  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.
