fix w37 math formatting
This commit is contained in:
KarlHenrik
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@@ -9,7 +9,7 @@
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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 37 -->"
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"<!-- dom:TITLE: Exercises week 37 -->\n"
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]
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},
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{
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@@ -20,9 +20,10 @@
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},
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"source": [
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"# Exercises week 37\n",
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"\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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"Date: **September 8-12, 2025**\n"
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]
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},
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{
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@@ -35,13 +36,14 @@
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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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"\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 Ridge 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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"4. Scale the data properly\n"
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]
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},
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{
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@@ -53,7 +55,7 @@
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"source": [
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"## Simple one-dimensional second-order polynomial\n",
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"\n",
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"We start with a very simple function"
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"We start with a very simple function\n"
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]
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},
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{
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@@ -65,7 +67,7 @@
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"source": [
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"$$\n",
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"f(x)= 2-x+5x^2,\n",
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"$$"
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"$$\n"
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]
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},
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{
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@@ -75,10 +77,10 @@
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"editable": true
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},
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"source": [
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"defined for $x\\in [-2,2]$. You can add noise if you wish. \n",
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"defined for $x\\in [-2,2]$. You can add noise if you wish.\n",
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"\n",
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"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",
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"Feel free to play around with higher-order polynomials."
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"Feel free to play around with higher-order polynomials.\n"
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]
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},
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{
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@@ -94,7 +96,7 @@
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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."
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"feature to have mean 0 and standard deviation 1.\n"
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]
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},
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{
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@@ -114,7 +116,7 @@
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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.)\n",
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"Choose $n=100$ data points and set up $\\boldsymbol{x}, $\\boldsymbol{y}$ and the design matrix $\\boldsymbol{X}$."
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"Choose $n=100$ data points and set up $\\boldsymbol{x}$, $\\boldsymbol{y}$ and the design matrix $\\boldsymbol{X}$.\n"
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]
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},
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{
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@@ -145,13 +147,13 @@
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"editable": true
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},
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"source": [
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"Fill in the necessary details. Do we need to center the $y$-values? \n",
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"Fill in the necessary details. Do we need to center the $y$-values?\n",
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"\n",
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"After this preprocessing, each column of $\\boldsymbol{X}_{\\mathrm{norm}}$ has mean zero and standard deviation $1$\n",
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"and $\\boldsymbol{y}_{\\mathrm{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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"\\theta_j^2$ in Ridge regression treats each coefficient fairly (since features are on the\n",
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"same scale)."
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"same scale).\n"
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]
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},
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{
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@@ -163,7 +165,7 @@
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"source": [
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"## Exercise 2, calculate the gradients\n",
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"\n",
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"Find the gradients for OLS and Ridge regression using the mean-squared error as cost/loss function."
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"Find the gradients for OLS and Ridge regression using the mean-squared error as cost/loss function.\n"
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]
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},
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{
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@@ -173,7 +175,7 @@
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"editable": true
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},
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"source": [
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"## Exercise 3, using the analytical formulae for OLS and Ridge regression to find the optimal paramters $\\boldsymbol{\\theta}$"
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"## Exercise 3, using the analytical formulae for OLS and Ridge regression to find the optimal paramters $\\boldsymbol{\\theta}$\n"
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]
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},
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{
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@@ -210,8 +212,8 @@
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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$. It adds $\\lambda$ to the diagonal of $X^T X$ for Ridge regression. 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 parameters $\\boldsymbol{\\theta}$."
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"for $\\boldsymbol{\\theta}$ is a NumPy array of shape (n$\\_$features,) containing the\n",
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"fitted parameters $\\boldsymbol{\\theta}$.\n"
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]
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},
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{
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@@ -223,7 +225,7 @@
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"source": [
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"### 3a)\n",
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"\n",
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"Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters $\\boldsymbol{\\theta}$."
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"Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters $\\boldsymbol{\\theta}$.\n"
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]
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},
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{
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@@ -235,7 +237,7 @@
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"source": [
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"### 3b)\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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"Explore the results as function of different values of the hyperparameter $\\lambda$. See for example exercise 4 from week 36.\n"
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]
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},
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{
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@@ -252,9 +254,9 @@
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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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"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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"Below is a template code for gradient descent implementation of ridge:\n"
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]
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},
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{
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@@ -301,7 +303,7 @@
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"### 4a)\n",
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"\n",
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"Write first a gradient descent code for OLS only using the above template.\n",
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"Discuss the results as function of the learning rate parameters and the number of iterations"
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"Discuss the results as function of the learning rate parameters and the number of iterations\n"
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]
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},
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{
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@@ -314,7 +316,7 @@
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"### 4b)\n",
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"\n",
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"Write then a similar code for Ridge regression using the above template.\n",
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"Try to add a stopping parameter as function of the number iterations and the difference between the new and old $\\theta$ values. How would you define a stopping criterion?"
