Mirrorless Mirror Descent: A Natural Derivation of Mirror Descent

We present a primal only derivation of Mirror Descent as a "partial" discretization of gradient flow on a Riemannian manifold where the metric tensor is the Hessian of the Mirror Descent potential. We contrast this discretization to Natural Gradient Descent, which is obtained by a "full" forward Euler discretization. This view helps shed light on the relationship between the methods and allows generalizing Mirror Descent to general Riemannian geometries, even when the metric tensor is {\em not} a Hessian, and thus there is no "dual."