Incremental Learning in Mirror Flows
2026-06-22 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study a mathematical process called mirror flows, which helps optimize functions with certain shapes. They focus on how these flows behave when starting near the edge of a specific area defined by a mirror potential. They find that when you zoom out, the paths stabilize and relate to a simpler problem defined by the domain's boundaries. This provides insight into how learning can happen step-by-step in these mirror flow systems.
mirror flowsconvex quadratic lossconvex lower semicontinuousmirror potentialrescaled trajectoriesindicator functionsubdifferentialsupport functiondual variableincremental learning
Authors
Raphaël Berthier, Loucas Pillaud-Vivien
Abstract
We study mirror flows generated by a convex quadratic loss and a general convex lower semicontinuous mirror potential. We show that, when initialized near the boundary of the domain of the mirror potential, their rescaled trajectories converge to a limiting mirror flow whose potential is the indicator function of the domain. In this limit, the primal variable minimizes the loss over a time-dependent hypothesis set: the subdifferential of the support function of the domain, evaluated at the dual variable. This characterization provides a general mechanism for incremental learning in mirror flows.