Statistical machine learning theory often tries to give generalization guarantees of machine learning models. Those models naturally underlie some fluctuation, as they are based on a data sample. If we were unlucky, and gathered a sample that is not representative of the underlying distribution, one cannot expect to construct a reliable machine learning model. Following that, statements made about the performance of machine learning models have to take the sampling process into account. The two common approaches for that are to generate statements that hold either in high-probability, or in-expectation, over the random sampling process. In this short note we show how one may transform one statement to another. As a technical novelty we address the case of unbounded loss function, where we use a fairly new assumption, called the witness condition.

We present new excess risk bounds for general unbounded loss functions including log loss and squared loss, where the distribution of the losses may be heavy-tailed. The bounds hold for general estimators, but they are optimized when applied to $\eta$-generalized Bayesian, MDL, and ERM estimators. When applied with log loss, the bounds imply convergence rates for generalized Bayesian inference under misspecification in terms of a generalization of the Hellinger metric as long as the learning rate $\eta$ is set correctly. For general loss functions, our bounds rely on two separate conditions: the $v$-GRIP (generalized reversed information projection) conditions, which control the lower tail of the excess loss; and the newly introduced witness condition, which controls the upper tail. The parameter $v$ in the $v$-GRIP conditions determines the achievable rate and is akin to the exponent in the well-known Tsybakov margin condition and the Bernstein condition for bounded losses, which the $v$-GRIP conditions generalize; favorable $v$ in combination with small model complexity leads to $\tilde{O}(1/n)$ rates. The witness condition allows us to connect the excess risk to an 'annealed' version thereof, by which we generalize several previous results connecting Hellinger and R\'enyi divergence to KL divergence.