Solutions to Chapter 3 The Linear Model.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercise 3.1\n",
    "\n",
    "No. PLA will keep updating if the data is not linearly separable.\n",
    "\n",
    "#### Exercise 3.2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Add lib input sys.path\n",
    "import os\n",
    "import sys\n",
    "nb_dir = os.path.split(os.getcwd())[0]\n",
    "if nb_dir not in sys.path:\n",
    "    sys.path.append(nb_dir)\n",
    "\n",
    "import libs.linear_models as lm\n",
    "import libs.data_util as data\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import sklearn\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.optimize import minimize\n",
    "import math\n",
    "from sklearn.preprocessing import normalize\n",
    "from functools import partial\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "def generate_random_coeffs(dim):\n",
    "    rn = data.generate_random_numbers(1, dim, 1000, -10, 10)\n",
    "    return rn\n",
    "\n",
    "def true_f(x, coeffs):\n",
    "    return coeffs.flatten()[0] + np.dot(coeffs.flatten()[1:], x.flatten())\n",
    "\n",
    "def generate_df(N, dim,true_func, rn_func):\n",
    "    cls1, cls2 = data.generate_two_classes(N/2, dim, true_func, rn_func)\n",
    "    cols = ['x' + str(i) for i in range(1, dim+1)]\n",
    "    df1 = pd.DataFrame(cls1, columns=cols)\n",
    "    df1['y'] = 1\n",
    "    df2 = pd.DataFrame(cls2, columns=cols)\n",
    "    df2['y'] = -1\n",
    "    df = pd.concat([df1, df2])\n",
    "    df['x0'] = 1\n",
    "    df = df[['x0'] + cols + ['y']]\n",
    "    return df\n",
    "\n",
    "def get_true_random_funcs(N, dim, num_grid_points, lb, ub, coeff_lb, coeff_ub):\n",
    "    rns = data.generate_random_numbers(N, dim, num_grid_points, lb, ub)\n",
    "    rn_func = partial(data.generate_random_numbers, num_grid_points = num_grid_points, lb = lb, ub = ub)\n",
    "    coeffs = data.generate_random_numbers(1, dim+1, num_grid_points, coeff_lb, coeff_ub)\n",
    "    norm_coeffs = coeffs.flatten()/coeffs.flatten()[-1]\n",
    "    true_func = partial(true_f, coeffs = norm_coeffs)\n",
    "    return true_func, rn_func, norm_coeffs\n",
    "\n",
    "def gen_random_2class_data(N, dim,  num_grid_points, lb, ub, coeff_lb, coeff_ub):\n",
    "    true_func, rn_func, norm_coeffs = get_true_random_funcs(N, dim, num_grid_points,\n",
    "                                                            lb, ub, coeff_lb, coeff_ub)\n",
    "    df = generate_df(N, dim, true_func, rn_func)\n",
    "    return df, norm_coeffs\n",
    "\n",
    "def plot_data(x1, df, norm_coeffs, norm_g, lb, ub):\n",
    "        figsize = plt.figaspect(1)\n",
    "        f, ax = plt.subplots(1, 1, figsize=figsize)\n",
    "        cls1_df = df.loc[df['y']==1]\n",
    "        cls2_df = df.loc[df['y']==-1]\n",
    "        line = ax.plot(x1, -(norm_coeffs[0]+norm_coeffs[1]*x1), label='True Function')\n",
    "        pluses = ax.scatter(cls1_df[['x1']].values, cls1_df[['x2']].values, marker='+', c= 'r', label='+1 labels')\n",
    "        minuses = ax.scatter(cls2_df[['x1']].values, cls2_df[['x2']].values, marker=r'$-$', c= 'b', label='-1 labels')\n",
    "        if norm_g:\n",
    "            hypothesis = ax.plot(x1, -(norm_g[0]+norm_g[1]*x1), c = 'r', label='Final Hypothesis')\n",
    "\n",
    "        ax.set_ylabel(r\"$x_2$\", fontsize=11)\n",
    "        ax.set_xlabel(r\"$x_1$\", fontsize=11)\n",
    "        ax.set_title('Data set size = %s'%N, fontsize=9)\n",
    "        ax.axis('tight')\n",
    "        legend_x = 2.0\n",
    "        legend_y = 0.5\n",
    "        ax.legend(['True Function', 'Final Hypothesis', \n",
    "                   '+1 labels', '-1 labels', ], \n",
    "                  loc='center right', bbox_to_anchor=(legend_x, legend_y))\n",
    "        #ax.legend(handles=[pluses, minuses], fontsize=9)\n",
    "        ax.set_ylim(bottom=lb, top=ub)\n",
    "        plt.show()\n",
    "        \n",
    "def plot_data_hypothesis(df, xs, hypotheses, des_s):\n",
    "    positives = df.loc[df['y']==1]\n",
    "    negatives = df.loc[df['y']==-1]\n",
    "    \n",
    "    figsize = plt.figaspect(1)\n",
    "    f, ax = plt.subplots(1, 1, figsize=figsize)\n",
    "\n",
    "    ps = ax.scatter(positives[['x1']].values, positives[['x2']].values, marker='+', c= 'b', label='+1 labels')\n",
    "    ns = ax.scatter(negatives[['x1']].values, negatives[['x2']].values, marker=r'$-$', c= 'r', label='-1 labels')\n",
    "    \n",
    "    labels = []\n",
    "    for g, des in zip(hypotheses, des_s):\n",
    "        label = 'Hypothesis: ' + des\n",
    "        labels.append(label)\n",
    "        norm_g = g/g[-1]\n",
    "        hypothesis = ax.plot(xs, -(norm_g[0]+norm_g[1]*xs), label=label)\n",
    "\n",
    "    ax.set_ylabel(r\"$x_2$\", fontsize=11)\n",
    "    ax.set_xlabel(r\"$x_1$\", fontsize=11)\n",
    "    ax.axis('tight')\n",
    "    legend_x = 2.0\n",
    "    legend_y = 0.5\n",
    "    ax.legend(labels + ['+1 labels', '-1 labels'], \n",
    "              loc='center right', bbox_to_anchor=(legend_x, legend_y))\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
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AEwBGAChBpKX/aLzXqepCVS1V1dJ+/fqlLV7quXy/D02hMJpCYbtDsY+bZ/sjorQxuiUvIlmIJPjfqurzAKCqn8TsfwrACpvCoxSJndo225dtczQOxhY8UcYzNsmLiAD4FYDNqrogZvtAq78eAGYCqLQjPkqdoJXkGxpD6J2XoUmeJXdKgXXr1vX3+XyLAIyFAyq5lJAwgMpQKHT9hAkTdrfdaWySR6TvfTaAjSKywdp2N4DLRaQEgALYCuCb9oRHqdKyEl0m9MsziVMa+Xy+RQMGDCjq16/fAY/Ho3bHQz0XDodlz549xbt27VoE4OK2+41N8qq6FoDE2fVKumOh9MrzZ1CS7wyTPyXXWCZ4d/F4PNqvX7+6Xbt2jY2339gkT5mrpVzv5svoeC072cPDBO8+1mcat/uFfTIu0JNpa02c8jZark/bZXSZOnI9U8+bKIOwJe8CPVmP3sS17IP+LABw99S2HFhHGWjXrl3eKVOmjAaAvXv3Znk8Hu3Tp08IADZs2LA5EAgkpcrwwgsv5M+ePXvE4MGDmwCgb9++x9auXfthMo4NAGvXrs3dtWuX75JLLvkUAJYsWXLihx9+6L/vvvs+6ey16cYkT8bJ83sBpKFcn6kl80w9b7LdgAEDmrds2VIFALfffvugYDDYfO+997ZKjOFwGKoKr9fbo/c666yz6l977bWaHh2kHe+8805uZWVlTjTJX3XVVQdT8T7JwHI9GScvO83lejtxoRoiVFZW+keNGjXmiiuuGDpmzJjimpqa7Pz8/JLo/oULF/a+7LLLhgHAtm3bfOeff/6IsWPHFo0bN67o9ddfz0v0fWbMmHHqs88+e2L0cW5u7hlApOX/hS98oeD8888fMXz48LEzZ84cHn3OG2+8kVdSUlI4evTo4tNPP73w008/9fz4xz8euHz58j6FhYXFTz/9dO8FCxb0vfbaa4cAwJYtW7LPOuusgoKCguIvfvGLo2pqarKi733NNdcMOeOMMwpPOeWUcUuWLDkRacCWPBnH4xEE/WlYiS5TS+aZet7Uyp3PvTfkg131uck8ZsGA/MM/vmT8tu68tqamJrBo0aJ/TZ48+T/Hjh1r93k33HDD0O9+97u7zjvvvEPV1dXZF1100agPP/xwU9vn/eMf/8gvLCwsBoCvf/3r+x988MFdHb3/pk2bcjdu3Lhp8ODBx0pKSopef/31vC984QuHZ8+efdrSpUtrJk2adHjfvn3e3Nzc8J133llbWVmZs3jx4m0AsGDBgr7R48ydO3fYnDlz9t544437H3nkkb4333zzkFdfffVjANi7d69v3bp1W959992cWbNmnZaOCgCTfCfKy4H581tvmzfPvH5st8nze909up6IWhkyZEjj5MmTD3f2vL/97W+9ampqAtHHdXV13oaGBgkGg63687tari8pKTk0bNiwYwAwduzYwzU1NdmBQCA8aNCgpkmTJh0GgJNOOqm5s+O89957eW+88caHAHDTTTfte+ihh1oWVrv44osPejwenHXWWUd2796dlpm+mOQ7YeLAtEyQ1uVmM7Ulm6nnTQCA7ra4UyUnJ6dlsQqPp3VP8tGjR1s2qGq3B+n5fD4NhyNvEwqF0Nzc3DIXS3Z2duz7aygUElW1Jl9NjtiYVdNzJSP75MlIwUAWJ8MhylBerxe9evVq3rhxo7+5uRkvvvhiS//12Wef/enDDz/csurYW2+9lZPocYcNG9ZUUVGRBwDPPvts7+bmjhvmEyZMOLJjx47stWvX5gLA/v37PaFQCPn5+c0NDQ1x82dJSUnDr371qz4A8OSTT540ceLE+kTjSwUmeTJS0O9lkk8WXg9PDlReXr59+vTpo774xS8WDBo0qKWTftGiRf95++23gwUFBcUjRowY88QTTyS8zOi3v/3tPatWreo1bty4ovXr1+dkZ2d32JzOycnRJUuWfHzTTTcNHT16dPHUqVMLjhw54vnKV75SX1VVlVtUVFT89NNP9459zRNPPPGfxYsX9y0oKCh+7rnnev/iF7+wtWIi6SoZ2Km0tFQrKirsDoO64JvPVmDr3sP439vOtTsU57NxgJ2IrFPV0rS/McX13nvvbR0/fvxeu+Og5Hvvvff6jh8/fnjb7WzJu4iJs9fFk0icQX83y/WpbLU6rUXMNemJMh4H3rmIUwYJJhJnfiCNA+/cgJfDEVEcTPJkpDyrT15VkdDo1lTO4ubUGeJ4PTxRxmOSJyMF/VloDiuOHgsjJ7tn01u6mlO/gBBRWjDJk5GCgc/WlE8oyaey1er0FrHT4iWipOHAO4dxyuC6ngpGF6lhv3zHonPfT54cuXEufCKK0aUkLyK5MfdPSn441JnyckC19c2dST6y3GyXp7ZNZZJjAiXqMa/XO6GwsLA4equurs5es2ZN7pw5c4Z095iDBw8eV1tbe1xluu32FStW5E+dOnVkd98nnrfeeitn6dKlJ0Qf33777YPuueeek3t63FtvvXXQCy+8kN/T4yRcrheRnwMYIiJVqno3gPsA3NTTALpLRC4A8FMAXgCLVPWHdsVCyRf0R1eia3+hCoooLwfmr14VeWCNUeT6CmQqv98fji43GzV69Oimc889t9N5601UUVGRW1FRkXfZZZfVJfO4jz322M5kHKcrLfkTVPVrANaKyD3JePPuEhEvgF8CuBBAMYDLRaTYzpicyOTSfzTJH2rsdD2IjJcp1R2yycSJozFx4uhUvkVsC/v2228fdOmllw6fOHHi6FNOOWXc/fff3z/6vGnTpo0YM2ZM0ciRI8c88sgjfds/Yseam5sxbNiwsTt37vRFHw8dOnRsbW2tr6ysbPgVV1wxdMKECaOHDx8+9ve///0JAHD48GG55JJLhhcUFBQXFRUVv/zyy/lHjx6Vhx56aNDLL7/cu7CwsPipp57qDQCbN2/OiRf/448/3mfcuHFFhYWFxVdcccWwUCiEUCiEsrKy4aNGjRpTUFBQPH/+/P4AUFZWNjw6m95NN900eMSIEWMKCgqK586de0pXzrUrA++aAEBVXxGR/gCuh30t+YkAPlLVjwFARP4AYAaAqg5fRa2YfF39ZwPv2JIncpPGxkZPdAnYIUOGNK5cufK4leI++uijwFtvvVV98OBBb1FR0dg777xzj9/v19/+9rdbTz755OaGhgY544wziq+88soDAwYM6LAlMHny5ILogjeHDx/2jBgx4qjX68Ull1yyb9GiRX3uueee3S+++GKvoqKiIwMHDgwBwLZt2/zvvPNOdVVVlX/atGmjZ8yYsfHhhx/uDwAffPBB1fr16wPTp08fVVNTU3nXXXftrKioyFuyZMl/AOD222/PiRf/pk2b/M8991yfioqKLX6/X6+88sqhTz755Enjx48/UltbmxVdLnfv3r2tRhp/8skn3ldeeaX3xx9/XOnxeI7b35lOk7yIlKjqBgBPR7ep6jMisqcrb5RkgwHEzge8HcBZsU8QkbkA5gLA0KFD0xcZJUW0Jc/lZolsEm29v/tusNXjd96p7slh45Xr2zr//PMP5uTkaE5OTqhPnz7Htm/f7hsxYsSxhx9++OQ///nPJwLArl27sjZt2hQYMGDAoY6OtXr16g+iyXvFihX5jz766MkAcOONN+69+OKLR95zzz27Fy9e3HfOnDkt0/2WlZXt93q9GDduXOOQIUMaN2zYEHjrrbeC3/rWt3YDwBlnnHF00KBBTRs3bgzEe8948b/66qv5lZWVuePHjy8CIivr9e/fP3TZZZcd3LZtm//qq68e8tWvfrVu5syZn8Yeq0+fPs1+vz88a9asYV/5ylfqutotkEi5fqmIfEFV/xbdICKTVfXPXXmjJIs3O0qrSfhVdaGqlqpqab9+Ca9fQIbIb2nJs1xPlGn8fn/L33Ov14tQKCQrVqzIX716dX5FRcWW6urqqqKioiNHjhzp9hViI0eOPNa3b9/QSy+9lL9+/fq8Sy+9tCV5tp2AS0S6tDRsvPhVVS699NJ9W7ZsqdqyZUvV1q1bKxcsWLCzX79+zZWVlVVTp06tf/zxx/vPmjVreOyxsrKysGHDhs1lZWUHX3jhhROnTJkyqivnmcgv6CIAi0RkmogUishLAJ7oypukwHYAsSMxTwGQlEEKZAa/zwOvRzKvXM/55ckU77xTjXfeqcbnPteAz32uoeWxTQ4ePOg94YQTmvPz88Pr168PvPfee3k9Pea111675/rrrz/14osv3u/zfVbYfv7553s3Nzdj06ZN/m3btvnHjx9/dNKkSQ2/+c1v+gDA+++/76+trc0+/fTTj/bq1avdZWdjXXDBBZ+uWLGi944dO3xApAz/wQcfZNfW1vqam5sxZ86cg/fff/+OjRs35sa+rq6uzrN//37vZZddVvfkk09u27x5c278d4iv03K9qn4oIv8HwJsAGgDMA/CrrrxJCrwLYJSInApgB4BZAK6wNyRKJhFB0O9juZ6IAABlZWV1Cxcu7GctMXt0/PjxHZbpE3H55ZfX3XLLLd65c+fui90+cuTIxokTJ47et29f1mOPPfbv3Nxc/c53vrN79uzZwwoKCoq9Xi/++7//e2tOTo5eeOGF9Y888sjAwsLC4jvuuKO2vfeaMGHC0R/84Ac7zjvvvIJwOIysrCz92c9+9p/c3NzwddddNzwcDgsA3HvvvdtjX3fw4EHvRRddNLKxsVEA4P777+/S0rWdLjVrjaS/AcASANMBzFPV5V15k1QQkekAHkPkErrFqvpAe8/lUrOpU14OzJ/feluyLt86+4dv4POnnYRHvzG+5wczXdvpaSdPjvx0+HX5XGrWLFxqtrU1a9bk3nbbbUPWrVvXUqEoKysbftFFF9Vdc801B+yMravaW2o2kdH1wwCUqupOEVkA4BUROUFVn0lyjF2iqq8AeMXOGCi1I/SDfl/mleu7w6nT7RLZ6O677x7wzDPP9Hv66af/ZXcsqZRIuf66mPu7RWQagJcBPJPCuIgQzKTlZp0+Pz6Rwzz44IO7HnzwwV1tty9btmyrDeGkTJcXqFHVgyJyfiqCIYoV9Ptw8HCT3WGYiyvQEVEnurUKnaoeSXYgRG0F/T5sP+DImS67z8QEzS8PRI7FpWbJWJE++Qwp13cHS/xE1AkmeTJWMMBL6GzF7gAix+N68mSsoN+HQ03NCIcTn2kqI3EJXHKRxYsX9x45cuQYj8czYc2aNXEnfqmurs4eNWrUmI6O051lZSdOnDi6vfd0KiZ5MlbLSnRNbM13S09nz4t+eZg8OXLjlwlKohUrVuSXlZUNb7u9pKTkyLJlyz4qLS1tsCEs12GSp6RK5vK1n61El0CS53SwRK5w5plnHh0/fnxjos+vrq7OnjBhwuji4uKi4uLiopUrV7ZMd1tfX+/90pe+NGLEiBFjrrjiiqHNzZG1MJ5//vleJSUlhcXFxUUXXnjhaXV1da1yYXvLvzoRk3wnTF5z3UTJXNucK9F1U/QLz+rVkVuyWvREBho0aFDozTff/KCqqmrz0qVLP77ttttalh3duHFj3k9/+tNt1dXVm7Zu3epfsmRJ79raWt+DDz44cM2aNR9UVVVtPvPMMw/fd999J8ce8+23386NLv/6wQcfVN188837jn9nZ+DAu06YvOa627Uk+Y5a8hwcRpQyt9+OQT/5CQbGbrvtNtQuWND9BcFOP/30wqamJs/hw4c9dXV1vuja8g888MD2srKyTzt7fVtNTU1y3XXXDauqqsrxeDz497//7Y/uGzdu3KHi4uImAPjGN76x/8033wwGAoFwTU1NYOLEiYUAcOzYMZkwYUKrroHCwsLGjpZ/dRImeTJWl8r19BleWkdJsmABdvYkocfz/vvvbwEiffJPP/30ST2dYe6BBx44uX///seWLVv2r3A4jJycnAnRfe0tGTtp0qRPX3755Xans40u/7p8+fJejz/+eP+lS5f2+dOf/tSjOO3Ccj0ZK6FyPQeHEWW0uro678CBA495vV48/vjjJ0X73