Аннотация
In recent years, an increasing number of neural network models have included
derivatives with respect to inputs in their loss functions, resulting in
so-called double backpropagation for first-order optimization. However, so far
no general description of the involved derivatives exists. Here, we cover a
wide array of special cases in a very general Hilbert space framework, which
allows us to provide optimized backpropagation rules for many real-world
scenarios. This includes the reduction of calculations for
Frobenius-norm-penalties on Jacobians by roughly a third for locally linear
activation functions. Furthermore, we provide a description of the
discontinuous loss surface of ReLU networks both in the inputs and the
parameters and demonstrate why the discontinuities do not pose a big problem in
reality.
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