Original: C OMBINING D IVERSE F EATURE P RIORS
Revised: C OMBINING D IVERSE F EATURE P RIORS

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: To improve model generalization, model designers often restrict the features that their models use, either implicitly or explicitly.
Revised: To improve model generalization, model designers often restrict the features that their models use, either implicitly or explicitly.

Original: In this work, we explore the design space of leveraging such feature priors by viewing them as distinct perspectives on the data.
Revised: In this work, we explore the design space of leveraging such feature priors by viewing them as distinct perspectives on the data.

Original: Speciﬁcally, we ﬁnd that models trained with diverse sets ofexplicit feature priors have less overlapping failure modes, and can thus be combined more effectively.
Revised: Speciﬁcally, we ﬁnd that models trained with diverse sets of feature priors have less overlapping failure modes, and can thus be combined more effectively.

Original: Moreover, we demonstrate that jointly training such models on additional (unlabeled) data allows them to correct each other’s mistakes, which, in turn, leads to better generalization and resilience to spurious correlations.
Revised: Moreover, we demonstrate that jointly training such models on additional (unlabeled) data allows them to correct each other’s mistakes, which, in turn, leads to better generalization and resilience to spurious correlations.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: The driving force behind deep learning’s success is its ability to automatically discover meaningful features in complex high-dimensional datasets.
Revised: The driving force behind deep learning’s success is its ability to automatically discover predictive features in complex high-dimensional datasets.

Original: In fact, these features can generalize beyond the speciﬁc task at hand, thus enabling models to transfer to other similar tasks (Donahue et al., 2014).
Revised: In fact, these features can generalize beyond the speciﬁc task at hand, thus enabling models to transfer to other (yet similar) tasks (Donahue et al., 2014).

Original: At the same time, the set of features that the model learns has a large impact on how well it will perform on unseen inputs, especially in the presence of distribution shift (Ponce et al., 2006; Torralba & Efros, 2011; Sagawa et al., 2020) or spurious correlations (Heinze-Deml & Meinshausen, 2017; Beery et al., 2018; Meinshausen, 2018).
Revised: At the same time, the set of features that the model learns has a large impact on how well it will perform on unseen inputs, especially in the presence of distribution shift (Ponce et al., 2006; Torralba & Efros, 2011; Sagawa et al., 2020) or spurious correlations (Heinze-Deml & Meinshausen, 2017; Beery et al., 2018; Meinshausen, 2018).

Original: Motivated by this, recent work focuses on encouraging speciﬁc modes ofmodel behavior by preventing the models from relying on certain features.
Revised: Motivated by this, recent work focuses on encouraging speciﬁc modes of behavior by preventing the models from relying on certain features.

Original: Examples include suppressing texture features (Geirhos et al., 2019; Wang et al., 2019), avoiding (cid:96) p -non-robust features (Tsipras et al., 2019; Engstrom et al., 2019), or utilizing different parts of the frequency spectrum (Yin et al., 2019).
Revised: Examples include suppressing texture features (Geirhos et al., 2019; Wang et al., 2019), avoiding (cid:96) p -non-robust features (Tsipras et al., 2019; Engstrom et al., 2019), or utilizing different parts of the frequency spectrum (Yin et al., 2019).

Original: At a high level, these methods can be thought of as ways of imposing a feature prior on the learning process, so as to bias the model towards acquiring features that generalize better.
Revised: At a high level, these methods can be thought of as ways of imposing a feature prior on the learning process, so as to bias the model towards acquiring features that generalize better.

Original: This makes the choice of the feature prior to impose a key design decision.
Revised: This makes the choice of the feature prior to impose a key design decision.

Original: The goal of this work is thus to explore the underlying design space of feature priors and, speciﬁcally, to understand:
Revised: The goal of this work is thus to explore the underlying design space of feature priors and, speciﬁcally, to understand:

Original: How can we effectively harness the diversity of feature priors?
Revised: How can we effectively harness the diversity of feature priors?

Original: O UR CONTRIBUTIONS
Revised: O UR CONTRIBUTIONS

Original: In this work, we cast different feature priors as diverse perspectives on the data and study how they can complement each other.
Revised: In this paper, we cast diverse feature priors as different perspectives on the data and study how they can complement each other.

Original: In particular, we aim to understand whether training with distinct priors result in models with non-overlapping failure modes and how such models can be combined to improve generalization.
Revised: In particular, we aim to understand whether training with distinct priors result in models with non-overlapping failure modes and how such models can be combined to improve generalization.

Original: This is particularly relevant in settings where the data is unreliable— e.g, when the training data contains a spurious correlation.
Revised: This is particularly relevant in settings where the data is unreliable— e.g, when the training data contains a spurious correlation.

Original: From this perspective, we focus our study on two priors that arise naturally in the context of image classiﬁcation—namely, shape and texture—and investigate the following:
Revised: From this perspective, we focus our study on two priors that arise naturally in the context of image classiﬁcation, shape and texture, and investigate the following:

Original: Feature diversity.
Revised: Feature diversity.

Original: We demonstrate that training models with diverse feature priors results in them making mistakes on different parts of the data distribution, even if their overall accuracy is similar.
Revised: We demonstrate that training models with diverse feature priors results in them making mistakes on different parts of the data distribution, even if they perform similarly in terms of overall accuracy.

Original: Further, one can harness this diversity to build model ensembles that are more accurate than those based on combining models which have the same feature prior.
Revised: Further, one can harness this diversity to build model ensembles that are more accurate than those based on combining models which have the same feature prior.

Original: Combining feature priors on unlabeled data.
Revised: Combining feature priors on unlabeled data.

Original: When learning from unlabeled data, the choice of feature prior can be especially important.
Revised: When learning from unlabeled data, the choice of feature prior can be especially important.

Original: For strategies such as self-training, sub-optimal prediction rules learned from sparse labeled data can be reinforced when pseudo-labeling the unlabeled data.
Revised: For strategies such as self-training, sub-optimal prediction rules learned from sparse labeled data can be reinforced when pseudo-labeling the unlabeled data.

Original: In this setting, we can leverage the diversity of feature priors to address such shortcomings.
Revised: We show that, in such settings, we can leverage the diversity of feature priors to address these issues.

Original: By  training models with different feature priorsjointly on the unlabeled data through the framework of co-training Blum & Mitchell (1998), we ﬁndwe ﬁnd that the models can correct each other’s mistakes to learn prediction rules that generalize better.
Revised: By jointly training models with different feature priors on the unlabeled data through the framework of co-training Blum & Mitchell (1998), we ﬁnd that the models can correct each other’s mistakes to learn prediction rules that generalize better.

Original: Learning in the presence of spurious correlations.
Revised: Learning in the presence of spurious correlations.

Original: Finally, we want to understand whether combining diverse priors during training, as described above, can prevent models from relying on correlations that are spurious, i.e., correlations that do not hold on the actual distribution of interest.
Revised: Finally, we want to understand whether combining diverse priors during training, as described above, can prevent models from relying on correlations that are spurious, i.e., correlations that do not hold on the actual distribution of interest.

Original: To model such scenarios, we consider a setting where a spurious correlation is present in the training data, but we also have access to (unlabeled) data where this correlation does not hold.
Revised: To model such scenarios, we consider a setting where a spurious correlation is present in the training data but we also have access to (unlabeled) data where this correlation does not hold.

Original: In this setting, we ﬁnd that co-training models with diverse feature priors can actually steer them away from such correlations and thus enable a successful generalization to the underlying distribution.
Revised: In this setting, we ﬁnd that co-training models with diverse feature priors can actually steer them away from such correlations and thus enable them to generalize to the underlying distribution.

Original: Overall, our ﬁndings highlight the potential of incorporating distinct feature priors into training.
Revised: Overall, our ﬁndings highlight the potential of incorporating distinct feature priors into the training process.

Original: We believe that further work along this direction will lead us to models that generalize more reliably.
Revised: We believe that further work along this direction will lead us to models that generalize more reliably.

Original:  B ACKGROUND : F EATURE P RIORS IN C OMPUTER V ISION
Revised:  B ACKGROUND : F EATURE P RIORS IN C OMPUTER V ISION

Original: When learning from structurally complex data, such as images, relying on raw input features alone (e.g., pixels) is not particularly useful.
Revised: When learning from structurally complex data, such as images, relying on raw input features alone (e.g., pixels) is not particularly useful.

Original: There has thus been a long line of work on extractingpredictive input patterns that can be more effective.
Revised: There has thus been a long line of work on extracting input patterns that can be more effective for prediction.

Original: While early approaches leveraged hand-crafted features, such as SIFT (Lowe, 1999) and HOG (Dalal & Triggs, 2005),  these have been  largely replaced by features that are automatically learned in an end-to-end fashion (Krizhevsky, 2009; Ciregan et al., 2012; Krizhevsky et al., 2012).
Revised: While early approaches, such as SIFT (Lowe, 1999) and HOG (Dalal & Triggs, 2005), leveraged hand-crafted features, these have been by now largely replaced by features that are automatically learned in an end-to-end fashion (Krizhevsky, 2009; Ciregan et al., 2012; Krizhevsky et al., 2012).

Original: Nevertheless,  model designers still tune their models to better suit a particular task via changes in the architecture or training methodology.
Revised: Nevertheless, even when features are learned, model designers still tune their models to better suit a particular task via changes in the architecture or training methodology.

Original: Such modiﬁcations can be thought of as imposing feature priors ,  that bias a model towards a particular set of features.
Revised: Such modiﬁcations can be thought of as imposing feature priors , i.e., priors that bias a model towards a particular set of features.

Original: One prominent example is convolutional neural networks, which are biased towards learning a hierarchy of localized features Fukushima (1980); LeCun et al.
Revised: One prominent example here are convolutional neural networks, which are biased towards learning a hierarchy of localized features Fukushima (1980); LeCun et al.

