{
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# [30 POINTS + 5 Bonus]\n",
        "# COS 324: Programming Assignment 2\n",
        "\n",
        "In this exercise, we'll use Gradient Descent to do predict house prices in Boston.\n",
        "\n",
        "## 2 POINTS Loading the Dataset\n",
        "Before we begin, let us load the dataset into the memory, and get to know it better."
      ],
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "%matplotlib notebook\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "from sklearn.datasets import load_boston\n",
        "dataset = load_boston()\n",
        "\nprint(dataset.DESCR)"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Boston House Prices dataset\n",
            "===========================\n",
            "\n",
            "Notes\n",
            "------\n",
            "Data Set Characteristics:  \n",
            "\n",
            "    :Number of Instances: 506 \n",
            "\n",
            "    :Number of Attributes: 13 numeric/categorical predictive\n",
            "    \n",
            "    :Median Value (attribute 14) is usually the target\n",
            "\n",
            "    :Attribute Information (in order):\n",
            "        - CRIM     per capita crime rate by town\n",
            "        - ZN       proportion of residential land zoned for lots over 25,000 sq.ft.\n",
            "        - INDUS    proportion of non-retail business acres per town\n",
            "        - CHAS     Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)\n",
            "        - NOX      nitric oxides concentration (parts per 10 million)\n",
            "        - RM       average number of rooms per dwelling\n",
            "        - AGE      proportion of owner-occupied units built prior to 1940\n",
            "        - DIS      weighted distances to five Boston employment centres\n",
            "        - RAD      index of accessibility to radial highways\n",
            "        - TAX      full-value property-tax rate per $10,000\n",
            "        - PTRATIO  pupil-teacher ratio by town\n",
            "        - B        1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town\n",
            "        - LSTAT    % lower status of the population\n",
            "        - MEDV     Median value of owner-occupied homes in $1000's\n",
            "\n",
            "    :Missing Attribute Values: None\n",
            "\n",
            "    :Creator: Harrison, D. and Rubinfeld, D.L.\n",
            "\n",
            "This is a copy of UCI ML housing dataset.\n",
            "http://archive.ics.uci.edu/ml/datasets/Housing\n",
            "\n\n",
            "This dataset was taken from the StatLib library which is maintained at Carnegie Mellon University.\n",
            "\n",
            "The Boston house-price data of Harrison, D. and Rubinfeld, D.L. 'Hedonic\n",
            "prices and the demand for clean air', J. Environ. Economics & Management,\n",
            "vol.5, 81-102, 1978.   Used in Belsley, Kuh & Welsch, 'Regression diagnostics\n",
            "...', Wiley, 1980.   N.B. Various transformations are used in the table on\n",
            "pages 244-261 of the latter.\n",
            "\n",
            "The Boston house-price data has been used in many machine learning papers that address regression\n",
            "problems.   \n",
            "     \n",
            "**References**\n",
            "\n",
            "   - Belsley, Kuh & Welsch, 'Regression diagnostics: Identifying Influential Data and Sources of Collinearity', Wiley, 1980. 244-261.\n",
            "   - Quinlan,R. (1993). Combining Instance-Based and Model-Based Learning. In Proceedings on the Tenth International Conference of Machine Learning, 236-243, University of Massachusetts, Amherst. Morgan Kaufmann.\n",
