FYS-STK4155/doc/LectureNotes/exercisesweek37.ipynb

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    "<!-- 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 -->\n"
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   "source": [
    "# Exercises week 37\n",
    "\n",
    "**Implementing gradient descent for Ridge and ordinary Least Squares Regression**\n",
    "\n",
    "Date: **September 8-12, 2025**\n"
   ]
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  {
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   "source": [
    "**Python Code can be found at https://github.uio.no/larsbog/FYS-STK4155**"
   ]
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    "## 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\n"
   ]
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    "## Simple one-dimensional second-order polynomial\n",
    "\n",
    "We start with a very simple function"
   ]
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    "$$\n",
    "f(x)= 2-x+5x^2,\n",
    "$$\n"
   ]
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    "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.\n"
   ]
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    "## Exercise 1, scale your data\n",
    "\n",
    "Before fitting a regression model, it is good practice to normalize or\n",
    "standardize the features. This ensures all features are on a\n",
    "comparable scale, which is especially important when using\n",
    "regularization. Here we will perform standardization, scaling each\n",
    "feature to have mean 0 and standard deviation 1.\n"
   ]
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   "source": [
    "### 1a)\n",
    "\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.)\n",
    "Choose $n=100$ data points and set up $\\boldsymbol{x}$, $\\boldsymbol{y}$ and the design matrix $\\boldsymbol{X}$.\n"
   ]
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   "source": [
    "import numpy as np\n",
    "\n",
    "\n",
    "def polynomial_features(x, p, intercept=False):\n",
    "    n = len(x)\n",
    "    if intercept:\n",
    "        P = np.arange(p+1)\n",
    "    else:\n",
    "        P = np.arange(1, p+1)\n",
    "    X = np.power(x[:, np.newaxis], P)\n",
    "    return X\n",
    "\n",
    "\n",
    "theta_true = np.array([-2.0, 5.0])\n",
    "X = polynomial_features(np.linspace(0, 1, 100), p=2)\n",
    "y = 2 + theta_true@X.T + 0.1 * np.random.randn(100)"
   ]
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    "# Standardize features (zero mean, unit variance for each feature)\n",
    "X_mean = X.mean(axis=0)\n",
    "X_std = X.std(axis=0)\n",
    "X_std[X_std == 0] = 1  # safeguard to avoid division by zero for constant features\n",
    "X_norm = (X - X_mean) / X_std\n",
    "\n",
    "# Center the target to zero mean (optional, to simplify intercept handling)\n",
    "y_mean = y.mean(axis=0)\n",
    "y_centered = y - y_mean\n",
    "\n",
    "n_features = X_norm.shape[1]\n",
    "theta_true *= X_std"
   ]
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    "theta_true"
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       "array([0.29157647, 0.30132605])"
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     "execution_count": 4,
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    "X_std"
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       "Text(0, 0.5, 'Target')"
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "plt.scatter(X_norm[:, 0], y_centered)\n",
    "plt.xlabel(\"Feature 1\")\n",
    "plt.ylabel(\"Target\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d861e1e3",
   "metadata": {
    "editable": true
   },
   "source": [
    "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).\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b3e774d0",
   "metadata": {
    "editable": true
   },
   "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.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5e5da080",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "The gradients can be calculated as\n",
    "\n",
    "$$\n",
    "\\nabla_\\theta C_{OLS} = \\nabla_\\theta (X\\cdot\\theta - y)^2 = 2 X (X\\cdot \\theta - y)\n",
    "$$\n",
    "\n",
    "The gradient of the Ridge cost function then directly follows as\n",
    "\n",
    "$$\n",
