Shortcut Features as Top Eigenfunctions of NTK: A Linear Neural Network Case and More
Authors: Jinwoo Lim
Suhyun Kim
Soo-Mook Moon
Venue: NeurIPS 2025 spotlight ()
Keywords: shortcut learning
neural tangent kernel
Primary area: deep_learning
TL;DR: We reveal that shortcut features correspond to top eigenfunctions of NTK and dominate the neural network output after convergence.

Abstract: One of the chronic problems of deep-learning models is shortcut learning. In a case where the majority of training data are dominated by a certain feature, neural networks prefer to learn such a feature even if the feature is not generalizable outside the training set. Based on the framework of Neural Tangent Kernel (NTK), we analyzed the case of linear neural networks to derive some important properties of shortcut learning. We defined a “feature” of a neural network as an eigenfunction of NTK. Then, we found that shortcut features correspond to features with larger eigenvalues when the shortcuts stem from the imbalanced number of samples in the clustered distribution. We also showed that the features with larger eigenvalues still have a large influence on the neural network output even after training, due to data variances in the clusters. Such a preference for certain features remains even when a margin of a neural network output is controlled, which shows that the max-margin bias is not the only major reason for shortcut learning. These properties of linear neural networks are empirically extended for more complex neural networks as a two-layer ReLU FC network and a ResNet-18.

Shortcut Features as Top Eigenfunctions of NTK: A Linear Neural Network Case and More
Jinwoo Lim Scoul National University j jinwoolimm 800snu. ac. kr
Suhyun Kim Kyung Hee University suhyun. kim@gmail com
Soo-Mook Moon Seoul National University smoon@snu. ac.
Abstract
One of the chronic problems of deep-learning models is shortcut learning. In a where the majority of training data dominated by certain feature, neural networks prefer to learn such a feature even the feature is not generalizable outside the training
Based on the framework of Neural TTngent Kernel (NTK), we analyzed the case of linear neural networks to derive some important properties of shortcut learning. We defined a "feature" of neural network as an eigenfunction of NTK. Then, we found that shortcut features correspond to features with larger eigenvalues when the shortcuts stem from the imbalanced number of samples the clustered distribution. We also showed that the features with larger eigenvalues still have large influence on the neural network output even after training, due to data variances in the clusters. Such a preference for certain features remains even when a margin of a neural network output is controlled, which shows that the margin bias is not the only major reason for shortcut learning. These properties of linear neural networks are empirically extended for more complex neural networks two-layer fully-connected ReLU network and ResNet-18.
Introduction
Based on various optimization algorithms, deep neural networks can learn rich features from data samples. Due to success in advance of neural network architectures, neural networks can perform complex tasks as natural language processing. However, when neural networks are optimized with suddient-descen--baseee methods, there often exists gap between feature that is desired to learn and a feature that neural networks rely on. Rather than core features which can predict labels with high accuracy, neural networks often rely certain features called shorteut features. Shortcut features or biased attributes are features that show a strong but non-generalizable correlation to the ground-truth labels B 17 Neural networks that learned shorteut features show minimum loss for the training set, but fail for data outside the training set. Such a phenomenon frequently occurs when neural networks are trained on biased dataset, of which the majority of training data samples are influenced by the biased attributes (28). This problem of shortcut learning is one of the most chronic problems of deep-learning models.
In this paper, we study the case where shorteut learning occurs due to the dominance of biased attributes the majority of samples. We analyze the cause and effect of shortcut learning upon the framework Neural Tangent Kernel (NTK) theory. We decompose the neural network outputs into eigenfunctions of NTK and measure the influence of each eigenfunction on neural network outputs. We study simple case: training a linear neural network on a dataset of an almost-separable Gaussian mixture model, of which the majority of the samples are clustered by certain feature. Under this assumption, we find interesting properties of shortcut learning.
cco-corresponing authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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First, shortcut features correspond to eigenfunctions with large eigenvalues. Due to larger eigenvalues, shorteut features are learned faster than other features, which has been empirically observed in previous works [14, 18 23 Second, shortcut features contribute more to the neural network output than other features after convergence. Therefore, while data samples with no shortcut features can also be learned with near-zero loss during training, such data outside the training set are still not well-predicted, which was also observed in previous works [20] 21]. In those previous works, the max-margin bias has been pointed out main underlying cause of shortcut learning. As an ablation study, we inspect the case where a margin of a neural network is penalized with a method called SD [19] or Marg- -trl [20]. We theoretically show that the decision boundary of neural network can still be dominated by shortcut features even when the margin is controlled, hence the max- mar-mn bias is not the only cause of shortcut learning. Another worth-noting thing is that an imbalance in feature contribution arises from the variance within groups of data samples.
Furthermore, we experimentally show that those properties of linear neural networks can be extended to the case of training complex neural networks such as ResNet-18 [ on real-world datasets. For experiments, we introduce metrics to estimate how much feature can predict the ground-truth labels, which we call predictability, and how much a feature aligned to top eigenfunctions NTK, which we call availability. We find that shortcut labels have lower predictability but higher availability in real-world data samples, which aligns with the results from linear neural networks.
Our main findings are as follows:
For linear networks, features corresponding clusters of larger weights have larger eigen- values. Shortcut features which correspond to clusters of larger weights converge faster.
Due to the data variance within each cluster, features corresponding to clusters of larger weights also have a larger influence on the output of a linear neural network. Thus, shortcut features which have larger eigenvalues also have a higher influence on the neural network output. Such a phenomenon can persist even when the margin of the neural networks is controlled with debiasing techniques as SD [9).
We introduce metrics to estimate the predictability and the availability of real-world data samples. Shortcut features have a lower predictability but higher availability.
2 Background
2.1 Notions of Neural Tangent Kernel
We briefly introduce some notions of Neural Tangent Kernel [I] in this section. The training set D is composed of input data X' fa (x. y) D} and ground-truth labels y {y R* (z., v) E 'D). Here, we train neural network f X' - Y of which the parameter is w. We assume a supervised learning setting where the neural network is trained with a loss function of L= E I(f (x, w). y). Most optimization algorithms in deep learning are based on gradient descent, which can be seen an Euler method of first order solving ODE called the gradient flow:
ش V.L of (X'V (C.
From the equation above, we can describe the evolution of the output of the neural network as
f(x'= = - VV..vv/fVVVVVv((((VVVv :=
(2)
where f(x² E [BA|D)x1 denotes the concatenated outputs from all data from the training set. Here, the kernel K K(X,)) x)) E (D) is called Neural Tangent Kernel (NTK), which determines the convergence behaviour of neural network outputs.
As the width of the neural network goes to infinity, the network falls into a lazy-training regime 6 and the kernel remains constant during training. Thus, the kernel becomes deterministic at initialization which is noted as K(X, X) > Ko(X, X). this regime, for MSE loss l(f. = 21lff y1l the ODE has rather simple solution shown in Lee et al. [5].
f(x) (I - nKot y,
(3)
when the initial neural network output zero. For a data point xr the neural network output is
f(x) Kolz, (x, e- -oot)
(4)

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Meanwhile, if the kernel can be eigendecomposed into (o Σ Aiaze where X is the i-th eigenvalue of Ko with its corresponding eigenvector vs the output of the training set can also be decomposed and its convergence is expressed as
(ni, y = (p,,
(5)
We can see that the convergence rate along the direction vi depends on the eigenvalue A; The neural network learns directions with larger eigenvalues much faster than other directions, which is called
the spectral bias [5]
Based on the cigendecomposition. it is possible to decompose the converged neural network output into eigenvectors of NTK as f(x) = Ko(r, 2) E "
[(x) Σ (c)) (x)
Ko(a, (vs, V/uu
(6)
and ( was defined as a feature of input space in Tsilivis et al. 127
2.2 Datasets
In this paper, we show our experimental observations shortcut learning using biased datasets as depicted in Figure Patched-MNIST [T] Colored-MNIST [3], Waterbirds 23 CelebA [16]. and
Dogs and Cats 12
The task is predict the digit: 0 4 are labelled as -1 and 5 9 are labelled 1. The core feature is the shape of digits, while the non-core feature is the x 3 patch at the corner of were patched, and others were not patched for the training set. The ratio is balanced as 50% for the test set.
Patched-MNIST.
image. 95% of digits labelled as -1 and only 5% of digits labelled
The task is to predict the digit: 0 are labelled as -1 and 5 9 are labelled as 1. The core feature is the shape of digits, while the non-core feature is the colour. 95% of digits labelled as -I and only 5% of digits labelled as were coloured red, and others were coloured blue for the training The ratio is balanced as 50% for the test set.
CNNITTd-MNIST.
Waterbirds. The task is distinguish between waterbirds and land-birds. The core feature the appearance of the bird, while the non-core feature is the background. 95% of waterbirds are in front of watery backgrounds, and 95% of landbirds are in front of land backgrounds for training set. The ratio is balanced as 50% for the test set.
The task is to determine the hair colour of the figure is blonde or not. The core feature the hair colour, while the non-core feature is other features such gender. The ratio of the biased samples is naturally determined by the dataset and most blonde people are female.
CelebA.
Dogs and Cats. The task is to distinguish dogs and cats. The core feature is the appearance of the animal, while the non-core feature the colour of the animal. 95% of cats and only 5% of dogs dark, while others are brightly coloured for the training set. The ratio is balanced as 50% for test set.
Bias
2
4
Blas-conFlicttin
Figure 1: Datasets containing biased and core features; Micche MNISST Waterbirds, CelebA, and Dogs and Cats.
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Shortcuts on Gaussian Mixture Model
For the ease of analysis. focus linear neural network with activation function. The setting seems somewhat extreme and not applicable to real cases, but linear networks can still manifest important general attributes of neural networks such the max-margin bias 20 26 and kernel
alignment 2 24].
thus being worthy of research.
We also focus on continuum limit where the number of discrete data samples is large and continuum approximation is possible. In such setting, rather than a matrix, NTK defined as a kernel function -). Under data distribution of X ~ p(z)). matrix multiplication of NTK and vector is defined as an application of an integral operator TK: (2) L2(() on a function, which
(xg(2) = J k(x, s)g(s)dp(s)).
Instead of eigenvector, continuous setting allows us to define an cigenfunction 6(z) of kernel operator as
xx(x) = Tx TxΦ(x)= Jfx k(x, s)p(s))p(s)
(8)
where the eigenvalue is À and the cigenfunction normalized to satisfy fxx (s(dd((ss)
For linear neural networks, the kernel function of NTK in the infinite-width limit for data x and is proportional to (ar, y) with a factor of the number of layers 2 4 Here, we define k( v) (x. v) for analysis hereafter.
3.1 Problem formulation
Here, we define simple setting of training data distribution. We assume binary setting where all samples are labelled as y E {-1,1}, Data samples are clustered as a simple Gaus- sian mixture model based on the attributes of samples. The clusters are as follows: By N(un,, BB.)) is the i-th cluster of samples with a biased attribute that were labelled as y, while Cy ~ N N(CC,..
Colored MNIST
Waterbirds
Y1
20/B:
Y/10
0-
1:00000
is the i-th cluster
-10
of bias bias-conflicting samples that were labelled as y. Since we study the case where biased attributes dominate the majority of samples, in a mixture model, we assume that weights of By are larger than the weights of Cg., as Σ, By, >> E, TC,.:" Though this setting studies continuum limit, the assumed distri-
-30
20
-20
10
(a) Colored-MNIST
(b) Waterbirds
Figure 2: t-SNE visualization of 1000 input data samples in datasets. Each input is marked in color corresponding to its label Y: ground-truth label, B: label from shorteut feature.
bution is biased and it can approximate the finite samples of the biased training set if the number of training samples is large. Therefore, we assume that training data follow a distribution
p(z)
Σ VE(-1,1)
By, N(B oi By.)) )+ Σ TCy. L E.
(9)
Many previous studies have focused on the case of uniformly distributed data to simplify the analysis On the other hand, we assume a case a clustered data distribution because the assumption on the uniform data distribution does not fully reflect real- world scenarios. In fact, in some cases, data are moderately clustered based on certain attributes data samples. In Figure 2 it is possible to observe that Waterbirds and Colored-MNIST dataset, which well-known datasets with shortcut features, are clustered by the biased attributes. This clustering is caused by the emergence of shortcut feature, highlighting how strongly the biased attribute can influence the data distribution.
5 22 29
Another thing to note that the data variance each cluster is not really small. For the ease of analysis, some previous works [10] used an assumption of small covariance for the clustered distribution. However, real cases, data samples are scattered and the variance is not small, which makes the small covariance assumption somewhat risky.

