@@ -378,21 +378,23 @@ document.write(`
@@ -406,8 +408,8 @@ document.write(`
-Exercises week 36#
+Exercises week 37#
Implementing gradient descent for Ridge and ordinary Least Squares Regression
Date: September 8-12, 2025
Scale the data properly
Ridge regression and a new Synthetic Dataset#
-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:
-Decide on the number of samples and features (e.g. 100 samples, 10 features). -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.
-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. -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).
-Below is the code to generate the dataset:
-
-
-
-
-
-
-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
-This code produces a dataset where only features 0, 1, and 6 -significantly influence \(\boldsymbol{y}\). The rest of the features have zero true -coefficient. 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:
+Simple one-dimensional second-order polynomial#
+We start with a very simple function
\[
-y \approx 5 \times x_0 \;-\; 3 \times x_1 \;+\; 2 \times x_6 \;+\; \text{noise}.
+\f(x)= 2-x+5x^2,
\]
-You can remove the noise if you wish to.
+defined for \(x\in [-2,2]\). You can add noise if you wish.
+We are going to fit this function with a polynomial ansatz. The easiest thing is to set up a second-order polynomial and see if you can fit the above function. +Feel free to play around with higher-order polynomials.
Exercise 1, scale your data#
@@ -477,13 +442,14 @@ regularization. Here we will perform standardization, scaling each feature to have mean 0 and standard deviation 1.1a)#
-Compute the mean and standard deviation of each column (feature) in \(\boldsymbol{X}\). +
Compute the mean and standard deviation of each column (feature) in your design/feature matrix \(\boldsymbol{X}\). Subtract the mean and divide by the standard deviation for each feature.
We will also center the target \(\boldsymbol{y}\) to mean \(0\). Centering \(\boldsymbol{y}\) (and each feature) means the model does not 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.)
+penalize it, but here we simplify by centering.) +Choose \(n=100\) data points and set up \(\boldsymbol{x}, \)\boldsymbol{y} and the design matrix \(\boldsymbol{X}\).# Standardize features (zero mean, unit variance for each feature)
@@ -507,8 +473,12 @@ nicer and ensures the regularization penalty
-Exercise 2, use the analytical formulae for OLS and Ridge regression to find the optimal paramters \(\boldsymbol{\theta}\)#
+
+Exercise 2, calculate the gradients#
+Find the gradients for OLS and Ridge regression using the mean-squared error as cost/loss function.
+
+
+Exercise 3, use the analytical formulae for OLS and Ridge regression to find the optimal paramters \(\boldsymbol{\theta}\)#
# Set regularization parameter, either a single value or a vector of values
@@ -531,16 +501,16 @@ then invert this matrix and multiply by \)X^T y\( is a NumPy array of shape (n\)_\(features,) containing the
fitted parameters \)\boldsymbol{\theta}$..
-2a)#
+3a)#
Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters \(\boldsymbol{\theta}\).
-2b)#
+3b)#
Explore the results as function of different values of the hyperparameter \(\lambda\). See for example exercise 4 from week 36.
-
-Exercise 3, Implementing the simplest form for gradient descent#
+
+Exercise 4, Implementing the simplest form for gradient descent#
Alternatively, we can fit the ridge regression model using gradient
descent. This is useful to visualize the iterative convergence and is
necessary if \(n\) and \(p\) are so large that the closed-form might be
@@ -587,19 +557,68 @@ print("Gradient Descent Ridge coefficients:", theta_gdRidge)
-3a)#
+4a)#
Discuss the results as function of the learning rate parameters and the number of iterations.
-3b)#
+4b)#
Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion?
+
+
+
+Exercise 5, Ridge regression and a new Synthetic Dataset#
+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:
+Decide on the number of samples and features (e.g. 100 samples, 10 features).
+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.
+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.
+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).
+Below is the code to generate the dataset:
+
+
+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
+
+
+
+
+This code produces a dataset where only features 0, 1, and 6
+significantly influence \(\boldsymbol{y}\). The rest of the features have zero true
+coefficient. 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:
+
+\[
+y \approx 5 \times x_0 \;-\; 3 \times x_1 \;+\; 2 \times x_6 \;+\; \text{noise}.
+\]
+You can remove the noise if you wish to.
+Try to fit the above data set using OLS and Ridge regression with the analytical expressions and your own gradient descent codes.
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. Which method (OLS or Ridge) gives the best results?
