Fixing linear regression problems with regularization

As we've seen, one outlier is enough to break the least-squares regression. Such instability is a manifestation of overfitting problems. Methods that help prevent models from overfitting are generally referred to as regularization techniques. Usually, regularization is achieved by imposing additional constraints on the model. This can be an additional term in a loss function, noise injection, or something else. We've already implemented one such technique previously, in Chapter 3, K-Nearest Neighbors Classifier. Locality constraint w in the DTW algorithm is essentially a way to regularize the result. In the case of linear regression, regularization imposes constraints on the weights vector values.

Ridge regression and Tikhonov regularization

Under the standard least squares method, the obtained regression coefficients can vary wildly. We can formulate the least squares regression as an optimization problem:

What we have on the right here...