  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.
