AI 中文总结
本文提出统一的理论框架，从逼近、优化、泛化等经典基础到过参数化、鲁棒性、生成建模、Transformer、上下文学习、缩放定律、可解释性、对齐和涌现等当代机制，系统梳理深度学习理论。
AI 中文摘要
深度学习已超越任何单一的数学解释。《从逼近到涌现》发展了一个统一的、以证明为导向的现代深度学习理论阐述，追踪了一条从逼近、优化和泛化的经典基础到过参数化、鲁棒性、生成建模、Transformer、上下文学习、缩放定律、可解释性、对齐和涌现等当代机制的路径。本书并非呈现孤立的结果，而是将广泛的文献组织成一个连贯的研究叙事：每个理论通过其控制的对象、使其成立的假设以及其未解释的现象来审视。本书面向研究人员、研究生和受过数学训练的从业者，提供了当今深度学习理论的严格地图：强大、不完整，且日益集中于学习机制如何从规模、数据、架构和训练中涌现的问题。
英文摘要
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.
CommentsWithdrawn by the author because the manuscript contains substantial theoretical errors in several core arguments and derivations, affecting the validity of multiple central claims. These problems require a comprehensive reassessment, and no corrected version is currently available. Readers should not rely on or cite this version
