AI & ML
Gradient Descent on Two ReLU Neurons: Global Landscape and Bifurcation Dynamics
arXiv stat.ML0 views
arXiv:2610.42118v1 Announce Type: cross 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.
Source:https://arxiv.org/abs/2610.42118PreviousCausalWAM: Learning the Causes of Actions in World Action ModelsNextTransport-Free Lower Bounds for Ollivier--Ricci Curvature in Higher-Order Networks: Scope, Tightness, and Computational Applications
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