---
title: "Gradient Descent on Two ReLU Neurons: Global Landscape and Bifurcation Dynamics"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2610.42118v1"
permalink: "https://feed.astralor.com/items/itm_f5195a9d36586569763e7be1"
published: "2026-10-30T17:51:35.000Z"
score: 54
channel: "ai"
tags: ["Gradient Descent","Two ReLU Neurons","Global Landscape","Bifurcation Dynamics Understanding","Understanding","ReLU","Gaussian","Our"]
---

# Gradient Descent on Two ReLU Neurons: Global Landscape and Bifurcation Dynamics

## At a glance

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: …

## Key points

- 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 popula…
- 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.

## Full text

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.

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