  PAC-Bayes learning is an established framework to both assess the
generalisation ability of learning algorithms, and design new learning
algorithm by exploiting generalisation bounds as training objectives. Most of
the exisiting bounds involve a \emph{Kullback-Leibler} (KL) divergence, which
fails to capture the geometric properties of the loss function which are often
useful in optimisation. We address this by extending the emerging
\emph{Wasserstein PAC-Bayes} theory. We develop new PAC-Bayes bounds with
Wasserstein distances replacing the usual KL, and demonstrate that sound
optimisation guarantees translate to good generalisation abilities. In
particular we provide generalisation bounds for the \emph{Bures-Wasserstein
SGD} by exploiting its optimisation properties.
