Generalized Pinsker Inequality for Bregman Divergences of Negative Tsallis Entropies

The Pinsker inequality lower bounds the Kullback–Leibler divergence $D_{\text{KL}}$ in terms of total variation and provides a canonical way to convert $D_{\text{KL}}$ control into $\parallel \cdot {\parallel }_1$-control. Motivated by applications to probabilistic prediction with Tsallis losses and online learning, we establish a generalized Pinsker inequality for the Bregman divergences $D_{\alpha }$ generated by the negative $\alpha$-Tsallis entropies – also known as $\beta$-divergences. Specifically, for any $p$, $q$ in the relative interior of the probability simplex ${\mathrm{\Delta }}^K$, we prove the sharp bound $$D_{\alpha }(p\mathrm{\parallel }q)\ge \frac{C_{\alpha ,K}}{2}\cdot \mathrm{\parallel }p-q{\mathrm{\parallel }}_1^2,$$ and we determine the optimal constant $C_{\alpha ,K}$ explicitly for every choice of $(\alpha ,K)$.