两层 ReLU 神经元梯度下降的全局景观与分岔动力学分析
Gradient Descent on Two ReLU Neurons: Global Landscape and Bifurcation Dynamics
一篇理论论文，用最简单的两神经元 ReLU 模型解释了梯度下降为什么会先合并再分化，做神经网络理论研究的可以看看证明思路。
一篇 arXiv 论文研究了带各向同性高斯输入的两神经元 ReLU teacher-student 模型，用 population gradient flow 从小初始化训练。论文证明损失景观中所有两个学生权重非零的临界点要么是全局最小值，要么是鞍点。轨迹分析显示学生神经元先坍缩到 bisector 鞍点，随后逃逸并向 teacher 神经元特化。结果从景观和动力学两个角度刻画了 multi-index 模型中的多阶段对称性破缺行为。
Gradient Descent on Two ReLU Neurons: Global Landscape and Bifurcation Dynamics
Understanding how gradient descent learns features remains a central challenge in neural network theory. We study this question in a minimal multi-index setting that already exhibits rich multi-phase gradient dynamics: a well-specified two-neuron ReLU teacher-student model under isotropic Gaussian inputs, trained by population gradient flow from small initialization. We show that the dynamics are organized by two directions: the "easy" bisector direction, which carries the leading signal, and the "hard" splitting direction, which governs specialization. To characterize the loss structure and learning behavior, we analyze the population squared-loss landscape and show that every critical point with both student weights nonzero is either a global minimum or a saddle. We then track the gradient-flow trajectory, showing that student neurons first collapse toward a bisector saddle before escaping and specializing to teacher neurons. Our results provide both a landscape and dynamics account of the multi-phase symmetry-breaking behavior that arises in a simple multi-index model and under standard algorithmic and architectural choices.