Updates and Boundaries

The Boundary Evolving

The same states can be drawn as boundary lines. This lesson overlays the exact zero-score line after each update and names the final line.

highlighted = computed this step

The boundary after each update

Every update gives a new line w*x+b=0. The legend lists the exact state after each update, ending at w=(2,-1), b=0.

wx+b=0w x+b=0
Boundary evolutionEach line is w*x+b equals zero after one update.Boundary after each updatePerceptron boundary evolutionTraining points with exact boundary lines after each update.0:+11:+12:-13:-1updatesu1: w=(1,0), b=1u2: w=(1,-1), b=0u3: w=(2,0), b=1u4: w=(2,-1), b=0final boundary: w=(2,-1), b=0

Why the final line works here

The final state makes the boundary 2x-y=0. Its sign matches all 4 displayed labels.

2x−y=02x-y=0
Boundary evolutionEach line is w*x+b equals zero after one update.Boundary after each updatePerceptron boundary evolutionTraining points with exact boundary lines after each update.0:+11:+12:-13:-1updatesu1: w=(1,0), b=1u2: w=(1,-1), b=0u3: w=(2,0), b=1u4: w=(2,-1), b=0final boundary: w=(2,-1), b=0

The sign rule on a number line

The same decision lives on one axis: place each final activation z=w*x+b and cut at z=0. The cut is inclusive on the positive side, so z=0 would read class 1; here the activations 2, 1, -1, -2 recompute to the same 4 labels the table already shows.

class=1  ⟺  z≥0; else −1\text{class}=1\iff z\ge 0;\ \text{else }-1
The sign rule on a number lineEach exact activation z=w*x+b is placed on the activation axis; the cut at z=0 is the sign rule.threshold t=0 on the exact activation axisclass -1 (z below 0)class +1 (z ≥ 0)-3-2-1012threshold z=0z=2z=1z=-1z=-2the activation axis and the cut at z=0 are exact (the perceptron has notranscendental, so every z=w*x+b is an exact integer); z=0 is the STATED decisionpolicy of the sign rule placed by the author, NOT a learned boundary: this doesnot claim the cut is fitted, NOT a probability, NOT a generalization claim, andseparating this shown run is not convergence

Summary

The plot is an overlay of exact boundary states, not an animation. The final state separates only the shown data in this run.

final shown boundary separates the shown rows\text{final shown boundary separates the shown rows}
Boundary evolutionEach line is w*x+b equals zero after one update.Boundary after each updatePerceptron boundary evolutionTraining points with exact boundary lines after each update.0:+11:+12:-13:-1updatesu1: w=(1,0), b=1u2: w=(1,-1), b=0u3: w=(2,0), b=1u4: w=(2,-1), b=0final boundary: w=(2,-1), b=0