YFIuX7Lli3Zzc3NeO655/qcc8459VOmTDlUUVERrKys9ANAfX295/333/fHHrOz5V+dhC15k7Dl1UpC5Xo6Hv8dkYMtWbLkxDvvvHPogQMHfDNnzhxVVFR0eO3atR+29/xbb4Lt890AAA72SURBVL11d1lZ2YgXXnih96RJk+pzcnLC0X0lJSUNd9xxxylbtmzJOeuss+pnz559MLpM7KxZs05ramoSAJg3b96O008/vWWw39atW7M6Wv7VSTpdatYNHLPUbHt/nDP0j/aBQ004476VmPfVYlxz9ql2h+McBv174VKzZuFSs+7Vk6VmKdU4eCyuPI6u7xr+OyKiNpjkTZbhf7SzfR74fR40dGcynAz7XRERxcMkb4L2RkO7YHKX8nJg/vzW2+bNS/yyxKCf89cnjKPqqXPhcDgsHo/H/f20GcQaOxCOt49J3mQu+KPd03kGgoEurkSX4dUPok5U7tmzp7hfv351TPTuEA6HZc+ePScAqIy338gkLyI/BvBVAE0AagBco6oHRWQ4gM0Aqq2n/l1Vb7AlyFRgIjpORrTkk/1FhP+OqB2hUOj6Xbt2Ldq1a9dY8BJqtwgDqAyFQtfH22lkkgewEsBdqhoSkYcB3AXgu9a+GlUtsS+09Ghd5l4FAJhXnnmz7+V1dU15F1Q/iFJlwoQJuwFcbHcclD5GJnlV/UvMw78DuMSuWOzC6XQj8v0+7Pr0qN1hpEYyuhb4ZYaIOuCEcs21AP4n5vGpIrJeRFaLyDntvUhE5opIhYhU7NmzJ/VRUkp0uU8+KjrzXZpXp+OCRkRkEtta8iLyGoABcXZ9X1VftJ7zfQAhAL+19tUCGKqq+0RkAoAXRGSMqh63eICqLgSwEIhMhpOKc6DUy/P7cMhBM951qQLTk64FDjAkogTYluRVdVpH+0XkagAXAThPrWn5VLURQKN1f52I1AAoAOCA6eyoO/L9PtR3Z+AdkyARkZl98iJyASID7Sar6uGY7f0A7FfVZhE5DcAoAB/bFCalQdDvQ2MojKZQGNk+J/QudUN3vnhwgCERJcDIJA/gFwD8AFZaSwVGL5U7F8C9IhIC0AzgBlXdb1+YlGrR5WYPNYaQ7ctO/IVMgkREZiZ5VR3ZzvZlAJalORyyUV7MSnS987qQ5DMFv7wQUQeMTPJEUfmxy812p1XOJEhEGcylnZzkFtFyPdeUJyLqOib5FOH10snRUq6/9Y7ISPnVq9N+7TsRkVMxyadIeTmg2vqWCUk+2V9uWsr1XvbHExF1FfvkKamSPR1vS7n+u98H9lnrErGfnYgoIWzJk9Gi5XonzXpHRGQKtuTJaHnZkX+i9UdDbMETEXURW/LUKTsHEXo9grxsb+aOrucgQyLqAbbkqVN2L3sbDPjQ0J3564mIMhyTPBkvz+9DQ1OGJXkusENEScByPRkv38+WPBFRd7AlT8YLBnyZ1yfPBXaIKAnYkifj5WX7eAkdEVE3sCVPxgsGfJFL6DIRW/BE1ANsyZPx8v0ZWK4nIkoCJnkyXp4/Uq5XVbtDISJyFCZ5Ml4w4EMorGgMhe0OhYjIUZjkDcRlaluLrkSXsf3yRETdZGSSF5FyEdkhIhus2/SYfXeJyEciUi0iX7YzzlTJ1GVq29OyEh375YmIusTk0fU/UdVHYjeISDGAWQDGABgE4DURKVDVZjsCpPSILlLDy+iIiLrGyJZ8B2YA+IOqNqrqvwB8BGCizTFRikVb8izXExF1jclJ/hYReV9EFotIb2vbYADbYp6z3dpGLpbvzwLAcj0RUVfZluRF5DURqYxzmwHgCQAjAJQAqAXwaPRlcQ4V97oqEZkrIhUiUrFnz56UnAOlR57fC4DleiKirrKtT15VpyXyPBF5CsAK6+F2AENidp8CYGc7x18IYCEAlJaW8gJrB2sp1zPJExF1iZHlehEZGPNwJoBK6/5LAGaJiF9ETgUwCsA76Y6P0qulXM8+eSKiLjF1dP2PRKQEkVL8VgDfBABV3SQifwRQBSAE4GaOrHe/QJYHHgEaGo/ZHQoRkaMYmeRVdXYH+x4A8EAawyGbiQiCfh8ONfL7HBFRVxhZridqKz+QxUvokmnKlM/Wqici12KSJ0cI+n3Hl+uTnaiY+IjIZYws1xO1lef3slyfDNEvMatXt37MdeuJXIlJ3lDl5cD8+a23zZuXuXPYBwNZqDtiteSTnaiY+IjIpZjkDVVenrkJPZ58vw87Dhy2Owzni35x4RcZoozAJE+O0Kpcn+xExcRHRC7FJE+OEPRnce76ZOIXGaKMwCRPjhAM+NDQGEI4rPB4rCUMkp2omPiIyGV4CR2lXzcuVQtGF6lpYmueiChRTPLkCEFr/npeRkdElDgmeQOVlwMirW+uGGkfbcGvXh25daFFH12Jzqj56zl5DhEZjn3yBuLlc8fL91vLzXJqWyKihDHJU/r04FK1PCvJG1Gu5+Q5ROQQLNeTIwT9BpbriYgMx5a8CzhuCtxutHjzAwaV6zl5DhE5BFvyLlBeDqi2vpmS4JM1iDCvpSVvQJInInIIJnkHceKo+2R9AcmLXidvUpJftYqteCIyGsv1DpLJo+79Pi+yfR7Um5TkiYgMZ2SSF5GlAEZbD08EcFBVS0RkOIDNAKqtfX9X1RvSHyHZIej3ocGEPnkiIocwMsmr6mXR+yLyKIC6mN01qlqS/qjIbkG/z6xyPRGR4YxM8lEiIgC+AeC/7I6F7Bf0+zjwjoioC0wfeHcOgE9U9cOYbaeKyHoRWS0i59gVGKVfMOAz4xI6IiKHsK0lLyKvARgQZ9f3VfVF6/7lAH4fs68WwFBV3SciEwC8ICJjVPXTOMefC2AuAAwdOjS5wZMtgn4fdtcftTuMxPE6eiKymW1JXlWndbRfRHwAvg5gQsxrGgE0WvfXiUgNgAIAFXGOvxDAQgAoLS3V5EVOdgn6ffh4D1vyRESJMrlPfhqALaq6PbpBRPoB2K+qzSJyGoBRAD62K0BKr2DAIX3ynNueiAxhcpKfhdalegA4F8C9IhIC0AzgBlXdn/bIyBZBP/vkiYi6wtgkr6pz4mxbBmBZ+qMhEwT9PjSGwjjWHEaW1+Axo5zbnogMYfBfSqLWgi3LzbI1T0SUCGNb8kRtRZN8/dEQTszNtjmaBLAFT0Q2Y0ueHCNoLTd7qIkt+WRw4oJHRNQ1TPKUNj1NKtGWPOevTw6TlygmouRguZ7Spqer6EVb8lyJjogoMWzJk2OwJU9E1DVM8hnCDf2vaR9dP2XKZ5fBERE5EMv1GaKnpXITRMv1jpj1jojIAEzy5Bh52Z9dQpdSnJaWiFyC5XpyDK9HkJvt5WQ4REQJYkueHCXoT8MiNZyWlohcgi15cpSg38dL6IiIEsSWPDlKMOBLX7meLXgicji25MlRgn4fr5MnIkoQkzw5Slr65ImIXIJJnhwl6Pel/hI6J+MEPkQUg0meHCUY8HEVOiKiBHHgHTlKtE9eVSEidodjDk7gQ0RxsCVPjpLn9yEUVjSGwnaHQkRkPFuTvIhcKiKbRCQsIqVt9t0lIh+JSLWIfDlm+wXWto9E5Hvpj5rslM/56+NbtSpymzw5cos+JqKMZndLvhLA1wGsid0oIsUAZgEYA+ACAI+LiFdEvAB+CeBCAMUALreeS13U2ap0pq5ax+VmiYgSZ2ufvKpuBhCvb3UGgD+oaiOAf4nIRwAmWvs+UtWPrdf9wXpuVXoido/OVqUzddW6liTPlnx8bL0TUQxTB94NBvD3mMfbrW0AsK3N9rPiHUBE5gKYaz1sEJHqBN63L4C9XQvVWG45l7jnMe5hGyLpGbd8HkDXzmVYKgMhoo6lPMmLyGsABsTZ9X1VfbG9l8XZpojfvaDxDqCqCwEsTCjI6JuKVKhqaefPNJ9bzoXnYR43nQuR26U8yavqtG68bDuAITGPTwGw07rf3nYiIiKKYffAu/a8BGCWiPhF5FQAowC8A+BdAKNE5FQRyUZkcN5LNsZJRERkLFv75EVkJoCfA+gH4M8iskFVv6yqm0Tkj4gMqAsBuFlVm63X3ALgfwF4ASxW1U1JDKlL5X3DueVceB7mcdO5ELmaqMbt0iYiIiKHM7VcT0RERD3EJE9ERORSGZvk3TilroiUi8gOEdlg3abH7It7TiYz/ffdERHZKiIbrc+hwtrWR0RWisiH1s/edscZj4gsFpHdIlIZsy1u7BLxM+szel9EzrQvciJqK2OTPNw7pe5PVLXEur0CtH9OdgbZGQf9vjsy1focol8ivwfgdVUdBeB167GJnkHk30ms9mK/EJGrX0YhMvnUE2mKkYgSkLFJXlU3q2q8WfBaptRV1X8BiE6pOxHWlLqq2gQgOqWuE7R3TiZz8u+7PTMA/Nq6/2sAX7Mxlnap6hoA+9tsbi/2GQCWaMTfAZwoIgPTEykRdSZjk3wHBuP4qXMHd7DdNLdYZdPFMeVgp8Qey4kxx1IAfxGRddYUywBwsqrWAoD1s79t0XVde7E7/XMicjVT565PCrum1E2ljs4JkVLpfYjEdR+ARwFci/bPyWROjDnW2aq6U0T6A1gpIlvsDihFnP45Ebmaq5O8G6fUTfScROQpACushx2dk6mcGHMLVd1p/dwtIssR6X74REQGqmqtVdLebWuQXdNe7I7+nIjcjuX64zl2St02faEzERlcCLR/TiYz/vfdHhHJE5H86H0A5yPyWbwE4GrraVcDaK+aZKL2Yn8JwFXWKPvPA6iLlvWJyH6ubsl3xMApdZPhRyJSgki5dCuAbwJAR+dkKlUNOeD33Z6TASwXESDyf+x3qvqqiLwL4I8ich2A/wC41MYY2yUivwcwBUBfEdkOYB6AHyJ+7K8AmI7IYM7DAK5Je8BE1C5Oa0tERORSLNcTERG5FJM8ERGRSzHJExERuRSTPBERkUsxyRMREbkUkzwREZFLMcmTUaw59+dZ90eJyAdcvpSIqHt4nTwZRUQGA1iHyBKmvwUwV1XX2hsVEZEzsSVPRlHVHYgsZfomgNtUda2InGC18Ld18nIiIorBJE9GsVZtmw7gEKwlTFW1TlWvBVBtZ2xERE6TsXPXk3lE5EQA/wOgHJE1BR4G8FU7YyIicjK25MkIIpKLyNK4j6vqMgCLEFmFbqq9kRERORcH3pEjiMgvAcxA5IvAw6r6L5tDIiIyHpM8ERGRS7FcT0RE5FJM8kRERC7FJE9ERORSTPJEREQuxSRPRETkUkzyRERELsUkT0RE5FJM8kRERC7FJE9ERORS/x9eptCJt5nEnAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "final Error Rate:  0.1\n",
      "final normalized w: [-0.64483871 19.90322581  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [-1.77688889 14.46666667  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [-0.35070175 20.          1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [ 2.16108108 11.54054054  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [ 1.11055556 20.55555556  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [ 1.31081967 15.08196721  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [-0.24679012 14.61728395  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [ 0.         15.76086957  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [-0.2315444  9.8030888  1.       ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [-0.71758974 10.96923077  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [ 0.         19.32352941  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [-0.10357513 11.29015544  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [ 0.         17.18604651  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [-4.18395349 12.20930233  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [ 1.08472868 17.54263566  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [0.35070175 9.68421053 1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [-0.35696429 11.17857143  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [ 0.20608247 12.08247423  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [-0.47595238 16.42857143  1.        ]\n",
      "final Error Rate:  0.1\n",
      "final normalized w: [ 1.05210526 11.21052632  1.        ]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### Exercise 3.2\n",
    "\n",
    "lb, ub = -100, 100\n",
    "N, dim = 100, 2 \n",
    "num_grid_points = 2000\n",
    "coeff_lb, coeff_ub = -10, 10\n",
    "eta = 1\n",
    "maxit = 1000\n",
    "test_N = 1000\n",
    "\n",
    "randomized, print_out = True, True\n",
    "x1 = np.arange(lb, ub, 0.01)\n",
    "#df, norm_coeffs = gen_random_2class_data(N, dim, num_grid_points, lb, ub, coeff_lb, coeff_ub)\n",
    "true_func, rn_func, norm_coeffs = get_true_random_funcs(N, dim, num_grid_points,\n",
    "                                           lb, ub, coeff_lb, coeff_ub)\n",
    "df = generate_df(N, dim, true_func, rn_func)\n",
    "test_df = generate_df(test_N, dim, true_func, rn_func)\n",
    "\n",
    "flips = np.random.randint(N, size=(int(N/10), 1))\n",
    "df.iloc[flips.flatten(), -1] = -df.iloc[flips.flatten(), ]['y']\n",
    "plot_data(x1, df, norm_coeffs, None, lb, ub)\n",
    "\n",
    "flips = np.random.randint(test_N, size=(int(test_N/10), 1))\n",
    "test_df.iloc[flips.flatten(), -1] = -test_df.iloc[flips.flatten(), ]['y']\n",
    "\n",
    "avg_w_ts, avg_what_ts = np.zeros(maxit), np.zeros(maxit)\n",
    "avg_test_w_ts, avg_test_what_ts = np.zeros(maxit), np.zeros(maxit)\n",
    "K = 20\n",
    "for it in range(K):\n",
    "    w, w_ts, what_ts, test_w_ts, test_what_ts, _ = lm.pocket_algo(df.values, dim, \n",
    "                                                                maxit,eta, randomized, \n",
    "                                                                print_out, test_df.values)\n",
    "    \n",
    "    #print(w_ts.shape, what_ts.shape, test_w_ts.shape, test_what_ts.shape)\n",
    "    avg_w_ts += w_ts\n",
    "    avg_what_ts += what_ts\n",
    "    avg_test_w_ts += test_w_ts\n",
    "    avg_test_what_ts += test_what_ts\n",
    "    \n",
    "plt.plot(np.arange(len(avg_w_ts)), avg_w_ts/K, 'b', label='Average E(w)')\n",
    "plt.plot(np.arange(len(avg_what_ts)), avg_what_ts/K, 'r', label='Average E(\\hat(W))')\n",
    "plt.legend()\n",
    "plt.show()\n",
    "\n",
    "plt.plot(np.arange(len(avg_test_w_ts)), avg_test_w_ts/K, 'b', label='Test: Average E(w)')\n",
    "plt.plot(np.arange(len(avg_test_what_ts)), avg_test_what_ts/K, 'r', label='Test: Average E(\\hat(W))')\n",
    "\n",
    "plt.legend()\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercise 3.3\n",
    "\n",
    "1. (a) $H$ is symmetric.\n",
    "\n",
    "\\begin{align*}\n",
    "H^T &= \\left(X(X^TX)^{-1}X^T\\right)^T\\\\\n",
    "&= X(X^TX)^{-T}X^T\\\\\n",
    "&= X(X^TX)^{-1}X^T\\\\\n",
    "&= H\\\\\n",
    "\\end{align*}\n",
    "\n",
    "1. (b) We show that $H^2=H$ first.\n",
    "\n",
    "\\begin{align*}\n",
    "H^2 &= \\left(X(X^TX)^{-1}X^T\\right)\\left(X(X^TX)^{-1}X^T\\right)\\\\\n",
    "&= X(X^TX)^{-1}(X^TX)(X^TX)^{-1}X^T\\\\\n",
    "&= X(X^TX)^{-1}X^T\\\\\n",
    "&= H\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Apply the above relationship repeatedly for $H^K$ we see that $H^K = H$.\n",
    "\n",
    "1. (c) First show $(I-H)^2=I-H$.\n",
    "\n",
    "\\begin{align*}\n",
    "(I-H)^2 &= (I-H)(I-H)\\\\\n",
    "&= II - IH -HI + H^2\\\\\n",
    "&= I - 2H + H^2\\\\\n",
    "&= I-2H + H\\\\\n",
    "&= I-H\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Apply above relationship repeatedly for $(I-H)^K$, we have $(I-H)^K = I-H$.\n",
    "\n",
    "1. (d) Let $A=X(X^TX)^{-1}$, $B=X^T$, apply the property $\\text{trace}(AB)=\\text{trace}(BA)$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\text{trace}(H) &= \\text{trace}\\left(X(X^TX)^{-1}X^T\\right)\\\\\n",
    "&= \\text{trace}\\left(AB\\right)\\\\\n",
    "&= \\text{trace}\\left(BA\\right)\\\\\n",
    "&=\\text{trace}\\left(X^TX(X^TX)^{-1}\\right)\\\\\n",
    "&=\\text{trace}\\left(I\\right)\\\\\n",
    "&= d+1\\\\\n",
    "\\end{align*}\n",
    "\n",
    "where the identity matrix $I$ is of size $d+1$.\n",
    "\n",
    "#### Exercise 3.4\n",
    "\n",
    "1. (a) We have $y = Xw^{*}+\\epsilon$, take this into the expression for in-sample estimate of $y$ is $\\hat{y}$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\hat{y} &= Hy\\\\\n",
    "&= H(Xw^{*} + \\epsilon)\\\\\n",
    "&= Hxw^{*} + H\\epsilon\\\\\n",
    "&= X(X^TX)^{-1}X^TXw^{*} + H\\epsilon\\\\\n",
