Guan-Ren Wang | Searcharxiv

arXiv2024

Linear Convergence in Hilbert's Projective Metric for Computing Augustin Information and a Rényi Information Measure

Consider the problems of computing the Augustin information and a Rényi 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 Rényi 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 $α\in (1/2, 1) \cup (1, 3/2)$ and Kamatsuka et al.'s algorithm for the Rényi information measure of order $α\in [1/2, 1) \cup (1, \infty)$, using Hilbert's projective metric.

math.OC↗

arXiv2024

Computing Augustin Information via Hybrid Geodesically Convex Optimization

We propose a Riemannian gradient descent with the Poincaré metric to compute the order-$α$ Augustin information, a widely used quantity for characterizing exponential error behaviors in information theory. We prove that the algorithm converges to the optimum at a rate of $\mathcal{O}(1 / T)$. As far as we know, this is the first algorithm with a non-asymptotic optimization error guarantee for all positive orders. Numerical experimental results demonstrate the empirical efficiency of the algorithm. Our result is based on a novel hybrid analysis of Riemannian gradient descent for functions that are geodesically convex in a Riemannian metric and geodesically smooth in another.

cs.IT↗