent, depending only on the exp-concavity parameter.
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.