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Morten Hjorth-Jensen
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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "7d56b2d5",
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"metadata": {
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"editable": true
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},
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"source": [
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"<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)\n",
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"doconce format html exercisesweek37.do.txt -->\n",
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"<!-- dom:TITLE: Exercises week 36 -->"
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]
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},
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{
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"cell_type": "markdown",
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"id": "c7a8e9c7",
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"metadata": {
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"editable": true
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},
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"source": [
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"# Exercises week 36\n",
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"**Implementing gradient descent for Ridge and ordinary Least Squares Regression**\n",
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"\n",
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"Date: **September 8-12, 2025**"
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]
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},
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{
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"cell_type": "markdown",
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"id": "cf8f0ecb",
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"metadata": {
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"editable": true
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},
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"source": [
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"## Learning goals\n",
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"\n",
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"After having completed these exercises you will have:\n",
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"1. Your own code for the implementation of the simplest gradient descent approach applied to ordinary least squares (OLS) and Ridge regression\n",
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"\n",
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"2. Be able to compare the analytical expressions for OLS and Rudge regression with the gradient descent approach\n",
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"\n",
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"3. Explore the role of the learning rate in the gradient descent approach and the hyperparameter $\\lambda$ in Ridge regression\n",
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"\n",
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"4. Scale the data properly"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a67ae548",
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"metadata": {
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"editable": true
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},
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"source": [
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"## Ridge regression and a new Synthetic Dataset\n",
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"\n",
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"We create a synthetic linear regression dataset with a sparse\n",
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"underlying relationship. This means we have many features but only a\n",
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"few of them actually contribute to the target. In our example, we’ll\n",
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"use 10 features with only 3 non-zero weights in the true model. This\n",
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"way, the target is generated as a linear combination of a few features\n",
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"(with known coefficients) plus some random noise. The steps we include are:\n",
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"\n",
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"Decide on the number of samples and features (e.g. 100 samples, 10 features).\n",
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"Define the **true** coefficient vector with mostly zeros (for sparsity). For example, we set $\\hat{\\boldsymbol{\\theta}} = [5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0]$, meaning only features 0, 1, and 6 have a real effect on y.\n",
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"\n",
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"Then we sample feature values for $\\boldsymbol{X}$ randomly (e.g. from a normal distribution). We use a normal distribution so features are roughly centered around 0.\n",
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"Then we compute the target values $y$ using the linear combination $\\boldsymbol{X}\\hat{\\boldsymbol{\\theta}}$ and add some noise (to simulate measurement error or unexplained variance).\n",
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"\n",
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"Below is the code to generate the dataset:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "f2d4a55d",
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"metadata": {
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"collapsed": false,
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"editable": true
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},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"\n",
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"# Set random seed for reproducibility\n",
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"np.random.seed(0)\n",
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"\n",
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"# Define dataset size\n",
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"n_samples = 100\n",
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"n_features = 10\n",
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"\n",
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"# Define true coefficients (sparse linear relationship)\n",
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"theta_true = np.array([5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0])\n",
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"\n",
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"# Generate feature matrix X (n_samples x n_features) with random values\n",
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"X = np.random.randn(n_samples, n_features) # standard normal distribution\n",
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"\n",
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"# Generate target values y with a linear combination of X and theta_true, plus noise\n",
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"noise = 0.5 * np.random.randn(n_samples) # Gaussian noise\n",
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"y = X.dot @ theta_true + noise"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a445583b",
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"metadata": {
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"editable": true
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},
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"source": [
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"This code produces a dataset where only features 0, 1, and 6\n",
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"significantly influence $\\boldsymbol{y}$. The rest of the features have zero true\n",
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"coefficient, so they only contribute noise. For example, feature 0 has\n",
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"a true weight of 5.0, feature 1 has -3.0, and feature 6 has 2.0, so\n",
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"the expected relationship is:"
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]
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},
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{
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"cell_type": "markdown",
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"id": "4a81ddf9",
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"metadata": {
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"editable": true
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},
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"source": [
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"$$\n",
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"y \\approx 5 \\times X_0 \\;-\\; 3 \\times X_1 \\;+\\; 2 \\times X_6 \\;+\\; \\text{noise}.\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ae590275",
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"metadata": {
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"editable": true
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},
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"source": [
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"## Exercise 1, scale your data\n",
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"\n",
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"Before fitting a regression model, it is good practice to normalize or\n",
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"standardize the features. This ensures all features are on a\n",
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"comparable scale, which is especially important when using\n",
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"regularization. Here we will perform standardization, scaling each\n",
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"feature to have mean 0 and standard deviation 1:\n",
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"\n",
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"Compute the mean and standard deviation of each column (feature) in $bm{X}$.\n",
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"Subtract the mean and divide by the standard deviation for each feature.\n",
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"\n",
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"We will also center the target $\\boldsymbol{y}$ to mean $0$. Centering $\\boldsymbol{y}$\n",
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"(and each feature) means the model won’t require a separate intercept\n",
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"term – the data is shifted such that the intercept is effectively 0\n",
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". (In practice, one could include an intercept in the model and not\n",
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"penalize it, but here we simplify by centering.)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "8b40c47a",
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"metadata": {
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"collapsed": false,
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"editable": true
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},
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"outputs": [],
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"source": [
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"# Standardize features (zero mean, unit variance for each feature)\n",
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"X_mean = X.mean(axis=0)\n",
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"X_std = X.std(axis=0)\n",
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"X_std[X_std == 0] = 1 # safeguard to avoid division by zero for constant features\n",
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"X_norm = (X - X_mean) / X_std\n",
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"\n",
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"# Center the target to zero mean (optional, to simplify intercept handling)\n",
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"y_mean = ?\n",
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"y_centered = ?"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ff9c0c81",
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"metadata": {
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"editable": true
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},
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"source": [
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"### 1a)\n",
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"\n",
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"Fill in the necessary details.\n",
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"\n",
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"After this preprocessing, each column of $\\boldsymbol{X}_norm$ has mean zero and standard deviation $1$\n",
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"and $\\boldsymbol{y}_centered$ has mean 0. This makes the optimization landscape\n",
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"nicer and ensures the regularization penalty $\\lambda \\sum_j\n",
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"\\beta_j^2$ treats each coefficient fairly (since features are on the\n",
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"same scale)."
