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 well-studied generalization of gradient descent for constrained convex optimization. Using new convex analysis tools, we show how relative smoothness and strong convexity analysis recovers known sublinear and linear convergence rates for Blahut-Arimoto algorithms. This mirror descent 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.

翻译：Blahut-Arimoto算法是计算经典信道容量与率失真函数的著名方法。近期研究已将该算法推广至计算这些量的各类量子对应形式。本文证明这些Blahut-Arimoto算法是镜像下降的特殊实例，后者是约束凸优化中梯度下降方法的成熟泛化形式。通过运用新的凸分析工具，我们展示了相对光滑性与强凸性分析如何恢复Blahut-Arimoto算法已知的次线性与线性收敛速率。这种镜像下降视角使我们能够推导出具有类似收敛保证的相关算法，用于求解信息论中Blahut-Arimoto类型算法无法直接适用的问题。我们将该框架应用于计算能量受限的经典与量子信道容量、经典与量子率失真函数，以及纠缠相对熵的近似值，所有计算均具有可证明的收敛保证。