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"Try to add a stopping parameter as function of the number iterations and the difference between the new and old $\\theta$ values. How would you define a stopping criterion?\n"
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]
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},
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{
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@@ -339,12 +341,12 @@
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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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"Below is the code to generate the dataset:\n"
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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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"execution_count": null,
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"id": "8be1cebe",
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"metadata": {
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"collapsed": false,
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@@ -368,7 +370,7 @@
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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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"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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@@ -383,7 +385,7 @@
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"significantly influence $\\boldsymbol{y}$. The rest of the features have zero true\n",
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"coefficient. 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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"the expected relationship is:\n"
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]
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},
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{
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@@ -395,7 +397,7 @@
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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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"$$"
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"$$\n"
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]
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},
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{
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@@ -405,7 +407,7 @@
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"editable": true
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},
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"source": [
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"You can remove the noise if you wish to. \n",
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"You can remove the noise if you wish to.\n",
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"\n",
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"Try to fit the above data set using OLS and Ridge regression with the analytical expressions and your own gradient descent codes.\n",
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"\n",
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@@ -413,11 +415,15 @@
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"close to the true values [5.0, -3.0, 0.0, …, 2.0, …] that we used to\n",
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"generate the data. Keep in mind that due to regularization and noise,\n",
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"the learned values will not exactly equal the true ones, but they\n",
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"should be in the same ballpark. Which method (OLS or Ridge) gives the best results?"
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"should be in the same ballpark. Which method (OLS or Ridge) gives the best results?\n"
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]
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}
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],
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"metadata": {},
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"metadata": {
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"language_info": {
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"name": "python"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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@@ -416,7 +416,8 @@ document.write(`
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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 37 --><section class="tex2jax_ignore mathjax_ignore" id="exercises-week-37">
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<!-- dom:TITLE: Exercises week 37 -->
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<section class="tex2jax_ignore mathjax_ignore" id="exercises-week-37">
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<h1>Exercises week 37<a class="headerlink" href="#exercises-week-37" title="Link to this heading">#</a></h1>
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<p><strong>Implementing gradient descent for Ridge and ordinary Least Squares Regression</strong></p>
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<p>Date: <strong>September 8-12, 2025</strong></p>
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@@ -457,18 +458,18 @@ Subtract the mean and divide by the standard deviation for each feature.</p>
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term, the data is shifted such that the intercept is effectively 0
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. (In practice, one could include an intercept in the model and not
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penalize it, but here we simplify by centering.)
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Choose <span class="math notranslate nohighlight">\(n=100\)</span> data points and set up <span class="math notranslate nohighlight">\(\boldsymbol{x}, \)</span>\boldsymbol{y}<span class="math notranslate nohighlight">\( and the design matrix \)</span>\boldsymbol{X}$.</p>
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Choose <span class="math notranslate nohighlight">\(n=100\)</span> data points and set up <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span>, <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> and the design matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>.</p>
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<div class="cell docutils container">
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<div class="cell_input docutils container">
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<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Standardize features (zero mean, unit variance for each feature)
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X_mean = X.mean(axis=0)
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X_std = X.std(axis=0)
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X_std[X_std == 0] = 1 # safeguard to avoid division by zero for constant features
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X_norm = (X - X_mean) / X_std
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<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Standardize features (zero mean, unit variance for each feature)</span>
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<span class="n">X_mean</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
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<span class="n">X_std</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
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<span class="n">X_std</span><span class="p">[</span><span class="n">X_std</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="c1"># safeguard to avoid division by zero for constant features</span>
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<span class="n">X_norm</span> <span class="o">=</span> <span class="p">(</span><span class="n">X</span> <span class="o">-</span> <span class="n">X_mean</span><span class="p">)</span> <span class="o">/</span> <span class="n">X_std</span>
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# Center the target to zero mean (optional, to simplify intercept handling)
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y_mean = ?
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y_centered = ?
