Convergence of the Exponentiated Gradient Method with Armijo Line Search

TL;DR

This paper proves that the exponentiated gradient method with Armijo line search converges to the optimal solution for convex differentiable functions on various constrained sets, under mild conditions, using properties of the log-partition function.

Contribution

It provides the first convergence proof for a mirror descent-type method relying solely on differentiability, leveraging self-concordant properties of the log-partition function.

Findings

Convergence is guaranteed if the iterates have a strictly positive limit point.

The proof utilizes the self-concordant likeness of the log-partition function.

This is the first such convergence result for this class of methods.

Abstract

Consider the problem of minimizing a convex differentiable function on the probability simplex, spectrahedron, or set of quantum density matrices. We prove that the exponentiated gradient method with Armjo line search always converges to the optimum, if the sequence of the iterates possesses a strictly positive limit point (element-wise for the vector case, and with respect to the Lowner partial ordering for the matrix case). To the best our knowledge, this is the first convergence result for a mirror descent-type method that only requires differentiability. The proof exploits self-concordant likeness of the log-partition function, which is of independent interest.