    "&= Xw^{*} + H\\epsilon\\\\\n",
    "\\end{align*}\n",
    "\n",
    "1. (b) The in-sample error vector is:\n",
    "\n",
    "\\begin{align*}\n",
    "\\hat{y} -  y &= Xw^{*} + H\\epsilon - (Xw^{*} + \\epsilon)\\\\\n",
    "&= (H-I)\\epsilon\\\\\n",
    "\\end{align*}\n",
    "\n",
    "1. (c) \n",
    "\n",
    "\\begin{align*}\n",
    "E_{in}(w_{lin}) &= \\frac{1}{N}\\|Xw_{lin}-y\\|^2\\\\\n",
    "&= \\frac{1}{N}\\|y-\\hat{y}\\|^2\\\\\n",
    "&= \\frac{1}{N}\\|(I-H)\\epsilon\\|^2\\\\\n",
    "&= \\frac{1}{N}\\epsilon^T(I-H)^T(I-H)\\epsilon\\\\\n",
    "&= \\frac{1}{N}\\epsilon^T(I-H^T)(I-H)\\epsilon\\\\\n",
    "&= \\frac{1}{N}\\epsilon^T(I-H)(I-H)\\epsilon\\\\\n",
    "&= \\frac{1}{N}\\epsilon^T(I-H)\\epsilon\\\\\n",
    "\\end{align*}\n",
    "\n",
    "1. (d)\n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\mathcal{D}}\\left[E_{in}(w_{lin})\\right] &= E_{\\mathcal{D}}\\left[\\frac{1}{N}\\epsilon^T(I-H)\\epsilon\\right]\\\\\n",
    "&= \\frac{1}{N}\\left(E_{\\mathcal{D}}[\\epsilon^T\\epsilon]-E_{\\mathcal{D}}[\\epsilon^TH\\epsilon]\\right)\\\\\n",
    "&= \\frac{1}{N}\\left(E_{\\mathcal{D}}[\\sum^N_{k=1}\\epsilon_k^2]-E_{\\mathcal{D}}[\\sum^N_{i=1}\\sum^N_{j=1}\\epsilon_ih_{ij}\\epsilon_j]\\right)\\\\\n",
    "&= \\frac{1}{N}\\left(\\sum^N_{k=1}E_{\\mathcal{D}}\\epsilon_k^2-\\sum^N_{i=1}\\sum^N_{j=1}E_{\\mathcal{D}}[\\epsilon_ih_{ij}\\epsilon_j]\\right)\\\\\n",
    "&= \\frac{1}{N}\\left(N\\sigma^2-\\sum^N_{i=1}E_{\\mathcal{D}}[\\epsilon_i^2h_{ii}]\\right)\\\\\n",
    "&= \\frac{1}{N}\\left(N\\sigma^2-\\sum^N_{i=1}h_{ii}E_{\\mathcal{D}}[\\epsilon_i^2]\\right)\\\\\n",
    "&= \\frac{1}{N}\\left(N\\sigma^2-\\sigma^2\\text{trace}(H)\\right)\\\\\n",
    "&= \\sigma^2\\left(1-\\frac{\\text{trace}(H)}{N}\\right)\\\\\n",
    "&= \\sigma^2\\left(1-\\frac{d+1}{N}\\right)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Here we used the independence of $\\epsilon_i$, and also we assume $H$ is not random variable, so we are not doing expectation w.r.t. $x$, that's how we can pull $h_{ii}$ out of $E_{\\mathcal{D}}[\\epsilon_i^2h_{ii}]$.\n",
    "\n",
    "1. (e) Since $X$ doesn't change, only $\\epsilon$ changes, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\hat{y} -  y^\\prime &= Xw^{*} + H\\epsilon - (Xw^{*} + \\epsilon^\\prime)\\\\\n",
    "&= H\\epsilon - \\epsilon^\\prime\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Following the procedure in problem (c) and (d), we have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\mathcal{D,\\epsilon'}}\\left[E_{test}(w_{lin})\\right] &= E_{\\mathcal{D},\\epsilon'}\\left[\\frac{1}{N}\\|y' - \\hat{y}\\|^2\\right]\\\\\n",
    "&= E_{\\mathcal{D},\\epsilon'}\\left[\\frac{1}{N}\\|\\epsilon' - H\\epsilon\\|^2\\right]\\\\\n",
    "&= \\frac{1}{N}E_{\\mathcal{D},\\epsilon'}\\left[(\\epsilon' - H\\epsilon)^T(\\epsilon' - H\\epsilon)\\right]\\\\\n",
    "&= \\frac{1}{N}E_{\\mathcal{D},\\epsilon'}\\left[(\\epsilon^{\\prime T} - \\epsilon^TH^T)(\\epsilon' - H\\epsilon)\\right]\\\\\n",
    "&= \\frac{1}{N}E_{\\mathcal{D},\\epsilon'}\\left[(\\epsilon^{\\prime T} - \\epsilon^TH)(\\epsilon' - H\\epsilon)\\right]\\\\\n",
    "&= \\frac{1}{N}E_{\\mathcal{D},\\epsilon'}\\left[\\epsilon^{\\prime T}\\epsilon' - \\epsilon^{\\prime T}H\\epsilon-\\epsilon^TH\\epsilon' + \\epsilon^THH\\epsilon\\right]\\\\\n",
    "&= \\frac{1}{N}E_{\\mathcal{D},\\epsilon'}\\left[\\epsilon^{\\prime T}\\epsilon' - \\epsilon^{\\prime T}H\\epsilon-\\epsilon^TH\\epsilon' + \\epsilon^TH\\epsilon\\right]\\\\\n",
    "&= \\frac{1}{N}E_{\\mathcal{D},\\epsilon'}\\left[\\epsilon^{\\prime T}\\epsilon' + \\epsilon^TH\\epsilon\\right]\\\\\n",
    "&= \\frac{1}{N}\\left(\\sum^N_{k=1}E_{\\mathcal{D}}\\epsilon_k^{\\prime 2} + \\sum^N_{i=1}\\sum^N_{j=1}E_{\\mathcal{D}}[\\epsilon_ih_{ij}\\epsilon_j]\\right)\\\\\n",
    "&= \\sigma^2\\left(1+\\frac{d+1}{N}\\right)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "Where we have used the fact that $\\epsilon$ and $\\epsilon^\\prime$ are independent of each other and each $\\epsilon_k$ and $\\epsilon_k^\\prime$ are independent among themselves. \n",
    "\n",
    "#### Exercise 3.5\n",
    "\n",
    "1. (a) \n",
    "\n",
    "\\begin{align*}\n",
    "\\tanh(s) &= \\frac{e^s - e^{-s}}{e^s + e^{-s}}\\\\\n",
    "&= \\frac{e^{2s} - 1}{1+e^{2s}}\\\\\n",
    "&= \\theta(2s) - \\frac{1}{1+e^{2s}}\\\\\n",
    "&= \\theta(2s) - (1-\\theta(2s))\\\\\n",
    "&= 2\\theta(2s) - 1\\\\\n",
    "\\end{align*}\n",
    "\n",
    "1. (b)\n",
    "\n",
    "\\begin{align*}\n",
    "\\tanh(s) &= \\frac{e^s - e^{-s}}{e^s + e^{-s}}\\\\\n",
    "&= \\frac{1-e^{-2s}}{1+e^{-2s}}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Let $s\\to \\infty$, we have $e^{-2s} \\to 0$, thus $\\tanh(s) \\to 1$.\n",
    "Similarly, when $s\\to -\\infty$, $\\tanh(s) \\to -1$.\n",
    "\n",
    "It's also easy to see that $\\tanh(s) \\lt 1$ and $\\tanh(s) \\gt -1$. So $-1$ and $1$ are hard thresholds for $\\tanh(s)$ when $|s|$ is large. \n",
    "\n",
    "When $|s|$ is small, consider Taylor expansion of $e^s = 1 + s + \\frac{s^2}{2}+O(s^3)$, then $e^{-s} = 1-s+\\frac{s^2}{2}-O(s^3)$, we have $\\tanh(s)=\\frac{e^s-e^{-s}}{e^s+e^{-s}} = \\frac{2s+2O(s^3)}{2+s^2} \\approx s$.\n",
    "\n",
    "So $\\tanh(s)$ is approximately linear when $|s|$ is small. There's no threshold in this case. \n",
    "\n",
    "#### Exercise 3.6\n",
    "\n",
    "1. (a) The probability to get $y_n$ is $P(y_n|x_n)$, by maximum likelihood method, we need maximize the likelihood, $\\prod^N_{n=1} P(y_n|x_n)$, this is equivalent to maximize the logrithm of it: $\\sum^N_{n=1}\\ln(P(y_n|x_n))$, or minimize the negative of it: $-\\sum^N_{n=1}\\ln(P(y_n|x_n))$\n",
    "\n",
    "When $y_n=+1$, $P(y_n|x_n)=h(x_n)$, and when $y_n=-1$, $P(y_n|x_n) = 1-h(x_n)$, separate the cases for $y_n=1$ and $y_n=-1$, we have:\n",
    "\n",
    "\\begin{align*}\n",
    "E_{in}(w) &= -\\sum^N_{n=1}\\ln(P(y_n|x_n)) \\\\\n",
    "&= -\\sum^N_{n=1} I(y_n=+1)\\ln h(x_n) + I(y_n=-1)\\ln(1-h(x_n))\\\\\n",
    "&= \\sum^N_{n=1} I(y_n=+1)\\ln\\frac{1}{h(x_n)} + I(y_n=-1)\\ln\\frac{1}{(1-h(x_n))}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "1. (b) For $h(x) = \\theta(w^Tx) = \\frac{e^{w^Tx}}{1+e^{w^Tx}}$, we have $\\ln \\frac{1}{h(x_n)} = \\ln (1+e^{-w^Tx_n})$ and $\\ln \\frac{1}{(1-h(x_n))} = \\ln (1+e^{w^Tx_n})$. Combine them together we have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{in}(w) &= \\sum^N_{n=1} I(y_n=+1)\\ln (1+e^{-w^Tx_n}) + I(y_n=-1)\\ln (1+e^{w^Tx_n})\\\\\n",
    "&= \\sum^N_{n=1}\\ln (1+e^{-y_nw^Tx_n})\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Which is equivalent of minimizing the one in equation (3.9). \n",
    "\n",
    "\n",
    "#### Exercise 3.7\n",
    "\n",
    "Take derivative of $E_{in}(w) = \\frac{1}{N}\\sum^N_{n=1}\\ln (1+e^{-y_nw^Tx_n})$ with respect to $w$, we have: \n",
    "\n",
    "\\begin{align*}\n",
    "\\nabla E_{in}(w) &= -\\frac{1}{N}\\sum^N_{n=1} \\frac{y_nx_n e^{-y_nw^Tx_n}}{1+e^{-y_nw^Tx_n}}\\\\\n",
    " &= -\\frac{1}{N}\\sum^N_{n=1} \\frac{y_nx_n }{1+e^{y_nw^Tx_n}}\\\\\n",
    "&= \\frac{1}{N}\\sum^N_{n=1} -y_nx_n \\theta(-y_nw^Tx_n)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "When a sample is misclassified, $y_nw^Tx_n \\lt 0$, so $\\theta(-y_nw^Tx_n) \\gt 0.5$, while when a sample is correctly classified, $\\theta(-y_nw^Tx_n) \\lt 0.5$, so the contribution of 'misclassified' example is more to the gradient than a correctly classified one. \n",
    "\n",
    "#### Exercise 3.8 \n",
    "\n",
    "$\\hat{v}$ is the direction which gives largest decrease in $E_{in}$ only holds for small $\\eta$, that's because when $\\eta$ is large, we can't ignore the squared term and smaller terms in the Taylor expansion. The lower bound can't be achieved.\n",
    "\n",
    "#### Exercise 3.9\n",
    "1. (a) See plot below\n",
    "1. (b) When $y = 1$\n",
    "  * If $s \\ge 0$, $e_{class}(s,y) = 0$, $e_{sq}(s,y) = (1-s)^2 \\ge 0$\n",
    "  * If $s \\lt 0$, $e_{class}(s,y) = 1$, $e_{sq}(s,y) = (1-s)^2 = 1 - 2s + s^2 \\gt 1$\n",
    "  \n",
    "  In both cases, we have $e_{sq}(s,y) \\ge e_{class}(s,y)$. \n",
    "Similarly, we can prove that when $y=-1$, we have $e_{sq}(s,y) \\ge e_{class}(s,y)$ as well.\n",
    "So $e_{sq}(s,y) \\ge e_{class}(s,y)$ for all cases of $s$ and $y$. The classification is upper bounded by the squared error. \n",
    "\n",
    "\n",
    "1. (c) When $y=1$, \n",
    "  * If $s \\ge 0$, $e_{class}(s,y) = 0$, $e_{log}(s,y) = \\ln(1+\\exp(-ys)) = \\ln(1+\\exp(-s)) \\ge \\ln1 = 0$, also $\\ln2 \\gt 0$, so we have $\\frac{1}{\\ln2}e_{log}(s,y) \\ge 0 = e_{class}(s,y)$.\n",
    "  * If $s \\lt 0$, $e_{class}(s,y) = 1$, $e_{log}(s,y) = \\ln(1+\\exp(-ys)) = \\ln(1+\\exp(-s)) \\gt \\ln2$, so $\\frac{1}{\\ln2}e_{log}(s,y) \\gt 1 = e_{class}(s,y)$\n",
    "  \n",
    "  Similaly, we can prove that when $y=-1$, we have $\\frac{1}{\\ln2}e_{log}(s,y) \\ge e_{class}(s,y)$ as well. \n",
    "The logistic regression is an upper bound to the classification error."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def tanh(s):\n",
    "    a = np.exp(s) - np.exp(-s)\n",
    "    b = np.exp(s) + np.exp(-s)\n",
    "    return a/b\n",
    "\n",
    "def classification_error(s, y):\n",
    "    signs = np.sign(s)\n",
    "    return signs != y\n",
    "\n",
    "def square_error(s, y):\n",
    "    return (y-s)**2\n",
    "\n",
    "def log_error(s, y):\n",
    "    return np.log(1.0+np.exp(-y*s))\n",
    "\n",
    "ss = np.arange(-10,10,0.1)\n",
    "cls_err = classification_error(ss, 1)\n",
    "sq_err = square_error(ss, 1)\n",
    "log_err = log_error(ss, 1)/np.log(2)\n",
    "plt.plot(ss, cls_err, label='Classification Error')\n",
    "plt.plot(ss, sq_err, label='Square Error')\n",
    "plt.plot(ss, log_err, label='Log Error')\n",
    "plt.ylim(-0.1, 2)\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercise 3.10\n",
    "\n",
    "1. (a) If $\\eta = 1$, then the SGD algorithm will update the $w$ by: $w(t+1) = w(t) -\\eta\\nabla e_n(w)  = w(t) -\\nabla e_n(w)$.\n",
    "\n",
    "When $e_n(w) = \\max(0, -y_n w^Tx_n)$, the derivative of $e_n(w)$ when $y_nw^Tx_n \\gt 0$ (when the sample is correctly classified) is zero, the derivative is $-y_nx_n$ when $y_nw^Tx_n \\lt 0$ (i.e. when the sample is misclassified). \n",
    "\n",
    "Take the derivatives into the SGD update equation, we see that's exactly PLA. \n",
    "\n",
    "1. (b) For logistic regression, we have $\\nabla e_n(w) = \\frac{-y_nx_n}{1+e^{y_nw^Tx_n}}$.\n",
    "If $w$ is very large:\n",
    "  * When $y_nw^Tx_n \\gt 0$, $\\nabla e_n(w) \\approx 0$.\n",
    "  * When $y_nw^Tx_n \\le 0$, $\\nabla e_n(w) \\approx -y_nx_n$. \n",
    "  \n",
    "The above results are consistent with the values used in PLA.\n",
    "\n",
    "This is another indication that the logistic regression weights can be used as a good approximation for classification. \n",
    "\n",
    "#### Exercise 3.11\n",
    "\n",
    "Feature transformation $\\Phi(x) = (1, x^2_1, x^2_2)$, a hyperplane in $\\mathcal{Z}$ can be expressed as $\\tilde{w}z = \\tilde{w}_0 z_0 + \\tilde{w}_1 z_1 + \\tilde{w}_2 z_2 = \\tilde{w}_0 + \\tilde{w}_1 x^2_1 + \\tilde{w}_2 x^2_2 = 0$. \n",
    "1. (a) $|\\tilde{w}_1| x^2_1 - |\\tilde{w}_1| x^2_2 = c$ : Hyperbola\n",
