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

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

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).

Inside analogy, removing \(s\) reduces the error to have a test delivery with high deviations out of no towards the next element, whereas deleting \(s\) advances the error to have a test distribution with a high deviations of no to 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) The newest projection off \(\theta^\star\) towards \(\beta^\star\) are confident about viewed assistance, but it is negative throughout the unseen recommendations; for this reason, deleting \(s\) reduces the error. (Right) The projection from \(\theta^\star\) into \(\beta^\star\) is comparable in both seen and you will unseen recommendations; therefore, deleting \(s\) boosts 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 key design assigns lbs \(0\) into unseen guidelines (lbs \(0\) towards 2nd and you will 3rd has actually within this analogy)

The new leftover front side ‘s the difference in brand new projection away from \(\theta^\star\) on the \(\beta^\star\) regarding seen assistance and their projection on unseen advice scaled by the test day covariance. Just the right top is the difference in 0 (i.e., staying away from spurious has) while the projection of \(\theta^\star\) with the \(\beta^\star\) throughout the unseen assistance scaled by decide to try time covariance. Removing \(s\) facilitate if your left front side is more than suitable front side.

While the idea is applicable merely to linear designs, we now reveal that within the non-linear habits instructed toward genuine-business datasets, deleting a good spurious function decreases the precision and impacts groups disproportionately.

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