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]
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},
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{
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"cell_type": "markdown",
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"id": "d27c70e4",
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"metadata": {
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"editable": true
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},
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"source": [
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"## Exercise 2, use the analytical formulae for OLS and Ridge regression to find the optimal paramters $\\boldsymbol{theta}$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "9f1e5184",
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"metadata": {
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||||
"collapsed": false,
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||||
"editable": true
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},
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"outputs": [],
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"source": [
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"# Set regularization parameter, either a single value or a vector of values\n",
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"lambda = ?\n",
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"\n",
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"# Analytical form for OLS and Ridge solution: theta_Ridge = (X^T X + lambda * I)^{-1} X^T y and theta_OLS = (X^T X)^{-1} X^T y\n",
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"I = np.eye(n_features)\n",
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"theta_closed_formRidge = ?\n",
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"theta_closed_formOLS = ?\n",
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"\n",
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"print(\"Closed-form Ridge coefficients:\", theta_closed_form)\n",
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"print(\"Closed-form OLS coefficients:\", theta_closed_form)"
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]
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||||
},
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{
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||||
"cell_type": "markdown",
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||||
"id": "2ec556b9",
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||||
"metadata": {
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||||
"editable": true
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||||
},
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||||
"source": [
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"This computes the ridge and OLS regression coefficients directly. The identity\n",
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"matrix $I$ has the same size as $X^T X$ (which is n_features x\n",
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"n_features), and lam * I adds $\\lambda$ to the diagonal of $X^T X. We\n",
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"then invert this matrix and multiply by $X^T y. The result\n",
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"for $\\boldsymbol{\\theta}$ is a NumPy array of shape (n_features,) containing the\n",
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"fitted weights."
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]
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},
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{
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||||
"cell_type": "markdown",
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||||
"id": "a821f0c5",
|
||||
"metadata": {
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||||
"editable": true
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||||
},
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||||
"source": [
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"### 2a)\n",
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"\n",
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"Finalize the OLS and Ridge regression determination of the optimal parameters $bm{\\theta}$."
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]
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||||
},
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{
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||||
"cell_type": "markdown",
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||||
"id": "d637130e",
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||||
"metadata": {
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||||
"editable": true
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||||
},
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"source": [
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"### 2b)\n",
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"\n",
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"Explore the results as function of different values of the hyperparameter $\\lambda$. See for example exercise 4 from week 36."