    "\n",
    "1. (b) $\\tilde{w}_0 + \\tilde{w}_1 x^2_1= 0$: It can be two vertical lines parallel to axis of $x1$.\n",
    "\n",
    "1. (c) Ellipse.\n",
    "1. (d) No solution. For any $x_1$ and $x_2$, $\\tilde{w}z \\gt 0$. So we can't find $\\tilde{w}$ that satisfies the condition.\n",
    "\n",
    "#### Exercise 3.12 TODO\n",
    " \n",
    "\n",
    "#### Exercise 3.13 \n",
    "\n",
    "$\\Phi_2(x) = (1, x_1, x_2, x^2_1, x_1x_2, x^2_x)$, make the corresponding $\\tilde{w}$ as $\\tilde{w}_0, \\dots, \\tilde{w}_5$, we have\n",
    "\n",
    "1. (a) $(x_1-3)^2 + x_2 -1 =-1 -6x_1 + x_2+ x^2_1$, we choose $\\tilde{w}_0 = -1$, $\\tilde{w}_1 = -6$, $\\tilde{w}_2 = 1$, $\\tilde{w}_3 = 1$, $\\tilde{w}_4 = 0$, $\\tilde{w}_5 = 0$.\n",
    "\n",
    "1. (b) $(x_1-3)^2 + (x_2-4)^2 -1 = 24 - 6x_1 -8x_2 + x^2_1 + x^2_2$ , we choose $\\tilde{w}_0 = 24$, $\\tilde{w}_1 = -6$, $\\tilde{w}_2 = -8$, $\\tilde{w}_3 = 1$, $\\tilde{w}_4 = 0$, $\\tilde{w}_5 = 1$\n",
    "\n",
    "1. (c) $2(x_1-3)^2 + (x_2-4)^2 -1 = 33 - 12x_1 -8x_2 + 2x^2_1 + x^2_2$, we choose $\\tilde{w}_0 = 33$, $\\tilde{w}_1 = -12$, $\\tilde{w}_2 = -8$, $\\tilde{w}_3 = 2$, $\\tilde{w}_4 = 0$, $\\tilde{w}_5 = 1$\n",
    "\n",
    "1. (d) $(x_1-3)^2 - (x_2-4)^2 -1 = -8- 6x_1 + 8x_2 + x^2_1 - x^2_2$, we choose $\\tilde{w}_0 = -8$, $\\tilde{w}_1 = -6$, $\\tilde{w}_2 = 8$, $\\tilde{w}_3 = 1$, $\\tilde{w}_4 = 0$,  $\\tilde{w}_5 = -1$\n",
    "\n",
    "1. (e) $2(x_1+x_2-3)^2 + (x_1-x_2-4)^2 -1 = 33- 20x_1 - 4x_2 + 3x^2_1+ 2x_1x_2 +3x^2_2$, we choose $\\tilde{w}_0 = 33$, $\\tilde{w}_1 = -20$, $\\tilde{w}_2 = -4$, $\\tilde{w}_3 = 1$, $\\tilde{w}_4 = 0$, $\\tilde{w}_5 = 3$\n",
    "\n",
    "1. (f) $2x_1 + x_2 -1$, we choose $\\tilde{w}_0 = -1$, $\\tilde{w}_1 = 2$, $\\tilde{w}_2 = 1$, $\\tilde{w}_3 = 0$, $\\tilde{w}_4 = 0$, $\\tilde{w}_5 = 0$.\n",
    "\n",
    "#### Exercise 3.14\n",
    "\n",
    "Consider the $Q$th order polynomial transform $\\Phi_{Q}$ for $\\mathcal{X} = \\mathcal{R}^d$. Let's count the terms with exactly $q$-degree that can be created with $x_1,x_2, \\dots, x_d$. \n",
    "There are total of $q+d-1 \\choose d-1$ such terms. See [Number of polynomial terms for certain degree and certain number of variables](https://mathoverflow.net/questions/225953/number-of-polynomial-terms-for-certain-degree-and-certain-number-of-variables/225963#225963).\n",
    "\n",
    "So the total terms for the $Q$th order polynomical transform (which is also the dimensionality $\\tilde{d}$ of the feature space $\\mathcal{Z}$) is (Excluding the constant $1$):\n",
    "\n",
    "\\begin{align*}\n",
    "\\tilde{d}(d, Q) &= \\sum^Q_{q=1}{ q+d-1 \\choose d-1}\\\\\n",
    "&= {d \\choose d-1} + {d+1 \\choose d-1} + \\dots + {Q+d-1 \\choose d-1}\\\\\n",
    "&= {d \\choose 1} + {d+1 \\choose 2} + \\dots + {Q+d-1 \\choose Q}\\\\\n",
    "&= ({d+1 \\choose 1} -{d \\choose 0}) + ({d+2 \\choose 2} -{d+1 \\choose 1}) + \\dots + ({d+Q \\choose Q} - {d+Q-1 \\choose Q-1})\\\\\n",
    "&= {d+Q \\choose Q} - 1\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where we have applied the equality: ${N \\choose k} = {N-1 \\choose k} + {N-1 \\choose k-1}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Feature Space Dimensionality\n",
      "d =  2  Q =  2 :  5\n",
      "d =  2  Q =  3 :  9\n",
      "d =  2  Q =  5 :  20\n",
      "d =  2  Q =  10 :  65\n",
      "d =  3  Q =  2 :  9\n",
      "d =  3  Q =  3 :  19\n",
      "d =  3  Q =  5 :  55\n",
      "d =  3  Q =  10 :  285\n",
      "d =  5  Q =  2 :  20\n",
      "d =  5  Q =  3 :  55\n",
      "d =  5  Q =  5 :  251\n",
      "d =  5  Q =  10 :  3002\n",
      "d =  10  Q =  2 :  65\n",
      "d =  10  Q =  3 :  285\n",
      "d =  10  Q =  5 :  3002\n",
      "d =  10  Q =  10 :  184755\n"
     ]
    }
   ],
   "source": [
    "from scipy.special import comb\n",
    "print('Feature Space Dimensionality')\n",
    "ds = [2,3,5,10]\n",
    "Qs = [2,3,5,10]\n",
    "for d in ds:\n",
    "    for Q in Qs:\n",
    "        r = comb(d+Q, Q) - 1\n",
    "        print('d = ', d, ' Q = ', Q, ': ', int(r))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercise 3.15 TODO\n",
    "\n",
    "#### Exercise 3.16\n",
    "\n",
    "Steps of the algorithm that combines $\\Phi_3$ with linear regression.\n",
    "\n",
    "1. For each $x \\in X$, apply $\\Phi_3$ to obtain the matrix $Z$\n",
    "1. Compute the pseudo-inverse $X^\\dagger$ of the matrix $Z$\n",
    "1. Return $w_{lin} = X^\\dagger y$\n",
    "\n",
    "If the dimension of data is large and the order of polynomial transformation is big, the resulting dimension in the feature space will grow in polynomial. The matrix $Z$ will be of size $N \\times \\tilde{d}$. The main computational cost will be spent on the calculation of the pseudo-inverse $X^\\dagger$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.1\n",
    "\n",
    "From the graphs below we can see that PLA and linear regression achieves very close solution. \n",
    "Consider a hyperplane $y = w_0 + w_1x_1 + w_2x_2$ for the $-1$ and $1$ points given $x$ in the space. \n",
    "Once we find the $w$ through linear regression, we let $w_0 + w_1x_1 + w_2x_2=0$, this hyperplane should approximately separate the points.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of positive points:  997\n",
      "Number of negatives points:  1003\n",
      "Final correctness:  2000 . Total iteration:  41\n",
      "Final w: [39.          2.64724348 52.26554305]\n",
      "Liner regression coefficients:  [ 0.24719806 -0.010201    0.07792978]\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### Problem 3.1 (a) \n",
    "N = 2000\n",
    "max_v = 10000\n",
    "rad, thk = 10, 5\n",
    "sep = 5\n",
    "eta = 1\n",
    "use_adaline=False\n",
    "maxit = 1000\n",
    "dim = 2\n",
    "\n",
    "radiuses, radians = data.generate_random_ring(N, rad, rad+thk, max_v)\n",
    "xs, ys, signs = data.move_bottom_ring_and_assign(radiuses, radians, rad + thk/2.0, -sep)\n",
    "df = pd.DataFrame({'x1':xs.flatten(), 'x2':ys.flatten(), 'y':signs.flatten()})\n",
    "df['x0'] = 1\n",
    "df = df[['x0','x1','x2','y']]\n",
    "positives = df.loc[df['y']==1]\n",
    "negatives = df.loc[df['y']==-1]\n",
    "\n",
    "figsize = plt.figaspect(1)\n",
    "f, ax = plt.subplots(1, 1, figsize=figsize)\n",
    "\n",
    "ps = ax.scatter(positives[['x1']].values, positives[['x2']].values, marker='+', c= 'b', label='+1 labels')\n",
    "ns = ax.scatter(negatives[['x1']].values, negatives[['x2']].values, marker=r'$-$', c= 'r', label='-1 labels')\n",
    "print('Number of positive points: ', len(positives))\n",
    "print('Number of negatives points: ', len(negatives))\n",
    "\n",
    "norm_g, num_its, _ = lm.perceptron(df.values, dim, maxit, use_adaline, eta, randomize=False, print_out = True)    \n",
    "x1 = np.arange(-(rad+thk), (rad+thk)+rad + thk/2)\n",
    "norm_g = norm_g/norm_g[-1]\n",
    "hypothesis = ax.plot(x1, -(norm_g[0]+norm_g[1]*x1), c = 'g', label='Final Hypothesis')\n",
    "\n",
    "w_lin = lm.linear_regression(df[['x0','x1','x2']].values, df['y'].values)\n",
    "print('Liner regression coefficients: ', w_lin)\n",
    "linear = ax.plot(x1, -(w_lin[0]+w_lin[1]*x1), c = 'r', label='Linear Regression')\n",
    "\n",
    "ax.set_ylabel(r\"$x_2$\", fontsize=11)\n",
    "ax.set_xlabel(r\"$x_1$\", fontsize=11)\n",
    "ax.set_title('Data set size = %s'%N, fontsize=9)\n",
    "ax.axis('tight')\n",
    "legend_x = 2.0\n",
    "legend_y = 0.5\n",
    "ax.legend(['PLA', 'Linear Regression', \n",
    "           '+1 labels', '-1 labels', ], \n",
    "          loc='center right', bbox_to_anchor=(legend_x, legend_y))\n",
    "#ax.set_ylim(bottom=lb, top=ub)\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.2\n",
    "\n",
    "It generally takes more iterations for PLA to converge when 'sep' is small. Looks like $\\|w\\|$ is increasing when 'sep' is decreasing. According to problem 1.3, the $R$ term is fixed due to $x$ are fixed. The $\\rho$ is the minimum of $y_nw^Tx_n$, which is less affected by the change of $\\|w\\|$. So the overall effect is to increase the time to converge for PLA."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Final correctness:  2000 . Total iteration:  199\n",
      "Final w: [ 25.          -1.28056494 133.40804478]\n",
      "Final correctness:  2000 . Total iteration:  113\n",
      "Final w: [ 47.          -3.44826697 124.55915407]\n",
      "Final correctness:  2000 . Total iteration:  82\n",
      "Final w: [54.         -3.87807228 98.26596635]\n",
      "Final correctness:  2000 . Total iteration:  85\n",
      "Final w: [ 55.          -3.98647007 102.80816471]\n",
      "Final correctness:  2000 . Total iteration:  18\n",
      "Final w: [10.          0.23283782 43.83870834]\n",
      "Final correctness:  2000 . Total iteration:  11\n",
      "Final w: [ 7.         -0.1939142  30.78323126]\n",
      "Final correctness:  2000 . Total iteration:  11\n",
      "Final w: [ 7.         -0.1939142  31.18323126]\n",
      "Final correctness:  2000 . Total iteration:  11\n",
      "Final w: [ 7.         -0.1939142  31.58323126]\n",
      "Final correctness:  2000 . Total iteration:  11\n",
      "Final w: [ 7.         -0.1939142  31.98323126]\n",
      "Final correctness:  2000 . Total iteration:  11\n",
      "Final w: [ 7.         -0.1939142  32.38323126]\n",
      "Final correctness:  2000 . Total iteration:  11\n",
      "Final w: [ 7.         -0.1939142  32.78323126]\n",
      "Final correctness:  2000 . Total iteration:  11\n",
      "Final w: [ 7.         -0.1939142  33.18323126]\n",
      "Final correctness:  2000 . Total iteration:  8\n",
      "Final w: [ 6.         -0.50151656 26.02788349]\n",
      "Final correctness:  2000 . Total iteration:  8\n",
      "Final w: [ 6.         -0.50151656 26.22788349]\n",
      "Final correctness:  2000 . Total iteration:  8\n",
      "Final w: [ 6.         -0.50151656 26.42788349]\n",
      "Final correctness:  2000 . Total iteration:  8\n",
      "Final w: [ 6.         -0.50151656 26.62788349]\n",
      "Final correctness:  2000 . Total iteration:  8\n",
      "Final w: [ 6.         -0.50151656 26.82788349]\n",
      "Final correctness:  2000 . Total iteration:  8\n",
      "Final w: [ 6.         -0.50151656 27.02788349]\n",
      "Final correctness:  2000 . Total iteration:  8\n",
      "Final w: [ 6.         -0.50151656 27.22788349]\n",
      "Final correctness:  2000 . Total iteration:  8\n",
      "Final w: [ 6.         -0.50151656 27.42788349]\n",
      "Final correctness:  2000 . Total iteration:  8\n",
      "Final w: [ 6.         -0.50151656 27.62788349]\n",
      "Final correctness:  2000 . Total iteration:  8\n",
      "Final w: [ 6.         -0.50151656 27.82788349]\n",
      "Final correctness:  2000 . Total iteration:  8\n",
      "Final w: [ 6.         -0.50151656 28.02788349]\n",
      "Final correctness:  2000 . Total iteration:  9\n",
      "Final w: [ 5.         -0.04332083 31.85258991]\n",
      "Final correctness:  2000 . Total iteration:  9\n",
      "Final w: [ 5.         -0.04332083 32.25258991]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1e8f988b860>]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### Problem 3.2\n",
