研究神经网络在零边界分类中的动态机制，揭示神经元聚类与块级信号协同演化规律。
Neuron Block Dynamics for XOR Classification with Zero-Margin
- 通过分析高斯XOR问题，提出神经元块动力学框架
- 发现神经元聚成四个方向，块信号协同演化
- 无需依赖边际假设，适用于高复杂度分类场景
神经网络通过随机梯度下降（SGD）学习有效特征的能力是其成功的关键。现有理论多关注回归或正边际分类任务，而本文研究零边际非线性分类，聚焦于高斯XOR问题——输入服从高斯分布，标签由异或决策边界决定。该设定下，部分数据点极度接近边界，破坏了传统基于边际的分析方法。基于Glasgow（2024）的工作，我们从离散输入扩展至高斯输入，构建神经元块动力学框架。研究表明，神经元聚集为四个方向，块级信号协同演化，这一现象在个体神经元信号波动显著的高斯设定中至关重要。借助块视角，我们不依赖边际假设分析泛化性能，采用平均情况观点区分可靠预测区与持续错误区。数值实验验证了预测的两阶段块动力学，并表明其在高斯以外设定中仍具鲁棒性。
原文摘要 · Abstract (English)
The ability of neural networks to learn useful features through stochastic gradient descent (SGD) is a cornerstone of their success. Most theoretical analyses focus on regression or on classification tasks with a positive margin, where worst-case gradient bounds suffice. In contrast, we study zero-margin nonlinear classification by analyzing the Gaussian XOR problem, where inputs are Gaussian and the XOR decision boundary determines labels. In this setting, a non-negligible fraction of data lies arbitrarily close to the boundary, breaking standard margin-based arguments. Building on Glasgow's (2024) analysis, we extend the study of training dynamics from discrete to Gaussian inputs and develop a framework for the dynamics of neuron blocks. We show that neurons cluster into four directions and that block-level signals evolve coherently, a phenomenon essential in the Gaussian setting where individual neuron signals vary significantly. Leveraging this block perspective, we analyze generalization without relying on margin assumptions, adopting an average-case view that distinguishes regions of reliable prediction from regions of persistent error. Numerical experiments confirm the predicted two-phase block dynamics and demonstrate their robustness beyond the Gaussian setting.
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