SlowGuess/NeurIPS_2023_OCR huggingface dataset:
With the smallest distribution shifts (leftmost case in Figure 8), for example training on ImageNet and testing on its replication ImageNet v2 [32], ID validation performance closely correlates with OOD test performance. This OOD setting is the easiest because one can focus on improving classical generalization and reap concurrent improvements OOD.
With a larger distribution shift, more features are likely to be spurious, which is likely to break the ID/OOD correlation. The task of improving OOD performance is likely to be under- or misspecified, i.e. there is not enough information to determine which features a model should rely on.
Valid approaches include modifying the training objective, injecting task-specific information (e.g., building-in invariance to rotations as in [43]), well-chosen data augmentations, or inhomogeneous training data such as multiple training environments [17] or counterfactuals [44, 1].
Positive transfer	Underspecification	Missspecification	No transfer
Distribution shift	Mild	→ (Too?) Severe
Typical pattern (toy representation)	Positive correlation	Vertical line/no clear trend	Negative correlation	Low horizontal line
OD Accuracy	OD Accuracy	OD Accuracy	OD Accuracy
ID Accuracy	ID Accuracy	ID Accuracy	ID Accuracy
Valid approaches for improving OOD performance	Simply focus on improving ID performance	Task-relevant inductive biases e.g., regularizers. Data augmentation with task-relevant transformationals. Non-i.i.d. training data e.g., multiple training domains.	Open question
Example datasets	ImageNet → ImageNet v2 [32]	PACS sketch → photograph [17]	WILDS Camelyon17 [15]	DomainNet infographic → quickdraw [15]
Figure 8: Overview of ID vs. OOD patterns occurring at different levels of distribution shift.
With extreme distribution shifts, most predictive features are overwhelmingly spurious and it is very difficult to learn any one relevant in OOD data (rightmost case in Figure 8).
The proposed ordering of patterns is rather informal and could be further developed following the two axes of diversity shifts and correlation shifts proposed by [58] (see also [52]). More recently, [48] showed that the suitability of various methods for OOD generalization depends on particularities of the underlying causal structure of the task - which must therefore be known to select a suitable method. Identifying which ID/OOD patterns occur with particular causal structures might serve as a tool to understand the type of OOD situation one is facing and identify a suitable method.
8 Revisiting advice from past studies
We have established that observations in past studies were incomplete. We now bring nuance to some recommendations and conclusions made in these studies.
- Focusing on a single metric.
"We see the following potential prescriptive outcomes (...) correlation between OOD and ID performance can simplify model development since we can focus on a single metric." [25]
We demonstrated that inverse correlations do occur, hence there exist scenarios where ID performance is misleading. Relying on a single metric for model development is ill-advised [45] especially if it cannot capture necessary trade-offs. We recommend tracking multiple metrics e.g. performance on multiple distributions or interpretable predictions on representative test points.
- Improving ID performance for OOD robustness.
"If practitioners want to make the model more robust on OOD data, the main focus should be to improve the ID classification error. (...) We speculate that the risk of overfitting large pretrained models to the downstream test set is minimal, and it seems to be not a good strategy to, e.g., reduce the capacity of the model in the hope of better OOD generalization." [50]
This recommendation assumes the persistence of a positive correlation. On the opposite, we saw that a positive correlation can precede a regime of inverse correlation (Figure 3, left panels). If the goal is to improve OOD, focusing on ID performance is a blind alley since this goal requires to increase ID performance at times and reduce it at others.
- Future achievable OOD performance.
As obvious as it is, it feels necessary to point out that empirical studies only chart regimes achievable with existing methods. Observations have limited predictive power, hence more care seems warranted when deriving prescriptive recommendations from empirical evidence.
The best possible performance e.g. on Camelyon17 is obviously not limited to the Pareto front of our experiments. The state of the art on this dataset [33, 51] injects additional task knowledge to bypass the under/misspecification of ERM, and exceeds both our highest ID and OOD performance.
The important message remains that a given hypothesis class (DenseNet architecture in our case) admits parametrizations on which ID and OOD metrics do not necessarily correlate.
- Possible invalidation of existing studies.
The possibility of inverse correlations may invalidate studies that implicitly assume a positive one. For example, Angarano et al. [3] evaluate the OOD robustness of computer vision backbones. They find modern architectures surpass domain generalization methods. However, they discard any model with submaximal ID performance by performing "training domain validation" as in [10]. Any model with high OOD performance but suboptimal ID is ignored. They also train every model for a fixed, large number of epochs. And this may additionally prevent from finding models with high OOD performance since robustness is often progressively lost during fine-tuning [2].
