For linear models operating on $$d$$-dimensional inputs, the Vapnik-Chervonenkis (VC) dimension is exactly $$d+1$$. For instance, a linear boundary can easily shatter, or assign any possible binary labeling to, three non-collinear points in two dimensions, but it cannot shatter four points.

Vapnik-Chervonenkis (VC) Dimension of Linear Models

A key result in statistical learning theory bounds the generalization gap between the empirical error $$R_	extrm{emp}$$ and the true error $$R$$ as a function of the Vapnik-Chervonenkis (VC) dimension and the dataset size $$n$$. With a probability of at least $$1-\delta$$, the generalization gap is strictly less than $$\alpha$$, provided that $$\alpha \geq c \sqrt{(	extrm{VC} - \log \delta)/n}$$, where $$c > 0$$ is a constant depending on the loss scale. While theoretically profound, this bound often provides a pessimistic estimate for complex models.

Generalization Error Bound via Vapnik-Chervonenkis Dimension

Following the foundational development of the Vapnik-Chervonenkis (VC) dimension, researchers introduced numerous alternative complexity measures to quantify the flexibility of model classes in statistical learning theory. Like the VC dimension, each of these advanced measures facilitates analogous mathematical guarantees for the generalization gap. However, despite their widespread utility in traditional statistical theory, these alternative function complexity metrics, when straightforwardly applied, similarly fail to explain the phenomenon of why highly parameterized deep neural networks generalize effectively in practice.

Alternative Function Complexity Measures

The Vapnik-Chervonenkis (VC) dimension is a metric that quantifies the complexity or flexibility of a model class. It is defined as the maximum number of data points for which the model class can perfectly classify any arbitrary binary labeling; that is, for any possible assignment of labels to these points, there exists some classifier $$f$$ in the class that correctly matches that labeling.

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Statistical learning theory is the mathematical subfield of machine learning that aims to elucidate the fundamental principles explaining why and when models trained on empirical data can generalize to unseen data. A primary objective is to bound the generalization gap by relating the properties of the chosen model class to the dataset size.

Statistical Learning Theory

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In machine learning, selecting a model class involves a fundamental trade-off. More flexible, or higher variance, model classes can perfectly fit the training data but carry a significant risk of overfitting and poor generalization. Conversely, more rigid, or higher bias, model classes generalize predictably but may underfit, failing to capture the underlying patterns in either the training or test data.

Trade-off Between Model Flexibility and Rigidity

Vapnik-Chervonenkis (VC) Dimension

In machine learning, models with continuously valued parameters, such as linear models, belong to an infinite class of functions (denoted as $$|\mathcal{F}| = \infty$$). Because the set of possible classifiers is infinite, it is impossible to simply test every function to guarantee that its empirical error matches its true population error without risking false discovery. This mathematical complexity directly motivates the need for advanced statistical learning theories, such as uniform convergence and the VC dimension, to bound the generalization gap.

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