  We present new PAC-Bayesian generalisation bounds for learning problems with
unbounded loss functions. This extends the relevance and applicability of the
PAC-Bayes learning framework, where most of the existing literature focuses on
supervised learning problems with a bounded loss function (typically assumed to
take values in the interval [0;1]). In order to relax this assumption, we
propose a new notion called HYPE (standing for \emph{HYPothesis-dependent
rangE}), which effectively allows the range of the loss to depend on each
predictor. Based on this new notion we derive a novel PAC-Bayesian
generalisation bound for unbounded loss functions, and we instantiate it on a
linear regression problem. To make our theory usable by the largest audience
possible, we include discussions on actual computation, practicality and
limitations of our assumptions.