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index
10 100
200
Figure 3: Original images and saliency maps from each feature of outputs from two-layer ReLU CNN networks. Saliency maps on the left side shows the spatial support of features with large eigenvalues, while saliency maps on the right side shows the spatial support of features with smaller eigenvalues. Indices indicate the ranks of the eigenvalues terms of magnitude. index indicates the saliency map from a feature with the (i + 1)-th largest eigenvalue. Features with larger eigenvalues focus on biased attributes of samples, i.e., in CelebA, features with larger eigenvalues focus the edges of the face, the background of an image rather than the hair itself.
saliency map from the i-th
3.2 Eigenvalues of features
For kernel function k(;-), we can obtain a simple result about the eigenfunction and eigenvalue of the kernel.
Proposition 3.1. Assume data E Bdd in Gaussian Mixture Model of =k TEN(pR of The kernel k(x,,) u) = (x,, v) has eigenfunctions oi and corresponding eigenvalues d as follows:
{
x velca = 1....,m
Φx(x)
(10)
x ot /ca
otherwise
Σx k
K
T th
+
a; ifi=l....,m
A
(11)
E Too
otherwise
when (E* TRPa )os = a,u, oft is a vector perpendicular to us for (l....,K), m and G = VXg.
rank 1= T(peP%)) ). ||0:|| = 1, ||
EK The result suggests that the eigenvalue of the inner product with eigenvector i of 2k=1 TaliHA depends on the eigenvalue a, Since eigenvectors close to a cluster with a larger weight Te have larger eigenvalues, inner product with vector from larger cluster has larger eigenvalue and converges faster due to the spectral bias. As assumed in Section 3.1 B is much larger than TC,,. Therefore, by Proposition 3.1 will converge faster than others. Also, if means of clusters are close to each other, eigenvectors v; nearly orthogonal the means pa have smaller eigenvalues a; In this case, features that help with distinguishing between clusters will have smaller eigenvalues and will converge more slowly. In a case where pB,.. HCC.... it harder to learn a neural network that distinguishes between biased samples and bias-conflicting samples.
expect that the inner products with data from these "shortcut clusters"
We empirically extend this result for the case of a two-layer ReLU CNN model. For the experiment, we generated a saliency map (25] for each feature of a neural network from Equation highlight the region of interest for each feature. We computed the magnitude of the loss gradient with respect to the input, C(C(fe y)) to measure the spatial support each feature ft. We use the same method the implementation done in Tsilivis al. (27).
Figure 3 shows the result. We can observe that features of larger eigenvalues rely on biased attributes. In Patched-MMISS features with large cigenvalues focus the patch of an image. In Waterbirds,
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features of large eigenvalues focus on the background of an image. Also in CelebA, features of large eigenvalues focus more on the edges on the face, or the background of an image rather than the hair itself. This is the reason why neural networks learn shortcut features faster.
3.3 Features after convergence
In Proposition 3.1 we inspected the convergence rate of neural network output during training. In this section, we inspect the influence of features on a neural network output after training. For linear neural networks, since the eigenfunctions are inner products, it is possible to dissect the neural network output into eigenfunctions of NTK. We will dissect the neural network output into a weighted sum of eigenfunctions of NTK, and investigate the magnitude of the weights to measure the influence of eigenfunctions on the neural network output. We assume that the neural network is trained with an MSE loss. Then we obtain the following result:
Proposition 3.2. Assume data x E Rd in Gaussian Mixture Model p( = A binary label function y(z) E {-1,1} nearly separates the mixture model that data from clusters with mean pee (c E C) are labelled as and otherwise -1. When the linear neural network optimized
( (f.. off).
for y(a) with the MSE loss and dissected into f(z) = => ||vel = 1),
mm (1) (fa(z) = x uk & dx(I),
Wk- Eyec JHZ Uk -jEEC TT2 uk
(12)
E ,O,, + (E Tipip Juk
If p. 11g Mt
for i + and on = Ma/ ////////lll then
liue lla
ik if E C
wk=
To pull
(13)
In the latter case, we assumed that means of clusters are orthogonal to each other so that a change in cluster weights me would not change the eigenfunction up to a constant factor.
Since Wig increases as T3 increases from 0 to I, eigenfunctions corresponding to clusters with larger weights have more influence the converged neural network output. Considering from Proposition 3.1 that eigenfunctions corresponding to larger clusters have larger cigenvalues, it could be said that eigenfunctions with larger eigenvalues have more influence on the converged neural network output. As assumed in Section 3.1 data samples with biased attributes, By. belong to a larger cluster (TB,, ) so biased features have large eigenvalues and large influence the output at the same time. This shows that shortcut features, which have large eigenvalues, also have a large influence on the neural network output when neural network is trained with the MSE loss.
One thing to note in Proposition 3.2 is that it not only applies to the lazy-training regime with constant NTK, but to general situations since this result is from the "optimal" function of neural networks in the distribution p(z).. Another thing to note is that such a dependency of ww on cluster weights Th originates from the existence of the variances, 0s, among data in the clusters. If very small data variances were assumed as in [00, there would be no change in Wig when there was a change in the distribution (TT)). The influence of shortcut features after convergence heavily depends not only on the weights of clusters, but on the variance of samples. Thus, a debiasing technique must consider the variance of samples if it aims to adjust the decision boundary of a neural network.
We empirically inspect the result Proposition 3.2 with a toy example in a 2D space. In Figure 4 we trained a two-layer fully-connected ReI.U network to classify four clusters of various weights and variances. The norms of the means of clusters were identical. The clusters on the x-axis become clusters of shortcut feature. In Figure the weight of clusters on the x-axis is denoted no and the decision boundary by the network was implicitly marked as a borderline between the regions of different colors. The results adhered to Proposition 3.2 When the variances in clusters were not small, the decision boundary was tilted towards the clusters with larger weights, and the inner products with samples from larger clusters had larger influence the decision boundary. However, when the variances of clusters were very small, the decision boundary did not depend on the weights of the clusters but solely on their positions, the same as the results from the linear kernel.
Another factor that affects the eigenvalues and the decision boundary the norm of the mean vectors. Proposition 3.1 indicates that eigenvalues scale with both mixing weights (TT) and the magnitude of

