THE MATHEMATICS OF GENERALIZATION IN DEEP NEURAL NETWORKS: FROM PAC LEARNING TO MODERN ARCHITECTURES

Abstract

Deep Neural Networks (DNNs) have revolutionized machine learning by achieving state-of-the-art performance across various domains. However, their ability to generalize performing well on unseen data remains a fundamental challenge. This paper explores the mathematical foundations of generalization in deep learning, beginning with the Probably Approximately Correct (PAC) learning framework and extending to modern architectures such as Deep Fuzzy Neural Networks (DFNNs). We analyze key theoretical concepts, including VC dimension, Radacher complexity, and the bias-variance tradeoff, while investigating how deep learning models balance memorization and generalization. Furthermore, we examine the role of regularization, optimization techniques, and hybrid architectures (neuro-fuzzy systems) in enhancing generalization under uncertainty. Empirical evaluations demonstrate that DFNNs achieve superior generalization (92.3% accuracy) compared to traditional models by integrating fuzzy logic for interpretability and deep learning for adaptive feature extraction. The study bridges theoretical learning theory with practical deep learning advancements, providing insights into the mathematical principles governing generalization in modern neural networks.