Optimal Bounds between $f$-Divergences and Integral Probability Metrics

The families of $f$-divergences (e.g. the Kullback-Leibler divergence) and Integral Probability Metrics (e.g. total variation distance or maximum mean discrepancies) are commonly used in optimization and estimation. In this work, we systematically study the relationship between these two families from the perspective of convex duality. Starting from a tight variational representation of the $f$-divergence, we derive a generalization of the moment generating function, which we show exactly characterizes the best lower bound of the $f$-divergence as a function of a given IPM. Using this characterization, we obtain new bounds on IPMs defined by classes of unbounded functions, while also recovering in a unified manner well-known results for bounded and subgaussian functions (e.g. Pinsker's inequality and Hoeffding's lemma). The variational representation also allows us to prove new results on the topological properties of the divergence which may be of independent interest.