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<span class="c1"># Center the target to zero mean (optional, to simplify intercept handling)</span>
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<span class="n">y_mean</span> <span class="o">=</span> <span class="err">?</span>
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<span class="n">y_centered</span> <span class="o">=</span> <span class="err">?</span>
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</pre></div>
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</div>
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</div>
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@@ -489,18 +490,18 @@ same scale).</p>
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<h2>Exercise 3, using 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-3-using-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta" title="Link to this heading">#</a></h2>
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<div class="cell docutils container">
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<div class="cell_input docutils container">
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<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Set regularization parameter, either a single value or a vector of values
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# Note that lambda is a python keyword. The lambda keyword is used to create small, single-expression functions without a formal name. These are often called "anonymous functions" or "lambda functions."
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lam = ?
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<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set regularization parameter, either a single value or a vector of values</span>
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<span class="c1"># Note that lambda is a python keyword. The lambda keyword is used to create small, single-expression functions without a formal name. These are often called "anonymous functions" or "lambda functions."</span>
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<span class="n">lam</span> <span class="o">=</span> <span class="err">?</span>
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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
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I = np.eye(n_features)
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theta_closed_formRidge = ?
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theta_closed_formOLS = ?
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<span class="c1"># 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</span>
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<span class="n">I</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">n_features</span><span class="p">)</span>
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<span class="n">theta_closed_formRidge</span> <span class="o">=</span> <span class="err">?</span>
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<span class="n">theta_closed_formOLS</span> <span class="o">=</span> <span class="err">?</span>
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print("Closed-form Ridge coefficients:", theta_closed_form)
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print("Closed-form OLS coefficients:", theta_closed_form)
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<span class="nb">print</span><span class="p">(</span><span class="s2">"Closed-form Ridge coefficients:"</span><span class="p">,</span> <span class="n">theta_closed_form</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="s2">"Closed-form OLS coefficients:"</span><span class="p">,</span> <span class="n">theta_closed_form</span><span class="p">)</span>
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</pre></div>
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</div>
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</div>
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@@ -508,7 +509,7 @@ print("Closed-form OLS coefficients:", theta_closed_form)
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<p>This computes the Ridge and OLS regression coefficients directly. The identity
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matrix <span class="math notranslate nohighlight">\(I\)</span> has the same size as <span class="math notranslate nohighlight">\(X^T X\)</span>. It adds <span class="math notranslate nohighlight">\(\lambda\)</span> to the diagonal of <span class="math notranslate nohighlight">\(X^T X\)</span> for Ridge regression. We
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then invert this matrix and multiply by <span class="math notranslate nohighlight">\(X^T y\)</span>. The result
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for <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span> is a NumPy array of shape (n<span class="math notranslate nohighlight">\(\_\)</span>features,) containing the
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for <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span> is a NumPy array of shape (n<span class="math notranslate nohighlight">\(\_\)</span>features,) containing the
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fitted parameters <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span>.</p>
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<section id="id1">
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<h3>3a)<a class="headerlink" href="#id1" title="Link to this heading">#</a></h3>
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@@ -526,32 +527,32 @@ descent. This is useful to visualize the iterative convergence and is
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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
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too slow or memory-intensive. We derive the gradients from the cost
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functions defined above. Use the gradients of the Ridge and OLS cost functions with respect to
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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>
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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>
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<p>Below is a template code for gradient descent implementation of ridge:</p>
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<div class="cell docutils container">
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<div class="cell_input docutils container">
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<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Gradient descent parameters, learning rate eta first
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eta = 0.1
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# Then number of iterations
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num_iters = 1000
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<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Gradient descent parameters, learning rate eta first</span>
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<span class="n">eta</span> <span class="o">=</span> <span class="mf">0.1</span>
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<span class="c1"># Then number of iterations</span>
|
||||
<span class="n">num_iters</span> <span class="o">=</span> <span class="mi">1000</span>
|
||||
|
||||
# Initialize weights for gradient descent
|
||||
theta = np.zeros(n_features)
|
||||
<span class="c1"># Initialize weights for gradient descent</span>
|
||||
<span class="n">theta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_features</span><span class="p">)</span>
|
||||
|
||||
# Gradient descent loop
|
||||
for t in range(num_iters):
|
||||
# Compute gradients for OSL and Ridge
|
||||
grad_OLS = ?
|
||||
grad_Ridge = ?
|
||||
# Update parameters theta
|
||||
theta_gdOLS = ?