Original: Indeed, such a convolutional prior can be quite powerful: it is sufﬁcient to enable many image synthesis tasks without any training Ulyanov et al.
Revised: Indeed, such a convolutional prior can be quite powerful: it is sufﬁcient to enable many image synthesis tasks without any training Ulyanov et al.

Original: More recently, there has been work exploring the impact of explicitly restricting the set of features utilized by the model.
Revised: More recently, there has been work exploring the impact of explicitly restricting the set of features utilized by the model.

Original: For instance, Geirhos et al.
Revised: For instance, Geirhos et al.

Original: (2019) demonstrate that training models on stylized inputs (and hence suppressing texture information) can improve model robustness to common corruptions.
Revised: (2019) demonstrate that training models on stylized inputs (and hence suppressing texture information) can improve model robustness to common corruptions.

Original: In a similar vein, Wang et al.
Revised: In a similar vein, Wang et al.

Original: (2019) penalize the predictive power of local features to learn shape-biased models that generalize better between image styles.
Revised: (2019) penalize the predictive power of local features to learn shape-biased models that generalize better between image styles.

Original: A parallel line of work focuses on training models to be robust to small, worst-case input perturbations—e.g., using adversarial training Goodfellow et al.
Revised: A parallel line of work focuses on training models to be robust to small, worst-case input perturbations using, for example, adversarial training Goodfellow et al.

Original: ; Madry et al.
Revised: ; Madry et al.

Original:  Such training biases these models away from non-robust features (Tsipras et al., 2019; Ilyas et al., 2019; Engstrom et al., 2019), which tends to result in them being more aligned with human perception (Tsipras et al.,  2019), more resilient to certain input corruptions (Ford et al., 2019;
Revised: (2018) or randomized smoothing (Lecuyer et al., 2019; Cohen et al., 2019). Such training biases these models away from non-robust features (Tsipras et al., 2019; Ilyas et al., 2019; Engstrom et al., 2019), which tends to result in them being more aligned with human perception (Tsipras et al., 2019; Kaur et al., 2019), more resilient to certain input corruptions (Ford et al., 2019;

Original: Kireev et al., 2021), and better suited for transfer to downstream tasks Utrera et al.
Revised: Kireev et al., 2021), and better suited for transfer to downstream tasks Utrera et al.

Original: Salman et al.
Revised: Salman et al.

Original:  F EATURE P RIORS AS D IFFERENT P ERSPECTIVES
Revised:  F EATURE P RIORS AS D IFFERENT P ERSPECTIVES

Original: As we discussed, the choice of feature prior can have a large effect on what features a model relies on and, by extension, on how well it generalizes to unseen inputs.
Revised: As we discussed, the choice of feature prior can have a large effect on what features a model relies on and, by extension, on how well it generalizes to unseen inputs.

Original: In fact, one can view such priors as distinct perspectives on the data which capture different information about the input.
Revised: In fact, one can view such priors as distinct perspectives on the data, capturing different information about the input.

Original: In this section, we provide evidence to support this view through a case study on a pair of feature priors that arise naturally in the context of image classiﬁcation: shape and texture.
Revised: In this section, we provide evidence to support this view; speciﬁcally, we examine a case study on a pair of feature priors that arise naturally in the context of image classiﬁcation: shape and texture.

Original: 3. T RAINING SHAPE - AND TEXTURE - BIASED MODELS
Revised: 3. T RAINING SHAPE - AND TEXTURE - BIASED MODELS

Original: In order to train shape- and texture-biased models, we either pre-process the model input or modify the model architecture. Speciﬁcally:
Revised: In order to train shape- and texture-biased models, we either pre-process the model input or modify the model architecture as follows:

Original: Shape-biased models.
Revised: Shape-biased models.

Original: To suppress texture information in the images, we pre-process our inputs by applying an edge detection algorithm.
Revised: To suppress texture information in the images, we pre-process our inputs by applying an edge detection algorithm.

Original: We consider two  canonicaledge detection algorithms: the Cannyedge detection algorithm Ding & Goshtasby (2001) which produces a binary edge mask, and the Sobeledge detection algorithm Sobel & Feldman (1968) which provide a softer edge detection, hence retaining some texture information (see Figures 1b and 1c).
Revised: We consider two such canonical algorithms: the Canny algorithm Ding & Goshtasby (2001) which produces a binary edge mask, and the Sobel algorithm Sobel & Feldman (1968) which provide a softer edge detection, hence retaining some texture information (see Figures 1b and 1c).

Original: Texture-biased models.
Revised: Texture-biased models.

Original: To prevent the model from relying on the global structure of the image we utilize a variant of the BagNet architecture Brendel & Bethge (2019).
Revised: To prevent the model from relying on the global structure of the image, we utilize a variant of the BagNet architecture Brendel & Bethge (2019).

Original: This architecture deliberately limits the receptive ﬁeld of the model, thus forcing it to rely on local features (see Figure 1d).
Revised: This architecture deliberately limits the receptive ﬁeld of the model, thus forcing it to rely on local features (see Figure 1d).

Original: We visualize all of these priors in Figure 1 and provide implementation details in Appendix A.
Revised: We visualize all of these priors in Figure 1 and provide implementation details in Appendix A.

Original: [Figure]
Revised: [Figure]

Revised: CIFARSTL-

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Original: [Table]
Revised: [Table]

Revised: -biased models (Sobel and Canny) are more aligned with each other, they are both quite different from the texture-biased model (BagNet).

Original: 3. D IVERSITY OF FEATURE - BIASED MODELS
Revised: 3. D IVERSITY OF FEATURE - BIASED MODELS

Original: After training models with shape and texture biases as outlined above, we evaluate whether these models  capture complementary information about the input.
Revised: After training models with shape and texture biases as outlined above, we evaluate whether these models indeed capture complementary information about the input.

Original: Speciﬁcally, we train models on a small subset (100 examples per class) of the CIFAR-10 (Krizhevsky, 2009) and STL-10 (Coates et al., 2011) datasets, and measure the correlation between which test examples they correctly classify.
Revised: Speciﬁcally, we train models on a small subset (100 examples per class) of the CIFAR-10 (Krizhevsky, 2009) and STL-10 (Coates et al., 2011) datasets, and measure the correlation between which test examples they correctly classify.

Original: We ﬁnd that pairs consisting of a shape-biased model and a texture-biased model (i.e Canny/BagNet or Sobel/BagNet) indeed have the least correlated predictions—cf.
Revised: We ﬁnd that pairs consisting of a shape-biased model and a texture-biased model (i.e., Canny and BagNet, or Sobel and BagNet) indeed have the least correlated predictions—cf.

Original: Table 2.
Revised: Table 2.

Original: In other words, the mistakes that these models make aresigniﬁcantly more diverse than those made by identical models trained from different random initializations.
Revised: In other words, the mistakes that these models make are more diverse than those made by identical models trained from different random initializations.

Original: At the same time, different shape-biased models (Sobel and Canny) are relatively well-correlated with each other, which corroborates  that models trained on similar features of the input are likely to make similar mistakes.
Revised: At the same time, different shape-biased models (Sobel and Canny) are relatively well-correlated with each other, which corroborates the fact that models trained on similar features of the input are likely to make similar mistakes.

Original: Model ensembles.
Revised: Model ensembles.

Original: Having shown that training models with these feature priors results in diverse prediction rules, we  can now combine them to improve our generalization.
Revised: Having shown that training models with these feature priors results in diverse prediction rules, we examine if we can now combine them to improve our generalization.

Original: The canonical approach for doing so is to incorporate these models into an ensemble.
Revised: The canonical approach for doing so is to incorporate these models into an ensemble.

Original: We ﬁnd that the diversity of models trained with different feature priors  directly translates into an improved performance when combining them into an ensemble—cf.
Revised: We ﬁnd that the diversity of models trained with different feature priors indeed directly translates into an improved performance when combining them into an ensemble—cf.

Original: Table 3.
Revised: Table 3.

Original: In fact, we ﬁnd that the performance of the ensemble is tightly connected to prediction similarity of its constituents (as measured in Table 2), i.e., more diverse ensembles tend to perform better.
Revised: In fact, we ﬁnd that the performance of the ensemble is tightly connected to prediction similarity of its constituents (as measured in Table 2), i.e., more diverse ensembles tend to perform better.

Original: For instance, the best ensemble for the STL-10 dataset is the one combining a shape-biased (Canny) and a texture-biased model (BagNet) which were the models with the least aligned predictions
Revised: For instance, the best ensemble for the STL-10 dataset is the one combining a shape-biased (Canny) and a texture-biased model (BagNet) which were the models with the least aligned predictions.

Original: [Table]
Revised: [Table]

Original: Different
Revised: Same

Original: [Table]
Revised: [Table]

Original: Different
Revised: Different

Revised: Standard + Sobel

Original: [Table]
Revised: [Table]

Revised: Sobel +

Original: [Figure]
Revised: [Table]

Revised: Same

Original: [Table]
Revised: [Table]

Original: Notice how models trained with different priors lead to ensembles with better performance.
Revised: Canny + Canny [Table]

Original: Moreover, when the accuracy of the two base models is comparable, models that are more diverse (as measured in Table 2) result in better ensembles.
Revised: BagNet + [Table] [Table] [Table]

Original: We describe the different methods of combining models in Appendix A. and provide the full results in Appendix B.2.
Revised: Different Standard + Canny 53. [Table] [Table]

Original:  C OMBINING D IVERSE P RIORS ON U NLABELED D ATA
Revised:  C OMBINING D IVERSE P RIORS ON U NLABELED D ATA

Original: In the previous section, we saw that training models with different feature priors (e.g., shape- and texture-biased models) can lead to prediction rules with less overlapping failure modes—which, in turn, can lead to more effective model ensembles.
Revised: In the previous section, we saw that training models with different feature priors (e.g., shape- and texture-biased models) can lead to prediction rules with less overlapping failure modes—which, in turn, can lead to more effective model ensembles.

