First, note that the smallest L2-norm vector that can fit the training data for the core model is \(>=[2,0,0]\)
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 this example, deleting \(s\) decreases the mistake to have a test delivery with high deviations regarding zero into 2nd element, while removing \(s\) advances the error to own a test shipping with high deviations out-of no towards the 3rd ability.
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 of \(\theta^\star\) into the \(\beta^\star\) are positive regarding seen recommendations, but it is bad on the unseen guidelines; hence, removing \(s\) decreases the error. (Right) The brand new projection off \(\theta^\star\) for the \(\beta^\star\) is comparable both in viewed and you may unseen directions; thus, deleting \(s\) advances the mistake.
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.
The new key model assigns lbs \(0\) into the unseen guidelines (lbs \(0\) with the next and you can 3rd provides inside example)
The newest Jersey City eros escort remaining front side ‘s the difference in the fresh new projection off \(\theta^\star\) for the \(\beta^\star\) throughout the viewed recommendations employing projection regarding unseen direction scaled of the decide to try day covariance. Best top is the difference between 0 (i.e., not using spurious enjoys) together with projection out of \(\theta^\star\) for the \(\beta^\star\) in the unseen guidance scaled by the try date covariance. Deleting \(s\) helps if your kept side is more than the proper front side.
While the principle applies merely to linear models, we have now demonstrate that in the low-linear models trained towards the real-business datasets, deleting an effective spurious function reduces the precision and you will influences teams disproportionately.