By design, this study [3] is incapable of finding OOD benefits of architectures or methods that require trading off some ID performance. Most importantly, once the assumption of a positive correlation is enacted by throwing away models with submaximal ID performance, there is no more opportunity to demonstrate its validity.
An even more recent example of this fallacy is found in [21]. The authors construct a new OOD benchmark, train and tune baselines for maximum ID performance, then observe:
"We find no [positive nor inverse] correlation between in- and out-of-distribution environment performance. All methods consistently achieve 98-99% ID test performance." [21]
But the authors did not give the chance to make any other observation. As in Figure 5, the ID criterion means that we only get to observe a thin vertical slice of the ID vs. OOD plot.
This paper highlighted that inverse correlations between ID/OOD performance are possible, not only theoretically but also with real-world data. It is difficult to estimate how frequent this situation is. Although we examined a single counterexample, we also showed that past studies may have systematically overlooked such cases. This suffices to show that one cannot know a priori where a task falls on the spectrum of Figure 8. It is clearly ill-advised to blindly assume a positive correlation.
Can we avoid inverse correlations with more training data? Scaling alone without data curation seems unlikely to prevent inverse correlations. [8] examined a more general question and determined that the impressive robustness of the large vision-and-language model CLIP is determined by the distribution of its training data rather than its quantity. Similarly, inverse correlations stem from biases in the training distribution (e.g. a class  $\mathcal{C}_1$  appearing more frequently with image background  $\mathcal{B}_1$  than any other). And biases in a distribution do not vanish with more i.i.d. samples. Indeed, more data can cover more of the support of the distribution, but this coverage will remain uneven, i.e. biased. The problem can become one of "subpopulation shift" [38] rather than distribution shift, but it remains similarly challenging.
Training full networks with a diversity-inducing method. We showed inverse correlations with standard ERM models and with linear classifiers trained with a diversity-inducing method [46]. To the best of our knowledge, this diversity method has not been applied to deep models because of its computational expense. It would be interesting to confirm our observations on networks trained entirely with diversity-inducing methods.
Qualitative differences along the Pareto frontier. Besides quantitative performance, interpretability methods could examine whether various ID/OOD trade-off models rely on different features and generalization strategies as done in NLP in [13].
Model selection for OOD generalization has recently seen promising advances. [7] get around selection based on either ID or OOD validation data with a tunable trade-off in their Quantile Risk Minimization method. And [48] examined existing approaches to OOD generalization from their suitability to various distribution shifts and causal structures.
Statements in [21] are factually correct but misleading. They are equivalent to "we did not detect  $X$ " but leave it to the reader to figure out "we designed our experiments such that there is no way of detecting  $X$ ".
[55] Huaxiu Yao, Caroline Choi, Bochuan Cao, Yoonho Lee, Pang Wei W Koh, and Chelsea Finn. Wild-time: A benchmark of in-the-wild distribution shift over time. Proc. Advances in Neural Inf. Process. Syst., 2022.
[56] Huaxiu Yao, Yu Wang, Sai Li, Linjun Zhang, Weixin Liang, James Zou, and Chelsea Finn. Improving out-of-distribution robustness via selective augmentation. In Proc. Int. Conf. Mach. Learn., 2022.
[57] Shingo Yashima, Teppei Suzuki, Kohta Ishikawa, Ikuro Sato, and Rei Kawakami. Feature space particle inference for neural network ensembles. arXiv preprint arXiv:2206.00944, 2022.
[58] Nanyang Ye, Kaican Li, Haoyue Bai, Runpeng Yu, Lanqing Hong, Fengwei Zhou, Zhenguo Li, and Jun Zhu. Ood-bench: Quantifying and understanding two dimensions of out-of-distribution generalization. In Proc. IEEE Conf. Comp. Vis. Patt. Recogn., 2022.
[59] Hongyang Zhang, Yaodong Yu, Jiantao Jiao, Eric Xing, Laurent El Ghaoui, and Michael Jordan. Theoretically principled trade-off between robustness and accuracy. In Proc. Int. Conf. Mach. Learn., 2019.
[60] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. arXiv preprint arXiv:1710.09412, 2017.
[61] Xiao Zhou, Yong Lin, Weizhong Zhang, and Tong Zhang. Sparse invariant risk minimization. In Proc. Int. Conf. Mach. Learn., 2022.
