The modern strategy for training deep neural networks for classification tasks includes optimizing the network's weights even after the training error vanishes to further push the training loss toward zero. Recently, a phenomenon termed "neural collapse" (NC) has been empirically observed in this training procedure. Specifically, it has been shown that the learned features (the output of the penultimate layer) of within-class samples converge to their mean, and the means of different classes exhibit a certain tight frame structure, which is also aligned with the last layer's weights. Recent papers have shown that minimizers with this structure emerge when optimizing a simplified "unconstrained features model" (UFM) with a regularized cross-entropy loss. In this paper, we further analyze and extend the UFM. First, we study the UFM for the regularized MSE loss, and show that the minimizers' features can have a more delicate structure than in the cross-entropy case. This affects also the structure of the weights. Then, we extend the UFM by adding another layer of weights as well as ReLU nonlinearity to the model and generalize our previous results. Finally, we empirically demonstrate the usefulness of our nonlinear extended UFM in modeling the NC phenomenon that occurs with practical networks.
The Ohio State University（俄亥俄州立大学）；Western Washington University（西华盛顿大学）
Neural collapse is an emergent phenomenon in deep learning that was recently discovered by Papyan, Han and Donoho. We propose a simple "unconstrained features model" in which neural collapse also emerges empirically. By studying this model, we provide some explanation for the emergence of neural collapse in terms of the landscape of empirical risk.
Peking University（北京大学）；Stanford University（斯坦福大学）；University of Pennsylvania（宾夕法尼亚大学）；Harvard University（哈佛大学）
Neural collapse is a highly symmetric geometric pattern of neural networks that emerges during the terminal phase of training, with profound implications on the generalization performance and robustness of the trained networks. To understand how the last-layer features and classifiers exhibit this recently discovered implicit bias, in this paper, we introduce a surrogate model called the unconstrained layer-peeled model (ULPM). We prove that gradient flow on this model converges to critical points of a minimum-norm separation problem exhibiting neural collapse in its global minimizer. Moreover…
Microsoft Research（微软研究院）；Google Research（谷歌研究院）；University of Denver（丹佛大学）；University of Michigan（密歇根大学）
When training deep neural networks for classification tasks, an intriguing empirical phenomenon has been widely observed in the last-layer classifiers and features, where (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. This phenomenon is called Neural Collapse (NC), which seems to take place regardless of the choice of loss functions. In this work, we justify NC under the mean squared error (MSE) loss, where recen…
Exploring Information-Theoretic Metrics Associated with Neural Collapse in Supervised Training
School of Computer and Communication Engineering, University of Science and Technology Beijing（北京科技大学计算机与通信工程学院）；Tsinghua University（清华大学）；Shanghai AI Laboratory（上海人工智能实验室）
University of Denver（丹佛大学）；Johns Hopkins University（约翰霍普金斯大学）；New York University（纽约大学）；Google Research（谷歌研究院）；University of Michigan（密歇根大学）
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\;featu…
Neural Collapse under Gradient Flow on Shallow ReLU Networks for Orthogonally Separable Data
Institute of Natural Sciences & School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学自然科学研究院与数学科学学院）；Computer Science and Engineering, Ohio State University（俄亥俄州立大学计算机科学与工程系）；Electrical and Systems Engineering & Department of Radiology, University of Pennsylvania（宾夕法尼亚大学电气与系统工程系及放射学系）
Multi-agent behavior modeling and trajectory forecasting are crucial for the safe navigation of autonomous agents in interactive scenarios. Variational Autoencoder (VAE) has been widely applied in multi-agent interaction modeling to generate diverse behavior and learn a low-dimensional representation for interacting systems. However, existing literature did not formally discuss if a VAE-based model can properly encode interaction into its latent space. In this work, we argue that one of the typical formulations of VAEs in multi-agent modeling suffers from an issue we refer to as social posteri…
