information density* between the hypothesis and a random variable that selects the data to be used for training, given all data samples. This exponential inequality is derived under the more stringent assumption of a bounded loss function. Then, we use this inequality to reobtain the average generalization bound in [@steinke20-a Cor. 5.2], and to derive novel PAC-Bayesian and single-draw bounds, both of data-dependent and of data-independent flavor. Similarly to the standard setting, we also obtain a bound that is explicit in the tail of the conditional information density by using tools from binary hypothesis testing. Finally, inspired by [@esposito19-12a], we derive a parametric inequality that can be used to obtain data-independent single-draw bounds. Using this inequality, we extend the results in [@esposito19-12a] for bounded loss functions to the random-subset setting, and obtain bounds in terms of the conditional versions of the $\alpha$-mutual information, the Rényi divergence, and the maximal leakage. Under some conditions, the conditional maximal leakage bound turns out to be stronger than its maximal leakage counterpart.

Preliminaries {#sec:preliminaries}
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In this section, we introduce some notation, define relevant information-theoretic quantities, and present some general results that will be used repeatedly in the remainder of this paper.

#### Standard 