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]
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||||
},
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||||
{
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||||
"cell_type": "markdown",
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||||
"id": "b455ce7e",
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||||
"metadata": {
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||||
"editable": true
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||||
},
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||||
"source": [
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||||
"## Implementing the simplest form for gradient descent\n",
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"\n",
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"Alternatively, we can fit the ridge regression model using gradient\n",
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"descent. This is useful to visualize the iterative convergence and is\n",
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"necessary if $n$ and $p$ are so large that the closed-form might be\n",
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"too slow or memory-intensive. We derive the gradients from the cost\n",
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"functions defined above. Use the gradients of the Ridge and OLS cost functions with respect to\n",
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"the parameters $\\boldsymbol{\\theta}$ and set up (using the template below) your own gradient descent code for OLS and Ridge regression.\n",
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"\n",
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"Below is a template code for gradient descent implementation of ridge:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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||||
"id": "cfa1eb29",
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||||
"metadata": {
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||||
"collapsed": false,
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||||
"editable": true
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||||
},
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||||
"outputs": [],
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"source": [
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"# Gradient descent parameters, learning rate eta first\n",
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"eta = 0.1\n",
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"# Then number of iterations\n",
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"num_iters = 1000\n",
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"\n",
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"# Initialize weights for gradient descent\n",
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"theta = np.zeros(n_features)\n",
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"\n",
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"# Arrays to store history for plotting\n",
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"cost_history = np.zeros(num_iters)\n",
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"\n",
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"# Gradient descent loop\n",
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"m = n_samples # number of examples\n",
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"for t in range(num_iters):\n",
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" # Compute prediction error\n",
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" error = X_norm.dot(theta) - y_centered \n",
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" # Compute cost for OLS and Ridge (MSE + regularization for Ridge) for monitoring\n",
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" cost_OLS = ?\n",
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" cost_Ridge = ?\n",
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" cost_history[t] = ?\n",
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" # Compute gradients for OSL and Ridge\n",
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" grad_OLS = ?\n",
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" grad_Ridge = ?\n",
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" # Update parameters theta\n",
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" theta_gdOLS = ?\n",
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" theta_gdRidge = ? \n",
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"\n",
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"# After the loop, theta contains the fitted coefficients\n",
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"theta_gdOLS = ?\n",
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"theta_gdRidge = ?\n",
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"print(\"Gradient Descent OLS coefficients:\", theta_gdOLS)\n",
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"print(\"Gradient Descent Ridge coefficients:\", theta_gdRidge)"
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]
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||||
},
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{
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||||
"cell_type": "markdown",
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||||
"id": "dc78d58d",
|
||||
"metadata": {
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||||
"editable": true
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||||
},
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"source": [
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"### 3a)\n",
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||||
"\n",
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"Discuss the results as function of the learning rate paramaters and the number of iterations."
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]
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||||
},
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||||
{
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||||
"cell_type": "markdown",
|
||||
"id": "15060acb",
|
||||
"metadata": {
|
||||
"editable": true
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||||
},
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||||
"source": [
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||||
"### 3b)\n",
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"\n",
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||||
"Add a stopping parameter as function of the number iterations. \n",
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||||
"\n",
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||||
"If everything worked correctly, the learned coefficients should be\n",
|
||||
"close to the true values [5.0, -3.0, 0.0, …, 2.0, …] that we used to\n",
|
||||
"generate the data. Keep in mind that due to regularization and noise,\n",
|
||||
"the learned values will not exactly equal the true ones, but they\n",
|
||||
"should be in the same ballpark."
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
@@ -229,6 +229,7 @@
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||||
<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Linear Regression and Gradient descent</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek37.html">Exercises week 36</a></li>
|
||||
</ul>
|
||||
|
||||
</div>
|
||||
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@@ -229,6 +229,7 @@
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||||
<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Linear Regression and Gradient descent</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek37.html">Exercises week 36</a></li>
|
||||
</ul>
|
||||
|
||||
</div>
|
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@@ -229,6 +229,7 @@
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||||
<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Linear Regression and Gradient descent</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek37.html">Exercises week 36</a></li>
|
||||
</ul>
|
||||
|
||||
</div>
|
||||
|
||||
@@ -229,6 +229,7 @@
|
||||
<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Linear Regression and Gradient descent</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek37.html">Exercises week 36</a></li>
|
||||
</ul>
|
||||
