Generalised Pinsker Inequalities

TL;DR

This paper extends the classical Pinsker inequality to a broader class of divergences and provides a tight, explicit bound relating KL divergence to variational divergence, solving a long-standing problem.

Contribution

It generalizes Pinsker inequalities to arbitrary f-divergences and develops the best possible bounds using a novel connection to Bayes risk.

Findings

Derived a tight explicit bound relating KL and variational divergence.

Generalized Pinsker inequality to arbitrary f-divergences.

Solved a 40-year-old problem posed by Vajda.

Abstract

We generalise the classical Pinsker inequality which relates variational divergence to Kullback-Liebler divergence in two ways: we consider arbitrary f-divergences in place of KL divergence, and we assume knowledge of a sequence of values of generalised variational divergences. We then develop a best possible inequality for this doubly generalised situation. Specialising our result to the classical case provides a new and tight explicit bound relating KL to variational divergence (solving a problem posed by Vajda some 40 years ago). The solution relies on exploiting a connection between divergences and the Bayes risk of a learning problem via an integral representation.