Backpropagation

Backpropagation is a supervised machine learning algorithm used in neural networks. It is an optimization algorithm used to adjust the weights of the neural network's parameters, such that the output produced is closer to the actual output. Backpropagation is also known as the reverse propagation of errors.

How Backpropagation Works

The algorithm works by calculating the error of the output layer and propagating it back to previous layers. The error of the output layer is calculated by comparing the predicted output with the actual output. The difference between the predicted and actual output is known as the loss, which is then used to adjust the weights of the neural network.

The error is then propagated back to the previous layer, and the weights are adjusted again. This process is repeated until the error is minimized, and the predicted output is as close as possible to the actual output.

Backpropagation Example

Let's say we have a neural network with one input layer, one hidden layer, and one output layer. The input layer has two neurons, the hidden layer has three neurons, and the output layer has one neuron. The neural network is trained to predict the price of a house based on its square footage.

The input layer takes the square footage of the house as input, and the output layer produces the predicted price. The weights of the neural network are adjusted using backpropagation, such that the predicted price is as close as possible to the actual price.

Code Example

Here is an example of backpropagation in Python:

import numpy as np

# Define the neural network
X = np.array([[0, 1, 0], [1, 0, 1]])
y = np.array([[1], [0]])

# Initialize the weights
w1 = np.random.randn(3, 4)
w2 = np.random.randn(4, 1)

# Forward propagation
z1 = np.dot(X, w1)
a1 = np.tanh(z1)
z2 = np.dot(a1, w2)
output = sigmoid(z2)

# Calculate the error
loss = y - output

# Backpropagation
deriv_loss = loss * sigmoid_deriv(output)
deriv_w2 = np.dot(a1.T, deriv_loss)
deriv_a1 = np.dot(deriv_loss, w2.T)
deriv_z1 = deriv_a1 * tanh_deriv(a1)
deriv_w1 = np.dot(X.T, deriv_z1)

# Update the weights
w1 += learning_rate * deriv_w1
w2 += learning_rate * deriv_w2
```.

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