Predicting and Enhancing the Fairness of DNNs with the Curvature of Perceptual Manifolds

TL;DR

The paper argues that fairness in DNNs cannot be understood solely through sample counts, proposing a geometric framework based on perceptual manifolds in embedding space. It introduces three metrics—volume, separation degree, and curvature—to quantify class geometry and shows that curvature imbalance correlates with model bias, even on balanced data. The authors propose Curvature Regularization and Dynamic Curvature Regularization to learn curvature-balanced, flatter manifolds, achieving consistent gains on multiple long-tailed and non-long-tailed datasets and improving convergence. This geometric perspective provides a new lens for diagnosing and mitigating bias in deep models and suggests curvature-aware training as a general strategy for fairer AI.

Abstract

To address the challenges of long-tailed classification, researchers have proposed several approaches to reduce model bias, most of which assume that classes with few samples are weak classes. However, recent studies have shown that tail classes are not always hard to learn, and model bias has been observed on sample-balanced datasets, suggesting the existence of other factors that affect model bias. In this work, we first establish a geometric perspective for analyzing model fairness and then systematically propose a series of geometric measurements for perceptual manifolds in deep neural networks. Subsequently, we comprehensively explore the effect of the geometric characteristics of perceptual manifolds on classification difficulty and how learning shapes the geometric characteristics of perceptual manifolds. An unanticipated finding is that the correlation between the class accuracy and the separation degree of perceptual manifolds gradually decreases during training, while the negative correlation with the curvature gradually increases, implying that curvature imbalance leads to model bias.Building upon these observations, we propose curvature regularization to facilitate the model to learn curvature-balanced and flatter perceptual manifolds. Evaluations on multiple long-tailed and non-long-tailed datasets show the excellent performance and exciting generality of our approach, especially in achieving significant performance improvements based on current state-of-the-art techniques. Our work opens up a geometric analysis perspective on model bias and reminds researchers to pay attention to model bias on non-long-tailed and even sample-balanced datasets.

Figures (17)

• Figure 1: Curvature regularization reduces the model bias present in multiple methods on CIFAR-100-LT and ImageNet-LT. The model bias is measured with the variance of the accuracy of all classes, and it is zero when the accuracy of each class is the same.
• Figure 2: The geometric perspective of data classification involves each class of data distributed around a submanifold. In image space, multiple submanifolds may be intertwined. Deep neural networks untangle these submanifolds and separate them from each other through layer-wise mappings, facilitating classification. The class perceptual manifolds in the embedding space are mapped into the decision space for classification, so the geometric complexity of the perceptual manifolds may directly affect the classification performance.
• Figure 3: The variation curve between the separation degree of two spherical point clouds and the distance between spherical centers.
• Figure 4: A The schematic diagram for calculating the curvature of class perceptual manifolds versus class accuracy. B Trained $13$ different models on three sample-balanced datasets, CIFAR-10, CIFAR-100, and SVHN, and calculated the correlation between the curvature of class perceptual manifolds generated by each model and class accuracy. The experimental settings for Figs 4, 6, 8, 9, and 11 are as follows: On the CIFAR-10, CIFAR-100, and SVHN datasets, we used SGD with a momentum of $0.9$, a batch size of $64$, and an initial learning rate of $0.1$. The difference is that on CIFAR-10 and SVHN, the model was trained for $60$ epochs, and the cosine annealing strategy was used for learning rate decay. On CIFAR-100, the model was trained for $200$ epochs, and the learning rate was adjusted to $0.02$, $0.004$, and $0.0008$ at epochs $60$, $120$, and $160$, respectively. All models were trained using Cross-Entropy (CE) loss.
• Figure 5: The surface equations in the first and second rows are $Z=w(X^2-Y^2)$ and $Z=\sin(\sin(0.5wX))+\cos(\cos(0.5wX))$, respectively. We increase the curvature of the surface by increasing $w$ and calculate the complexity of the two-dimensional point cloud surface. Also, we investigate the effect of the number of neighbors $k$ on the complexity of the manifold.

Theorems & Definitions (1)

• proof