arXiv:2509.18025v2 Announce Type: replace-cross Abstract: One can see deep-learning models as compositions of functions within the so-called tame geometry. In this expository note, we give an overview of some topics at the interface of tame geometry (also known as o-minimality), optimization theory, and deep learning theory and practice. To do so, we gradually introduce the concepts and tools used to build convergence guarantees for stochastic gradient descent in a general nonsmooth nonconvex, but tame, setting. This illustrates some ways in which tame geometry is a natural mathematical framew
The paper demonstrates the growing theoretical rigor applied to deep learning, moving beyond empirical success to foundational mathematical understanding.
A deeper mathematical understanding of AI models, particularly in optimization and convergence, can lead to more robust, reliable, and interpretable systems.
The theoretical underpinnings of deep learning are becoming more formalized, potentially enabling new algorithmic design principles and performance guarantees.
- · AI researchers
- · Deep learning framework developers
- · AI safety and interpretability initiatives
- · Ad-hoc deep learning development methods
Increased theoretical understanding of AI model behavior and stability.
Development of new AI models with provable properties and enhanced real-world reliability.
Broader adoption of AI in safety-critical domains due to improved theoretical foundations and trust.
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