Conformal Mirror Descent with Logarithmic Divergences

TL;DR

This paper introduces conformal mirror descent, a new optimization method based on logarithmic divergence, extending mirror descent with applications in online estimation and optimal transport.

Contribution

It generalizes continuous-time mirror descent using logarithmic divergence and explores its dynamics, convergence, and applications in exponential families and optimal transport.

Findings

Derived the dynamics of conformal mirror descent under generalized mirror maps.

Proved convergence of the proposed method in continuous time.

Applied the method to online estimation and gradient flows on the simplex.

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.