How do you implement a neural network from scratch?

Implementing a neural network from scratch involves designing its architecture, coding the forward and backward propagation steps, and training the model using optimization techniques. Start by defining the network’s layers, activation functions, and loss metric. For example, a basic feedforward network might include an input layer, one hidden layer with ReLU activation, and an output layer with a sigmoid for binary classification. Initialize weights randomly and set biases to zero or small values to break symmetry. The forward pass computes predictions by passing input data through these layers, applying matrix multiplications and activation functions. For instance, if your input is a vector of size 2 and the hidden layer has 3 neurons, the weights for the first layer would be a 2x3 matrix.

Next, implement backpropagation to update the weights based on the loss gradient. Calculate the loss (e.g., mean squared error or cross-entropy) between predictions and true labels. Then, compute gradients using the chain rule, starting from the output layer backward. For example, if using sigmoid output and cross-entropy loss, the gradient of the loss with respect to the output weights simplifies to (predicted - true) * activation_derivative. Update weights using an optimizer like stochastic gradient descent (SGD), where weights = weights - learning_rate * gradients. Repeat this process for a fixed number of epochs or until convergence. Tools like numerical gradient checking can help verify correctness during development.

Finally, test the network on a simple dataset to validate functionality. For instance, train it on the XOR problem, where inputs are (0,0), (0,1), (1,0), (1,1) and outputs are 0, 1, 1, 0. Use a learning rate of 0.1 and 10,000 epochs. Monitor loss to ensure it decreases over time. Practical challenges include tuning hyperparameters (e.g., learning rate, layer sizes) and avoiding pitfalls like vanishing gradients. Extensions like adding more layers, dropout, or momentum-based optimizers can improve performance but increase complexity. This hands-on approach builds a foundational understanding of how neural networks operate internally, which is valuable for debugging and customizing models later.