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Figure 4: Classification of 4 clusters with two-layer ReLU fully-connected network. The network was trained to classify yellow and purple data samples by their colors. The decision boundary by the network was implicitly marked as borderline between the regions of different colors. For small variances, there is no change in borderline though there is change of weights in the clusters. Only for large variances, there is a change borderline when there change of cluster weights.
the mean vectors (1lx)12). In contrast, Proposition 3.2 implies that the contribution of feature on the decision boundary (Wx) scales inversely with the magnitude of the mean vectors. While the scope of paper is restricted to cases where the norms of the mean vectors comparable (e.g. normalized), in scenarios where cluster with a large weight has a very small-norm mean, its direction may no longer dominate the spectral bias, yet it can still exert stronger influence the decision boundary.
3.4 Discussion on the max-margin bias
Puli al. 20 have also obtained similar result under a case where linear networks are trained with the cross-entropy loss. In this work, the max-margin bias has been pointed out as main cause of the imbalance in the feature contribution. In order solve the problem, they inspected the case where the margin of the neural network is controlled and the parameter does not converge to the max-margin solution. Such control done famous debiasing method known as SD [19] or Marg-CC Cirl (20], which regularizes the parameter by adding a penalty on the norm of the neural network output. Thus a neural network is trained under loss function below if the original loss is the cross-entropy loss:
(sD(f(x), y) = log(1 + exp(-uf(z)) + fxx))1
(14)
In order to know if "maximization" " margins itself is the only problem, we inspect the case where the strength of the regularization is strong in SD.
Corollary 3.3. Assume the linear neural network converges to fSD(#) -E (wk) (x) (fi(2) = T uk x Φx(x)), ||vell = 1) when the network trained with SD, and the linear neural network converges to fMSE(E) = Em (Wa)MsEJin (x) when the network is trained with the MSE loss function. When the Lagrange multiplier SD becomes infinite as - 00, then
lim (w))sD (w))sSS
(15)
(w))sD (W) MSE
when (us)ss + 0 and (W)) o (w:) (S)) (w)) MSF does not change although SD is applied to the MSE loss function.
First, if the original loss function is the MSE loss, the decision boundary does not change even when the regularization is applied. Second, when the regularization is strong, the decision boundary of SD converges to the decision boundary of a neural network trained with MSE loss. Since neural network trained with the MSE loss is affected by shorteut learning, this means that the neural network can still be affected by shortcut learning even when the margin minimized. The max-margin bias not the only cause of shortcut learning, but the shortcut bias arises from learning the label itself.
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We empirically show Corollary 3.3 with toy example in a 2D space in Figure 5 We trained two-layer fully-connected ReLU network to classify four clusters with various loss functions. We used the MSE loss function with and without SD regularization, and the cross-entropy (CE) loss with and without SD. The regularization strength À was varied into two values: 0.1 and 1.0. The clusters on the x-axis become clusters of shortcut feature.
As in Corollary 3.3 the decision boundary under the MSE loss did not change when SD was applied. The decision boundary under CE loss changed when SD was ap- plied. When the value of À was quite large as 1.0, the decision boundary converged to the one under the MSE loss.
(a) MSE
(b) MSE (A 0.1) (c) MSE (A = 1.0)
Experiments
4.1 Predictability and availability
Here, we empirically extend theoretical re- sults on linear neural networks to the case of more complex networks. For the exper- iments, propose some new metrics. In Hermann et al. [10], some interesting meth- ods were proposed to quantify how well a feature aligned to the ground-trune truth and how much feature is "easy" learn for a neural network. These metrics are called predictability and availability of feature, respectively. While they were based on synthetic datasets, we propose metrics for realistic datasets for experiments.
(d) CE
(0) CE (A 0.1) (f) CE 1.0)
Figure 5: Classification of 4 clusters with two-layer ReLU fully-connected network. The network was trained classify yellow and purple data samples by their colors. SD was respectively applied to the MSE loss and CE loss. The decision boundary under SD con- verges to the one under the MSE loss.
Definition 4.1. Predictability measures how well a feature g is aligned with the ground-truth Predictabilityy is y (/', which is Sx y(z)g(() (=))((((=))))))) for continuous case. y = [y(x:) for X]"
vector of ground-trute labels, and g [g(x) for X]T vector of labels assigned by certain feature gzz).
Definition 4.2. Availability measures how "easy" it is to learn label for neural network. For discrete case, availability an alignment of label g empirical NTK (eNTK) of K(X X)
A(K,g)
g K X)g
Σ
A
(U
(16)
g ((2
X
|g]1
when K(X, X') can be eigendecomposed into Σ, A.auR.
Thus, availability measures how much top eigenspaces of NTK are aligned to a feature, which shows the convergence speed feature. By measuring availability of shortcut labels, we check if top eigenfunctions of NTK also have larger influence after convergence as noted in Proposition 3.2
4.2 Empirical results
Here, we measure the availability of shortcut labels on real-world datasets to compare to the availability of the ground-truth labels. We used pretrained 18 model for real-world datasets, which is a more complex network. Last-layer weights of both models were initialized as zeros and the bias parameter was removed. The model was trained with an SGD of a learning rate 0.001 and weight decay 0.0001. In the main paper, we only show the results from training with the CE loss, but we also show results from the MSE loss and SD in Appendix A.3
Finding shorteut label. Though a neural network dependent on shortcut features, a label that neural network prefers to learn can be actually different from the label solely from shortcut feature.
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Waterbirds
CelebA
ColebA
1.04
0.:5
0.30
0.25
AQQUSCCe
10.00
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Fitas (groundimmhh
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Grouss
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Teat lahertest
101
107
101
10t
10'
161 10
101
(a)
(b)
(c)
(d)
Figure 6: Availability and test accuracy of ground-truth labels and shortcut labels in two realistic
datasets: Waterbirds [23] and CelebA [16]. The tested model was a pretrained ResNet-18 18. Availability was measured from 500 randomly sampled training data. Both the availability and the accuracy in the test sets were higher for shortcut labels than the ones for ground ground-truth labels. Smoothing was applied to the graphs for visual clarity. Results from five independent runs provided in Appendix (A.4
By inspecting the predictions of a neural network on the test set, we manually found a 'shortcul label' that a neural network actually learns. The shorteut labels from the datasets and their match rates to the test predictions are listed in Appendix[A.1 [A.] We measured the availability of these manually-found shortcut labels for the experiments. Predictability of shorteut label is lower than the ground-truth label by the way the dataset was composed.
Shortcut label has higher availability. The application of these metrics is shown in Figure 6 By the way the shorteut labels were found, the predictions of neural network were closer to shortcut labels in real-world datasets. Mcanwhile, the availability of these shortcut labels was larger than the availability of the ground-truth labels. Since the availability measures the alignment of label top eigenvectors of empirical NTK (eNTK). it is possible to say that the shorteut labels were more aligned with top eigenfunctions of eNTK. The empirical results were in line with the theoretical results of a linear neural network from Proposition 3.1 and Proposition 3.2 More results on synthetic datasets such as Colored-MNIST, Patched-MNIST, and Dogs vs Cats are in Appendix A.2 and A.5
Patched-MMM -NNISIT
Pelxh size
7x7
Avaliabiitty
0.3
5x5
2x33
0.2
0.0
10a
10a
102
102
10*
Ieraaion
Figure 7: The availability of shortcut labels in Patched- MNIST. The size of patch is controlled to control the strength of the shorteut feature. The size varies from x I to x pixels. The availability of the shortcut label was larger when the strength of the short- cut was larger.
Availability can reflect the strength of shorteut. This notion of availability is in line with the one from experiments of Hermann et al. [0]. Hermann et al. artificially manipulated the portion of images that were related to a certain feature to control availability. Similarly, in a linear kernel, the eigenvalues of shortcut features increase when the norms of means from shortcut clusters are large, so availability measures how much portion of input vectors is related to shortcut features. Thus, availability can reflect the strength of shorteut.
We empirically extended this result to the case of other networks. In Batched-MNIST we measured the availability of a two-layer ReLU fully connecco network varying the size of a patch in -Anche x 1, 3 x 3, 5 x 5, and x 7 pixels. The result is shown in Figure 7 When patch was large and the shortcut strength was strong, the availability of the shortcut label was large. This shows that availability can reflect the strength of a shortcut even in a case where
the NTK kernel is not linear.
Conclusion
In this paper. we used the NTK framework to analyze shorteut learning caused by the imbalance in data distribution using a case of linear neural networks. We showed why shortcut features are learned quickly and are more influential on the neural network output than other features even after training. We discussed that the max-margin bias is not the only reason for the dominance of shortcut features on a neural network output. Finally, we proposed metrics to measure how a feature can predict ground-truit -ruth labels and how a feature can be easily learned by network.

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On the other hand, our work has limitations. First, our theoretical analysis is based on a simple case of training linear neural network on Gaussian Mixture model. Second, Proposition 3.1 is based on the NTK framework, which assumes an infinite width of the model. However, we empirically confirmed our results can be extended to general situations. Third, while Proposition 3.1 indicates that eigenvalues scale with the magnitude of the mean vectors, Proposition 3.2 implies that the contribution a feature on the decision boundary scales inversely with the magnitude the mean vectors. This could potentially limit the ability of using the NTK spectrum assess the influence of shortcut bias after network convergence. Another limitation that, as shown in Appendix [A.7] availability might not detect certain shorteut labels if the ground-truth label is predominated by stronger shortcut. These challenges are left for future works and we hope our work could be useful for future works that study the phenomenon of shortcut learning.
6 Acknowledgement
This work was supported in part by the National Research Foundation of Korea (NRF) grant funded by Korean Government [Ministry of Science and ICT (MSIT) under Grant AS-2003-00208224 50%; in part by the Institute of Information and Communications Technology Planning and Evaluation (IITP) grant funded by Korean Government (MSIT) under Grant 2021-0-00180, 10% and Grant 2021-0-00136, 10%; in part by the Information Technology Research Center (ITRC) support Program Supervised by ITTP under Grant ITTP-2021-00 2021-0-01835, 20%; and in part by ITTP under Artificial Intelligence Semiconductor Support Program to Nurture the Best Talents under Grant ITTP-2023-RS- 2023-0025608,, 10%.
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---

A Additional experiments
A.1 Experimental details
Shorteut labels. Though a neural network is dependent on shortcut features, a label that a neural network prefers to learn can be actually different from the label solely from shortcut feature. For the experiments, by looking at the predictions of neural network on the test set, we manually found 'shortcut label' that neural network actually learns using both shorteut features and core features.
First, we grouped the samples into four groups: (Bias-aligned samples labelled as 1], (Bias-aligne samples labelled as -1], (Bias-confficting samples labelled IJ. and [Bias-conflicting samples labelled as -1]. When the network converged, we observed how each group was predicted in the test sets and assigned shortcut label each group the label predicted by the majority of samples in the group. For synthetic datasets, since the task is so simple that the neural network was able to learn core features as well, we composed the label solely from the shortcut features of the datasets.
If the colors of digits are green, then those samples are shortcut-labelled as 1. The rest labelled -1. Predictability of the shortcut label is 0.950.
Colored-MNIST.
MANISTd-MNIST. If the digits are not patched, then those samples are shortcut-labelled as 1. The rest are labelled as 1.. Predictability of the shortcut label is 0.950.
Waterbirds. If the birds are waterbirds and the backgrounds are from water, then those samples are shortcut-labelled as 1. The rest are labelled as -1. Predictability of the shorteut label is 0.987.
CelebA. If the person is blonde and is a woman, then the sample is shortcut-labelled as 1. The rest labelled 11 Predictability of the shortcut label is 0.991.
Dogs vs Cats. the sample contains a cat and a cat dark-colored, then the sample is shortcut- labelled as -1. The rest labelled as 1. Predictability of the shortcut label is 0.975.
Ground-truth
Bias
Waterbirds
CelebA
Dogs vs Cats
1
1
0.9268 = 0.0031
0.8268 + 0.0117
0.9384 + 0.0129
1
-1
0.4720 0.0137
0.4044 0.0133
0.9378 + 0.0112
-1
1
0.2396 = 0.0143
0.0398 = 0.0028
0.7742 + 0.0128
-1
0.0063 = 0.0007
0.0081 = 0.0012
0.0940 + 0.0169
Table The ratio samples predicted as label for each group in the datasets across 5 runs. 'Bias' denotes the label assigned solely from the shortcut feature. Shortcut label is assigned according to the label predicted by the majority of samples in each group.
Experimental settings. Two models were trained for datasets: two-layer ReLU FC network with the width of 256 for synthetic datasets, and a ResNet-18 pretrained with ImageNet for real-world datasets. The two-layer CNN model used for Figure was composed of two convolutional layers with the kernel size of 3. padding of 1, ReLU activation, and fully-connected layer. The models were trained with SGD of a learning for real- world datasets and 1000 for synthetic datasets. The models were learned with various loss functions: a cross-entropy loss with and without SD, and an MSE loss. For SD, the hyperparameter was set 0.1 as in Puli et al. 20 The model was learned for 130 epochs on CelebA and 200 epochs on the rest of the datasets. Availability of ground-truth -ruth labels and shorteut labels was measured by 500 randomly sampled training data. The experiments were run on 2 RTX 3090 GPUs and AMD Ryzen Threadripper PRO 5955WX 16-Cores, for 6 hours for experiments with real- cal-world data, hour for synthetic datasets, and 12 hours for experiments with CelebA.
0.001 and weight decay 0.0001. The batch size was 128
12

---

A.2 Synthetic datasets
Colorod-MNIST
MNIST
Potchod-MINiST
Futche-RRNIST
1.2
0.171
0.5
0.9
0:152
0.4
0.2
Taan
Jest (groard buth
0.125
0.3
Jest (shomour)
0:102
Traie
Ground-r
Ground trutt
Text (grunnd-ruuhy
0.2
Shoftout
Text (shertzuk)
0.4
100
10
3CP
200
20
Baration
Bersston
Raratton
(a) CE
(b)
(c) CE
(d)
Colorad-ANIST
Colared-MNIST
Patched- MNISS
Patched-MNIST
Ground truth Quarteut
0.25
0.0-
0.4
0.8
iground butt)
0.20
6.5-
Rest iground (ues)
(shertcul)
0.35
Rest isheroou
Shortest
Ground-rrtth
0.5
0.20
0.5
305
100
10
107
10"
10?
109
107
10:
Seration
Berstton
teradon
Bersttnn
(e) MSE
(I)
(g) MSE
(h)
Colored-MNIST
Colored-MNiS
Potchee--MNNIC
Patched-MNIIT
Ground-truth Shartest
0.5
0.2
0.8
tnsin
0.35
.round-tr.th
0.8
0.20
0.2
Sho/tbut
0.5
0.55
0.5
10f
201
127
102
107
10'
30
101
100
20
10
10'
10a
101
Rerston
Ecraton
Canation
Moration
(i) SD (2 = 0.1)
(k) SD (2 = 0.1)
(l)
Figure& 8: Availability and test accuracy of ground-truth labels and shorteut labels in two synthetic datasets: Colored-MNIST and PatShed-MNIST. The tested model was two-layer ReLU FC network. Availability was measured from 500 randomly sampled training data. Availability was higher for shortcut labels than the ones for ground- truth labels.
We measured the availability and test accuracies of ground-trutt -tuuh labels and shortcut labels in two synthetic datasets: ColOred-MNIST and Patched-MNIST. The tested model was two-layer ReLU FC network. Availability was measured from 500 randomly sampled training data.
As result, availability of shortcut labels was larger than the availability of ground-truie truth labels, no matter what the loss function was. This also shows that shorteut labels are learned faster due to higher alignment to NTK, which extends the result of Proposition 3.1 to the case of ReLU network. The model was able to partially learn the core feature as well and the test accuracy in shortcut labels was larger only in the early training phase. However, the test accuracies of the model (=s 0.92 for Colored-M -NIIST under all losses, x 0.94 for Matched-MNISS under all losses) were lower than the test accuracy when the model was trained with the original MNIST (> 0.98). This implies that the model was affected by shortcut features and showed lower test accuracies.
Meanwhile, in such simple settings with a two-layer ReLU network, the trajectories of availability in SD resembled the one in MSE. This implies that the dynamics of SD could be mildly approximated by the dynamics of MSE if the regularization strength is high enough and the setting is simple. Another notable thing is that shortcut learning also occurred under the MSE loss and the SD loss, which means that shorteut learning can occur no matter if the margin is maximized or not. This shows that the max- max-margin bias is not the only main cause of shortcut learning even when the activation function is not linear.
13

