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

Computer Science > Machine Learning [Submitted on 12 Mar 2026] A Geometrically-Grounded Drive for MDL-Based Optimization in Deep Learning Ming Lei, Shufan Wu, Christophe Baehr 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. Comments: 8 pages, 9 figures, submitted to a journal and under review Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI) Cite as: arXiv:2603.12304 [cs.LG]   (or arXiv:2603.12304v1 [cs.LG] for this version)   https://doi.org/10.48550/arXiv.2603.12304 Focus to learn more Submission history From: Ming Lei PhD [view email] [v1] Thu, 12 Mar 2026 08:31:00 UTC (667 KB) Access Paper: HTML (experimental) view license Current browse context: cs.LG < prev   |   next > new | recent | 2026-03 Change to browse by: cs cs.AI References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)