Convergence guarantees without pathwise mirror-descent equivalence

Develop checkable conditions that ensure convergence of generalized quantum Arimoto–Blahut trajectories to globally optimal fixed points without requiring pathwise equivalence between the Arimoto–Blahut and mirror-descent algorithms.

Background

The paper characterizes when a full-rank fixed point of a generalized quantum Arimoto–Blahut algorithm is globally optimal and shows that this condition need only hold at the terminal fixed point, whereas pathwise equivalence with mirror descent requires compatibility at every iterate. Consequently, the paper establishes optimality criteria for candidate fixed points but does not provide general convergence conditions ensuring that an arbitrary Arimoto–Blahut trajectory reaches a globally optimal fixed point.

The unresolved direction is to identify verifiable, problem-level or trajectory-level conditions that guarantee convergence to a globally optimal fixed point while allowing the Arimoto–Blahut and mirror-descent trajectories to differ.