A Additional results on Camelyon17
Figure 9: As in Figure 3, we show that higher OOD accuracy can be sometimes be traded off for a lower ID accuracy. Each panel shows results from a different pretrained model (i.e. pretrained with a different random seed). Each dot represents a linear classifier re-trained on features from this pretrained model with standard ERM (red dots  $\bullet$ ) or with a diversity-inducing method [46] (gray dots  $\odot$ ). The latter set includes models with higher OOD/ lower ID accuracies.
Figure 10: Same as in Figure 9, using val-ood instead of test-ood as the OOD evaluation set.
Figure 11: Same as in Figure 10, zoomed-in on ERM models (red dots  $\bullet$ ).
B Results on other datasets
B.1 CivilComments
Data. The CivilComments dataset [15] is another widely-used dataset in OOD research. It contains text comments from internet forums to be classified as toxic or not. Each example is labeled with a topical attribute (e.g. Christian, male, LGBT, etc.) that is spuriously associated with ground truth labels in the training data. The target metric is again worst-group accuracy, where a group is any label/attribute combination.
Methods. We follow the same procedure as described above. We experiment two classes of architectures: pretrained BERT-tiny fine-tuned on CivilComments, and linear classifiers trained of features from the same frozen BERT-tiny models (a.k.a. linear probing).
Results. We observe in Figures 12-13 different patterns with the two classes of architectures. With linear probing, the ID vs. OOD trade-off is minimal, and the model of highest ID performance within a run as well as across methods is very similar to the model of highest OOD performance. With fine-tuning however, the trade-off is more pronounced. The ID and OOD performance usually peak then diminish at different epochs during the fine-tuning. This agrees with previous reports [2] of OOD robustness being progressively lost during fine-tuning.
Figure 12: Results on CivilComments with fine-tuned BERT.
Figure 13: Results on CivilComments with linear probing on frozen BERT embeddings.
B.2 WildTime-MIMIC-Readmission
Data. The WildTime-MIMIC-Readmission [55] dataset contains hospital records (sequences of codes representing diagnoses and treatments) to be classified into two classes, corresponding to the readmission of the patient within a short time. ID and OOD splits contain records from different time periods.
Methods. We follow the same procedure as described above. We train a standard bag-of-embeddings architecture, which associate each diagnosis/treatment with a learned embedding, then summed and fed to a linear classifier. We train this model with standard ERM, and with a resampling to balance the classes in the training data [12], which is a standard approach for imbalanced datasets. We also
train models with a "mild balancing", where classes are sampled according to a distribution half-way between the original one of the training data, and a uniform  $(50\% - 50\%)$  one.
Results. In Figure 14 we observe that ID and OOD performance are mostly positively correlated across methods. The best models are obtained with uniform balancing of classes, in which case model selection based on OOD performance could give a small advantage, but it is marginal compared to the improvement over the ERM baseline, which can be clearly detected on both the ID and OOD performance.
Figure 14: Results on WildTime-MIMIC-Readmission.
B.3 WildTime-Yearbook
Data. The WildTime-Yearbook [55] dataset contains yearbook portraits. Each image is to be classified as male or female, and the ID and OOD splits contain images from different time periods.
Methods. We follow the same procedure as described above. We train the simple CNN architecture described in [55]. We report in Figure 15 both the "average-group" accuracy (over the entire OOD test set) and the "worst-group" accuracy (where a group is any 5-year period within the OOD test period.
Results. The patterns are slightly different in the two cases but similar conclusions can be drawn from both. There is a mostly-positive correlation, but at the highest accuracies (upper-right quadrant), some small trade-off exists. This suggests that fine-grained differences exist that are useful for either ID or OOD generalization, but not both. Although differences are small, this "pointy end" of the spectrum is where the state-of-the-art models compete, hence the relevance of this observation.
Figure 15: Results on WildTime-Yearbook (left: average-group accuracy, right: worst-group accuracy).
