  A well-known result across information theory, machine learning, and
statistical physics shows that the maximum entropy distribution under a mean
constraint has an exponential form called the Gibbs-Boltzmann distribution.
This is used for instance in density estimation or to achieve excess risk
bounds derived from single-scale entropy regularizers (Xu-Raginsky '17). This
paper investigates a generalization of these results to a multiscale setting.
We present different ways of generalizing the maximum entropy result by
incorporating the notion of scale. For different entropies and arbitrary scale
transformations, it is shown that the distribution maximizing a multiscale
entropy is characterized by a procedure which has an analogy to the
renormalization group procedure in statistical physics. For the case of
decimation transformation, it is further shown that this distribution is
Gaussian whenever the optimal single-scale distribution is Gaussian. This is
then applied to neural networks, and it is shown that in a teacher-student
scenario, the multiscale Gibbs posterior can achieve a smaller excess risk than
the single-scale Gibbs posterior.
