Stochastic

The problem, from first principles

02-mini-batch showed that a gradient computed from a few rows is a cheap, noisy stand-in for the full gradient. Push that idea as far as it goes and use a single row per update. That is stochastic gradient descent: look at one training example, nudge the weights to fit it a little better, move on to the next one. The model updates m times per pass through the data instead of once, and every one of those updates is as cheap as an update can be. This question fits the same linear model as before, one example at a time.

From theory to code

Implement sample_gradient(x_i, y_i, w), the gradient of the squared error on a single example, then stochastic_gradient_descent(X, y, w0, learning_rate, num_epochs, rng), which runs the full loop and returns every intermediate weight vector. The signatures and docstrings are already in the editor.

Constraints

• X has shape (m, d), y has shape (m,), and w0 has shape (d,). x_i is one row of X with shape (d,) and y_i is its scalar target. The model predicts x_i @ w.
• stochastic_gradient_descent draws a fresh shuffle at the start of every epoch with rng.permutation(m) and makes one update per example, so each example is used exactly once per epoch.
• It returns a list of length 1 + num_epochs * m: w0 followed by the weights after each update.
• The inputs X, y and w0 must not be modified.

Hints