What does it mean for a machine to learn?

The model

A straight line with slope w and intercept b turns a temperature x into a guess ŷ:

ŷ = w · x + b

On this page the machine’s line is not learned yet.

The score

For each of the n days, square the miss, then average. This is the mean squared error, and it is the number the machine tries to shrink:

L = 1n Σ (ŷ_i − y_i)²

The “typical miss” in the figure is its square root, √L, which brings it back to plain coffees. Why square the misses at all? That’s the next puff.

The nudge

Each nudge moves both knobs a small step η (the learning rate) against the slope of the score:

w ← w − η ∂L∂w     b ← b − η ∂L∂b

∂L∂w = 2n Σ (ŷ_i − y_i) x_i     ∂L∂b = 2n Σ (ŷ_i − y_i)

The page first rescales temperatures to sit around zero, a standard trick that lets both knobs learn at a similar pace. That’s why the machine needs only a few dozen nudges here.

Why not just solve it?

For a straight line there is a formula, least squares, that gives the best line in one go: w = 2.87, b = −21.9. The machine’s nudging lands on the same line. But a neural network has millions or billions of knobs and no such formula. Nudging is the only way, and the rest of this course is built on it.