Statistics and Learning Theory
L.10.2
Lecture
Learning Theory I

# Robust Generalization via f-Mutual Information

Amedeo Roberto Esposito, Michael Gastpar, Ibrahim Issa

## Date & Time

01:00 am – 01:00 am

## Abstract

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.

## Presenters

EPFL

#### Michael Gastpar

École Polytechnique Fédérale de Lausanne

#### Ibrahim Issa

American University of Beirut

Paper

## Session Chair

#### Lizhong Zheng

Massachusetts Institute of Technology