I(W;{\boldsymbol{Z}})}.$$ Note that the supremization in  is over the argument of ${{\imath}(W,{\boldsymbol{Z}})}$, whereas $I(W;{\boldsymbol{Z}})$ is a constant.

The data-independent relaxation in Corollary \[cor:singledraw\_uncond\] is not as tight as the one obtained in Corollary \[cor:pacb\_uncond\]. Indeed, since ${{\imath}(W,{\boldsymbol{Z}})}$ can be negative, we had to use a weaker version of Markov’s inequality (compare  with ). In the following corollary, we provide two alternative data-independent bounds, both of which are novel. The first bound depends on $\esssup_{P_{{\boldsymbol{Z}}}}\Exop_{P_W}[\exp({{\imath}(W,{\boldsymbol{Z}})})]$, which is a lower bound to the maximal leakage $\mathcal{L}({\boldsymbol{Z}}\rightarrow W)$ defined in . The second bound is in terms of the Rényi divergence .

\[cor:singledraw\_uncond\_alphadiv\_leakage\] Under the setting of Theorem \[thm:unconditional\_subgaussian\], the following inequalities hold with probability at least $1-\delta$ under $P_{W\!{\boldsymbol{Z}}}$: $$\label{eq:cor_singledraw_uncond_leakage}
\abs{ {{\textnormal{gen}}(W,{\boldsymbol{Z}})}} \leq \sqrt{\frac{2\sigma^2}{n}\left(\log \esssup_{P_{{\boldsymbol{Z}}}}\Exop_{P_{W} }\lefto[\exp(\imath(W,{\boldsymbol{Z}})) \right] +2\log \frac{2}{\delta}\right)}$$ and, for all $\alpha,\gamma> 1$ such that $1/\alpha+1/\gamma=1$,

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

We start by noting that, with probability $1$ under $P_{W\!{\boldsymbol{Z}}}$, $$\imath(W,{\boldsymbol{Z}}) = \log \frac{\dv P_{W\! {\boldsymbol{Z}}}}{\dv P_W\!P_{{\boldsymbol{Z}}}} \leq \log \esssup_{P_{{\boldsymbol{Z}}\vert W}} \frac{\dv P_{W\! {\boldsymbol{Z}}}}{\dv P_W\!P_{{\boldsymbol{Z}}}}.$$ The assumption that $P_{W\!{\boldsymbol{Z}}}\ll P_W\!P_{{\boldsymbol{Z}}}$ means that any set in the support of $P_{W\!{\boldsymbol{Z}}}$ is also in the support of $P_W\!P_{{\boldsymbol{Z}}}$. We can therefore upper-bound the $\esssup$ as $$\label{eq:singledraw_uncond_leakage_esssup_weakening}
\log \esssup_{P_{{\boldsymbol{Z}}\vert W}} \frac{\dv P_{W\! {\boldsymbol{Z}}}}{\dv P_W\!P_{{\boldsymbol{Z}}}} \leq \log 