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The number of unique labellings of the training data that can be achieved with linear classifiers is, in fact, finite. A line can bedefined by picking any pair of training points, as illustrated in [link] . Two classifiers can be defined from each such line: one that outputs a label “1” for everything on or abovethe line, and another that outputs “0” for everything on or above. There exist such pairs of training points, and these define all possible unique labellings of the training data.Therefore, there are at most unique linear classifiers for any random set of 2-dimensional features (the factor of 2 is due to the fact that for each linear classifier thereare 2 possible assignments of the labelling).
Thus, instead of infinitely many linear classifiers, we realize that as far as a random sample of training data is concerned, there are at most
unique linear classifiers. That is, using linear classification rules, there are at most unique label assignments for data points. If we like, we can encode each possibility with bits. In dimensions there are hyperplane classification rules which can be encoded in roughly bits. Roughly speaking, the number of bits required for encoding each model is the VC dimension. Theremarkable aspect of the VC dimension is that it is often finite even when is infinite (as in this example).
If has dimensions in total, we might consider linear classifiers based on features at a time. Lower dimensional hyperplanes are less complex than higher dimensionalones. Suppose we set
These spaces have increasing VC dimensions, and we can try to balance the empirical risk and a cost function depending on the VC dimension.Such procedures are often referred to as Structural Risk Minimization . This gives you a glimpse of what the VC dimension is all about. In future lectures we will revisit this topic in greaterdetail.
The basic idea of “hold-out” methods is to split the samples into a training set, , and a test set, .
Now, suppose we have a collection of different model spaces indexed by (e.g., is the set of polynomials of degree , with ), or suppose that we have a collection of complexity penalization criteria indexed by ( e.g., let , with ). We can obtain candidate solutions using the training set as follows. Define
and take
or
This provides us with a set of candidate solutions . Then we can define the hold-out error estimate using the test set:
and select the “best” model to be where
This type of procedure has many nice theoretical guarantees, provided both the training and test set grow with .
A very popular hold-out method is the so call “leaving-one-out cross-validation” studied in depth by Grace Wahba (UW-Madison,Statistics). For each we compute
or
Then we have cross-validation function
To summarize, this lecture gave a brief and incomplete survey of different methods for dealing with the issues of overfitting and modelselection. Given a set of training data, , our overall goal is to find
from some collection of functions, . Because we do not know the true distribution underlyingthe data points , it is difficult to get an exact handle on the risk, . If we only focus on minimizing the empirical risk we end up overfitting to the training data. Two general approaches were presented.
If an estimator or classifier satisfies
then we say that is -consistent with respect to the risk . When the context is clear, we will simply say that is consistent.
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