See a gradient step succeed—and fail

Before you begin

• Algebra prerequisite; inspect the worked fixture before beginning.
• A text editor/spreadsheet for the manual route. Hosted Python/Jupyter for code exercises; applicable dependencies are in requirements.txt.
• For actual AI calls, a legitimate available cloud service and authorised learner credentials; no particular provider or country is mandatory.

Free original, non-accredited practice. Self-review is not independent assessment. No particular provider, payment platform or country is mandatory; use a legitimate available route or explicitly labelled manual practice.

Inspect a simple loss curve

Gradient descent changes parameters using a derivative and a learning rate. A downhill direction does not guarantee improvement with an arbitrarily large step. Inspect a scalar example before moving to matrices and higher-dimensional objectives. Calculate three losses by hand and identify the minimiser. Inspect the supplied input before using any tool. Write the decision rules in your own words and calculate one small result independently. The reference output is an answer key for this deliberately bounded fixture, not proof of general capability. Policy for this project: Use f(x)=(x-3)^2 and derivative 2(x-3). Apply exactly one step. Round reported values consistently; calculations are deterministic, not an LLM judgement. For learners across regions underserved or excluded by mainstream providers: verify country eligibility, language, account/identity requirements, data terms, cost, quota and connectivity before choosing a cloud service. Use only legitimate available alternatives; never bypass restrictions or invent a country/account. No service access? Complete the labelled manual/fixture route without claiming a live AI run.

Worked example

SOURCE (authored fixture):
{
  "start": 0,
  "learning_rate": 0.25
}
INDEPENDENT FIXTURE TARGET:
{
  "gradient": -6,
  "next_x": 1.5,
  "loss_before": 9,
  "loss_after": 2.25
}

Your practical task

1. Calculate three losses by hand and identify the minimiser.
2. Open input.json and expected.json from the practice pack; keep an unmodified original.
3. Record a manual prediction and the relevant rule before consulting the answer key.
4. Perform one small change and predict its effect before rerunning.
5. Keep the result and your reasoning in the lesson notes; do not mark a simulated run as live.

Evidence to keep

• Inspect a simple loss curve — a saved, inspectable artefact tied to the supplied rule.
• A baseline or change-test record with expected/observed results and a limitation.

Mistakes to watch for

• Learning rate 1.5: next x is 9 and loss increases to 36.
• Treating a supplied answer key, an authored example or self-review as independent field validation.

Check your understanding

Which exact input, rule and expected value would let another learner falsify your conclusion in this activity, and what remains untested?

No-service / offline route

Use the supplied fictional JSON, expected results and paper/text evidence log without a provider account. Manual or constructed completion is labelled; it is not a live cloud integration or trained-model run.

Derive the update rule

Gradient descent changes parameters using a derivative and a learning rate. A downhill direction does not guarantee improvement with an arbitrarily large step. Inspect a scalar example before moving to matrices and higher-dimensional objectives. Connect the derivative sign to the direction of movement. Establish a baseline before changing prompts or code. Keep the original input immutable. Record expected values, observed values and whether each comparison passed. If the expected file conflicts with a justified policy, investigate rather than forcing a convenient match. Policy for this project: Use f(x)=(x-3)^2 and derivative 2(x-3). Apply exactly one step. Round reported values consistently; calculations are deterministic, not an LLM judgement.

Worked example

MANUAL BASELINE / compare every named field
{
  "gradient": -6,
  "next_x": 1.5,
  "loss_before": 9,
  "loss_after": 2.25
}
Keep a separate observed-result column; the answer key is not a model response.

Your practical task

1. Connect the derivative sign to the direction of movement.
2. Open input.json and expected.json from the practice pack; keep an unmodified original.
3. Record expected and observed values separately; explain every mismatch.
4. Perform one small change and predict its effect before rerunning.
5. Keep the result and your reasoning in the lesson notes; do not mark a simulated run as live.

Evidence to keep

• Derive the update rule — a saved, inspectable artefact tied to the supplied rule.
• A baseline or change-test record with expected/observed results and a limitation.

Mistakes to watch for

• Start at 3: gradient is zero and the point stays fixed.
• Treating a supplied answer key, an authored example or self-review as independent field validation.

Check your understanding

Which exact input, rule and expected value would let another learner falsify your conclusion in this activity, and what remains untested?

No-service / offline route

Use the supplied fictional JSON, expected results and paper/text evidence log without a provider account. Manual or constructed completion is labelled; it is not a live cloud integration or trained-model run.

