Backpropagation Algorithm

import numpy as np

class NeuralNetwork:
    def __init__(self, input_size, hidden_size, output_size):
        # Initialize weights
        self.W1 = np.random.randn(input_size, hidden_size) * 0.01
        self.b1 = np.zeros((1, hidden_size))
        self.W2 = np.random.randn(hidden_size, output_size) * 0.01
        self.b2 = np.zeros((1, output_size))
    
    def sigmoid(self, z):
        return 1 / (1 + np.exp(-np.clip(z, -500, 500)))
    
    def sigmoid_derivative(self, z):
        return z * (1 - z)
    
    def forward(self, X):
        # Hidden layer
        self.z1 = np.dot(X, self.W1) + self.b1
        self.a1 = self.sigmoid(self.z1)
        
        # Output layer
        self.z2 = np.dot(self.a1, self.W2) + self.b2
        self.a2 = self.sigmoid(self.z2)
        
        return self.a2
    
    def backward(self, X, y, output, learning_rate=0.1):
        m = X.shape[0]
        
        # Output layer gradients
        dz2 = output - y
        dW2 = (1/m) * np.dot(self.a1.T, dz2)
        db2 = (1/m) * np.sum(dz2, axis=0, keepdims=True)
        
        # Hidden layer gradients
        dz1 = np.dot(dz2, self.W2.T) * self.sigmoid_derivative(self.a1)
        dW1 = (1/m) * np.dot(X.T, dz1)
        db1 = (1/m) * np.sum(dz1, axis=0, keepdims=True)
        
        # Update weights
        self.W2 -= learning_rate * dW2
        self.b2 -= learning_rate * db2
        self.W1 -= learning_rate * dW1
        self.b1 -= learning_rate * db1
    
    def train(self, X, y, epochs=1000):
        for epoch in range(epochs):
            # Forward pass
            output = self.forward(X)
            
            # Backward pass
            self.backward(X, y, output)
            
            if epoch % 100 == 0:
                loss = np.mean(np.square(output - y))
                print(f"Epoch {epoch}, Loss: {loss:.4f}")

# XOR problem
X = np.array([[0, 0], [0, 1], [1, 0], [1, 1]])
y = np.array([[0], [1], [1], [0]])

# Train network
nn = NeuralNetwork(input_size=2, hidden_size=4, output_size=1)
nn.train(X, y, epochs=1000)

# Test
print("\nPredictions:")
predictions = nn.forward(X)
for i, xi in enumerate(X):
    print(f"Input: {xi} -> Predicted: {predictions[i][0]:.4f}, Actual: {y[i][0]}")