|
||||
</div>
|
||||
|
||||
@@ -229,6 +229,7 @@
|
||||
<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Linear Regression and Gradient descent</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek37.html">Exercises week 36</a></li>
|
||||
</ul>
|
||||
|
||||
</div>
|
||||
|
||||
@@ -229,6 +229,7 @@
|
||||
<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Linear Regression and Gradient descent</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek37.html">Exercises week 36</a></li>
|
||||
</ul>
|
||||
|
||||
</div>
|
||||
|
||||
@@ -229,6 +229,7 @@
|
||||
<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
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Applied Data Analysis and Machine Learning
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<li class="toctree-l1"><a class="reference internal" href="chapter11.html">15. Solving Differential Equations with Deep Learning</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="chapter12.html">16. Convolutional Neural Networks</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="chapter13.html">17. Recurrent neural networks: Overarching view</a></li>
|
||||
|
||||
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|
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Weekly material, notes and exercises</span></p>
|
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<ul class="current nav bd-sidenav">
|
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek34.html">Exercises week 34</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week34.html">Week 34: Introduction to the course, Logistics and Practicalities</a></li>
|
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek35.html">Exercises week 35</a></li>
|
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<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
|
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
|
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<li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Linear Regression and Gradient descent</a></li>
|
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<li class="toctree-l1 current active"><a class="current reference internal" href="#">Exercises week 36</a></li>
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<h1>Exercises week 36</h1>
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<nav aria-label="Page">
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<ul class="visible nav section-nav flex-column">
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning goals</a></li>
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ridge-regression-and-a-new-synthetic-dataset">Ridge regression and a new Synthetic Dataset</a></li>
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-1-scale-your-data">Exercise 1, scale your data</a><ul class="nav section-nav flex-column">
|
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#a">1a)</a></li>
|
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</ul>
|
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</li>
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-2-use-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta">Exercise 2, use the analytical formulae for OLS and Ridge regression to find the optimal paramters <span class="math notranslate nohighlight">\(\boldsymbol{theta}\)</span></a><ul class="nav section-nav flex-column">
|
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">2a)</a></li>
|
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#b">2b)</a></li>
|
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</ul>
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</li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#implementing-the-simplest-form-for-gradient-descent">Implementing the simplest form for gradient descent</a><ul class="nav section-nav flex-column">
|
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">3a)</a></li>
|
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">3b)</a></li>
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<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
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doconce format html exercisesweek37.do.txt -->
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<!-- dom:TITLE: Exercises week 36 --><section class="tex2jax_ignore mathjax_ignore" id="exercises-week-36">
|
||||
<h1>Exercises week 36<a class="headerlink" href="#exercises-week-36" title="Link to this heading">#</a></h1>
|
||||
<p><strong>Implementing gradient descent for Ridge and ordinary Least Squares Regression</strong></p>
|
||||
<p>Date: <strong>September 8-12, 2025</strong></p>
|
||||
<section id="learning-goals">
|
||||
<h2>Learning goals<a class="headerlink" href="#learning-goals" title="Link to this heading">#</a></h2>
|
||||
<p>After having completed these exercises you will have:</p>
|
||||
<ol class="arabic simple">
|
||||
<li><p>Your own code for the implementation of the simplest gradient descent approach applied to ordinary least squares (OLS) and Ridge regression</p></li>
|
||||
<li><p>Be able to compare the analytical expressions for OLS and Rudge regression with the gradient descent approach</p></li>
|
||||
<li><p>Explore the role of the learning rate in the gradient descent approach and the hyperparameter <span class="math notranslate nohighlight">\(\lambda\)</span> in Ridge regression</p></li>
|
||||
<li><p>Scale the data properly</p></li>
|
||||
</ol>
|
||||
</section>
|
||||
<section id="ridge-regression-and-a-new-synthetic-dataset">
|
||||
<h2>Ridge regression and a new Synthetic Dataset<a class="headerlink" href="#ridge-regression-and-a-new-synthetic-dataset" title="Link to this heading">#</a></h2>
|
||||
<p>We create a synthetic linear regression dataset with a sparse
|
||||
underlying relationship. This means we have many features but only a
|
||||
few of them actually contribute to the target. In our example, we’ll
|
||||
use 10 features with only 3 non-zero weights in the true model. This
|
||||
way, the target is generated as a linear combination of a few features
|
||||
(with known coefficients) plus some random noise. The steps we include are:</p>
|
||||
<p>Decide on the number of samples and features (e.g. 100 samples, 10 features).
|
||||
Define the <strong>true</strong> coefficient vector with mostly zeros (for sparsity). For example, we set <span class="math notranslate nohighlight">\(\hat{\boldsymbol{\theta}} = [5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0]\)</span>, meaning only features 0, 1, and 6 have a real effect on y.</p>
|
||||
<p>Then we sample feature values for <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> randomly (e.g. from a normal distribution). We use a normal distribution so features are roughly centered around 0.
|
||||
Then we compute the target values <span class="math notranslate nohighlight">\(y\)</span> using the linear combination <span class="math notranslate nohighlight">\(\boldsymbol{X}\hat{\boldsymbol{\theta}}\)</span> and add some noise (to simulate measurement error or unexplained variance).</p>
|
||||
<p>Below is the code to generate the dataset:</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import numpy as np
|
||||
|
||||
# Set random seed for reproducibility
|
||||
np.random.seed(0)
|
||||
|
||||
# Define dataset size
|
||||
n_samples = 100
|
||||
n_features = 10
|
||||
|
||||
# Define true coefficients (sparse linear relationship)
|
||||
theta_true = np.array([5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0])
|
||||
|
||||
# Generate feature matrix X (n_samples x n_features) with random values
|
||||
X = np.random.randn(n_samples, n_features) # standard normal distribution
|
||||
|
||||
# Generate target values y with a linear combination of X and theta_true, plus noise
|
||||
noise = 0.5 * np.random.randn(n_samples) # Gaussian noise
|
||||
y = X.dot @ theta_true + noise
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>This code produces a dataset where only features 0, 1, and 6
|
||||
significantly influence <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span>. The rest of the features have zero true
|
||||
coefficient, so they only contribute noise. For example, feature 0 has
|
||||
a true weight of 5.0, feature 1 has -3.0, and feature 6 has 2.0, so
|
||||
the expected relationship is:</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
y \approx 5 \times X_0 \;-\; 3 \times X_1 \;+\; 2 \times X_6 \;+\; \text{noise}.