|
||||
theta_gdRidge = ?
|
||||
<span class="c1"># Gradient descent loop</span>
|
||||
<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_iters</span><span class="p">):</span>
|
||||
<span class="c1"># Compute gradients for OSL and Ridge</span>
|
||||
<span class="n">grad_OLS</span> <span class="o">=</span> <span class="err">?</span>
|
||||
<span class="n">grad_Ridge</span> <span class="o">=</span> <span class="err">?</span>
|
||||
<span class="c1"># Update parameters theta</span>
|
||||
<span class="n">theta_gdOLS</span> <span class="o">=</span> <span class="err">?</span>
|
||||
<span class="n">theta_gdRidge</span> <span class="o">=</span> <span class="err">?</span>
|
||||
|
||||
# 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)
|
||||
<span class="c1"># After the loop, theta contains the fitted coefficients</span>
|
||||
<span class="n">theta_gdOLS</span> <span class="o">=</span> <span class="err">?</span>
|
||||
<span class="n">theta_gdRidge</span> <span class="o">=</span> <span class="err">?</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Gradient Descent OLS coefficients:"</span><span class="p">,</span> <span class="n">theta_gdOLS</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Gradient Descent Ridge coefficients:"</span><span class="p">,</span> <span class="n">theta_gdRidge</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -582,24 +583,24 @@ Then we compute the target values <span class="math notranslate nohighlight">\(y
|
||||
<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
|
||||
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
|
||||
|
||||
# Set random seed for reproducibility
|
||||
np.random.seed(0)
|
||||
<span class="c1"># Set random seed for reproducibility</span>
|
||||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
|
||||
|
||||
# Define dataset size
|
||||
n_samples = 100
|
||||
n_features = 10
|
||||
<span class="c1"># Define dataset size</span>
|
||||
<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">100</span>
|
||||
<span class="n">n_features</span> <span class="o">=</span> <span class="mi">10</span>
|
||||
|
||||
# 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])
|
||||
<span class="c1"># Define true coefficients (sparse linear relationship)</span>
|
||||
<span class="n">theta_true</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">5.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">3.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span>
|
||||
|
||||
# Generate feature matrix X (n_samples x n_features) with random values
|
||||
X = np.random.randn(n_samples, n_features) # standard normal distribution
|
||||
<span class="c1"># Generate feature matrix X (n_samples x n_features) with random values</span>
|
||||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">)</span> <span class="c1"># standard normal distribution</span>
|
||||
|
||||
# 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
|
||||
<span class="c1"># Generate target values y with a linear combination of X and theta_true, plus noise</span>
|
||||
<span class="n">noise</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span> <span class="c1"># Gaussian noise</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">dot</span> <span class="o">@</span> <span class="n">theta_true</span> <span class="o">+</span> <span class="n">noise</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -619,7 +620,7 @@ y \approx 5 \times x_0 \;-\; 3 \times x_1 \;+\; 2 \times x_6 \;+\; \text{noise}.
|
||||
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?</p>
|
||||
should be in the same ballpark. Which method (OLS or Ridge) gives the best results?</p>
|
||||
</section>
|
||||
</section>
|
||||
|
||||
|
||||
+3
-3
@@ -2,7 +2,7 @@
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "b2bc44f6",
|
||||
"id": "2c4ae3e0",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Notebooks with MyST Markdown\n",
|
||||
@@ -19,7 +19,7 @@
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "4ddd8e91",
|
||||
"id": "bd842239",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
@@ -28,7 +28,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "8e6cbd45",
|
||||
"id": "0b607518",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"When your book is built, the contents of any `{code-cell}` blocks will be\n",
|
||||
|
||||
@@ -9,7 +9,7 @@
|
||||
"source": [
|
||||
"<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)\n",
|
||||
"doconce format html exercisesweek37.do.txt -->\n",
|
||||
"<!-- dom:TITLE: Exercises week 37 -->"
|
||||
"<!-- dom:TITLE: Exercises week 37 -->\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -20,9 +20,10 @@
|
||||
},
|
||||
"source": [
|
||||
"# Exercises week 37\n",
|
||||
"\n",
|
||||
"**Implementing gradient descent for Ridge and ordinary Least Squares Regression**\n",
|
||||
"\n",
|
||||
"Date: **September 8-12, 2025**"
|
||||
"Date: **September 8-12, 2025**\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -35,13 +36,14 @@
|
||||
"## Learning goals\n",
|
||||
"\n",
|
||||
"After having completed these exercises you will have:\n",
|
||||
"\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",
|
||||
"\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"
|
||||
"4. Scale the data properly\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -53,7 +55,7 @@
|
||||
"source": [
|
||||
"## Simple one-dimensional second-order polynomial\n",
|
||||
"\n",
|
||||
"We start with a very simple function"
|
||||
"We start with a very simple function\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -65,7 +67,7 @@
|
||||
"source": [
|
||||
"$$\n",
|
||||
"f(x)= 2-x+5x^2,\n",
|
||||
"$$"
|
||||
"$$\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -75,10 +77,10 @@
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"defined for $x\\in [-2,2]$. You can add noise if you wish. \n",
|
||||
"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."
