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This is the reason why backpropagation requires that the activation function be differentiable. (Nevertheless, the ReLU activation function, which is non-differentiable at 0, has become quite popular, e.g. in AlexNet) The first factor is straightforward to evaluate if the neuron is in the output layer, because then = and
Neural backpropagation is the phenomenon in which, after the action potential of a neuron creates a voltage spike down the axon (normal propagation), another impulse is generated from the soma and propagates towards the apical portions of the dendritic arbor or dendrites (from which much of the original input current originated).
Back_Propagation_Through_Time(a, y) // a[t] is the input at time t. y[t] is the output Unfold the network to contain k instances of f do until stopping criterion is met: x := the zero-magnitude vector // x is the current context for t from 0 to n − k do // t is time. n is the length of the training sequence Set the network inputs to x, a[t ...
Next we rewrite in the last term as the sum over all weights of each weight times its corresponding input : = ′ [] Because we are only concerned with the i {\\displaystyle i} th weight, the only term of the summation that is relevant is x i w j i {\\displaystyle x_{i}w_{ji}} .
Hardware advances have meant that from 1991 to 2015, computer power (especially as delivered by GPUs) has increased around a million-fold, making standard backpropagation feasible for networks several layers deeper than when the vanishing gradient problem was recognized.
Almeida–Pineda recurrent backpropagation is an extension to the backpropagation algorithm that is applicable to recurrent neural networks. It is a type of supervised learning . It was described somewhat cryptically in Richard Feynman 's senior thesis, and rediscovered independently in the context of artificial neural networks by both Fernando ...
In 1970, Seppo Linnainmaa published the modern form of backpropagation in his master thesis (1970). [23] [24] [13] G.M. Ostrovski et al. republished it in 1971. [25] [26] Paul Werbos applied backpropagation to neural networks in 1982 [7] [27] (his 1974 PhD thesis, reprinted in a 1994 book, [28] did not yet describe the algorithm [26]).
Universal approximation theorems are existence theorems: They simply state that there exists such a sequence ,,, and do not provide any way to actually find such a sequence. They also do not guarantee any method, such as backpropagation, might actually find such a sequence. Any method for searching the space of neural networks, including ...