A Bregman Proximal Perspective on Classical and Quantum Blahut-Arimoto Algorithms
Kerry He, James Saunderson, Hamza Fawzi
备注：
30 pages. v2: Revised introduction and numerical experiments; strengthened proof for Theorem 4.7; other minor edits throughout. v3: Accepted into IEEE Transactions of Information Theory
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.
Improved Capacity Upper Bounds for the Deletion Channel using a Parallelized Blahut-Arimoto Algorithm
Instituto Superior Técnico, Universidade de Lisboa（里斯本技术学院，里斯本大学）；Instituto de Telecomunicações and Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa（电信研究所与数学系，里斯本技术学院，里斯本大学）
A Constrained BA Algorithm for Rate-Distortion and Distortion-Rate Functions
Tsinghua University（清华大学）；Yangtze Delta Region Institute (Huzhou), University of Electronic Science and Technology of China（电子科技大学长三角研究院（湖州））；Huawei Technologies Co. Ltd.（华为技术有限公司）；University of Science and Technology of China（中国科学技术大学）
本文改进 Blahut-Arimoto 算法，通过 Newton 求根动态更新乘子，直接计算给定失真下的 RD 及 DR 函数，并证明其 O(1/n) 收敛率与运算复杂度，实验显示显著加速。
