Classical and Quantum Iterative Optimization Algorithms Based on Matrix Legendre-Bregman Projections

TL;DR

This paper introduces iterative optimization algorithms based on matrix Legendre-Bregman projections, establishing duality and convergence results, and explores quantum algorithmic techniques for potential speedups in matrix-based optimization tasks.

Contribution

It develops a general duality theorem for Bregman divergences on Hermitian matrices and connects classical algorithms with quantum techniques for improved efficiency.

Findings

Established a duality theorem for Bregman divergences on Hermitian matrices.

Designed iterative algorithms including exact and approximate Bregman projections.

Linked the algorithms to quantum techniques for potential computational speedups.

Abstract

We consider Legendre-Bregman projections defined on the Hermitian matrix space and design iterative optimization algorithms based on them. A general duality theorem is established for Bregman divergences on Hermitian matrices, and it plays a crucial role in proving the convergence of the iterative algorithms. We study both exact and approximate Bregman projection algorithms. In the particular case of Kullback-Leibler divergence, our approximate iterative algorithm gives rise to the non-commutative versions of both the generalized iterative scaling (GIS) algorithm for maximum entropy inference and the AdaBoost algorithm in machine learning as special cases. As the Legendre-Bregman projections are simple matrix functions on Hermitian matrices, quantum algorithmic techniques are applicable to achieve potential speedups in each iteration of the algorithm. We discuss several quantum algorithmic design techniques applicable in our setting, including the smooth function evaluation technique, two-phase quantum minimum finding, and NISQ Gibbs state preparation.