  The batch exponentiated gradient (EG) method provides a principled approach
to convex smooth minimization on the probability simplex or the space of
quantum density matrices. However, it is not always guaranteed to converge.
Existing convergence analyses of the EG method require certain quantitative
smoothness conditions on the loss function, e.g., Lipschitz continuity of the
loss function or its gradient, but those conditions may not hold in important
applications. In this paper, we prove that the EG method with Armijo line
search always converges for any convex loss function with a locally Lipschitz
continuous gradient. Because of our convergence guarantee, the EG method with
Armijo line search becomes the fastest guaranteed-to-converge algorithm for
maximum-likelihood quantum state estimation, on the real datasets we have.
