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 \(<\theta^\text>=[2,0,0]\)

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 analogy, removing \(s\) decreases the mistake to have a test shipment with high deviations out of no with the second element, whereas deleting \(s\) escalates the error to possess a test shipping with a high deviations away from zero on 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 new projection of \(\theta^\star\) toward \(\beta^\star\) is actually confident throughout the seen direction, but it is bad on the unseen assistance; hence, deleting \(s\) reduces the mistake. (Right) The newest projection regarding \(\theta^\star\) towards the \(\beta^\star\) is similar both in viewed and you may unseen directions; therefore, removing \(s\) increases 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.

The brand new key design assigns weight \(0\) towards the unseen recommendations (weight \(0\) to your 2nd and 3rd keeps in this analogy)

The left top is the difference in brand new projection from \(\theta^\star\) to the \(\beta^\star\) in the viewed direction and their projection from the unseen guidelines scaled of the try date covariance. The proper front side ‘s the difference in 0 (i.elizabeth., not using spurious enjoys) together with projection regarding \(\theta^\star\) toward \(\beta^\star\) from the unseen assistance scaled from the try big date covariance. Deleting \(s\) assists when your left top try greater than the best front.

Just like the concept enforce just to linear activities, we have now reveal that inside low-linear activities taught toward real-industry datasets, removing a good spurious feature reduces the accuracy and you will influences teams disproportionately.

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