Original: To build p φ ( (cid:96) | y , z , r γ ( x )) , we stack two feedforward layers of 16 hidden units with ReLU activations on top of a concatenation of y , r γ ( x ) ,and the scalar representation s ψ ( z ) as described above; the parameters φ contain ψ and the parameters for the two hidden-layer neural network.
Revised: To build p φ ( (cid:96) | y , z , r γ ( x )) , we stack two feedforward layers of 16 hidden units with ReLU activations on top of a concatenation of y , r γ ( x ) ,and the scalar representation s ψ ( z ) as described above; the parameters φ contain ψ and the parameters for the two hidden-layer neural network.

Original: We use 5 cross-ﬁtting folds in estimating the weights in reweighting-N U RD.
Revised: We use 5 cross-ﬁtting folds in estimating the weights in reweighting-N U RD.

Original: We use binary cross entropy as the loss in training ˆ p tr ( y | z ) , p θ ( y | r γ ( x )) , and p φ ( (cid:96) | y , z , r γ ( x ))) using the Adam (Kingma and Ba, 2014) optimizer with a learning rate of 10 − 3 .
Revised: We use binary cross entropy as the loss in training ˆ p tr ( y | z ) , p θ ( y | r γ ( x )) , and p φ ( (cid:96) | y , z , r γ ( x ))) using the Adam (Kingma and Ba, 2014) optimizer with a learning rate of 10 − 3 .

Original: We use a batch size of 1000 for both stages of N U RD.
Revised: We use a batch size of 1000 for both stages of N U RD.

Original: We optimized the model for ˆ p tr ( y | z ) for 150 epochs and ran the distillation step for 100 epochs with the Adam optimizer with the default learning rate.
Revised: We optimized the model for ˆ p tr ( y | z ) for 150 epochs and ran the distillation step for 100 epochs with the Adam optimizer with the default learning rate.

Original: Only the optimization for ˆ p tr ( y | z ) has a weight decay of 1 e − 2 .
Revised: Only the optimization for ˆ p tr ( y | z ) has a weight decay of 1 e − 2 .

Original: We run the distillation step with a ﬁxed λ = 1 and two epoch’s worth of gradient steps ( 20 ) for the critic model for each gradient step of the predictive model and the representation.
Revised: We run the distillation step with a ﬁxed λ = 1 and two epoch’s worth of gradient steps ( 20 ) for the critic model for each gradient step of the predictive model and the representation.

Original: To prevent the critic model from overﬁtting, we re-initialize φ, ψ after every gradient step of the predictive model.
Revised: To prevent the critic model from overﬁtting, we re-initialize φ, ψ after every gradient step of the predictive model.

Original: B. A DDITIONAL EXPERIMENTS
Revised: B. A DDITIONAL EXPERIMENTS

Original: Excluding the boundary from the images (covariates) does not improve ERM in general.
Revised: Excluding the boundary from the images (covariates) does not improve ERM in general.

Original: In both Waterbirds and chest X-rays, we use the easy-to-acquire border as a nuisance in N U RD.
Revised: In both Waterbirds and chest X-rays, we use the easy-to-acquire border as a nuisance in N U RD.

Original: Models trained on the central (non-border) regions of the image can exploit the nuisances in the center and consequently fail to generalize when the nuisance-label relationship changes.
Revised: Models trained on the central (non-border) regions of the image can exploit the nuisances in the center and consequently fail to generalize when the nuisance-label relationship changes.

Original: In fact, classiﬁers produced by ERM on border-less images do not generalize well to the test data, producing test accuracies of 39 ± 0 .
Revised: In fact, classiﬁers produced by ERM on border-less images do not generalize well to the test data, producing test accuracies of 39 ± 0 .

Original: 5% on chest X-rays and 65 ± 2 .
Revised: 5% on chest X-rays and 65 ± 2 .

Original: 3% on Waterbirds averaged over 10 seeds.
Revised: 3% on Waterbirds averaged over 10 seeds.

Original: However, as independence properties that hold for the border also hold for nuisances in the central region that are determined by the border, N U RD can use the border to control for certain nuisances in the center of the image.
Revised: However, as independence properties that hold for the border also hold for nuisances in the central region that are determined by the border, N U RD can use the border to control for certain nuisances in the center of the image.

Original: Additional experiments with N U RD.
Revised: Additional experiments with N U RD.

Original: We evaluate reweighting-N U RD further in the following ways:
Revised: We evaluate reweighting-N U RD further in the following ways:

Original: Run N U RD on data from the training data distribution deﬁned in section 5 and evaluate on data from test distributions p te with different nuisance-label relationships.
Revised: Run N U RD on data from the training data distribution deﬁned in section 5 and evaluate on data from test distributions p te with different nuisance-label relationships.

Original: Train N U RD with different-sized borders as nuisances.
Revised: Train N U RD with different-sized borders as nuisances.

Original: Train N U RD without a nuisance where the training and the test data have the same nuisance-label relationship; we implement this by setting the nuisance z = 0 wherever it is passed as input in the weight model or critic model.
Revised: Train N U RD without a nuisance where the training and the test data have the same nuisance-label relationship; we implement this by setting the nuisance z = 0 wherever it is passed as input in the weight model or critic model.

Original: Run the distillation step with different λ .
Revised: Run the distillation step with different λ .

Original: Different test distributions.
Revised: Different test distributions.

Original: For this experiment, we compute the test accuracies of the models trained in the experiments in section 5 on data with different nuisance-label relationships.
Revised: For this experiment, we compute the test accuracies of the models trained in the experiments in section 5 on data with different nuisance-label relationships.

