First, note that the smallest L2-norm vector that can fit the training data for the core model is \(<\theta^\text>=[2,0,0]\)

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 \(s\) (\(|<\theta^\text>|_2^2 = 4\) while \(|<\theta^\text>|_2^2 + w^2 = 2 < 4\)).

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 \(^\top z\), 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).

In this example, deleting \(s\) decreases the error to own an examination distribution with a high deviations off zero into the next ability, whereas removing \(s\) escalates the mistake getting a test shipping with a high deviations out of no into the 3rd element.

Drop in accuracy in test time depends on the relationship between the true target parameter (\(\theta^\star\)) and the true spurious feature parameters (\(\)) in the seen directions and unseen direction

As we saw in the previous example, by using the spurious feature, the full model incorporates \(\) into its estimate. The true target parameter (\(\theta^\star\)) and the true spurious feature parameters (\(\)) 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 \(\) on the seen directions. If this number is close to the projection of \(\theta^\star\) on \(\) 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 latest projection off \(\theta^\star\) into the \(\beta^\star\) try self-confident in the viewed recommendations, however it is bad on unseen recommendations; hence, deleting \(s\) decreases the mistake. (Right) This new projection regarding \(\theta^\star\) towards \(\beta^\star\) is comparable in seen and you may unseen information; ergo, deleting \(s\) advances the error.

Let’s now formalize the conditions under which removing the spurious feature (\(s\)) increases the error. Let \(\Pi = Z(ZZ^\top)^Z\) 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 key model assigns weight \(0\) on the unseen information (lbs \(0\) on second and you can third provides within analogy)

The latest left front side ‘s the difference between the projection regarding \(\theta^\star\) towards the \(\beta^\star\) regarding the seen guidelines with regards to projection on the unseen advice scaled by the sample go out covariance. Best side is the difference in 0 (i.elizabeth., not using spurious enjoys) as well as the projection of \(\theta^\star\) toward \(\beta^\star\) in the unseen assistance scaled by the test date covariance. Removing \(s\) assists in the event your kept front side is actually higher than suitable top.

While the concept is applicable merely to linear activities, we now demonstrate that from inside the low-linear habits coached with the genuine-globe datasets, removing a beneficial spurious element reduces the precision and you can has an effect on communities disproportionately.