examples/machine_learning/ridge_regression.ipynb

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    "# Machine Learning: Ridge Regression\n",
    "\n",
    "Ridge regression is a regression technique that is quite similar to unadorned least squares linear regression: simply adding an $\\ell_2$ **penalty** on the parameters $\\beta$ to the objective function for linear regression yields the objective function for ridge regression.\n",
    "\n",
    "Our goal is to find an assignment to $\\beta$ that minimizes the function\n",
    "\n",
    "$$f(\\beta) = \\|X\\beta - Y\\|_2^2 + \\lambda \\|\\beta\\|_2^2,$$\n",
    "\n",
    "where $\\lambda$ is a hyperparameter and, as usual, $X$ is the training data and $Y$ the observations. In practice, we tune $\\lambda$ until we find a model that generalizes well to the test data.\n",
    "\n",
    "Ridge regression is an example of a **shrinkage method**: compared to least squares, it shrinks the parameter estimates in the hopes of **reducing variance, improving prediction accuracy, and aiding interpetation**.\n",
    "\n",
    "In this notebook, we show how to fit a ridge regression model using CVXPY, how to evaluate the model, and how to tune the hyper-parameter $\\lambda$.\n"
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import cvxpy as cp\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
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   "cell_type": "markdown",
   "metadata": {},
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    "### Writing the objective function\n",
    "\n",
    "We can decompose the **objective function** as the sum of a **least squares loss function** and an $\\ell_2$ **regularizer**."
   ]
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   "metadata": {},
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   "source": [
    "def loss_fn(X, Y, beta):\n",
    "    return cp.pnorm(cp.matmul(X, beta) - Y, p=2)**2\n",
    "\n",
    "def regularizer(beta):\n",
    "    return cp.pnorm(beta, p=2)**2\n",
    "\n",
    "def objective_fn(X, Y, beta, lambd):\n",
    "    return loss_fn(X, Y, beta) + lambd * regularizer(beta)\n",
    "\n",
    "def mse(X, Y, beta):\n",
    "    return (1.0 / X.shape[0]) * loss_fn(X, Y, beta).value"
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   "cell_type": "markdown",
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   "source": [
    "### Generating data\n",
    "Because ridge regression encourages the parameter estimates to be small, and as such tends to lead to models with **less variance** than those fit with vanilla linear regression. We generate a small dataset that will illustrate this."
   ]
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   "cell_type": "code",
   "execution_count": 3,
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   "source": [
    "def generate_data(m=100, n=20, sigma=5):\n",
    "    \"Generates data matrix X and observations Y.\"\n",
    "    np.random.seed(1)\n",
    "    beta_star = np.random.randn(n)\n",
    "    # Generate an ill-conditioned data matrix\n",
    "    X = np.random.randn(m, n)\n",
    "    # Corrupt the observations with additive Gaussian noise\n",
    "    Y = X.dot(beta_star) + np.random.normal(0, sigma, size=m)\n",
    "    return X, Y\n",
    "\n",
    "m = 100\n",
    "n = 20\n",
    "sigma = 5\n",
    "\n",
    "X, Y = generate_data(m, n, sigma)\n",
    "X_train = X[:50, :]\n",
    "Y_train = Y[:50]\n",
    "X_test = X[50:, :]\n",
    "Y_test = Y[50:]"
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   "source": [
    "### Fitting the model\n",
    "\n",
    "All we need to do to fit the model is create a CVXPY problem where the objective is to minimize the the objective function defined above. We make $\\lambda$ a CVXPY parameter, so that we can use a single CVXPY problem to obtain estimates for many values of $\\lambda$."
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   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
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   "source": [
    "beta = cp.Variable(n)\n",
    "lambd = cp.Parameter(nonneg=True)\n",
    "problem = cp.Problem(cp.Minimize(objective_fn(X_train, Y_train, beta, lambd)))\n",
    "\n",
    "lambd_values = np.logspace(-2, 3, 50)\n",
    "train_errors = []\n",
    "test_errors = []\n",
    "beta_values = []\n",
    "for v in lambd_values:\n",
    "    lambd.value = v\n",
    "    problem.solve()\n",
    "    train_errors.append(mse(X_train, Y_train, beta))\n",
    "    test_errors.append(mse(X_test, Y_test, beta))\n",
    "    beta_values.append(beta.value)"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Evaluating the model\n",
    "\n",
    "Notice that, up to a point, penalizing the size of the parameters reduces test error at the cost of increasing the training error, trading off higher bias for lower variance; in other words, this indicates that, for our example, a properly tuned ridge regression **generalizes better** than a least squares linear regression."
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   "cell_type": "code",
   "execution_count": 5,
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   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "\n",
    "def plot_train_test_errors(train_errors, test_errors, lambd_values):\n",
    "    plt.plot(lambd_values, train_errors, label=\"Train error\")\n",
    "    plt.plot(lambd_values, test_errors, label=\"Test error\")\n",
    "    plt.xscale(\"log\")\n",
    "    plt.legend(loc=\"upper left\")\n",
    "    plt.xlabel(r\"$\\lambda$\", fontsize=16)\n",
    "    plt.title(\"Mean Squared Error (MSE)\")\n",
    "    plt.show()\n",
    "    \n",
    "plot_train_test_errors(train_errors, test_errors, lambd_values)"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Regularization path\n",
    "As expected, increasing $\\lambda$ drives the parameters towards $0$. In a real-world example, those parameters that approach zero slower than others might correspond to the more **informative** features. It is in this sense that ridge regression can be considered **model selection.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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   "source": [
    "def plot_regularization_path(lambd_values, beta_values):\n",
    "    num_coeffs = len(beta_values[0])\n",
    "    for i in range(num_coeffs):\n",
    "        plt.plot(lambd_values, [wi[i] for wi in beta_values])\n",
    "    plt.xlabel(r\"$\\lambda$\", fontsize=16)\n",
    "    plt.xscale(\"log\")\n",
    "    plt.title(\"Regularization Path\")\n",
    "    plt.show()\n",
    "    \n",
    "plot_regularization_path(lambd_values, beta_values)"
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