Original: However, ensembles only combine model predictions post hoc and thus cannot take advantage of diversity during the training process.
Revised: However, ensembles only combine model predictions post hoc and thus cannot take advantage of diversity during the training process.

Original: In this section, we instead focus on utilizing diversity during training.
Revised: In this section, we instead focus on utilizing diversity during training.

Original: Speciﬁcally, we will leverage the diversity introduced through these priors in the context of self-training Lee et al.
Revised: Speciﬁcally, we will leverage the diversity introduced through feature priors in the context of self-training Lee et al.

Original: (2013): a framework commonly used when the labeled data is insufﬁcient to learn a well-generalizing model.
Revised: (2013)—a framework commonly used when the labeled data is insufﬁcient to learn a well-generalizing model.

Original: This framework utilizes unlabeled data, which are then pseudo-labeled using an existing model and used for further training.
Revised: This framework utilizes unlabeled data, which are then pseudo-labeled using an existing model and used for further training.

Original: While such methods can often improve the overall model performance, they suffer from a signiﬁcant drawback: models tend to reinforce suboptimal prediction rules even when these  do not generalize to the underlying distribution Arazo et al.
Revised: While such methods can often improve the overall model performance, they suffer from a signiﬁcant drawback: models tend to reinforce suboptimal prediction rules even when these rules do not generalize to the underlying distribution Arazo et al.

Original: Our goal here is  to leverage diverse feature priors to address this exact shortcoming.
Revised: Our goal here is thus to leverage diverse feature priors to address this exact shortcoming.

Original: Speciﬁcally, we will jointly train models with different priors on the unlabeled data through the framework of co-training Blum & Mitchell (1998).
Revised: Speciﬁcally, we will jointly train models with different priors on the unlabeled data through the framework of co-training Blum & Mitchell (1998).

Original: Since these models capture complementary information about the input (cf.
Revised: Since these models capture complementary information about the input (cf.

Original: Table 2), we expect them to correct each other’s mistakes and improve their prediction rules.
Revised: Table 2), we expect them to correct each other’s mistakes and improve their prediction rules.

Original: As we will see in this section, this approach can indeed have a signiﬁcant impact  on the performance of the resulting model, outperforming ensembles that combine such models only at evaluation time—see summary in Figure 4.
Revised: As we will see in this section, this approach can indeed have a signiﬁcant impact [Figure] on the performance of the resulting model, outperforming ensembles that combine such models only at evaluation time—see summary in Figure 4.

Original: Setup.
Revised: Setup.

Original: We base our analysis on the CIFAR-10 and STL-10 datasets.
Revised: We base our analysis on the CIFAR-10 and STL-10 datasets.

Original: Speciﬁcally, we treat a small fraction of the training set as labeled examples (100 examples per class), another fraction as our validation set for tuning hyperparameters (10% of the total training examples), and the rest as un- [Figure] labeled data.
Revised: Speciﬁcally, we treat a small fraction of the training set as labeled examples (100 examples per class), another fraction as our validation set for tuning hyperparameters (10% of the total training examples), and the rest as unlabeled data.

Original: We report our results on the standard test set of each dataset.
Revised: We report our results on the standard test set of each dataset.

Original: (See Appendix A for experimental details, and Appendix B.5 forfurther experiments with varying levels of labeled data.)
Revised: (See Appendix A for experimental details, and Appendix B.6 for experiments with varying levels of labeled data.)

Original: 4. S ELF - TRAINING AND ENSEMBLES
Revised: 4. S ELF - TRAINING AND ENSEMBLES

Original: Before outlining our method for jointly training models with multiple priors on unlabeled data, we ﬁrst describe the standard approach for self-training a single model.
Revised: Before outlining our method for jointly training models with multiple priors, we ﬁrst describe the standard approach to self-training a single model.

Original: At a high level, the predictions of the model on the unlabeled data are treated as correct labels and are then used to re-train the same model Lee et al.
Revised: At a high level, the predictions of the model on the unlabeled data are treated as correct labels and are then used to further train the same model Lee et al.

Original: ; Iscen et al.
Revised: ; Iscen et al.

Original: ; Zou et al.
Revised: ; Zou et al.

Original: ; Xie et al.
Revised: ; Xie et al.

Original: The underlying intuition is that the classiﬁer will predict the correct labels for that data better than chance, and thus these pseudo-labels can be used to expand the training set.
Revised: The underlying intuition is that the classiﬁer will predict the correct labels for that data better than chance, and thus these pseudo-labels can be used to expand the training set.

Original: In practice, however, these pseudo-labels tend to be noisy.
Revised: In practice, however, these pseudo-labels tend to be noisy.

Original: Thus, a common approach is to only use the labels to which the model assigns the highest probability Lee et al.
Revised: Thus, a common approach is to only use the labels to which the model assigns the highest probability Lee et al.

Original: This process is repeated, self-training on increasingly larger fractions of the unlabeled data until all of it is used.
Revised: This process is repeated, self-training on increasingly larger fractions of the unlabeled data until all of it is used.

Original: We refer to each such training phase as an era .
Revised: We refer to each such training phase as an era .

Original: Ensembles of diverse self-trained models.
Revised: Ensembles of diverse self-trained models.

Original: Similarly to our results in Table 3, we ﬁnd that ensembles comprised of self-trained models with diverse feature priors outperform those that use the same prior from different random initializations (see Figure 4 for a summary and Appendix B. for the full results).
Revised: Similarly to our results in Table 3, we ﬁnd that ensembles comprised of self-trained models with diverse feature priors outperform those that use the same prior from different random initializations (see Figure 4 for a summary and Appendix B. for the full results).

Original: This demonstrates that, after self-training, these models continue to capture complementary information about the input that can be leveraged to improve performance.
Revised: This demonstrates that, after self-training, these models continue to capture complementary information about the input that can be leveraged to improve performance.

Original: 4. C O - TRAINING MODELS WITH DIFFERENT FEATURE PRIORS
Revised: 4. C O - TRAINING MODELS WITH DIFFERENT FEATURE PRIORS

Original: Moving beyond self-training with a single feature prior, our goal in this section is to leverage multiple feature priors by jointly training them on the unlabeled data.
Revised: Moving beyond self-training with a single feature prior, our goal in this section is to leverage multiple feature priors by jointly training them on the unlabeled data.

Original: This idea naturally ﬁts into the framework of co-training : a method used to learn from unlabeled data when inputs correspond to multiple independent sets of features Blum & Mitchell (1998).
Revised: This idea naturally ﬁts into the framework of co-training : a method used to learn from unlabeled data when inputs correspond to multiple independent sets of features Blum & Mitchell (1998).

Original: In this framework, each set of features is used to train a different model.

Original: Then, these models are used to pseudo-label the unlabeled data, from which the most conﬁdent examples are added to a common pool used for further training.

Original: Concretely, we ﬁrst train a model for each feature prior.
Revised: Concretely, we ﬁrst train a model for each feature prior.

Original: Then, we combine the pseudo-labels on the unlabeled data that were assigned the highest probability for each model—including duplicates with potentially different labels—to form a new training set which we use for further training.
Revised: Then, we collect the pseudo-labels on the unlabeled data that were assigned the highest probability for each model—including duplicates with potentially different labels—to form a new training set which we use for further training.

Original: Similarly to the self-training case, we repeat this process over several eras, increasing the fraction of the unlabeled dataset used at each era.
Revised: Similarly to the self-training case, we repeat this process over several eras, increasing the fraction of the unlabeled dataset used at each era.

Original: Intuitively, this iterative process allows the models to bootstrap off of each other’s predictions, learning correlations that they might fail to learn from the labeled data alone.
Revised: Intuitively, this iterative process allows the models to bootstrap off of each other’s predictions, learning correlations that they might fail to learn from the labeled data alone.

Original: At the end of this process, we are left with two models, one for each prior, which we combine into a single classiﬁer by training a standard model from scratch on the combined pseudo-labels.
Revised: At the end of this process, we are left with two models, one for each prior, which we combine into a single classiﬁer by training a standard model from scratch on the combined pseudo-labels.

Original: We provide a more detailed explanation of the methodology in Appendix A.5.
Revised: We provide a more detailed explanation of the methodology in Appendix A.

Original: [Table]
Revised: [Table]

Original: Recall that, for the case of co

Original: -training pseudo-labeling is performed by combining the predictions of both models

Original: Finally, we report the performance of a standard model trained from scratch on the resulting pseudo-labels (Column 5).

Original: We provide 95% conﬁdence intervals computed via bootstrap with 5000 iterations

Original: Co-training performance.
Revised: Co-training performance.

Original: We ﬁnd that co-training with shape- and texture-based priors can signiﬁcantly improve the test accuracy of the ﬁnal model compared to self-training with any of the priors alone (Table 5).
Revised: We ﬁnd that co-training with shape- and texture-based priors can signiﬁcantly improve the test accuracy of the ﬁnal model compared to self-training with any of the priors alone (Table 5).

Original: This is despite the fact that, when using self-training alone, the standard model outperforms all other models (Column 4, Table 5).
Revised: This is despite the fact that, when using self-training alone, the standard model outperforms all other models (Column 4, Table 5).

Original: Moreover, co-training models with diverse priors improves upon simply combining them in an ensemble (Appendix B.3).
Revised: Moreover, co-training models with diverse priors improves upon simply combining them in an ensemble (Appendix B.3).

Original: In Appendix B.4, we report the performance of co-training with every pair of priors.
Revised: In Appendix B.5, we report the performance of co-training with every pair of priors.

