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.
The /suppress lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems
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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.