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A Geometrically-Grounded Drive for MDL-Based Optimization in Deep Learning

arXiv:2603.12304v1 Announce Type: cross Abstract: This paper introduces a novel optimization framework that fundamentally integrates the Minimum Description Length (MDL) principle into the training dynamics of deep neural networks. Moving beyond its conventional role as a model selection criterion, we reformulate MDL as an active, adaptive driving force within the optimization process itself. The core of our method is a geometrically-grounded cognitive manifold whose evolution is governed by a \textit{coupled Ricci flow}, enriched with a novel \textit{MDL Drive} term derived from first principles. This drive, modulated by the task-loss gradient, creates a seamless harmony between data fidelity and model simplification, actively compressing the internal representation during training. We establish a comprehensive theoretical foundation, proving key properties including the monotonic decrease of description length (Theorem~\ref{thm:convergence}), a finite number of topological phase tra

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Ming Lei, Shufan Wu, Christophe Baehr · · 1 min read · 22 views
#cs.LG #cs.AI

arXiv:2603.12304v1 Announce Type: cross Abstract: This paper introduces a novel optimization framework that fundamentally integrates the Minimum Description Length (MDL) principle into the training dynamics of deep neural networks. Moving beyond its conventional role as a model selection criterion, we reformulate MDL as an active, adaptive driving force within the optimization process itself. The core of our method is a geometrically-grounded cognitive manifold whose evolution is governed by a \textit{coupled Ricci flow}, enriched with a novel \textit{MDL Drive} term derived from first principles. This drive, modulated by the task-loss gradient, creates a seamless harmony between data fidelity and model simplification, actively compressing the internal representation during training. We establish a comprehensive theoretical foundation, proving key properties including the monotonic decrease of description length (Theorem~\ref{thm:convergence}), a finite number of topological phase transitions via a geometric surgery protocol (Theorems~\ref{thm:surgery}, \ref{thm:ultimate_fate}), and the emergence of universal critical behavior (Theorem~\ref{thm:universality}). Furthermore, we provide a practical, computationally efficient algorithm with $O(N \log N)$ per-iteration complexity (Theorem~\ref{thm:complexity}), alongside guarantees for numerical stability (Theorem~\ref{thm:stability}) and exponential convergence under convexity assumptions (Theorem~\ref{thm:convergence_rate}). Empirical validation on synthetic regression and classification tasks confirms the theoretical predictions, demonstrating the algorithm's efficacy in achieving robust generalization and autonomous model simplification. This work provides a principled path toward more autonomous, generalizable, and interpretable AI systems by unifying geometric deep learning with information-theoretic principles.

Executive Summary

This article introduces a novel optimization framework for deep learning, integrating the Minimum Description Length (MDL) principle into the training process. The framework utilizes a geometrically-grounded cognitive manifold and a coupled Ricci flow to achieve a balance between data fidelity and model simplification. The authors provide a comprehensive theoretical foundation, including proofs of key properties and guarantees for numerical stability and convergence. Empirical validation demonstrates the algorithm's efficacy in achieving robust generalization and autonomous model simplification, paving the way for more autonomous and interpretable AI systems.

Key Points

  • ▸ Integration of MDL principle into deep learning training process
  • ▸ Geometrically-grounded cognitive manifold and coupled Ricci flow
  • ▸ Theoretical foundation with proofs of key properties and guarantees
  • ▸ Empirical validation on synthetic regression and classification tasks

Merits

Unification of Geometric Deep Learning and Information-Theoretic Principles

The framework provides a principled path toward more autonomous, generalizable, and interpretable AI systems by combining geometric deep learning with information-theoretic principles.

Theoretical Foundation

The authors provide a comprehensive theoretical foundation, including proofs of key properties and guarantees for numerical stability and convergence, which strengthens the validity of the proposed framework.

Demerits

Computational Complexity

The algorithm's per-iteration complexity of O(N log N) may be a limitation for large-scale applications, potentially hindering its adoption in practice.

Limited Empirical Validation

The empirical validation is limited to synthetic regression and classification tasks, and further experimentation on real-world datasets is necessary to fully assess the framework's efficacy.

Expert Commentary

The proposed framework represents a significant advancement in the field of deep learning, as it provides a principled path toward more autonomous, generalizable, and interpretable AI systems. The integration of the MDL principle into the training process is a novel and innovative approach that can lead to more efficient and effective model training. However, further research is necessary to fully explore the potential of this framework and to address the limitations and challenges associated with its implementation.

Recommendations

  • ✓ Further experimentation on real-world datasets to assess the framework's efficacy and robustness.
  • ✓ Investigation of the framework's potential applications in various domains, such as computer vision and natural language processing.
  • ✓ Development of more efficient and scalable algorithms to reduce the computational complexity and facilitate large-scale adoption.

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