Original: One natural way to minimize the effect of statistical skews is to use (cid:96) 2 weight decay while learning the classiﬁer: weight decay is known to exponentially speed up the convergence rate of gradient descent, leading to the max-margin solution in polynomial time.
Revised: One natural way to minimize the effect of statistical skews is to use (cid:96) 2 weight decay while learning the classiﬁer: weight decay is known to exponentially speed up the convergence rate of gradient descent, leading to the max-margin solution in polynomial time.

Original: Note that this doesn’t require any information about which datapoint belongs to the minority group and which to the majority.
Revised: Note that this doesn’t require any information about which datapoint belongs to the minority group and which to the majority.

Original: In the setting where we do have such information, we can consider another solution: simply oversample the minority dataset while running gradient descent.
Revised: In the setting where we do have such information, we can consider another solution: simply oversample the minority dataset while running gradient descent.

Original: This would result in a dataset with no statistical skews, and should hence completely nullify the effect of statistical skews.
Revised: This would result in a dataset with no statistical skews, and should hence completely nullify the effect of statistical skews.

Original: Thus, our argument provides a theoretical justiﬁcation for importance sampling for OoD generalization.
Revised: Thus, our argument provides a theoretical justiﬁcation for importance sampling for OoD generalization.

Original: E D EMONSTRATING SKEWS ON A NON - IMAGE - CLASSIFICATION DATASET
Revised: E D EMONSTRATING SKEWS ON A NON - IMAGE - CLASSIFICATION DATASET

Original: So far, we have demonstrated our insights in the context of image classiﬁcation tasks.
Revised: So far, we have demonstrated our insights in the context of image classiﬁcation tasks.

Original: However, our theoretical framework is abstract enough for us to apply these insights even in non-image classiﬁcation tasks.
Revised: However, our theoretical framework is abstract enough for us to apply these insights even in non-image classiﬁcation tasks.

Original: To demonstrate this, we consider an obesity estimation task based on the dataset from
Revised: To demonstrate this, we consider an obesity estimation task based on the dataset from

Original: Palechor & de la Hoz Manotas (2019).
Revised: Palechor & de la Hoz Manotas (2019).

Original: Dataset details.
Revised: Dataset details.

Original: The dataset consists of 16 features including height, weight, gender and habits of a person.
Revised: The dataset consists of 16 features including height, weight, gender and habits of a person.

Original: The label takes one of six different values corresponding to varying levels of obesity.
Revised: The label takes one of six different values corresponding to varying levels of obesity.

Original: We convert this to a binary classiﬁcation task by considering the ﬁrst three levels as one class and the last[Figure] three levels as another; we ignore any datapoint from the middle level, resulting in a dataset ofpoints.
Revised: We convert this to a binary classiﬁcation task by considering the ﬁrst three levels as one class and the last three levels as another; we ignore any datapoint from the middle level, resulting in a dataset ofpoints.

Original: We randomly split the data to extract 729 test datapoints.
Revised: We randomly split the data to extract 729 test datapoints.

Original: The dataset also has a categorical variable corresponding to the preferred mode of transport of the individual, which we convert to ﬁve binary features (corresponding to automobile, motorbike, bike, public transport and walking).
Revised: The dataset also has a categorical variable corresponding to the preferred mode of transport of the individual, which we convert to ﬁve binary features (corresponding to automobile, motorbike, bike, public transport and walking).

Original: We then scale all the resulting 20 features to lie between − 1 and 1 .
Revised: We then scale all the resulting 20 features to lie between − 1 and 1 .

Original: For all our experiments, we will consider ﬁtting a linear classiﬁer directly on these features.
Revised: For all our experiments, we will consider ﬁtting a linear classiﬁer directly on these features.

Original: Next, we construct a “biased” training set by sampling from the above training set in a way that the public transport feature becomes spuriously correlated with the label.
Revised: Next, we construct a “biased” training set by sampling from the above training set in a way that the public transport feature becomes spuriously correlated with the label.

Original: For convenience we denote this feature as x sp and the remaining features as x inv .
Revised: For convenience we denote this feature as x sp and the remaining features as x inv .

