Derive the stochastic gradient descent algorithm

The stochastic gradient descent (SGD) algorithm is a popular optimization algorithm used in machine learning. It is an iterative algorithm that updates the model parameters in small steps based on the gradient of the loss function with respect to the parameters. The algorithm works as follows:

1. Initialize the model parameters randomly.
2. Set the learning rate, which determines the step size of the updates.
3. For each training example:
- Compute the gradient of the loss function with respect to the parameters using the current example.
- Update the model parameters by subtracting the gradient multiplied by the learning rate.

The key difference between SGD and regular gradient descent is that in SGD, the gradient is computed and the parameters are updated for each training example, rather than for the entire training set. This makes the algorithm faster and more scalable for large datasets.

The stochastic aspect of the algorithm comes from the fact that the training examples are sampled randomly from the training set, rather than being processed in a fixed order. This randomness can help the algorithm escape from local minima and find better solutions.

Here is the pseudocode for the SGD algorithm:

Input: Training set (X, Y), learning rate α, number of iterations T
Output: Model parameters θ

Initialize θ randomly
for t = 1 to T do
    Sample a training example (x, y) from (X, Y) randomly
    Compute the gradient ∇θ L(θ; x, y) using the current example
    Update the parameters: θ ← θ - α * ∇θ L(θ; x, y)
end for
return θ