Morten Hjorth-Jensen
424 lines
14 KiB
Plaintext
424 lines
14 KiB
Plaintext
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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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"source": [
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"# Exercises week 37\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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"source": [
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"## Learning goals\n",
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"\n",
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"After having completed these exercises you will have:\n",
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"1. Your own code for the implementation of the simplest gradient descent approach applied to ordinary least squares (OLS) and Ridge regression\n",
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"\n",
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"2. Be able to compare the analytical expressions for OLS and 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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"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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"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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]
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"cell_type": "markdown",
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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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"\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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]
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"source": [
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"## Exercise 1, scale your data\n",
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"\n",
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"Before fitting a regression model, it is good practice to normalize or\n",
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"standardize the features. This ensures all features are on a\n",
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"comparable scale, which is especially important when using\n",
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"regularization. Here we will perform standardization, scaling each\n",
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"feature to have mean 0 and standard deviation 1."
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]
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"source": [
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"### 1a)\n",
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"\n",
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"Compute the mean and standard deviation of each column (feature) in your design/feature matrix $\\boldsymbol{X}$.\n",
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"Subtract the mean and divide by the standard deviation for each feature.\n",
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"\n",
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"We will also center the target $\\boldsymbol{y}$ to mean $0$. Centering $\\boldsymbol{y}$\n",
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"(and each feature) means the model does not require a separate intercept\n",
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"term, the data is shifted such that the intercept is effectively 0\n",
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". (In practice, one could include an intercept in the model and not\n",
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"penalize it, but here we simplify by centering.)\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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]
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"cell_type": "code",
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"source": [
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"# Standardize features (zero mean, unit variance for each feature)\n",
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"X_mean = X.mean(axis=0)\n",
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"X_std = X.std(axis=0)\n",
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"X_std[X_std == 0] = 1 # safeguard to avoid division by zero for constant features\n",
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"X_norm = (X - X_mean) / X_std\n",
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"\n",
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"# Center the target to zero mean (optional, to simplify intercept handling)\n",
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"y_mean = ?\n",
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"y_centered = ?"
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]
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},
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"cell_type": "markdown",
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"id": "d861e1e3",
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"metadata": {
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"editable": true
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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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"\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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]
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"cell_type": "markdown",
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"id": "b3e774d0",
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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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]
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"cell_type": "markdown",
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"id": "d5dc7708",
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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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]
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"cell_type": "code",
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"execution_count": 2,
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"id": "4c9c86ac",
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"metadata": {
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"collapsed": false,
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"outputs": [],
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"source": [
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"# Set regularization parameter, either a single value or a vector of values\n",
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"# 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.\"\n",
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"lam = ?\n",
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"\n",
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"\n",
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"# Analytical form for OLS and Ridge solution: theta_Ridge = (X^T X + lambda * I)^{-1} X^T y and theta_OLS = (X^T X)^{-1} X^T y\n",
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"I = np.eye(n_features)\n",
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"theta_closed_formRidge = ?\n",
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"theta_closed_formOLS = ?\n",
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"\n",
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"print(\"Closed-form Ridge coefficients:\", theta_closed_form)\n",
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"print(\"Closed-form OLS coefficients:\", theta_closed_form)"
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]
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},
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"cell_type": "markdown",
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"id": "eeae00fd",
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"metadata": {
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"editable": true
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"source": [
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"This computes the Ridge and OLS regression coefficients directly. The identity\n",
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"matrix $I$ has the same size as $X^T X$. 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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"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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]
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"cell_type": "markdown",
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"id": "587dd3dc",
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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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]
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},
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"cell_type": "markdown",
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"source": [
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"## Exercise 4, Implementing the simplest form for gradient descent\n",
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"\n",
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"Alternatively, we can fit the ridge regression model using gradient\n",
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"descent. This is useful to visualize the iterative convergence and is\n",
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"necessary if $n$ and $p$ are so large that the closed-form might be\n",
