# Hessian informed mirror descent

arXiv: https://arxiv.org/abs/2106.13477 · HTML: https://mlsanity.com/paper/2106.13477/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jun 2021.

## Built on by (most cited first, 6 of 6)

- [Geometry and convergence of natural policy gradient methods](https://mlsanity.com/paper/2211.02105/) (2022)
- [Achieving High Accuracy with PINNs via Energy Natural Gradients](https://mlsanity.com/paper/2302.13163/) (2023)
- [Fisher-Rao Gradient Flows of Linear Programs and State-Action Natural Policy Gradients](https://mlsanity.com/paper/2403.19448/) (2024)
- [Approximate Newton policy gradient algorithms](https://mlsanity.com/paper/2110.02398/) (2021)
- [Conjugate Natural Selection](https://mlsanity.com/paper/2208.13898/) (2022)
- [Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers](https://mlsanity.com/paper/2607.22004/) (2026)

## Abstract (arXiv)

Inspired by the recent paper (L. Ying, Mirror descent algorithms for minimizing interacting free energy, Journal of Scientific Computing, 84 (2020), pp. 1-14),we explore the relationship between the mirror descent and the variable metric method. When the metric in the mirror decent is induced by a convex function, whose Hessian is close to the Hessian of the objective function, this method enjoys both robustness from the mirror descent and superlinear convergence for Newton type methods. When applied to a linearly constrained minimization problem, we prove the global and local convergence, both in the continuous and discrete settings. As applications, we compute the Wasserstein gradient flows and Cahn-Hillard equation with degenerate mobility. When formulating these problems using a minimizing movement scheme with respect to a variable metric, our mirror descent algorithm offers a fast convergent speed for the underlining optimization problem while maintaining the total mass and bounds of the solution.