铰链损耗分类的浅网络动力学解析理论
An analytic theory of shallow networks dynamics for hinge loss classification
摘要：
神经网络已经被证明在结构化高维数据集的分类任务中表现得非常好。然而，这种网络的学习动态仍然知之甚少。在本文中，我们详细研究了一个简单类型的神经网络的训练动态：一个单一的隐藏层训练执行分类任务。我们表明，在一个合适的平均场的限制，这种情况下映射到一个单节点的学习问题与时间相关的数据集确定自相一致的平均节点人口。我们专门我们的理论的原型情况下，线性可分离的数据集和线性铰链损失，可以明确解决的动态。这使我们能够在一个简单的设置中解决现代网络中出现的几种现象，例如训练动态减慢，丰富和懒惰学习之间的交叉以及过拟合。最后，我们通过研究大量但有限数量的节点和训练样本的情况来评估平均场理论的局限性。
英文摘要：
Neural networks have been shown to perform incredibly well in classification tasks over structured high-dimensional datasets. However, the learning dynamics of such networks is still poorly understood. In this paper we study in detail the training dynamics of a simple type of neural network: a single hidden layer trained to perform a classification task. We show that in a suitable mean-field limit this case maps to a single-node learning problem with a time-dependent dataset determined self-consistently from the average nodes population. We specialize our theory to the prototypical case of a linearly separable dataset and a linear hinge loss, for which the dynamics can be explicitly solved. This allow us to address in a simple setting several phenomena appearing in modern networks such as slowing down of training dynamics, crossover between rich and lazy learning, and overfitting. Finally, we asses the limitations of mean-field theory by studying the case of large but finite number of nodes and of training samples.
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