We perform an average case analysis of the generalization dynamics of large
neural networks trained using gradient descent. We study the
practically-relevant "high-dimensional" regime where the number of free
parameters in the network is on the order of or even larger than the number of
examples in the dataset. Using random matrix theory and exact solutions in
linear models, we derive the generalization error and training error dynamics
of learning and analyze how they depend on the dimensionality of data and
signal to noise ratio of the learning problem. We find that the dynamics of
gradient descent learning naturally protect against overtraining and
overfitting in large networks. Overtraining is worst at intermediate network
sizes, when the effective number of free parameters equals the number of
samples, and thus can be reduced by making a network smaller or larger.
Additionally, in the high-dimensional regime, low generalization error requires
starting with small initial weights. We then turn to non-linear neural
networks, and show that making networks very large does not harm their
generalization performance. On the contrary, it can in fact reduce
overtraining, even without early stopping or regularization of any sort. We
identify two novel phenomena underlying this behavior in overcomplete models:
first, there is a frozen subspace of the weights in which no learning occurs
under gradient descent; and second, the statistical properties of the
high-dimensional regime yield better-conditioned input correlations which
protect against overtraining. We demonstrate that naive application of
worst-case theories such as Rademacher complexity are inaccurate in predicting
the generalization performance of deep neural networks, and derive an
alternative bound which incorporates the frozen subspace and conditioning
effects and qualitatively matches the behavior observed in simulation.

本文通过随机矩阵理论和线性模型中的准确解，研究了使用梯度下降训练的大型神经网络的泛化动态，发现梯度下降学习的动态自然地保护了大型网络免受过度训练和过拟合的影响，当自由参数的有效数量等于样本数量时，网络过度训练最严重，大小的适当调整可以减少网络过度训练，另外，高维域下，低泛化误差需要从小的初始权重开始。此外，本文还发现了两个新的现象：在梯度下降过程中存在一个冻结的权重子空间，而高维状态的统计特性可保护免受过度训练的影响。