# Convergence of the Exponentiated Gradient Method with Armijo Line Search

arXiv: https://arxiv.org/abs/1712.08480 · HTML: https://mlsanity.com/paper/1712.08480/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Dec 2017.

## Built on by (most cited first, 7 of 7)

- [Generalized Exponentiated Gradient Algorithms and Their Application to On-Line Portfolio Selection](https://mlsanity.com/paper/2406.00655/) (2024)
- [Federated Hypergradient Descent](https://mlsanity.com/paper/2211.02106/) (2022)
- [Faster Stochastic First-Order Method for Maximum-Likelihood Quantum State Tomography](https://mlsanity.com/paper/2211.12880/) (2022)
- [Fast Minimization of Expected Logarithmic Loss via Stochastic Dual Averaging](https://mlsanity.com/paper/2311.02557/) (2023)
- [Non-KKT Accumulation in Entropic Mirror Descent](https://mlsanity.com/paper/2608.01658/) (2026)
- [Convex optimization over a probability simplex](https://mlsanity.com/paper/2305.09046/) (2023)
- [Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization](https://mlsanity.com/paper/2608.07248/) (2026)

## Abstract (arXiv)

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.