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"too slow or memory-intensive. We derive the gradients from the cost\n",
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"functions defined above. Use the gradients of the Ridge and OLS cost functions with respect to\n",
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"the parameters $\\boldsymbol{\\theta}$ and set up (using the template below) your own gradient descent code for OLS and Ridge regression.\n",
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"\n",
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"Below is a template code for gradient descent implementation of ridge:"
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]
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"source": [
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"# Gradient descent parameters, learning rate eta first\n",
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"eta = 0.1\n",
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"# Then number of iterations\n",
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"num_iters = 1000\n",
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"\n",
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"# Initialize weights for gradient descent\n",
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"theta = np.zeros(n_features)\n",
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"\n",
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"# Gradient descent loop\n",
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"for t in range(num_iters):\n",
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" # Compute gradients for OSL and Ridge\n",
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" grad_OLS = ?\n",
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" grad_Ridge = ?\n",
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" # Update parameters theta\n",
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" theta_gdOLS = ?\n",
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" theta_gdRidge = ? \n",
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"\n",
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"# After the loop, theta contains the fitted coefficients\n",
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"theta_gdOLS = ?\n",
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"theta_gdRidge = ?\n",
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"print(\"Gradient Descent OLS coefficients:\", theta_gdOLS)\n",
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"print(\"Gradient Descent Ridge coefficients:\", theta_gdRidge)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "f3f43f2c",
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"editable": true
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"source": [
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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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]
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},
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{
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"cell_type": "markdown",
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"id": "9ba303be",
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"metadata": {
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"editable": true
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},
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"source": [
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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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]
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"source": [
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"## Exercise 5, Ridge regression and a new Synthetic Dataset\n",
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"\n",
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"We create a synthetic linear regression dataset with a sparse\n",
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"underlying relationship. This means we have many features but only a\n",
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"few of them actually contribute to the target. In our example, we’ll\n",
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"use 10 features with only 3 non-zero weights in the true model. This\n",
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"way, the target is generated as a linear combination of a few features\n",
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"(with known coefficients) plus some random noise. The steps we include are:\n",
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"\n",
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"Decide on the number of samples and features (e.g. 100 samples, 10 features).\n",
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"Define the **true** coefficient vector with mostly zeros (for sparsity). For example, we set $\\hat{\\boldsymbol{\\theta}} = [5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0]$, meaning only features 0, 1, and 6 have a real effect on y.\n",
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"\n",
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"Then we sample feature values for $\\boldsymbol{X}$ randomly (e.g. from a normal distribution). We use a normal distribution so features are roughly centered around 0.\n",
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"Then we compute the target values $y$ using the linear combination $\\boldsymbol{X}\\hat{\\boldsymbol{\\theta}}$ and add some noise (to simulate measurement error or unexplained variance).\n",
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"\n",
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"Below is the code to generate the dataset:"
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]
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"cell_type": "code",
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"execution_count": 4,
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"collapsed": false,
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"editable": true
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"source": [
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"import numpy as np\n",
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"\n",
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"# Set random seed for reproducibility\n",
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"np.random.seed(0)\n",
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"\n",
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"# Define dataset size\n",
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"n_samples = 100\n",
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"n_features = 10\n",
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"\n",
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"# Define true coefficients (sparse linear relationship)\n",
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"theta_true = np.array([5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0])\n",
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"\n",
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"# Generate feature matrix X (n_samples x n_features) with random values\n",
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"X = np.random.randn(n_samples, n_features) # standard normal distribution\n",
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"\n",
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"# Generate target values y with a linear combination of X and theta_true, plus noise\n",
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"noise = 0.5 * np.random.randn(n_samples) # Gaussian noise\n",
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"y = X.dot @ theta_true + noise"
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]
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"cell_type": "markdown",
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"source": [
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"This code produces a dataset where only features 0, 1, and 6\n",
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"significantly influence $\\boldsymbol{y}$. The rest of the features have zero true\n",
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"coefficient. For example, feature 0 has\n",
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"a true weight of 5.0, feature 1 has -3.0, and feature 6 has 2.0, so\n",
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"the expected relationship is:"
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]
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"source": [
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"$$\n",
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"y \\approx 5 \\times x_0 \\;-\\; 3 \\times x_1 \\;+\\; 2 \\times x_6 \\;+\\; \\text{noise}.\n",
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"$$"
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]
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"cell_type": "markdown",
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"source": [
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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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"If everything worked correctly, the learned coefficients should be\n",
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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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]
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}
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