Computing Augustin Information via Hybrid Geodesically Convex Optimization

Author:

Guan-Ren Wang, Chung-En Tsai, Hao-Chung Cheng, Yen-Huan Li

Keyword:

Computer Science, Information Theory, Information Theory (cs.IT), Optimization and Control (math.OC)

journal:

date:

2024-02-05 00:00:00

Abstract

We propose a Riemannian gradient descent with the Poincar\'e metric to compute the order-$\alpha$ 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.

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