New 'Entropic Learnability Horizon' theory explains why deep networks generalize

A new theoretical paper from Srinivasa Rao P. and Vangmayi P. Reddy (arXiv:2606.30512) takes on one of machine learning's most stubborn puzzles: why overparameterized deep networks generalize so well when classical frameworks like VC dimension and Rademacher complexity predict catastrophic overfitting. The authors bridge this gap by unifying information theory, topology, and statistical mechanics into a single framework. Their central contribution is the Entropic Learnability Horizon (ELH), a fundamental law stating that a network can truly learn a target function only if the Shannon entropy of the data manifold outpaces the topological entropy of the function's decision boundary—balanced by the von Neumann entropy of the network's weight space.

The paper's Shannon-Topological Bottleneck Theorem formalizes this: when a target boundary's geometric complexity exceeds the informational horizon, the system undergoes an entropic phase transition into a state of 'Informational Frustration'—a glassy, rigid memorization phase where generalization becomes thermodynamically impossible. The authors also reinterpret the enigmatic 'grokking' phenomenon as an Entropic Release, where weights abruptly reorganize to unlock the bottleneck. To translate theory into practice, they introduce Entropic Gradient Descent (EGD), an optimization algorithm that dynamically manages weight entropy to keep learning on track. This work repositions entropy as the fundamental physical currency governing machine learnability, potentially reshaping how researchers diagnose overfitting, design architectures, and schedule training.