COSMO · STUDY + TEACHING DECK

Computer Science & AI

July 22, 2026

A question worth sharing

Nearest Neighbors and Evaluation

Use local neighborhoods to retrieve examples and test whether an embedding preserves useful relationships.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use nearest neighbor, k-nearest neighbors, neighborhood agreement 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

A query is at (0,0). Which point is nearest: A=(1,1), B=(2,0), or C=(-1,2)?

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.

nearest neighbor

item with smallest distance to a query

k-nearest neighbors

set of the k closest items

neighborhood agreement

fraction of expected neighbors that are recovered

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

The core idea

Keep the idea. Explain the conditions.

\widehat y(q)=\operatorname{majority}\{y_i:i\in N_k(q)\}

The idea

Use local neighborhoods to retrieve examples and test whether an embedding preserves useful relationships.

What to remember

Evaluate an embedding with the relationships and decisions it is supposed to support.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Classify q=(2,2) using the three nearest labeled points: red (1,2), red (2,1), blue (3,3).

Predict each next step

All three are among the local neighbors.
Two of the three labels are red.
Majority vote predicts red.
The prediction depends on the chosen distance and k.

Which step needs a reason, not just a calculation?

See the structure

The structure behind nearest neighbors and evaluation

\widehat y(q)=\operatorname{majority}\{y_i:i\in N_k(q)\}

The chosen representation and metric define the k nearest neighbors; label ties need a rule.

Read the representation

Use local neighborhoods to retrieve examples and test whether an embedding preserves useful relationships.

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 nearest neighbors and evaluation

Your shared task

Teams create a small labeled dataset, compare two metrics, and write a test set before declaring one representation better.

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

Evaluate an embedding with the relationships and decisions it is supposed to support.
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

Compare two embeddings on five known related pairs.

Predict each next step

List the expected partner for each query.
Retrieve the nearest neighbor in each embedding.
Count how often the expected partner appears.
Prefer the embedding with stronger held-out agreement for this task.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

Nearest-neighbor results depend on the representation, metric, scaling, and value of k.

Build the argument

The representation chooses coordinates.
The metric defines closeness.
Scaling changes which dimensions dominate, and k changes how local the decision remains.

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

Stretch together

Take the idea somewhere new.

Transfer challenge

Why can an embedding look good in a two-dimensional plot but fail retrieval?

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

Nearest neighbors power retrieval, recommendation, few-shot classification, anomaly detection, and embedding benchmarks.

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

A query is at (0,0). Which point is nearest: A=(1,1), B=(2,0), or C=(-1,2)?

Question 2

Explain why this claim is true: Nearest-neighbor results depend on the representation, metric, scaling, and value of k.

Question 3

Why can an embedding look good in a two-dimensional plot but fail retrieval?

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

Evaluate an embedding with the relationships and decisions it is supposed to support.

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

A good question belongs to everyone in the room.