---

A.3 Real-world datasets
Vidlethirds
1.5
Waterbirds
CelebA
CelebA
0.0
0.50
0.23
0.00
0.25
244
Tran
Trin
Oroanskkrush Stortntt
04
(groandkrutll Tea Ghhrltal)
0.45
Ground mnth
Shorteat
0.85
Iod Igrouns truti
lest IshentcuD
3.
101
10'
101
10'
1D'
107
10
30'
10'
10'
70'
10"
nicraiinn
Beration
Iaration
(a) CE
(b)
(c) CE
Woterrbinds
3.2
Waterbirrs
0.00
Calebsk
Groand brt
1.50
Ca Bebb
0.355
0.55
0:525
Aoursty
0:40
0.09
0:550
0275
0.500
Cround buth
Startor
0:45
0.40
0.32
00000
0.05
Test iground buth
Thein
Teat rotortock)
301
100
107
12
127
30+
107
30
507
10
(c) MSE
( MSE
Watcrbirds
Celob4
Waterbiidd
3.2
0.302
Acurssy
05
0.00
2240040
04
0252
Train (groudd
Ground Duth
0222
Shartou
0.05
(shemroutt
301
127
10
10
10
107
107
16 10
10
(i) SD (A 0.1)
(k) SD (A 0.1)
Figure 9: Availability and test accuracy of ground-truth labels and shorteut labels in two woal-world datasets: Waterbirds and CelebA. The tested model was a pretrained ResNet-18. Availability was measured from 500 randomly sampled training data. Both the availability and the accuracy in the test sets were higher for shortcut labels than the ones for ground-truth labels.
Next, we measured the availability and test accuracies of ground-truth labels and shortcut labels in two real-world datasets: Waterbirds and CelebA. The tested model was pretrained ResNet-18. 18. Availability was measured from 500 randomly sampled training data.
As result, availability of shortcut labels was larger than the availability of ground-truth labels. This also shows that shorteut labels are learned faster due to higher alignment to NTK. The test accuracy of shortcut labels (=s 0.83 for Waterbirds under all losses, 0.953 for CelebA under all losses) was larger than that of ground-truth labels (+= 0.82 for Waterbirds under all losses, z 0.951 for CelebA under all losses), which shows that the output of model was dominated by shortcut features. These empirical results extend the theoretical findings in the case of a linear neural network in Proposition and 3.2
A notable thing is that shortcut learning again occurred under the MSE loss and the SD loss, which shows that shortcut learning can occur matter if the margin is maximized or not. This implies that the max- max-margin bias is not the only main cause of shortcut learning even when the model is complex.
14

---

A.4 Availability from multiple runs
Coltroc-MNIS
PeSChec-MMMTT
Wateroirds
CelebA
0.05
boat Difheetc
0.35
0.61
0.54
0.6
0.03
0.25
0.03
0.30
0.05
2.01
0.03
0.03
107
10a
100
10
201
304
102
102
127
107
10
101
20
20*
10a
102
20
CE
(h)
(c)
0.20
0.05
0.05
Wattesbirds
0.00
Do eeebb
Antibbito Offencc
0.35
0.04
2.03
0.00
0.00
0.30
Antibbatty
0.00
0.06
2.01
0.03
0.01
30"
10'
10"
10"
304
324
10:
10'
301
10
0.50
30"
10'
30'
201
0.00
101
10'
10-
Karabon
Banation
(c) MSE
(f)
@
(h)
0.20
Colered-WNIST
0.05
Petchec-ANIST
0.05
Waterbirds
0.03
Coloba
0.04
0.04
0.00
2.03
0.20
0.30
0.02
0.00
0.31
0.02
0.50
10t
20
20
20
204
0.00
324
1.2
10a
0.50
20'
30
201
-0:23
teracon
Eoratin
Reraton
Eeration
(i) SD (A = 0.1)
(k)
(I)
Figure 10: Availability of shortcut labels minus(-) availability of ground-truth labels in datasets: Colored-MNIST, Patched-MNIST, Waterbirds and CelebA. Availability was measured from 500 randomly sampled training data. The availability was higher for shorteut labels than the one for ground-truth labels.
Though the availability of a shortcut label was consistently higher than the availability of the ground- truth label, the values of availability varied due to the randomness in picking samples for measuring NTK. Thus, in Section[A.3] we did not include error bars for the clarity of the graphs. Instead, we show the results from 5 runs in this section. Since the values of the availability varied, we show the availability of shortcut labels minus(-) availability of ground-truth labels. It is possible to observe that the availability was higher for shortcut labels than the one for ground-truth labels. We also show the results in a tabular form: we show availability of shortcut label availability of ground-truth label the 1000-th SGD update across other runs. As a result, the availability of shorteut label was consistently higher than the availability of ground-truth label.
Loss CE
Waterbirds
CelebA
MNISS
Patched-MNIST
0.0153 = 0.045
0.0203 + 0.0162
0.0977 = 0.0129
0.0160 + 0.0061
MSE
0.0182 = 0.0079
0.0258 + 0.0086
0.0832 = 0.0040
0.0121 - 0.0048
SD Availability of shortcut label availability of ground label at the 1000-th SGD update
0.0182 = 0.0064
0.0164 + 0.0057
0.0728 + 0.0220
0.0140 + 0.0040
Table 2:
across 5 runs. Availability of shortcut label is higher than the availability of ground-truth label.
15

---

A.5 Effect of pretraining
Dogs vs
Dogs vs Cats
Dogs
Ground truth
0.00
Shorteut
0.25-
0.8
0.06
0.20
0.15
Trah
(orsund-truth
04
Text (prund-truth
(ahortzut))
Shortout
Text (shortout)
0.00
100
Paration
Reration
Reration
(a) Un-pretrained model
(b)
(c) Pretrained model
(d)
Figure 11: Availability and test accuracy of ground-truth labels and shortcut labels in Dogs vs Cats dataset. The tested model was an un un-pretrained and a pretrained ResNet-18 18. Availability was measured from 500 randomly sampled training data. Since the pretrained model was already trained with samples of Dogs and Cats in ImageNet, availability was higher for ground-truth labels than the ones for shortcut labels when trained with a pretrained model.
In order to inspect the effect pretraining, we trained ResNet- with both versions that were un- pretrained and pretrained with ImageNet (7]. Here we used Dogs vs Cats, which aims to distinguish dogs and cats but spuriously correlated with colors animals. ImageNet includes images of dogs and cats, therefore pretrained ResNet- 18 already contains information of dogs and cats. The model was trained with the cross-entropy loss.
As a result, availability of shortcut labels for an un un-pretrained model was higher than the one of ground-truth labels the early and late phase of training. Though this result aligns with previous experiments, availability of shortcut labels for pretrained model was consistently lower than the one of ground-truth labels. The prediction of models in the test set was also heavily affected by pretraining. While an un-pretrained model was dependent shortcut labels, pretrained model was more dependent on ground-truth labels. This shows that the initial parameter of a model a critical component of shortcut learning.
A.6 Effect of correlation availability
0.6
Patched-NNIST
We measured the availability of the shorteut la- bel varying the correlation between the shortcut label and the ground-trutt label. Here, the cor- relation was controlled by manipulating the pre- dictability of the shorteut label. As a result, the
Prediiuatilyyy
0.4
0.55
0,02
availability of a shortcut label was higher when the correlation between the labels was higher. This implies that it is easier to learn a shortcut label if the spurious correlation is stronger in the dataset.
10a
107
107
101
10
307
107
(a) Colored-MNIST
(b) Patched-MNIST
Figure 12: Availability of model when the spuri- ous correlation (predictability) is varied. When the label more correlated, availability higher.
16

---

A.7 Limitation of availability due to multiple shortcuts
Vattrbiids
Viatorbircs
Vaterbirds
Waterbirds
0.40
1.5
0.0
0.40
0.55
0.8
0.30
67
0.30
Thtt Teat (sherbouli
0.25
0.5
Iteration
Peraion
(a) First label
(b)
(c) Second label
(d)
Figure 13: Availability and test accuracy of ground-truth labels and shortcut labels in Waterbirds dataset. The tested model was an un un-pretrained ResNet-18. 18. Availability was measured from 500
randomly sampled training data.
We also trained an un-pretrained ResNet-18 on the modified Waterbirds dataset. Here, the sizes of the birds were 40% smaller than the original Waterbirds dataset so that the model would be more affected by the shortcut. Also, the un- un-pretrained model was less affected by the core feature and the shortcut label showed a different accuracy on the test sets. Here, the shortcut labels were composed in two ways. The first shortcut label is identical to the one from the experiment with pretrained model. The second shortcut label is the label solely from the biased attribute: if the bird is on the watery background, then it is labelled as
Both labels showed higher test accuracies than the ground-truth label, however, the speeds at which the labels were learned were different. The first shortcut label was learned faster, while the second label was learned slower than the ground-truth in the early phase of learning. Since the first label was learned quickly, the availability of the first label was larger than the ground-trus truth label. The second label was learned more slowly in the early phase, and the availability of the second label was consistently lower than the ground-truth. However, the test predictions of the model were closer to the second label.
Waterbirds
3.0
0.9
Acoussay
0.5
0.7
Trann
Test Iground-trun (ruth))
06
10a
10'
10?
10
Reestion
We found that this is because the model first learned to classify all the samples as land-bird, due to the majority of label -1. The majority of the land-birds itself became another shortcut. Since the first label and the ground- -truth were closer to the label uniformly composed as -1, the availability of the first label and the ground-trum -truh label was actually larger than the one of the second label. If the ground-truth is affected by multiple shortcuts, then availability cannot detect the shorteut that learned later than the other shortcut. We suspect that the difference in the predictability the reason why the second label more influential than the uniform label in the test predictions of an un un-pretrained model. As implied in Section A.6 the predictability, or spurious correlation other words, heavily affects the learning process. The predictability of the second label, 0.95, is much higher than the one of the uniform label, 0.78, and the test predictions were more affected by the second label. More studies are left for future work.
Figure 14: Comparing
test predictions of model to label uniformly com- posed as -1. The early predictions of model is uniformly composed as -1, which is affected by the shortcut.
17

