Two parameters, one loss

Prediction = 2w + b · L = (2w + b − 6)²

3D surface of weight w, bias b and squared loss L

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Loss surfaceSlice b = 0Sampled loss pathCurrent pointGround −gradient direction

Prediction2

Loss16

Updates0

Weight w

Bias b

Learning rate η

Start at w=1, b=0. Choose a learning rate, then compare the two updates.

Calculation, update log and notes

The input stays at 2 and the target at 6. The ground axes are parameters w and b; height is the same squared loss. The valley satisfies 2w+b=6, so one problem cannot uniquely determine both parameters.

Only the endpoints perform a parameter update. The green path samples 64 intervals along the straight parameter segment between old and new values, recalculating L at every sample. It shows loss while crossing the valley; it is not a straight line between endpoint losses. The yellow arrow shows the direction of −∇L in the parameter plane, with its length scaled for display.

From w=1, b=0, rate 0.1 gives L=0.64 for w only, or L=0 for both. Rate 0.3 gives L=31.36 for w only, or L=64 for both. An oversized step crosses the valley. Both derivatives are calculated at the same old point.

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Plot domain (linear height scale)

Updates

Step	Changed	Parameters (w, b)	Loss L

No update yet.

Data and formula: code/artifacts/lessons/l04.json · Open the original static figure ↗

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