-
-
+
@@ -229,7 +229,7 @@
Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
Exercises week 36
Week 36: Linear Regression and Gradient descent
-Exercises week 36
+Exercises week 37
@@ -2410,7 +2410,7 @@ plt.show()
title="next page">
diff --git a/doc/LectureNotes/_build/jupyter_execute/exercisesweek35.ipynb b/doc/LectureNotes/_build/jupyter_execute/exercisesweek35.ipynb
index 6b559d91e..e88352077 100644
--- a/doc/LectureNotes/_build/jupyter_execute/exercisesweek35.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/exercisesweek35.ipynb
@@ -323,7 +323,7 @@
"source": [
"n = 100\n",
"x = np.linspace(-3, 3, n)\n",
- "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + np.random.normal(0, 0.1)"
+ "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + np.random.normal(0, 1.0)"
]
},
{
diff --git a/doc/LectureNotes/_build/jupyter_execute/exercisesweek37.ipynb b/doc/LectureNotes/_build/jupyter_execute/exercisesweek37.ipynb
index 373a3f02a..4857c84a5 100644
--- a/doc/LectureNotes/_build/jupyter_execute/exercisesweek37.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/exercisesweek37.ipynb
@@ -2,24 +2,24 @@
"cells": [
{
"cell_type": "markdown",
- "id": "d3aa801d",
+ "id": "b0268cb1",
"metadata": {
"editable": true
},
"source": [
"\n",
- ""
+ ""
]
},
{
"cell_type": "markdown",
- "id": "7c64e6da",
+ "id": "700a1d0b",
"metadata": {
"editable": true
},
"source": [
- "# Exercises week 36\n",
+ "# Exercises week 37\n",
"**Implementing gradient descent for Ridge and ordinary Least Squares Regression**\n",
"\n",
"Date: **September 8-12, 2025**"
@@ -27,7 +27,7 @@
},
{
"cell_type": "markdown",
- "id": "51e35698",
+ "id": "dbe5809a",
"metadata": {
"editable": true
},
@@ -46,98 +46,44 @@
},
{
"cell_type": "markdown",
- "id": "74fb184e",
+ "id": "ac99e9c0",
"metadata": {
"editable": true
},
"source": [
- "## Ridge regression and a new Synthetic Dataset\n",
+ "## Simple one-dimensional second-order polynomial\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": "9e6acfef",
- "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"
+ "We start with a very simple function"
]
},
{
"cell_type": "markdown",
- "id": "f2d03ca8",
- "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. 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": "d2d64f9b",
+ "id": "6d71a32d",
"metadata": {
"editable": true
},
"source": [
"$$\n",
- "y \\approx 5 \\times x_0 \\;-\\; 3 \\times x_1 \\;+\\; 2 \\times x_6 \\;+\\; \\text{noise}.\n",
+ "\\f(x)= 2-x+5x^2,\n",
"$$"
]
},
{
"cell_type": "markdown",
- "id": "b4248e9d",
+ "id": "c6496768",
"metadata": {
"editable": true
},
"source": [
- "You can remove the noise if you wish to."
+ "defined for $x\\in [-2,2]$. You can add noise if you wish. \n",
+ "\n",
+ "We are going to fit this function with a polynomial ansatz. The easiest thing is to set up a second-order polynomial and see if you can fit the above function.\n",
+ "Feel free to play around with higher-order polynomials."
]
},
{
"cell_type": "markdown",
- "id": "5fed181f",
+ "id": "24678181",
"metadata": {
"editable": true
},
@@ -153,27 +99,28 @@
},
{
"cell_type": "markdown",
- "id": "6ec0227c",
+ "id": "6b1bd90a",
"metadata": {
"editable": true
},
"source": [
"### 1a)\n",
"\n",
- "Compute the mean and standard deviation of each column (feature) in $\\boldsymbol{X}$.\n",
+ "Compute the mean and standard deviation of each column (feature) in your design/feature matrix $\\boldsymbol{X}$.\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 does not 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.)"
+ "penalize it, but here we simplify by centering.)\n",
+ "Choose $n=100$ data points and set up $\\boldsymbol{x}, $\\boldsymbol{y} and the design matrix $\\boldsymbol{X}$."