Original: Note that in the original dataset there is not much correlation between x sp and y : a max-margin classiﬁer w trained on the original dataset does not rely on this feature.
Revised: Note that in the original dataset there is not much correlation between x sp and y . In particular Pr D train [ x s p · y > 0] ≈ 0 . 47 (a value close to 0 .implies no spurious correlation).

Revised: Indeed, a max-margin classiﬁer w trained on the original dataset does not rely on this feature.

Original: In particular, w sp / (cid:107) w (cid:107) is as small as 0 . 
Revised: In particular, w sp / (cid:107) w (cid:107) is as small as 0 . 008 .

Original: Furthermore,the accuracy of this classiﬁer on a on a test dataset of 400 points where x sp = y is 100% while the accuracy on a test dataset of328 points where x sp = − y is only 99 . 7% , thus conﬁrming that the reliance on the spurious feature is negligible.
Revised: Furthermore, this classiﬁer achieves perfect accuracy on all of the test dataset.

Original: However, as we discuss below, the classiﬁer learned onin the biased set does rely on the spurious feature to a concerning level.
Revised: However, as we discuss below, the classiﬁer learned on the biased set does rely on the spurious feature.

Original: Geometric skews.
Revised: Geometric skews.

Original: We sample a biased training dataset that has 575 datapoints for which x sp = y and 10 datapoints for which x sp = − y .
Revised: We sample a biased training dataset that has 581 datapoints for which x sp = y and 10 datapoints for which x sp = − y .

Original: We then train a max-margin classiﬁer on this dataset, and observe that its spurious component w sp / (cid:107) w (cid:107) is as large as 0 .[-
-]
Revised: We then train a max-margin classiﬁer on this dataset, and observe that its spurious component w sp / (cid:107) w (cid:107) is as large as 0 .[-
-]

Original:  (about 15 times larger than the component of the max-margin on the original dataset).
Revised: 12 (about 15 times larger than the component of the max-margin on the original dataset).

Original: But more importantly, the accuracy of this model on the test dataset where x sp = y is 99 .
Revised: But more importantly, the accuracy of this model on the test dataset where x sp = y is 99 .

Original: 7% while the accuracy on a test dataset where x sp = − y is only 78 .
Revised: 5% while the accuracy on a test dataset where x sp = − y is only 57 .

Original: In other words, the classiﬁer learned here relies on the spurious feature, and suffers from poor accuracy on datapoints where the spurious feature does not align with the label.
Revised: In other words, the classiﬁer learned here relies on the spurious feature, and suffers from poor accuracy on datapoints where the spurious feature does not align with the label.

Original: Why does this happen?
Revised: Why does this happen?

Original: Our theory says that this must be because of the fact that as we increase the number of training datapoints, it requires greater and greater (cid:96) 2 max-margin norm to ﬁt the data using only the invariant feature space (and ignoring the spurious feature).
Revised: Our theory says that this must be because of the fact that as we increase the number of training datapoints, it requires greater and greater (cid:96) 2 max-margin norm to ﬁt the data using only the invariant feature space (and ignoring the spurious feature).

Original: Indeed, we verify that this increase does happen, in Fig 16a.
Revised: Indeed, we verify that this increase does happen, in Fig 16a.

Original: Statistical skews.
Revised: Statistical skews.

Original: To demonstrate the effect of statistical skews, we take a similar approach as in earlier sections.
Revised: To demonstrate the effect of statistical skews, we take a similar approach as in earlier sections.

Original: We consider a dataset where we have the same number of unique data in the majority group (where x sp = y ) and in the minority group (where x sp = − y ).
Revised: We consider a dataset where we have the same number of unique data in the majority group (where x sp = y ) and in the minority group (where x sp = − y ).

Original: However, the majority group contains many duplicates of its unique points, outnumbering the minority group.
Revised: However, the majority group contains many duplicates of its unique points, outnumbering the minority group.

Original: In particular, we consider two different datasets of 500 datapoints each, and in one dataset, the majority group forms 0 .
Revised: In particular, we consider two different datasets of 500 datapoints each, and in one dataset, the majority group forms 0 .

Original: 75 fraction of the data, and in the other it forms a 0 . 99 fraction of the data.
Revised: 75 fraction of the data, and in the other it forms a 0 . 85 fraction of the data.

Original: Note that in both these datasets, the max-margin classiﬁer does not rely on the spurious feature since there’s no geometric skew.
Revised: Note that in both these datasets, the max-margin classiﬁer does not rely on the spurious feature since there’s no geometric skew.