            "   - many more! (see http://archive.ics.uci.edu/ml/datasets/Housing)\n",
            "\n"
          ]
        }
      ],
      "execution_count": 1,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "# 2 POINTS \n",
        "# ACT (1) Append a column filled with 1s to the var `dataset.data`. \n",
        "# This will act as an intercept for the regression problem we're about the solve. \n",
        "# When tyring to model y ~ wx+b; b is the intercept.\n",
        "dataset.data = np.hstack((np.ones((len(dataset.target),1)), dataset.data))\n",
        "\n",
        "assert(dataset.data.shape[1] == 14)\n",
        "\n",
        "print(\"Shape of Feature Matrix \", dataset.data.shape)\n",
        "print(\"Shape of Target Variable \", dataset.target.shape)\n",
        "print(\"Sample (feature, target) \", (dataset.data[0], dataset.target[0]))"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Shape of Feature Matrix  (506, 14)\n",
            "Shape of Target Variable  (506,)\n",
            "Sample (feature, target)  (array([  1.00000000e+00,   6.32000000e-03,   1.80000000e+01,\n",
            "         2.31000000e+00,   0.00000000e+00,   5.38000000e-01,\n",
            "         6.57500000e+00,   6.52000000e+01,   4.09000000e+00,\n",
            "         1.00000000e+00,   2.96000000e+02,   1.53000000e+01,\n",
            "         3.96900000e+02,   4.98000000e+00]), 24.0)\n"
          ]
        }
      ],
      "execution_count": 2,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 3 POINTS Train-Test Split\n",
        "\nAs we've learned in the class, evaluating the performance of our predictor on the same dataset it was trained on isn't the best idea. For this reason, we'll split the dataset into `(X_train, y_train)` and `(X_test, y_test)` datasets. We want the number of examples in the former to be roughly twice that of the later. "
      ],
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 3 POINTS\n",
        "# ACT (2) Split the variables (dataset.data, dataset.target) into (X_train, y_train) and (X_test, y_test).\n",
        "tot_samples = len(dataset.target)\n",
        "num_train = int(2/3*tot_samples)\n",
        "indices = np.random.permutation(tot_samples)\n",
        "X_train = dataset.data[indices[:num_train]]\n",
        "y_train = dataset.target[indices[:num_train]]\n",
        "X_test = dataset.data[indices[num_train:]]\n",
        "y_test = dataset.target[indices[num_train:]]\n",
        "\n",
        "assert(len(y_train) == len(X_train))\n",
        "assert(len(y_test) == len(X_test))\n",
        "assert(len(y_test)+len(y_train) == len(dataset.target))\n",
        "assert(len(y_train) > 1.8*len(y_test))\n",
        "assert(len(y_train) < 2.1*len(y_test))\n",
        "\n",
        "print(\"Number of Training Samples \", len(y_train))\n",
        "print(\"Number of Test Samples \", len(y_test))"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Number of Training Samples  337\n",
            "Number of Test Samples  169\n"
          ]
        }
      ],
      "execution_count": 3,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 4 POINTS The Gradient and the Loss\n",
        "\n",
        "Here we'll define a ``LossSq`` function that takes in\n",
        "+ a weight vector $w\\in\\mathbb{R}^d$\n",
        "+ a feature matrix $X\\in\\mathbb{R}^{n\\times d}$\n",
        "+ a vector of outcomes $y\\in\\mathbb{R}^{n}$\n",
        "\n",
        "and outputs $\\sum_{i=1}^n (x_{(i)}^Tw-y_i)^2$, where $x_{(i)}$ is the $i^{th}$ row.\n",
        "\n\n",
        "We'll also define a ``GradientSq`` function that takes in\n",
        "+ a weight vector $w\\in\\mathbb{R}^d$\n",
        "+ a feature matrix $X\\in\\mathbb{R}^{n\\times d}$\n",
        "+ a vector of outcomes $y\\in\\mathbb{R}^{n}$\n",
        "\nand outputs $\\nabla_w \\left(\\sum_{i=1}^n (x_{(i)}^Tw-y_i)^2\\right)$ - this is a vector in $\\mathbb{R}^d$."
