PAC-Bayesian learning is a branch of learning theory recently highlighted for its non-vacuous generalisation guarantees of deep neural networks, yielding a sharper understanding of their practical efficiency on a novel, unseen example.  However, most of the existing PAC-Bayes bounds holds simultaneously for any learning algorithm and thus do not exploit the benefits acquired through the learning phase. Recently, it has been empirically unveiled that learners reaching a \emph{flat minimum} (i.e. a minimum whose neighbourhood nearly minimizes the loss as well) correlate well to a good generalisation ability on a large number of learning problems. Then a question arises: is it possible to transform correlation in causality?

 
In this presentation, I provide positive elements of answer to this question, starting with a general introduction to PAC-Bayes learning, presenting the most classical results alongside modern extensions. Then, I show that it is possible to involve the benefits of flat minima in PAC-Bayes by involving Poincaré and log-Sobolev assumptions at the cost of a main technical assumption, which is promisingly verified empirically for a small neural network on MNIST.