|
||||
\]</div>
|
||||
</section>
|
||||
<section id="exercise-1-scale-your-data">
|
||||
<h2>Exercise 1, scale your data<a class="headerlink" href="#exercise-1-scale-your-data" title="Link to this heading">#</a></h2>
|
||||
<p>Before fitting a regression model, it is good practice to normalize or
|
||||
standardize the features. This ensures all features are on a
|
||||
comparable scale, which is especially important when using
|
||||
regularization. Here we will perform standardization, scaling each
|
||||
feature to have mean 0 and standard deviation 1:</p>
|
||||
<p>Compute the mean and standard deviation of each column (feature) in <span class="math notranslate nohighlight">\(bm{X}\)</span>.
|
||||
Subtract the mean and divide by the standard deviation for each feature.</p>
|
||||
<p>We will also center the target <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> to mean <span class="math notranslate nohighlight">\(0\)</span>. Centering <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span>
|
||||
(and each feature) means the model won’t require a separate intercept
|
||||
term – the data is shifted such that the intercept is effectively 0
|
||||
. (In practice, one could include an intercept in the model and not
|
||||
penalize it, but here we simplify by centering.)</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Standardize features (zero mean, unit variance for each feature)
|
||||
X_mean = X.mean(axis=0)
|
||||
X_std = X.std(axis=0)
|
||||
X_std[X_std == 0] = 1 # safeguard to avoid division by zero for constant features
|
||||
X_norm = (X - X_mean) / X_std
|
||||
|
||||
# Center the target to zero mean (optional, to simplify intercept handling)
|
||||
y_mean = ?
|
||||
y_centered = ?
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<section id="a">
|
||||
<h3>1a)<a class="headerlink" href="#a" title="Link to this heading">#</a></h3>
|
||||
<p>Fill in the necessary details.</p>
|
||||
<p>After this preprocessing, each column of <span class="math notranslate nohighlight">\(\boldsymbol{X}_norm\)</span> has mean zero and standard deviation <span class="math notranslate nohighlight">\(1\)</span>
|
||||
and <span class="math notranslate nohighlight">\(\boldsymbol{y}_centered\)</span> has mean 0. This makes the optimization landscape
|
||||
nicer and ensures the regularization penalty <span class="math notranslate nohighlight">\(\lambda \sum_j
|
||||
\beta_j^2\)</span> treats each coefficient fairly (since features are on the
|
||||
same scale).</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="exercise-2-use-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta">
|
||||
<h2>Exercise 2, use the analytical formulae for OLS and Ridge regression to find the optimal paramters <span class="math notranslate nohighlight">\(\boldsymbol{theta}\)</span><a class="headerlink" href="#exercise-2-use-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta" title="Link to this heading">#</a></h2>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Set regularization parameter, either a single value or a vector of values
|
||||
lambda = ?
|
||||
|
||||
# Analytical form for OLS and Ridge solution: theta_Ridge = (X^T X + lambda * I)^{-1} X^T y and theta_OLS = (X^T X)^{-1} X^T y
|
||||
I = np.eye(n_features)
|
||||
theta_closed_formRidge = ?
|
||||
theta_closed_formOLS = ?
|
||||
|
||||
print("Closed-form Ridge coefficients:", theta_closed_form)
|
||||
print("Closed-form OLS coefficients:", theta_closed_form)
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>This computes the ridge and OLS regression coefficients directly. The identity
|
||||
matrix <span class="math notranslate nohighlight">\(I\)</span> has the same size as <span class="math notranslate nohighlight">\(X^T X\)</span> (which is n_features x
|
||||
n_features), and lam * I adds <span class="math notranslate nohighlight">\(\lambda\)</span> to the diagonal of <span class="math notranslate nohighlight">\(X^T X. We
|
||||
then invert this matrix and multiply by \)</span>X^T y. The result
|
||||
for <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span> is a NumPy array of shape (n_features,) containing the
|
||||
fitted weights.</p>
|
||||
<section id="id1">
|
||||
<h3>2a)<a class="headerlink" href="#id1" title="Link to this heading">#</a></h3>
|
||||
<p>Finalize the OLS and Ridge regression determination of the optimal parameters <span class="math notranslate nohighlight">\(bm{\theta}\)</span>.</p>
|
||||
</section>
|
||||
<section id="b">
|
||||
<h3>2b)<a class="headerlink" href="#b" title="Link to this heading">#</a></h3>
|
||||
<p>Explore the results as function of different values of the hyperparameter <span class="math notranslate nohighlight">\(\lambda\)</span>. See for example exercise 4 from week 36.</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="implementing-the-simplest-form-for-gradient-descent">
|
||||
<h2>Implementing the simplest form for gradient descent<a class="headerlink" href="#implementing-the-simplest-form-for-gradient-descent" title="Link to this heading">#</a></h2>
|
||||
<p>Alternatively, we can fit the ridge regression model using gradient
|
||||
descent. This is useful to visualize the iterative convergence and is
|
||||
necessary if <span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(p\)</span> are so large that the closed-form might be
|
||||
too slow or memory-intensive. We derive the gradients from the cost
|
||||
functions defined above. Use the gradients of the Ridge and OLS cost functions with respect to
|
||||
the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span> and set up (using the template below) your own gradient descent code for OLS and Ridge regression.</p>
|
||||
<p>Below is a template code for gradient descent implementation of ridge:</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Gradient descent parameters, learning rate eta first
|
||||
eta = 0.1
|
||||
# Then number of iterations
|
||||
num_iters = 1000
|
||||
|
||||
# Initialize weights for gradient descent
|
||||
theta = np.zeros(n_features)
|
||||
|
||||
# Arrays to store history for plotting
|
||||
cost_history = np.zeros(num_iters)
|
||||
|
||||
# Gradient descent loop
|
||||
m = n_samples # number of examples
|
||||
for t in range(num_iters):
|
||||
# Compute prediction error
|
||||
error = X_norm.dot(theta) - y_centered
|
||||
# Compute cost for OLS and Ridge (MSE + regularization for Ridge) for monitoring
|
||||
cost_OLS = ?