      ],
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 2 PONITS\n",
        "# ACT (3) Design a function LossSq matching the above specifications.\n",
        "def LossSq(w, X, y):\n",
        "    return np.linalg.norm( X.dot(w)-y )**2\n",
        "\nassert(LossSq(np.array(2), np.matrix(3), np.array(9))==9)"
      ],
      "outputs": [],
      "execution_count": 4,
      "metadata": {
        "collapsed": true
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 2 POINTS\n",
        "# ACT (4) Design a function GradientSq matching the above specifications.\n",
        "def GradientSq(w, X, y):\n",
        "    return 2*np.dot( ( X.dot(w)-y ), X)\n",
        "\n",
        "assert(np.sum(GradientSq(np.array([2, 1]), \n",
        "                      np.matrix([[2, 5], [3, 0], [3, 1]]), \n",
        "                      np.array([9, 6, 7]))) == 0)"
      ],
      "outputs": [],
      "execution_count": 5,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 4 POINTS Gradient Descent\n",
        "\n",
        "Now that the hard work is done, we'll code up the `GradD` algorithm which takes in\n",
        "+ the dimension of the weight vector $d\\in\\mathbb{N}$\n",
        "+ a function to compute the gradient $g:w\\to \\nabla_w(\\cdot)$\n",
        "+ a function to compute the loss $l:w\\to \\mathbb{R}$\n",
        "+ the learning rate $\\eta\\in\\mathbb{R}$ - we're using a constant step size\n",
        "+ the number of steps $t\\in\\mathbb{N}$\n",
        "\n",
        "and returns \n",
        "+ a weight vector $w\\in\\mathbb{R}^d$\n",
        "+ a vector of loss at all iterates $l_t\\in\\mathbb{R}^{t}$"
      ],
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 3 POINTS\n",
        "# ACT (5) Design a function GradD does Gradient Descent.\n",
        "def GradD(d, g, l, eta, t):\n",
        "    weight, l_t = np.zeros(d), np.zeros(t)\n",
        "    for it in range(t):\n",
        "        weight -= eta*g(weight)\n",
        "        l_t[it] = l(weight)\n",
        "    return weight, l_t\n",
        "\nassert(GradD(1, lambda _: 1, lambda _: 0, 0.1, 1)[0] == -0.1)"
      ],
      "outputs": [],
      "execution_count": 6,
      "metadata": {
        "collapsed": true
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 1 POINT\n",
        "# ACT (6) Set the value of eta and t.\n",
        "# After completing the next section, you may need to revise this (multiple times).\n",
        "eta, t = 1e-9, 10000\n",
        "\n",
        "w, l_t  = GradD(len(X_train[0]),\n",
        "               lambda w: GradientSq(w, X_train, y_train),\n",
        "               lambda w: LossSq(w, X_train, y_train),\n",
        "               eta, t)"
      ],
      "outputs": [],
      "execution_count": 35,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 4 POINTS Did it work?\n",
        "\n",
        "Write a function `LostPlot` that takes in a vector of loss at all iterates\n",
        "$l_t\\in\\mathbb{R}^{t}$, and plots the loss values against the number of steps.\n",
        "\n",
        "Design a function `DebugInfo` that takes in \n",
        "+ a weight vector $w\\in\\mathbb{R}^d$\n",
        "+ the variables `X_train`, `y_train`, `X_test`, `y_test`\n",
        "\n",
        "and prints\n",
        "+ the range of the target variable\n",
        "+ the [RMSE](https://en.wikipedia.org/wiki/Root-mean-square_deviation) loss on the training dataset\n",
        "+ the [RMSE](https://en.wikipedia.org/wiki/Root-mean-square_deviation) loss on the test dataset\n",
        "\n",
        "Finally, code up `TvF` a function that takes in \n",
        "+ a weight vector $w\\in\\mathbb{R}^d$\n",
        "+ the variables `X_test`, `y_test`\n",
        "\n",
        "and outputs a scatter plot with predicted prices on the $y$-axis and true prices on the $x$ axis.\n",
        "On the same plot draw a line indicating what a perfect predictor would look like - this is \n",
        "a 45 degree line passing through $(5,5)$ and $(50,50)$."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "# 1 POINT\n",
        "# ACT (7) LossPlot to plot the loss against time.\n",
        "def LostPlot(l_t):\n",
        "    plt.plot(l_t)\n",
        "    plt.xlabel(\"Number of Steps\")\n",
        "    plt.ylabel(\"Loss\")\n",
        "    plt.show()\n",
        "    \n",
        "LostPlot(l_t)"
      ],
      "outputs": [
        {
          "output_type": "display_data",
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              "hfv/HKK5vdSusdcAAAAASUVORK5CYII=\n"
            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x10c9cf3c8>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 36,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 1 POINT\n",
        "# ACT (8) Code for DebugInfo.\n",
        "def DebugInfo(w, X_train, y_train, X_test, y_test):\n",
        "    print(\"Range of Target \", min(y_train), max(y_train))\n",
        "    print(\"Train RMSE Loss \", \n",
        "      np.sqrt(LossSq(w, X_train, y_train)/len(y_train)))\n",
