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Ridge Regression

Implement Ridge Regression (L2 regularization) using the regularized normal equation.

Given feature matrix $X$ (without bias column), target vector $y$ , and regularization parameter $\lambda$ , compute: $w = (X_a^T X_a + \lambda I')^{-1} X_a^T y$

where $X_a$ has a column of ones prepended, and $I'$ is the identity matrix with the top-left element set to 0 (we don't regularize the bias).

Return the weight vector as a list, rounded to 4 decimal places.

Example:

Input:

X = [[1], [2], [3]]
y = [2, 4, 6]
lambda_reg = 0.0

Output:

[0.0, 2.0]

Reasoning:

First, we prepend a column of ones to the feature matrix $X$ to get $X_a$ : $X_a = [[1, 1], [1, 2], [1, 3]]$
Then, we calculate $X_a^T X_a$ and $X_a^T y$ : $X_a^T X_a = [[3, 6], [6, 14]]$ and $X_a^T y = [[2 + 4 + 6], [2 + 8 + 12]] = [[12], [22]]$
Next, we compute the regularized matrix $X_a^T X_a + \lambda I'$ , where $\lambda = 0.0$ and $I'$ is the identity matrix with the top-left element set to 0: $X_a^T X_a + \lambda I' = [[0, 6], [6, 14]]$
Finally, we calculate the weight vector $w = (X_a^T X_a + \lambda I')^{-1} X_a^T y$ : $w = [[0.0], [2.0]]$ , which when rounded to 4 decimal places gives the output $[0.0, 2.0]$

Constraints:

X is a 2D list (n x m), y is a 1D list of n targets
lambda_reg is a positive float
Return list of (m+1) weights rounded to 4 decimal places
First weight is the intercept (not regularized)
Implement without numpy

Editor

Python 3.13.1

Test Results

0/0

Run code to see test results.