|
||||
"Feel free to play around with higher-order polynomials.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -94,7 +96,7 @@
|
||||
"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."
|
||||
"feature to have mean 0 and standard deviation 1.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -114,7 +116,7 @@
|
||||
"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.)\n",
|
||||
"Choose $n=100$ data points and set up $\\boldsymbol{x}, $\\boldsymbol{y}$ and the design matrix $\\boldsymbol{X}$."
|
||||
"Choose $n=100$ data points and set up $\\boldsymbol{x}$, $\\boldsymbol{y}$ and the design matrix $\\boldsymbol{X}$.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -145,13 +147,13 @@
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"Fill in the necessary details. Do we need to center the $y$-values? \n",
|
||||
"Fill in the necessary details. Do we need to center the $y$-values?\n",
|
||||
"\n",
|
||||
"After this preprocessing, each column of $\\boldsymbol{X}_{\\mathrm{norm}}$ has mean zero and standard deviation $1$\n",
|
||||
"and $\\boldsymbol{y}_{\\mathrm{centered}}$ has mean 0. This makes the optimization landscape\n",
|
||||
"nicer and ensures the regularization penalty $\\lambda \\sum_j\n",
|
||||
"\\theta_j^2$ in Ridge regression treats each coefficient fairly (since features are on the\n",
|
||||
"same scale)."
|
||||
"same scale).\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -163,7 +165,7 @@
|
||||
"source": [
|
||||
"## Exercise 2, calculate the gradients\n",
|
||||
"\n",
|
||||
"Find the gradients for OLS and Ridge regression using the mean-squared error as cost/loss function."
|
||||
"Find the gradients for OLS and Ridge regression using the mean-squared error as cost/loss function.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -173,7 +175,7 @@
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"## Exercise 3, using the analytical formulae for OLS and Ridge regression to find the optimal paramters $\\boldsymbol{\\theta}$"
|
||||
"## Exercise 3, using the analytical formulae for OLS and Ridge regression to find the optimal paramters $\\boldsymbol{\\theta}$\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -210,8 +212,8 @@
|
||||
"This computes the Ridge and OLS regression coefficients directly. The identity\n",
|
||||
"matrix $I$ has the same size as $X^T X$. It adds $\\lambda$ to the diagonal of $X^T X$ for Ridge regression. 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 parameters $\\boldsymbol{\\theta}$."
|
||||
"for $\\boldsymbol{\\theta}$ is a NumPy array of shape (n$\\_$features,) containing the\n",
|
||||
"fitted parameters $\\boldsymbol{\\theta}$.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -223,7 +225,7 @@
|
||||
"source": [
|
||||
"### 3a)\n",
|
||||
"\n",
|
||||
"Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters $\\boldsymbol{\\theta}$."
|
||||
"Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters $\\boldsymbol{\\theta}$.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -235,7 +237,7 @@
|
||||
"source": [
|
||||
"### 3b)\n",
|
||||
"\n",
|
||||
"Explore the results as function of different values of the hyperparameter $\\lambda$. See for example exercise 4 from week 36."
|
||||
"Explore the results as function of different values of the hyperparameter $\\lambda$. See for example exercise 4 from week 36.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -252,9 +254,9 @@
|
||||
"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",
|
||||
"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:"
|
||||
"Below is a template code for gradient descent implementation of ridge:\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -301,7 +303,7 @@
|
||||
"### 4a)\n",
|
||||
"\n",
|
||||
"Write first a gradient descent code for OLS only using the above template.\n",
|
||||
"Discuss the results as function of the learning rate parameters and the number of iterations"
|
||||
"Discuss the results as function of the learning rate parameters and the number of iterations\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -314,7 +316,7 @@
|
||||
"### 4b)\n",
|
||||
"\n",
|
||||
"Write then a similar code for Ridge regression using the above template.\n",
|
||||
"Try to add a stopping parameter as function of the number iterations and the difference between the new and old $\\theta$ values. How would you define a stopping criterion?"
