Scratch Implementation in Python: Linear Regression

🎯 Core Idea

Implementing linear regression from scratch using NumPy means writing the fit (training) and predict (inference) methods yourself—without relying on scikit-learn. The essence is:

• Fit: learn parameters $\mathbf{w}, b$ by minimizing the loss function using gradient descent.
• Predict: use learned parameters to estimate outputs.

This exercise ensures you truly understand the mechanics—how math translates into code.

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🌱 Intuition & Real-World Analogy

Think of this as teaching a robot to throw darts at a target:

• Each dart = prediction.
• Target = true value.
• Miss distance = error (loss).
• Robot adjusts hand position slightly after each throw (gradient update) until darts consistently land near the bullseye.

Another analogy: cooking by tasting. You try a recipe, taste it (loss), adjust seasoning (parameters), repeat. Gradient descent is just systematic tasting and adjusting.

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📐 Mathematical Foundation

Model:

$$ \hat{y} = X \mathbf{w} + b $$

• $X \in \mathbb{R}^{n \times d}$: feature matrix (n samples, d features)
• $\mathbf{w} \in \mathbb{R}^{d}$: weight vector
• $b \in \mathbb{R}$: bias
• $\hat{y} \in \mathbb{R}^{n}$: predictions

Loss Function (MSE):

$$ J(\mathbf{w}, b) = \frac{1}{2n} \sum_{i=1}^n \left( \hat{y}_i - y_i \right)^2 $$

Factor $\frac{1}{2n}$ simplifies derivative math.

Gradients:

• Gradient w.r.t. weights:

$$ \nabla_{\mathbf{w}} J = \frac{1}{n} X^\top (X \mathbf{w} + b - y) $$

• Gradient w.r.t. bias:

$$ \nabla_{b} J = \frac{1}{n} \sum_{i=1}^n (\hat{y}_i - y_i) $$

Update Rule: For learning rate $\eta$:

$$ \mathbf{w} := \mathbf{w} - \eta \cdot \nabla_{\mathbf{w}} J $$$$ b := b - \eta \cdot \nabla_{b} J $$

This is the heart of gradient descent. Each line in your code should map directly to one of these equations.

📉 Step-by-Step Gradient Derivation

1. Expand loss: $J = \frac{1}{2n}|X\mathbf{w} + b - y|^2$
2. Differentiate wrt $\mathbf{w}$: use chain rule → $\frac{1}{n}X^\top(X\mathbf{w}+b-y)$
3. Differentiate wrt $b$: → $\frac{1}{n}\sum (\hat{y}_i - y_i)$
4. Plug into update rule.

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⚖️ Strengths, Limitations & Trade-offs

Strengths:

• Forces you to understand the mechanics, not just use libraries.
• NumPy vectorization makes training efficient.
• Debugging loss progression teaches convergence behavior.

Limitations:

• Not production-ready (scikit-learn or PyTorch is far more optimized).
• Numerical stability issues if features not normalized.
• Sensitive to learning rate $\eta$: too high → divergence, too low → slow.

Trade-offs:

• Batch vs. Stochastic Gradient Descent:

  • Batch GD: stable, but expensive for large datasets.
  • Stochastic GD: faster updates, noisier convergence.
  • Mini-batch GD: compromise—common in practice.

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🔍 Variants & Extensions

• Stochastic Gradient Descent (SGD): Updates after each sample.

• Mini-Batch Gradient Descent: Updates on chunks of data.

• Momentum / Adam Optimizers: Speed up convergence with adaptive updates.

• Closed-form Solution (Normal Equation):

  $$ \mathbf{w} = (X^\top X)^{-1}X^\top y $$

  Efficient for small d, but impractical for high-dimensional data due to matrix inversion cost.

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🚧 Common Challenges & Pitfalls

• Naive Python loops: Inefficient—always prefer vectorized NumPy operations.

• Unscaled features: Large magnitude features dominate updates → normalize/standardize.

• Debugging: If loss isn’t decreasing, check:

  • Learning rate too high/low.
  • Gradient formula signs.
  • Data preprocessing (bias term handling).

• Overfitting: If weights grow large, consider adding regularization (L2 penalty).

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📚 Reference Pointers

• Andrew Ng’s Notes on Gradient Descent
• Koller & Friedman’s PGM Text (for deeper optimization background)
• Wikipedia: Gradient Descent
• Deep Learning Book (Goodfellow et al.) — Chapter 8: Optimization