Unveiling and Mitigating Generalized Biases of DNNs through the Intrinsic Dimensions of Perceptual Manifolds

TL;DR

A geometric perspective for analyzing the fairness of DNNs is established, comprehensively exploring how DNNs internally shape the intrinsic geometric characteristics of datasets—the intrinsic dimensions (IDs) of perceptual manifolds, and the impact of IDs on the fairness of DNNs.

Abstract

Building fair deep neural networks (DNNs) is a crucial step towards achieving trustworthy artificial intelligence. Delving into deeper factors that affect the fairness of DNNs is paramount and serves as the foundation for mitigating model biases. However, current methods are limited in accurately predicting DNN biases, relying solely on the number of training samples and lacking more precise measurement tools. Here, we establish a geometric perspective for analyzing the fairness of DNNs, comprehensively exploring how DNNs internally shape the intrinsic geometric characteristics of datasets-the intrinsic dimensions (IDs) of perceptual manifolds, and the impact of IDs on the fairness of DNNs. Based on multiple findings, we propose Intrinsic Dimension Regularization (IDR), which enhances the fairness and performance of models by promoting the learning of concise and ID-balanced class perceptual manifolds. In various image recognition benchmark tests, IDR significantly mitigates model bias while improving its performance.

Figures (6)

• Figure 1: a On a dataset containing the same number of samples per category, the trained model still has a significant bias. b Geometric Perspectives and Quantitative Results of Information Compression and Classification by DNN, with d representing the depth of the DNN. The process of information compression by DNN can be understood as gradually obtaining low-dimensional perceptual manifolds along the internal layers of the DNN. Low-dimensional perceptual manifolds benefit the performance of subsequent tasks. c Each category corresponds to a data manifold, which is mapped to perceptual manifolds in the DNN. As the depth of the DNN increases, the perceptual manifolds gradually separate from each other and the dimensions decrease, facilitating the correct identification of each category by the model. d Data manifolds are constructed on a dataset basis rather than by category. As the depth of the DNN increases, the intrinsic dimensions of perceptual manifolds corresponding to four benchmark image datasets show a trend of initially increasing and then decreasing.
• Figure 2: The Relationship between Intrinsic Dimensions of Perceptual Manifolds and Model Performance, and the Learning Process.a The intrinsic dimensions of perceptual manifolds generated by the last hidden layer of DNN exhibit a negative correlation with the overall performance of the DNN across four benchmark image datasets. b The correlation between the intrinsic dimensions of perceptual manifolds generated by different layers of DNN and the overall performance of the DNN. Across the four benchmark datasets, there is an increasing negative correlation between the intrinsic dimensions of perceptual manifolds and the overall performance of the DNN as the depth increases. c As training progresses, the intrinsic dimensions of perceptual manifolds corresponding to datasets generated by multiple benchmark networks gradually decrease.
• Figure 3: The negative correlation between the intrinsic dimensions of class perceptual manifolds generated by the last hidden layer of the DNN and the class accuracy.a Extracting image embeddings for each class from the last hidden layer of the DNN and calculating the intrinsic dimensions of class perceptual manifolds for each embedding set. b The scatter plot visualizes the distribution of class accuracy against the intrinsic dimensions of class perceptual manifolds, while the bar chart further illustrates their Pearson correlation coefficients (PCCs).
• Figure 4: The impact of different layers of the DNN and the learning process on the correlation between the intrinsic dimensions of class perceptual manifolds and class accuracy.a Extracting image embeddings for each class from different layers of well-trained DNNs and calculating the intrinsic dimensions of class perceptual manifolds. b The correlation between the intrinsic dimensions of class perceptual manifolds generated by different layers and class accuracy. c The existing optimization objectives are almost unable to reduce the imbalance of intrinsic dimensions among perceptual manifolds.
• Figure 5: a&b The computation process of Intrinsic Dimension Regularization (IDR) and its corresponding training scheme. a We present two examples ($ID_1=10, ID_2=8, ID_C=2$ and $ID_1=ID_2=ID_C=8$) to verify whether $L_{ID}$ satisfies the first and second design principles. It can be observed that when $ID_1>ID_2>\dots>ID_C$, then $L_{ID_1}>L_{ID_2}>\dots>L_{ID_C}$. $L_{ID}$ is a decreasing function with respect to $ID$, thus fulfilling principle (1). When $ID_1=ID_2=\dots=ID_C$, then $L_{ID_1}=L_{ID_2}=\dots=L_{ID_C}=0$. Clearly, $L_{ID}$ conforms to principle (2). When $ID_i=\min\{ID_1,\dots,ID_C\}$, then $L_{ID_i}=0$, thus the purpose of $L_{ID}$ is to make the intrinsic dimensions of all class perceptual manifolds close to $min\{ID_1,\dots,ID_C\}$. Obviously, ${ \sum_{i=1}^{C}}ID_i \ge C\cdot \min\{ID_1,\dots,ID_C\}$, therefore $L_{ID}$ satisfies principle (3). b End-to-end training scheme. We construct a queue to store the covariance matrices generated at each iteration, which is capable of storing at most the covariance matrix generated in a complete epoch. As training progresses, the latest generated covariance matrix is continuously used to update the oldest covariance matrix in the queue. When applying $L_{ID}$, the stored covariance matrices in the queue can be used for its calculation. c The bias of multiple DNN models is significantly reduced by measuring the standard deviation of class accuracy, demonstrating the efficacy of intrinsic dimension regularization.d Using ResNet-50 as the backbone network, model performance under different values of $\alpha$ when employing $L_{ID}$. e IDR improves the overall performance of the model on multiple datasets.