|
||||
"Try to add a stopping parameter as function of the number iterations and the difference between the new and old $\\theta$ values. How would you define a stopping criterion?\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -339,12 +341,12 @@
|
||||
"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:"
|
||||
"Below is the code to generate the dataset:\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"execution_count": null,
|
||||
"id": "8be1cebe",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
@@ -368,7 +370,7 @@
|
||||
"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",
|
||||
"noise = 0.5 * np.random.randn(n_samples) # Gaussian noise\n",
|
||||
"y = X.dot @ theta_true + noise"
|
||||
]
|
||||
},
|
||||
@@ -383,7 +385,7 @@
|
||||
"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:"
|
||||
"the expected relationship is:\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -395,7 +397,7 @@
|
||||
"source": [
|
||||
"$$\n",
|
||||
"y \\approx 5 \\times x_0 \\;-\\; 3 \\times x_1 \\;+\\; 2 \\times x_6 \\;+\\; \\text{noise}.\n",
|
||||
"$$"
|
||||
"$$\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -405,7 +407,7 @@
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"You can remove the noise if you wish to. \n",
|
||||
"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",
|
||||
@@ -413,11 +415,15 @@
|
||||
"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. Which method (OLS or Ridge) gives the best results?"
|
||||
"should be in the same ballpark. Which method (OLS or Ridge) gives the best results?\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {},
|
||||
"metadata": {
|
||||
"language_info": {
|
||||
"name": "python"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
@@ -9,7 +9,7 @@
|
||||
"source": [
|
||||
"<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)\n",
|
||||
"doconce format html exercisesweek37.do.txt -->\n",
|
||||
"<!-- dom:TITLE: Exercises week 37 -->"
|
||||
"<!-- dom:TITLE: Exercises week 37 -->\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -20,9 +20,10 @@
|
||||
},
|
||||
"source": [
|
||||
"# Exercises week 37\n",
|
||||
"\n",
|
||||
"**Implementing gradient descent for Ridge and ordinary Least Squares Regression**\n",
|
||||
"\n",
|
||||
"Date: **September 8-12, 2025**"
|
||||
"Date: **September 8-12, 2025**\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -35,13 +36,14 @@
|
||||
"## Learning goals\n",
|
||||
"\n",
|
||||
"After having completed these exercises you will have:\n",
|
||||
"\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",
|
||||
"\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"
|
||||
"4. Scale the data properly\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -53,7 +55,7 @@
|
||||
"source": [
|
||||
"## Simple one-dimensional second-order polynomial\n",
|
||||
"\n",
|
||||
"We start with a very simple function"
|
||||
"We start with a very simple function\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -65,7 +67,7 @@
|
||||
"source": [
|
||||
"$$\n",
|
||||
"f(x)= 2-x+5x^2,\n",
|
||||
"$$"
|
||||
"$$\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -75,10 +77,10 @@
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"defined for $x\\in [-2,2]$. You can add noise if you wish. \n",
|
||||
"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."
|
||||
"Feel free to play around with higher-order polynomials.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -94,7 +96,7 @@
|
||||
"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."
|
||||
"feature to have mean 0 and standard deviation 1.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -114,7 +116,7 @@
|
||||
"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.)\n",
|
||||
"Choose $n=100$ data points and set up $\\boldsymbol{x}, $\\boldsymbol{y}$ and the design matrix $\\boldsymbol{X}$."
|
||||
"Choose $n=100$ data points and set up $\\boldsymbol{x}$, $\\boldsymbol{y}$ and the design matrix $\\boldsymbol{X}$.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -145,13 +147,13 @@
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"Fill in the necessary details. Do we need to center the $y$-values? \n",
|
||||
"Fill in the necessary details. Do we need to center the $y$-values?\n",
|
||||
"\n",
|
||||
"After this preprocessing, each column of $\\boldsymbol{X}_{\\mathrm{norm}}$ has mean zero and standard deviation $1$\n",
|
||||
"and $\\boldsymbol{y}_{\\mathrm{centered}}$ has mean 0. This makes the optimization landscape\n",
|
||||
"nicer and ensures the regularization penalty $\\lambda \\sum_j\n",
|
||||
"\\theta_j^2$ in Ridge regression treats each coefficient fairly (since features are on the\n",
|
||||
"same scale)."