Original: For both classifying Waterbirds and Pneumonia, a scalar parameter ρ controls nuisance-label relationships in the data generating process.
Revised: For both classifying Waterbirds and Pneumonia, a scalar parameter ρ controls nuisance-label relationships in the data generating process.

Original: In waterbirds, ρ = p ( y = waterbird | background = land ) = p ( y = landbird | background = water ) .
Revised: In waterbirds, ρ = p ( y = waterbird | background = land ) = p ( y = landbird | background = water ) .

Original: In chest X-rays, ρ corresponds to the fraction of Pneumonia cases that come from CheXpert and normal cases that come from MIMIC in the data; in this task, hospital
Revised: In chest X-rays, ρ corresponds to the fraction of Pneumonia cases that come from CheXpert and normal cases that come from MIMIC in the data; in this task, hospital

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Original: Unlike N U RD, ERM’s performance quickly degrades as the difference between the train and the increases.
Revised: Unlike N U RD, ERM’s performance quickly degrades as the difference between the train and the increases.

Original: differences are one source of nuisance-induced spurious correlations.
Revised: differences are one source of nuisance-induced spurious correlations.

Original: In both tasks, ρ = 0 . 1 in the training data; as test ρ increases, the nuisance-label relationship changes and becomes more different from the training data.
Revised: In both tasks, ρ = 0 . 1 in the training data; as test ρ increases, the nuisance-label relationship changes and becomes more different from the training data.

Original: We plot the average and standard error of accuracies aggregated over 10 seeds for different test
Revised: We plot the average and standard error of accuracies aggregated over 10 seeds for different test

Original: ρ ∈ { 0 . 5 , 0 . 7 , 0 . 9 } in ﬁg. 4.
Revised: ρ ∈ { 0 . 5 , 0 . 7 , 0 . 9 } in ﬁg. 4.

Original: Nuisance speciﬁcation with different borders.
Revised: Nuisance speciﬁcation with different borders.

Original: For Waterbirds, we ran NuRD with the pixels outside the central 168x168 patch (a 56 pixel border) as the nuisance.
Revised: For Waterbirds, we ran NuRD with the pixels outside the central 168x168 patch (a 56 pixel border) as the nuisance.

Original: Averaged over 10 seeds, reweighting-NuRD produced a model with 81% test accuracy which is similar to the accuracy achieved by N U RD using a 28-pixel border as the nuisance.
Revised: Averaged over 10 seeds, reweighting-NuRD produced a model with 81% test accuracy which is similar to the accuracy achieved by N U RD using a 28-pixel border as the nuisance.

Original: In comparison, ERM achieves an accuracy of 66% .
Revised: In comparison, ERM achieves an accuracy of 66% .

Original: N U RD without a nuisance and no nuisance-induced spurious correlations in classifying Waterbirds.
Revised: N U RD without a nuisance and no nuisance-induced spurious correlations in classifying Waterbirds.

Original: We performed an additional experiment on classifying Waterbirds where N U RD is given a constant nuisance which is equivalent to not using the nuisance.
Revised: We performed an additional experiment on classifying Waterbirds where N U RD is given a constant nuisance which is equivalent to not using the nuisance.

Original: We generated training and test data with independence between the nuisance and the label; the nuisance-label relationship does not change between training and test.
Revised: We generated training and test data with independence between the nuisance and the label; the nuisance-label relationship does not change between training and test.

Original: Averaged over 10 seeds, ERM achieved a test accuracy of 89 ± 0 .
Revised: Averaged over 10 seeds, ERM achieved a test accuracy of 89 ± 0 .

Original: and N U RD achieved a test accuracy of 88 ± 1% .
Revised: and N U RD achieved a test accuracy of 88 ± 1% .

Original: Reweighting-N U RD with different λ .
Revised: Reweighting-N U RD with different λ .

Original: Large λ s may make optimization unstable by penalizing even small violations of joint independence.
Revised: Large λ s may make optimization unstable by penalizing even small violations of joint independence.

Original: Such instabilities can lead N U RD to build predictive models that do not do better than marginal prediction, resulting in large distillation loss (loglikelihood + information loss) on the validation subset of the training data.
Revised: Such instabilities can lead N U RD to build predictive models that do not do better than marginal prediction, resulting in large distillation loss (loglikelihood + information loss) on the validation subset of the training data.

Original: However, a small λ may result in N U RD learning non-uncorrelating representations which can also perform worse than chance.
Revised: However, a small λ may result in N U RD learning non-uncorrelating representations which can also perform worse than chance.

Original: We ran N U RD on the waterbirds and class-conditional Gaussians experiments with λ = 5 (instead of λ = 1 like in section 5) and found that, on a few seeds, N U RD produces models with close to 50% accuracy (which is the same as majority prediction) or large information loss or both.
Revised: We ran N U RD on the waterbirds and class-conditional Gaussians experiments with λ = 5 (instead of λ = 1 like in section 5) and found that, on a few seeds, N U RD produces models with close to 50% accuracy (which is the same as majority prediction) or large information loss or both.

Original: Excluding seeds with large validation loss, reweighting-N U RD achieves an average test accuracy of 76% on waterbirds and 61% on class-conditional Gaussians.
Revised: Excluding seeds with large validation loss, reweighting-N U RD achieves an average test accuracy of 76% on waterbirds and 61% on class-conditional Gaussians.

Original: Annealing the λ during training could help stabilize optimization.
Revised: Annealing the λ during training could help stabilize optimization.

Original: In setting the hyperparameter λ in general, a practitioner should choose the largest λ such that optimization is still stable over different seeds and the validation loss is bounded away from that of predicting without any features.
Revised: In setting the hyperparameter λ in general, a practitioner should choose the largest λ such that optimization is still stable over different seeds and the validation loss is bounded away from that of predicting without any features.