$t$th moments of the relative entropy $\relent{P_{W\vert {\boldsymbol{Z}}}}{P_W}$ and of the information density, respectively. The dependence of these bounds on the confidence parameter $\delta$ is of order $1/\delta^t$. This is more favorable than that of similar bounds reported in [@xu17-05a] and [@bassily18-02a], which have a dependence of order $1/\delta$. The moment bounds that we obtain illustrate that tighter estimates of the generalization error are available with higher confidence if the higher moments of the information measures that the bounds depend on are sufficiently small. Through a more refined analysis, we also derive a high-probability data-independent single-draw bound that is as simple to compute as the maximal leakage bound in [@esposito19-12a Cor. 10], but tighter in some cases. Finally, by using a different approach that relies on tools from binary hypothesis testing, we obtain a data-independent single-draw bound in terms of the tail of the information density. Similarly to the moment bounds, this bound illustrates that the faster the decay of the tail of the information density random variable, the more benign the dependence of the bound on $\delta$.

Moving to the random-subset setting, we establish an exponential inequality, similar to that for the standard setting, in terms of the *conditional 