# Conformal Mirror Descent with Logarithmic Divergences

arXiv: https://arxiv.org/abs/2209.02938 · HTML: https://mlsanity.com/paper/2209.02938/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Sep 2022.

## Built on by (most cited first, 3 of 3)

- [Mirror Descent and Novel Exponentiated Gradient Algorithms Using Trace-Form Entropies and Deformed Logarithms](https://mlsanity.com/paper/2503.08748/) (2025)
- [Information Geometry for the Working Information Theorist](https://mlsanity.com/paper/2310.03884/) (2023)
- [A Closed-Form Upper Bound for Admissible Learning-Rate Steps in Belief-Space Dynamics](https://mlsanity.com/paper/2605.06741/) (2026)

## Abstract (arXiv)

The logarithmic divergence is an extension of the Bregman divergence motivated by optimal transport and a generalized convex duality, and satisfies many remarkable properties. Using the geometry induced by the logarithmic divergence, we introduce a generalization of continuous time mirror descent that we term the conformal mirror descent. We derive its dynamics under a generalized mirror map, and show that it is a time change of a corresponding Hessian gradient flow. We also prove convergence results in continuous time. We apply the conformal mirror descent to online estimation of a generalized exponential family, and construct a family of gradient flows on the unit simplex via the Dirichlet optimal transport problem.