Your task is to implement the Stochastic Gradient Descent (SGD) optimization algorithm from scratch in Python and use it to train a linear regression model. SGD is a widely used optimization method in machine learning. In this problem, you will write a Python function to perform a single update step of SGD for a given set of model parameters. Additionally, you will implement the computation of gradients using backpropagation.

For this purpose, you will use a linear regression model with a single feedforward layer. The input to your method will be a set of feature-output pairs, and your implementation should handle the parameter updates accordingly.

Inputs:

• X (ndarray): The input data matrix (m x n), where m is the number of examples and n is the number of features.
• y (ndarray): The target values (m x 1) corresponding to the input data.
• w_init (ndarray): The initial weights (n x 1) for the linear regression model.
• alpha (float): The learning rate, a positive scalar that determines the step size in the gradient descent.
• num_iters (int): The number of iterations to run the gradient descent algorithm.

Outputs:

• w (ndarray): The optimized weights (n x 1) after running gradient descent.
• cost_history (list): A list of the cost function values at each iteration, useful for monitoring convergence.