|
||||
"same scale).\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -163,7 +165,7 @@
|
||||
"source": [
|
||||
"## Exercise 2, calculate the gradients\n",
|
||||
"\n",
|
||||
"Find the gradients for OLS and Ridge regression using the mean-squared error as cost/loss function."
|
||||
"Find the gradients for OLS and Ridge regression using the mean-squared error as cost/loss function.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -173,7 +175,7 @@
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"## Exercise 3, using the analytical formulae for OLS and Ridge regression to find the optimal paramters $\\boldsymbol{\\theta}$"
|
||||
"## Exercise 3, using the analytical formulae for OLS and Ridge regression to find the optimal paramters $\\boldsymbol{\\theta}$\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -210,8 +212,8 @@
|
||||
"This computes the Ridge and OLS regression coefficients directly. The identity\n",
|
||||
"matrix $I$ has the same size as $X^T X$. It adds $\\lambda$ to the diagonal of $X^T X$ for Ridge regression. 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 parameters $\\boldsymbol{\\theta}$."
|
||||
"for $\\boldsymbol{\\theta}$ is a NumPy array of shape (n$\\_$features,) containing the\n",
|
||||
"fitted parameters $\\boldsymbol{\\theta}$.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -223,7 +225,7 @@
|
||||
"source": [
|
||||
"### 3a)\n",
|
||||
"\n",
|
||||
"Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters $\\boldsymbol{\\theta}$."
|
||||
"Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters $\\boldsymbol{\\theta}$.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -235,7 +237,7 @@
|
||||
"source": [
|
||||
"### 3b)\n",
|
||||
"\n",
|
||||
"Explore the results as function of different values of the hyperparameter $\\lambda$. See for example exercise 4 from week 36."
|
||||
"Explore the results as function of different values of the hyperparameter $\\lambda$. See for example exercise 4 from week 36.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -252,9 +254,9 @@
|
||||
"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",
|
||||
"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:"
|
||||
"Below is a template code for gradient descent implementation of ridge:\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -301,7 +303,7 @@
|
||||
"### 4a)\n",
|
||||
"\n",
|
||||
"Write first a gradient descent code for OLS only using the above template.\n",
|
||||
"Discuss the results as function of the learning rate parameters and the number of iterations"
|
||||
"Discuss the results as function of the learning rate parameters and the number of iterations\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -314,7 +316,7 @@
|
||||
"### 4b)\n",
|
||||
"\n",
|
||||
"Write then a similar code for Ridge regression using the above template.\n",
|
||||
"Try to add a stopping parameter as function of the number iterations and the difference between the new and old $\\theta$ values. How would you define a stopping criterion?"
|
||||
"Try to add a stopping parameter as function of the number iterations and the difference between the new and old $\\theta$ values. How would you define a stopping criterion?\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -339,12 +341,12 @@
|
||||
"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:"
|
||||
"Below is the code to generate the dataset:\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"execution_count": null,
|
||||
"id": "8be1cebe",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
@@ -368,7 +370,7 @@
|
||||
"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",
|
||||
"noise = 0.5 * np.random.randn(n_samples) # Gaussian noise\n",
|
||||
"y = X.dot @ theta_true + noise"
|
||||
]
|
||||
},
|
||||
@@ -383,7 +385,7 @@
|
||||
"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:"
|
||||
"the expected relationship is:\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -395,7 +397,7 @@
|
||||
"source": [
|
||||
"$$\n",
|
||||
"y \\approx 5 \\times x_0 \\;-\\; 3 \\times x_1 \\;+\\; 2 \\times x_6 \\;+\\; \\text{noise}.\n",
|
||||
"$$"
|
||||
"$$\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -405,7 +407,7 @@
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"You can remove the noise if you wish to. \n",
|
||||
"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",
|
||||
@@ -413,11 +415,15 @@
|
||||
"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. Which method (OLS or Ridge) gives the best results?"
|
||||
"should be in the same ballpark. Which method (OLS or Ridge) gives the best results?\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {},
|
||||
"metadata": {
|
||||
"language_info": {
|
||||
"name": "python"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
|
||||
Reference in New Issue
Block a user