Conformal Mirror Descent with Logarithmic Divergences
Optimization and Control
2022-09-08 v1 Statistics Theory
Statistics Theory
Abstract
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.
Cite
@article{arxiv.2209.02938,
title = {Conformal Mirror Descent with Logarithmic Divergences},
author = {Amanjit Singh Kainth and Ting-Kam Leonard Wong and Frank Rudzicz},
journal= {arXiv preprint arXiv:2209.02938},
year = {2022}
}