Informational Frustration in Neural Manifolds: Shannon Bottlenecks and the Limits of Learnability

Abstract: Why overparameterised deep networks generalise so remarkably well remains one of the most stubborn open questions in machine learning theory. Classical frameworks like VC dimension and Rademacher complexity predict catastrophic overfitting in modern models, leaving a massive theoretical gap between theory and reality. In this paper, we bridge this divide by introducing a unified framework that links information theory, topology, and statistical mechanics to map the hard limits of deep learning. Central to our approach is the Entropic Learnability Horizon (ELH): a fundamental law stating that a network can only truly learn a target function 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. We establish the Shannon-Topological Bottleneck Theorem, proving that when a target boundary's geometric complexity exceeds this informational horizon, the system undergoes a sudden entropic phase transition. It falls into a state of Informational Frustration - a glassy, rigid memorization phase where generalization becomes thermodynamically impossible. Using this lens, we show that the enigmatic phenomenon of "grokking" is actually an Entropic Release, where weights abruptly reorganise to unlock the bottleneck. Finally, we translate this theory into practice with Entropic Gradient Descent (EGD), an optimization algorithm that dynamically manages weight entropy to keep learning on track. Ultimately, this work repositions entropy not just as a tool for tracking uncertainty but as the fundamental physical currency that dictates whether a machine can learn.

• The paper establishes that learnability is constrained by an interplay among Shannon, topological, and von Neumann entropies.
• It introduces the Entropic Learnability Horizon and the Shannon-topological bottleneck theorem to quantify generalization limits.
• It presents Entropic Gradient Descent, an entropy-driven optimization method to overcome informational frustration and promote robust learning.

Entropic Constraints and the Limits of Learnability in Neural Networks

Introduction

"Informational Frustration in Neural Manifolds: Shannon Bottlenecks and the Limits of Learnability" (2606.30512) addresses the foundational dissonance between empirical generalization in overparameterized neural models and the deficit of satisfactory theoretical explanations within classical statistical learning paradigms. Departing from conventional analyses predicated on VC-dimension and Rademacher complexity, the paper constructs a unified theoretical framework in which the learnability of a function is dictated by a triad of entropies—Shannon entropy of the data manifold, topological entropy of the decision boundary, and the von Neumann entropy of the weight distribution—culminating in the Entropic Learnability Horizon (ELH).

Theoretical Framework: The Entropic Triad

The paper formalizes three critical entropic quantities:

1. Shannon Entropy ($H_S$): Quantifies the intrinsic informational density of the data distribution, functioning as the baseline informational resource.
2. Topological Entropy ($H_T$): Captures the geometric and fractal complexity of the target decision boundary. High $H_T$ indicates highly entangled or non-smooth labeling structure, presenting inherent obstacles to generalization.
3. Von Neumann Entropy ($S_{vN}$): Represents the structural flexibility of the weight space by treating the optimized weight distribution as a density matrix. This allows direct measurement of entropy within the ensemble of solutions induced by stochastic optimization dynamics.

The central premise is that learnability is governed not simply by parameter scaling, but by the interplay among these entropic factors, constraining the information transmission and storage potential of the model.

The Entropic Learnability Horizon and Bottlenecks

The Entropic Learnability Horizon, $\Lambda$, is posited as the sum of the data and network entropies:

$\Lambda(D, \rho) = H_S(D) + S_{vN}(\rho)$

The main postulate asserts that reliable learning is feasible if and only if the topological entropy of the decision boundary does not exceed this horizon ($H_T(\partial F) \leq \Lambda$). If the complexity of the function's boundary overshoots $\Lambda$, the model is unable to internalize the mapping in a structured, generalizable fashion.

Shannon-Topological Bottleneck Theorem: The theorem rigorously bounds the learnability of a function by establishing that

$$H_T(\partial F) \leq \frac{1}{\gamma(d)} \left[ H_S(D) + S_{vN}(\rho) \right] + \mathcal{O}\left(\frac{1}{N}\right)$$

where $\gamma(d)$ is a geometric constant and $H_T$0 is the dataset size. The proof leverages geometric measure theory and the quantum data processing inequality, tightly integrating information theory with topological data analysis. This result links task hardness (as a function of decision boundary fractality) to the sum total of informative data content and entropy in the model weights.

Informational Frustration: When the condition $H_T$1 is met, the network transitions into a "Glassy Memorization Phase," characterized by high-loss landscape ruggedness and loss of generalization. This phenomenon is conceptualized as a phase transition analogous to physical systems undergoing geometric frustration—here, referred to as Informational Frustration—where the optimization process can only minimize loss via brittle memorization rather than principled generalization.

Explaining Grokking as Entropic Phase Transition

The paper provides a thermodynamic account of grokking, interpreting it as an instance of Entropic Release. Grokking, previously noted as a sharp, delayed improvement in generalization following prolonged overfitting, is here described as a sudden spike in the von Neumann entropy, expanding the ELH and allowing the network to resolve the target’s topological complexity. The free energy of the loss landscape, modeled as $H_T$2, elucidates how continued stochastic optimization can enable the system to overcome entropic bottlenecks and reorganize the weight space for improved generalization.

Entropic Gradient Descent: Algorithmic Implications

The Entropic Learnability Horizon is operationalized in the form of Entropic Gradient Descent (EGD). EGD directly integrates an entropic regularizer into the optimization objective, actively manipulating $H_T$3 through corrections in weight updates. The method estimates the weight covariance matrix online and applies an entropy-driven step to steer the distribution away from Informational Frustration. This is formalized as:

$H_T$4

with explicit covariance- and entropy-based updates to the weights. This approach is designed to maintain the network within a regime where the ELH is not violated, preserving generalization and precluding the glassy memorization phase.

Discussion: Implications and Future Trajectories

This theoretical synthesis repositions core machine learning problems—generalization, overparameterization, adversarial robustness, and scaling limits—as fundamentally thermodynamic phenomena. It imposes stricter conditions on model scaling, explicitly stating that hardware and architectural constraints on achievable von Neumann entropy can bottleneck performance regardless of parameter proliferation. The account of adversarial vulnerability as an entropic deficit in the presence of informational frustration offers a new explanatory axis for model fragility.

The approach suggests fertile ground for new algorithmic mechanisms explicitly targeting increases in stable weight entropy, as well as the architectural design of models based on optimizing entropic flexibility rather than mere statistical depth or width.

Conclusion

This work advances an integrated theory of neural network learnability grounded in the conservation and distribution of entropy across data, decision boundaries, and parameter spaces. The results, anchored by the Shannon-Topological Bottleneck Theorem, identify hard boundaries for generalization capacity, rigorously characterize phase transitions in learning dynamics, and propose entropy-driven optimization as a practical remedy for informational bottlenecks. The implications are profound, recasting learning in neural models as a process with strict thermodynamic and topological constraints, thereby setting new directions for theoretical and practical advancements in AI.