Original: We ﬁnd that co-training with shape- and texture-based priors together (Canny + BagNet for STL-10 and Sobel + BagNet for CIFAR-10) outperform every other prior combination.
Revised: We ﬁnd that co-training with shape- and texture-based priors together (Canny + BagNet for STL-10 and Sobel + BagNet for CIFAR-10) outperform every other prior combination.

Original: Note that this is the case even though, when only ensembling models with different priors (c.f Table 3 and Appendix B.3), Standard + Sobel is consistently the best performing pair for CIFAR-10.
Revised: Note that this is the case even though, when only ensembling models with different priors (c.f Table 3 and Appendix B.3), Standard + Sobel is consistently the best performing pair for CIFAR-10.

Original: Overall, these results indicate that the diversity of shape- and texture-biased models allows them to improve each other over training.
Revised: Overall, these results indicate that the diversity of shape- and texture-biased models allows them to improve each other over training.

Original: Additionally, we ﬁnd that, even when training a single model on the pseudo-labels of another model, prior diversity can  help.
Revised: Additionally, we ﬁnd that, even when training a single model on the pseudo-labels of another model, prior diversity can still help.

Original: Speciﬁcally, we compare the performance of a standard model trained from scratch using pseudo-labels from various self-trained models (Column 5, Table 5).
Revised: Speciﬁcally, we compare the performance of a standard model trained from scratch using pseudo-labels from various self-trained models (Column 5, Table 5).

Original: In this setting, using a self-trained shape- or texture-biased model for pseudo-labeling outperforms using a selftrained standard model.
Revised: In this setting, using a self-trained shape- or texture-biased model for pseudo-labeling outperforms using a self-trained standard model.

Original: This is despite the fact that, in isolation, the standard model has higher accuracy than the shape- or texture-biased ones (Column 4, Table 5).
Revised: This is despite the fact that, in isolation, the standard model has higher accuracy than the shape- or texture-biased ones (Column 4, Table 5).

Original: Model alignment over co-training.
Revised: Model alignment over co-training.

Original: To further explore the dynamics of co-training, we evaluate how the correlation between the predictionsof the two models evolves as the eras progress in Figure 6 (using the prediction alignment measure of Table 2).
Revised: To further explore the dynamics of co-training, we evaluate how the correlation between model predictions evolves as the eras progress in Figure 6 (using the prediction alignment measure of Table 2).

Original: We ﬁnd that shape- and texture-biased models exhibit low correlation at the start of co-training, but this correlation increases as co-training progresses.
Revised: We ﬁnd that shape- and texture-biased models exhibit low correlation at the start of co-training, but this correlation increases as co-training progresses.

Original: This is in contrast to self-training each model on its own, where the correlation remains relatively low.
Revised: This is in contrast to self-training each model on its own, where the correlation remains relatively low.

Original: It is also worth noting that the correlation appears to plateau at a lower value when co-training models with distinct feature priors as opposed to co-training two standard models.
Revised: It is also worth noting that the correlation appears to plateau at a lower value when co-training models with distinct feature priors as opposed to co-training two standard models.

Original: Finally [Figure] we ﬁnd that a standard model trained on the pseudo-labels of other models correlates well with the models themselves (see Appendix B.6).
Revised: Finally, we ﬁnd that a standard model trained on the pseudo-labels of other models correlates well with the models themselves (see Appendix B.7).

Original: Overall, these ﬁndings indicate that models trained on each other’s pseudo-labels end up behaving more similarly.
Revised: Overall, these ﬁndings indicate that models trained on each other’s pseudo-labels end up behaving more similarly.

Original: [Figure]
Revised: [Figure]

Revised: -trained together.

Original:  U SING C O -T RAINING TO A VOID S PURIOUS C ORRELATIONS
Revised:  U SING C O -T RAINING TO A VOID S PURIOUS C ORRELATIONS

Original: A major challenge when training models for real-world deployment is avoiding spurious correlations: associations which are predictive on the training data but not valid for the actual task.
Revised: A major challenge when training models for real-world deployment is avoiding spurious correlations: associations which are predictive on the training data but not valid for the actual task.

Original: Since models are typically trained to maximize train accuracy, they are quite likely to rely on such spurious correlations Gururangan et al.
Revised: Since models are typically trained to maximize train accuracy, they are quite likely to rely on such spurious correlations Gururangan et al.

Original: ; Beery et al.
Revised: ; Beery et al.

Original: ; Geirhos et al.
Revised: ; Geirhos et al.

Original: Xiao et al.
Revised: Xiao et al.

Original: In this section, our goal is to leverage diverse feature priors to control the sensitivity of the training process to such spurious correlations.
Revised: In this section, our goal is to leverage diverse feature priors to control the sensitivity of the training process to such spurious correlations.

Original: Speciﬁcally, we will assume that the spurious correlation does not hold on the unlabeled data (which is likely since unlabeled data is often  collected through a more diverse process).
Revised: Speciﬁcally, we will assume that the spurious correlation does not hold on the unlabeled data (which is likely since unlabeled data can often be collected at a larger scale).

Revised: Without this assumption, the unlabeled contains no examples that could (potentially) contradict the spurious correlation (we investigate the setting where the unlabeled data is also similarly skewed in Appendix B.10).

Original: As we will see, if the problematic correlation is not easily captured by one of the priors, the corresponding model generates pseudo-labels that are inconsistent with this association, thus steering other models away from this correlation during co-training.
Revised: As we will see, if the problematic correlation is not easily captured by one of the priors, the corresponding model generates pseudo-labels that are inconsistent with this correlation, thus steering other models away from this correlation during co-training.

Original: Setup.
Revised: Setup.

Original: We study spurious correlations in two settings.
Revised: We study spurious correlations in two settings.

Original: First, we create a synthetic dataset by tinting each image of the STL-10 labeled dataset in a class-speciﬁc way.
Revised: First, we create a synthetic dataset by tinting each image of the STL-10 labeled dataset in a class-speciﬁc way.

Original: This encourages models to predict using the tint, as it is highly predictive on the training set.
Revised: This encourages models to rely on the tint, as it is highly predictive on the training set.

Original: At the same time, such a prediction rule does not generalize to the test set where this correlation is absent.
Revised: However, this prediction rule does not generalize to the test set where this correlation is absent.

Original: Second,  we consider arealworld gender classiﬁcation task based on CelebA (Liu et al., 2015) where hair color (“blond” vs.
Revised: Second, similar to Sagawa et al. (2020), we consider a gender classiﬁcation task based on CelebA (Liu et al., 2015) where hair color (“blond” vs.

Original: “non-blond”) is predictive on the labeled data but not on the unlabeled and test data.
Revised: “non-blond”) is predictive on the labeled data but not on the unlabeled and test data.

Original: Speciﬁcally, gender and hair color are independent attributes on the unlabeled dataset,while the labeled dataset consists only of blond females and non-blond males.
Revised: While gender and hair color are independent attributes on the unlabeled dataset, the labeled dataset consists only of blond females and non-blond males.

Original: Similarly to the synthetic case, the labeled data encourages a prediction rule based only on hair color.
Revised: Similarly to the synthetic case, the labeled data encourages a prediction rule based only on hair color.

Original: See Appendix A.1 for details.
Revised: See Appendix A.1 for details.

Original: Performance on datasets with spurious features.
Revised: Performance on datasets with spurious features.

Original: We ﬁnd that, when trained only on the labeled data (where the correlation is fully predictive), both the standard and BagNet models generalizepoorly in comparison to the shape-biased models (see Table 7). This behavior is expected: the spurious attribute in both datasets is color-related and hence mostly suppressed by the edge detection algorithms used to train shape-based models. Moreover, even after self-training on the unlabeled data (where the correlation is absent), the performance of the standard and BagNet models does not improve signiﬁcantly. Finally, simply ensembling self-trained models post hoc does not improve
Revised: We ﬁnd that, when trained only on the labeled data (where the correlation is fully predictive), both the standard and BagNet models generalize

Original: [Table]
Revised: [Table]

Original: self-training and co-training on tinted STL-10 and CelebA, two datasets with spurious features (table structure is identical Table 5).
Revised: self-training and co-training on tinted STL-10 and CelebA, two datasets with spurious features (table structure is identical Table 5).

Original: In both datasets, the spurious correlation is more easily captured by the BagNet and Standard models over the shape-based ones.
Revised: In both datasets, the spurious correlation is more easily captured by the BagNet and Standard models over the shape-based ones.

Original: Nevertheless, when co-trained with a shaped-biased model, BagNets are able to signiﬁcantly improve their performance, indicating that they rely less on this spurious correlation.
Revised: Nevertheless, when co-trained with a shaped-biased model, BagNets are able to signiﬁcantly improve their performance, indicating that they rely less on this spurious correlation.

Original: CI: 95% bootstrap.
Revised: CI: 95% bootstrap.

Original:  their performance.
Revised: poorly in comparison to the shape-biased models (see Table 7). This behavior is expected: the spurious attribute in both datasets is color-related and hence mostly suppressed by the edge detection algorithms used to train shape-based models. Even after self-training on the unlabeled data (where the correlation is absent), the performance of the standard and BagNet models does not improve signiﬁcantly. Finally, simply ensembling self-trained models post hoc does not improve their performance.

Original: Since the texture-biased and standard models are signiﬁcantly less accurate than the shape-biased one, they end up lowering the overall accuracy of the ensemble (see Appendix B.7).
Revised: Indeed as the texture-biased and standard models are signiﬁcantly less accurate than the shape-biased one, they end up lowering the overall accuracy of the ensemble (see Appendix B.8).