---

B Shortcuts in Gaussian Mixture Model
Now we provide the proofs for propositions in Section
B.1 Eigenvalues of features
Assume Gaussian Mixture Model of input data
Rd below:
K
p(x) Th (JuR; Ex)
Σ
T
exp exp(- 1
(ux)) [xx
(s - vkk))
(17)
(2
/2/)
For kernel function )(;;), an eigenfunction ((x) must be an inner product with a vector,
<(z) = =x Tv
(18)
Since (xz) an eigenfunction for k(.,)), the eigenfunction with eigenvalue must satisfy
Ax(x)= To TKg(x)= / k(z, (()))))(
(19)
Thi
exp (- 2- m1)) T -xx)))d
(20)
(2Th )d/ Σx 11/2
B4
K
T (s + (s){s 8 + mx)) T
k
Eイ -1/2 exp (- - 21 s)ds
(21)
(2T) d/T)
k= Rd
1 T s
(x 8 x (s T uxx) (2T)d//1E*
exp (
(-25
(22)
-
2
k= Rd
K
(x To T88 T u + T mge s + Me vx T 8 + TT 1u1l
Rd
TTK
1/2 exp (-5 2 Ex s)ds
(23)
(2 (2m))d/2
K
K
E
[s8 "Iu x mhit v
Σ
Tyat Ex + Nxa parte v
(24)
(0,
k=1
For simplicity, we take Σg = I, then
k
AΦ(x) Ax 0
Σ
uax T voo + Txat
(25)
1=
To satisfy the equation above, u should be an eigenvector of Σ Te or perpendicular to μk for k {1...., K}. If D k= THARH ac, then eigenvalue À becomes
A
T +
(26)
18

---

When v is perpendicular to μą for l E (l..... K,, then = O. Now we obtain the normalization factor c. Here we assume that eigenvector v normalized.
c²
| Rd
x
K
k
expl
(x - pa) - pu)
20
)do
(27)
K
Tkk
(xr + us)
TunTTe(z u1x + um) ''epp (-
xt x 2
)d
(28)
(2TOT d/2
Rdd
TTk
x. xt (xJvuT x v maNE u) exp (-
dr
(29)
(2 d/12
_gd
20
k!
K
TROKU v v+a=
(30)
1=1
Thus, the normalization factor c= VX
B.2 Features after convergence
We first need to obtain the general form of the optimal function from Hermann et al. The process identical to the process from Hermann et except that we do not decompose the function into normalized cigenfunctions but into f(x) = - wafu (fx(F) = x ew). The function needs to
minimize the following loss function:
L= / 2
((z))) -(())
/ 1 wxfx(I) p(z)))
(31)
We need to obtain Wy that minimizes the loss function above, which satisfies
① C
m wxfu(x) - ((())((((((
(32)
Owj
1=1
wk S [((=)1((())((o)) J ()())(o)d((()=== =
(33)
and it can be interpreted as
Σ wk / x
f)(=)f(())dd(t)) J (((()(()dp((z).
(34)
Thus, the weights are as follows:
wk = -H where H₁n - [x()) and y, = fx I
(35)
Now assume data of Idd in a Gaussian Mixture Model of p(z) E (ENV of.). A binary label function y(x: E {-1, 1} nearly separates the mixture model that data from clusters with mean Me (c E C) are labelled as 1. Under such assumptions, we obtain H and y.
HGK 1x (1()))((())d((()
Uk
K
Pmo?)d/ exp (- (I m4) (x Ms)
T
dr
(36)
20
K
(x TI Touoo x v/ pale uk) exp (- TT.
T
(2TT 2//2
Jee
20 do
(37)
K
Too 85k + 13
Tilllt 1ok
(38)
19

---

[)(z)u(z)dd(())
(2ToR)aa2 exp (- (x usx) (xr mes)) )dl
Σ
x Uj
20
BEC
(x - pk) T (x - pu))
Σ
1/
(2 exp
The
)dx )r
(39)
20
kEC
Σ
TE
TEl2 Uj
(40)
kEC
RECO
Meanwhile, Js and fa are orthogonal to each other if j * k. therefore we can write as
Ejec Uk - EjEc= Tyll Uk K Σ प.o. + 0i ( Tiplp yox
wkk
(41)
In the case where the means of clusters, μkx, are orthogonal to each other, the direction of Uh does not change over the change of the weight ma- us always becomes μa wa/lluu / l|2 unit vector orthogonal to the means. In this case, only diagonal elements of H remain and the weights become
me lla moo7 +oo7+mmlllu m
if E C otherwise
Wh
Te l
(42)
Due the existence of k== T weight wı becomes larger when गą. becomes larger. This means that features corresponding to heavier clusters have more influence on the output of neural network.
K
B.3 Margin controlling in CE loss
Here, we employ the same decomposition of a neural network output and inspect the case where the margin of the output is controlled. We use the same decomposition of a neural network output as vk). Since SD [19] is originally based cross-entropp loss,
f(z)= k= W&f& (fx(z) first investigate the case where the regularization is done on CE loss. Then, the function needs to minimize the following loss function:
C = / [log(1 + +exp(= +
2 11x))) lddp(x)
/ [log(1 [lo([1+ + exp(-xp(a Σ we (x)x
wh (x.f(()))())((z))
(43)
We need obtain wy that minimizes the loss function above, which satisfies
0 c / 1+ exsp((vv(=)((t) ow
(x)) + m
wn. w,.f(a)((())(doo))
(44)
and it can be interpreted
wxfk(=) = Aua/ / ((x)dp(())
(45)
/ 1+ cxp(- Σ (x) ((e
En (xfa(z))
(46)
since fo and fy are orthogonal to each other if # in the first line. Thus, the weights are as follows:
exp( u((r)) D We ((())
expl v(x) Er_ ((
(47)
Ix (7())pp(z)
20

---

Here, for simplicity, we define a function gzz, A) as
exp( ( k= (xff(z)) g(z, A) = 1 expl 2k= wa (x)x()))
(48)
Now consider the case where À +oo. In this case, all weights wy go 0. Though all weights go to zero, the ratio between those weights, which actually determines the decision boundary, can be different. First, take the limit the function g(x, A), then
lim g(z, A) A-+
(49)
2
We take the limit on the ratio between w, and wy if w/ + O.
Wi
Ix (x))((())
lim
lim
(50)
+00
Wi
2++00
Jx fj
(x))dd lima-++00 fx g(x, (x)d))(
(51)
(2 limx-- (x, (1
Sx (z()dp((1)) lima (0x, ())
(52)
Sx (x())p((z) Jx lim g(x (=)
Sx (x))p((2 Sxx [1mx-+++ 00. ())dp))
(53)
Sx (x))p((z) Jx g( (lle))))))
Sx (x)d((()) Sxx (z)=)f((z)dp(zz) (z)ddd)) fxx (z))(()))
(54)
Sx (=))d((z fx (=))){(())dp(() f(=)dd()) fx v(z)ff(z)dp(l)
Equation 52 is from the dominated convergence theorem, since
exp(-y(<) Ik= (x)x(x))
(55)
+ exp(-y(( En wxfx(z))
and the sequence is dominated by an integrable function.
If you look closer, Equation 54 is equivalent to the ratio of weights trained under an MSE loss. Therefore.
(w))sD (w:)aSE
lim > (W))sD
(56)
(w) MSE
When the original loss function is an MSE loss, need to obtain we that minimizes the loss function,
L = x 1=
1 -(()))(()))² v(z))² + - (
f Σ - 2
A m
Σ wk (x((=))(dp()))
(57)
We need to obtain wy that minimizes the loss function above, which satisfies
i C
wxfx(x) v(x) + wxfh(=))f
(58)
Dw;
k=1
(1 + Njwkk / f((=)f(())d(())) / f)(e)(()dd(()) (59)
=
Therefore, when wy f O, the ratio between weights is
fx (w)d)(p)
Sx (x))dd(() Ix (z))
wi
(1+ Ix (2()))p(z)
(60)
W L (x)d))))))
(((c)d(()) Ix (()))
(1+A) Ix ((())(())
which does not change though there is change in the value of X.
21

---

C
Broader impacts
The goal of our work is understand the reason for shorteut learning and subsidiary phenomena accompanied by shortcut learning. As our work does not propose a new method and only aims to understand gradient-descent-like optimization algorithms, there no downside risk of the research. The upside benefit would be to better understand the principles behind shortcut learning, which has been pointed out as socially problematic [10].
22