C Theoretical analysis
We present a minimal case of a trade-off between ID and OOD performance to aid understanding its cause and show it can occur even in a simple linear setting. Let  $y \in \mathbb{R}$  be the target variable predicted by the model and  $\pmb{x}$  the features used as input. Without loss of generality, we consider a regression setting for mathematical convenience, though the results are valid for classification as well. The purpose of this analysis is to investigate theoretically how variations in certain input features correspond to changes in risk (i.e. expected loss) on OOD data. We consider the input features to be a concatenation of invariant and spurious elements:  $\pmb{x} = [x_{\mathrm{inv}}; x_{\mathrm{spu}}]$  with  $x_{\mathrm{inv}} \in \mathbb{R}^{d_{\mathrm{inv}}}$  and  $x_{\mathrm{spu}} \in \mathbb{R}^{d_{\mathrm{spu}}}$ . Following [4, 34, 61], we consider the simple data-generating process defined by the following structural equations:  $y^{e} := \gamma^{\top} x_{\mathrm{inv}}^{e} + \epsilon_{\mathrm{inv}}$ , and  $x_{\mathrm{spu}}^{e} := y^{e} \mathbf{1}^{\mathbf{s}} + \alpha^{e} \circ \epsilon_{\mathrm{spu}}$  where  $e \in \{e_{\mathrm{ID}}, e_{\mathrm{OOD}}\}$  is an environment index referring to ID or OOD data,  $\mathbf{1}^{\mathbf{s}}$  is a vector of ones and  $\circ$  denotes the element-wise product. The random variables  $\epsilon_{\mathrm{inv}}$  and  $\epsilon_{\mathrm{spu}}$  represent symmetric independent random noise with zero-mean, sub-Gaussian tail probabilities such that  $\operatorname{Var}(\epsilon_{\mathrm{inv}}) > 0$ ,  $\operatorname{Var}(\epsilon_{\mathrm{spu},i}) > 0$ ,  $\forall i \in [1, d_{\mathrm{spu}}]$ . The vector  $\gamma \in \mathbb{R}^{d_{\mathrm{inv}}}$  determines the linear relation between the target variable and the invariant features, which is identical across environments. In contrast, the vector  $\alpha^{e}$  acts on the spurious features and is environment-specific. Invariant features are similarly predictive on ID and OOD data while spurious ones are not.
Let us now study the relationship between ID and OOD performance of a predictive model that relies on a subset  $\Phi$  of the features  $\pmb{x}$  (not that the subset could be spurious or invariant). This subset is represented with a binary mask  $\Phi \in \{0,1\}^{d_{\mathrm{inv}} + d_{\mathrm{spu}}}$ . Suppose we have selected  $\hat{d}_{\mathrm{inv}}$  invariant features and  $\hat{d}_{\mathrm{spu}}$  spurious features, with  $(\hat{d}_{\mathrm{inv}} + \hat{d}_{\mathrm{spu}}) = \hat{d} = ||\Phi_{\hat{d}}||_1$  the number of selected features. We denote the selected features by  $\Phi_{\hat{d}}$  with  $[x_{\mathrm{inv},1}, \ldots, x_{\mathrm{inv},\hat{d}_{\mathrm{inv}}}, x_{\mathrm{spu},1}, \ldots, x_{\mathrm{spu},\hat{d}_{\mathrm{spu}}}]$ . To investigate the changes in model performance, we consider a simple linear regression model. This enables us to theoretically measure the sensitivity of predictions when additional spurious features are added without complications from non-linearities. We denote with  $\mathbb{E}^{\mathrm{ID}}$  and  $\mathbb{E}^{\mathrm{OOD}}$  the in- and out-of-domain expectation, and with  $\beta$  the optimal parameters of the linear regression for a certain domain, i.e.  $\beta_{\hat{d}} = \mathbb{E}[\Phi_{\hat{d}}(\pmb{x})^\top \Phi_{\hat{d}}(\pmb{x})]\mathbb{E}[\Phi_{\hat{d}}(\pmb{x})^\top y]$ . The MSE loss of the fitted linear regressor is then  $\mathbb{E}[y - \Phi_{\hat{d}}(\pmb{x})^\top \beta_{\hat{d}}]^2$ . Given the feature mask  $\Phi_{\hat{d}}$ , we denote with  $\mathcal{L}_{\mathrm{ID}}(\Phi_{\hat{d}})$  and  $\mathcal{L}_{\mathrm{OOD}}(\Phi_{\hat{d}})$  the risk of the model on the in- and out-of-domain distribution, respectively. Theorem 1 below (see complete form including assumptions and proof in Appendix D) examines how the ID and OOD risks of the model vary as additional spurious features are included into  $\Phi_{\hat{d}}$  to obtain  $\Phi_{\hat{d} + 1}$ .
