The score is a signed distance once it is divided by the length of the weight vector.

Demonstration

The raw score $s = \langle \mathbf{w}, \mathbf{x}\rangle + b$ is the only quantity a perceptron computes. It is tempting to read it as a confidence, and tempting for a reason: it does grow as a point moves away from the boundary. But it is not yet a distance, and the difference matters.

Drop a perpendicular from x to the boundary, meeting it at a point p. Since w is normal to the boundary (Proposition 1), the segment from p to x points along w, so

x = p + t · w/‖w‖

for some signed scalar t, which is the quantity wanted. Take the score of x:

s = ⟨w, x⟩ + b
  = ⟨w, p⟩ + b  +  t · ⟨w, w⟩/‖w‖
  =      0      +  t · ‖w‖

using $\langle \mathbf{w},\mathbf{p}\rangle + b = 0$ because p is on the boundary, and $\langle \mathbf{w},\mathbf{w}\rangle = \lVert\mathbf{w}\rVert^2$. Therefore

t = s / ‖w‖

The score is the distance, multiplied by ‖w‖.

Why the division cannot be skipped

Multiply both w and b by 10. Every score becomes ten times larger. The boundary has not moved, no prediction has changed, and no point is further from anything — yet a reader looking at raw scores would conclude the model became ten times more confident.

This makes the raw score useless for any purpose that compares:

• comparing two points under one model is fine, since ‖w‖ cancels;
• comparing two models is not, because their weight norms differ arbitrarily;
• putting a score into a theorem is not, because the theorem could be satisfied by rescaling rather than by learning.

The scale-free quantity — the geometric margin — is what survives. It is the one that appears in the convergence theorem, and it is the one worth reporting.

Corollary

The same distinction reappears in a different costume in almost every later chapter. A logit is a raw score; a temperature is a rescaling of it; softmax sharpness is a statement about scale rather than about content. When Chapter II.3 divides attention scores by √d_k, it is doing exactly what the division by ‖w‖ does here — removing a scale that would otherwise grow with dimension and be mistaken for information.

The habit to build: whenever a number is about to be interpreted, ask what rescaling of the parameters would change it. If the answer is “any”, the number is not yet a measurement.