Abstract:Learning in structured, multi-context, or non-stationary environments involves two orthogonal difficulties. The first is \emph{metric}: once the correct context is known, how hard is prediction within it? This is the domain of Statistical Learning Theory (SLT). The second is \emph{structural}: how many local contexts are required, and how can they be discovered from data? This paper develops \emph{Structural Learning Theory} (StrLT) for the structural axis. We introduce \emph{width}, the minimum number of jointly contractive and low-risk cells needed to cover a learning problem. Width is incomparable with VC dimension: either can diverge while the other remains bounded. We show that width induces a \emph{phase transition}: if the allocated number of cells \(K<w\), learning suffers an irreducible structural error floor; if \(K\ge w\), the problem reduces to ordinary within-cell statistical learning. To estimate width, we introduce the \emph{contractive-similarity} (CS) operator, a task-adaptive graph kernel combining geometric locality with predictive compatibility. Its CS Laplacian exposes contractive basins through spectral separation. We further develop the \emph{metric slingshot}, which reuses low-dimensional latent contraction maps to reduce funnel-learning cost. Together, width, CS estimation, and the slingshot decompose learning into trap discovery and funnel generalization, with deep implications for continual and lifelong learning in an open-ended environment.
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2602.07974 [cs.LG] |
| (or arXiv:2602.07974v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2602.07974 arXiv-issued DOI via DataCite |
From: Xin Li [view email]
[v1]
Sun, 8 Feb 2026 13:59:22 UTC (202 KB)
[v2]
Wed, 6 May 2026 19:48:05 UTC (64 KB)
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