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Backpropagation then consists essentially of evaluating this expression from right to left (equivalently, multiplying the previous expression for the derivative from left to right), computing the gradient at each layer on the way; there is an added step, because the gradient of the weights is not just a subexpression: there's an extra ...
Consider an example of a neural network that contains a recurrent layer and a feedforward layer . There are different ways to define the training cost, but the aggregated cost is always the average of the costs of each of the time steps. The cost of each time step can be computed separately.
The perceptron uses the Heaviside step function as the activation function (), and that means that ′ does not exist at zero, and is equal to zero elsewhere, which makes the direct application of the delta rule impossible.
The backpropagation step then repeatedly pops off vertices, which are naturally sorted by their distance from , descending. For each popped node v {\displaystyle v} , we iterate over its predecessors u ∈ p ( v ) {\displaystyle u\in p(v)} : the contribution of v {\displaystyle v} towards δ s ( u ) {\displaystyle \delta _{s}(u)} is added, that is,
An example is the BFGS method which consists in calculating on every step a matrix by which the gradient vector is multiplied to go into a "better" direction, combined with a more sophisticated line search algorithm, to find the "best" value of .
Backpropagation training algorithms fall into three categories: steepest descent (with variable learning rate and momentum, resilient backpropagation); quasi-Newton (Broyden–Fletcher–Goldfarb–Shanno, one step secant);
Backpropagation allowed researchers to train supervised deep artificial neural networks from scratch, initially with little success. Hochreiter 's diplom thesis of 1991 formally identified the reason for this failure in the "vanishing gradient problem", [ 2 ] [ 3 ] which not only affects many-layered feedforward networks , [ 4 ] but also ...
The step function is often used as an activation function, and the outputs are generally restricted to -1, 0, or 1. The weights are updated with w new = w old + η ( t − o ) x i {\displaystyle w_{\text{new}}=w_{\text{old}}+\eta (t-o)x_{i}} where "t" is the target value and " o" is the output of the perceptron, and η {\displaystyle \eta } is ...