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 escort service Aurora CO 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 error to have an examination delivery with a high deviations of zero to the second element, whereas removing \(s\) boosts the mistake having a test delivery with high deviations of no toward 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 \(\theta^\star\) for the \(\beta^\star\) is actually positive in the viewed guidelines, but it is bad about unseen assistance; hence, removing \(s\) reduces the mistake. (Right) Brand new projection of \(\theta^\star\) to your \(\beta^\star\) is similar both in viewed and you will unseen tips; therefore, 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 brand new center model assigns lbs \(0\) into the unseen guidelines (pounds \(0\) into the next and you will 3rd has actually contained in this example)
The fresh new leftover side ‘s the difference between the latest projection regarding \(\theta^\star\) towards \(\beta^\star\) regarding seen assistance with the projection regarding the unseen advice scaled because of the shot date covariance. The proper side ‘s the difference between 0 (we.elizabeth., not using spurious has actually) together with projection of \(\theta^\star\) on \(\beta^\star\) about unseen recommendations scaled of the test go out covariance. Removing \(s\) facilitate whether your kept front side is actually greater than best top.
Given that theory enforce just to linear habits, we have now show that within the non-linear designs educated towards genuine-industry datasets, removing an excellent spurious function decreases the accuracy and has an effect on communities disproportionately.