# What is the VC dimension of a neural network?

## What is the VC dimension of a neural network?

Not surprisingly, the VC-dimension of a neural network is related to the number of training examples that are needed in order to train N to compute—or approximate—a specific target function h : D → {0, 1}.

What does VC dimension measure?

In Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the capacity (complexity, expressive power, richness, or flexibility) of a set of functions that can be learned by a statistical binary classification algorithm. A much simpler alternative is to threshold a linear function.

What is the VC dimension of H?

Definition 3 (VC Dimension). The VC-dimension of a hypothesis class H, denoted VCdim(H) is the size of the largest set C ⊂ X that can be shattered by H. If H can shatter sets of arbitrary size, then VCdim(H) = ∞.

### What is VC dimension in SVM?

functions. The VC dimension of {f(α)} is the maximum number of. training points that can be shattered by {f(α)} For example, the VC dimension of a set of oriented lines in R2 is three. In general, the VC dimension of a set of oriented hyperplanes in Rn is n+1.

What is a VC class?

If there is a largest, finite k such that C shatters at least one set of cardinality k, then C is called a Vapnik–Chervonenkis class, or VC class, of sets and S(C)=k its Vapnik–Chervonenkis index. …

Why is VC dimension useful?

VC dimension is useful in formal analysis of learnability, however. This is because VC dimension provides an upper bound on generalization error. So if we have some notion of how many generalization errors are possible, VC dimension gives an indication of how many could be made in any given context.

#### Is a higher VC dimension better?

The images shows that a higher VC dimension allows for a lower empirical risk (the error a model makes on the sample data), but also introduces a higher confidence interval. This interval can be seen as the confidence in the model’s ability to generalize.

Why is VC dimension important?

What is the VC dimension for an n dimensional linear classifier?

For a configuration of N points, there are 2^N possible assignments of positive or negative, so the classifier must be able to properly separate the points in each of these. In the below example, we show that the VC dimension for a linear classifier is at least 3, since it can shatter this configuration of 3 points.

## How do you get a VC dimension?

If you can find a set of n points, so that it can be shattered by the classifier (i.e. classify all possible 2n labelings correctly) and you cannot find any set of n+1 points that can be shattered (i.e. for any set of n+1 points there is at least one labeling order so that the classifier can not seperate all points …

Why VC dimension is important?

What does infinite VC dimension mean?

From your linked notes: The VC dimension of H is the size of the largest set of examples that can be shattered by H. The VC dimension is infinite if for all m, there is a set of m examples shattered by H. Usually, one considers a set of points in “general position” and shows that they can be shattered.