Generalized Pinsker Inequality for Bregman Divergences of Negative Tsallis Entropies
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917251265855488 |
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| author | Beretta, Guglielmo Cesari, Tommaso Colomboni, Roberto |
| author_facet | Beretta, Guglielmo Cesari, Tommaso Colomboni, Roberto |
| contents | 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 $(α,K)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05744 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalized Pinsker Inequality for Bregman Divergences of Negative Tsallis Entropies Beretta, Guglielmo Cesari, Tommaso Colomboni, Roberto Information Theory 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 $(α,K)$. |
| title | Generalized Pinsker Inequality for Bregman Divergences of Negative Tsallis Entropies |
| topic | Information Theory |
| url | https://arxiv.org/abs/2602.05744 |