Original: In contrast, when we co-train a texture-biased model with a shape-biased one, the texture-biased model improves substantially.
Revised: In contrast, when we co-train a texture-biased model with a shape-biased one, the texture-biased model improves substantially.

Original: For instance, when co-trained with a Canny model, theresulting BagNet model outperforms its self-trainingperformance by 42% on the tinted STL-10 dataset and 27% on the CelebA dataset.
Revised: For instance, when co-trained with a Canny model, the BagNet model improves over self-training by 42% on the tinted STL-10 dataset and 27% on the CelebA dataset.

Original: This improvement can be attributed to the fact that the predictions of the shape-biased model are not consistent with the spurious correlation on the unlabeled data.
Revised: This improvement can be attributed to the fact that the predictions of the shape-biased model are not consistent with the spurious correlation on the unlabeled data.

Original: Hence, by being trained on pseudo-labels from that model, the BagNet model is forced to rely on alternative, non-spurious features for its predictions.
Revised: Hence, by being trained on pseudolabels from that model, the BagNet model is forced to rely on alternative, non-spurious features.

Original: Moreover, particularly on the CelebA dataset, the shape-biased model also improves when cotrained with a texture-biased model.
Revised: Moreover, particularly on CelebA, the shape-biased model also improves when co-trained with a texture-biased model.

Original: This indicates that even though the texture-biased model reliesheavily on the spurious correlation, it also captures non-spurious features that, through pseudolabeling, improve the performance of the shape-based model.
Revised: This indicates that even though the texture-biased model relies on the spurious correlation, it also captures non-spurious features that, through pseudo-labeling, improve the performance of the shape-based model.

Original: In Appendix B.8, we ﬁnd that these improvements are concentrated on input samples where the spurious correlation does not hold.
Revised: In Appendix B.9, we ﬁnd that these improvements are concentrated on inputs where the spurious correlation does not hold.

Original:  A DDITIONAL R ELATED W ORK
Revised:  A DDITIONAL R ELATED W ORK

Original: In Section 2, we discussed the most relevant prior work on implicit or explicit feature priors.
Revised: In Section 2, we discussed the most relevant prior work on implicit or explicit feature priors.

Original: Here, we discuss additional related work and how it connects to our approach.
Revised: Here, we discuss additional related work and how it connects to our approach.

Original: Shape-biased models.
Revised: Shape-biased models.

Original: Several works have focused on biasing models towards shape-based features through methods such as: stylization Geirhos et al.
Revised: Several other methods aim to bias models towards shape-based features: input stylization Geirhos et al.

Original: ; Somavarapu et al.
Revised: ; Somavarapu et al.

Original: ; Li et al.
Revised: ; Li et al.

Original: (2021), penalizing early layers Wang et al.
Revised: (2021), penalizing early layer predictiveness Wang et al.

Original: , jigsaw puzzles Carlucci et al.
Revised: , jigsaw puzzles Carlucci et al.

Original: ; Asadi et al.
Revised: ; Asadi et al.

Original: (2019), dropout Shi et al.
Revised: dropout Shi et al.

Original: (2020), or data augmentation Hermann et al.
Revised: (2020), or data augmentation Hermann et al.

Original: While, in our work, we choose to suppress texture information via edge detection algorithms, any of these methods can be substituted to generate the shape-based model for our analysis.
Revised: While, in our work, we choose to suppress texture information via edge detection algorithms, any of these methods can be substituted to generate the shape-based model for our analysis.

Original: Avoiding spurious correlations.
Revised: Avoiding spurious correlations.

Original: This line of work aims to avoid learning spurious correlations by utilizing methods such as: learning representations that are simultaneously optimal across domains (Arjovsky et al., 2019), enforcing robustness to group shifts (Sagawa et al., 2020), and utilizing multiple data points corresponding to a single physical entity (Heinze-Deml & Meinshausen, 2017).
Revised: Other methods that can prevent models from learning spurious correlations include: learning representations that are simultaneously optimal across domains (Arjovsky et al., 2019), enforcing robustness to group shifts (Sagawa et al., 2020), and utilizing multiple data points corresponding to a single physical entity (Heinze-Deml & Meinshausen, 2017).

Original: Similar  to our work, these methods aim to learn prediction rules that are supported by multiple views of the data.
Revised: Similar in spirit to our work, these methods aim to learn prediction rules that are supported by multiple views of the data.

Original: However, we do not rely on annotations or multiple sources and instead impose feature priors through the model architecture and input preprocessing.
Revised: However, we do not rely on annotations or multiple sources and instead impose feature priors through the model architecture and input preprocessing.

Original: Pseudo-labeling.
Revised: Pseudo-labeling.

Original: Since the initial proposal of pseudo-labeling for neural networks Lee et al.
Revised: Since the initial proposal of pseudo-labeling for neural networks Lee et al.

Original: (2013), there has been a number of more sophisticated pseudo-labeling schemes aimed at improving the accuracy and diversity of the labels Iscen et al.
Revised: (2013), there has been a number of more sophisticated pseudo-labeling schemes aimed at improving the accuracy and diversity of the labels Iscen et al.

Original: ; Augustin & Hein (2020); Xie et al.
Revised: ; Augustin & Hein (2020); Xie et al.

Original: (2020); Rizve et al.
Revised: (2020); Rizve et al.

Original: ; Huang et al.
Revised: ; Huang et al.

Original: In our work, we focus on the simplest scheme for self-labeling—i.e., conﬁdence based example selection.
Revised: In our work, we focus on the simplest scheme for self-labeling—i.e., conﬁdence based example selection.

Original: Nevertheless, most of these schemes can be directly incorporated into our framework to potentially improve its overall performance.
Revised: Nevertheless, most of these schemes can be directly incorporated into our framework to potentially improve its overall performance.

Original: A recent line of work exploresthe theoretical under-pinnings of self-training by analyzing such schemes under different assumptions on the data (Mobahi et al., 2020; Wei et al., 2021; Allen-Zhu & Li, 2020; Kumar et al., 2020).
Revised: A recent line of work explores self-training by analyzing it under different assumptions on the data (Mobahi et al., 2020; Wei et al., 2021; Allen-Zhu & Li, 2020; Kumar et al., 2020).

Original: Closest to our work, Chen et al.
Revised: Closest to our work, Chen et al.

Original: (2020b) show that self-training on unlabeled data can reduce reliance on spurious correlations under certain assumptions.
Revised: (2020b) show that self-training on unlabeled data can reduce reliance on spurious correlations under certain assumptions.

Original: In contrast, we demonstrate that by leveraging multiple feature priors, we can avoid spurious correlations even if one of the models heavily relies on them.
Revised: In contrast, we demonstrate that by leveraging diverse feature priors, we can avoid spurious correlations even if a model heavily relies on them.

Original: Consistency regularization.
Revised: Consistency regularization.

Original: In parallel to pseudo-labeling, consistency regularization is another canonical technique for leveraging unlabeled data.
Revised: In parallel to pseudo-labeling, consistency regularization is another canonical technique for leveraging unlabeled data.

Original: Here, a model is trained to be invariant to a set of  transformations.
Revised: Here, a model is trained to be invariant to a set of input transformations.

Original: These transformations might stem from data augmentations and architecture stochasticity Laine & Aila (2017);
Revised: These transformations might stem from data augmentations and architecture stochasticity Laine & Aila (2017);

Original: Berthelot et al.
Revised: Berthelot et al.

Original: ; Chen et al.
Revised: ; Chen et al.

Original: (2020a)
Revised: (2020a)

Original: ; Sohn et al.
Revised: ; Sohn et al.

Revised: Prabhu et al.

Original: or using adversarial examples Miyato et al.
Revised: or using adversarial examples Miyato et al.

Original: Co-training.
Revised: Co-training.

Original: One line of work studies co-training from a theoretical perspective (Nigam & Ghani, 2000; Balcan et al., 2005; Goldman & Zhou, 2000).
Revised: One line of work studies co-training from a theoretical perspective (Nigam & Ghani, 2000; Balcan et al., 2005; Goldman & Zhou, 2000).

Original: Other work aims to improve co-training by either expanding the settings where it can be applied (Chen et al., 2011) or by improving its stability (Ma et al., 2020; Zhang & Zhou, 2011).
Revised: Other work aims to improve co-training by either expanding the settings where it can be applied (Chen et al., 2011) or by improving its stability (Ma et al., 2020; Zhang & Zhou, 2011).

Original: Finally, a third line of work applies co-training to images.
Revised: Finally, a third line of work applies co-training to images.

Original: Since images cannot be separated into disjoint feature sets, one would apply co-training by training multiple models Han et al.
Revised: Since images cannot be separated into disjoint feature sets, one would apply co-training by training multiple models Han et al.

Original: (2018), either regularized to be diverse  Qiao et al.
Revised: (2018), either regularized to be diverse through adversarial examples Qiao et al.

Original: (2018) or each trained using a different method Yang et al.
Revised: (2018) or each trained using a different method Yang et al.

Original: Our method is complementary to these approaches as it relies onusing a set of explicit feature priors to obtain different views on the data.
Revised: Our method is complementary to these approaches as it relies on explicit feature priors to obtain different views.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this work, we explored the beneﬁts of combining feature priors with non-overlapping failure modes.
Revised: In this work, we explored the beneﬁts of combining feature priors with non-overlapping failure modes.

Original: By capturing complementary perspectives on the data, models trained with diverse feature priors can offset each others mistakes when combined through methods such as ensembles.
Revised: By capturing complementary perspectives on the data, models trained with diverse feature priors can offset each others mistakes when combined through methods such as ensembles.

Original: Moreover, in the presence of unlabeled data, we can leverage prior diversity by jointly boostrapping models with different priors through co-training.
Revised: Moreover, in the presence of unlabeled data, we can leverage prior diversity by jointly boostrapping models with different priors through co-training.

