$f$-divergence Inequalities

TL;DR

This paper systematically develops inequalities relating various $f$-divergences, providing tight bounds and extensions of classical inequalities like Pinsker's, with applications to relative entropy, total variation, and Rényi divergence.

Contribution

It introduces new methods for deriving $f$-divergence inequalities, including functional domination and moment-based bounds, extending classical results and providing tight bounds under boundedness assumptions.

Findings

Derived tight bounds among $f$-divergences.

Extended Pinsker's inequality to $E_eta$ divergence.

Established integral expressions for Rényi divergence.

Abstract

This paper develops systematic approaches to obtain $f$-divergence inequalities, dealing with pairs of probability measures defined on arbitrary alphabets. Functional domination is one such approach, where special emphasis is placed on finding the best possible constant upper bounding a ratio of $f$-divergences. Another approach used for the derivation of bounds among $f$-divergences relies on moment inequalities and the logarithmic-convexity property, which results in tight bounds on the relative entropy and Bhattacharyya distance in terms of $\chi^2$ divergences. A rich variety of bounds are shown to hold under boundedness assumptions on the relative information. Special attention is devoted to the total variation distance and its relation to the relative information and relative entropy, including "reverse Pinsker inequalities," as well as on the $E_\gamma$ divergence, which generalizes the total variation distance. Pinsker's inequality is extended for this type of $f$-divergence, a result which leads to an inequality linking the relative entropy and relative information spectrum. Integral expressions of the R\'enyi divergence in terms of the relative information spectrum are derived, leading to bounds on the R\'enyi divergence in terms of either the variational distance or relative entropy.