|
||||
cost_Ridge = ?
|
||||
cost_history[t] = ?
|
||||
# Compute gradients for OSL and Ridge
|
||||
grad_OLS = ?
|
||||
grad_Ridge = ?
|
||||
# Update parameters theta
|
||||
theta_gdOLS = ?
|
||||
theta_gdRidge = ?
|
||||
|
||||
# After the loop, theta contains the fitted coefficients
|
||||
theta_gdOLS = ?
|
||||
theta_gdRidge = ?
|
||||
print("Gradient Descent OLS coefficients:", theta_gdOLS)
|
||||
print("Gradient Descent Ridge coefficients:", theta_gdRidge)
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<section id="id2">
|
||||
<h3>3a)<a class="headerlink" href="#id2" title="Link to this heading">#</a></h3>
|
||||
<p>Discuss the results as function of the learning rate paramaters and the number of iterations.</p>
|
||||
</section>
|
||||
<section id="id3">
|
||||
<h3>3b)<a class="headerlink" href="#id3" title="Link to this heading">#</a></h3>
|
||||
<p>Add a stopping parameter as function of the number iterations.</p>
|
||||
<p>If everything worked correctly, the learned coefficients should be
|
||||
close to the true values [5.0, -3.0, 0.0, …, 2.0, …] that we used to
|
||||
generate the data. Keep in mind that due to regularization and noise,
|
||||
the learned values will not exactly equal the true ones, but they
|
||||
should be in the same ballpark.</p>
|
||||
</section>
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning goals</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ridge-regression-and-a-new-synthetic-dataset">Ridge regression and a new Synthetic Dataset</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-1-scale-your-data">Exercise 1, scale your data</a><ul class="nav section-nav flex-column">
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#a">1a)</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-2-use-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta">Exercise 2, use the analytical formulae for OLS and Ridge regression to find the optimal paramters <span class="math notranslate nohighlight">\(\boldsymbol{theta}\)</span></a><ul class="nav section-nav flex-column">
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">2a)</a></li>
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#b">2b)</a></li>
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</li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#implementing-the-simplest-form-for-gradient-descent">Implementing the simplest form for gradient descent</a><ul class="nav section-nav flex-column">
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">3a)</a></li>
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">3b)</a></li>
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{
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"id": "7d56b2d5",
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||||
"source": [
|
||||
"# Exercises week 36\n",
|
||||
"**Implementing gradient descent for Ridge and ordinary Least Squares Regression**\n",
|
||||
"\n",
|
||||
"Date: **September 8-12, 2025**"
|
||||
]
|
||||
},
|
||||
{
|
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"cell_type": "markdown",
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"id": "cf8f0ecb",
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"metadata": {
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"editable": true
|
||||
},
|
||||
"source": [
|
||||
"## Learning goals\n",
|
||||
"\n",
|
||||
"After having completed these exercises you will have:\n",
|
||||
"1. Your own code for the implementation of the simplest gradient descent approach applied to ordinary least squares (OLS) and Ridge regression\n",
|
||||
"\n",
|
||||
"2. Be able to compare the analytical expressions for OLS and Rudge regression with the gradient descent approach\n",
|
||||
"\n",
|
||||
"3. Explore the role of the learning rate in the gradient descent approach and the hyperparameter $\\lambda$ in Ridge regression\n",
|
||||
"\n",
|
||||
"4. Scale the data properly"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "a67ae548",
|
||||
"metadata": {
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"editable": true
|
||||
},
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||||
"source": [
|
||||
"## Ridge regression and a new Synthetic Dataset\n",
|
||||
"\n",
|
||||
"We create a synthetic linear regression dataset with a sparse\n",
|
||||
"underlying relationship. This means we have many features but only a\n",
|
||||
"few of them actually contribute to the target. In our example, we’ll\n",
|
||||
"use 10 features with only 3 non-zero weights in the true model. This\n",
|
||||
"way, the target is generated as a linear combination of a few features\n",
|
||||
"(with known coefficients) plus some random noise. The steps we include are:\n",
|
||||
"\n",
|
||||
"Decide on the number of samples and features (e.g. 100 samples, 10 features).\n",
|
||||
