Backpropagation · Explore together

Follow the gradient

Follow values forward, sensitivities backward, and then make one weight update.

One person controls the experiment. The other predicts and questions. Swap roles after a trial.

y^=wx,L=(y^−y)2\hat y=wx,\qquad L=(\hat y-y)^2

One input. One weight. Follow the chain rule.

Scroll across the diagram to follow the whole calculation.

Input x2 Weight w1 × Prediction? Target y6 − Loss L? error e = ŷ − y? then L = e² Backward pass: multiply local derivatives
Ready to traceŷ = wx, e = ŷ − y, L = e²
Choose the step size before updating the weight.
Predict with a partner

Will increasing w increase or decrease the loss at w = 1? Explain before tracing.

The weight stays fixed throughout the forward and backward passes.

Check the chain rule

dL/dw = (dL/de)(de/dŷ)(dŷ/dw) = 2e · 1 · x. With e = −4 and x = 2, the gradient is −16.

The update is separate: w ← w − η(dL/dw). The graph’s weight changes only when you choose Apply update.

Compare your evidence

What can you explain now?

Ask your partner to restate your reasoning. An experiment suggests a pattern; your explanation should say why it holds and when it applies.

  1. Why does each backward step multiply two derivatives?
  2. What stays fixed while the backward pass runs?
  3. Can a correct gradient still lead to a worse loss after an update?