# Convergence Rate Analysis of the Multiplicative Gradient Method for PET-Type Problems

arXiv: https://arxiv.org/abs/2109.05601 · HTML: https://mlsanity.com/paper/2109.05601/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Sep 2021.

## Built on by (most cited first, 1 of 1)

- [Optimistic Online Learning in Symmetric Cone Games](https://mlsanity.com/paper/2504.03592/) (2025)

## Abstract (arXiv)

We analyze the convergence rate of the multiplicative gradient (MG) method for PET-type problems with m component functions and an n-dimensional optimization variable. We show that the MG method has an O(ln(n)/t) convergence rate, in both the ergodic and the non-ergodic senses. Furthermore, we show that the distances from the iterates to the set of optimal solutions converge (to zero) at rate O(1/√t). Our results show that, in the regime n=O(exp(m)), to find an varepsilon-optimal solution of the PET-type problems, the MG method has a lower computational complexity compared with the relatively-smooth gradient method and the Frank-Wolfe method for convex composite optimization involving a logarithmically-homogeneous barrier.