    "N = 2000\n",
    "max_v = 10000\n",
    "rad, thk = 10, 5\n",
    "eta = 1\n",
    "use_adaline=False\n",
    "maxit = 1000\n",
    "dim = 2\n",
    "seps = np.arange(0.2, 5.2, 0.2)\n",
    "\n",
    "radiuses, radians = data.generate_random_ring(N, rad, rad+thk, max_v)\n",
    "its, threoticals_ts = [], []\n",
    "for sep in seps:\n",
    "    xs, ys, signs = data.move_bottom_ring_and_assign(radiuses, radians, rad + thk/2.0, -sep)\n",
    "    df = pd.DataFrame({'x1':xs.flatten(), 'x2':ys.flatten(), 'y':signs.flatten()})\n",
    "    df['x0'] = 1\n",
    "    df = df[['x0','x1','x2','y']]\n",
    "    positives = df.loc[df['y']==1]\n",
    "    negatives = df.loc[df['y']==-1]\n",
    "    norm_g, num_its, theoretical_t = lm.perceptron(df.values, dim, maxit, use_adaline, eta, \n",
    "                                                   randomize=False, print_out = True)\n",
    "    its.append(num_its)\n",
    "    threoticals_ts.append(theoretical_t)\n",
    "\n",
    "plt.plot(seps, its, marker='.')\n",
    "#plt.plot(seps, threoticals_ts, marker='.') # The theoretical ts are too large, in the order of 1e7"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.3\n",
    "\n",
    "1. (a) If $sep=-5$, that means the points are not separable by a line. If we run PLA on those examples, the PLA algorithm will never stop.\n",
    "\n",
    "1. (d) It takes much much longer to run pocket algorithm than linear regression (in this case, about 30000 times slower). This is because at each iteration we need to recompute the in sample error in pocket algorithm. \n",
    "\n",
    "The quality of pocket solution seems to be better than linear regression though. It has less mis-classified points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "final Error Rate:  0.079\n",
      "final normalized w: [-1.56463472 -0.14400001  1.        ]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### Problem 3.3 (b)-(e)\n",
    "N = 2000\n",
    "max_v = 10000\n",
    "rad, thk = 10, 5\n",
    "eta = 1\n",
    "use_adaline=False\n",
    "maxit = 100000\n",
    "dim = 2\n",
    "sep = -5\n",
    "randomized, print_out = True, True\n",
    "\n",
    "radiuses, radians = data.generate_random_ring(N, rad, rad+thk, max_v)\n",
    "xs, ys, signs = data.move_bottom_ring_and_assign(radiuses, radians, rad + thk/2.0, -sep)\n",
    "df = pd.DataFrame({'x1':xs.flatten(), 'x2':ys.flatten(), 'y':signs.flatten()})\n",
    "df['x0'] = 1\n",
    "df = df[['x0','x1','x2','y']]\n",
    "    \n",
    "w_pocket, w_ts, what_ts, test_w_ts, test_what_ts, _ = lm.pocket_algo(df.values, dim, \n",
    "                                                           maxit,eta, randomized, \n",
    "                                                           print_out, None)\n",
    "\n",
    "plt.plot(w_ts)\n",
    "\n",
    "w_lin = lm.linear_regression(df[['x0','x1','x2']].values, df['y'].values)\n",
    "xs = np.arange(-(rad+thk), (rad+thk)+rad + thk/2)\n",
    "plot_data_hypothesis(df, xs, [w_pocket, w_lin], ['pocket_algo', 'linear regression'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "final Error Rate:  0.0345\n",
      "final normalized w: [-3.87198093 -3.88535272 -1.33406246  5.49714086  6.89048836 -7.47225709\n",
      " -0.80807891  0.36033478 -0.46447154  1.        ]\n",
      "Pocket algorithm time:  275.1863398551941\n",
      "Linear Regression time:  0.001995086669921875\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### Problem 3.3 (e)\n",
    "import time\n",
    "\n",
    "\n",
    "def polynomial_transformation_2v(df):\n",
    "    df['x1x2'] = df['x1']*df['x2']\n",
    "    df['x1_sq'] = df['x1']*df['x1']\n",
    "    df['x2_sq'] = df['x2']*df['x2']\n",
    "    df['x2x1_sq'] = df['x2']*df['x1_sq']\n",
    "    df['x1x2_sq'] = df['x1']*df['x2_sq']\n",
    "    df['x1_cu'] = df['x1_sq']*df['x1']\n",
    "    df['x2_cu'] = df['x2_sq']*df['x2']\n",
    "    df = df[['x0','x1','x2', \n",
    "             'x1x2', 'x1_sq', 'x2_sq', \n",
    "             'x1x2_sq', 'x2x1_sq', 'x1_cu', 'x2_cu', \n",
    "             'y']]\n",
    "    return df\n",
    "\n",
    "df_poly = polynomial_transformation_2v(df)\n",
    "dim_poly = 9\n",
    "start = time.time()\n",
    "w_pocket, w_ts, what_ts, test_w_ts, test_what_ts, _ = lm.pocket_algo(df_poly.values, dim_poly, \n",
    "                                                                  maxit,eta, randomized, \n",
    "                                                                  print_out, None)\n",
    "end = time.time()\n",
    "print(\"Pocket algorithm time: \", end-start)\n",
    "\n",
    "plt.plot(w_ts)\n",
    "start = time.time()\n",
    "w_lin = lm.linear_regression(df_poly[['x0','x1','x2', 'x1x2', 'x1_sq', 'x2_sq', \n",
    "                                      'x1x2_sq', 'x2x1_sq', 'x1_cu', 'x2_cu']].values, \n",
    "                             df_poly['y'].values)\n",
    "end = time.time()\n",
    "print(\"Linear Regression time: \", end-start)\n",
    "xs = np.arange(-(rad+thk), (rad+thk)+rad + thk/2, 0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [],
   "source": [
    "def poly3(x2, x1, w):\n",
    "    return (w[0] + w[1]*x1 + w[2]*x2 + w[3]*x1*x2\n",
    "            + w[4]*x1**2 + w[5]*x2**2 + w[6]*x1*x2**2\n",
    "            + w[7]*x2*x1**2 + w[8]*x1**3+w[9]*x2**3\n",
    "           )\n",
    "\n",
    "def get_coeffs(w, x1):\n",
    "    return [w[9], \n",
    "            w[6]*x1+w[5], \n",
    "            w[7]*x1**2+w[3]*x1+w[2], \n",
    "            w[8]*x1**3+w[4]*x1**2+w[1]*x1+w[0]]\n",
    "\n",
    "def deriv(x2, x1, w):\n",
    "    return (3*w[9]*x2**2 + w[7]*x1**2+2*w[6]*x1*x2+2*w[5]*x2+w[3]*x1+w[2])\n",
    "\n",
    "from functools import partial\n",
    "import scipy.optimize\n",
    "from scipy.optimize import minimize\n",
    "#p3 = partial(poly3, x1= x, w= w_lin)\n",
    "#d3 = partial(deriv, x1= x, w= w_lin)\n",
    "#res = minimize(p3, 0, method='BFGS', jac=d3, tol=1e-12)\n",
    "\n",
    "def find_poly_roots(xs, w):\n",
    "    x2s1, x2s2, x2s3, vs = [], [], [], []\n",
    "    for x in xs:\n",
    "        coeffs = get_coeffs(w, x)\n",
    "        roots = np.roots(coeffs)\n",
    "        r1, r2, r3 = roots\n",
    "        if np.iscomplex(roots[0]):\n",
    "            if np.iscomplex(roots[1]):\n",
    "                r1 = r3\n",
    "                r2 = r3\n",
    "            elif np.iscomplex(roots[2]):\n",
    "                r1 = r2\n",
    "                r3 = r2\n",
    "        elif np.iscomplex(roots[1]):\n",
    "            r2 = r1\n",
    "            r3 = r1\n",
    "\n",
    "        x2s1.append(r1)\n",
    "        x2s2.append(r2)\n",
    "        x2s3.append(r3)\n",
    "    return x2s1, x2s2, x2s3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### Problem 3.3 (e) Polynomial transformation with Pocket algorithm\n",
    "\n",
    "positives = df_poly.loc[df_poly['y']==1]\n",
    "negatives = df_poly.loc[df_poly['y']==-1]\n",
    "\n",
    "figsize = plt.figaspect(1)\n",
    "f, ax = plt.subplots(1, 1, figsize=figsize)\n",
    "ps = ax.scatter(positives[['x1']].values, positives[['x2']].values, marker='+', c= 'b', label='+1 labels')\n",
    "ns = ax.scatter(negatives[['x1']].values, negatives[['x2']].values, marker=r'$-$', c= 'r', label='-1 labels')\n",
    "\n",
    "labels= ['Hypothesis: Pocket_Algo (root1)', \n",
    "         'Hypothesis: Pocket_Algo (root2)', \n",
    "         'Hypothesis: Pocket_Algo (root3)']\n",
    "\n",
    "x2s1, x2s2, x2s3 = find_poly_roots(xs, w_pocket)\n",
    "hypothesis = ax.plot(xs, x2s1, marker='.', markersize='2', color='g', linestyle=\"\")\n",
    "hypothesis = ax.plot(xs, x2s2, marker='.', markersize='2', color='g', linestyle=\"\")\n",
    "hypothesis = ax.plot(xs, x2s3, marker='.', markersize='2', color='g', linestyle=\"\")\n",
    "\n",
    "ax.set_ylabel(r\"$x_2$\", fontsize=11)\n",
    "ax.set_xlabel(r\"$x_1$\", fontsize=11)\n",
    "#ax.axis('tight')\n",
    "legend_x = 2.0\n",
    "legend_y = 0.5\n",
    "ax.legend(labels + ['+1 labels', '-1 labels'], \n",
    "          loc='center right', bbox_to_anchor=(legend_x, legend_y))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### Problem 3.3 (e) Polynomial transformation with Linear Regression\n",
    "positives = df_poly.loc[df_poly['y']==1]\n",
    "negatives = df_poly.loc[df_poly['y']==-1]\n",
    "\n",
    "figsize = plt.figaspect(1)\n",
    "f, ax = plt.subplots(1, 1, figsize=figsize)\n",
    "ps = ax.scatter(positives[['x1']].values, positives[['x2']].values, marker='+', c= 'b', label='+1 labels')\n",
    "ns = ax.scatter(negatives[['x1']].values, negatives[['x2']].values, marker=r'$-$', c= 'r', label='-1 labels')\n",
    "\n",
    "labels= ['Hypothesis: LR (root1)', \n",
    "         'Hypothesis: LR (root2)', \n",
    "         'Hypothesis: LR (root3)']\n",
    "\n",
    "x2s1, x2s2, x2s3 = find_poly_roots(xs, w_lin)\n",
    "hypothesis = ax.plot(xs, x2s1, marker='.', markersize='2', color='g', linestyle=\"\")\n",
    "hypothesis = ax.plot(xs, x2s2, marker='.', markersize='2', color='g', linestyle=\"\")\n",
    "hypothesis = ax.plot(xs, x2s3, marker='.', markersize='2', color='g', linestyle=\"\")\n",
    "\n",
    "ax.set_ylabel(r\"$x_2$\", fontsize=11)\n",
    "ax.set_xlabel(r\"$x_1$\", fontsize=11)\n",
    "#ax.axis('tight')\n",
    "legend_x = 2.0\n",
    "legend_y = 0.5\n",
    "ax.legend(labels + ['+1 labels', '-1 labels'], \n",
    "          loc='center right', bbox_to_anchor=(legend_x, legend_y))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.4\n",
    "\n",
    "1. (a) \n",
    "\n",
    "  * If $y_nw^Tx_n \\gt 1$, then we have $E_n(w) = 0$, which is continus and differentiable.\n",
    "  * If $y_nw^Tx_n \\lt 1$, then we have $E_n(w) = (1-y_nw^Tx_n)^2 $, a polynomial function on $w$, which is also continuous and differentiable. \n",
    "  * If $y_nw^Tx_n = 1$, we have $w^Tx_n=y_n$, both above functions converge to $E_n(w^*) = 0$ at this point. So $E_n(w)$ is continuous at this $w$.\n",
    "  \n",
    "  Take the derivative of $(1-y_nw^Tx_n)^2 $, we have $\\frac{\\partial{E_n(w)}}{\\partial{w}} = -2y_nx_n(1-y_nw^Tx_n) $, which equals to zero when $w^Tx_n=y_n$. This proves that the function $E_n(w)$ is differentiable everywhere. \n",
    "\n",
    "1. (b) \n",
    "  * If $y_n \\gt 0$ and $w^Tx_n \\ge 0$, then $E_n(w) \\ge 0 = sign(w^Tx_n\\ne y_n)$.\n",
    "  * If $y_n \\gt 0$ and $w^Tx_n \\lt 0$, then $E_n(w) = \\left(\\max(0,1-y_nw^Tx_n)\\right)^2 = \\left(1-y_nw^Tx_n\\right)^2\\gt 1 = sign(w^Tx_n \\ne y_n)$ because $-y_nw^Tx_n \\gt 0$.\n",
    "  \n",
    "    Similarly we can prove the same thing when $y_n \\le 0$. So $E_n(w)$ is an upper bound for $sign(w^Tx_n)\\ne y_n$.\n",
    "\n",
    "    Hence $\\frac{1}{N}\\sum^N_{n=1}E_n(w)$ is an upper bound for the in-sample classification error $E_in(w)$.\n",
    "\n",
    "1. (c) From problem (a), we have $\\frac{\\partial{E_n(w)}}{\\partial{w}} = -2y_nx_n(1-y_nw^Tx_n)$ when \n",
    "$y_nw^Tx_n \\lt 1$ and $\\frac{\\partial{E_n(w)}}{\\partial{w}} = 0$ when $y_nw^Tx_n \\ge 1$. Let $x_n$ and $y_n$ be the example picked at time $t$ and $s_t = w^T(t)x_n$. The update formula for SGD is thus: \n",
