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.