Incremental Learning in Mirror Flows

2026-06-22

AI Summary

This work investigates the mechanism by which mirror flows achieve incremental learning in dynamically evolving hypothesis spaces. Focusing on mirror flows driven by convex quadratic losses and general convex, lower semicontinuous mirror potentials, the study shows that when initialized near the boundary of the potential’s domain, the rescaled trajectory converges to a limiting mirror flow whose potential is the indicator function of the domain. In this regime, the primal variable continuously minimizes the loss over a time-varying hypothesis set induced by the dual variable. By leveraging tools from convex analysis, subdifferential calculus, and mirror descent dynamics, the authors establish a theoretical framework linking mirror flows to incremental learning dynamics. This reveals a geometric optimization mechanism—activated under specific initialization—that naturally enables continual learning, offering a novel theoretical perspective on incremental learning systems.

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.

Problem

• Incremental Learning
• Mirror Flows
• Convex Optimization
• Limiting Dynamics
• Time-dependent Hypothesis Set

Innovation

• incremental learning
• mirror flows
• convex optimization
• limiting dynamics
• support function