        "    print(\"Test RMSE Loss \", \n",
        "      np.sqrt(LossSq(w, X_test, y_test)/len(y_test)))\n",
        "    \n",
        "DebugInfo(w, X_train, y_train, X_test, y_test)"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Range of Target  5.0 50.0\n",
            "Train RMSE Loss  7.99424850305\n",
            "Test RMSE Loss  8.39354780964\n"
          ]
        }
      ],
      "execution_count": 37,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 2 POINT\n",
        "# ACT (9) Code for TvF.\n",
        "def TvF(w, X_test, y_test):\n",
        "    plt.scatter(y_test, X_test.dot(w), edgecolors=(0,0,0))\n",
        "    plt.plot([y_test.min(), y_test.max()], \n",
        "             [y_test.min(), y_test.max()], 'k--', lw=4)\n",
        "    plt.xlabel(\"True Prices\")\n",
        "    plt.ylabel(\"Predicted Prices\")\n",
        "    plt.show()\n",
        "    \n",
        "TvF(w, X_test, y_test)"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x10c3abf98>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 38,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 4 POINTS Normalizing the Features\n",
        "Normalize each _feature_ so that it is zero mean and unit variance. (Hint: Look at the asserts.) \n",
        "As a consequence of this action, we'll have 2 new matrices `X_train_norm` and `X_test_norm`.\n",
        "Note that in section 1 (**Loading the Dataset**) we set the feature indexed at $0$ to be $1$; we\n",
        "wish to leave this feature as it is.\n",
        "\nWe re-run GD on this transformed space of features. Does it perform better?"
      ],
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 3 POINT\n",
        "# ACT (10) Normalize the features.\n",
        "mean, std = np.mean(X_train, axis=0), np.std(X_train, axis=0)\n",
        "std[0] = 1e-12\n",
        "X_train_norm = (X_train-mean)/std\n",
        "X_train_norm[:, 0] = 1\n",
        "X_test_norm = (X_test-mean)/std\n",
        "X_test_norm[:, 0] = 1\n",
        "\n",
        "assert( np.linalg.norm(X_train_norm[:,0])**2 == 337 )\n",
        "assert( np.linalg.norm(np.mean(X_train_norm, axis=0)) == 1.0 )\n",
        "assert( 12.9< np.linalg.norm(np.std(X_train_norm, axis=0))**2 < 13.1 )\n",
        "assert(X_train_norm.shape == X_train.shape)\n",
        "assert(X_test_norm.shape == X_test.shape)"
      ],
      "outputs": [],
      "execution_count": 11,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 1 POINT\n",
        "# ACT (11) Choose eta, t again.\n",
        "eta, t = 1e-4, 400\n",
        "w, l_t  = GradD(len(X_train_norm[0]),\n",
        "               lambda w: GradientSq(w, X_train_norm, y_train),\n",
        "               lambda w: LossSq(w, X_train_norm, y_train),\n",
        "               eta, t)\n",
        "\n",
        "print(\"Gradient Descent on Normalized Features\")\n",
        "LostPlot(l_t)\n",
        "DebugInfo(w, X_train_norm, y_train, X_test_norm, y_test)\n",
        "TvF(w, X_test_norm, y_test)"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Gradient Descent on Normalized Features\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
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            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x10ace4cc0>"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Range of Target  5.0 50.0\n",
            "Train RMSE Loss  4.59803888299\n",
            "Test RMSE Loss  4.90765603271\n"
          ]
        },
        {
          "output_type": "display_data",
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              "AABJRU5ErkJggg==\n"
            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x10b03fd68>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 12,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# OPTIONAL ACT -- No credits bonus or otherwise.\n",
        "# Observe.\n",
        "print(\"Original Condition Number \", np.linalg.cond(X_train))\n",
        "print(\"New Condition Number \", np.linalg.cond(X_train_norm))"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Original Condition Number  15106.3649261\n",
            "New Condition Number  9.15428277989\n"
          ]
        }
      ],
      "execution_count": 13,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 3 POINT Stochastic Gradient Descent\n",
        "\n",
        "We'll code up Stochastic Gradient Descent. But we'll do it cleverly.\n",
        "\n",
        "The idea is that we'll design a function `StoGradientSq` (in the spirit of `GradientSq`) \n",
        "which calculates the gradient on **one** randomly sampled data point, rather than calculaing it\n",
        "on the entire dataset. Concretely, it takes in\n",
        "+ a weight vector $w\\in\\mathbb{R}^d$\n",
        "+ a feature matrix $X\\in\\mathbb{R}^{n\\times d}$\n",
        "+ a vector of outcomes $y\\in\\mathbb{R}^{n}$\n",
        "\nand outputs $\\nabla_w  (x_{(i)}^Tw-y_i)^2$, where $i\\sim Unif(\\lbrace1,2,\\dots n\\rbrace)$."
