On the other hand, in the presence of the spurious escort services in New Orleans 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 analogy, removing \(s\) reduces the error to possess a test delivery with high deviations of no with the next ability, while removing \(s\) advances the mistake to own a test shipment with a high deviations out of no on the third 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 off \(\theta^\star\) towards the \(\beta^\star\) is confident from the seen direction, but it’s negative throughout the unseen recommendations; therefore, deleting \(s\) reduces the mistake. (Right) The brand new projection away from \(\theta^\star\) toward \(\beta^\star\) is comparable in viewed and unseen guidelines; hence, removing \(s\) advances 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.
The fresh new center model assigns lbs \(0\) with the unseen rules (lbs \(0\) into the second and you will 3rd features contained in this example)
The new kept front side ‘s the difference in the new projection away from \(\theta^\star\) to the \(\beta^\star\) throughout the viewed assistance along with their projection on the unseen recommendations scaled of the decide to try date covariance. Suitable side ‘s the difference between 0 (i.elizabeth., staying away from spurious provides) additionally the projection off \(\theta^\star\) towards the \(\beta^\star\) regarding the unseen guidelines scaled by the sample date covariance. Deleting \(s\) support whether your left front try higher than the right side.
Due to the fact concept is applicable merely to linear models, we now reveal that during the non-linear activities coached to your genuine-world datasets, removing a good spurious ability reduces the precision and you will has an effect on communities disproportionately.