Implement and verify one step

Gradient descent changes parameters using a derivative and a learning rate. A downhill direction does not guarantee improvement with an arbitrarily large step. Inspect a scalar example before moving to matrices and higher-dimensional objectives. Record both parameter change and loss change. Build one inspectable transformation or bounded AI prompt. Keep calculation and validation deterministic where possible. If you make a permitted hosted call, save the actual response separately from the authored examples and verify its material claims; the supplied reference never simulates a live model. Policy for this project: Use f(x)=(x-3)^2 and derivative 2(x-3). Apply exactly one step. Round reported values consistently; calculations are deterministic, not an LLM judgement.

Worked example

PROJECT CONTRACT:
Use f(x)=(x-3)^2 and derivative 2(x-3). Apply exactly one step. Round reported values consistently; calculations are deterministic, not an LLM judgement.
HOSTED REFERENCE: open hosted-lab.ipynb or run reference.py with all fixture files together.
EXPECTED CONSTRUCTED RESULT:
{
  "gradient": -6,
  "next_x": 1.5,
  "loss_before": 9,
  "loss_after": 2.25
}

Your practical task

1. Record both parameter change and loss change.
2. Open hosted-lab.ipynb and the reference code from the practice pack; keep an unmodified original.
3. Record expected and observed values separately; explain every mismatch.
4. Perform Negative learning rate: label invalid for this descent policy.
5. Keep the result and your reasoning in the lesson notes; do not mark a simulated run as live.

Evidence to keep

• Implement and verify one step — a saved, inspectable artefact tied to the supplied rule.
• A baseline or change-test record with expected/observed results and a limitation.

Mistakes to watch for

• Negative learning rate: label invalid for this descent policy.
• Treating a supplied answer key, an authored example or self-review as independent field validation.

Check your understanding

Which exact input, rule and expected value would let another learner falsify your conclusion in this activity, and what remains untested?

No-service / offline route

Use the supplied fictional JSON, expected results and paper/text evidence log without a provider account. Manual or constructed completion is labelled; it is not a live cloud integration or trained-model run.

Make the learning rate fail

Gradient descent changes parameters using a derivative and a learning rate. A downhill direction does not guarantee improvement with an arbitrarily large step. Inspect a scalar example before moving to matrices and higher-dimensional objectives. Use the deliberate overshoot as evidence, not an unexplained software error. Change one assumption at a time and predict the result first. Retain the failure, the revision and the rerun evidence. A failure is useful information: hiding a bad output or editing a test target after seeing the answer destroys the value of the experiment. Policy for this project: Use f(x)=(x-3)^2 and derivative 2(x-3). Apply exactly one step. Round reported values consistently; calculations are deterministic, not an LLM judgement.

Worked example

CHANGE TESTS — predict before running:
1. Learning rate 1.5: next x is 9 and loss increases to 36.
2. Start at 3: gradient is zero and the point stays fixed.
3. Negative learning rate: label invalid for this descent policy.
Keep the original expected.json unchanged.

Your practical task

1. Use the deliberate overshoot as evidence, not an unexplained software error.
2. Open failure-tests.md from the practice pack; keep an unmodified original.
3. Record expected and observed values separately; explain every mismatch.
4. Perform Learning rate 1.5: next x is 9 and loss increases to 36.
5. Keep the result and your reasoning in the lesson notes; do not mark a simulated run as live.

Evidence to keep

• Make the learning rate fail — a saved, inspectable artefact tied to the supplied rule.
• A baseline or change-test record with expected/observed results and a limitation.

Mistakes to watch for

• Learning rate 1.5: next x is 9 and loss increases to 36.
• Treating a supplied answer key, an authored example or self-review as independent field validation.

Check your understanding

Which exact input, rule and expected value would let another learner falsify your conclusion in this activity, and what remains untested?

No-service / offline route

Use the supplied fictional JSON, expected results and paper/text evidence log without a provider account. Manual or constructed completion is labelled; it is not a live cloud integration or trained-model run.

Connect vectors and gradients

Gradient descent changes parameters using a derivative and a learning rate. A downhill direction does not guarantee improvement with an arbitrarily large step. Inspect a scalar example before moving to matrices and higher-dimensional objectives. Apply the same reasoning coordinate by coordinate without pretending a scalar demo proves deep learning. Review what the fixture cannot establish. Explain who owns the decision, where human approval is needed and which access or professional constraints remain. Do not describe a small educational success as production readiness, certification, legal compliance or a guarantee. Policy for this project: Use f(x)=(x-3)^2 and derivative 2(x-3). Apply exactly one step. Round reported values consistently; calculations are deterministic, not an LLM judgement. For learners across regions underserved or excluded by mainstream providers: verify country eligibility, language, account/identity requirements, data terms, cost, quota and connectivity before choosing a cloud service. Use only legitimate available alternatives; never bypass restrictions or invent a country/account. No service access? Complete the labelled manual/fixture route without claiming a live AI run.

