For deep feedforward networks with piecewise-smooth activations, the autodiff gradient is shown to be the unique limit of gradients of smoothed activations, a limiting Frechet subgradient, and equal to the true gradient wherever the loss is continuously differentiable.
Approximation and Gradient Descent Training with Neural Networks
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abstract
It is well understood that neural networks with carefully hand-picked weights provide powerful function approximation and that they can be successfully trained in over-parametrized regimes. Since over-parametrization ensures zero training error, these two theories are not immediately compatible. Recent work uses the smoothness that is required for approximation results to extend a neural tangent kernel (NTK) optimization argument to an under-parametrized regime and show direct approximation bounds for networks trained by gradient flow. Since gradient flow is only an idealization of a practical method, this paper establishes analogous results for networks trained by gradient descent.
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Mathematical analysis of the gradients in deep learning
For deep feedforward networks with piecewise-smooth activations, the autodiff gradient is shown to be the unique limit of gradients of smoothed activations, a limiting Frechet subgradient, and equal to the true gradient wherever the loss is continuously differentiable.