      ],
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 2 POINT\n",
        "# ACT (12) Fill in the code for StoGradientSq.\n",
        "def StoGradientSq(w, X, y):\n",
        "    index = np.random.randint(len(y))\n",
        "    return GradientSq(w, X[index], y[index])\n",
        "\n",
        "# 1 POINT\n",
        "# ACT (13) Set eta, t.\n",
        "eta, t = 1e-2, 400\n",
        "w, l_t  = GradD(len(X_train[0]),\n",
        "               lambda w: StoGradientSq(w, X_train_norm, y_train),\n",
        "               lambda w: LossSq(w, X_train_norm, y_train),\n",
        "               eta, t)\n",
        "\n",
        "print(\"Stochastic Gradient Descent on Normalized Features\")\n",
        "LostPlot(l_t)\n",
        "DebugInfo(w, X_train_norm, y_train, X_test_norm, y_test)\n",
        "TvF(w, X_test_norm, y_test)"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Stochastic Gradient Descent on Normalized Features\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
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            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x10abcbe48>"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Range of Target  5.0 50.0\n",
            "Train RMSE Loss  4.80898844265\n",
            "Test RMSE Loss  5.12971368746\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
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            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x10ad13a20>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 14,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 4 POINT Is SGD faster?\n",
        "\n",
        "We'll attempt to investigate the performance of SGD against GD. This will involve _trail-error_.\n",
        "1. Calculate the number of steps it takes for GD (on normalized features) to reach a training RMSE of 5.0 or less.\n",
        "2. Calculate the number of examples GD computes the gradient on to reach a training RMSE of 5.0 or less.\n",
        "3. Calculate the number of steps it takes for SGD (on normalized features) to reach a training RMSE of 5.0 or less.\n",
        "4. Calculate the number of the number of examples SGD computes the gradient on to reach a training RMSE of 5.0 or less.\n",
        "\nWhat do you conclude?"
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "# 2 POINT (1 for each)\n",
        "# ACT (14) Number of steps and number of examples it takes for GD.\n",
        "eta_GD, t_GD = 4e-4, 8\n",
        "n_GD = t_GD*len(y_train)\n",
        "w, l_t  = GradD(len(X_train_norm[0]),\n",
        "               lambda w: GradientSq(w, X_train_norm, y_train),\n",
        "               lambda w: LossSq(w, X_train_norm, y_train),\n",
        "               eta_GD, t_GD)\n",
        "\n",
        "print(\"Gradient Descent on Normalized Features\")\n",
        "DebugInfo(w, X_train_norm, y_train, X_test_norm, y_test)\n",
        "print(\"Number of steps for GD\", t_GD) \n",
        "print(\"Number of examples for GD\", n_GD)"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Gradient Descent on Normalized Features\n",
            "Range of Target  5.0 50.0\n",
            "Train RMSE Loss  5.13762604887\n",
            "Test RMSE Loss  5.28371476129\n",
            "Number of steps for GD 8\n",
            "Number of examples for GD 2696\n"
          ]
        }
      ],
      "execution_count": 15,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 2 POINT (1 for each)\n",
        "# ACT (15) Number of steps and number of examples it takes for SGD.\n",
        "# Being random, this will involve some amount of variability.\n",
        "# Choose t_SGD so that RMSE<5.0 on 10 or so consecutive trails.\n",
        "eta_SGD, t_SGD = 1e-2, 400\n",
        "n_SGD = t_SGD\n",
        "w, l_t  = GradD(len(X_train_norm[0]),\n",
        "               lambda w: StoGradientSq(w, X_train_norm, y_train),\n",
        "               lambda w: LossSq(w, X_train_norm, y_train),\n",
        "               eta_SGD, t_SGD)\n",
        "\n",
        "print(\"Stochastic Gradient Descent on Normalized Features\")\n",
