Distribution-Dependent Improvements of McDiarmid's Inequality via Rényi Divergence

Source abstract

We investigate whether Rényi divergence can sharpen McDiarmid's inequality and its distribution-dependent refinement induced by KL divergence. Starting from Marton's transportation proof, we show that its transport inequality fails in general when KL divergence is replaced by Rényi divergence of order $0<α<1$. For sums of independent bounded random variables, however, the additivity of Rényi divergence allows us to adapt the argument and derive concentration bounds. For Bernoulli sums, these bounds strictly improve on the distribution-dependent bound induced by KL divergence over a range of deviation thresholds. We also establish a distribution-dependent refinement of Gilardoni's inequality relating total variation to Rényi divergence, providing explicit constants for our general concentration bounds.