]
},
{
"cell_type": "code",
- "execution_count": 2,
- "id": "a140aac7",
+ "execution_count": 1,
+ "id": "5e751a16",
"metadata": {
"collapsed": false,
"editable": true
@@ -193,7 +140,7 @@
},
{
"cell_type": "markdown",
- "id": "57ad18f5",
+ "id": "50a68a52",
"metadata": {
"editable": true
},
@@ -209,18 +156,30 @@
},
{
"cell_type": "markdown",
- "id": "2886697d",
+ "id": "db74a970",
"metadata": {
"editable": true
},
"source": [
- "## Exercise 2, use the analytical formulae for OLS and Ridge regression to find the optimal paramters $\\boldsymbol{\\theta}$"
+ "## Exercise 2, calculate the gradients\n",
+ "\n",
+ "Find the gradients for OLS and Ridge regression using the mean-squared error as cost/loss function."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f8feaa49",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Exercise 3, use the analytical formulae for OLS and Ridge regression to find the optimal paramters $\\boldsymbol{\\theta}$"
]
},
{
"cell_type": "code",
- "execution_count": 3,
- "id": "97ac6cb6",
+ "execution_count": 2,
+ "id": "e6eca7a7",
"metadata": {
"collapsed": false,
"editable": true
@@ -241,7 +200,7 @@
},
{
"cell_type": "markdown",
- "id": "3efb067b",
+ "id": "a2dff0cf",
"metadata": {
"editable": true
},
@@ -255,36 +214,36 @@
},
{
"cell_type": "markdown",
- "id": "53be2bf8",
+ "id": "d7542c3e",
"metadata": {
"editable": true
},
"source": [
- "### 2a)\n",
+ "### 3a)\n",
"\n",
"Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters $\\boldsymbol{\\theta}$."
]
},
{
"cell_type": "markdown",
- "id": "e4126591",
+ "id": "b922fd54",
"metadata": {
"editable": true
},
"source": [
- "### 2b)\n",
+ "### 3b)\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": "642d0850",
+ "id": "92268a9a",
"metadata": {
"editable": true
},
"source": [
- "## Exercise 3, Implementing the simplest form for gradient descent\n",
+ "## Exercise 4, 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",
@@ -298,8 +257,8 @@
},
{
"cell_type": "code",
- "execution_count": 4,
- "id": "a67af634",
+ "execution_count": 3,
+ "id": "81660fa3",
"metadata": {
"collapsed": false,
"editable": true
@@ -342,26 +301,119 @@
},
{
"cell_type": "markdown",
- "id": "1c8c35dc",
+ "id": "95149551",
"metadata": {
"editable": true
},
"source": [
- "### 3a)\n",
+ "### 4a)\n",
"\n",
"Discuss the results as function of the learning rate parameters and the number of iterations."
]
},
{
"cell_type": "markdown",
- "id": "899fec5c",
+ "id": "09b5400e",
"metadata": {
"editable": true
},
"source": [
- "### 3b)\n",
+ "### 4b)\n",
"\n",
- "Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion? \n",
+ "Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion?"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c635ca9e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Exercise 5, 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": 4,
+ "id": "4ca4ce05",
+ "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": "af39d8bc",
+ "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. 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": "d35d3438",
+ "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": "b28bf122",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "You can remove the noise if you wish to. \n",
+ "\n",
+ "Try to fit the above data set using OLS and Ridge regression with the analytical expressions and your own gradient descent codes.\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",
diff --git a/doc/LectureNotes/exercisesweek37.ipynb b/doc/LectureNotes/exercisesweek37.ipynb
index 7a5dcfc68..a23fb43b2 100644
--- a/doc/LectureNotes/exercisesweek37.ipynb
+++ b/doc/LectureNotes/exercisesweek37.ipynb
@@ -2,24 +2,24 @@
"cells": [
{
"cell_type": "markdown",
- "id": "d3aa801d",
+ "id": "b0268cb1",
"metadata": {
"editable": true
},
"source": [
"\n",
- ""
+ ""
]
},
{
"cell_type": "markdown",
- "id": "7c64e6da",
+ "id": "700a1d0b",
"metadata": {
"editable": true
},
"source": [
- "# Exercises week 36\n",
+ "# Exercises week 37\n",
"**Implementing gradient descent for Ridge and ordinary Least Squares Regression**\n",
"\n",
"Date: **September 8-12, 2025**"
@@ -27,7 +27,7 @@
},
{
"cell_type": "markdown",
- "id": "51e35698",
+ "id": "dbe5809a",
"metadata": {
"editable": true
},
@@ -37,7 +37,7 @@
"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 Ridge regression with the gradient descent approach\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",
@@ -46,101 +46,44 @@
},
{
"cell_type": "markdown",
- "id": "74fb184e",
+ "id": "ac99e9c0",
"metadata": {
"editable": true
},
"source": [
- "## Ridge regression and a new Synthetic Dataset\n",
+ "## Simple one-dimensional second-order polynomial\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 will\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": "9e6acfef",
- "metadata": {
- "collapsed": false,
- "editable": true,
- "jupyter": {
- "outputs_hidden": false
- }
- },
- "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"
+ "We start with a very simple function"
]
},
{
"cell_type": "markdown",
- "id": "f2d03ca8",
- "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. 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": "d2d64f9b",
+ "id": "6d71a32d",
"metadata": {
"editable": true
},
"source": [
"$$\n",
- "y \\approx 5 \\times x_0 \\;-\\; 3 \\times x_1 \\;+\\; 2 \\times x_6 \\;+\\; \\text{noise}.\n",
+ "\\f(x)= 2-x+5x^2,\n",
"$$"
]
},
{
"cell_type": "markdown",
- "id": "b4248e9d",
+ "id": "c6496768",
"metadata": {
"editable": true
},
"source": [
- "You can remove the noise if you wish to."
