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Title: Toward Understanding Generalization of Over-parameterized Deep ReLU network trained with SGD in Student-teacher Setting

Paper content:
To analyze deep ReLU network, we adopt a student-teacher setting in which an over-parameterized student network learns from the output of a fixed teacher network of the same depth, with Stochastic Gradient Descent (SGD).
Our contributions are two-fold.
First, we prove that when the gradient is zero (or bounded above by a small constant) at every data point in training, a situation called  \emph{interpolation setting}, there exists many-to-one \emph{alignment} between student and teacher nodes in the lowest layer under mild conditions.
This suggests that generalization in unseen dataset is achievable, even the same condition often leads to zero training error.
Second, analysis of noisy recovery and training dynamics in 2-layer network shows that strong teacher nodes (with large fan-out weights) are learned first and subtle teacher nodes are left unlearned until late stage of training.
As a result, it could take a long time to converge into these small-gradient critical points.
Our analysis shows that over-parameterization plays two roles: (1) it is a necessary condition for alignment to happen at the critical points, and (2) in training dynamics, it helps student nodes cover more teacher nodes with fewer iterations.
Both improve generalization.
Experiments justify our finding.
Deep Learning has achieved great success in the recent years (Silver et al., 2016; He et al., 2016; Devlin et al., 2018) .
Although networks with even one-hidden layer can fit any function (Hornik et al., 1989) , it remains an open question how such networks can generalize to new data.
Different from what traditional machine learning theory predicts, empirical evidence (Zhang et al., 2017) shows more parameters in neural network lead to better generalization.
How over-parameterization yields strong generalization is an important question for understanding how deep learning works.
In this paper, we analyze deep ReLU networks with teacher-student setting: a fixed teacher network provides the output for a student to learn via SGD.
Both teacher and student are deep ReLU networks.
Similar to (Goldt et al., 2019) , the student is over-realized compared to the teacher: at each layer l, the number n l of student nodes is larger than the number m l of teacher (n l > m l ).
Although over-realization is different from over-parameterization, i.e., the total number of parameters in the student model is larger than the training set size N , over-realization directly correlates with the width of networks and is a measure of over-parameterization.
The student-teacher setting has a long history (Saad & Solla, 1996; 1995; Freeman & Saad, 1997; Mace & Coolen, 1998) and recently gains increasing interest (Goldt et al., 2019; Aubin et al., 2018) in analyzing 2-layered network.
While worst-case performance on arbitrary data distributions may not be a good model for real structured dataset and can be hard to analyze, using a teacher network implicitly enforces an inductive bias and could potentially lead to better generalization bound.
Specialization, that is, a student node becomes increasingly correlated with a teacher node during training (Saad & Solla, 1996) , is one of the important topic in this setup.
If all student nodes are specialized to the teacher, then student tends to output the same as the teacher and generalization performance can be expected.
Empirically, it has been observed in 2-layer networks (Saad & Solla, 1996; Goldt et al., 2019) and multi-layer networks (Tian et al., 2019; Li et al., 2016) , in both synthetic and real dataset.
In contrast, theoretical analysis is limited with strong assumptions (e.g., Gaussian inputs, infinite input dimension, local convergence, 2-layer setting, small number of hidden nodes).
In this paper, with arbitrary training distribution and finite input dimension, we show rigorously that when gradient at each training sample is small (i.e., the interpolation setting as suggested in (Ma
In this paper, we use student-teacher setting to analyze how an (over-parameterized) deep ReLU student network trained with SGD learns from the output of a teacher.
When the magnitude of gradient per sample is small (student weights are near the critical points), the teacher can be proven to be covered by (possibly multiple) students and thus the teacher network is recovered in the lowest layer.
By analyzing training dynamics, we also show that strong teacher node with large v * is reconstructed first, while weak teacher node is reconstructed slowly.
This reveals one important reason why the training takes long to reconstruct all teacher weights and why generalization improves with more training.
As the next step, we would like to extend our analysis to finite sample case, and analyze the training dynamics in a more formal way.
Verifying the insights from theoretical analysis on a large dataset (e.g., ImageNet) is also the next step.
Figure 8: Mean of the max teacher correlation ρmean with student nodes over epochs in CIFAR10.
More over-realization gives better student specialization across all layers and achieves strong generalization (higher evaluation accuracy on CIFAR-10 evaluation set).

assistant: text: Study of over-parametrization in student-teacher multilayer ReLU networks, a theoretical part about SGD critical points for the teacher-student setting, and a heuristic and empirical part on dynamics of the SDG algorithm as a function of teacher networks.