"Define the **true** coefficient vector with mostly zeros (for sparsity). For example, we set $\\hat{\\boldsymbol{\\theta}} = [5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0]$, meaning only features 0, 1, and 6 have a real effect on y.\n",
|
||||
"\n",
|
||||
"Then we sample feature values for $\\boldsymbol{X}$ randomly (e.g. from a normal distribution). We use a normal distribution so features are roughly centered around 0.\n",
|
||||
"Then we compute the target values $y$ using the linear combination $\\boldsymbol{X}\\hat{\\boldsymbol{\\theta}}$ and add some noise (to simulate measurement error or unexplained variance).\n",
|
||||
"\n",
|
||||
"Below is the code to generate the dataset:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 1,
|
||||
"id": "f2d4a55d",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import numpy as np\n",
|
||||
"\n",
|
||||
"# Set random seed for reproducibility\n",
|
||||
"np.random.seed(0)\n",
|
||||
"\n",
|
||||
"# Define dataset size\n",
|
||||
"n_samples = 100\n",
|
||||
"n_features = 10\n",
|
||||
"\n",
|
||||
"# Define true coefficients (sparse linear relationship)\n",
|
||||
"theta_true = np.array([5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0])\n",
|
||||
"\n",
|
||||
"# Generate feature matrix X (n_samples x n_features) with random values\n",
|
||||
"X = np.random.randn(n_samples, n_features) # standard normal distribution\n",
|
||||
"\n",
|
||||
"# Generate target values y with a linear combination of X and theta_true, plus noise\n",
|
||||
"noise = 0.5 * np.random.randn(n_samples) # Gaussian noise\n",
|
||||
"y = X.dot @ theta_true + noise"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "a445583b",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"This code produces a dataset where only features 0, 1, and 6\n",
|
||||
"significantly influence $\\boldsymbol{y}$. The rest of the features have zero true\n",
|
||||
"coefficient, so they only contribute noise. For example, feature 0 has\n",
|
||||
"a true weight of 5.0, feature 1 has -3.0, and feature 6 has 2.0, so\n",
|
||||
"the expected relationship is:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "4a81ddf9",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"y \\approx 5 \\times X_0 \\;-\\; 3 \\times X_1 \\;+\\; 2 \\times X_6 \\;+\\; \\text{noise}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "ae590275",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"## Exercise 1, scale your data\n",
|
||||
"\n",
|
||||
"Before fitting a regression model, it is good practice to normalize or\n",
|
||||
"standardize the features. This ensures all features are on a\n",
|
||||
"comparable scale, which is especially important when using\n",
|
||||
"regularization. Here we will perform standardization, scaling each\n",
|
||||
"feature to have mean 0 and standard deviation 1:\n",
|
||||
"\n",
|
||||
"Compute the mean and standard deviation of each column (feature) in $bm{X}$.\n",
|
||||
"Subtract the mean and divide by the standard deviation for each feature.\n",
|
||||
"\n",
|
||||
"We will also center the target $\\boldsymbol{y}$ to mean $0$. Centering $\\boldsymbol{y}$\n",
|
||||
"(and each feature) means the model won’t require a separate intercept\n",
|
||||
"term – the data is shifted such that the intercept is effectively 0\n",
|
||||
". (In practice, one could include an intercept in the model and not\n",
|
||||
"penalize it, but here we simplify by centering.)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"id": "8b40c47a",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# Standardize features (zero mean, unit variance for each feature)\n",
|
||||
"X_mean = X.mean(axis=0)\n",
|
||||
"X_std = X.std(axis=0)\n",
|
||||
"X_std[X_std == 0] = 1 # safeguard to avoid division by zero for constant features\n",
|
||||
"X_norm = (X - X_mean) / X_std\n",
|
||||
"\n",
|
||||
"# Center the target to zero mean (optional, to simplify intercept handling)\n",
|
||||
"y_mean = ?\n",
|
||||
"y_centered = ?"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "ff9c0c81",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### 1a)\n",
|
||||
"\n",
|
||||
"Fill in the necessary details.\n",
|
||||
"\n",
|
||||
"After this preprocessing, each column of $\\boldsymbol{X}_norm$ has mean zero and standard deviation $1$\n",
|
||||
"and $\\boldsymbol{y}_centered$ has mean 0. This makes the optimization landscape\n",
|
||||
"nicer and ensures the regularization penalty $\\lambda \\sum_j\n",
|
||||
"\\beta_j^2$ treats each coefficient fairly (since features are on the\n",
|
||||
"same scale)."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "d27c70e4",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"## Exercise 2, use the analytical formulae for OLS and Ridge regression to find the optimal paramters $\\boldsymbol{theta}$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"id": "9f1e5184",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# Set regularization parameter, either a single value or a vector of values\n",