        "DebugInfo(w, X_train_norm, y_train, X_test_norm, y_test)\n",
        "print(\"Number of steps for SGD\", t_SGD) \n",
        "print(\"Number of examples for SGD\", n_SGD)"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Stochastic Gradient Descent on Normalized Features\n",
            "Range of Target  5.0 50.0\n",
            "Train RMSE Loss  4.81684909027\n",
            "Test RMSE Loss  5.24352209639\n",
            "Number of steps for SGD 400\n",
            "Number of examples for SGD 400\n"
          ]
        }
      ],
      "execution_count": 16,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 2 POINT Least Squares Regression thorugh Matrix Inversion\n",
        "\n",
        "We want to minimize $\\min_w \\|Xw-y\\|^2$. If we set the gradient to zero, we get \n",
        "$$\\nabla_w \\|Xw-y\\|^2 = 2X^T(Xw-y),$$\n",
        "$$\\implies w^* = (X^TX)^{-1}X^Ty$$."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "# 2 POINT\n",
        "# ACT (16) Choose w as prescribed by the equation.\n",
        "w = np.linalg.pinv(X_train).dot(y_train)\n",
        "\n",
        "print(\"Analytic Solution\")\n",
        "DebugInfo(w, X_train, y_train, X_test, y_test)\n",
        "TvF(w, X_test, y_test)"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Analytic Solution\n",
            "Range of Target  5.0 50.0\n",
            "Train RMSE Loss  4.596321851\n",
            "Test RMSE Loss  4.90302946321\n"
          ]
        },
        {
          "output_type": "display_data",
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            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x10aeb8b38>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 17,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 5 POINTS BONUS: Polynomial Combination of Features\n",
        "Any row of the feature matrix (`X_train` or `X_test`) can be represented as $(1, x_1, x_2, \\dots x_{13})$. \n",
        "For any such row, once can create a new representation (and hence richer feature set)\n",
        "$$(1,x_1,x_2,\\dots x_{13}, x_1^2, x_1 x_2, \\dots x_1 x_{13}, x_2^2, x_2 x_3,\\dots x_{13}, \\dots, \\dots x_{13}^2) $$\n",
        "Intuitively, having cross terms like $x_ix_j$ captures how features $x_i$ and $x_j$ interact.\n",
        "\n",
        "Create matrices `X_train_2` and `X_test_2` from `X_train` and `X_test` where each row of the original is expanded to include these interaction terms.\n",
        "\n",
        "As an example, the matrix `[[1, 2, 3], [1, 4, 5]]` should transform to `[[1, 2, 3, 4, 6, 9], [1, 4, 5, 16, 20, 25]]`.\n",
        "\nNow solve the new regression problem. Note that the weight vector in this case will not be $14$ dimensional."
      ],
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 3 POINTS\n",
        "# ACT (17) Create X_train_2, X_test_2.\n",
        "from sklearn.preprocessing import PolynomialFeatures\n",
        "pf = PolynomialFeatures(2)\n",
        "X_train_2 = pf.fit_transform(X_train)\n",
        "X_test_2 = pf.transform(X_test)\n",
        "\n",
        "# 2 POINTS\n",
        "# ACT (18) Choose your favorite way to solve this regression problem.\n",
        "w = np.linalg.pinv(X_train_2).dot(y_train)\n",
        "\n",
        "print(\"With Polynomial Features\")\n",
        "DebugInfo(w, X_train_2, y_train, X_test_2, y_test)\n",
        "TvF(w, X_test_2, y_test)\n"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "With Polynomial Features\n",
            "Range of Target  5.0 50.0\n",
            "Train RMSE Loss  2.2752104443\n",
            "Test RMSE Loss  3.94182331871\n"
          ]
        },
        {
          "output_type": "display_data",
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            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x10a954588>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 18,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    }
  ],
  "metadata": {
    "kernel_info": {
      "name": "python3"
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