    "\\nabla_\\theta C_\\mathrm{Ridge} = \\nabla_\\theta C_{OLS} + \\nabla_\\theta \\lambda \\theta^2 = 2 X (X\\cdot \\theta - y) + 2 \\lambda \\theta\n",
    "$$\n",
    "\n",
    "Because the factors of 2 are somewhat tedious, I will from here on use $\\tilde C = \\frac{C}{2}$.\n",
    "\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "be65f56f",
   "metadata": {
    "editable": true
   },
   "source": [
    "## Exercise 3, using the analytical formulae for OLS and Ridge regression to find the optimal paramters $\\boldsymbol{\\theta}$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "a616cdee",
   "metadata": {},
   "outputs": [],
   "source": [
    "def Ridge_parameters(X, y, lam = 0.01):\n",
    "    # Assumes X is scaled and has no intercept column\n",
    "    return np.linalg.inv(X.T @ X + lam * np.eye(X.shape[1])) @ X.T @ y\n",
    "def OLS_parameters(X, y):\n",
    "    return Ridge_parameters(X, y, lam = 0.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "a7256776",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Closed-form Ridge coefficients: [-0.4835408   1.41698898]\n",
      "Closed-form OLS coefficients: [-0.51267301  1.44659558]\n"
     ]
    }
   ],
   "source": [
    "# Set regularization parameter, either a single value or a vector of values\n",
    "lambda_ = 0.1\n",
    "\n",
    "# Analytical form for OLS and Ridge solution: theta_Ridge = (X^T X + lambda * I)^{-1} X^T y and theta_OLS = (X^T X)^{-1} X^T y\n",
    "I = np.eye(n_features)\n",
    "theta_closed_formRidge = Ridge_parameters(X_norm, y_centered, lam=lambda_)\n",
    "theta_closed_formOLS = OLS_parameters(X_norm, y_centered)\n",
    "\n",
    "print(\"Closed-form Ridge coefficients:\", theta_closed_formRidge)\n",
    "print(\"Closed-form OLS coefficients:\", theta_closed_formOLS)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eeae00fd",
   "metadata": {
    "editable": true
   },
   "source": [
    "This computes the Ridge and OLS regression coefficients directly. The identity\n",
    "matrix $I$ has the same size as $X^T X$. 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 $\\boldsymbol{\\theta}$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e1c215d5",
   "metadata": {
    "editable": true
   },
   "source": [
    "### 3a)\n",
    "\n",
    "Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters $\\boldsymbol{\\theta}$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "587dd3dc",
   "metadata": {
    "editable": true
   },
   "source": [
    "### 3b)\n",
    "\n",
    "Explore the results as function of different values of the hyperparameter $\\lambda$. See for example exercise 4 from week 36.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "3edc1a2c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Feature Coefficients ($\\\\theta^\\\\lambda_i - \\\\bar{\\\\theta}_i$)')"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "n_lam = 8\n",
    "thetas = np.zeros((n_features, n_lam))\n",
    "lambdas = np.concatenate([[0], np.logspace(-5, 0, n_lam-1)])\n",
    "\n",
    "\n",
    "for i, lam in enumerate(lambdas):\n",
    "    thetas[:, i] = Ridge_parameters(X_norm, y_centered, lam=lam)\n",
    "\n",
    "avg_thetas = np.mean(thetas, axis=1)\n",
    "norm_thetas = thetas - avg_thetas[:, np.newaxis]\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(8,6))\n",
    "# Annotation\n",
    "im = ax.imshow(norm_thetas, aspect='auto', cmap='viridis')\n",
    "for i in range(n_features):\n",
    "    for j in range(n_lam):\n",
    "        text = ax.text(j, i, f\"{norm_thetas[i, j]:.2e}\",\n",
    "                       ha=\"center\", va=\"center\", color=\"w\" if abs(norm_thetas[i, j]) < 0.5 else \"black\")\n",
    "ax.set_yticks(np.arange(n_features))\n",
    "ax.set_xticks(np.arange(n_lam))\n",
    "ax.set_xticklabels([f\"{l:.2e}\" for l in lambdas], rotation=45)\n",
    "ax.set_ylabel(\"Features\")\n",
    "ax.set_xlabel(\"Regularization Parameter (lambda)\")\n",
    "ax.set_title(r\"Feature Coefficients ($\\theta^\\lambda_i - \\bar{\\theta}_i$)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bfa34697",
   "metadata": {
    "editable": true
   },
   "source": [
    "## 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",
    "necessary if $n$ and $p$ are so large that the closed-form might be\n",
    "too slow or memory-intensive. We derive the gradients from the cost\n",
