Sharp Continuity of Petz and Sandwiched Rényi Conditional Entropies

What the math gives to ML

The paper supplies a sharp, dimension-aware continuity modulus for optimized Rényi conditional entropy under trace-distance perturbations. Its transferable asset is not the quantum formalism itself, but the explicit worst-case entropy budget \(\Gamma_{\alpha,D}(\delta)\), including saturation at \(\log D\) and an equality construction showing that the bound is tight. This can turn entropy regularization in classifiers, attention maps, or discrete latent modules into a perturbation-aware objective with a mathematically calibrated margin rather than an arbitrary robustness coefficient. The most practical initial test is to apply the bound to normalized neural probability vectors or attention distributions under augmentation, quantization, or dropout perturbations.