Original: More precisely, the value of w sp / (cid:107) w (cid:107) is less than 0 .

Original: 005 for the max-margin classiﬁer w on both these datasets.

Original: To demonstrate the effect of the statistical skew, we consider training a linear classiﬁer to minimize the logistic loss using SGD with a learning rate of 0 .
Revised: To demonstrate the effect of the statistical skew, we train a linear classiﬁer to minimize the logistic loss using SGD with a learning rate of 0 .

Original: 01 and batch size of 32 for as many as 10 k epochs (which is well beyond the number of epochs required to ﬁt the dataset to zero error).
Revised: 01 and batch size of 32 for as many as 10 k epochs (which is well beyond the number of epochs required to ﬁt the dataset to zero error).

Original: We then verify in Fig 16b that gradient descent takes a long time to let the spurious component of the classiﬁer get close to its ﬁnal value which is close to zero.
Revised: We then verify in Fig 16b that gradient descent takes a long time to let the spurious component of the classiﬁer get close to its ﬁnal value which is close to zero.

Original: This convergence rate, as we saw in our earlier experiments, is slower when the statistical skew is more prominent (i.e., for the dataset that has 0 .
Revised: This convergence rate, as we saw in our earlier experiments, is slower when the statistical skew is more prominent.

Original: 99 fraction of majority datapoints).

Original: Important note.
Revised: Important note.

Original: We must caution the reader that this demonstration is merely intended to showcase that our theoretical insights can help understand the workings of a classiﬁer in a practically important, non-image classiﬁcation dataset.
Revised: We must caution the reader that this demonstration is merely intended to showcase that our theoretical insights can help understand the workings of a classiﬁer in a practically important, non-image classiﬁcation dataset.

Original: However, we make no recommendations about how to deal with such high-risk tasks in practice.
Revised: However, we make no recommendations about how to deal with such high-risk tasks in practice.

Original: Such tasks require the practitioner to make careful ethical and social considerations which are beyond the scope of this paper.
Revised: Such tasks require the practitioner to make careful ethical and social considerations which are beyond the scope of this paper.

Original: Indeed, empirical benchmarks for evaluating OoD generalization (Gulrajani & Lopez-Paz, 2020) are largely based on low-risk image classiﬁcation tasks as it provides a safe yet reasonable test-bed for developing algorithms.
Revised: Indeed, empirical benchmarks for evaluating OoD generalization (Gulrajani & Lopez-Paz, 2020) are largely based on low-risk image classiﬁcation tasks as it provides a safe yet reasonable test-bed for developing algorithms.

Original: Remark on synthetic vs. natural spurious feature shifts.
Revised: Remark on synthetic vs. natural spurious feature shifts.

Original: It would be a valuable exercise to validate our insights on datasets with naturally-embedded spurious features.
Revised: It would be a valuable exercise to validate our insights on datasets with naturally-embedded spurious features.

Original: However, in order to test our insights, it is necessary to have datasets where one can explicitly quantify and manipulate the spurious features e.g., we need this power to be able to discard the spurious feature and examine the (cid:96) 2 norms of the max-margin in the invariant feature space.
Revised: However, in order to test our insights, it is necessary to have datasets where one can explicitly quantify and manipulate the spurious features e.g., we need this power to be able to discard the spurious feature and examine the (cid:96) 2 norms of the max-margin in the invariant feature space.

Original: Currently though, OoD research lacks such datasets: we either have MNIST-like datasets with synthetic but quantiﬁable spurious features, or realistic datasets like PACS (Asadi et al., 2019), VLCS (Fang et al., 2013) where it is not even clear what the spurious feature is.
Revised: Currently though, OoD research lacks such datasets: we either have MNIST-like datasets with synthetic but quantiﬁable spurious features, or realistic datasets like PACS (Asadi et al., 2019), VLCS (Fang et al., 2013) where it is not even clear what the spurious feature is.

Original: The lack of datasets with natural yet quantiﬁable spurious features is a gap that is beyond the scope of this paper, and is worth being bridged by the OoD community in the future.
Revised: The lack of datasets with natural yet quantiﬁable spurious features is a gap that is beyond the scope of this paper, and is worth being bridged by the OoD community in the future.