Despite extensive research, the fundamental reasons why certain architectural or optimization choices lead to improved generalization in deep networks remain largely unknown. Resolving this discrepancy between practical success and classical statistical theory is considered a massive open problem in the field.

Claude

While guarantees from classical learning theory (such as bounds based on the Vapnik-Chervonenkis (VC) dimension or Rademacher complexity) can be conservative even for classical models, they appear powerless to explain why deep neural networks generalize. For classification problems, deep models are typically expressive enough to perfectly fit arbitrary labels even for datasets consisting of millions of examples. In the classical picture, such extreme model complexity—even when utilizing familiar methods like $$\ell_2$$ regularization—should lead to severe overfitting. Paradoxically, despite perfectly fitting the training data with zero training error, these highly expressive models often generalize remarkably well to unseen data, contradicting traditional complexity-based generalization bounds.

Generalization Paradox in Deep Learning

Dive into Deep Learning

In deep learning, the relationship between a model's complexity (such as network width or depth) and its generalization gap can be non-monotonic. This counterintuitive behavior is known as the 'double-descent' pattern. Strangely, for many tasks where models perfectly fit the training data and achieve zero training error, practitioners can often reduce the generalization error further by making the model even more expressive—such as by adding layers, nodes, or training for more epochs. In this pattern, greater model complexity initially hurts generalization performance but subsequently helps mitigate overfitting.

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