From Approximation to Emergence: A Theory of Deep Learning
A new monograph by Zhilin Zhao presents a unified, proof-oriented account of modern deep learning theory, bridging classical approximation, optimization, and generalization with contemporary topics like overparameterization, transformers, in-context learning, scaling laws, and emergence.
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[Submitted on 1 Jul 2026]
Title:From Approximation to Emergence: A Theory of Deep Learning
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Abstract:Deep learning has outgrown any single mathematical explanation. From Approximation to Emergence develops a unified, proof-oriented account of modern deep learning theory, tracing a path from the classical foundations of approximation, optimization, and generalization to the contemporary mechanisms of overparameterization, robustness, generative modeling, transformers, in-context learning, scaling laws, interpretability, alignment, and emergence. Rather than presenting isolated results, the book organizes a broad literature into a coherent research narrative: each theory is examined through the object it controls, the assumptions that make it valid, and the phenomena it leaves unexplained. Written for researchers, graduate students, and mathematically trained practitioners, this monograph offers a rigorous map of deep learning theory as it stands today: powerful, incomplete, and increasingly centered on the question of how learned mechanisms arise from scale, data, architecture, and training.
Subjects:
Machine Learning (cs.LG); Machine Learning (stat.ML)
Cite as: arXiv:2607.01311 [cs.LG]
(or arXiv:2607.01311v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.01311
arXiv-issued DOI via DataCite
Submission history
From: Zhilin Zhao [view email] [v1] Wed, 1 Jul 2026 17:26:19 UTC (2,634 KB)
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