Original: This allows the models to correct each other during training, thus improving pseudo-labeling and controlling for correlations that do not generalize well.
Revised: This allows the models to correct each other during training, thus improving pseudo-labeling and controlling for correlations that do not generalize well.

Original: We believe that our work is only the ﬁrst step in exploring the design space of creating, manipulating, and combining feature priors to improve generalization.
Revised: We believe that our work is only the ﬁrst step in exploring the design space of creating, manipulating, and combining feature priors to improve generalization.

Original: In particular, our framework is quite ﬂexible and allows for a number of different design choices, such as choosing other feature priors (cf.
Revised: In particular, our framework is quite ﬂexible and allows for a number of different design choices, such as choosing other feature priors (cf.

Original: Sections 2 and 6), using other methods for pseudo-label selection (e.g., using uncertainty estimation (Lee et al., 2018; Rizve et al., 2021)), and combining pseudo-labels via different ensembling methods.
Revised: Sections 2 and 6), using other methods for pseudo-label selection (e.g., using uncertainty estimation (Lee et al., 2018; Rizve et al., 2021)), and combining pseudo-labels via different ensembling methods.

Original: More broadly, we believe that exploring the synthesis of explicit feature priors in new applications is an exciting avenue for further research.
Revised: More broadly, we believe that exploring the synthesis of explicit feature priors in new applications is an exciting avenue for further research.

Original: A E XPERIMENTAL D ETAILS A. D ATASETS
Revised: A E XPERIMENTAL D ETAILS A. D ATASETS

Original: For our ﬁrst set of experiments (Section 4), we focus on a canonical setting where a small portion of the training set if labeled and we have access to a pool of unlabeled data.
Revised: For our ﬁrst set of experiments (Section 4), we focus on a canonical setting where a small portion of the training set if labeled and we have access to a pool of unlabeled data.

Original: STL-10.
Revised: STL-10.

Original: The STL-10 Coates et al.
Revised: The STL-10 Coates et al.

Original: (2011) dataset contains 5,000 training and 8,000 test images of size 96 × 96 from 10 classes.
Revised: (2011) dataset contains 5,000 training and 8,000 test images of size 96 × 96 from 10 classes.

Original: We designate 1,000 of the 5,000 (20%) training examples to be the labeled training set, 500 (10%) to be the validation set, and the rest are used as unlabeled data.
Revised: We designate 1,000 of the 5,000 (20%) training examples to be the labeled training set, 500 (10%) to be the validation set, and the rest are used as unlabeled data.

Original: CIFAR-10.
Revised: CIFAR-10.

Original: The CIFAR-10 Krizhevsky (2009) dataset contains 50,000 training and 8,000 test images of size 32 × 32 from 10 classes.
Revised: The CIFAR-10 Krizhevsky (2009) dataset contains 50,000 training and 8,000 test images of size 32 × 32 from 10 classes.

Original: We designate 1,000 of the 50,000 (2%) training examples to be the labeled training set, 5000 (10%) to be the validation set, and the rest as unlabeled data.
Revised: We designate 1,000 of the 50,000 (2%) training examples to be the labeled training set, 5000 (10%) to be the validation set, and the rest as unlabeled data.

Original: In both cases, we report the ﬁnal performance on the standard test set of that dataset.
Revised: In both cases, we report the ﬁnal performance on the standard test set of that dataset.

Original: We also create two datasets that each contain a different spurious correlation.
Revised: We also create two datasets that each contain a different spurious correlation.

Original: Tinted STL-10.
Revised: Tinted STL-10.

Original: We reuse the STL-10 setup described above, but we add a class-speciﬁc tint to each image in the (labeled) training set.
Revised: We reuse the STL-10 setup described above, but we add a class-speciﬁc tint to each image in the (labeled) training set.

Original: Speciﬁcally, we hand-pick a different color for each of the 10 classes and then add this color to each of the pixels (ensuring that each RGB channel remains within the valid range)—see Figure 8 for examples.
Revised: Speciﬁcally, we hand-pick a different color for each of the 10 classes and then add this color to each of the pixels (ensuring that each RGB channel remains within the valid range)—see Figure 8 for examples.

Original: This tint is only present in the labeled part of the training set, the unlabeled and test parts of the dataset are left unaltered.
Revised: This tint is only present in the labeled part of the training set, the unlabeled and test parts of the dataset are left unaltered.

Original: [Figure]
Revised: [Figure]

Original: Biased CelebA.
Revised: Biased CelebA.

Original: We consider the task of predicting gender in the CelebA Liu et al.
Revised: We consider the task of predicting gender in the CelebA Liu et al.

Original: (2015) dataset.
Revised: (2015) dataset.

Original: In order to create a biased training set, we choose a random sample of 500 non-blond males andblond females.
Revised: In order to create a biased training set, we choose a random sample of 500 non-blond males andblond females.

Original: We then use a balanced unlabeled dataset consisting of 1,000 random samples for each of: blond males, blond females, non-blond males, and non-blond females.
Revised: We then use a balanced unlabeled dataset consisting of 1,000 random samples for each of: blond males, blond females, non-blond males, and non-blond females.

Original: We use the standard
Revised: We use the standard

Original: CelebA test set which consists of 12.41% blond females, 48.92% non-blond females, 0.90% blond males, and 37.77% non-blond males.
Revised: CelebA test set which consists of 12.41% blond females, 48.92% non-blond females, 0.90% blond males, and 37.77% non-blond males.

Original: (Note that a classiﬁer predicting purely based on hair color with have an accuracy of 50.18% on that test set.)
Revised: (Note that a classiﬁer predicting purely based on hair color with have an accuracy of 50.18% on that test set.)

Original: All of the datasets that we use are freely available for non-commercial research purposes.
Revised: All of the datasets that we use are freely available for non-commercial research purposes.

Original: Moreover, to the best of our knowledge, they do not contain offensive content or identiﬁable information (other than publicly available celebrity photos).
Revised: Moreover, to the best of our knowledge, they do not contain offensive content or identiﬁable information (other than publicly available celebrity photos).

Original: A. M ODEL ARCHITECTURES AND INPUT PREPROCESSING
Revised: A. M ODEL ARCHITECTURES AND INPUT PREPROCESSING

Original: For both the standard model and the models trained on images processed by edge detection algorithm, we use a standard model architecture—namely, VGG16 Simonyan & Zisserman (2015) with the addition of batch normalization Ioffe & Szegedy (2015) (often referred to as VGG16-BN).
Revised: For both the standard model and the models trained on images processed by edge detection algorithm, we use a standard model architecture—namely, VGG16 Simonyan & Zisserman (2015) with the addition of batch normalization Ioffe & Szegedy (2015) (often referred to as VGG16-BN).

Original: We describe the exact edge detection process as well as the architecture of the BagNet model (texture prior) below.
Revised: We describe the exact edge detection process as well as the architecture of the BagNet model (texture prior) below.

Original: We visualize these priors in Figure 10.
Revised: We visualize these priors in Figure 10.

Original: Canny edge detection.
Revised: Canny edge detection.

Original: Given an image, we ﬁrst smooth it with a 5 pixel bilateral ﬁlter Tomasi & Manduchi (1998), with ﬁlter σ in the coordinate and color space set to 75.
Revised: Given an image, we ﬁrst smooth it with a 5 pixel bilateral ﬁlter Tomasi & Manduchi (1998), with ﬁlter σ in the coordinate and color space set to 75.

Original: After smoothing, the image is converted to gray-scale.
Revised: After smoothing, the image is converted to gray-scale.

Original: Finally, a Canny ﬁlter Canny (1986) is applied to the image, with hysteresis thresholds 100 and 200, to extract the edges.
Revised: Finally, a Canny ﬁlter Canny (1986) is applied to the image, with hysteresis thresholds 100 and 200, to extract the edges.

Original: Sobel edge detection.
Revised: Sobel edge detection.

Original: Given an image, we ﬁrst upsample it to 128 × 128 pixels.
Revised: Given an image, we ﬁrst upsample it to 128 × 128 pixels.

Original: Then we convert it to gray-scale and apply a Gaussian blur (kernel size=5, σ = 5 ).
Revised: Then we convert it to gray-scale and apply a Gaussian blur (kernel size=5, σ = 5 ).

Original: The image is then passed through a Sobel ﬁlter Sobel & Feldman (1968) with a kernel size of 3 in both the horizontal and the vertical direction to extract the image gradients.
Revised: The image is then passed through a Sobel ﬁlter Sobel & Feldman (1968) with a kernel size of 3 in both the horizontal and the vertical direction to extract the image gradients.

Original: BagNet.
Revised: BagNet.

Original: For our texture-biased model, we use a slimmed down version of the BagNet architecture from Brendel & Bethge (2019).
Revised: For our texture-biased model, we use a slimmed down version of the BagNet architecture from Brendel & Bethge (2019).

Original: The goal of this architecture is to limit the receptive ﬁeld of the model, hence forcing it to make predictions based on local features.
Revised: The goal of this architecture is to limit the receptive ﬁeld of the model, hence forcing it to make predictions based on local features.

Original: The exact architecture we used is shown in Figure 9.
Revised: The exact architecture we used is shown in Figure 9.

Original: Intuitively, the top half of the network—i.e., the green and blue blocks— construct features on patches of size 20 × 20 for 96 × 96 images and 10 × 10 for 32 × 32 images.
Revised: Intuitively, the top half of the network—i.e., the green and blue blocks— construct features on patches of size 20 × 20 for 96 × 96 images and 10 × 10 for 32 × 32 images.

Original: The rest of the network consists only of 1 × 1 convolutions and max-pooling, hence not utilizing the image’s spatial structure.
Revised: The rest of the network consists only of 1 × 1 convolutions and max-pooling, hence not utilizing the image’s spatial structure.

