  The Blahut-Arimoto algorithm is a well-known method to compute classical
channel capacities and rate-distortion functions. Recent works have extended
this algorithm to compute various quantum analogs of these quantities. In this
paper, we show how these Blahut-Arimoto algorithms are special instances of
mirror descent, which is a type of Bregman proximal method, and a well-studied
generalization of gradient descent for constrained convex optimization. Using
recently developed convex analysis tools, we show how analysis based on
relative smoothness and strong convexity recovers known sublinear and linear
convergence rates for Blahut-Arimoto algorithms. This Bregman proximal
viewpoint allows us to derive related algorithms with similar convergence
guarantees to solve problems in information theory for which
Blahut-Arimoto-type algorithms are not directly applicable. We apply this
framework to compute energy-constrained classical and quantum channel
capacities, classical and quantum rate-distortion functions, and approximations
of the relative entropy of entanglement, all with provable convergence
guarantees.
