PAC-Bayes-Chernoff bounds for unbounded losses

We present a new high-probability PAC-Bayes oracle bound for unbounded losses. This result can be understood as a PAC-Bayes version of the Chernoff bound. The proof technique relies on uniformly bounding the tail of certain random variable based on the Cram\'er transform of the loss. We highlight two applications of our main result. First, we show that our bound solves the open problem of optimizing the free parameter on many PAC-Bayes bounds. Finally, we show that our approach allows working with flexible assumptions on the loss function, resulting in novel bounds that generalize previous ones and can be minimized to obtain Gibbs-like posteriors.