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: A. T RAINING SETUP A.3. B ASIC TRAINING
Revised: A. T RAINING SETUP A.3. B ASIC TRAINING

Original: We train all our models using stochastic gradient descent (SGD) with momentum (a coefﬁcient of 0.9) and a decaying learning rate.
Revised: We train all our models using stochastic gradient descent (SGD) with momentum (a coefﬁcient of 0.9) and a decaying learning rate.

Original: We add weight decay regularization with a coefﬁcient of 10 − 4 .
Revised: We add weight decay regularization with a coefﬁcient of 10 − 4 .

Original: In terms of data augmentation, we apply random cropping with a padding of 4 pixels, random horizontal ﬂips, and a random rotation of ± 2 degrees.
Revised: In terms of data augmentation, we apply random cropping with a padding of 4 pixels, random horizontal ﬂips, and a random rotation of ± 2 degrees.

Original: These transformations are applied after the edge detection processing.
Revised: These transformations are applied after the edge detection processing.

Original: We train all models with a batch size of 64 for 96 × 96-sized images and 128 for 32 × 32-sized images for a total of 300 epochs.
Revised: We train all models with a batch size of 64 for 96 × 96-sized images and 128 for 32 × 32-sized images for a total of 300 epochs.

Original: All our experiments are performed using our internal cluster which mainly consists of NVIDIA 

Revised: All our experiments are performed using our internal cluster which mainly consists of NVIDIA

Original: Ti GTX GPUs.
Revised: Ti GTX GPUs.

Original: Hyperparameter tuning.
Revised: Hyperparameter tuning.

Original: To ensure a fair comparison across feature priors, we selected the hyperparameters for each dataset-prior pair separately, using the held-out validation set (separate from the ﬁnal test used for reporting performance).
Revised: To ensure a fair comparison across feature priors, we selected the hyperparameters for each dataset-prior pair separately, using the held-out validation set (separate from the ﬁnal test used for reporting performance).

Original: Speciﬁcally, we performed a grid search choosing the learning rate (LR) from [0 . 1 , 0 . 05 , 0 . 02 , 0 . 01 , 0 . 005] , the number of epochs between each learning rate drop ( K ) from [50 , 100 , 300] and the factor with which the learning rate is multiplied ( γ ) from [0 . 5 , 1] .
Revised: Speciﬁcally, we performed a grid search choosing the learning rate (LR) from [0 . 1 , 0 . 05 , 0 . 02 , 0 . 01 , 0 . 005] , the number of epochs between each learning rate drop ( K ) from [50 , 100 , 300] and the factor with which the learning rate is multiplied ( γ ) from [0 . 5 , 1] .

Original: The parameters chosen are shown in Table 11.
Revised: The parameters chosen are shown in Table 11.

Original: We found that all models achieved nearoptimal performance strictly within the range of each hyperparameters.
Revised: We found that all models achieved nearoptimal performance strictly within the range of each hyperparameters.

Original: Thus, we did not consider a wider grid.
Revised: Thus, we did not consider a wider grid.

Original: Dataset
Revised: Dataset

Original: [Table]
Revised: [Table]

Original: A. E NSEMBLES
Revised: A. E NSEMBLES

Original: In order to leverage prior diversity, we ensemble models trained with (potentially) different priors.
Revised: In order to leverage prior diversity [Table] we ensemble models trained with (potentially) different priors.

Original: We use the following ensembles:
Revised: We use the following ensembles:

Original: Take Max:
Revised: Take Max:

Original: Predict based on the model assigning the highest probability on this example.
Revised: Predict based on the model assigning the highest probability on this example.

Original: Average:
Revised: Average:

Original: Average the (softmax) output probabilities of the models, predict the class assigned the highest probability.
Revised: Average the (softmax) output probabilities of the models, predict the class assigned the highest probability.

Original: Rank:
Revised: Rank:

Original: Each model ranks all test examples based on the probability assigned to their predicted labels.
Revised: Each model ranks all test examples based on the probability assigned to their predicted labels.

Original: Then, for each example, we predict using the model which has a lower rank on this example.
Revised: Then, for each example, we predict using the model which has a lower rank on this example.

Original: We then report the maximum of these ensemble methods in Table 3.
Revised: We then report the maximum of these ensemble methods in Table 3.

Original: A. S ELF - TRAINING AND CO - TRAINING SCHEMES
Revised: A. S ELF - TRAINING AND CO - TRAINING SCHEMES

Original: In the setting that we are focusing on, we are provided with a labeled dataset X and an unlabeled dataset U , where typically there is much more unlabeled data ( | U | (cid:29) | X | ).
Revised: In the setting that we are focusing on, we are provided with a labeled dataset X and an unlabeled dataset U , where typically there is much more unlabeled data ( | U | (cid:29) | X | ).

Original: We are then choosing a set of (one or more) feature priors each of which corresponds to a different way of training a model (e.g., using edge detection preprocessing).
Revised: We are then choosing a set of (one or more) feature priors each of which corresponds to a different way of training a model (e.g., using edge detection preprocessing).

Original: General methodology.
Revised: General methodology.

Original: We start by training each of these models on the labeled dataset.
Revised: We start by training each of these models on the labeled dataset.

Original: Then, we combine the predictions of these models to produce pseudo-labels for the unlabeled dataset.
Revised: Then, we combine the predictions of these models to produce pseudo-labels for the unlabeled dataset.

Original: Finally, we choose a fraction of the unlabeled data and train the models on that set using the produced pseudo-labels (in additional to the original labeled set X ).
Revised: Finally, we choose a fraction of the unlabeled data and train the models on that set using the produced pseudo-labels (in additional to the original labeled set X ).

Original: This process is repeated using increasing fractions of the unlabeled dataset until, eventually, models are trained on its entirety.
Revised: This process is repeated using increasing fractions of the unlabeled dataset until, eventually, models are trained on its entirety.

Original: We refer to each such phase as an era .
Revised: We refer to each such phase as an era .

Original: We include an additional 5% of the unlabeled data per era, resulting in a total of 20 eras.
Revised: We include an additional 5% of the unlabeled data per era, resulting in a total of 20 eras.

Original: During each era, we use the training process described in Appendix A.3.
Revised: During each era, we use the training process described in Appendix A.3.

Original: without re-initializing the models (warm start).
Revised: without re-initializing the models (warm start).

Original: After completing this process, we train a standard model from scratch using both the labeled set and resulting pseudo-labels.
Revised: After completing this process, we train a standard model from scratch using both the labeled set and resulting pseudo-labels.

Original: The methodology used for choosing and combining pseudo-labels is described below for each scheme.
Revised: The methodology used for choosing and combining pseudo-labels is described below for each scheme.

Original: Self-training.
Revised: Self-training.

Original: Since we are only training one model, we only need to decide how to choose the pseudo-labels to use for each era.
Revised: Since we are only training one model, we only need to decide how to choose the pseudo-labels to use for each era.

Original: We do this in the simplest way: at ear t , we pick the subset
Revised: We do this in the simplest way: at ear t , we pick the subset

Original: U t ⊆
Revised: U t ⊆

Original: U of examples that are assigned the highest probability on their predicted label.
Revised: U of examples that are assigned the highest probability on their predicted label.

Original: We attempt to produce a class-balanced training set by applying this process separately on each class (as predicted by the model).
Revised: We attempt to produce a class-balanced training set by applying this process separately on each class (as predicted by the model).

Original: The pseudocode for the method is provided in Algorithm 1.
Revised: The pseudocode for the method is provided in Algorithm 1.

Original: Algorithm
Revised: Algorithm

Original: Parameters: Number of eras T .
Revised: Parameters: Number of eras T .

Original: Fraction added per era k .
Revised: Fraction added per era k .

Original: Input : Labeled data X with n classes, unlabeled data U , model trained on X . for era t ∈ 1 ...
Revised: Input : Labeled data X with n classes, unlabeled data U , model trained on X . for era t ∈ 1 ...

Original: T do forward-pass U through the model
Revised: T do forward-pass U through the model

Original: Standard co-training.
Revised: Standard co-training.

Original: Here, we train multiple models (in our experiments two) based on a common pool of pseudo-labeled examples in each era.
Revised: Here, we train multiple models (in our experiments two) based on a common pool of pseudo-labeled examples in each era.

Original: In each era t , each model labels the unlabeled dataset U .
Revised: In each era t , each model labels the unlabeled dataset U .

Original: Then, for each class, we alternate between models, adding the next most conﬁdent example predicted as that class for that model to U t , until we reach a ﬁxed number of unique examples have been added for that class (5% of the size of the unlabeled dataset per era).
Revised: Then, for each class, we alternate between models, adding the next most conﬁdent example predicted as that class for that model to U t , until we reach a ﬁxed number of unique examples have been added for that class (5% of the size of the unlabeled dataset per era).

Original: Note that this process allows both conﬂicts and duplicates: if multiple models are conﬁdent about a speciﬁc example, that example may be added more than once (potentially with a different label each time).
Revised: Note that this process allows both conﬂicts and duplicates: if multiple models are conﬁdent about a speciﬁc example, that example may be added more than once (potentially with a different label each time).

Original: Finally, we train each model (without re-initializing) on X ∪ U t .
Revised: Finally, we train each model (without re-initializing) on X ∪ U t .

Original: The pseudocode for this method can be found in Algorithm 2.
Revised: The pseudocode for this method can be found in Algorithm 2.