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Reviews:
[{"review_id": "ZNbnyHY2Ta", "reviewer": "Reviewer_uhAz", "summary": "Propose a toy model where the problem of shortcut features can be investigated, using a mixture of two Gaussians with biased clusters (unbalanced probabilities) and considerable variance. Demonstrate, using an NTK setting in a linear network, that in this setup, shortcut features get learned earlier and influence the learned predictions. Furthermore, it demonstrates that a previously suggested measure to avoid shortcut features, controlling the classification margin, does not seem to alleviate the problem. Propose measures of availability and predictability to identify shortcut features in real-world datasets and demonstrate that they exhibit similar properties.", "questions": "1. Can you show that the Gaussian mixture model exhibits shortcut features without using the NTK analysis?\n2. Can you quantitatively show the parameters of the GMM under which shortcut feature do or do not show, based on the NTK analysis?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "## Strengths \n 1. A toy model where the problem of shortcut features can be reproduced and against which possible solutions can be tested.\n 2. Use an existing theoretic framework (NTK) in a simplified manner to better understand the learned features which correspond to shortcut features.\n 3. The predictability and availability metrics seem reasonable and useful.\n 4. Test to see if a previously proposed mitigation for shortcut features work on the model, and found it does not.\n\n## Weaknesses\n 1. Unclear logical structure. The manuscript would benefit from a more layered structured, which makes clearer claims. Specifically, it is unclear which results depend on the Gaussian mixture model, which on the NTK analysis, and which do not. I think the structure of the logical argument is:  \n   ** Propose a model of biased Gaussian mixture to test shortcut features (with equations).  \n   ** Show when the model demonstrates shortcut features and when it does not (e.g., yes when biased larger than something and variance larger than something, show figure).  \n   **Apply a previously suggested method to understand the learned features in this model, demonstrating that they exist, are learned faster, and contribute to task performance.  \n   ** Show that a previously suggested solution for shortcut features, controlling prediction margin, is not helpful in this model.\n   **Based on the analysis, suggest measures for identifying shortcut features, availability, and predictability.    \n   ** Replicate the results of real-world datasets, without the model's assumptions.   \n 2. The predictability and availability metrics seem reasonable, to the point that I would guess they already have a name or previous usage.\n 3. The NTK analysis is relevant only in the lazy regime, which is not what happens when neural networks get trained on real-world samples. The derived results provide only an intuitive understanding that needs to be tested.\n4. While it is note that \"the data variance of each cluster is not really small\", the contribution of this phenomenon is not well investigated. The point of Figure 4 is unclear.", "was_revised": true, "final_justification": "The authors have properly answered my questions and clarified things I misunderstood, so I increased my score. The manuscript would still benefit from being updated to include clarifications, fix typos, and correct the experimental results, which contained a mistake.", "rebuttal": "Thank you very much for your constructive review! \nWe first notify you that our equations 21, 22, 23 in Appendix B have typos: $\\exp{(-\\frac{1}{2} (s - \\mu_k)^\\top \\Sigma^{-1}_k (s - \\mu_k))}$ must be $\\exp{(-\\frac{1}{2} s^\\top \\Sigma^{-1}_k s)}$. We are sorry for the inconvenience.\n\n**1. Unclear logical structure. The manuscript would benefit from a more layered structured, which makes clearer claims.**\n\nWe are sorry for our unclear statements. Your statements you wrote about the logical argument are correct. For more information, we clarify our results as below:\nTheorem 3.1 and Theorem 3.2 are based on data following the Gaussian mixture model, while Corollary 3.3 does not necessarily need data to be on a Gaussian mixture model. Those three results need an assumption that the trained model is a linear neural network. Meanwhile, the toy experiments from Fig. 4, Fig. 5, and Fig. 7 are trained on a two-layer ReLU FC network. Saliency maps (Fig. 3) were generated from a two-layer ReLU CNN network and predictability and availability (Fig 6) were measured from a pretrained ResNet-18.\n\n**2. The predictability and availability metrics seem reasonable, to the point that I would guess they already have a name or previous usage.**\n\nYes, especially the availability metric has been widely used to measure the alignment of NTK to the ground-truth label [1], but it has not been used to investigate the phenomenon of shortcut learning.\n\n**3. The NTK analysis is relevant only in the lazy regime, which is not what happens when neural networks get trained on real-world samples. The derived results provide only an intuitive understanding that needs to be tested.**\n\nTheorem 3.1 is relevant in the lazy regime and our analysis on the convergence speeds of features is dependent on the lazy regime assumption. We acknowledge that it is our limitation. However, it is known that the neural network can sometimes exhibit the lazy regime behaviour in the early phase of training, which is our main focus of our analysis on the convergence speeds of features [2]. Also, Theorem 3.2 and Corollary 3.3 do not need an assumption of lazy regime.\n\n**4. While it is note that \"the data variance of each cluster is not really small\", the contribution of this phenomenon is not well investigated. The point of Figure 4 is unclear.**\n\nThe fact that the data variances of the clusters are not small is used in Theorem 3.2, which implies that shortcut features dominate the decision boundary when the variances are not zero. The point of Fig. 4 is to empirically prove Theorem 3.2 and show the effect of data variances on the decision boundary.\n\nWe also answer your questions below:\n\n**1. Can you show that the Gaussian mixture model exhibits shortcut features without using the NTK analysis?**\n\nIn Theorem 3.2, we show that the Gaussian mixture model exhibits shortcut features after convergence without using any NTK analysis. (Unfortunately, we currently do not provide an analysis on convergence speeds of features during convergence without NTK analysis.)\n\n**2. Can you quantitatively show the parameters of the GMM under which shortcut feature do or do not show, based on the NTK analysis?**\n\nIf the data distribution follows the Gaussian mixture model of $p(x) = \\sum^K_{k=1} \\pi_k \\mathcal{N}(\\mu_k, \\sigma^2_K I)$ and $\\sigma_k > 0$, then the imbalance in the weights $\\pi_k$ can trigger shortcut learning after convergence. Spectral bias can occur even when $\\sigma_k = 0$.\n\n\n[1] Baratin, Aristide, et al. \"Implicit regularization via neural feature alignment.\" International Conference on Artificial Intelligence and Statistics. PMLR, 2021.\n\n[2] Lyu, Kaifeng, et al. \"Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking.\" The Twelfth International Conference on Learning Representations."}, {"review_id": "NYOOcuO0CG", "reviewer": "Reviewer_jAtx", "summary": "Introduces an NTK-based theory explaining why shortcut (spurious) features dominate training: clusters with higher sample weight π acquire larger NTK eigen-values, are fitted fastest, and retain the largest post-training influence. \n\nProves (Theorems 3.1–3.2) these properties for linear networks; then shows empirically that two-layer ReLU nets and ResNet-18 exhibit the same behaviour on Patched-/Colored-MNIST, Waterbirds, CelebA, Dogs vs Cats.\n\nDefines predictability (alignment to ground-truth) and availability (alignment to top-NTK eigen-vectors) and demonstrates that shortcut labels have low predictability but high availability across five biased benchmarks. \n\nArgues, both theoretically (Cor. 3.3) and with 2-D toy data, that margin-control methods (SD / Marg-Ctrl) cannot fully remove shortcut bias.", "questions": "Finite-width drift – Could you run a width-sweep (e.g. 256-2 k channels) on Patched-MNIST to show when NTK approximation breaks?\n\nStatistical confidence – Please repeat each benchmark over ≥5 seeds and report mean ± s.d.; how stable is the availability ranking?\n\nShortcut-label sensitivity – How do availability curves change if shortcut labels are derived automatically (e.g. k-means on top eigen-projections) instead of manual inspection?\n\nVariance term in Thm 3.2 – Provide a concrete bound showing when a small but high-variance cluster can outweigh a large low-variance one.\n\nLate-epoch NTK – Have you measured availability using the empirical NTK after feature learning (e.g. epoch 50 ResNet-18) to see if shortcut alignment grows?", "limitations": "Authors acknowledge linear-kernel and infinite-width assumptions but societal-impact section is brief; suggest expanding on potential mis-use of availability for data filtering.", "rating": 5, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Quality**\nStrength\n1. Solid linear-kernel proofs, clearly laid out in Appendix B.\n2. Comprehensive experiments (5 real, 2 synthetic datasets) validate theory.\n\nWeakness\n\n3. All curves come from single runs; no confidence intervals reported.\n\n4. Manual construction of shortcut labels may inject subjectivity.\n\n**Clarity**\n\nStrength \n\n1 Helpful visuals (Fig. 3 saliency, Fig. 4-5 decision boundaries) illuminate ideas.\n\nWeakness\n\n2 Main text is heavy with notation; many key plots relegated to appendix.\n\n**Significance**\n\nStrength\n\n1. Bridges spectral bias and shortcut learning, highlighting cluster variance as a new factor.\n\nWeakness\n\n2. Results remain in infinite-width/lazy-training regime; impact on feature-learning settings uncertain.\n\n**Originality**\n\nStrength\n\n1. Novel availability metric linking shortcut strength to NTK spectrum; challenges margin-bias narrative.\n\nWeakness\n\n2. Core insight “large eigen-values ⇒ faster learning” is rooted in prior spectral-bias work, limiting novelty. \n3. Core contribution also overlaps with Learnability in the Context of NTK: because mathematically Tr K = ∑ λᵢ, their large-k(x,x) “easy” samples are exactly authors large-λ “shortcuts”.", "was_revised": true, "final_justification": "I appreciate the authors' efforts to address my concerns. I raised my score to 5.", "rebuttal": "Thank you very much for your constructive review! \nWe first notify you that our equations 21, 22, 23 in Appendix B have typos: $\\exp{(-\\frac{1}{2} (s - \\mu_k)^\\top \\Sigma^{-1}_k (s - \\mu_k))}$ must be $\\exp{(-\\frac{1}{2} s^\\top \\Sigma^{-1}_k s)}$. We are sorry for the inconvenience.\n\n**1. All curves come from single runs; no confidence intervals reported.**\n\nThough the availability of a shortcut label was consistently higher than the availability of the ground-truth label, the values of availability varied due to the randomness in picking samples for measuring NTK. Thus, we did not include error bars for the clarity of the graphs. Due to the limited time, we cannot run all experiments with multiple runs during our rebuttal period: but we will try to run important experiments such as Waterbirds and CelebA on CE loss to show you that our results are consistent. Though the values can vary due to the randomness of picking training samples for measurement of NTK, the availability of a shortcut label was consistently higher than that of ground-truth label. Also, in the experiment measuring the strength of a shortcut (Fig. 7), a stronger shortcut always exhibited higher availability. The experiments below were run with CE loss. The availability was measured at the 1000-th iteration to observe the convergence behaviour in the early phase of training (which was also our focus in Fig. 6, 9, and 10)\n## Waterbirds \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.3669 | 0.2700 | 0.3480 | 0.4049 | 0.2996 | \n| Ground-truth label | 0.3496 | 0.2576 | 0.3325 | 0.3827 | 0.2906 | \n## CelebA \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.6300 | 0.5895 | 0.5886 | 0.5767 | 0.5958 | \n| Ground-truth label | 0.6257 | 0.5382 | 0.5707 | 0.5611 | 0.5833 | \n## Colored-MNIST \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.5317 | 0.5360 | 0.5170 | 0.5347 | 0.5191 | \n| Ground-truth label | 0.4370 | 0.4248 | 0.4032 | 0.4472 | 0.4380 | \n## Patched-MNIST \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.1628 | 0.1686 | 0.1751 | 0.1739 | 0.1684 | \n| Ground-truth label | 0.1555 | 0.1578 | 0.1533 | 0.1567 | 0.1454 |\n\nOn the other hand, for the experiments measuring the strength of a shortcut in Patched-MNIST (Fig. 7), the variance of availability was small (across 5 runs):\n| Size of patch | 1x1 | 3x3 | 5x5 | 7x7 | \n|---------------|----------------|----------------|----------------|----------------| \n| Shortcut label | 0.0605 ± 0.0042 | 0.1698 ± 0.0049 | 0.2861 ± 0.0038 | 0.3721 ± 0.0085 |\n\nIf given a chance to revise our paper, we will reflect your comment and include an error bar for this experiment.\nThe experiments below were run with MSE loss.\n## Waterbirds \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.2040 | 0.2342 | 0.1910 | 0.2406 | 0.2777 | \n| Ground-truth label | 0.1830 | 0.2078 | 0.1745 | 0.2173 | 0.2739 | \n## CelebA \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.4466 | 0.4572 | 0.3853 | 0.3861 | 0.4841 | \n| Ground-truth label | 0.4140 | 0.4230 | 0.3715 | 0.3691 | 0.4527 | \n## Colored-MNIST \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.5334 | 0.5529 | 0.5379 | 0.5529 | 0.5550 | \n| Ground-truth label | 0.4479 | 0.4670 | 0.4599 | 0.4742 | 0.4672 | \n## Patched-MNIST \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.1695 | 0.1954 | 0.1528 | 0.1648 | 0.1718 | \n| Ground-truth label | 0.1544 | 0.1786 | 0.1431 | 0.1610 | 0.1566 |\n\n**2. Manual construction of shortcut labels may inject subjectivity.