Implementing XOR Problem with Backpropagation Algorithm

Implementation of XOR Problem Using Backpropagation Algorithm

The XOR (exclusive OR) problem is a classic linearly non-separable challenge in neural networks where single-layer perceptrons fail, requiring multi-layer feedforward networks and backpropagation (BP) algorithm for resolution.

Solution Approach: Network Architecture: A two-layer network (including one hidden layer) with 2 input nodes (corresponding to XOR's two inputs), at least 2 hidden nodes (key for nonlinear separation), and 1 output node (result 0/1). In code implementation, this typically involves initializing weight matrices with appropriate dimensions using random values.

Activation Function: Sigmoid function compresses output to (0,1) range, enabling nonlinear transformation while maintaining differentiability (core requirement for BP algorithm). Code implementation requires defining sigmoid(x) = 1/(1+exp(-x)) and its derivative sigmoid_derivative(x) = sigmoid(x)*(1-sigmoid(x)) for gradient calculations.

Backpropagation Process: Forward Propagation: Input samples compute layer outputs through matrix multiplication and activation functions, generating final predictions. Code typically involves looping through layers with z = w*x + b and a = sigmoid(z). Error Calculation: Compare predictions with true values using squared error function E = 0.5*(target-output)^2. Weight Adjustment: Reverse layer-by-layer gradient computation using chain rule, updating weights via gradient descent to minimize error. Code implementation requires storing intermediate values during forward pass for efficient backward pass calculations.

Key Challenges: Weight Initialization Sensitivity: Random initialization may lead to local optima; code solutions often use Xavier/Glorot initialization. Learning Rate Selection: Too large causes oscillation, too small slows convergence; momentum term implementation (e.g., w_update = learning_rate*gradient + momentum*previous_update) can optimize this.

Extended Considerations: Hidden layer neuron count, adaptive learning rate strategies (like Adam optimizer), and differences between batch training (updating after full dataset) versus online training (per-sample updates) all impact XOR problem solving efficiency. This case serves as a fundamental starting point for understanding BP algorithms in solving nonlinear problems.