两个 ReLU 神经元上的梯度下降：全局景观与分岔动力学
Gradient Descent on Two ReLU Neurons: Global Landscape and Bifurcation Dynamics
Binghua Li · Mengzhe Li · Denny Wu · Tianhao Wang
中文摘要
理解梯度下降如何学习特征仍是神经网络理论的核心挑战。研究一个最小多指标（multi-index）设定，该设定已呈现丰富的多阶段梯度动力学：在 isotropic Gaussian 输入下，一个良设定的双神经元 ReLU teacher-student 模型，从小初始化出发用 population 梯度流训练。动力学由两个方向组织：携带主导信号的“容易”平分线（bisector）方向，以及支配特化的“困难”分裂方向。为刻画损失结构与学习行为，分析 population 平方损失景观，证明所有学生权重均非零的临界点要么是全局最小要么是鞍点。随后追踪梯度流轨迹，显示学生神经元先坍缩向平分线鞍点，再逃离并特化到教师神经元。结果从景观与动力学两方面解释了简单多指标模型在标准算法与架构选择下出现的多阶段对称破缺行为。
关键要点
- 01研究双神经元 ReLU teacher-student 模型，揭示梯度下降的多阶段动力学。
- 02动力学由携带信号的平分线方向与支配特化的分裂方向共同组织。
- 03证明所有学生权重非零的临界点要么是全局最小要么是鞍点。
- 04学生神经元先坍缩至平分线鞍点，再逃离并特化到教师神经元。
解读
尚无解读。
原始英文摘要
arXiv:2610.42118v1 Announce Type: new Abstract: 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.