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Shortcut Features as Top Eigenfunctions of NTK: A Linear Neural Network Case and More

Jinwoo Lim, Suhyun Kim, Soo-Mook Moon

TL;DR

This work analyzes shortcut learning through the Neural Tangent Kernel (NTK) lens, showing that shortcut features align with NTK eigenfunctions possessing large eigenvalues in imbalanced, clustered data. Through a linear network on a Gaussian Mixture Model and extensions to two-layer ReLU networks and ResNet-18, it demonstrates that shortcut features converge quickly and retain substantial influence on predictions, even when margins are debiased. It introduces predictability and availability metrics to quantify how learnable and NTK-aligned features are, and finds that shortcut labels exhibit higher availability and can persist across real-world datasets like Waterbirds and CelebA. The findings suggest that max-margin bias is not the sole driver of shortcut learning and highlight the role of data variance and NTK spectrum in shaping feature influence, with practical implications for debiasing strategies and evaluation metrics.

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 fully-connected ReLU network and a ResNet-18.

Shortcut Features as Top Eigenfunctions of NTK: A Linear Neural Network Case and More

TL;DR

This work analyzes shortcut learning through the Neural Tangent Kernel (NTK) lens, showing that shortcut features align with NTK eigenfunctions possessing large eigenvalues in imbalanced, clustered data. Through a linear network on a Gaussian Mixture Model and extensions to two-layer ReLU networks and ResNet-18, it demonstrates that shortcut features converge quickly and retain substantial influence on predictions, even when margins are debiased. It introduces predictability and availability metrics to quantify how learnable and NTK-aligned features are, and finds that shortcut labels exhibit higher availability and can persist across real-world datasets like Waterbirds and CelebA. The findings suggest that max-margin bias is not the sole driver of shortcut learning and highlight the role of data variance and NTK spectrum in shaping feature influence, with practical implications for debiasing strategies and evaluation metrics.

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 fully-connected ReLU network and a ResNet-18.
Paper Structure (42 sections, 3 theorems, 42 equations, 14 figures, 2 tables)

This paper contains 42 sections, 3 theorems, 42 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Notions of Neural Tangent Kernel
  4. Datasets
  5. Patched-MNIST.
  6. Colored-MNIST.
  7. Waterbirds.
  8. CelebA.
  9. Dogs and Cats.
  10. Shortcuts on Gaussian Mixture Model
  11. Problem formulation
  12. Eigenvalues of features
  13. Features after convergence
  14. Discussion on the max-margin bias
  15. Experiments
  16. ...and 27 more sections

Key Result

Proposition 3.1

Assume data $x \in \mathbb{R}^d$ in a Gaussian Mixture Model of $p(x) = \sum^K_{k=1} \pi_k \mathcal{N}(\mu_k, \sigma^2_k I)$. The kernel $k(x,y) = \langle x, y \rangle$ has eigenfunctions $\phi_i$ and corresponding eigenvalues $\lambda_i$ as follows: when $(\sum^K_{k=1} \pi_k \mu_k \mu^\top_k) v_i = a_i v_i$, $v^\bot_i$ is a vector perpendicular to $\mu_k$ for $k \in \{1, \dots, K\}$, $m = \text{

Figures (14)

  • Figure 1: Datasets containing biased and core features: Patched-MNIST, Colored-MNIST, Waterbirds, CelebA, and Dogs and Cats.
  • 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 a shortcut feature.
  • 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 in terms of magnitude. A saliency map from the $i$-th 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 on the edges of the face, or the background of an image rather than the hair itself.
  • Figure 4: Classification of 4 clusters with a 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 a borderline between the regions of different colors. For small variances, there is no change in borderline though there is a change of weights in the clusters. Only for large variances, there is a change in borderline when there is a change of cluster weights.
  • Figure 5: Classification of 4 clusters with a two-layer ReLU fully-connected network. The network was trained to 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 converges to the one under the MSE loss.
  • ...and 9 more figures

Theorems & Definitions (5)

  • Proposition 3.1
  • Proposition 3.2
  • Corollary 3.3
  • Definition 4.1
  • Definition 4.2