Convergence of the Exponentiated Gradient Method with Armijo Line Search

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 Armijo 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 of 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.