On the other hand, in the presence of the spurious feature, the full model can fit the training data perfectly with a smaller norm by assigning weight \(1\) for the feature \(s\) (\(|<\theta^\text<-s>>|_2^2 = 4\) while \(|<\theta^\text<+s>>|_2^2 + w^2 = 2 < 4\)).
Generally, in the overparameterized regime, since the number of training examples is less than the number of features, there are some directions of data variation that are not observed in the training data. In this example, we do not observe any information about the second and third features. However, the non-zero weight for the spurious feature leads to a different assumption for the unseen directions. In particular, the full model does not assign weight \(0\) to the unseen directions. Indeed, by substituting \(s\) with \(<\beta^\star>^\top z\), we can view the full model as not using \(s\) but implicitly assigning weight \(\beta^\star_2=2\) to the second feature and \(\beta^\star_3=-2\) to the third feature (unseen directions at training).
Within example, deleting \(s\) reduces the error having an examination shipping with a high deviations out of no on next feature, while removing \(s\) advances the error getting a test distribution with a high deviations out-of no to the 3rd feature.
Drop in accuracy in test time depends on the relationship between the true target parameter (\(\theta^\star\)) and the true spurious feature parameters (\(<\beta^\star>\)) in the seen directions and unseen direction
As we saw in the previous example, by using the spurious feature, the full model incorporates \(<\beta^\star>\) into its estimate. The true target parameter (\(\theta^\star\)) and the true spurious feature parameters (\(<\beta^\star>\)) agree on some of the unseen directions and do not agree on the others. Thus, depending on which unseen directions are weighted heavily in the test time, removing \(s\) can increase or decrease the error.
More formally, the weight assigned to the spurious feature is proportional to the projection of \(\theta^\star\) on \(<\beta^\star>\) on the seen directions. If this number is close to the projection of \(\theta^\star\) on \(<\beta^\star>\) on the unseen directions (in comparison to 0), removing \(s\) increases the error, and it decreases the error otherwise. Note that since we are assuming noiseless linear regression and choose models that fit training data, the model predicts perfectly in the seen directions and only variations in unseen directions contribute to the error.
(Left) The fresh new projection out of \(\theta^\star\) on \(\beta^\star\) are positive regarding the seen direction, but it’s bad about unseen guidance; hence, deleting \(s\) reduces the mistake. (Right) The latest projection out of \(\theta^\star\) with the \(\beta^\star\) is similar in both viewed and you may unseen advice; ergo, removing \(s\) escalates the error.
Let’s now formalize the conditions under which removing the spurious feature (\(s\)) increases the error. Let \(\Pi = Z(ZZ^\top)^<-1>Z\) denote the column space of training data (seen directions), thus \(I-\Pi\) denotes the null space of training data (unseen direction). The below equation determines when removing the spurious feature decreases the error.
New key design assigns pounds \(0\) into unseen directions (weight \(0\) to your next and you will third have contained in this example)
The brand new remaining front side ‘s the difference between the projection of \(\theta^\star\) for the \(\beta^\star\) regarding the viewed assistance with their projection about unseen guidelines scaled because of the try time covariance. Ideal front side ‘s the difference between 0 (i.e., staying away from spurious have) while the projection out-of \(\theta^\star\) into \(\beta^\star\) on unseen direction scaled because of the attempt go out covariance. Deleting \(s\) assists if your kept front side was higher than best top.
Due to the fact principle can be applied only to linear habits, we have now show that into the low-linear patterns educated to your real-industry datasets, deleting