    "\n",
    "\\begin{align*}\n",
    "w(t+1) &= w(t) - \\eta \\frac{\\partial{E_n(w)}}{\\partial{w}}\\\\\n",
    "&= \\left\\{\\begin{array}{l}\n",
    "w(t) - \\eta \\times 0 &\\mbox{if $y_nw^T(t)x_n \\gt 1$}\\\\\n",
    "w(t) - \\eta\\left(-2y_nx_n(1-y_nw^T(t)x_n)\\right) &\\mbox{if $y_nw^T(t)x_n \\le 1$}\n",
    "\\end{array}\\right.\\\\\n",
    "&= \\left\\{\\begin{array}{l}\n",
    "w(t) &\\mbox{if $y_ns_t \\gt 1$}\\\\\n",
    "w(t)+2\\eta y_nx_n(1-y_ns_t) &\\mbox{if $y_ns_t \\le 1$}\n",
    "\\end{array}\\right.\\\\\n",
    "&= \\left\\{\\begin{array}{l}\n",
    "w(t) &\\mbox{if $y_ns_t \\gt 1$}\\\\\n",
    "w(t)+2\\eta(y_n-y_ny_ns_t)x_n &\\mbox{if $y_ns_t \\le 1$}\n",
    "\\end{array}\\right.\\\\\n",
    "&= \\left\\{\\begin{array}{l}\n",
    "w(t) &\\mbox{if $y_ns_t \\gt 1$}\\\\\n",
    "w(t)+2\\eta(y_n-s_t)x_n &\\mbox{if $y_ns_t \\le 1$}\n",
    "\\end{array}\\right.\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Note we used the fact that $s_t$ is scalar, and moved $x_n$ outside of the brackets.\n",
    "Take $\\eta^{'} = 2\\eta$,we recover the Adaline algorithm. \n",
    "\n",
    "#### Problem 3.5\n",
    "1. (a)   \n",
    "  * If $y_nw^Tx_n \\gt 1$, then we have $E_n(w) = 0$, which is continus and differentiable.\n",
    "  * If $y_nw^Tx_n \\lt 1$, then we have $E_n(w) = 1-y_nw^Tx_n$, a linear function in $w$, which is also continuous and differentiable. \n",
    "  * If $y_nw^Tx_n = 1$, we have $w^Tx_n=y_n$, both above functions converge to $E_n(w^*) = 0$ at this point. So $E_n(w)$ is continuous at this $w$.\n",
    "  \n",
    "  Take the derivative of $1-y_nw^Tx_n$, we have $\\frac{\\partial{E_n(w)}}{\\partial{w}} = -y_nx_n$, which doesn't equal to zero when $w^Tx_n=y_n$. This says that the function $E_n(w)$ is  differentiable everywhere except when $w^Tx_n=y_n$. \n",
    "  \n",
    "1. (b) \n",
    "  * If $y_n \\gt 0$ and $w^Tx_n \\ge 0$, then $E_n(w) = \\max(0,1-y_nw^Tx_n) \\ge 0 = sign(w^Tx_n\\ne y_n)$.\n",
    "  * If $y_n \\gt 0$ and $w^Tx_n \\lt 0$, then $E_n(w) =\\max(0,1-y_nw^Tx_n) = 1-y_nw^Tx_n \\gt 1 = sign(w^Tx_n \\ne y_n)$ because $-y_nw^Tx_n \\gt 0$.\n",
    "  \n",
    "    Similarly we can prove the same thing when $y_n \\le 0$. So $E_n(w)$ is an upper bound for $sign(w^Tx_n)\\ne y_n$.\n",
    "\n",
    "    Hence $\\frac{1}{N}\\sum^N_{n=1}E_n(w)$ is an upper bound for the in-sample classification error $E_in(w)$.\n",
    "    \n",
    "    \n",
    "1. (c) From problem (a), we have $\\frac{\\partial{E_n(w)}}{\\partial{w}} = -y_nx_n$ when \n",
    "$y_nw^Tx_n \\lt 1$ and $\\frac{\\partial{E_n(w)}}{\\partial{w}} = 0$ when $y_nw^Tx_n \\ge 1$. Let $x_n$ and $y_n$ be the example picked at time $t$. The update formula for SGD is thus: \n",
    "\n",
    "\\begin{align*}\n",
    "w(t+1) &= w(t) - \\eta \\frac{\\partial{E_n(w)}}{\\partial{w}}\\\\\n",
    "&= \\left\\{\\begin{array}{l}\n",
    "w(t) - \\eta \\times 0 &\\mbox{if $y_nw^T(t)x_n \\gt 1$}\\\\\n",
    "w(t) - \\eta\\left(-y_nx_n\\right) &\\mbox{if $y_nw^T(t)x_n \\le 1$}\n",
    "\\end{array}\\right.\\\\\n",
    "&= \\left\\{\\begin{array}{l}\n",
    "w(t) &\\mbox{if $y_ns_t \\gt 1$}\\\\\n",
    "w(t)+\\eta y_nx_n &\\mbox{if $y_ns_t \\le 1$}\n",
    "\\end{array}\\right.\\\\\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "Take $\\eta =1$,we recover the Perceptron Learning Algorithm.     "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.6\n",
    "\n",
    "* (a) For linearly separable data, there exists a plane that can separate them. Suppose the norm of the plane is $w_0$, then for all points $x_n$, we have $y_nw_0^Tx_n \\gt 0$. We can scale $w_0$ up by a scalar $a \\gt 1$ that makes $y_n(aw_0)^Tx_n \\ge 1$. Thus $w = aw_0$ exists.\n",
    "\n",
    "* (b) To find the $w$, we want the sum of $y_n(w^Tx_n)$ to be as large as possible but as we see from problem (a), we can increase $w$ to make the sum as large as possible. To find a solution, we instead minimize the sum while looking for solutions that satisfy the constraints, which will make sure that we find a solution. So we have following linear program:\n",
    "\n",
    "\\begin{align*}\n",
    "\\min_{w} \\sum^N_{n=1} y_n(w^Tx_n)\\\\\n",
    "\\text{subject to} \\;\\; -y_nw^Tx_n \\le -1 \\text{ for } n=1,\\dots,N\n",
    "\\end{align*}\n",
    "\n",
    "The minimal sum will be $N$. \n",
    "\n",
    "Here $z=(w_1,\\dots,w_d)^T$, $c^T = \\left(\\sum^N_{k=1}y_kx_{k1},\\dots, \\sum^N_{k=1}y_kx_{kd},\\right)$, $b=-1$ and \n",
    "\n",
    "$A=\\begin{bmatrix} \n",
    "-y_1x_{11} & \\dots & -y_1x_{1d}\\\\ \n",
    "\\dots &\\dots \\dots\\\\\n",
    "-y_kx_{k1} & \\dots & -y_kx_{kd}\\\\ \n",
    "\\dots &\\dots \\dots\\\\\n",
    "-y_Nx_{N1} & \\dots & -y_Nx_{Nd}\\\\ \n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "* (c) Re-formulate the problem as a linear program.\n",
    "\n",
    "\\begin{align*}\n",
    "\\min_{w,\\xi_n}\\sum^N_{n=1}\\xi_n\\\\\n",
    "\\text{subject to} \\\\\n",
    "-\\xi_n -y_n(w^Tx_n) \\le -1\\\\\n",
    "-\\xi_n \\le 0\n",
    "\\end{align*}\n",
    "\n",
    "Then we have \n",
    "\n",
    "\\begin{align*}\n",
    "z &= \\left[\\xi_1,\\dots,\\xi_n,w_1,\\dots,w_d\\right]^T\\\\\n",
    "c &= \\left[1,\\dots,1,0,\\dots,0\\right]^T\\\\\n",
    "b &= \\left[-1,\\dots,-1,0,\\dots,0\\right]^T_{(n+d)\\times 1}\\\\\n",
    "A &= \\begin{bmatrix}\n",
    "-1 & 0 & \\dots & 0 & -y_1x_{11} & \\dots & -y_1x_{1d}\\\\\n",
    "0 & -1 & \\dots & 0 & -y_2x_{21} & \\dots & -y_2x_{2d}\\\\\n",
    "\\dots &\\dots & \\dots & \\dots &\\dots & \\dots & \\dots\\\\\n",
    "0 & 0 & \\dots & -1 & -y_nx_{n1} & \\dots & -y_nx_{nd}\\\\\n",
    "-1 & 0 & \\dots & 0 & 0 & \\dots & 0\\\\\n",
    "0 & -1 & \\dots & 0 & 0 & \\dots & 0\\\\\n",
    "\\dots &\\dots & \\dots & \\dots &\\dots & \\dots & \\dots\\\\\n",
    "0 & 0 & \\dots & -1 & 0 & \\dots & 0\\\\\n",
    "\\end{bmatrix}_{(n+n,n+d)}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (d) Suppose $w$ and $\\xi$ are the optimal values found by the linear program in problem (c), we claim that $w$ also minimize the equation in problem (3.5), i.e. $\\frac{1}{N}\\sum^N_{n=1}E_n(w) = \\frac{1}{N}\\sum^N_{n=1}max(0, 1-y_nw^Tx_n)$. Otherwise, there exists another $w'$ such that $\\frac{1}{N}\\sum^N_{n=1}E_n(w) \\gt \\frac{1}{N}\\sum^N_{n=1}E_n(w')$.\n",
    "\n",
    "On the other hand, since the optimal $\\xi$ satisfies the two contraints, we have $y_n(w^Tx_n) \\ge 1-\\xi_n$ and $\\xi_n \\ge 0$, which is equivalent to $\\xi_n \\ge max(0, 1-y_n(w^Tx_n))$. So we have $\\sum^N_{n=1}\\xi_n = \\sum^N_{n=1}max(0, 1-y_n(w^Tx_n))$ which is minimial by the set up of problem (c). However, this contradicts the assumption above that there exists another $w'$ that can achieves lower sum. So we conclude that the solutions found by problem (c) also minimizes equation in problem 3.5.\n",
    "\n",
    "Now we assume $w$ minimizes equation in problem 3.5, as shown in above, it minimizes $\\sum^N_{n=1}max(0, 1-y_n(w^Tx_n)) =\\sum^N_{n=1}\\xi_n$ where $\\xi_n$ satisfies the two constraints in problem (c).\n",
    "\n",
    "Combine the two arguments, we conclude that problem (c) and problem 3.5 optimizations are equivalent.\n",
    "\n",
    "#### Problem 3.7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of positive points:  1023\n",
      "Number of negatives points:  977\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     con: array([], dtype=float64)\n",
      "     fun: 8337.750936051727\n",
      " message: 'Optimization terminated successfully.'\n",
      "     nit: 12\n",
      "   slack: array([4.0237619 , 2.58730323, 3.26863103, ..., 3.41969709, 4.25790331,\n",
      "       2.06756635])\n",
      "  status: 0\n",
      " success: True\n",
      "       x: array([9.92419060e-01, 6.04367241e-05, 3.98226621e-01])\n",
      "(2000,) (2000, 3)\n"
     ]
    }
   ],
   "source": [
    "#### Problem 3.7\n",
    "from scipy.optimize import linprog\n",
    "N = 2000\n",
    "max_v = 10000\n",
    "rad, thk = 10, 5\n",
    "sep = 5\n",
    "eta = 1\n",
    "use_adaline=False\n",
    "maxit = 1000\n",
    "dim = 2\n",
    "\n",
    "radiuses, radians = data.generate_random_ring(N, rad, rad+thk, max_v)\n",
    "xs, ys, signs = data.move_bottom_ring_and_assign(radiuses, radians, rad + thk/2.0, -sep)\n",
    "df = pd.DataFrame({'x1':xs.flatten(), 'x2':ys.flatten(), 'y':signs.flatten()})\n",
    "df['x0'] = 1\n",
    "df = df[['x0','x1','x2','y']]\n",
    "positives = df.loc[df['y']==1]\n",
    "negatives = df.loc[df['y']==-1]\n",
    "\n",
    "figsize = plt.figaspect(1)\n",
    "f, ax = plt.subplots(1, 1, figsize=figsize)\n",
    "\n",
    "ps = ax.scatter(positives[['x1']].values, positives[['x2']].values, marker='+', c= 'b', label='+1 labels')\n",
    "ns = ax.scatter(negatives[['x1']].values, negatives[['x2']].values, marker=r'$-$', c= 'r', label='-1 labels')\n",
    "print('Number of positive points: ', len(positives))\n",
    "print('Number of negatives points: ', len(negatives))\n",
    "\n",
    "#### problem 3.7 (b)(d) Separable Points\n",
    "df['yx0'] = df['y']*df['x0']\n",
    "df['yx1'] = df['y']*df['x1']\n",
    "df['yx2'] = df['y']*df['x2']\n",
    "\n",
    "def prepare_seperable_c(df):\n",
    "    c = [np.sum(df['yx0']), np.sum(df['yx1']), np.sum(df['yx2'])]\n",
    "    return c\n",
    "\n",
    "def prepare_seperable_A_b(df):\n",
    "    b = -np.ones(len(df))\n",
    "    A = -df[['yx0', 'yx1', 'yx2']].values\n",
    "    return A, b\n",
    "\n",
    "c = prepare_seperable_c(df)\n",
    "A,b = prepare_seperable_A_b(df) \n",
    "\n",
    "w0_bounds = (None, None)\n",
    "w1_bounds = (None, None)\n",
    "w2_bounds = (None, None)\n",
    "res = linprog(c, A_ub=A, b_ub=b, bounds=[w0_bounds, w1_bounds, w2_bounds], method='interior-point')\n",
    "\n",
    "x1 = np.arange(-(rad+thk), (rad+thk)+rad + thk/2)\n",
    "norm_g = res.x/res.x[-1]\n",
    "hypothesis = ax.plot(x1, -(norm_g[0]+norm_g[1]*x1), c = 'g', label='Final Hypothesis')\n",
    "\n",
    "ax.set_ylabel(r\"$x_2$\", fontsize=11)\n",
    "ax.set_xlabel(r\"$x_1$\", fontsize=11)\n",
    "ax.set_title('Data set size = %s'%N, fontsize=9)\n",
    "ax.axis('tight')\n",
    "legend_x = 2.0\n",
    "legend_y = 0.5\n",
    "ax.legend(['Linear Programming (Separable)', \n",
    "           '+1 labels', '-1 labels', ], \n",
    "          loc='center right', bbox_to_anchor=(legend_x, legend_y))\n",
    "plt.show()\n",
    "print(res)\n",
    "print(b.shape, A.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of positive points:  981\n",
      "Number of negatives points:  1019\n",
      "(4000, 2003) (4000,)\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.optimize import linprog\n",
    "N = 2000\n",
    "max_v = 10000\n",
    "rad, thk = 10, 5\n",
    "sep = -5\n",
    "eta = 1\n",
    "use_adaline=False\n",
    "maxit = 1000\n",
    "dim = 2\n",
    "num_feature = 3\n",
    "\n",
    "radiuses, radians = data.generate_random_ring(N, rad, rad+thk, max_v)\n",
    "xs, ys, signs = data.move_bottom_ring_and_assign(radiuses, radians, rad + thk/2.0, -sep)\n",
    "df = pd.DataFrame({'x1':xs.flatten(), 'x2':ys.flatten(), 'y':signs.flatten()})\n",
    "df['x0'] = 1\n",
    "df = df[['x0','x1','x2','y']]\n",
