Generalized Pinsker Inequality for Bregman Divergences of Negative Tsallis Entropies

Abstract: The Pinsker inequality lower bounds the Kullback--Leibler divergence $D_{\textrm{KL}}$ in terms of total variation and provides a canonical way to convert $D_{\textrm{KL}}$ control into $\lVert \cdot \rVert_1$-control. Motivated by applications to probabilistic prediction with Tsallis losses and online learning, we establish a generalized Pinsker inequality for the Bregman divergences $D_α$ generated by the negative $α$-Tsallis entropies -- also known as $β$-divergences. Specifically, for any $p$, $q$ in the relative interior of the probability simplex $Δ^K$, we prove the sharp bound [ D_α(p\Vert q) \ge \frac{C_{α,K}}{2}\cdot |p-q|1^2, ] and we determine the optimal constant $C{α,K}$ explicitly for every choice of $(α,K)$.