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.

翻译：Blahut-Arimoto算法是计算经典信道容量和率失真函数的著名方法。近期研究将该算法推广至计算这些量的各种量子对应物。本文证明这些Blahut-Arimoto算法是镜像下降法的特例——镜像下降法属于Bregman邻近方法，是约束凸优化中对梯度下降法的一种成熟推广。通过利用近期发展的凸分析工具，我们展示了基于相对平滑性和强凸性的分析如何恢复Blahut-Arimoto算法已知的次线性与线性收敛速率。这种Bregman邻近视角使我们能推导出具有类似收敛保证的相关算法，以解决信息论中Blahut-Arimoto型算法无法直接应用的问题。我们应用该框架计算了能量约束下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。