Optimization and Generalization of Shallow Neural Networks with Quadratic Activation Functions
This work addresses the theoretical understanding of neural network training dynamics for researchers in machine learning theory, but it is incremental as it focuses on a specific activation function and setup.
The authors studied the optimization and generalization of shallow neural networks with quadratic activations in an over-parametrized teacher-student setup, establishing conditions for no spurious minima and showing that gradient descent converges with small generalization error, as confirmed by numerical experiments.
We study the dynamics of optimization and the generalization properties of one-hidden layer neural networks with quadratic activation function in the over-parametrized regime where the layer width $m$ is larger than the input dimension $d$. We consider a teacher-student scenario where the teacher has the same structure as the student with a hidden layer of smaller width $m^*\le m$. We describe how the empirical loss landscape is affected by the number $n$ of data samples and the width $m^*$ of the teacher network. In particular we determine how the probability that there be no spurious minima on the empirical loss depends on $n$, $d$, and $m^*$, thereby establishing conditions under which the neural network can in principle recover the teacher. We also show that under the same conditions gradient descent dynamics on the empirical loss converges and leads to small generalization error, i.e. it enables recovery in practice. Finally we characterize the time-convergence rate of gradient descent in the limit of a large number of samples. These results are confirmed by numerical experiments.