Add Problem #40: Linear Regression

[ThierryHJ]

Add Problem #40: Linear Regression

Summary

Adds a new Medium-difficulty problem where users implement linear regression using three approaches:

1. Closed-form (Normal Equation) — solve w = (X^T X)^{-1} X^T y via torch.linalg.lstsq
2. Gradient Descent from scratch — manual gradient computation, no autograd
3. PyTorch nn.Linear — standard training loop with loss.backward() + optimizer.step()

All three methods return (w, b) so that y_pred = X @ w + b.

Motivation

• Linear regression is the most common ML interview warm-up question — asked at Meta, Google, Amazon, and others
• Implementing it three ways tests understanding of the math (normal equation), optimization fundamentals (manual GD), and PyTorch training conventions (nn.Linear + autograd)
• The "three approaches" format is a popular interview structure that tests depth of understanding

Files Changed

File	Status	Description
torch_judge/tasks/linear_regression.py	NEW	Task definition with 6 test cases
templates/40_linear_regression.ipynb	NEW	Template notebook (blank stub)
solutions/40_linear_regression_solution.ipynb	NEW	Reference solution
templates/00_welcome.ipynb	Modified	Added row #40 to Training & Optimization table, updated count 39→40
README.md	Modified	Added row #40 to problem table, updated badge 39→40

Test Cases

#	Test Name	What It Verifies
1	Closed-form returns correct shapes	(w, b) with w: (D,), b: ()
2	Closed-form finds correct weights	Exact recovery on noiseless data (atol=1e-4)
3	Gradient descent converges	Near-correct weights after 2000 steps (atol=0.1)
4	nn.Linear approach works	Training loop produces correct weights (atol=0.1)
5	All three methods agree	Cross-method consistency on noisy data (atol=0.15)
6	Closed-form uses no autograd	Verifies requires_grad=False on output

Conventions Followed

• Auto-discovery via _registry.py — no manual registration needed
• Standard 5-cell notebook format (markdown → import → implementation → debug → submit)
• Colab badge on template notebook
• {fn} placeholder in all test code
• torch.allclose() for numerical comparison with appropriate tolerances
• Difficulty emoji consistent with existing Medium problems (🟡)

How to Test

# From repo root
python3 -c "
from torch_judge.tasks._registry import TASKS
print('linear_regression' in TASKS)  # True
"

Open solutions/40_linear_regression_solution.ipynb in JupyterLab and run all cells — the final check('linear_regression') should show 6/6 tests passed.