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Consider data points, each characterized by [math]\featurelen=10[/math] features [math]\featurevec \in \mathbb{R}^{\featurelen}[/math] and a single numeric label [math]\truelabel[/math]. We want to learn a linear hypothesis [math]h(\featurevec) = \weights^{T} \featurevec[/math] by minimizing the average squared error on the training set [math]\dataset[/math] of size [math]\samplesize=4[/math]. We could learn such a hypothesis by two approaches. The first approach is to split the dataset into a training set and a validation set. Then we consider all models that consists of linear hypotheses with weight vectors having at most two non-zero weights. Each of these models corresponds to a different subset of two weights that might be non-zero.

Find the model resulting in the smallest validation errors (see Algorithm).

Compute the average loss of the resulting optimal linear hypothesis on some data points that have neither been used in the training set nor the validation set.

Compare this average loss (“test error”) with the average loss obtained on the same data points by the hypothesis learnt by ridge regression .