A Thermodynamic Theory of Learning Part II: Critical Period Closure and Continual Learning Failure

TL;DR

This paper develops a thermodynamic and geometric framework to understand the limits of continual learning, showing that irreversibility causes capacity loss and catastrophic forgetting through compositional constraints on model reconfiguration.

Contribution

It introduces a capacity-threshold criterion based on the Jacobian semigroup and effective rank, formalizing how irreversibility limits continual learning.

Findings

Irreversibility imposes geometric restrictions on adaptability.

Finite-time learning reduces reconfiguration capacity over time.

A threshold criterion predicts when forgetting occurs.

Abstract

Learning performed over finite time is inherently irreversible. In Part~I of this series, we modeled learning as a transport process in the space of parameter distributions and derived the Epistemic Speed Limit (ESL), which lower-bounds entropy production under finite-time dynamics. In this work (Part~II), we show that irreversibility imposes a geometric restriction on future adaptability through the compositional structure of learning dynamics. Successive learning phases compose multiplicatively as transport maps, and their Jacobians form a semigroup whose rank and singular values are submultiplicative. As a result, dynamically usable degrees of reconfiguration can only decrease under composition. We formalize the remaining adaptability of a model in terms of compatible effective rank, defined as the log-volume of task-preserving directions that remain dynamically accessible. Although task performance may remain unchanged, finite-time learning can progressively reduce this reconfiguration capacity. We prove a capacity-threshold criterion for continual learning: let m_B denote the stable rank of the Hessian of a new task B restricted to the task-preserving manifold of a previously learned task A. If m_B exceeds the residual compatible effective rank, then task B is trajectory-level incompatible with task A; any sufficient adaptation necessarily induces forgetting. Thus catastrophic forgetting arises not from the absence of multi-task solutions, but from irreversible loss of reconfiguration capacity under compositional learning dynamics. This establishes a trajectory-level capacity limit for continual learning.