|
||||
"lambda = ?\n",
|
||||
"\n",
|
||||
"# Analytical form for OLS and Ridge solution: theta_Ridge = (X^T X + lambda * I)^{-1} X^T y and theta_OLS = (X^T X)^{-1} X^T y\n",
|
||||
"I = np.eye(n_features)\n",
|
||||
"theta_closed_formRidge = ?\n",
|
||||
"theta_closed_formOLS = ?\n",
|
||||
"\n",
|
||||
"print(\"Closed-form Ridge coefficients:\", theta_closed_form)\n",
|
||||
"print(\"Closed-form OLS coefficients:\", theta_closed_form)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "2ec556b9",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"This computes the ridge and OLS regression coefficients directly. The identity\n",
|
||||
"matrix $I$ has the same size as $X^T X$ (which is n_features x\n",
|
||||
"n_features), and lam * I adds $\\lambda$ to the diagonal of $X^T X. We\n",
|
||||
"then invert this matrix and multiply by $X^T y. The result\n",
|
||||
"for $\\boldsymbol{\\theta}$ is a NumPy array of shape (n_features,) containing the\n",
|
||||
"fitted weights."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "a821f0c5",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### 2a)\n",
|
||||
"\n",
|
||||
"Finalize the OLS and Ridge regression determination of the optimal parameters $bm{\\theta}$."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "d637130e",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### 2b)\n",
|
||||
"\n",
|
||||
"Explore the results as function of different values of the hyperparameter $\\lambda$. See for example exercise 4 from week 36."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "b455ce7e",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"## Implementing the simplest form for gradient descent\n",
|
||||
"\n",
|
||||
"Alternatively, we can fit the ridge regression model using gradient\n",
|
||||
"descent. This is useful to visualize the iterative convergence and is\n",
|
||||
"necessary if $n$ and $p$ are so large that the closed-form might be\n",
|
||||
"too slow or memory-intensive. We derive the gradients from the cost\n",
|
||||
"functions defined above. Use the gradients of the Ridge and OLS cost functions with respect to\n",
|
||||
"the parameters $\\boldsymbol{\\theta}$ and set up (using the template below) your own gradient descent code for OLS and Ridge regression.\n",
|
||||
"\n",
|
||||
"Below is a template code for gradient descent implementation of ridge:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"id": "cfa1eb29",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# Gradient descent parameters, learning rate eta first\n",
|
||||
"eta = 0.1\n",
|
||||
"# Then number of iterations\n",
|
||||
"num_iters = 1000\n",
|
||||
"\n",
|
||||
"# Initialize weights for gradient descent\n",
|
||||
"theta = np.zeros(n_features)\n",
|
||||
"\n",
|
||||
"# Arrays to store history for plotting\n",
|
||||
"cost_history = np.zeros(num_iters)\n",
|
||||
"\n",
|
||||
"# Gradient descent loop\n",
|
||||
"m = n_samples # number of examples\n",
|
||||
"for t in range(num_iters):\n",
|
||||
" # Compute prediction error\n",
|
||||
" error = X_norm.dot(theta) - y_centered \n",
|
||||
" # Compute cost for OLS and Ridge (MSE + regularization for Ridge) for monitoring\n",
|
||||
" cost_OLS = ?\n",
|
||||
" cost_Ridge = ?\n",
|
||||
" cost_history[t] = ?\n",
|
||||
" # Compute gradients for OSL and Ridge\n",
|
||||
" grad_OLS = ?\n",
|
||||
" grad_Ridge = ?\n",
|
||||
" # Update parameters theta\n",
|
||||
" theta_gdOLS = ?\n",
|
||||
" theta_gdRidge = ? \n",
|
||||
"\n",
|
||||
"# After the loop, theta contains the fitted coefficients\n",
|
||||
"theta_gdOLS = ?\n",
|
||||
"theta_gdRidge = ?\n",
|
||||
"print(\"Gradient Descent OLS coefficients:\", theta_gdOLS)\n",
|
||||
"print(\"Gradient Descent Ridge coefficients:\", theta_gdRidge)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "dc78d58d",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### 3a)\n",
|
||||
"\n",
|
||||
"Discuss the results as function of the learning rate paramaters and the number of iterations."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "15060acb",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### 3b)\n",
|
||||
"\n",
|
||||
"Add a stopping parameter as function of the number iterations. \n",
|
||||
"\n",
|
||||
"If everything worked correctly, the learned coefficients should be\n",
|
||||
"close to the true values [5.0, -3.0, 0.0, …, 2.0, …] that we used to\n",
|
||||
"generate the data. Keep in mind that due to regularization and noise,\n",
|
||||
"the learned values will not exactly equal the true ones, but they\n",
|
||||
"should be in the same ballpark."
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
@@ -46,4 +46,5 @@ parts:
|
||||
- file: exercisesweek35.ipynb
|
||||
- file: week35.ipynb
|
||||
- file: exercisesweek36.ipynb
|
||||
- file: week36.ipynb
|
||||
- file: week36.ipynb
|
||||
- file: exercisesweek37.ipynb
|
||||
Reference in New Issue
Block a user