Structural Learning Theory: A Metric-Topology Factorization Approach
For researchers in continual and lifelong learning, this work provides a theoretical framework to quantify and overcome structural challenges in non-stationary environments.
This paper introduces Structural Learning Theory (StrLT), which formalizes the structural difficulty of learning in multi-context environments via a new complexity measure called width. The authors show that width induces a phase transition: if the number of allocated cells is below the width, an irreducible structural error floor occurs; otherwise, learning reduces to standard statistical learning. They also propose the contractive-similarity operator to estimate width and the metric slingshot to reduce learning cost.
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.