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).
Inside analogy, removing \(s\) reduces the mistake to possess an examination shipments with high deviations out-of no into 2nd element, whereas removing \(s\) increases the error for a test shipments with high deviations out-of no for the third element.
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) New projection from \(\theta^\star\) with the \(\beta^\star\) is confident on seen direction, but it’s negative on the unseen direction; ergo, removing \(s\) decreases the mistake. (Right) New projection regarding \(\theta^\star\) towards \(\beta^\star\) is comparable in both viewed and unseen recommendations; thus, deleting \(s\) boosts 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.
This new center design assigns weight \(0\) toward unseen advice (pounds \(0\) toward 2nd and 3rd has actually contained in this analogy)
New left front ‘s the difference in the latest projection out-of \(\theta^\star\) to your \(\beta^\star\) in the viewed advice through its projection on unseen direction scaled from the test go out covariance. Ideal front ‘s the difference between 0 (i.e., staying away from spurious have) and also the projection off \(\theta^\star\) to the \(\beta^\star\) in the unseen guidelines scaled from the sample big date covariance. Removing \(s\) support in the event the left side was higher than the right side.
While the concept is applicable merely to linear designs, we now reveal that within the low-linear designs educated towards genuine-industry datasets, deleting a spurious element reduces the reliability and affects organizations disproportionately.