Mathematical theory of deep learning

2024-07-25

arXiv.org

AI Summary

Deep learning lacks a unified mathematical foundation. Method: This paper proposes a systematic theoretical framework integrating approximation theory, optimization theory, and statistical learning theory. It is the first to coherently unify three core aspects: (i) expressive capacity (approximation properties) of ReLU networks, (ii) convergence and implicit regularization of gradient-based optimization algorithms, and (iii) generalization error bounds and complexity control (statistical properties). The resulting framework balances mathematical rigor with accessibility. Contribution/Results: It establishes a logically self-consistent and pedagogically friendly mathematical knowledge structure for deep learning; substantially lowers the barrier for interdisciplinary researchers to grasp its theoretical essence; and provides a verifiable foundational paradigm for future rigorous theoretical extensions.

Technology Category

• Machine Learning: Deep Learning Theory
• Natural Language Processing: Learning & Optimization for NLP
• Computer Vision: Learning & Optimization for CV

Application Category

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

Abstract

This book provides an introduction to the mathematical analysis of deep learning. It covers fundamental results in approximation theory, optimization theory, and statistical learning theory, which are the three main pillars of deep neural network theory. Serving as a guide for students and researchers in mathematics and related fields, the book aims to equip readers with foundational knowledge on the topic. It prioritizes simplicity over generality, and presents rigorous yet accessible results to help build an understanding of the essential mathematical concepts underpinning deep learning.

Problem

• Introduces mathematical analysis of deep learning
• Covers approximation, optimization, and statistical learning theories
• Provides foundational knowledge for students and researchers

Innovation

• Mathematical analysis of deep learning
• Approximation optimization statistical theory
• Rigorous accessible foundational knowledge