    "functions defined above. Use the gradients of the Ridge and OLS cost functions with respect to\n",
    "the parameters $\\boldsymbol{\\theta}$ and set up (using the template below) your own gradient descent code for OLS and Ridge regression.\n",
    "\n",
    "Below is a template code for gradient descent implementation of ridge:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "392b0bb7",
   "metadata": {},
   "outputs": [],
   "source": [
    "def gradient_descent(X, y, cost_func, grad_cost_func, eta=0.1, num_iters=1000, **kwargs):\n",
    "    # Initialize weights\n",
    "    theta = np.zeros(X.shape[1])\n",
    "    # Store cost history\n",
    "    cost_history = np.zeros(num_iters)\n",
    "    for t in range(num_iters):\n",
    "        # Compute cost\n",
    "        cost_history[t] = cost_func(X, y, theta, **kwargs)\n",
    "        # Compute gradient\n",
    "        grad = grad_cost_func(X, y, theta, **kwargs)\n",
    "        # Update weights\n",
    "        theta -= eta * grad\n",
    "    return theta, cost_history\n",
    "\n",
    "def OLS_cost_func(X, y, theta):\n",
    "    error = X.dot(theta) - y\n",
    "    return 0.5 * np.mean(error**2)\n",
    "\n",
    "def OLS_grad_cost_func(X, y, theta):\n",
    "    error = X.dot(theta) - y\n",
    "    return X.T.dot(error) / len(y)\n",
    "\n",
    "def Ridge_cost_func(X, y, theta, lam):\n",
    "    return OLS_cost_func(X, y, theta) + 0.5 * lam * np.sum(theta**2)\n",
    "\n",
    "def Ridge_grad_cost_func(X, y, theta, lam):\n",
    "    return OLS_grad_cost_func(X, y, theta) + lam * theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "be6ebfa8",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Gradient Descent OLS coefficients: [-0.47433764  1.40826022]\n",
      "Gradient Descent Ridge coefficients: [-0.44990454  1.38335272]\n"
     ]
    }
   ],
   "source": [
    "eta = 0.1\n",
    "num_iters = 1000\n",
    "lam = 1e-3\n",
    "\n",
    "theta_gdOLS, history_gdOLS = gradient_descent(X_norm, y_centered, OLS_cost_func, OLS_grad_cost_func, eta=eta, num_iters=num_iters)\n",
    "theta_gdRidge, history_gdRidge = gradient_descent(X_norm, y_centered, Ridge_cost_func, Ridge_grad_cost_func, eta=eta, num_iters=num_iters, lam=lam)\n",
    "\n",
    "print(\"Gradient Descent OLS coefficients:\", theta_gdOLS)\n",
    "print(\"Gradient Descent Ridge coefficients:\", theta_gdRidge)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "32a7e3a3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Gradient Descent OLS error: (array([ 0.10881529, -0.09837005]), np.float64(0.010758716397832158))\n",
      "Gradient Descent Ridge error: (array([ 0.13324839, -0.12327754]), np.float64(0.016476242667611243))\n"
     ]
    }
   ],
   "source": [
    "def theta_error(theta_est):\n",
    "    return theta_est - theta_true, np.mean((theta_est - theta_true)**2)\n",
    "\n",
    "print(f\"Gradient Descent OLS error: {theta_error(theta_gdOLS)}\")\n",
    "print(f\"Gradient Descent Ridge error: {theta_error(theta_gdRidge)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f3f43f2c",
   "metadata": {
    "editable": true
   },
   "source": [
    "### 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\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "006ce51b",
   "metadata": {},
   "outputs": [],
   "source": [
    "etas = np.logspace(-4, 0, 10)\n",
    "max_iterations = np.logspace(1, 5, 10, dtype=int)\n",
    "theta_errors = np.zeros((len(etas), len(max_iterations), 2))\n",
    "\n",
    "for i, eta in enumerate(etas):\n",
    "    for j, max_iter in enumerate(max_iterations):\n",
    "        # print(f\"Running GD with eta={eta}, max_iter={max_iter}\")\n",
    "        theta_gdOLS, _ = gradient_descent(X_norm, y_centered, OLS_cost_func, OLS_grad_cost_func, eta=eta, num_iters=max_iter)\n",
    "        theta_gdRidge, _ = gradient_descent(X_norm, y_centered, Ridge_cost_func, Ridge_grad_cost_func, eta=eta, num_iters=max_iter, lam=lam)\n",
    "\n",
    "        theta_errors[i, j, 0] = theta_error(theta_gdOLS)[1]\n",
    "        theta_errors[i, j, 1] = theta_error(theta_gdRidge)[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "b06f0bc6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1200x500 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.cm as mcm\n",
