Deep Learning as the Disciplined Construction of Tame Objects

2025-09-22

AI Summary

Deep learning lacks a rigorous mathematical foundation, particularly concerning the theoretical understanding of optimization dynamics in non-convex, non-smooth settings. Method: This work introduces, for the first time, the *o*-minimal geometry (tame geometry) framework to model deep neural networks as compositions of tame functions. It integrates *o*-minimality theory, nonsmooth optimization analysis, and stochastic gradient descent (SGD) dynamical modeling to characterize loss landscapes that are non-convex and non-smooth yet structurally well-behaved. Contributions/Results: (1) It establishes the first rigorous convergence guarantee for SGD in *o*-minimal non-convex environments; (2) it identifies the geometric origins of predictability and structural stability in deep models; and (3) it constructs a unified mathematical framework linking geometric structure, optimization dynamics, and empirical deep learning practice—substantially enhancing the theoretical interpretability and analytical tractability of training dynamics.

Technology Category

• Machine Learning: Deep Learning Theory
• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 framework for the study of AI systems, especially within Deep Learning.

Problem

• Providing convergence guarantees for stochastic gradient descent
• Studying deep learning models within tame geometry framework
• Analyzing optimization in nonsmooth nonconvex but tame settings

Innovation

• Deep learning models as tame geometry compositions
• Convergence guarantees for stochastic gradient descent
• Tame geometry framework for nonsmooth nonconvex optimization