Deep neural networks (DNNs) at convergence consistently represent the training data in the last layer via a highly symmetric geometric structure referred to as neural collapse. This empirical evidence has spurred a line of theoretical research aimed at proving the emergence of neural collapse, mostly focusing on the unconstrained features model. Here, the features of the penultimate layer are free variables, which makes the model data-agnostic and, hence, puts into question its ability to capture DNN training. Our work addresses the issue, moving away from unconstrained features and studying DNNs that end with at least two linear layers. We first prove generic guarantees on neural collapse that assume (i) low training error and balancedness of the linear layers (for within-class variability collapse), and (ii) bounded conditioning of the features before the linear part (for orthogonality of class-means, as well as their alignment with weight matrices). We then show that such assumptions hold for gradient descent training with weight decay: (i) for networks with a wide first layer, we prove low training error and balancedness, and (ii) for solutions that are either nearly optimal or stable under large learning rates, we additionally prove the bounded conditioning. Taken together, our results are the first to show neural collapse in the end-to-end training of DNNs.

本研究解决了深度神经网络收敛时神经崩溃的理论证明问题，通过研究终止于至少两个线性层的神经网络，而不是之前的无约束特征模型。关键发现是，证明了在带权重衰减的梯度下降训练下，这种神经崩溃现象可以得到普遍性保证，从而为神经网络训练的理解提供了新的视角。

带权重衰减训练的宽神经网络显著展现神经崩溃的现象