Regression, Regularization and Gradient-Based Learning Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Regression, Regularization and Gradient-Based Learning notes for GATE DA: 62 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Regression, Regularization and Gradient-Based Learning

Chapter Journey

1. Linear Least-Squares Regression

The foundation. Minimizing the sum of squared residuals to find the best-fit line. Closed-form solution and its geometric interpretation.

2. Ridge Regression and Regularization

The problem with OLS: overfitting and ill-conditioned matrices. Introducing the L2 penalty to shrink weights. The bias-variance tradeoff.

3. Gradient-Based Parameter Updates

Moving beyond closed-form solutions. Batch, Stochastic, and Mini-batch Gradient Descent. Learning rates and convergence.

By the end of this chapter, you will be able to analytically solve linear models, regularize them to prevent overfitting, and optimize them iteratively.

The Core Idea of Ridge Regression

The Intuition Behind Regularization

Ordinary Least Squares (OLS) finds weights that minimize the training error. However, when data is noisy or features are highly correlated, OLS assigns excessively large weights to fit the noise. This leads to overfitting.

Ridge Regression modifies the objective by adding a penalty term proportional to the square of the magnitude of the weights.

Goal: Minimize training error while keeping the weights as small as possible.
Effect: Shrinks the coefficients towards zero, reducing model complexity and variance.
Result: A model that generalizes much better to unseen data, even if it slightly underfits the training data.

The Ridge Regression Objective and Closed Form

Mathematical Formulation

Let $X \in R^{n \times d}$ be the design matrix, $y \in R^{n}$ be the target vector, and $w \in R^{d}$ be the weight vector.

Objective Function

J (w) = ∣∣ y - X w ∣ ∣_{2}^{2} + λ ∣∣ w ∣ ∣_{2}^{2}

where $λ \geq 0$ is the regularization hyperparameter.

Closed-Form Solution

To find the optimal $w$ , we take the gradient of $J (w)$ with respect to $w$ and set it to zero:

\nabla_{w} J (w) = - 2 X^{T} (y - X w) + 2 λ w = 0

X^{T} X w + λ w = X^{T} y

(X^{T} X + λ I) w = X^{T} y

w_{r i d g e} = (X^{T} X + λ I)^{- 1} X^{T} y

Why Ridge Solves the Singular Matrix Problem

The Algebraic Rescue

In OLS, we need to invert $X^{T} X$ . If $n < d$ or if features are highly collinear, $X^{T} X$ is singular or ill-conditioned.

The Eigenvalue Shift

Let the eigenvalues of $X^{T} X$ be $σ_{1}^{2}, σ_{2}^{2}, \dots, σ_{d}^{2}$ . The eigenvalues of $(X^{T} X + λ I)$ are simply $(σ_{1}^{2} + λ), \dots, (σ_{d}^{2} + λ)$ .

Since $λ > 0$ , every eigenvalue is strictly greater than zero.
The matrix $(X^{T} X + λ I)$ is guaranteed to be strictly positive definite.
It is always invertible, providing a unique, stable solution.

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Regression, Regularization and Gradient-Based Learning Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Regression, Regularization and Gradient-Based Learning notes for GATE DA: 62 study cards covering concepts, formulas, shortcuts and exam traps, plus solved pr

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