Hessian informed mirror descent

TL;DR

This paper investigates a Hessian-informed mirror descent method that combines robustness and superlinear convergence, applicable to constrained minimization problems like Wasserstein gradient flows and Cahn-Hilliard equations.

Contribution

It introduces a novel mirror descent algorithm with a Hessian-based metric that achieves both robustness and superlinear convergence, extending to constrained problems with proven convergence.

Findings

Achieves superlinear convergence when Hessian of metric approximates the objective's Hessian.

Proves global and local convergence for linearly constrained problems.

Demonstrates fast convergence in computing Wasserstein gradient flows and Cahn-Hilliard equations.

Abstract

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.