Divergence Inequalities from Multivariate Taylor's Theorem
Ian M. George, Alice Zheng, Akshay Bansal · 2024
Divergences are a fundamental framework for measuring dissimilarity in information theory and statistics. Here, we use the multivariate Taylor's theorem to obtain new integral representations of twice-differentiable Bregman and$\boldsymbol{f}$-divergences in finite dimensions. This results in input-dependent bounds on many$f$-divergences in terms of the$\chi^{2}$-divergence as well as reverse Pinsker inequalities. As an application, we show how this provides upper bounds on the mixing time of irreducible, scrambling Markov chains under a large class of$f$divergences in terms of the input-dependent$\chi^{2}$contraction coefficient.