Backpropagation: Assigning Blame for an Error · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Blame travels backward

Computing · Machine Learning · ages 21-22
Name ______________________   Date ____________

  1. A layer computes u = 2x + 1 and the next layer applies y = u^3, so y = (2x + 1)^3. Find dy/dx at x = 1.

    Answer: ______________

  2. A tiny two-layer network first computes u = 2x + 1, then squares the result, so the output is y = (2x + 1)^2. Find the rate of change dy/dx at x = 1. This is the same calculation backpropagation does at one layer boundary.

    Answer: ______________

  3. What happens on the forward pass?

    • Inputs flow to a prediction
    • Errors walk backward at once
    • Weights delete themselves

  4. Static backpropagation also rewires neurons and layers.

    Circle one:   True   False

  5. A network has input x, hidden value h, output o, and loss L. In Leibniz notation, which expression gives the blame dL/dx as a fully expanded product with one local derivative for each of the three adjacent links?

    • dL/do plus do/dh plus dh/dx
    • dL/do divided by (do/dh times dh/dx)
    • dL/do times do/dh times dh/dx
    • dL/dh times dh/dx

  6. Which direction does the error travel on the backward pass?

    • From input straight to the printer
    • In a circle around one neuron
    • From output back to input

  7. A three-layer network computes y = (x^2 + 1)^2. Find dy/dx at x = 1 by applying the chain rule once per layer boundary.

    Answer: ______________

  8. If a neuron's activation sits in the flat tail of a sigmoid squashing function, the chain rule makes that weight learn slowly, because the derivative multiplied into the error signal is near zero.

    Circle one:   True   False

  9. Early layers barely move while later ones learn. The squash has flat tails. Why?

    • The signal shrinks going back
    • The loss function ran out of ink
    • Forward passes skip early layers

  10. A network output is y = x(x + 1)^2, where the input x is multiplied by a squashed signal (x + 1)^2. Find dy/dx at x = 1.

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Blame travels backward W1-mt_5bwaSWfF96-s1

  1. 54 · Inner derivative 2 times outer derivative 27 gives 54, one multiplication per layer boundary.
  2. 12 · The chain rule multiplies the derivative of the outer function by the derivative of the inner function: 6 times 2 gives 12.
  3. Inputs flow to a prediction · The forward pass propagates inputs layer by layer into an output.
  4. False · Static means weights and biases only, never the wiring.
  5. dL/do times do/dh times dh/dx · The chain rule says rates of change multiply along a path. The fully expanded form shows one factor for each link: dL/do, do/dh, and dh/dx. The shorter product groups the first two links as dL/dh, so it does not show the requested expanded form.
  6. From output back to input · Errors propagate recursively from the output back through the layers.
  7. 8 · Each layer boundary contributes one derivative, and backpropagation multiplies them all together: 4 times 2 gives 8.
  8. True · In the flat tail the sigmoid barely changes, so its derivative is nearly zero, and multiplying by nearly zero shrinks the error signal reaching every weight behind it.
  9. The signal shrinks going back · Near-zero local rates multiply the backward signal down to almost nothing.
  10. 8 · Use the product rule with the chain rule inside it: the derivative is 1 times 4 plus 1 times 4, which is 8.
Worksheet · LightMySky