The rule moves the boundary toward the point it got wrong, and only then.

Demonstration

The rule is one line:

if y(⟨w, x⟩ + b) ≤ 0:   w ← w + y·x ,  b ← b + y

Its shape is not arbitrary, and the quickest way to see that is to ask what the update does to the very score that triggered it. Write w′ = w + yx and b′ = b + y, and recompute the margin of the same example:

y(⟨w′, x⟩ + b′) = y(⟨w, x⟩ + b)  +  y²(⟨x, x⟩ + 1)
                = m  +  (‖x‖² + 1)

using y² = 1. The margin increases by ‖x‖² + 1, which is strictly positive. Every update strictly improves the example that caused it. The improvement may not be enough to fix it in one step, and fixing it may break another example, but the direction is never wrong.

Geometrically: adding yx to w tilts the normal toward x when y = +1 and away from it when y = −1. Since the normal fixes the orientation (Proposition 1), the boundary swings with it.

The idleness is the substance

The clause that looks like a mere efficiency — do nothing when the point is already right — is the load-bearing part.

In the convergence proof (I.2.B03) the norm of w is bounded by expanding

‖w + y·x‖² = ‖w‖² + 2y⟨w, x⟩ + ‖x‖²

and discarding the middle term because it is non-positive. It is non-positive precisely because the update only happens when y(⟨w, x⟩ + b) ≤ 0. An algorithm that updated on every example would have that term unsigned, the norm could grow faster than √k, and the proof would collapse.

So “only when wrong” is not a shortcut. It is the hypothesis that makes the theorem true.

What kind of algorithm this is

The rule is exactly subgradient descent, at learning rate 1, on

ℓ = max(0, −y(⟨w, x⟩ + b))

— a loss that is zero on the correct side and linear on the wrong side. That identification is worked out in full in problem I.2.B07, and it is worth having because it places the perceptron inside a family rather than leaving it as a curiosity. Change the loss to max(0, 1 − m) and the same derivation yields the support-vector machine’s update. Change it to the logistic loss of Chapter I.4 and the idle region disappears entirely: every example, however well classified, contributes a small gradient forever.

That last difference is the one to carry forward. The perceptron has a region where it does nothing; almost nothing after Chapter I.4 does. A loss with no flat region is what allows a model to keep improving a point it already gets right — which is how margins get large without anyone asking for it.