Original: Input : for era t
Revised: Input : for era t

Original: .T do forward-pass U through each model to create pseudo-labels
Revised: .T do forward-pass U through each model to create pseudo-labels

Original: U t =
Revised: U t =

Original: [] for each class c do
Revised: [] for each class c do

Original: U ( c ) t = [] while the number of unique examples in U ( c ) t < kt | U | n do for each model m do
Revised: U ( c ) t = [] while the number of unique examples in U ( c ) t < kt | U | n do for each model m do

Original: Addthe next mostconﬁdent predicted m c
Revised: Add next most c

Original: Add
Revised: Add

Original: B A DDITIONAL E XPERIMENTS B. E XPERIMENT ORGANIZATION
Revised: B A DDITIONAL E XPERIMENTS B. E XPERIMENT ORGANIZATION

Original: We now provide the full experimental results used to create the plots in the main body as well as additional analysis.
Revised: We now provide the full experimental results used to create the plots in the main body as well as additional analysis.

Original: Speciﬁcally, in Appendix B.2 and B.3 we present the performance of individual ensemble schemes for pre-trained and self-trained models respectively.
Revised: Speciﬁcally, in Appendix B.2 and B.3 we present the performance of individual ensemble schemes for pre-trained and self-trained models respectively.

Original: Then, in Appendix B.4 we present the performance of co-training for each combination of feature priors.
Revised: Then, in Appendix B.5 we present the performance of co-training for each combination of feature priors.

Original: In Appendix B.6 we analyse the effect that co-training has on model similarity after training.
Revised: In Appendix B.7 we analyse the effect that co-training has on model similarity after training.

Original: Finally, in Appendix B.7 we evaluate model ensembles on datasets with spurious correlations and in Appendix B.8 we breakdown the performance of co-training on the skewed CelebA dataset according to different input attributes.
Revised: Finally, in Appendix B.8 we evaluate model ensembles on datasets with spurious correlations and in Appendix B.9 we breakdown the performance of co-training on the skewed CelebA dataset according to different input attributes.

Original: B. 
Revised: B.

Original: [Table]
Revised: [Table]

Original: NSEMBLE R ESULTS
Revised: NSEMBLE R ESULTS

Original: In Table
Revised: In Table

Original: [Table]
Revised: [Table]

Original: we reported the best ensemble method for each pair of models trained with different priors on the labeled data.
Revised: we reported the best ensemble method for each pair of models trained with different priors on the labeled data.

Original: In Table
Revised: In Table

Original: [Table]
Revised: [Table]

Original: we report the full results over the individual ensembles.
Revised: we report the full results over the individual ensembles.

Original: [Table]
Revised: [Table]

Original: B. E NSEMBLING S ELF -T RAINED M ODELS
Revised: B. E NSEMBLING S ELF -T RAINED M ODELS

Original: In Table
Revised: In Table

Original: [Table]
Revised: [Table]

Original: we report the best ensemble method for pairs of self-trained models with different priors.
Revised: we report the best ensemble method for pairs of self-trained models with different priors.

Original: In Table
Revised: In Table

Original: [Table]
Revised: [Table]

Original: we report the full results over the individual ensembles.
Revised: we report the full results over the individual ensembles.

Original: We ﬁnd that, similar to the ensembles of models trained on the labeled data, models with diverse priors gain more from ensembling.
Revised: We ﬁnd that, similar to the ensembles of models trained on the labeled data, models with diverse priors gain more from ensembling.

Original: However, co-training models with diverse priors together still outperforms ensembling self-trained models.
Revised: However, co-training models with diverse priors together still outperforms ensembling self-trained models.

Original: [Table]
Revised: [Table]

Original: Different
Revised: Different

Original: [Table]
Revised: [Table]

Original: Different
Revised: Different

Original: [Table]
Revised: [Table]

Original: [Figure] 
Revised: [Figure] [Table] B. S TACKED E NSEMBLING Here we consider an ensembling technique that leverages a validation set.

Original: [Table]
Revised: We implement stacking (also called blending) Töscher et al. ; Sill et al. (2009), which takes in the outputs of the member models as input

Original: [Table]
Revised: [Table]

Revised: and then trains a second model to produce the ﬁnal layer.

Revised: Here, we take the logits of each model in the ensemble

Original: [Table]
Revised: [Table]

Revised: and train the secondary model using logistic regression on the validation set for the dataset.

Revised: We report accuracies of the ensemble on the test set below.

Revised: We again ﬁnd that prior diversity is important for the performance of the ensemble

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Original: B. C O -T RAINING WITH VARYING AMOUNTS OF LABELED DATA .
Revised: B. C O -T RAINING WITH VARYING AMOUNTS OF LABELED DATA .

Original: In Table 17, we study how the efﬁcacy of combining diverse priors through cotraining changes as the number of labeled examples increase for STL-10.
Revised: In Table 19, we study how the efﬁcacy of combining diverse priors through cotraining changes as the number of labeled examples increase for STL-10.

Original: As one might expect, when labeled data is sparse, the feature priors learned by the models alone are relatively brittle: thus, leveraging diverse priors against each other on unlabeled data improves generalization.
Revised: As one might expect, when labeled data is sparse, the feature priors learned by the models alone are relatively brittle: thus, leveraging diverse priors against each other on unlabeled data improves generalization.

Original: As the number of labeled examples increases, the models with single feature priors learn more reliable prediction rules that can already generalize, so the additional beneﬁt of combining feature priors diminishes.
Revised: As the number of labeled examples increases, the models with single feature priors learn more reliable prediction rules that can already generalize, so the additional beneﬁt of combining feature priors diminishes.

Original: However, even in settings with plentiful data, combining diverse feature priors can aid generalization if there is a spurious correlation in the labeled data (see Section 5.)
Revised: However, even in settings with plentiful data, combining diverse feature priors can aid generalization if there is a spurious correlation in the labeled data (see Section 5.)

Original: [Table]
Revised: [Table]

Original: B. C ORRELATION BETWEEN THE INDIVIDUAL FEATURE - BIASED MODELS AND THE FINAL STANDARD MODEL
Revised: B. C ORRELATION BETWEEN THE INDIVIDUAL FEATURE - BIASED MODELS AND THE FINAL STANDARD MODEL

Original: [Table] [Table]
Revised: [Table] [Table]

Original: B. E NSEMBLES FOR S PURIOUS D ATASETS
Revised: B. E NSEMBLES FOR S PURIOUS D ATASETS

Original: In Table 19 (full table in Table 20)
Revised: In Table 21 (full table in Table 22)

Original: [Table]
Revised: [Table]

Original: we ensemble the self-trained priors for the Tinted STL-10 dataset and the CelebA dataset as in Section 5.
Revised: we ensemble the self-trained priors for the Tinted STL-10 dataset and the CelebA dataset as in Section 5.

Original: Both of these datasets have a spurious correlation base on color, which results in a weak Standard and BagNet model.
Revised: Both of these datasets have a spurious correlation base on color, which results in a weak Standard and BagNet model.

Original: As a result, the ensembles with the
Revised: As a result, the ensembles with the

Original: Standard or BagNet models do not perform well on the test set.
Revised: Standard or BagNet models do not perform well on the test set.

Original: However, in Section 7, we ﬁnd that co-training in this setting allows the BagNet model to improve when jointly trained with a shape model, thus boosting the ﬁnal performance 
Revised: However, in Section 7, we ﬁnd that co-training in this setting allows the BagNet model to improve when jointly trained with a shape model, thus boosting the ﬁnal performance [Table] [Table] [Figure]

Revised: [Table]

Original: [Table]
Revised: [Table]

Revised: B. W HAT IF THE UNLABELED DATA ALSO CONTAINED THE SPURIOUS CORRELATION ?

Revised: In Section

Revised: [Table]

Revised: we assume that the unlabeled data does not contain the spurious correlation present in the labeled data.

Revised: This is often the case when unlabeled data can be collected through a more diverse process than labeled data (for example, by scraping the web large scales or by passively collecting data during deployment).

Revised: This assumption is important: in order to successfully steer models away from the spurious correlation during co-training

Original: [Table]
Revised: [Table]

Revised: the process needs to surface examples which contradict the spurious correlation.

Original: [Figure]
Revised: However, if the unlabeled data is also heavily skewed, such examples might be rare or non-existent. What happens if the unlabeled data is as heavily skewed as the labeled data? We return the setting of a spurious association between hair color and gender in CelebA. However, unlike in Section 5, we use an unlabeled dataset that also perfectly correlates hair color and gender – it containsnon-blond males and 2000 blond females.

Revised: The unlabeled data thus has the same distribution as the labeled data

Original: [Table]
Revised: [Table]

Revised: and contains no examples that reject the spurious correlation (blond males or non-blond females).

Revised: [Table]

Revised: shown.

Revised: Self-Training: Since the unlabeled data follows the spurious correlation between hair color and gender, the standard and BagNet models almost perfectly pseudo-label the unlabeled data.

Revised: Thus, they are simply increasing the number of examples in the training dataset but maintaining the same overall distribution.

Revised: Self-training thus does not change the accuracy for models with these priors signiﬁcantly.

Revised: In contrast, in the setting in Section 5, there were examples in the unlabeled data which did not align with the spurious correlation (blond males and non-blond females).

Revised: Since they relied mostly on hair color, the standard and BagNet models actively mislabeled these examples (i.e, by labeling a blond male as female).

Revised: Training on these erroneous pseudo-labels actively suppressed any features that were not hair color, causing the standard and Bagnet models to perform worse after self-training.

Revised: Co-Training:

Revised: In contrast, when performing co-training with the Canny and BagNet priors, the

Revised: Canny model (which cannot detect hair color) will make mistakes on the unlabeled dataset.

Revised: These mistakes help are inconsistent with a reliance on hair color: due to this regularization, the BagNet’s accuracy improves from 69.35% to 76.52%.

Revised: Overall, though the gain is not as signiﬁcant as the setting with a balanced unlabeled dataset, the Canny + BagNet co-trained model can mitigate the pitfalls of the BagNet’s reliance on hair color and outperform even the canny self-trained model.