+ "defined for $x\\in [-2,2]$. You can add noise if you wish. \n",
+ "\n",
+ "We are going to fit this function with a polynomial ansatz. The easiest thing is to set up a second-order polynomial and see if you can fit the above function.\n",
+ "Feel free to play around with higher-order polynomials."
]
},
{
"cell_type": "markdown",
- "id": "5fed181f",
+ "id": "24678181",
"metadata": {
"editable": true
},
@@ -156,33 +99,31 @@
},
{
"cell_type": "markdown",
- "id": "6ec0227c",
+ "id": "6b1bd90a",
"metadata": {
"editable": true
},
"source": [
"### 1a)\n",
"\n",
- "Compute the mean and standard deviation of each column (feature) in $\\boldsymbol{X}$.\n",
+ "Compute the mean and standard deviation of each column (feature) in your design/feature matrix $\\boldsymbol{X}$.\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 does not 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.)"
+ "penalize it, but here we simplify by centering.)\n",
+ "Choose $n=100$ data points and set up $\\boldsymbol{x}, $\\boldsymbol{y} and the design matrix $\\boldsymbol{X}$."
]
},
{
"cell_type": "code",
- "execution_count": 2,
- "id": "a140aac7",
+ "execution_count": 1,
+ "id": "5e751a16",
"metadata": {
"collapsed": false,
- "editable": true,
- "jupyter": {
- "outputs_hidden": false
- }
+ "editable": true
},
"outputs": [],
"source": [
@@ -199,7 +140,7 @@
},
{
"cell_type": "markdown",
- "id": "57ad18f5",
+ "id": "50a68a52",
"metadata": {
"editable": true
},
@@ -215,24 +156,33 @@
},
{
"cell_type": "markdown",
- "id": "2886697d",
+ "id": "db74a970",
"metadata": {
"editable": true
},
"source": [
- "## Exercise 2, use the analytical formulae for OLS and Ridge regression to find the optimal paramters $\\boldsymbol{\\theta}$"
+ "## Exercise 2, calculate the gradients\n",
+ "\n",
+ "Find the gradients for OLS and Ridge regression using the mean-squared error as cost/loss function."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f8feaa49",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Exercise 3, use the analytical formulae for OLS and Ridge regression to find the optimal paramters $\\boldsymbol{\\theta}$"
]
},
{
"cell_type": "code",
- "execution_count": 3,
- "id": "97ac6cb6",
+ "execution_count": 2,
+ "id": "e6eca7a7",
"metadata": {
"collapsed": false,
- "editable": true,
- "jupyter": {
- "outputs_hidden": false
- }
+ "editable": true
},
"outputs": [],
"source": [
@@ -250,7 +200,7 @@
},
{
"cell_type": "markdown",
- "id": "3efb067b",
+ "id": "a2dff0cf",
"metadata": {
"editable": true
},
@@ -264,36 +214,36 @@
},
{
"cell_type": "markdown",
- "id": "53be2bf8",
+ "id": "d7542c3e",
"metadata": {
"editable": true
},
"source": [
- "### 2a)\n",
+ "### 3a)\n",
"\n",
"Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters $\\boldsymbol{\\theta}$."
