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

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

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).

Within this analogy, removing $$s$$ reduces the error to have an examination shipment with a high deviations regarding no with the second ability, whereas deleting $$s$$ increases the error having an examination delivery with high deviations out of zero for 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 () 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 projection regarding $$\theta^\star$$ towards the $$\beta^\star$$ is actually self-confident on viewed guidance, however it is negative regarding the unseen guidance; therefore, removing $$s$$ decreases the mistake. (Right) This new projection from $$\theta^\star$$ on the $$\beta^\star$$ is comparable in seen and you may unseen advice; therefore, deleting $$s$$ boosts the mistake.

Let’s now formalize the conditions under which removing the spurious feature ($$s$$) increases the error. Let $$\Pi = Z(ZZ^\top)^Z$$ 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.

## Brand new center design assigns lbs $$0$$ into the unseen instructions (pounds $$0$$ on next and you will third has actually contained in this example)

The fresh new remaining top ‘s the difference in this new projection from $$\theta^\star$$ toward $$\beta^\star$$ regarding the seen guidelines with the projection about unseen direction scaled by the try date covariance. Just the right front is the difference in 0 (we.age., staying away from spurious possess) and the projection out-of $$\theta^\star$$ to the $$\beta^\star$$ regarding unseen Providence backpage escort direction scaled from the test time covariance. Deleting $$s$$ helps if the left front was higher than the proper front.

Once the theory applies just to linear designs, we currently reveal that from inside the non-linear patterns educated on the genuine-business datasets, removing a great spurious function reduces the reliability and you can impacts groups disproportionately.

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