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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.07950 |
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| _version_ | 1866914320836722688 |
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| author | Okanohara, Daisuke |
| author_facet | Okanohara, Daisuke |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07950 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Thermodynamic Theory of Learning Part II: Critical Period Closure and Continual Learning Failure Okanohara, Daisuke Machine Learning 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. |
| title | A Thermodynamic Theory of Learning Part II: Critical Period Closure and Continual Learning Failure |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2602.07950 |