]
},
{
"cell_type": "markdown",
- "id": "e4126591",
+ "id": "b922fd54",
"metadata": {
"editable": true
},
"source": [
- "### 2b)\n",
+ "### 3b)\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": "642d0850",
+ "id": "92268a9a",
"metadata": {
"editable": true
},
"source": [
- "## Exercise 3, Implementing the simplest form for gradient descent\n",
+ "## Exercise 4, 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",
@@ -307,14 +257,11 @@
},
{
"cell_type": "code",
- "execution_count": 4,
- "id": "a67af634",
+ "execution_count": 3,
+ "id": "81660fa3",
"metadata": {
"collapsed": false,
- "editable": true,
- "jupyter": {
- "outputs_hidden": false
- }
+ "editable": true
},
"outputs": [],
"source": [
@@ -354,26 +301,119 @@
},
{
"cell_type": "markdown",
- "id": "1c8c35dc",
+ "id": "95149551",
"metadata": {
"editable": true
},
"source": [
- "### 3a)\n",
+ "### 4a)\n",
"\n",
"Discuss the results as function of the learning rate parameters and the number of iterations."
]
},
{
"cell_type": "markdown",
- "id": "899fec5c",
+ "id": "09b5400e",
"metadata": {
"editable": true
},
"source": [
- "### 3b)\n",
+ "### 4b)\n",
"\n",
- "Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion? \n",
+ "Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion?"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c635ca9e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Exercise 5, 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": 4,
+ "id": "4ca4ce05",
+ "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": "af39d8bc",
+ "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. 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": "d35d3438",
+ "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": "b28bf122",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "You can remove the noise if you wish to. \n",
+ "\n",
+ "Try to fit the above data set using OLS and Ridge regression with the analytical expressions and your own gradient descent codes.\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",
@@ -383,25 +423,7 @@
]
}
],
- "metadata": {
- "kernelspec": {
- "display_name": "Python 3 (ipykernel)",
- "language": "python",
- "name": "python3"
- },
- "language_info": {
- "codemirror_mode": {
- "name": "ipython",
- "version": 3
- },
- "file_extension": ".py",
- "mimetype": "text/x-python",
- "name": "python",
- "nbconvert_exporter": "python",
- "pygments_lexer": "ipython3",
- "version": "3.9.15"
- }
- },
+ "metadata": {},
"nbformat": 4,
"nbformat_minor": 5
}
diff --git a/doc/src/week37/exercisesweek37.do.txt b/doc/src/week37/exercisesweek37.do.txt
index d03f4b781..556724301 100644
--- a/doc/src/week37/exercisesweek37.do.txt
+++ b/doc/src/week37/exercisesweek37.do.txt
@@ -1,4 +1,4 @@
-TITLE: Exercises week 36
+TITLE: Exercises week 37
AUTHOR: Implementing gradient descent for Ridge and ordinary Least Squares Regression
DATE: September 8-12, 2025
@@ -11,7 +11,149 @@ o Be able to compare the analytical expressions for OLS and Rudge regression wit
o Explore the role of the learning rate in the gradient descent approach and the hyperparameter $\lambda$ in Ridge regression
o Scale the data properly
-===== Ridge regression and a new Synthetic Dataset =====
+
+===== Simple one-dimensional second-order polynomial =====
+
+We start with a very simple function
+!bt
+\[
+\f(x)= 2-x+5x^2,
+\]
+!et
+
+defined for $x\in [-2,2]$. You can add noise if you wish.
+
+We are going to fit this function with a polynomial ansatz. The easiest thing is to set up a second-order polynomial and see if you can fit the above function.
+Feel free to play around with higher-order polynomials.
+
+===== Exercise 1, scale your data =====
+
+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.
+
+=== 1a) ===
+
+Compute the mean and standard deviation of each column (feature) in your design/feature matrix $\bm{X}$.
+Subtract the mean and divide by the standard deviation for each feature.
+
+
+We will also center the target $\bm{y}$ to mean $0$. Centering $\bm{y}$
+(and each feature) means the model does not 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.)
+Choose $n=100$ data points and set up $\bm{x}, $\bm{y} and the design matrix $\bm{X}$.
+
+!bc pycod
+# 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 = ?
+!ec
+
+Fill in the necessary details.
+
+After this preprocessing, each column of $\bm{X}_{\mathrm{norm}}$ has mean zero and standard deviation $1$
+and $\bm{y}_{\mathrm{centered}}$ has mean 0. This makes the optimization landscape
+nicer and ensures the regularization penalty $\lambda \sum_j
+\theta_j^2$ in Ridge regression treats each coefficient fairly (since features are on the
+same scale).
