Linear Convergence in Hilbert's Projective Metric for Computing Augustin Information and a R\'{e}nyi Information Measure

TL;DR

This paper proves linear convergence of two iterative algorithms for computing Augustin and Re9nyi information measures using Hilbert's projective metric, advancing understanding of their efficiency and convergence rates.

Contribution

It establishes the first rigorous proof of linear convergence for these algorithms within specific order ranges, previously only known to converge asymptotically.

Findings

Proves linear convergence of Augustin's fixed-point iteration for certain b5 values.

Establishes linear convergence of Kamatsuka et al.'s algorithm for specific b5 ranges.

Uses Hilbert's projective metric to analyze and demonstrate convergence rates.

Abstract

Consider the problems of computing the Augustin information and a R\'{e}nyi information measure of statistical independence, previously explored by Lapidoth and Pfister (IEEE Information Theory Workshop, 2018) and Tomamichel and Hayashi (IEEE Trans. Inf. Theory, 64(2):1064--1082, 2018). Both quantities are defined as solutions to optimization problems and lack closed-form expressions. This paper analyzes two iterative algorithms: Augustin's fixed-point iteration for computing the Augustin information, and the algorithm by Kamatsuka et al. (arXiv:2404.10950) for the R\'{e}nyi information measure. Previously, it was only known that these algorithms converge asymptotically. We establish the linear convergence of Augustin's algorithm for the Augustin information of order $\alpha \in (1/2, 1) \cup (1, 3/2)$ and Kamatsuka et al.'s algorithm for the R\'{e}nyi information measure of order $\alpha \in [1/2, 1) \cup (1, \infty)$, using Hilbert's projective metric.