Worked example

REVIEW BOUNDARY:
Use f(x)=(x-3)^2 and derivative 2(x-3). Apply exactly one step. Round reported values consistently; calculations are deterministic, not an LLM judgement.

ACCESS / NOT TESTED: unavailable provider, unsupported language and absent external feedback must remain explicit.

Your practical task

1. Apply the same reasoning coordinate by coordinate without pretending a scalar demo proves deep learning.
2. Open README.md and your evidence log from the practice pack; keep an unmodified original.
3. Record the human decision owner, limitations and any untested service access.
4. Perform one small change and predict its effect before rerunning.
5. Keep the result and your reasoning in the lesson notes; do not mark a simulated run as live.

Evidence to keep

• Connect vectors and gradients — a saved, inspectable artefact tied to the supplied rule.
• A baseline or change-test record with expected/observed results and a limitation.

Mistakes to watch for

• Start at 3: gradient is zero and the point stays fixed.
• Treating a supplied answer key, an authored example or self-review as independent field validation.

Check your understanding

Which exact input, rule and expected value would let another learner falsify your conclusion in this activity, and what remains untested?

No-service / offline route

Use the supplied fictional JSON, expected results and paper/text evidence log without a provider account. Manual or constructed completion is labelled; it is not a live cloud integration or trained-model run.

Ship an inspectable optimisation notebook

Gradient descent changes parameters using a derivative and a learning rate. A downhill direction does not guarantee improvement with an arbitrarily large step. Inspect a scalar example before moving to matrices and higher-dimensional objectives. Include the curve, failed step and assumptions. Deliver a reproducible bundle rather than a screenshot alone. Include inputs, rules, output, failure tests, a short README and residual limitations. Ask another person to follow the instructions if possible; self-review and actual external review must be labelled separately. Policy for this project: Use f(x)=(x-3)^2 and derivative 2(x-3). Apply exactly one step. Round reported values consistently; calculations are deterministic, not an LLM judgement.

Worked example

HANDOFF MANIFEST:
input.json — immutable source
expected.json — fixture answer key
actual.json — result actually observed
evidence-log.csv — baseline plus three changes
README.md — reproduction and access route
limitations.md — unresolved failures, rights, approvals and scope
Never bundle a credential.

Your practical task

1. Include the curve, failed step and assumptions.
2. Open README.md and your evidence log from the practice pack; keep an unmodified original.
3. Record the human decision owner, limitations and any untested service access.
4. Perform one small change and predict its effect before rerunning.
5. Keep the result and your reasoning in the lesson notes; do not mark a simulated run as live.

Evidence to keep

• Ship an inspectable optimisation notebook — a saved, inspectable artefact tied to the supplied rule.
• A baseline or change-test record with expected/observed results and a limitation.

Mistakes to watch for

• Negative learning rate: label invalid for this descent policy.
• Treating a supplied answer key, an authored example or self-review as independent field validation.

Check your understanding

Which exact input, rule and expected value would let another learner falsify your conclusion in this activity, and what remains untested?

No-service / offline route

Use the supplied fictional JSON, expected results and paper/text evidence log without a provider account. Manual or constructed completion is labelled; it is not a live cloud integration or trained-model run.

Capstone self-review

Score each criterion 0 (missing), 1 (partial), or 2 (demonstrated). A self-review pass needs at least 8/10 and 2/2 on the critical safety/limits criterion. This is learner practice feedback, not verified assessment.

• Defined task and preserved input: Original input and the project policy are retained; the actual question is clear.
• Checked baseline: The supplied baseline reconciles field by field; unsupported live-run claims are absent.
• Three failure/change tests: All three specified changes have predictions, observed results and explanations.
• Repeatable handoff: Another person can follow the README; supporting practice evidence and unresolved limits are included.
• Safety, rights and access honesty — critical: Use f(x)=(x-3)^2 and derivative 2(x-3). Apply exactly one step. Round reported values consistently; calculations are deterministic, not an LLM judgement. Regional access route is legitimate; credentials and private data are absent; manual/constructed work is clearly labelled.

Review questions

Practice files

• README.md
• evidence-log.csv
• expected.json
• failure-tests.md
• hosted-lab.ipynb
• input.json
• lab_engine.py
• reference.py
• requirements.txt

References and editorial limits

Original authored practice material; deterministic fixtures build-tested. Not independently expert-reviewed or learner-validated. Time estimates unvalidated.

• NumPy linear algebra documentation

Browse all build paths → · Curriculum overview