Student Specialization in Deep ReLU Networks With Finite Width and Input Dimension
Abstract
We consider a deep ReLU / Leaky ReLU student network trained from the output of a fixed teacher network of the same depth, with Stochastic Gradient Descent (SGD). The student network is \emph{overrealized}: at each layer $l$, the number $n_l$ of student nodes is more than that ($m_l$) of teacher. Under mild conditions on dataset and teacher network, we prove that when the gradient is small at every data sample, each teacher node is \emph{specialized} by at least one student node \emph{at the lowest layer}. For twolayer network, such specialization can be achieved by training on any dataset of \emph{polynomial} size $\mathcal{O}( K^{5/2} d^3 \epsilon^{1})$. until the gradient magnitude drops to $\mathcal{O}(\epsilon/K^{3/2}\sqrt{d})$. Here $d$ is the input dimension, $K = m_1 + n_1$ is the total number of neurons in the lowest layer of teacher and student. Note that we require a specific form of data augmentation and the sample complexity includes the additional data generated from augmentation. To our best knowledge, we are the first to give polynomial sample complexity for student specialization of training twolayer (Leaky) ReLU networks with finite depth and width in teacherstudent setting, and finite complexity for the lowest layer specialization in multilayer case, without parametric assumption of the input (like Gaussian). Our theory suggests that teacher nodes with large fanout weights get specialized first when the gradient is still large, while others are specialized with small gradient, which suggests inductive bias in training. This shapes the stage of training as empirically observed in multiple previous works. Experiments on synthetic and CIFAR10 verify our findings. The code is released in https://github.com/facebookresearch/luckmatters.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.13458
 Bibcode:
 2019arXiv190913458T
 Keywords:

 Computer Science  Machine Learning;
 Statistics  Machine Learning
 EPrint:
 Accepted in ICML 2020