While cross entropy (CE) is the most commonly used loss function to train deep neural networks for classification tasks, many alternative losses have been developed to obtain better empirical performance. Among them, which one is the best to use is still a mystery, because there seem to be multiple factors affecting the answer, such as properties of the dataset, the choice of network architecture, and so on. This paper studies the choice of loss function by examining the last-layer features of deep networks, drawing inspiration from a recent line work showing that the global optimal solution of CE and mean-square-error (MSE) losses exhibits a Neural Collapse phenomenon. That is, for sufficiently large networks trained until convergence, (i) all features of the same class collapse to the corresponding class mean and (ii) the means associated with different classes are in a configuration where their pairwise distances are all equal and maximized. We extend such results and show through global solution and landscape analyses that a broad family of loss functions including commonly used label smoothing (LS) and focal loss (FL) exhibits Neural Collapse. Hence, all relevant losses (i.e., CE, LS, FL, MSE) produce equivalent features on training data. In particular, based on the unconstrained feature model assumption, we provide either the global landscape analysis for LS loss or the local landscape analysis for FL loss and show that the (only!) global minimizers are neural collapse solutions, while all other critical points are strict saddles whose Hessian exhibit negative curvature directions either in the global scope for LS loss or in the local scope for FL loss near the optimal solution. The experiments further show that Neural Collapse features obtained from all relevant losses (i.e., CE, LS, FL, MSE) lead to largely identical performance on test data as well, provided that the network is sufficiently large and trained until convergence.

Graph neural networks (GNNs) have become increasingly popular for classification tasks on graph-structured data. Yet, the interplay between graph topology and feature evolution in GNNs is not well understood. In this paper, we focus on node-wise classification, illustrated with community detection on stochastic block model graphs, and explore the feature evolution through the lens of the Neural Collapse (NC) phenomenon. When training instance-wise deep classifiers (e.g. for image classification) beyond the zero training error point, NC demonstrates a reduction in the deepest features' within-class variability and an increased alignment of their class means to certain symmetric structures. We start with an empirical study that shows that a decrease in within-class variability is also prevalent in the node-wise classification setting, however, not to the extent observed in the instance-wise case. Then, we theoretically study this distinction. Specifically, we show that even an optimistic mathematical model requires that the graphs obey a strict structural condition in order to possess a minimizer with exact collapse. Furthermore, by studying the gradient dynamics of this model, we provide reasoning for the partial collapse observed empirically. Finally, we present a study on the evolution of within-and between-class feature variability across layers of a well-trained GNN and contrast the behavior with spectral methods.

Deep Equilibrium Model (DEQ), which serves as a typical implicit neural network, emphasizes their memory efficiency and competitive performance compared to explicit neural networks. However, there has been relatively limited theoretical analysis on the representation of DEQ. In this paper, we utilize the Neural Collapse ( \mathcal{NC}) as a tool to systematically analyze the representation of DEQ under both balanced and imbalanced conditions. While extensively studied in traditional explicit neural networks, the \mathcal{NC} phenomenon has not received substantial attention in the context of implicit neural networks. We theoretically show that \mathcal{NC} exists in DEQ under balanced conditions.

While cross entropy (CE) is the most commonly used loss function to train deep neural networks for classification tasks, many alternative losses have been developed to obtain better empirical performance. Among them, which one is the best to use is still a mystery, because there seem to be multiple factors affecting the answer, such as properties of the dataset, the choice of network architecture, and so on. This paper studies the choice of loss function by examining the last-layer features of deep networks, drawing inspiration from a recent line work showing that the global optimal solution of CE and mean-square-error (MSE) losses exhibits a Neural Collapse phenomenon. That is, for sufficiently large networks trained until convergence, (i) all features of the same class collapse to the corresponding class mean and (ii) the means associated with different classes are in a configuration where their pairwise distances are all equal and maximized. We extend such results and show through global solution and landscape analyses that a broad family of loss functions including commonly used label smoothing (LS) and focal loss (FL) exhibits Neural Collapse.

Graph neural networks (GNNs) have become increasingly popular for classification tasks on graph-structured data. Yet, the interplay between graph topology and feature evolution in GNNs is not well understood. In this paper, we focus on node-wise classification, illustrated with community detection on stochastic block model graphs, and explore the feature evolution through the lens of the "Neural Collapse" (NC) phenomenon. When training instance-wise deep classifiers (e.g. for image classification) beyond the zero training error point, NC demonstrates a reduction in the deepest features' within-class variability and an increased alignment of their class means to certain symmetric structures. We start with an empirical study that shows that a decrease in within-class variability is also prevalent in the node-wise classification setting, however, not to the extent observed in the instance-wise case.

Label smoothing loss is a widely adopted technique to mitigate overfitting in deep neural networks. This paper studies label smoothing from the perspective of Neural Collapse (NC), a powerful empirical and theoretical framework which characterizes model behavior during the terminal phase of training. We first show empirically that models trained with label smoothing converge faster to neural collapse solutions and attain a stronger level of neural collapse. Additionally, we show that at the same level of NC1, models under label smoothing loss exhibit intensified NC2. These findings provide valuable insights into the performance benefits and enhanced model calibration under label smoothing loss. We then leverage the unconstrained feature model to derive closed-form solutions for the global minimizers for both loss functions and further demonstrate that models under label smoothing have a lower conditioning number and, therefore, theoretically converge faster. Our study, combining empirical evidence and theoretical results, not only provides nuanced insights into the differences between label smoothing and cross-entropy losses, but also serves as an example of how the powerful neural collapse framework can be used to improve our understanding of DNNs.