Concepts and steps of error backpropagation

What is error back propagation

The error back propagation method, also known as the Backpropagation algorithm, is a commonly used method for training neural networks method. It uses the chain rule to calculate the error between the neural network output and the label, and backpropagates the error to each node layer by layer to calculate the gradient of each node. These gradients can be used to update the weights and biases of the neural network, bringing the network gradually closer to the optimal solution. Through backpropagation, the neural network can automatically learn and adjust parameters to improve the performance and accuracy of the model.

In error backpropagation, we use the chain rule to calculate the gradient.

We have a neural network with an input x, an output y and a hidden layer. We calculate the gradient of each node in the hidden layer through backpropagation.

First, we need to calculate the error of each node. For the output layer, the error is the difference between the actual value and the predicted value; for the hidden layer, the error is the error of the next layer multiplied by the weight of the current layer. These errors will be used to adjust weights to minimize the difference between predictions and actual values.

Then, we use the chain rule to calculate the gradient. For each weight, we calculate its contribution to the error and then backpropagate this contribution to the previous layer.

Specifically, assume that our neural network has a weight w that connects two nodes. Then, the contribution of this weight to the error is the product of the weight and the error. We backpropagate this contribution to the previous layer by multiplying this contribution by the product of the output of the previous layer and the input of the current layer.

In this way, we can calculate the gradient of each node and then use these gradients to update the weights and biases of the network.

Detailed steps of error back propagation

Suppose we have a neural network with an input layer, a hidden layer and an output layer. The activation function of the input layer is a linear function, the activation function of the hidden layer is a sigmoid function, and the activation function of the output layer is also a sigmoid function.

Forward propagation

1. Input the training set data into the input layer of the neural network and obtain the activation value of the input layer.

2. Pass the activation value of the input layer to the hidden layer, and obtain the activation value of the hidden layer through non-linear transformation of the sigmoid function.

3. Pass the activation value of the hidden layer to the output layer, and obtain the activation value of the output layer through nonlinear transformation of the sigmoid function.

Calculate the error

The error is calculated using the cross-entropy loss between the activations of the output layer and the actual labels. Specifically, for each sample, the cross entropy between the predicted label and the actual label is calculated, and then this cross entropy is multiplied by the corresponding sample weight (the sample weight is usually determined based on the importance and distribution of the sample).

Backpropagation

1. Calculate the gradient of each node of the output layer

According to Chain rule, for each node, we calculate its contribution to the error, and then backpropagate this contribution to the previous layer. Specifically, for each node, we calculate its contribution to the error (i.e., the node's weight times the error), and then multiply this contribution by the product of the previous layer's output and the current layer's input. In this way, we get the gradient of each node of the output layer.

2. Calculate the gradient of each node in the hidden layer

Similarly, according to the chain rule, for each node, we calculate it contribution to the error, and then backpropagates this contribution to the previous layer. Specifically, for each node, we calculate its contribution to the error (i.e., the node's weight times the error), and then multiply this contribution by the product of the previous layer's output and the current layer's input. In this way, we get the gradient of each node in the hidden layer.

3. Update the weights and biases of the neural network

According to the gradient descent algorithm, for each weight, we calculate its contribution to the error The gradient is then multiplied by a learning rate (that is, a parameter that can control the update speed) to obtain the update amount of the weight. For each bias, we also need to calculate its gradient on the error, and then multiply this gradient by a learning rate to get the update amount for that bias.

Iterative training

Repeat the above process (forward propagation, calculation error, back propagation, update parameters) until the stopping criterion is met ( For example, the preset maximum number of iterations is reached or the error reaches the preset minimum value).

This is the detailed process of error backpropagation. It should be noted that in practical applications, we usually use more complex neural network structures and activation functions, as well as more complex loss functions and learning algorithms to improve the performance and generalization ability of the model.

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