Balanced Data, Imbalanced Spectra: Unveiling Class Disparities with Spectral Imbalance

TL;DR

This work develops a theoretical framework for studying class disparities and derive exact expressions for the per-class error in a high-dimensional mixture model setting and provides new insights into how state-of-the-art pretrained features may have unknown biases that can be diagnosed through their spectra.

Abstract

Classification models are expected to perform equally well for different classes, yet in practice, there are often large gaps in their performance. This issue of class bias is widely studied in cases of datasets with sample imbalance, but is relatively overlooked in balanced datasets. In this work, we introduce the concept of spectral imbalance in features as a potential source for class disparities and study the connections between spectral imbalance and class bias in both theory and practice. To build the connection between spectral imbalance and class gap, we develop a theoretical framework for studying class disparities and derive exact expressions for the per-class error in a high-dimensional mixture model setting. We then study this phenomenon in 11 different state-of-the-art pretrained encoders and show how our proposed framework can be used to compare the quality of encoders, as well as evaluate and combine data augmentation strategies to mitigate the issue. Our work sheds light on the class-dependent effects of learning, and provides new insights into how state-of-the-art pretrained features may have unknown biases that can be diagnosed through their spectra.

Figures (10)

• Figure 1: Spectral imbalance in different pretrained models. (A) We plot the spectrum of the top and bottom performing classes and find that classes that achieve lower accuracy have larger eigenvalues. (B) Histogram of the $k = 5$ eigenvalue across classes for two encoders (ResNet-50, ViT-B) with the quartiles indicated. If we look at classes with similarly ranked or "adjacent" eigenvalues (upper quartile) in ResNet-50, we find that they can be mapped to very different positions in the distribution of eigenvalues in another encoder (ViT-B).
• Figure 1: Interplay of sample and spectral imbalance: Heatmap of the class gap across different amounts of sample imbalance and spectral imbalance. Settings (A), (B), and (C) correspond to the same three types of spectral imbalance considered in Section \ref{['sec:theory-insights']} and the simulation details provided in Appendix \ref{['app:sim_details']}.
• Figure 2: Spectral imbalance in the GMM for the settings in Section \ref{['sec:theory-insights']}. Top Row: Visualization of the spectra of both classes. Bottom Row: Theoretical predictions of Theorem \ref{['thm:main_thm']} (solid lines) and numerical simulations (average over 50 trials $\pm 1$ std.) for the per-class error.
• Figure 2: Eigenspectrum visualization plots in normal scale and log scale for VIT-B.
• Figure 3: Examining the relationship between class-dependent spectra and performance. (left) The spectral offset of each class vs. their classification accuracy, computed for ResNet-50 on the validation set of ImageNet. (right) Pearson correlation coefficient between class accuracy and individual eigenvalues (blue) and the power law offset (red).

Theorems & Definitions (5)

• Theorem 1
• Lemma 2: Expression for the POE in the imbalanced GMM
• proof
• proof : Proof of Theorem \ref{['thm:main_thm']}
• Definition 3: Effective Rank