Analysis of the Gradient Descent Algorithm for a Deep Neural Network Model with Skip-connections
This provides theoretical guarantees for training deep networks with skip-connections, but it is incremental as it extends existing analysis to this architecture.
The paper analyzes gradient descent for deep neural networks with skip-connections, proving that in the over-parametrized regime, it can find a global minimum exponentially fast with high probability and establish generalization error estimates, showing that early-stopping solutions exist when the target function is in a specific RKHS.
The behavior of the gradient descent (GD) algorithm is analyzed for a deep neural network model with skip-connections. It is proved that in the over-parametrized regime, for a suitable initialization, with high probability GD can find a global minimum exponentially fast. Generalization error estimates along the GD path are also established. As a consequence, it is shown that when the target function is in the reproducing kernel Hilbert space (RKHS) with a kernel defined by the initialization, there exist generalizable early-stopping solutions along the GD path. In addition, it is also shown that the GD path is uniformly close to the functions given by the related random feature model. Consequently, in this "implicit regularization" setting, the deep neural network model deteriorates to a random feature model. Our results hold for neural networks of any width larger than the input dimension.