our attention to single-draw bounds of the form given in . We will derive generalization bounds by using two different approaches. Our first approach relies on the exponential inequality from Theorem \[thm:unconditional\_subgaussian\], which we use to get a data-dependent bound in terms of the information density ${{\imath}(W,{\boldsymbol{Z}})}$. We relax this result in different ways to obtain several data-independent bounds, some of which are tighter than existing results. Our second approach, which yields a generalization bound that is explicit in the tail of the information density, relies on the change of measure result stated in Lemma \[lem:strong\_converse\_lemma\]. This bound can be relaxed to obtain essentially the same data-independent bounds obtained using the first approach.

### Generalization Bounds from the Exponential Inequality

We begin by using Theorem \[thm:unconditional\_subgaussian\] to derive a data-dependent single-draw generalization bound and a data-independent relaxation, similar to the PAC-Bayesian results in Corollary \[cor:pacb\_uncond\]. Both of these bounds are novel.

\[cor:singledraw\_uncond\] Under the setting of Theorem \[thm:unconditional\_subgaussian\], with probability at least $1-\delta$ under $P_{W\!{\boldsymbol{Z}}}$, the following inequalities hold for all $t>0$:[^5]

[rCl]{}\[eq:cor\_singledraw\_uncond\] &&\
\[eq:cor\_singledraw\_uncond\_second\] & & .

Here, the first inequality provides a data-dependent bound and the second inequality is a data-independent relaxation. In , $M_t(W;{\boldsymbol{Z}})$ is the $t$th 