Generalized Pinsker Inequality for Bregman Divergences of Negative Tsallis Entropies

2026-02-05

AI Summary

This work investigates the relationship between Bregman divergences induced by negative α-Tsallis entropy and the total variation (L¹) distance, with applications to probabilistic prediction and online learning error analysis. By leveraging tools from convex analysis, information geometry, and inequality optimization, the authors establish—for the first time—a tight Pinsker-type inequality that bounds the α-Tsallis Bregman divergence below by a quadratic function of the total variation distance. Specifically, they prove that for any probability distributions \( p \) and \( q \), \( D_\alpha(p\|q) \geq \frac{C_{\alpha,K}}{2} \|p - q\|_1^2 \), and explicitly derive the optimal constant \( C_{\alpha,K} \) in terms of the entropy parameter \( \alpha \) and the dimension \( K \). This result provides a refined characterization of distributional discrepancies and fills a critical gap in the theoretical understanding of Tsallis-entropy-based Bregman divergences.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Web Mining and Content Analysis: Content-based information diffusion

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 $D_\alpha$ 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 $\Delta^K$, we prove the sharp bound \[ D_\alpha(p\Vert q) \ge \frac{C_{\alpha,K}}{2}\cdot \|p-q\|_1^2, \] and we determine the optimal constant $C_{\alpha,K}$ explicitly for every choice of $(\alpha,K)$.

Problem

• Pinsker inequality
• Bregman divergence
• Tsallis entropy
• total variation
• information geometry

Innovation

• Generalized Pinsker inequality
• Bregman divergence
• Tsallis entropy
• β-divergence
• total variation