FX-DARTS: Designing Topology-unconstrained Architectures with Differentiable Architecture Search and Entropy-based Super-network Shrinking
Beijing Normal University（北京师范大学）；Southern University of Science and Technology（南方科技大学）；University of Illinois Chicago（伊利诺伊大学香槟分校）；Politecnico di Milano（米兰理工大学）；Universita’ Della Svizzera Italiana（瑞士意大利大学）
Peking University（北京大学）；University of Pennsylvania（宾夕法尼亚大学）
In this paper, we introduce the \textit{Layer-Peeled Model}, a nonconvex yet analytically tractable optimization program, in a quest to better understand deep neural networks that are trained for a sufficiently long time. As the name suggests, this new model is derived by isolating the topmost layer from the remainder of the neural network, followed by imposing certain constraints separately on the two parts of the network. We demonstrate that the Layer-Peeled Model, albeit simple, inherits many characteristics of well-trained neural networks, thereby offering an effective tool for explaining …
Mitigating Convergence Collapse in Fixed-Target Anomaly Detectors via Kernel-Anchored Locality Regularization
Université Paris-Saclay（巴黎-萨克雷大学）；CNRS（法国国家科学研究中心）；CentraleSupélec（中央理工-高等电力学院）；LISN UMR 9015（数字科学跨学科实验室）；Laboratoire Interdisciplinaire des Sciences du Numérique（数字科学跨学科实验室）
The Implicit Bias of Depth: From Neural Collapse to Softmax Codes
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Department of Electrical and Computer Engineering, University of British Columbia（不列颠哥伦比亚大学电气与计算机工程系）
University of British Columbia（不列颠哥伦比亚大学）；University of California, Santa Barbara（加州大学圣巴巴拉分校）
Neural Collapse refers to the remarkable structural properties characterizing the geometry of class embeddings and classifier weights, found by deep nets when trained beyond zero training error. However, this characterization only holds for balanced data. Here we thus ask whether it can be made invariant to class imbalances. Towards this end, we adopt the unconstrained-features model (UFM), a recent theoretical model for studying neural collapse, and introduce Simplex-Encoded-Labels Interpolation (SELI) as an invariant characterization of the neural collapse phenomenon. Specifically, we prove …
University of Michigan（密歇根大学）；Ohio State University（俄亥俄州立大学）
When training overparameterized deep networks for classification tasks, it has been widely observed that the learned features exhibit a so-called "neural collapse" phenomenon. More specifically, for the output features of the penultimate layer, for each class the within-class features converge to their means, and the means of different classes exhibit a certain tight frame structure, which is also aligned with the last layer's classifier. As feature normalization in the last layer becomes a common practice in modern representation learning, in this work we theoretically justify the neural coll…
The Prevalence of Neural Collapse in Neural Multivariate Regression
New York University Abu Dhabi（纽约大学阿布扎克分校）；SFSC of AI and DL, NYU Shanghai（人工智能与深度学习学院（SFSC），纽约大学上海）；New York University Shanghai（纽约大学上海）；New York University（纽约大学）
Progressive Feedforward Collapse of ResNet Training
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
The Exploration of Neural Collapse under Imbalanced Data
Huazhong University of Science and Technology（华中科技大学）；School of Mathematics and Statistics（数学与统计学学院）；Institute of Interdisciplinary Research for Mathematics and Applied Science（数学与应用科学交叉研究院）
Neural Collapse in Deep Linear Networks: From Balanced to Imbalanced Data
FPT Software AI Center（FPT软件人工智能中心）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Deakin University（迪肯大学）；University of Texas at Austin（得克萨斯大学奥斯汀分校）；National University of Singapore（新加坡国立大学）
Ohio State University（俄亥俄州立大学）；Google Research（谷歌研究院）；University of Michigan（密歇根大学）；New York University（纽约大学）
While cross entropy (CE) is the most commonly used loss 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 …
Why Do LLM Agents Fail in Exploring New Environments? A World-Modeling Perspective
City University of Hong Kong（香港城市大学）；Northwestern University（西北大学）；The Hong Kong University of Science and Technology（香港科学与技术大学）；Oxford University（牛津大学）；Allen Institute for AI (AI2)（人工智能研究所）；University of Washington（华盛顿大学）；National University of Singapore（新加坡国立大学）；The Hong Kong Polytechnic University（香港理工大学）