**\n\nThough we manually constructed shortcut labels, the assignment of labels does not depend on our subjective decision: for real-world datasets, we grouped the samples into four groups: [Bias-aligned samples labelled as 1], [Bias-aligned samples labelled as -1], [Bias-conflicting samples labelled as 1], and [Bias-conflicting samples labelled as -1]. We observed how each group was predicted in the test sets and assigned a shortcut label to each group as **the predicted label of the majority of samples in the group** (if the majority of the group was predicted as 1, then the shortcut label becomes 1 for this group). In Fig. 6, 9, and 10, the test predictions of the network were closer to our constructed shortcut labels rather than the original ground-truth labels, which shows that our constructed shortcut labels are valid. More explanation is in Appendix A.1.\nThis is purely based on the statistics of the predictions not our subjective decisions. As in an example in Waterbirds, when the label was manually found, the match rate of the test predictions to the found label was **0.7819 ± 0.0018** (measured at > 100 epochs across 5 runs). On the other hand, when the label was composed solely from the shortcut feature, the match rate of the test predictions to the composed label was **0.7488 ± 0.0029** (across 5 runs) which was lower than the aforementioned one. In an example in CelebA, when the label was manually found, the match rate of the test predictions to the found label was **0.9527 ± 0.0003** (measured at > 100 epochs across 5 runs). On the other hand, when the label was the ground-truth label, the match rate of the test predictions to the ground-truth label was **0.9510 ± 0.0004** (across 5 runs) which was lower than the aforementioned one. In CelebA, we compared the match ratio of the shortcut label to the match ratio of the ground-truth label since the match rate to the label solely from the shortcut feature was much lower than the one to the ground-truth label. This shows that our shortcut label is reliable.\n\n**3. Results remain in infinite-width/lazy-training regime; impact on feature-learning settings uncertain.**\n\nTheorem 3.1 assumes the infinite-width/lazy-training regime, which is the limitation of our work. However, it is known that the neural network can sometimes exhibit the lazy regime behaviour in the early phase of training, which is the reason our analysis focuses on the convergence speeds of features [1]. On the other hand, Theorem 3.2 and Corollary 3.3, which show the behaviour of shortcut features after convergence of the network, are not based on the constant NTK assumption.\n\nWe also answer your questions below:\n\n**1. Finite-width drift – Could you run a width-sweep (e.g. 256-2 k channels) on Patched-MNIST to show when NTK approximation breaks?**\n\nWe measured availability for two-layer ReLU FC networks with various widths: 32, 64, 256, 512, 1024, and 2048. Availability of the shortcut label was consistently higher than the availability of the ground-truth label.\n| Width | 32 | 64 | 256 | 512 | 1024 | 2048 | \n|-------------------|--------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.1726 | 0.1625 | 0.1686 | 0.1684 | 0.1494 | 0.1138 | \n| Ground-truth label | 0.1637 | 0.1539 | 0.1578 | 0.1571 | 0.1430 | 0.1050 |\n\n**2. Statistical confidence – Please repeat each benchmark over ≥5 seeds and report mean ± s.d.; how stable is the availability ranking?**\n\nWe showed the partial results above: though the value of the availability itself could be unstable, the availability of a shortcut label was consistently larger than the availability of the ground-truth label.\n\n**3. Shortcut-label sensitivity – How do availability curves change if shortcut labels are derived automatically (e.g. k-means on top eigen-projections) instead of manual inspection?**\n\nWe are not sure if we understood your question correctly, but if you mean automatic derivation by composing the label from k-means clustering on data projected to principal components, then the availability varied across runs. We experimented on Waterbirds and the label was constructed from 2-means clustering on data projected to principal components. Below shows the result:\n## Waterbirds \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------------|--------|--------|--------|--------|--------| \n| Ground-truth label | 0.3631 | 0.3481 | 0.2902 | 0.3498 | 0.3466 | \n| K-means label | 0.0878 | 0.0836 | 0.0460 | 0.0419 | 0.0409 |\nWe suspect that the labels from k-means clustering were far from the ground-truth label.\n\n**4. Variance term in Thm 3.2 – Provide a concrete bound showing when a small but high-variance cluster can outweigh a large low-variance one.**\n\nIf the means of the clusters are orthogonal to each other and the norms of the means are identical, a small but high-variance cluster cannot outweigh a large low-variance one. The variance term $\\sum^K_{i=1} \\pi_i \\sigma^2_i $ in $w_k$ includes the variance of all clusters and $w_k$ is not dependent only on the variance of a single cluster. If at least one cluster has a positive variance, then a large cluster always outweighs small clusters.\n\n**5. Late-epoch NTK – Have you measured availability using the empirical NTK after feature learning (e.g. epoch 50 ResNet-18) to see if shortcut alignment grows?**\n\nAll the experiments in Fig. 6, 9, and 10 show the alignment of empirical NTK after epoch 100 (except CelebA that was run for 130 epochs, all the experiments were run for 200 epochs). The availability grew in experiments on synthetic datasets, but it did not in experiments on real-world datasets.\n\n[1] Lyu, Kaifeng, et al. \"Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking.\" The Twelfth International Conference on Learning Representations."}, {"review_id": "ZKhBGdMWCw", "reviewer": "Reviewer_TcGE", "summary": "The paper proposes an analysis of shortcut learning resulting from the dominance of biased attributes in the data for the case of linear networks using NTK theory, by tying the network features to the eigenfunctions of the NTK and measuring the influence of each eigenfunction to the neural network outputs.\n\nThe central parts of the theoretical argument are two theorems, formulated for the case of linear neural networks and datasets generated from mixtures of Gaussians with isotropic covariances:\n\n**Theorem 3.1** quantitatively inspects the spectral bias by relating the convergence speed of each feature direction to the eigenvalues​ of the between-means covariance matrix M. It states that the eigenfunctions of the integral operator defined for the inner product kernel are linear projections on the eigenvectors of M and the eigenvalues decompose in the sum of average within-cluster variances, which is a constant term, and the eigenvalues associated with the eigenvectors of M. The theorem provides insight on why shortcut features (which, in the considered scenario are aligned with large cluster mixing probabilities), corresponding to larger M eigenvalues, are fitted faster: their kernel eigenvalues are large. In contrast, discriminative features that correspond to either small or similar clusters show a slower learning speed.\n\n**Theorem 3.2** further shows that, under training with MSE loss, eigenfunctions of the kernel operator corresponding to clusters with larger mixing weights have a higher influence on the network output. Coupled with theorem 3.1, it states that shortcut features (which correspond to larger kernel eigenvalues) are fitted early and dominate the network predictions after convergence. One important note from the authors is that the impact of bias feature after convergence depend not only on the cluster mixing weights but also on the variances of the samples, highlighted empirically on a toy example.\n\nThe theoretical results and experiments on toy data additionally challenge the idea that maximal margin control (for example via SD) alone can eliminate the shortcut bias, by stating that the SD decision boundary converges to the MSE decision boundary in the infinite-limit of the regularization coefficient, which is affected by shortcut learning. Further, the authors provide metrics for the concepts of predictability (how well a feature is aligned with the ground truth) and availability (the alignment of the feature with the top eigenspace of NTK) and empirically apply them on real datasets.", "questions": "* Given the spectral bias analysis, I am curious whether you have thought about how the new spectral bias insights (and the metrics for availability and predictability) might inform designing further debiasing strategies?\n* Since the theory assumes a static NTK (which holds for infinite-width limit), have you observed a particular width threshold beyond which the spectral bias observations tend to emerge in the experiments?\n* As mentioned in the Weaknesses section, for the toy experiments highlighting theorems 3.1 and 3.2, has the norm of the cluster means been controlled in some way in order to observe its effect on the eigenvalue scales?\n* Can some aspects from the theoretical argument be directly applied from linear NTK to other types of kernels?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**:\n* The authors take an interesting perspective on shortcut features through the lens of NTK theory and advances the theoretical understanding\n* The theoretical argument (restricted to the case of linear networks and GMM data) provides explicit forms for the eigenfunctions and weights as functions of the dataset mixing probabilities\n* The results improve understanding on debiasing, by first challenging the idea that maximal margin alone cannot eliminate shortcut biases and by additionally highlighting the impact of within-cluster variance on both spectral bias and feature weights\n* The experimental setup is extended beyond the limited regime considered in the theoretical argument (linear networks, GMM data) to real data and networks and empirically validates the spectral bias properties through newly introduced metrics for predictability and availability\n* The results might potentially provide insights applicable for the study of interpretability, as shown in Figure 3 where saliency maps for directions with high and low eigenvalues are tied to human-interpretable features\n\n**Weaknesses**:\n* In the follow up analysis of the theorem 3.1, it is implied that eigenvalues depend primarily on the cluster mixing weights (\"Since eigenvectors close to a cluster with a larger weight pk have larger eigenvalues, an inner product with a vector from a larger cluster has a larger eigenvalue and converges faster due to the spectral bias\"). However, it follows from Theorem 3.1 that eigenvalues scale with both mixing weights and the magnitude of the mean vectors (concretely, with pk * ||uk||). Therefore, in scenarios where a cluster with a high mixing weight has a small-norm mean, its direction might not dominate the spectral bias anymore. I recommend clarifying this aspect in the analysis, as it potentially restricts the theoretical setup even further to datasets where the mean vectors have approximately equal norms, further drifting away from practice.\n* Some of the limitations already pointed out by the authors: highly limiting regime in the theoretical analysis, with respect to data generation and class of models, as well as limitation of availability due to multiple shortcuts. While the empirical results show agreement with theory, the gap between the two is still considerable.\n* Theory relies on static NTK assumption that only holds in the infinite-width limit", "was_revised": true, "final_justification": "Key updates in the rebuttal include the following points :\n* reiterates where the GMM-data and constant-NTK premises are assumed in Theorems 3.1 and 3.2 and Corollary 3.3\n* discussions on the case of small, high-variance clusters (discussion can be found in the replies to reviewers uhAz and jAtx)\n* supplies preliminary confidence-interval experiments across multiple runs, strengthening the empirical claims (discussion can be found in the replies to reviewers jAtx and o9y3)\n* tests on how measures of availability varies with network width, confirming that the availability of the shortcut label is consistently higher than the availability of the ground-truth.\n\nHowever, I also emphasized in my discussion with the authors that, even if the empirical evaluation regarding Theorem 3.2 focuses exclusively on the role of the mixing weights, the paper should still discuss more clearly that eigenvalues are influenced by both the mixing weights and the magnitude of the mean vectors, a consequence deriving from Theorem 3.1. I recommended including this discussion such that the influence of mean-vector norms is not overlooked, as equal mean vectors constitutes another limiting assumption of the setup. I appreciate that the authors plan to detail this discussion for the final version.\n\nFollowing the rebuttal, I have decided to increase my rating.", "rebuttal": "Thank you very much for your constructive review! \nWe first notify you that our equations 21, 22, 23 in Appendix B have typos: $\\exp{(-\\frac{1}{2} (s - \\mu_k)^\\top \\Sigma^{-1}_k (s - \\mu_k))}$ must be $\\exp{(-\\frac{1}{2} s^\\top \\Sigma^{-1}_k s)}$. We are sorry for the inconvenience.\n\n**1. I recommend clarifying this aspect in the analysis, as it potentially restricts the theoretical setup even further to datasets where the mean vectors have approximately equal norms, further drifting away from practice.**\n\nFirstly, both toy experiments in Fig. 4 and 5 were conducted with orthogonal clusters with means of norm 1. The norms of the means of clusters were identical and the effect of the norm was decoupled. While we were also aware of the effect of the norm of the mean, we did not elaborate on the subject since our focus was on the weight $\\pi_k$. If given a chance to revise our paper, we will reflect your comment and emphasize more on the effect of norms of the means to the decision boundary of a neural network.\n\n**2. Theory relies on static NTK assumption that only holds in the infinite-width limit.