Theorem 1. Under Assumption 1 and sufficient changes in  $\alpha$ , including an additional spurious feature leads to:
$$
\mathcal {L} _ {\mathrm {I D}} (\Phi_ {\hat {d} + 1}) - \mathcal {L} _ {\mathrm {I D}} (\Phi_ {\hat {d}}) = \mathbb {E} ^ {\mathrm {I D}} [ y - \Phi_ {\hat {d}} (\pmb {x}) ^ {\top} \pmb {\beta} _ {\hat {d}} ^ {\mathrm {I D}} ] ^ {2} - \mathbb {E} ^ {\mathrm {I D}} [ y - \Phi_ {\hat {d} + 1} (\pmb {x}) ^ {\top} \pmb {\beta} _ {\hat {d} + 1} ^ {\mathrm {I D}} ] ^ {2} <   0
$$
$$
\mathcal {L} _ {\mathrm {O O D}} (\Phi_ {\hat {d} + 1}) - \mathcal {L} _ {\mathrm {O O D}} (\Phi_ {\hat {d}}) = \mathbb {E} ^ {\mathrm {O O D}} [ y - \Phi_ {\hat {d}} (\pmb {x}) ^ {\top} \pmb {\beta} _ {\hat {d}} ^ {\mathrm {O O D}} ] ^ {2} - \mathbb {E} ^ {\mathrm {O O D}} [ y - \Phi_ {\hat {d} + 1} (\pmb {x}) ^ {\top} \pmb {\beta} _ {\hat {d} + 1} ^ {\mathrm {O O D}} ] ^ {2} > 0.
$$
Theorem 1 shows that adding a spurious feature to those used by the model can affect its ID and OOD losses in opposite directions, implying a trade-off between ID and OOD accuracy. In other words, this minimal case shows that a simple model without/with an extra (spurious) feature can exhibit an inverse correlation between its ID and OOD performance.
Theorem 1. Including an additional spurious feature leads to the following change in the risks:
$$
\mathcal {L} _ {\mathrm {I D}} (\Phi_ {\hat {d} + 1}) - \mathcal {L} _ {\mathrm {I D}} (\Phi_ {\hat {d}}) = \mathbb {E} ^ {\mathrm {I D}} [ y - \Phi_ {\hat {d}} (\pmb {x}) ^ {\top} \beta_ {\hat {d}} ^ {\mathrm {I D}} ] ^ {2} - \mathbb {E} ^ {\mathrm {I D}} [ y - \Phi_ {\hat {d} + 1} (\pmb {x}) ^ {\top} \beta_ {\hat {d} + 1} ^ {\mathrm {I D}} ] ^ {2} <   0
$$
$$
\mathcal {L} _ {\mathrm {O O D}} (\Phi_ {\hat {d} + 1}) - \mathcal {L} _ {\mathrm {O O D}} (\Phi_ {\hat {d}}) = \mathbb {E} ^ {\mathrm {O O D}} [ y - \Phi_ {\hat {d}} (\pmb {x}) ^ {\top} \pmb {\beta} _ {\hat {d}} ^ {\mathrm {O O D}} ] ^ {2} - \mathbb {E} ^ {\mathrm {O O D}} [ y - \Phi_ {\hat {d} + 1} (\pmb {x}) ^ {\top} \pmb {\beta} _ {\hat {d} + 1} ^ {\mathrm {O O D}} ] ^ {2} > 0
$$
$$
\mathcal {L} _ {\mathrm {O O D}} \left(\Phi_ {\hat {d} + 1}\right) - \mathcal {L} _ {\mathrm {O O D}} \left(\Phi_ {\hat {d}}\right) = Q _ {1} + Q _ {2} + Q _ {3}
$$
with  $Q_{1}, Q_{2}, Q_{3}$  defined as:
$$
\begin{array}{l} Q _ {1} = \mathbb {E} ^ {\mathrm {O O D}} [ y - \Phi_ {\hat {d}} (\pmb {x}) ^ {\top} \beta_ {\hat {d}} ^ {\mathrm {O O D}} ] ^ {2} - \mathbb {E} [ y - \Phi_ {\hat {d} + 1} (\pmb {x}) ^ {\top} \beta_ {\hat {d} + 1} ^ {\mathrm {O O D}} ] ^ {2} \\ Q _ {2} = \sum_ {i = 1} ^ {\hat {d}} \left[ \left(\mathbb {E} ^ {\mathrm {O O D}} [ \Phi_ {\hat {d}} (\boldsymbol {x}) y ] ^ {\top} \boldsymbol {v} _ {i} ^ {\mathrm {O O D}, \hat {d}}\right) ^ {2} \left(\lambda_ {i} ^ {\mathrm {O O D}, \hat {d}}\right) \left(\frac {1}{\lambda_ {i} ^ {\mathrm {I D} , \hat {d}}} - \frac {1}{\lambda_ {i} ^ {\mathrm {O O D} , \hat {d}}}\right) ^ {2} \right. \\ \left. \right.