VC dimension
|
|
The VC dimension (for Vapnik Chervonenkis dimension) is a measure of the capacity of a statistical classification algorithm. It is one of the core concepts in Vapnik Chervonenkis theory. It was originally defined by Vladimir Vapnik and Alexey Chervonenkis.
Intuitively, the capacity of a classification model is related to how complicated it can be. Think of thresholding a high-degree polynomial, where if the polynomial evaluates above zero, we classify that point into a positive class, negative otherwise. If we use a very high-degree polynomial, it can be very wiggly, and can fit a training set exactly. But, we should expect that outside of the training points, the classifier would not generalize well, because it is too wiggly. Such a polynomial has a high capacity. Alternatively, we can think about thresholding a linear polynomial. This polynomial may not fit the entire training set, because it has a low capacity. This notion of capacity can be made more rigorous.
First, we need the concept of shattering. Consider a classification model <math>f<math> with some parameter vector <math>\theta<math>. The model <math>f<math> can shatter a set of data points (<math>x_1,x_2,\ldots,x_n<math>) if, for all assignments of labels to those data points, there exists a <math>\theta<math> such that the model <math>f<math> makes no errors when evaluating that set of data points.
Now, we are ready to define a mathematical notion of capacity, called the VC dimension. The VC dimension of a model <math>f<math> is the maximum <math>h<math> such that some data point set of cardinality <math>h<math> can be shattered by <math>f<math>.
The VC dimension has utility in statistical learning theory, because it can predict a probabilistic upper bound on the test error of a classification model.
The bound on the test error of a classification model (on data that is drawn i.i.d. from the same distribution as the training set) is given by
- Training error + <math>\sqrt{h(\log(2N/h)+1)-\log(\eta/4)\over N}<math>
with probability <math>1-\eta<math>, where <math>h<math> is the VC dimension of the classification model, and <math>N<math> is the size of the training set.
References and sources
- Andrew Moore's VC dimension tutorial (http://www-2.cs.cmu.edu/~awm/tutorials/vcdim.html)
- V. Vapnik and A. Chervonenkis. "On the uniform convergence of relative frequencies of events to their probabilities." Theory of Probability and its Applications, 16(2):264--280, 1971.
- A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth. "Learnability and the Vapnik-Chervonenkis dimension." Journal of the ACM, 36(4):929--865, 1989.