Operator Learning for Algorithmic Equivalence: From Mirror Flows to Finite Steps

Mirror descent and Euclidean gradient methods use different geometries, but in continuous time they can sometimes describe the same dynamics after a nonlinear change of coordinates. We ask how far this equivalence extends when both methods take finite steps of the same size. For a mirror regularizer $R$, we first construct the continuous coordinate map $q_R$ and study operator learning of the map $R\mapsto q_R$ across regularizer families. At finite step size, requiring one objective-independent nonlinear coordinate map to match the two updates for a neighborhood of gradient directions forces the map to be affine and the mirror geometry to be quadratic. We therefore allow the finite-step coordinate map to depend on the objective. For fixed $R$, $f$, and step size $\eta$, we solve for $q_{R,f}(\eta ,\cdot )$; across problem families, we study operator learning of $(R,f)\mapsto q_{R,f}(\eta ,\cdot )$. We then study how the continuous and finite-step coordinate maps are related as $\eta$ varies. Finally, we ask whether the objective or regularizer can be modified so that the continuous and finite-step formulations produce the same coordinate map. Our numerical experiments include one-dimensional examples with exact solutions and genuinely coupled two-dimensional function families.