PAC-Bayes unleashed: generalisation bounds with unbounded losses
This work addresses a theoretical limitation in machine learning for researchers and practitioners dealing with unbounded losses, though it is incremental as it builds on the established PAC-Bayes framework.
The paper tackles the problem of extending PAC-Bayesian generalization bounds to learning problems with unbounded loss functions, which are not covered by existing literature focused on bounded losses, and it demonstrates this by deriving a new bound and applying it to linear regression.
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.