18. Regularization Mathematics: L1 Lasso (Sparsity) vs L2 Ridge (Weight Decay) Penalties • Mathematics, Statistics & Machine Learning (Big Tech Prep) • Rishabh Ki Kaksha

Mathematics, Statistics & Machine Learning (Big Tech Prep) › 18. Regularization Mathematics: L1 Lasso (Sparsity) vs L2 Ridge (Weight Decay) Penalties

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Course Syllabus

Module 1: Linear Algebra for Machine Learning & Deep Learning

01. Vectors, Vector Spaces, Dot Products & Norms (L1, L2, L-Infinity) in High Dimensions 02. Matrices, Matrix Multiplication & Linear Transformations (Rotation, Scaling, Shear) 03. Eigenvalues & Eigenvectors: Geometric Intuition, Diagonalization & Spectral Theorem 04. Principal Component Analysis (PCA) & Singular Value Decomposition (SVD) Math

Module 2: Multivariable Calculus, Gradients & Optimization Algorithms

05. Partial Derivatives, Directional Derivatives & The Gradient Vector (Del Operator) 06. The Jacobian Matrix & Hessian Matrix (Second Derivative Curvature Analysis) 07. Multivariate Chain Rule & The Mathematical Backbone of Neural Backpropagation 08. Optimization Algorithms: Gradient Descent, SGD with Momentum, RMSprop & Adam Optimizer Math

Module 3: Probability Theory & Random Variables

09. Probability Foundations: Axioms of Probability, Conditional Probability & Independence 10. Bayes' Theorem & Bayesian Inference: Prior, Likelihood, Posterior & Evidence 11. Probability Distributions: Gaussian (Normal), Bernoulli, Binomial, Poisson & Exponential Distributions

Module 4: Inferential Statistics & Hypothesis Testing

12. Expectation, Variance, Covariance & Pearson Correlation Coefficient 13. Law of Large Numbers & Central Limit Theorem (CLT) with Monte Carlo Simulations 14. Hypothesis Testing: Null Hypothesis, p-Values, Z-Test, Student's t-Test & Chi-Square Test 15. Maximum Likelihood Estimation (MLE) vs Maximum A Posteriori (MAP) Estimation

Module 5: Classical Machine Learning Algorithms from Scratch (NumPy)

16. Linear Regression from Scratch: Ordinary Least Squares (OLS) Closed Form vs Gradient Descent 17. Logistic Regression & Binary Cross-Entropy Loss from Scratch with NumPy 18. Regularization Mathematics: L1 Lasso (Sparsity) vs L2 Ridge (Weight Decay) Penalties

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# 18. Regularization Mathematics: L1 Lasso (Sparsity) vs L2 Ridge (Weight Decay) Penalties

> Mathematics & ML Foundations Notes by Rishabh - Topic 18

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### 1. Fighting Overfitting with Regularization Penalties 🥊

$$\text{Total Loss} = \text{Data Loss}(w) + \lambda \cdot \Omega(w)$$

1. **L2 Ridge Regularization (Tikhonov / Weight Decay):**
   $$\Omega(w) = \frac{1}{2} \|w\|_2^2 = \frac{1}{2} \sum w_j^2 \implies \text{Gradient: } w \leftarrow (1 - \eta \lambda) w - \eta \nabla L$$
   *Weights ko smoothly zero ke paas shrink karta hai.*
2. **L1 Lasso Regularization (Least Absolute Shrinkage):**
   $$\Omega(w) = \|w\|_1 = \sum |w_j| \implies \text{Gradient: } \text{sign}(w)$$
   *Irrelevant weights ko **exactly zero** bana deta hai (Automatic Feature Selection!).*

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### 2. Geometric Diamond vs Circle Intuition 💎⭕

*   **L1 Constraint ($\|w\|_1 \le C$):** Diamond shape with sharp corners on axes $\rightarrow$ Loss contours touch diamond at **axes corners where $w_j = 0$ (Sparsity!)**.
*   **L2 Constraint ($\|w\|_2^2 \le C$):** Smooth circular contour $\rightarrow$ Touches without forcing exact zeros.

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### 3. Master Comparison Matrix

| Regularization | Penalty Term | Produces Sparse Weights (Zeros)? | Mathematical Equivalent |
| :--- | :--- | :--- | :--- |
| **L1 Lasso** | $\lambda \sum |w_i|$ | 👑 **YES (Feature Selection)** | Laplace Prior MAP |
| **L2 Ridge** | $\frac{\lambda}{2} \sum w_i^2$| ❌ NO (Shrinks towards zero) | Gaussian Prior MAP (Weight Decay)|
| **ElasticNet** | $\lambda_1 \|w\|_1 + \lambda_2 \|w\|_2^2$ | ✅ Yes (Balances both) | Correlated features handling |

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### 4. Interview Ke 3 Sawal Is Topic Se

1. **L1 Lasso non-differentiable $w = 0$ par kaise optimize hoti hai?** -> Subgradient Descent ya Proximal Gradient Methods (Soft-Thresholding operator).
2. **Weight Decay AdamW me Adam se mathematically alag kaise hai?** -> Traditional Adam me weight decay gradient me add hota hai jo scaling matrix se divide ho jata hai; AdamW direct weight update step me $(1 - \eta \lambda) w_t$ multiply karta hai.
3. **Hyperparameter $\lambda$ (Alpha) cross-validation me kaise select kiya jata hai?** -> K-Fold Cross-Validation run karke Validation Error vs $\log(\lambda)$ curve ka minima pick karna.

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**Course Completed:** Mathematics, Statistics & Machine Learning Foundations (All 18 Lessons Completed!) 🎉🚀📐📊🤖💎

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