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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).

In this example, deleting $$s$$ reduces the mistake getting a test delivery with a high deviations out of no into next element, whereas removing $$s$$ boosts the mistake to possess a test shipment with a high deviations of zero for 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) The projection from $$\theta^\star$$ towards $$\beta^\star$$ was confident about seen direction, but it is negative regarding the unseen assistance; for this reason, removing $$s$$ reduces the error. (Right) The new projection away from $$\theta^\star$$ towards the $$\beta^\star$$ is comparable in seen and you will unseen tips; ergo, deleting $$s$$ escalates 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 newest core design assigns pounds $$0$$ to your unseen guidelines (weight $$0$$ on 2nd and you Richmond escort service can 3rd possess contained in this analogy)

The newest remaining front ‘s the difference between the new projection regarding $$\theta^\star$$ into $$\beta^\star$$ regarding the viewed direction the help of its projection on the unseen guidelines scaled by try big date covariance. Best side ‘s the difference between 0 (i.e., not using spurious provides) while the projection of $$\theta^\star$$ for the $$\beta^\star$$ regarding unseen guidelines scaled because of the attempt time covariance. Deleting $$s$$ facilitate in case the leftover front side try more than ideal top.

Given that principle can be applied in order to linear patterns, we have now show that during the non-linear patterns trained towards actual-world datasets, removing good spurious ability reduces the precision and you may affects organizations disproportionately.