Is the Skip Connection Provable to Reform the Neural Network Loss Landscape?
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The residual network is now one of the most effective structures in deep learning, which utilizes the skip connections to “guarantee" the performance will not get worse. However, the non-convexity of the neural network makes it unclear whether the skip connections do provably improve the learning ability since the nonlinearity may create many local minima. In some previous works <cit.>, it is shown that despite the non-convexity, the loss landscape of the two-layer ReLU network has good properties when the number m of hidden nodes is very large. In this paper, we follow this line to study the topology (sub-level sets) of the loss landscape of deep ReLU neural networks with a skip connection and theoretically prove that the skip connection network inherits the good properties of the two-layer network and skip connections can help to control the connectedness of the sub-level sets, such that any local minima worse than the global minima of some two-layer ReLU network will be very “shallow". The “depth" of these local minima are at most O(m^(η-1)/n), where n is the input dimension, η<1. This provides a theoretical explanation for the effectiveness of the skip connection in deep learning.
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