Given two probability measures $P$ and $Q$ and an event $E$, we provide bounds on $P(E)$ in terms of $Q(E)$ and $f-$divergences. In particular, the bounds are instantiated when the measures considered are a joint distribution and the corresponding product of marginals. This allows us to control the measure of an event under the joint, using the product of the marginals (typically easier to compute) and a measure of how much the two distributions differ, \textit{i.e.,} an $f-$divergence between the joint and the product of the marginals, also known in the literature as $f-$Mutual Information. The result is general enough to induce, as special cases, bounds involving $\chi^2$-divergence, Hellinger distance, Total Variation, etc. Moreover, it also recovers a result involving R\'enyi's $\alpha-$divergence. As an application, we provide bounds on the generalization error of learning algorithms via $f-$divergences.


Michael Gastpar

École Polytechnique Fédérale de Lausanne

Ibrahim Issa

American University of Beirut

Session Chair

Lizhong Zheng

Massachusetts Institute of Technology