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.

翻译：我们提出了一种采用庞加莱度量的黎曼梯度下降算法，用于计算阶数为$\alpha$的奥古斯丁信息——这一信息论中刻画指数误差行为的常用量。我们证明该算法以$\mathcal{O}(1 / T)$的速率收敛至最优解。据我们所知，这是首个对所有正阶数具有非渐近优化误差保证的算法。数值实验结果表明了该算法的实证效率。我们的研究基于一种新型的混合分析框架，该框架针对在一种黎曼度量下测地凸、而在另一种黎曼度量下测地光滑的函数，进行黎曼梯度下降分析。