+
+===== Exercise 2, calculate the gradients =====
+
+Find the gradients for OLS and Ridge regression using the mean-squared error as cost/loss function.
+
+
+===== Exercise 3, use the analytical formulae for OLS and Ridge regression to find the optimal paramters $\bm{\theta}$ =====
+
+!bc pycod
+# 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)
+!ec
+
+This computes the Ridge and OLS regression coefficients directly. The identity
+matrix $I$ has the same size as $X^T X$. It adds $\lambda$ to the diagonal of $X^T X for Ridge regression. We
+then invert this matrix and multiply by $X^T y$. The result
+for $\bm{\theta}$ is a NumPy array of shape (n$\_$features,) containing the
+fitted parameters $\bm{\theta}$..
+
+=== 3a) ===
+Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters $\bm{\theta}$.
+
+=== 3b) ===
+Explore the results as function of different values of the hyperparameter $\lambda$. See for example exercise 4 from week 36.
+
+===== Exercise 4, Implementing the simplest form for gradient descent =====
+
+Alternatively, we can fit the ridge regression model using gradient
+descent. This is useful to visualize the iterative convergence and is
+necessary if $n$ and $p$ 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 $\bm{\theta}$ and set up (using the template below) your own gradient descent code for OLS and Ridge regression.
+
+
+
+Below is a template code for gradient descent implementation of ridge:
+!bc pycod
+# 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)
+!ec
+
+=== 4a) ===
+Discuss the results as function of the learning rate parameters and the number of iterations.
+
+=== 4b) ===
+Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion?
+
+
+===== Exercise 5, Ridge regression and a new Synthetic Dataset =====
We create a synthetic linear regression dataset with a sparse
@@ -62,128 +204,8 @@ y \approx 5 \times x_0 \;-\; 3 \times x_1 \;+\; 2 \times x_6 \;+\; \text{noise}.
!et
You can remove the noise if you wish to.
-===== Exercise 1, scale your data =====
-
-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.
-
-=== 1a) ===
-
-Compute the mean and standard deviation of each column (feature) in $\bm{X}$.
-Subtract the mean and divide by the standard deviation for each feature.
-
-
-We will also center the target $\bm{y}$ to mean $0$. Centering $\bm{y}$
-(and each feature) means the model does not 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.)
-
-!bc pycod
-# 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 = ?
-!ec
-
-Fill in the necessary details.
-
-After this preprocessing, each column of $\bm{X}_{\mathrm{norm}}$ has mean zero and standard deviation $1$
-and $\bm{y}_{\mathrm{centered}}$ has mean 0. This makes the optimization landscape
-nicer and ensures the regularization penalty $\lambda \sum_j
-\theta_j^2$ in Ridge regression treats each coefficient fairly (since features are on the
-same scale).
-
-
-===== Exercise 2, use the analytical formulae for OLS and Ridge regression to find the optimal paramters $\bm{\theta}$ =====
-
-!bc pycod
-# 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)
-!ec
-
-This computes the Ridge and OLS regression coefficients directly. The identity
-matrix $I$ has the same size as $X^T X$. It adds $\lambda$ to the diagonal of $X^T X for Ridge regression. We
-then invert this matrix and multiply by $X^T y$. The result
-for $\bm{\theta}$ is a NumPy array of shape (n$\_$features,) containing the
-fitted parameters $\bm{\theta}$..
-
-=== 2a) ===
-Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters $\bm{\theta}$.
-
-=== 2b) ===
-Explore the results as function of different values of the hyperparameter $\lambda$. See for example exercise 4 from week 36.
-
-===== Exercise 3, Implementing the simplest form for gradient descent =====
-
-Alternatively, we can fit the ridge regression model using gradient
-descent. This is useful to visualize the iterative convergence and is
-necessary if $n$ and $p$ 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 $\bm{\theta}$ and set up (using the template below) your own gradient descent code for OLS and Ridge regression.
-
-
-
-Below is a template code for gradient descent implementation of ridge:
-!bc pycod
-# 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)
-!ec
-
-=== 3a) ===
-Discuss the results as function of the learning rate parameters and the number of iterations.
-
-=== 3b) ===
-Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion?
-
+Try to fit the above data set using OLS and Ridge regression with the analytical expressions and your own gradient descent codes.
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