# The Information Geometry of Mirror Descent

arXiv: https://arxiv.org/abs/1310.7780 · HTML: https://mlsanity.com/paper/1310.7780/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2013.

## Built on by (most cited first, 30 of 54)

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- [The Information Geometry of Unsupervised Reinforcement Learning](https://mlsanity.com/paper/2110.02719/) (2021)
- [A Fisher-Rao gradient flow for entropy-regularised Markov decision processes in Polish spaces](https://mlsanity.com/paper/2310.02951/) (2023)
- [Sampling with Mirrored Stein Operators](https://mlsanity.com/paper/2106.12506/) (2021)
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- [Partitioned Variational Inference: A Framework for Probabilistic Federated Learning](https://mlsanity.com/paper/2202.12275/) (2022)
- [Entropy annealing for policy mirror descent in continuous time and space](https://mlsanity.com/paper/2405.20250/) (2024)
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- [Understanding Stochastic Natural Gradient Variational Inference](https://mlsanity.com/paper/2406.01870/) (2024)
- [FGOT: Graph Distances based on Filters and Optimal Transport](https://mlsanity.com/paper/2109.04442/) (2021)
- [Robust Imitation via Mirror Descent Inverse Reinforcement Learning](https://mlsanity.com/paper/2210.11201/) (2022)
- [Variational Inference with Mixtures of Isotropic Gaussians](https://mlsanity.com/paper/2506.13613/) (2025)
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- [Approximate Newton policy gradient algorithms](https://mlsanity.com/paper/2110.02398/) (2021)
- [Constrained Structure Learning for Scene Graph Generation](https://mlsanity.com/paper/2201.11697/) (2022)
- [Provably efficient variational generative modeling of quantum many-body systems via quantum-probabilistic information geometry](https://mlsanity.com/paper/2206.04663/) (2022)
- [Bregman Power k-Means for Clustering Exponential Family Data](https://mlsanity.com/paper/2206.10860/) (2022)
- [Mirror Descent and Novel Exponentiated Gradient Algorithms Using Trace-Form Entropies and Deformed Logarithms](https://mlsanity.com/paper/2503.08748/) (2025)
- [Federated ADMM from Bayesian Duality](https://mlsanity.com/paper/2506.13150/) (2025)
- [Natural Gradient VI: Guarantees for Non-Conjugate Models](https://mlsanity.com/paper/2510.19163/) (2025)
- [Mirror descent of Hopfield model](https://mlsanity.com/paper/2211.15880/) (2022)

## Abstract (arXiv)

Information geometry applies concepts in differential geometry to probability and statistics and is especially useful for parameter estimation in exponential families where parameters are known to lie on a Riemannian manifold. Connections between the geometric properties of the induced manifold and statistical properties of the estimation problem are well-established. However developing first-order methods that scale to larger problems has been less of a focus in the information geometry community. The best known algorithm that incorporates manifold structure is the second-order natural gradient descent algorithm introduced by Amari. On the other hand, stochastic approximation methods have led to the development of first-order methods for optimizing noisy objective functions. A recent generalization of the Robbins-Monro algorithm known as mirror descent, developed by Nemirovski and Yudin is a first order method that induces non-Euclidean geometries. However current analysis of mirror descent does not precisely characterize the induced non-Euclidean geometry nor does it consider performance in terms of statistical relative efficiency. In this paper, we prove that mirror descent induced by Bregman divergences is equivalent to the natural gradient descent algorithm on the dual Riemannian manifold. Using this equivalence, it follows that (1) mirror descent is the steepest descent direction along the Riemannian manifold of the exponential family; (2) mirror descent with log-likelihood loss applied to parameter estimation in exponential families asymptotically achieves the classical Cram'er-Rao lower bound and (3) natural gradient descent for manifolds corresponding to exponential families can be implemented as a first-order method through mirror descent.