**\n\nTheorem 3.1 assumes the static NTK assumption, which is the limitation of our work. However, it is known that the neural network can sometimes exhibit the lazy regime behaviour in the early phase of training, which is a focus of our analysis on the convergence speeds of features [3]. On the other hand, Theorem 3.2 and Corollary 3.3, which show the decision boundary of a network after convergence of the network, are not based on the constant NTK assumption.\n\nWe also answer your questions below:\n\n**1. Given the spectral bias analysis, I am curious whether you have thought about how the new spectral bias insights (and the metrics for availability and predictability) might inform designing further debiasing strategies?**\n\nOur new spectral bias insights might aid finding possible shortcut labels inherent in the dataset. Since shortcut features are learned faster than core features, one can construct shortcut labels by looking at the predictions of the network in the early phase of training, which is already done in a famous debiasing method called LfF [1].\n\n**2. Since the theory assumes a static NTK (which holds for infinite-width limit), have you observed a particular width threshold beyond which the spectral bias observations tend to emerge in the experiments?**\n\nWe could not particularly find a case where our observations were not valid. We show our results on Patched-MNIST and a two-layer ReLU FC network when the width of the network is small as 32 or 64: availability of the shortcut label was consistently higher than the availability of the ground-truth label.\n| Width | 32 | 64 | 256 | 512 | 1024 | 2048 | \n|-------------------|--------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.1726 | 0.1625 | 0.1686 | 0.1684 | 0.1494 | 0.1138 | \n| Ground-truth label | 0.1637 | 0.1539 | 0.1578 | 0.1571 | 0.1430 | 0.1050 |\n\n**3. As mentioned in the Weaknesses section, for the toy experiments highlighting theorems 3.1 and 3.2, has the norm of the cluster means been controlled in some way in order to decouple its effect on the eigenvalue scales?**\n\nBoth toy experiments in Fig. 4 and 5 were conducted with orthogonal clusters with means of norm 1. The norms of the means of clusters were identical and the effect of the norm was decoupled.\n\n**4. Can some aspects from the theoretical argument be directly applied from linear NTK to other types of kernels?**\n\nYou cannot directly apply the results to NTK from a ReLU network, but in some cases where eigenfunctions of a ReLU NTK include linear functions [2], our results might be valid for linear eigenfunctions in those simple cases.\n\n[1] Nam, Junhyun, et al. \"Learning from failure: De-biasing classifier from biased classifier.\" Advances in Neural Information Processing Systems 33 (2020): 20673-20684.\n[2] Hermann, Katherine L., et al. \"On the foundations of shortcut learning.\" arXiv preprint arXiv:2310.16228 (2023).\n[3] Lyu, Kaifeng, et al. \"Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking.\" The Twelfth International Conference on Learning Representations."}, {"review_id": "ubDSfN7pSE", "reviewer": "Reviewer_o9y3", "summary": "The paper investigates shortcut learning in deep learning models, where networks overly rely on dominant but non-generalizable features in the training data. Using the Neural Tangent Kernel (NTK) framework, the authors analyze linear neural networks and define a “feature” as an eigenfunction of the NTK. \nThey find that shortcut features align with large NTK eigenvalues, particularly in settings with imbalanced or clustered data distributions. These dominant features significantly affect the model output even after training, suggesting that shortcut learning isn't solely due to max-margin biases. The insights are also shown to extend to nonlinear networks, like two-layer ReLU networks and ResNet-18.", "questions": "1. The theoretical analysis relies heavily on the NTK framework (infinite-width, lazy training). In practice, finite-width networks often exhibit feature learning.  Is there any empirical observations or theoretical intuitions about how the findings generalize to feature-learning regimes?\n\n2. The shortcut labels in real-world datasets are manually constructed based on model predictions. How consistent are these across runs or architectures? Could a data-driven or automated method for shortcut label discovery be developed using availability scores?", "limitations": "Theoretical results rely on idealized settings (linear networks, infinite width, Gaussian mixture data), which limits applicability.\n\nThe empirical results do not provide statistical significance analysis.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The paper presents a thorough theoretical analysis using the Neural Tangent Kernel (NTK) framework to dissect shortcut learning, starting from linear models and extending the insights to more complex architectures. It provides formal theorems and proofs (e.g., Theorem 3.1 and 3.2) that explain how shortcut features arise due to spectral properties of NTK and data distribution characteristics.\n\n2. Introduces a clean interpretation of features as NTK eigenfunctions and relates shortcut features to eigenvalues, thereby linking shortcut learning to fundamental learning dynamics like spectral bias.\n\n3. The paper empirically shows that findings from linear settings extend to two-layer ReLU networks and ResNet-18, increasing the real-world relevance of the theoretical insights.\n\nWeaknesses:\n\n1. Core theoretical results are derived under simplifying assumptions: linear networks, Gaussian mixture models, infinite-width limit. While extensions to complex models are attempted empirically, theoretical generality is limited.\n\n2. The paper does not include error bars or statistical tests for empirical results. This undermines confidence in the generality of findings, especially on small or noisy datasets.", "was_revised": true, "final_justification": "The authors provided a bunch of empirical results provide more solid evidence on the claims. Also, the authors clarified most of the questions and concerns I have. Overall, I think it is a solid paper with all the extended results.", "rebuttal": "Thank you very much for your constructive review! \nWe first notify you that our equations 21, 22, 23 in Appendix B have typos: $\\exp{(-\\frac{1}{2} (s - \\mu_k)^\\top \\Sigma^{-1}_k (s - \\mu_k))}$ must be $\\exp{(-\\frac{1}{2} s^\\top \\Sigma^{-1}_k s)}$. We are sorry for the inconvenience.\n\n**1. The paper does not include error bars or statistical tests for empirical results. This undermines confidence in the generality of findings, especially on small or noisy datasets.**\n\nThough the availability of a shortcut label was consistently higher than the availability of the ground-truth label, the values of availability varied due to the randomness in picking samples for measuring NTK. Thus, we did not include error bars for the clarity of the graphs. But we will try to run important experiments such as Waterbirds and CelebA on CE loss to show you that our results are consistent.\nThough the values can vary due to randomness of picking training samples for measurement of NTK, the availability of shortcut labels was consistently higher than the one of ground-truth labels. Also, in the experiment measuring the strength of a shortcut (Fig. 7), a stronger shortcut always exhibited higher availability. The experiments below were run with CE loss. The availability was measured at the 1000-th iteration to observe the convergence behaviour in the early phase of training (which was also our focus in Fig. 6, 9, and 10)\n## Waterbirds \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.3669 | 0.2700 | 0.3480 | 0.4049 | 0.2996 | \n| Ground-truth label | 0.3496 | 0.2576 | 0.3325 | 0.3827 | 0.2906 | \n## CelebA \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.6300 | 0.5895 | 0.5886 | 0.5767 | 0.5958 | \n| Ground-truth label | 0.6257 | 0.5382 | 0.5707 | 0.5611 | 0.5833 | \n## Colored-MNIST \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.5317 | 0.5360 | 0.5170 | 0.5347 | 0.5191 | \n| Ground-truth label | 0.4370 | 0.4248 | 0.4032 | 0.4472 | 0.4380 | \n## Patched-MNIST \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.1628 | 0.1686 | 0.1751 | 0.1739 | 0.1684 | \n| Ground-truth label | 0.1555 | 0.1578 | 0.1533 | 0.1567 | 0.1454 |\n\nOn the other hand, for the experiments measuring the strength of a shortcut in Patched-MNIST (Fig. 7), the variance of availability was small (across 5 runs):\n| Size of patch | 1x1 | 3x3 | 5x5 | 7x7 | \n|---------------|----------------|----------------|----------------|----------------| \n| Shortcut label | 0.0605 ± 0.0042 | 0.1698 ± 0.0049 | 0.2861 ± 0.0038 | 0.3721 ± 0.0085 |\n\nIf given a chance to revise our paper, we will reflect your comment and include an error bar for this experiment.\nThe experiments below were run with MSE loss.\n## Waterbirds \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.2040 | 0.2342 | 0.1910 | 0.2406 | 0.2777 | \n| Ground-truth label | 0.1830 | 0.2078 | 0.1745 | 0.2173 | 0.2739 | \n## CelebA \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.4466 | 0.4572 | 0.3853 | 0.3861 | 0.4841 | \n| Ground-truth label | 0.4140 | 0.4230 | 0.3715 | 0.3691 | 0.4527 | \n## Colored-MNIST \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.5334 | 0.5529 | 0.5379 | 0.5529 | 0.5550 | \n| Ground-truth label | 0.4479 | 0.4670 | 0.4599 | 0.4742 | 0.4672 | \n## Patched-MNIST \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.1695 | 0.1954 | 0.1528 | 0.1648 | 0.1718 | \n| Ground-truth label | 0.1544 | 0.1786 | 0.1431 | 0.1610 | 0.1566 |\nThe experiments below were run with CE loss and SD regularization.\n## Waterbirds \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.2263 | 0.2237 | 0.1935 | 0.2318 | 0.2362 | \n| Ground-truth label | 0.2003 | 0.2122 | 0.1818 | 0.2155 | 0.2108 | \n## CelebA \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.4824 | 0.4093 | 0.4607 | 0.4243 | 0.4203 | \n| Ground-truth label | 0.4662 | 0.4015 | 0.4350 | 0.4068 | 0.4053 | \n## Colored-MNIST \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.5217 | 0.5533 | 0.5590 | 0.5293 | 0.5465 | \n| Ground-truth label | 0.4499 | 0.4548 | 0.4928 | 0.4934 | 0.4551 | \n## Patched-MNIST \n| | 1 | 2 | 3 | 4 | 5 | \n|--------------------|--------|--------|--------|--------|--------| \n| Shortcut label | 0.1732 | 0.1542 | 0.1616 | 0.1572 | 0.1742 | \n| Ground-truth label | 0.1620 | 0.1453 | 0.1431 | 0.1447 | 0.1553 |\n\nWe also answer your questions below:\n\n**1. The theoretical analysis relies heavily on the NTK framework (infinite-width, lazy training). In practice, finite-width networks often exhibit feature learning. Is there any empirical observations or theoretical intuitions about how the findings generalize to feature-learning regimes?**\n\nAll the plots in Fig. 6, 9, and 10 are measurements from empirical NTK of finite-width networks. They were measured at >= 100 epoch (130 epoch for CelebA, 200 epoch for the rest). This empirically implies that our findings could be generalizable to feature-learning regimes. Also, one might extend our results with perturbation theory and dynamical mean-field theory, which might need more effort.\n\n**2. The shortcut labels in real-world datasets are manually constructed based on model predictions. How consistent are these across runs or architectures? Could a data-driven or automated method for shortcut label discovery be developed using availability scores?**\n\nThe shortcut labels were consistent across runs. As in an example in Waterbirds, when the label was manually found, the match rate of the test predictions to the found label was **0.7819 ± 0.0018** (measured at > 100 epochs across 5 runs). On the other hand, when the label was composed solely from the shortcut feature, the match rate of the test predictions to the composed label was **0.7488 ± 0.0029** (across 5 runs) which was lower than the aforementioned one. In an example in CelebA, when the label was manually found, the match rate of the test predictions to the found label was **0.9527 ± 0.0003** (measured at > 100 epochs across 5 runs). On the other hand, when the label was the ground-truth label, the match rate of the test predictions to the ground-truth label was **0.9510 ± 0.0004** (across 5 runs) which was lower than the aforementioned one. In CelebA, we compared the match ratio of the shortcut label to the match ratio of the ground-truth label since the match rate to the label solely from the shortcut feature was much lower than the one to the ground-truth label. This shows that our shortcut label is reliable. But the shortcut labels were not consistent across parameters or architectures. For example, as shown in A.4, the shortcut labels might be different if the network is pretrained. \nAnd yes, it might be possible to construct a shortcut label from availability scores or from top eigenvectors of NTK (by constructing top eigenvectors of NTK as shortcut labels). But there is a limitation which was also noted in Appendix A.6: if there are multiple shortcuts in the datasets, it might not be possible to detect all the shortcut labels from the datasets."}]

Author rebuttal:

Meta-review: The paper formally connects the widely observed phenomenon of shortcut learning to spectral bias, proposing that shortcut features correspond to NTK eigenfunctions with the largest eigenvalues.  The authors supported their theory with extensive experiments across a wide range of datasets and models. The work also argues that max-margin bias is not the only cause of shortcut learning, and how that the phenomenon persists even when the network's margin is controlled. Further, the authors demonstrate that data variance within clusters also plays an important role in learning shortcut features.

Originally, the results were reported for a single run, but during the rebuttal the authors extended the results and repeated the experiments for more runs. Another concern noted by the reviewers is that the theory relies on a linear network in an infinite-width regime. To address this concern, the authors showed that their findings empirically hold across different architecture sizes.
Decision: Accept (spotlight)