Analysis of the Gradient Descent Algorithm for a Deep Neural Network Model with Skip-connections

Abstract

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.

29 pages, 4 figures

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• Kolmogorov Width Decay and Poor Approximators in Machine Learning: Shallow Neural Networks, Random Feature Models and Neural Tangent Kernels
• Global Convergence of Gradient Descent for Deep Linear Residual Networks
• A Revision of Neural Tangent Kernel-based Approaches for Neural Networks