    "positives = df.loc[df['y']==1]\n",
    "negatives = df.loc[df['y']==-1]\n",
    "\n",
    "\n",
    "#### problem 3.7 (c)(d) Non-Separable Points\n",
    "df['yx0'] = df['y']*df['x0']\n",
    "df['yx1'] = df['y']*df['x1']\n",
    "df['yx2'] = df['y']*df['x2']\n",
    "\n",
    "def prepare_nonseperable_c(df, num_feature):\n",
    "    num_instances = len(df)\n",
    "    c1 = np.ones(num_instances)\n",
    "    c2 = np.zeros(num_feature)\n",
    "    c = np.hstack([c1,c2])\n",
    "    return c\n",
    "\n",
    "def prepare_nonseperable_A_b(df, num_feature):\n",
    "    b1 = -np.ones(len(df))\n",
    "    b2 = np.zeros(len(df))\n",
    "    b = np.hstack([b1,b2])\n",
    "    \n",
    "    A11 = -np.identity(len(df))\n",
    "    A21 = A11\n",
    "    A12 = -df[['yx0', 'yx1', 'yx2']].values\n",
    "    A22 = np.zeros((len(df), num_feature))\n",
    "    Atop = np.hstack([A11, A12])\n",
    "    Abottom = np.hstack([A21,A22])\n",
    "    A = np.vstack([Atop, Abottom])\n",
    "    return A, b\n",
    "\n",
    "c = prepare_nonseperable_c(df, num_feature)\n",
    "A,b = prepare_nonseperable_A_b(df, num_feature) \n",
    "print(A.shape, b.shape)\n",
    "\n",
    "bounds = []\n",
    "xb = (0, None)\n",
    "for k in np.arange(len(df)):\n",
    "    bounds.append(xb)\n",
    "\n",
    "wb = (None, None)    \n",
    "for k in np.arange(num_feature):\n",
    "    bounds.append(wb)    \n",
    "    \n",
    "res = linprog(c, A_ub=A, b_ub=b, bounds=bounds, method='interior-point')\n",
    "res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "     con: array([], dtype=float64)\n",
       "     fun: 419.0163285412144\n",
       " message: 'The iteration limit was reached before the algorithm converged.'\n",
       "     nit: 1000\n",
       "   slack: array([ 1.23543948e+04, -6.71717584e+03,  2.93459476e+04, ...,\n",
       "        3.68409824e-03,  3.57578030e-20,  1.53483653e-02])\n",
       "  status: 1\n",
       " success: False\n",
       "       x: array([ 3.57578052e-20,  2.12569169e-03,  3.57578058e-20, ...,\n",
       "       -4.32866842e+03, -3.60749905e+02,  3.02254645e+03])"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of positive points:  981\n",
      "Number of negatives points:  1019\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figsize = plt.figaspect(1)\n",
    "f, ax = plt.subplots(1, 1, figsize=figsize)\n",
    "\n",
    "ps = ax.scatter(positives[['x1']].values, positives[['x2']].values, marker='+', c= 'b', label='+1 labels')\n",
    "ns = ax.scatter(negatives[['x1']].values, negatives[['x2']].values, marker=r'$-$', c= 'r', label='-1 labels')\n",
    "print('Number of positive points: ', len(positives))\n",
    "print('Number of negatives points: ', len(negatives))\n",
    "\n",
    "x1 = np.arange(-(rad+thk), (rad+thk)+rad + thk/2)\n",
    "\n",
    "ws = res.x[-3:]\n",
    "norm_g = ws/ws[-1]\n",
    "hypothesis = ax.plot(x1, -(norm_g[0]+norm_g[1]*x1), c = 'g', label='Final Hypothesis')\n",
    "\n",
    "ax.set_ylabel(r\"$x_2$\", fontsize=11)\n",
    "ax.set_xlabel(r\"$x_1$\", fontsize=11)\n",
    "ax.set_title('Data set size = %s'%N, fontsize=9)\n",
    "ax.axis('tight')\n",
    "legend_x = 2.0\n",
    "legend_y = 0.5\n",
    "ax.legend(['Linear Programming (Non-Separable)', \n",
    "           '+1 labels', '-1 labels', ], \n",
    "          loc='center right', bbox_to_anchor=(legend_x, legend_y))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.7 (d)\n",
    "Compare with the linear regression approach with and without the 3rd order polynomial feature transform. \n",
    "\n",
    "* Without polynomial transform:\n",
    "  * For separable data, both can find a solution that separate the data. Both computations are similar in their performance.\n",
    "* Linear regression with polynomial: \n",
    "  * When deal with non-separable data, linear programming is very slow (several hours), it has to add extra $N$ $\\xi$ into the unknown variables, even though we only need $3$ $w$ components. Compared to 3rd order polynomial transform, it only increases the dimension from 3 to 10. \n",
    "  * Also, linear programming was not able to generate line that can separate the points. Although it tries to minimize the errors."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.8\n",
    "\n",
    "For $E_{out}(h)=E_{x,y}\\left[(h(x)-y)^2\\right]$, take derivative w.r.t. $h$ and let it to be zero, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{\\partial{E_{out}(h)}}{\\partial{h}}&=E_{x,y}\\left[2(h(x)-y)\\right]\\\\\n",
    "&=2E_{x,y}[h(x)-y]\\\\\n",
    "&=2E_xE_{y|x}\\left[h(x)-y\\right]\\\\\n",
    "\\end{align*}\n",
    "\n",
    "To make this derivative equal to zero, it's suffice to have $E_{y|x}\\left[h(x)-y\\right]$ equal to 0. That is $E_{y|x}\\left[h(x)-y\\right] = E_{y|x}[h(x)]-E_{y|x}[y] = h(x) - E[y|x] = 0$.\n",
    "\n",
    "Thus we obtain an optimal $h$, i.e. $h^{*}(x) = E[y|x]$.\n",
    "\n",
    "If we treat $h^{*}$ as a deterministic target function, we can write $y=h^{*}(x) + \\epsilon(x)$ where $\\epsilon(x)$ is a noise variable. Take expectation on both sides of the equation, we have\n",
    "\n",
    "\\begin{align*}\n",
    "E[y] &= E[h^{*}(x)] + E[\\epsilon(x)]\\\\\n",
    "&= E[E[y|x]] + E[\\epsilon(x)]\\\\\n",
    "&= E[y] + E[\\epsilon(x)]\\\\\n",
    "\\end{align*}\n",
    "\n",
    "From which we derive that $E[\\epsilon(x)] = 0$.\n",
    "\n",
    "#### Problem 3.9\n",
    "\n",
    "\\begin{align*}\n",
    "E_{in}(w) &= \\left(w-(X^TX)^{-1}X^Ty\\right)^T\\left(X^TX\\right)\\left(w-(X^TX)^{-1}X^Ty\\right) + y^T\\left(I-X(X^TX)^{-1}X^T\\right)y\\\\\n",
    "&= \\left(w^T-y^TX(X^TX)^{-T}\\right)\\left(X^TX\\right)\\left(w-(X^TX)^{-1}X^Ty\\right) + y^T\\left(I-X(X^TX)^{-1}X^T\\right)y\\\\\n",
    "&= \\left(w^TX^TX-y^TX(X^TX)^{-T}(X^TX)\\right)\\left(w-(X^TX)^{-1}X^Ty\\right) + y^T\\left(I-X(X^TX)^{-1}X^T\\right)y\\\\\n",
    "&= \\left(w^TX^TX-y^TX\\right)\\left(w-(X^TX)^{-1}X^Ty\\right) + y^T\\left(I-X(X^TX)^{-1}X^T\\right)y\\\\\n",
    "&= w^TX^TXw-y^TXw-w^T(X^TX)(X^TX)^{-1}X^Ty + y^TX(X^TX)^{-1}X^Ty + y^T\\left(I-X(X^TX)^{-1}X^T\\right)y\\\\\n",
    "&= w^TX^TXw-y^TXw-w^TX^Ty + y^TX(X^TX)^{-1}X^Ty + y^Ty -y^TX(X^TX)^{-1}X^Ty\\\\\n",
    "&= w^TX^TXw-2w^TX^Ty + y^Ty\\\\\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "The second to last step is because $y^TXw$ is a scalar, so $y^TXw=w^TX^Ty$. \n",
    "The last step shows that this formula agress with Equation (3.4).\n",
    "\n",
    "From this equation, if we take derivative w.r.t. $w$, it's easy to see that the second term has no $w$, so it gives $0$, we only need take care of first term, let $A(w) = w - (X^TX)^{-1}X^Ty$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{\\partial{E_{in}(w)}}{\\partial{w}} &= \\frac{\\partial{\\left(A^T(w)(X^TX)A(w)\\right)}}{\\partial{w}}\\\\\n",
    "&= \\left((X^TX)+(X^TX)^T\\right)A(w)\\\\\n",
    "&= 2(X^TX)\\left(w - (X^TX)^{-1}X^Ty\\right)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Since $X^TX$ is positive definite, if we let the above derivative equal to zero, We obtain $w_{lin} = (X^TX)^{-1}X^Ty$. \n",
    "\n",
    "The in-sample error is thus:\n",
    "\n",
    "$E_{in}(w_{lin}) =  y^T\\left(I-X(X^TX)^{-1}X^T\\right)y$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.10\n",
    "\n",
    "1. (a) Suppose $\\lambda$ is an eigenvalue of $H$ and $v$ is the corresponding eigenvector. Then we have $Hv = \\lambda v$, and $HHv=H\\lambda v = \\lambda Hv = \\lambda^2 v$. Since we know $H^k = H$, on the left side, we have $HHv = Hv = \\lambda v$, so we get $\\lambda v = \\lambda^2 v$. It's easy to see that $\\lambda$ can only either $0$ or $1$.\n",
    "\n",
    "1. (b) According to Spectral theorem, for a symmetric matrix $A$, we can write it as $A=U\\Lambda U^T$, where $U$ is an orthonomal matrix and $\\Lambda$ is a diagonial matrix with entries of eigenvalues from matrix $A$. Apply trace on both sides, we have\n",
    "\n",
    "    $trace(A) = trace(U\\Lambda U^T) = trace(U^TU\\Lambda) = trace(\\Lambda)$, where the last part is just sume of eigenvalues of $A$.\n",
    "    \n",
    "1. (c) From exercise 3.3(d), we have $trace(H)=d+1$, also from previous problem (b), we have $trace(H) = \\sum^N_{k=1}\\lambda_k$. So we have $\\sum^N_{k=1}\\lambda_k = d+1$. \n",
    "\n",
    "Now from problem (a) we know that $\\lambda_k$'s are either $0$ or $1$, so there are exactly $d+1$ eigenvalues of $H$ are $1$.\n",
    "\n",
    "The rank of a matrix can be defined as the number of non-zero eigenvalues of the matrix. So $H$ has a rank of $d+1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.11\n",
    "\n",
    "1. (a) Following exercise 3.4 and use the fact that $w_{lin} = (X^TX)^{-1}X^Ty$, for a given test point $x_t$, we have \n",
    "\n",
    "\\begin{align*}\n",
    "g(x_t) &= x_t^Tw_{lin} \\\\\n",
    "&= x_t^T(X^TX)^{-1}X^Ty \\\\\n",
    "&= x_t^T(X^TX)^{-1}X^T(Xw^*+\\epsilon) \\\\\n",
    "&= x_t^T(X^TX)^{-1}X^TXw^*+ x_t^T(X^TX)^{-1}X^T\\epsilon \\\\\n",
    "&= x_t^Tw^*+ x_t^T(X^TX)^{-1}X^T\\epsilon\\\\\n",
    "\\end{align*}\n",
    "\n",
    "On the other side, the $y$ at test point $x_t$ is: $y=x_t^Tw^* + \\epsilon_{t}$, so we have\n",
    "\n",
    "\\begin{align*}\n",
    "y - g(x_t) &= \\epsilon_{t} -  x_t^T(X^TX)^{-1}X^T\\epsilon\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where $\\epsilon_t$ is the noise realization for the test point and $\\epsilon$ is the vector of noise realizations on the data.\n",
    "\n",
    "1. (b) Take the expectation w.r.t. to the test point, i.e. $x_t$ and $\\epsilon_t$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{out} &= E\\left[(y-g(x_t))^2\\right]\\\\\n",
    "&= E\\left[\\left(\\epsilon_{t} -  x_t^T(X^TX)^{-1}X^T\\epsilon\\right)^2\\right]\\\\\n",
    "&= E\\left[\\epsilon_{t}^2 - 2\\epsilon_{t}x_t^T(X^TX)^{-1}X^T\\epsilon + (x_t^T(X^TX)^{-1}X^T\\epsilon)(x_t^T(X^TX)^{-1}X^T\\epsilon)^T\\right]\\text{Note the last term is a scalar}\\\\\n",
    "&= E[\\epsilon_t^2] - 2E\\left[\\epsilon_{t}x_t^T(X^TX)^{-1}X^T\\epsilon\\right] + E\\left[x_t^T(X^TX)^{-1}X^T\\epsilon\\epsilon^TX(X^TX)^{-T}x_t\\right]\\\\\n",
    "&= \\sigma^2-2E[\\epsilon_{t}]E\\left[x_t^T(X^TX)^{-1}X^T\\epsilon\\right] + E\\left[trace\\left(x_t^T(X^TX)^{-1}X^T\\epsilon\\epsilon^TX(X^TX)^{-T}x_t\\right)\\right]\\\\\n",
    "&\\text{In the last term we use the fact that trace on a scalar equals to the scalar}\\\\\n",
    "&\\text{We also apply the independence between }\\;\\epsilon_t\\;\\text{and}\\;x_t. \\\\\n",
    "&\\text{Also note that }X\\text{ and } \\epsilon \\text{ are non-random in this expectation}\\\\\n",
    "&= \\sigma^2 + E\\left[trace\\left(x_tx_t^T(X^TX)^{-1}X^T\\epsilon\\epsilon^TX(X^TX)^{-T}\\right)\\right]\\\\\n",
    "&= \\sigma^2 + trace\\left(E\\left[x_tx_t^T(X^TX)^{-1}X^T\\epsilon\\epsilon^TX(X^TX)^{-T}\\right]\\right)\\\\\n",
    "&= \\sigma^2 + trace\\left(E[x_tx_t^T]E\\left[(X^TX)^{-1}X^T\\epsilon\\epsilon^TX(X^TX)^{-1}\\right]\\right)\\\\\n",
    "&= \\sigma^2 + trace\\left(\\Sigma (X^TX)^{-1}X^T\\epsilon\\epsilon^TX(X^TX)^{-1}\\right)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "1. (c) $\\epsilon \\epsilon^T$ is a $N \\times N$ matrix, with entries $\\epsilon_i\\epsilon_j$. So $E_{\\epsilon}[\\epsilon \\epsilon^T] = \\sigma^2I$ where the expectation of $E[\\epsilon_i\\epsilon_j] = 0$ when $i\\ne j$, otherwise $\\sigma^2$.\n",
