COSMO · STUDY + TEACHING DECK

Computer Science & AI

August 8, 2026

A question worth sharing

Training with Backpropagation

Connect loss, gradients, gradient descent, and backpropagation into one transparent training step.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use loss, gradient, backpropagation to name what you notice.

Convince

Build an argument a partner can check.

The goal is a method someone else can use.

The opener

Start with a question.

Opening problem

For prediction yhat=wx with x=2, target y=6, and w=1, what is the squared loss?

Think · 2 min

Sketch, test a small case, or make a first guess.

Pair · 3 min

Compare methods. Find one thing you agree on and one question.

Be ready to share your route, even if it is unfinished.

Compare before checking

Put two attempts in conversation.

Explain

Tell a partner how you started. Let them finish before responding.

Question

Which step feels certain? Which step still needs evidence?

Build

Combine one useful idea from each attempt into a new route.

Can you explain your partner’s method in your own words?

Board conversation

Notice before naming.

Look for structure

Try a small case and record every step.
Separate what changes from what stays fixed.
State the pattern in a sentence before naming the theorem.

Make a conjecture

We think ___ because ___. A case that might break this is ___.

What changed? What stayed the same?

Useful language

Now the language has a job.

loss

number measuring prediction error

gradient

vector of partial derivatives of loss

backpropagation

reverse-mode chain-rule method for computing gradients

Use one of these words to explain something you already noticed.

The core idea

Keep the idea. Explain the conditions.

\frac{dL}{dw}=2(wx-y)x,\qquad w_{\mathrm{new}}=w-\eta\frac{dL}{dw}

The idea

Connect loss, gradients, gradient descent, and backpropagation into one transparent training step.

What to remember

Training is a measured cycle: predict, score the error, propagate responsibility, update, and test again.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Take one gradient-descent step for L=(wx-y)^2 with x=2, y=6, w=1, and learning rate 0.1.

Predict each next step

Differentiate: dL/dw=2(wx-y)x.
At w=1 the gradient is 2(2-6)(2)=-16.
Update w to 1-0.1(-16).
The new weight is 2.6.

Which step needs a reason, not just a calculation?

See the structure

The structure behind training with backpropagation

\frac{dL}{dw}=2(wx-y)x,\qquad w_{\mathrm{new}}=w-\eta\frac{dL}{dw}

For squared loss, backpropagation finds the gradient and gradient descent chooses the update.

Read the representation

Connect loss, gradients, gradient descent, and backpropagation into one transparent training step.

Look together

Point to the part that explains the relationship. Change one input and predict what changes.

What can you see here that was harder to see in words?

Two approaches

Compare the reasoning.

Structural method

Name the objects and conditions.
Use a reusable relationship.
Explain why the method must work.

Brute force

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Which route can you justify? What would you test first?

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Repair together

Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.

A useful correction explains why the original step fails.

Work together

Investigate training with backpropagation

Your shared task

Students annotate a computational graph with forward values, local derivatives, backward gradients, and one parameter update.

Explore

4 min - test cases

Compare

5 min - form a rule

Defend

4 min - defend the rule

Everyone should be able to explain the group’s final claim.

Share → connect

What did the room discover?

Bring your evidence

Share a conjecture, a useful example, or a route that changed.

Connections to investigate

Training is a measured cycle: predict, score the error, propagate responsibility, update, and test again.
A complete solution names the condition that makes the method valid.
A counterexample is useful when it exposes a missing condition.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Backpropagate through yhat=w2(w1x) with x=1, y=6, w1=2, and w2=2.

Predict each next step

The prediction is 4 and dL/dyhat=2(4-6)=-4.
Since dyhat/dw2=w1x=2, dL/dw2=-8.
Since dyhat/dw1=w2x=2, dL/dw1=-8.
The same upstream derivative is reused efficiently.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

Backpropagation computes gradients; gradient descent uses those gradients to update parameters.

Build the argument

The chain rule moves sensitivity backward through the computational graph.
Each node combines an upstream derivative with its local derivative.
The optimizer then chooses how far to move using the resulting gradients.

What does each step rely on? Where are the conditions used?

Stretch together

Take the idea somewhere new.

Transfer challenge

Why can an excessively large learning rate increase the loss?

Choose a route

Name the objects, the conditions, and a relationship that might help. Start with one test case.

What transfers from the earlier example? What is different?

Peer review

Check the argument together.

Restate

Explain the other person’s route before evaluating it.

Test

Try a small case or boundary case. Does every step still make sense?

Revise

Fix an assumption, add a reason, or find a second method.

Where did your thinking change?

Beyond this problem

Where might this idea help?

A connection

The same training logic appears in deep learning, differentiable simulation, inverse problems, and scientific parameter fitting.

Discuss the model

What objects or quantities are represented?
Which relationship from today is preserved?
What assumption would you check before trusting it?

Invent one more situation with the same structure.

Exit ticket

Leave with an explanation.

Question 1

For prediction yhat=wx with x=2, target y=6, and w=1, what is the squared loss?

Question 2

Explain why this claim is true: Backpropagation computes gradients; gradient descent uses those gradients to update parameters.

Question 3

Why can an excessively large learning rate increase the loss?

Choose one. Write an answer with a reason, then compare with a partner.

Reflect together

Make your thinking visible.

I used to think…

Name a claim or approach that changed.

Now I can explain…

Give the reason, not only the result.

I still wonder…

Write a question worth bringing to the next circle.

Share one sentence. Ask one follow-up question.

The idea to keep

Training is a measured cycle: predict, score the error, propagate responsibility, update, and test again.

Use the matching practice sets to move from direct examples to proof and transfer.

A good question belongs to everyone in the room.