We provide the first global optimization landscape analysis of Neural Collapse -- an intriguing empirical phenomenon that arises in the last-layer classifiers and features of neural networks during the terminal phase of training. As recently reported by Papyan et al., this phenomenon implies that (i) the class means and the last-layer classifiers all collapse to the vertices of a Simplex Equiangular Tight Frame (ETF) up to scaling, and (ii) cross-example within-class variability of last-layer activations collapses to zero. We study the problem based on a simplified unconstrained feature model, which isolates the topmost layers from the classifier of the neural network. In this context, we show that the classical cross-entropy loss with weight decay has a benign global landscape, in the sense that the only global minimizers are the Simplex ETFs while all other critical points are strict saddles whose Hessian exhibit negative curvature directions. Our analysis of the simplified model not only explains what kind of features are learned in the last layer, but also shows why they can be efficiently optimized, matching the empirical observations in practical deep network architectures. These findings provide important practical implications. As an example, our experiments demonstrate that one may set the feature dimension equal to the number of classes and fix the last-layer classifier to be a Simplex ETF for network training, which reduces memory cost by over 20% on ResNet18 without sacrificing the generalization performance.

Neural Collapse (NC) is a recently observed phenomenon in neural networks that characterises the solution space of the final classifier layer when trained until zero training loss. Specifically, NC suggests that the final classifier layer converges to a Simplex Equiangular Tight Frame (ETF), which maximally separates the weights corresponding to each class. By duality, the penultimate layer feature means also converge to the same simplex ETF. Since this simple symmetric structure is optimal, our idea is to utilise this property to improve convergence speed. Specifically, we introduce the notion of \textit{nearest simplex ETF geometry} for the penultimate layer features at any given training iteration, by formulating it as a Riemannian optimisation. Then, at each iteration, the classifier weights are implicitly set to the nearest simplex ETF by solving this inner-optimisation, which is encapsulated within a declarative node to allow backpropagation. Our experiments on synthetic and real-world architectures on classification tasks demonstrate that our approach accelerates convergence and enhances training stability.

Recent studies on Neural Collapse (NC) reveal that, under class-balanced conditions, the class feature means and classifier weights spontaneously align into a simplex equiangular tight frame (ETF). In long-tailed regimes, however, severe sample imbalance tends to prevent the emergence of the NC phenomenon, resulting in poor generalization performance. Current efforts predominantly seek to recover the ETF geometry by imposing constraints on features or classifier weights, yet overlook a critical problem: There is a pronounced misalignment between the feature and the classifier weight spaces. In this paper, we theoretically quantify the harm of such misalignment through an optimal error exponent analysis. Built on this insight, we propose three explicit alignment strategies that plug-and-play into existing long-tail methods without architectural change. Extensive experiments on the CIFAR-10-L T, CIFAR-100-L T, and ImageNet-L T datasets consistently boost examined baselines and achieve the state-of-the-art performances.

We provide the first global optimization landscape analysis of Neural Collapse -- an intriguing empirical phenomenon that arises in the last-layer classifiers and features of neural networks during the terminal phase of training. As recently reported in [1], this phenomenon implies that (i) the class means and the last-layer classifiers all collapse to the vertices of a Simplex Equiangular Tight Frame (ETF) up to scaling, and (ii) cross-example within-class variability of last-layer activations collapses to zero. We study the problem based on a simplified unconstrained feature model, which isolates the topmost layers from the classifier of the neural network. In this context, we show that the classical cross-entropy loss with weight decay has a benign global landscape, in the sense that the only global minimizers are the Simplex ETFs while all other critical points are strict saddles whose Hessian exhibit negative curvature directions. Our analysis of the simplified model not only explains what kind of features are learned in the last layer, but also shows why they can be efficiently optimized, matching the empirical observations in practical deep network architectures. These findings provide important practical implications. As an example, our experiments demonstrate that one may set the feature dimension equal to the number of classes and fix the last-layer classifier to be a Simplex ETF for network training, which reduces memory cost by over 20% on ResNet18 without sacrificing the generalization performance.

Then the same solution in the Lemma 1 is obtained. We will prove that if Assumptions 1 and 2 hold, the stochastic gradients cannot be uniformly bounded. However, FedGELA might reach better local optimal by adapting the feature structure. Here we complete the proof. "existing angle" as the angle of classifier vectors belonging to classes that exist in a local client In Fed-ISIC2019, there exists a true PCDD situation that needs to be solved.