PAC-Bayes learning is an established framework to assess the generalisation
ability of learning algorithm during the training phase. However, it remains
challenging to know whether PAC-Bayes is useful to understand, before training,
why the output of well-known algorithms generalise well. We positively answer
this question by expanding the \emph{Wasserstein PAC-Bayes} framework, briefly
introduced in \cite{amit2022ipm}. We provide new generalisation bounds
exploiting geometric assumptions on the loss function. Using our framework, we
prove, before any training, that the output of an algorithm from
\citet{lambert2022variational} has a strong asymptotic generalisation ability.
More precisely, we show that it is possible to incorporate optimisation results
within a generalisation framework, building a bridge between PAC-Bayes and
optimisation algorithms