# 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).

Contained in this analogy, deleting $$s$$ decreases the mistake to possess a test shipping with a high deviations of no into the second element, whereas deleting $$s$$ boosts the mistake to own an examination shipments with a high deviations off no on the 3rd Savannah escort service function.

## 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 new projection of $$\theta^\star$$ to your $$\beta^\star$$ are confident on seen direction, but it is bad regarding the unseen guidelines; therefore, deleting $$s$$ decreases the mistake. (Right) The latest projection off $$\theta^\star$$ to the $$\beta^\star$$ is similar in viewed and you can 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 brand new key model assigns weight $$0$$ with the unseen guidelines (pounds $$0$$ toward 2nd and you may 3rd possess within this analogy)

The leftover front side is the difference in the latest projection out of $$\theta^\star$$ towards the $$\beta^\star$$ in the seen direction and their projection on unseen advice scaled because of the take to big date covariance. Ideal front ‘s the difference in 0 (we.e., not using spurious have) and also the projection out-of $$\theta^\star$$ for the $$\beta^\star$$ from the unseen recommendations scaled by test date covariance. Deleting $$s$$ facilitate if for example the remaining front side was higher than just the right front.

Due to the fact theory enforce only to linear designs, we currently reveal that in low-linear designs trained towards the real-globe datasets, deleting an effective spurious element reduces the precision and has an effect on organizations disproportionately.