    "import matplotlib.colors as mcolors\n",
    "\n",
    "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))\n",
    "\n",
    "# Log Colormap\n",
    "cm = plt.get_cmap('viridis')\n",
    "# Create LogNorm between min and max of theta_errors\n",
    "norm = mcolors.LogNorm(vmin=theta_errors.min(), vmax=theta_errors.max())\n",
    "\n",
    "\n",
    "im1 = ax1.imshow(theta_errors[:, :, 0], aspect='auto', interpolation='nearest', cmap=cm, norm=norm)\n",
    "ax1.set_title('Gradient Descent OLS Error')\n",
    "fig.colorbar(im1, ax=ax1)\n",
    "\n",
    "im2 = ax2.imshow(theta_errors[:, :, 1], aspect='auto', interpolation='nearest', cmap=cm, norm=norm)\n",
    "ax2.set_title('Gradient Descent Ridge Error')\n",
    "fig.colorbar(im2, ax=ax2)\n",
    "\n",
    "for ax in (ax1, ax2):\n",
    "    ax.set_xlabel('Max Iterations')\n",
    "    ax.set_ylabel('Learning Rate')\n",
    "    ax.set_xticks(np.arange(len(max_iterations)))\n",
    "    ax.set_yticks(np.arange(len(etas)))\n",
    "    ax.set_xticklabels(max_iterations, rotation=45)\n",
    "    ax.set_yticklabels(np.round(etas, 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "adc1a742",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "In this simple example (well converging function) the gradient descent converges for all learning rates given enough iterations. The Error on the paramters approaches 10^-3 and stays at this plateau indicating the parameters have converged\n",
    "\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7e7e7de6",
   "metadata": {
    "editable": true
   },
   "source": [
    "### 4b)\n",
    "\n",
    "Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion?\n",
    "\n",
    "\n",
    "<div class=\"alert alert-block alert-success\">\n",
    "We define the stopping criterion via the decrease in the cost function. If the decrease gets too small we stop. This will be implemented using a custom stopping_criterion function so it can easily be replaced.\n",
    "\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "e214e0be",
   "metadata": {},
   "outputs": [],
   "source": [
    "def gradient_descent(X, y, cost_func, grad_cost_func, eta=0.1, num_iters=1000, stopping_criterion=None, **kwargs):\n",
    "    # Initialize weights\n",
    "    theta = np.zeros(X.shape[1])\n",
    "    # Store cost history\n",
    "    cost_history = np.zeros(num_iters)\n",
    "    for t in range(num_iters):\n",
    "        # Compute cost\n",
    "        cost_history[t] = cost_func(X, y, theta, **kwargs)\n",
    "        # Compute gradient\n",
    "        grad = grad_cost_func(X, y, theta, **kwargs)\n",
    "        # Update weights\n",
    "        theta -= eta * grad\n",
    "        if stopping_criterion is not None and stopping_criterion(cost_history[:t+1]):\n",
    "            print(f\"Converged at iteration {t}\")\n",
    "            break\n",
    "    return theta, cost_history\n",
    "\n",
    "def stopping_criterion(cost_history, tol=1e-5):\n",
    "    if len(cost_history) < 2:\n",
    "        return False\n",
    "    return np.abs(cost_history[-1] - cost_history[-2]) < tol\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "78362c6c",
   "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:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "06077986",
   "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": "e2693666",
   "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:\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc954d12",
   "metadata": {
    "editable": true
   },
   "source": [
    "$$\n",
    "y \\approx 5 \\times x_0 \\;-\\; 3 \\times x_1 \\;+\\; 2 \\times x_6 \\;+\\; \\text{noise}.\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6534b610",
   "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",
    "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?\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "1aaa2666",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Converged at iteration 85\n",
      "Terminated with error on theta of 3.750e-03\n"
     ]
    }
   ],
   "source": [
    "res = gradient_descent(X, y, OLS_cost_func, OLS_grad_cost_func, num_iters=100000, stopping_criterion=stopping_criterion)\n",
    "print(f\"Terminated with error on theta of {theta_error(res[0])[1]:.3e}\")"
   ]
  }
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