When the Nudges Stop Mattering

In Gradient Descent by Hand the loop ran exactly 500 times. Nobody knows in advance how many steps a line will need, so real training loops decide for themselves when to stop. They stop when the nudges stop mattering.

Fit the line $\hat y = w x + b$ to $n$ points by gradient descent on the mean squared error. This is the same loop as in the lesson:

\text{error}_i = w x_i + b - y_i

\text{grad}_w = \frac{1}{n}\sum_{i=1}^{n} 2\, x_i\, \text{error}_i, \qquad \text{grad}_b = \frac{1}{n}\sum_{i=1}^{n} 2\, \text{error}_i

A nudge is how far a knob moves in one step: $\Delta w = \text{rate} \times \text{grad}_w$ and $\Delta b = \text{rate} \times \text{grad}_b$ . The step itself is $w \leftarrow w - \Delta w$ and $b \leftarrow b - \Delta b$ .

Task: write fit_until_settled(xs, ys, rate, tol, max_steps).

Start at $w = 0$ and $b = 0$ .
In every step, compute all the errors with the current $w$ and $b$ , then work out both nudges from those errors, and only then move both knobs.
After a step, if both nudges were smaller than tol in size (ignore their sign), stop. That step counts.
Never take more than max_steps steps, even if the nudges have not settled.

Return a tuple (w, b, steps). w and b are the final values, each rounded to 4 decimal places, and steps is the number of steps taken, as an integer. Return plain Python numbers.

A cap on the number of steps is not optional in real code. With a learning rate that is far too large, the nudges grow instead of shrinking, and without a cap the loop would never end.

When the Nudges Stop Mattering

Examples

Hints