Convergence and Implicit Regularization Properties of Gradient Descent for Deep Residual Networks
This provides theoretical guarantees for training deep residual networks, addressing convergence and regularization issues in deep learning.
The authors proved that gradient descent converges linearly to a global optimum for training deep residual networks with constant width and smooth activations, and showed that weight scaling limits have finite 2-variation as depth increases, supported by numerical experiments on supervised learning tasks.
We prove linear convergence of gradient descent to a global optimum for the training of deep residual networks with constant layer width and smooth activation function. We show that if the trained weights, as a function of the layer index, admit a scaling limit as the depth increases, then the limit has finite $p-$variation with $p=2$. Proofs are based on non-asymptotic estimates for the loss function and for norms of the network weights along the gradient descent path. We illustrate the relevance of our theoretical results to practical settings using detailed numerical experiments on supervised learning problems.