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

A General Convergence Result for the Exponentiated Gradient Method