\left. - \left(\mathbb {E} ^ {\mathrm {O O D}} \left[ \Phi_ {\hat {d}} (\boldsymbol {x}) y \right] ^ {\top} \boldsymbol {v} _ {i} ^ {\mathrm {O O D}, \hat {d} + 1}\right) ^ {2} \left(\lambda_ {i} ^ {\mathrm {O O D}, \hat {d} + 1}\right)\left(\frac {1}{\lambda_ {i} ^ {\mathrm {I D} , \hat {d} + 1}} - \frac {1}{\lambda_ {i} ^ {\mathrm {O O D} , \hat {d} + 1}}\right) ^ {2} \right] \\ \end{array}
$$
$$
Q _ {3} = \left(\mathbb {E} ^ {\mathrm {O O D}} \left[ \Phi_ {\hat {d} + 1} (\boldsymbol {x}) y \right] ^ {\top} \boldsymbol {v} _ {\hat {d} + 1} ^ {\mathrm {O O D}, \hat {d} + 1}\right) ^ {2} \frac {\left(\left(\alpha_ {\hat {d} + 1} ^ {\mathrm {I D}}\right) ^ {2} - \left(\alpha_ {\hat {d} + 1} ^ {\mathrm {O O D}}\right) ^ {2}\right) ^ {2}}{\left(\lambda_ {\hat {d} + 1} ^ {\mathrm {I D}, \hat {d} + 1}\right) ^ {2} \lambda_ {\hat {d} + 1} ^ {\mathrm {O O D}, \hat {d} + 1}} > 0.
$$
Further, if the new feature is sufficiently unstable in the test domain, i.e. if  $(\alpha_{\hat{d} +1}^{\mathrm{ID}})^2 -(\alpha_{\hat{d} +1}^{\mathrm{OOD}})^2$  is sufficiently large such that:
$$
| \left(\alpha_ {\hat {d} + 1} ^ {\mathrm {I D}}\right) ^ {2} - \left(\alpha_ {\hat {d} + 1} ^ {\mathrm {O O D}}\right) ^ {2} | > \sqrt {\frac {\left(\lambda_ {\hat {d} + 1} ^ {\mathrm {I D} , \hat {d} + 1}\right) ^ {2} \lambda_ {\hat {d} + 1} ^ {\mathrm {O O D} , \hat {d} + 1}}{\left(\mathbb {E} ^ {\mathrm {O O D}} [ \boldsymbol {\Phi} (\boldsymbol {x}) y ] ^ {\top} \boldsymbol {v} _ {\hat {d} + 1} ^ {\mathrm {O O D} , \hat {d} + 1}\right) ^ {2}}} \left| Q _ {1} + Q _ {2} \right|},
$$
then we have  $Q_{3} > |Q_{1} + Q_{2}|$  and therefore  $\mathcal{L}_{\mathrm{OOD}}(\Phi_{\hat{d} +1}) - \mathcal{L}_{\mathrm{OOD}}(\Phi_{\hat{d}}) > 0$
Let  $\pmb{x}_{\hat{d}} \coloneqq \Phi_{\hat{d}}(\pmb{m}x)[\pmb{x}_{\mathrm{inv},1}, \dots, \pmb{x}_{\mathrm{inv},\hat{d}_{\mathrm{inv}}}, \pmb{x}_{\mathrm{spu},1}, \dots, \pmb{x}_{\mathrm{spu},\hat{d}_{\mathrm{spu}}}]$  be the  $\hat{d}$  features already selected,  $\pmb{x}_{\hat{d} + 1} \coloneqq \Phi_{\hat{d} + 1}(\pmb{m}x)$  the features after adding a new spurious feature  $\pmb{x}_{\mathrm{spu},\hat{d}_{\mathrm{spu}} + 1}$  to  $\pmb{x}_{\hat{d}}$
$[\lambda_1^{\hat{d}}, \lambda_2^{\hat{d}}, \dots, \lambda_{\hat{d}}^{\hat{d}}]$  the eigenvalues of  $\mathbb{E}[\pmb{x}_{\hat{d}}^{\top}\pmb{x}_{\hat{d}}]$  and  $[\pmb{v}_1^{\hat{d}}, \pmb{v}_2^{\hat{d}}, \dots, \pmb{v}_{\hat{d}}^{\hat{d}}]$  the corresponding eigenvectors.