  We present a framework to derive bounds on the test loss of randomized
learning algorithms for the case of bounded loss functions. Drawing from
Steinke & Zakynthinou (2020), this framework leads to bounds that depend on the
conditional information density between the the output hypothesis and the
choice of the training set, given a larger set of data samples from which the
training set is formed. Furthermore, the bounds pertain to the average test
loss as well as to its tail probability, both for the PAC-Bayesian and the
single-draw settings. If the conditional information density is bounded
uniformly in the size $n$ of the training set, our bounds decay as $1/n$. This
is in contrast with the tail bounds involving conditional information measures
available in the literature, which have a less benign $1/\sqrt{n}$ dependence.
We demonstrate the usefulness of our tail bounds by showing that they lead to
nonvacuous estimates of the test loss achievable with some neural network
architectures trained on MNIST and Fashion-MNIST.