    "\n",
    "1. (d) Take the expectation w.r.t. $\\epsilon$, which is a $N\\times 1$ vector. We have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{out} &= \\sigma^2 + E_{\\epsilon}\\left[trace\\left(\\Sigma (X^TX)^{-1}X^T\\epsilon\\epsilon^TX(X^TX)^{-1}\\right)\\right]\\\\\n",
    "&= \\sigma^2 + trace\\left(E_{\\epsilon}\\left[\\Sigma (X^TX)^{-1}X^T\\epsilon\\epsilon^TX(X^TX)^{-1}\\right]\\right)\\\\\n",
    "&= \\sigma^2 + trace\\left(E_{\\epsilon}\\left[\\Sigma (X^TX)^{-1}X^T\\right]E_{\\epsilon}\\left[\\epsilon\\epsilon^T\\right]E_{\\epsilon}\\left[X(X^TX)^{-1}\\right]\\right)\\\\\n",
    "&= \\sigma^2 + trace\\left(\\Sigma (X^TX)^{-1}X^T\\sigma^2 IX(X^TX)^{-1}\\right)\\\\\n",
    "&= \\sigma^2 + \\sigma^2 trace\\left(\\Sigma (X^TX)^{-1}X^TX(X^TX)^{-1}\\right)\\\\\n",
    "&= \\sigma^2 + \\sigma^2 trace\\left(\\Sigma (X^TX)^{-1}\\right)\\\\\n",
    "&= \\sigma^2 + \\frac{\\sigma^2}{N} trace\\left(\\Sigma (\\frac{1}{N}X^TX)^{-1}\\right)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Note that $\\frac{1}{N}X^TX=\\frac{1}{N}\\sum^N_{n=1}x_nx^T_n$ is an $N$-sample estimate of $\\Sigma$. So $\\frac{1}{N}X^TX \\approx \\Sigma$, in such case, we have\n",
    "\n",
    "$E_{out} = \\sigma^2 + \\frac{\\sigma^2}{N} trace(I) = \\sigma^2 + \\frac{\\sigma^2 (d+1)}{N}= \\sigma^2(1+\\frac{d+1}{N})$\n",
    "\n",
    "\n",
    "1. (e) By law of large numbers $\\frac{1}{N}X^TX$ converges in probability to $\\Sigma$, so by continuity of the inverse at $\\Sigma$, $\\left(\\frac{1}{N}X^TX\\right)^{-1}$ converges in probability to $\\Sigma^{-1}$. $trace\\left(\\Sigma (\\frac{1}{N}X^TX)^{-1}\\right) = trace(I) + o(1)$, so we have $E_{out} = \\sigma^2 + \\frac{\\sigma^2}{N}(d+1+o(1)) = \\sigma^2\\left(1+\\frac{d+1}{N} + o(\\frac{1}{N})\\right)$\n",
    "\n",
    "#### Problem 3.12\n",
    "\n",
    "$H^2 = X(X^TX)^{-1}X^TX(X^TX)^{-1}X^T = H$ so $H$ is a projection matrix. \n",
    "\n",
    "The $\\hat{y}$ is the projection of $y$ onto the space spanned by $X$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.13\n",
    "\n",
    "* (a) Suppose $x$ has a dimension of $d$, when we convert to classification problem, the new $x_{class}$ has a dimension of $d+1$, with $y$ in the extra dimension. So the classification problem has $d+2$ weights, where we count the weight for constant. \n",
    "  On the other hand, the linear regression needs only $d+1$ weights. \n",
    "  \n",
    "* (b) The linear fit wegiths $w$, where $h(x) = w^Tx$. The weights of solving the classification is $w_{class} = (w^c_0, w^c_1, \\dots, w^c_d, w^c_{d+1})$, where $w^c_{d+1}$ is the weight corresponding to the original $y$s. Since it solves the classification problem, it defines the line that separting the points, i.e. $w_{class}^T\\hat{x} = 0$, where $\\hat{x} = \\begin{bmatrix}x\\\\y\\end{bmatrix}$, so we have\n",
    "\n",
    "\\begin{align*}\n",
    "w_{class}^T\\hat{x} &= w_{class}^T\\begin{bmatrix}x\\\\y\\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix}w^c & w^c_{d+1}\\end{bmatrix} \\begin{bmatrix}x\\\\y\\end{bmatrix}\\\\\n",
    "&= (w^c)^Tx + w^c_{d+1}y \\\\\n",
    "&= 0\n",
    "\\end{align*}\n",
    "\n",
    "  Move $y$ to the left hand side of the equation, and divide both sides by $w^c_{d+1}$, we have $y = -(w^c)^T x / w^c_{d+1}$, so the linear regression weights $w = -\\frac{(w^c)^T}{w^c_{d+1}}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Liner regression coefficients:  [[-0.18553388]\n",
      " [ 1.01062413]]\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### Problem 3.13 (c) (d)\n",
    "import libs.linear_models as lm\n",
    "import libs.data_util as data\n",
    "\n",
    "N = 50\n",
    "sigma = 0.1\n",
    "dim = 1\n",
    "mu = 0\n",
    "gaussian_sigma = 1\n",
    "delta = np.array([0, 0.1])\n",
    "\n",
    "def generate_y(x, sigma, epsilons):\n",
    "    y = x**2 + sigma * epsilons\n",
    "    return y\n",
    "\n",
    "def move_data(xys, delta):\n",
    "    return xys + delta, xys - delta\n",
    "    \n",
    "xs = data.generate_random_numbers01(N, dim, 1000) #uniform on [0,1]\n",
    "epsilons = np.random.normal(mu, gaussian_sigma, N)\n",
    "ys = generate_y(xs, sigma, epsilons.reshape(-1, 1))\n",
    "xys = np.hstack([xs, ys])\n",
    "dps, dms = move_data(xys, delta)\n",
    "df1 = pd.DataFrame({'x0':np.ones(N)})\n",
    "df1['x1'] = dps[:,0]\n",
    "df1['x2'] = dps[:,1]\n",
    "df1['y'] = 1\n",
    "\n",
    "df2 = pd.DataFrame({'x0':np.ones(N)})\n",
    "df2['x1'] = dms[:,0]\n",
    "df2['x2'] = dms[:,1]\n",
    "df2['y'] = -1\n",
    "df = pd.concat([df1,df2])\n",
    "\n",
    "figsize = plt.figaspect(1)\n",
    "f, ax = plt.subplots(1, 1, figsize=figsize)\n",
    "\n",
    "original_points = ax.scatter(xs, ys, marker='o', c= 'y', label='Original Points')\n",
    "ps = ax.scatter(dps[:,0], dps[:, 1], marker='+', c= 'b', label='+1 labels')\n",
    "ns = ax.scatter(dms[:,0], dms[:, 1], marker=r'$-$', c= 'r', label='-1 labels')\n",
    "\n",
    "x1 = np.arange(0,1,0.01)\n",
    "x1s = np.zeros((xs.shape[0], xs.shape[1]+1))\n",
    "x1s[:, 0] = 1\n",
    "x1s[:, 1] = xs.flatten()\n",
    "\n",
    "w_lin = lm.linear_regression(x1s, ys)\n",
    "print('Liner regression coefficients: ', w_lin)\n",
    "linear = ax.plot(x1, (w_lin[0]+w_lin[1]*x1), c = 'r', label='Linear Regression')\n",
    "\n",
    "\n",
    "maxit = 1000\n",
    "eta = 0.5\n",
    "randomized = False\n",
    "print_out = False\n",
    "w_pocket, w_ts, what_ts, test_w_ts, test_what_ts, _ = lm.pocket_algo(df.values, 2, \n",
    "                                                           maxit,eta, randomized, \n",
    "                                                           print_out, None)\n",
    "\n",
    "\n",
    "linear_pocket = ax.plot(x1, -(w_pocket[0]+w_pocket[1]*x1), c = 'g', label='Pocket Classification')\n",
    "\n",
    "\n",
    "\n",
    "ax.set_ylabel(r\"$x_2$\", fontsize=11)\n",
    "ax.set_xlabel(r\"$x_1$\", fontsize=11)\n",
    "ax.set_title('Data set size = %s'%N, fontsize=9)\n",
    "ax.axis('tight')\n",
    "legend_x = 2.0\n",
    "legend_y = 0.5\n",
    "ax.legend(['Linear Regression', 'Pocket Classification', 'Original Points',\n",
    "           '+1 labels', '-1 labels', ], \n",
    "          loc='center right', bbox_to_anchor=(legend_x, legend_y))\n",
    "#ax.set_ylim(bottom=lb, top=ub)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.13 (d)\n",
    "From above graph, we can see that the classification line sometimes can be close to the linear regression line in this case. However, as the $a$ increases, clearly the classification will a worse job. Even when $a$ is not so big, the classification's job is to separate the points, but this is not necessarily along the line that fit the original data. It can be a good start point for linear regression though."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.14\n",
    "\n",
    "* (a) The average function is:\n",
    "\n",
    "\\begin{align*}\n",
    "\\bar{g}(x) &= E_{\\mathcal{D}}\\left[g^{\\mathcal{D}}(x)\\right]\\\\\n",
    "&= E_{\\mathcal{D}}\\left[x^Tw_{lin}\\right]\\\\\n",
    "&= E_{\\mathcal{D}}\\left[x^T(X^TX)^{-1}X^Ty\\right]\\\\\n",
    "&= E_{\\mathcal{D}}\\left[x^T(X^TX)^{-1}X^T(Xw+\\epsilon)\\right]\\\\\n",
    "&= E_{\\mathcal{D}}\\left[x^T(X^TX)^{-1}X^TXw+x^T(X^TX)^{-1}X^T\\epsilon\\right]\\\\\n",
    "&= E_{\\mathcal{D}}\\left[x^Tw+x^T(X^TX)^{-1}X^T\\epsilon\\right]\\\\\n",
    "&= x^Tw+E_{\\mathcal{D}}\\left[x^T(X^TX)^{-1}X^T\\epsilon\\right]\\\\\n",
    "&= x^Tw+x^TE_{\\mathcal{D}}\\left[(X^TX)^{-1}X^T\\right]E_{\\mathcal{D}}[\\epsilon]\\\\\n",
    "&= x^Tw\\\\\n",
    "&= f(x)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Note the $\\epsilon$ in the data set is independent of $X$ by assumption.\n",
    "\n",
    "* (b) The variance is:\n",
    "\n",
    "\\begin{align*}\n",
    "\\text{variance} &= E_{x}E_{\\mathcal{D}}\\left[\\left(g^{\\mathcal{D}}(x)-\\bar{g}(x)\\right)^2\\right]\\\\\n",
    "&= E_{x}E_{\\mathcal{D}}\\left[\\left(g^{\\mathcal{D}}(x)-f(x)\\right)^2\\right]\\\\\n",
    "&= E_{x}E_{\\mathcal{D}}\\left[\\left(g^{\\mathcal{D}}(x)-(y_x-\\epsilon_x)\\right)^2\\right]\\\\\n",
    "&= E_{x}E_{\\mathcal{D}}\\left[\\left(g^{\\mathcal{D}}(x)-y_x+\\epsilon_x\\right)^2\\right]\\\\\n",
    "&= E_{x}E_{\\mathcal{D}}\\left[\\left(y_x-g^{\\mathcal{D}}(x)-\\epsilon_x\\right)^2\\right]\\\\\n",
    "&= E_{\\mathcal{D}}E_{x}\\left[\\left(y_x-g^{\\mathcal{D}}(x)\\right)^2-2\\epsilon_x\\left(y_x-g^{\\mathcal{D}}(x)\\right) + \\epsilon_x^2\\right]\\\\\n",
    "&= E_{\\mathcal{D}}E_{x}\\left[\\left(y_x-g^{\\mathcal{D}}(x)\\right)^2\\right]-2E_{\\mathcal{D}}E_{x}\\left[\\epsilon_x\\left(y_x-g^{\\mathcal{D}}(x)\\right)\\right] + E_{\\mathcal{D}}E_{x}\\left[\\epsilon_x^2\\right]\\\\\n",
    "&= E_{\\mathcal{D}}E_{x}\\left[\\left(y_x-g^{\\mathcal{D}}(x)\\right)^2\\right]-2E_{\\mathcal{D}}E_{x}\\left[\\epsilon_x\\left(f(x)+\\epsilon_x-g^{\\mathcal{D}}(x)\\right)\\right]+\\sigma^2\\\\\n",
    "&= E_{\\mathcal{D}}E_{x}\\left[\\left(y_x-g^{\\mathcal{D}}(x)\\right)^2\\right]-2E_{\\mathcal{D}}E_{x}[\\epsilon_xf(x)]-2E_{\\mathcal{D}}E_{x}[\\epsilon_x^2]+2E_{\\mathcal{D}}E_{x}[\\epsilon_x g^{\\mathcal{D}}(x)]+\\sigma^2\\\\\n",
    "&= E_{\\mathcal{D}}E_{x}\\left[\\left(y_x-g^{\\mathcal{D}}(x)\\right)^2\\right] - 2\\sigma^2 + \\sigma^2\\\\\n",
    "&= E_{\\mathcal{D}}E_{x}\\left[\\left(y_x-g^{\\mathcal{D}}(x)\\right)^2\\right] - \\sigma^2\\\\\n",
    "&= \\sigma^2\\left(1+\\frac{d+1}{N} + o(\\frac{1}{N})\\right)  - \\sigma^2\\\\\n",
    "&= \\sigma^2\\left(\\frac{d+1}{N} + o(\\frac{1}{N})\\right)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where we have used the result of problem 3.11 for the first term.\n",
    "\n",
    "Notice that the bias is zero due to result (a). So we recovered the bias-variance decomposition for noisy data, where $E_{\\mathcal{D}}\\left[E_{out}(g^{\\mathcal{D}}\\right] = \\sigma^2 + bias + variance$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.15\n",
    "\n",
    "* (a) We know that $rank(X) = \\rho$, by the property of rank, we have $rank(X) = rank(X^TX)$. \n",
    "Note the dimension of $X^TX$ is $(d+1)\\times (d+1)$, and if $X^TX$ is not invertible, then $rank(X^TX) \\lt d+1$, we thus have $\\rho \\lt d+1$.\n",
    "\n",
    "* (b) \n",
    "\n",
    "\\begin{align*}\n",
    "X^TXw_{lin} &= X^TXV\\Gamma^{-1}U^Ty \\\\\n",
    "&= (U\\Gamma V^T)^TU\\Gamma V^TV\\Gamma^{-1}U^Ty \\\\\n",
    "&= V\\Gamma^T U^T U\\Gamma V^TV\\Gamma^{-1}U^Ty \\\\\n",
    "&= V\\Gamma^2\\Gamma^{-1}U^Ty\\\\\n",
    "&= V\\Gamma U^Ty\\\\\n",
    "&= X^Ty\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (c) TODO\n",
    "\n",
    "#### Problem 3.16\n",
    "\n",
    "* (a)\n",
    "  * The expected cost if you accept the person is: $g(x)\\times 0 + (1-g(x))c_a= (1-g(x))c_a$\n",
    "  * The expected cost if you reject the person is: $g(x)c_r + (1-g(x))\\times 0= g(x)c_r$\n",
    "\n",
    "  * Question: Why use the predicted probabilities to calculate the expected cost instead of the true probabilities? \n",
    "  \n",
    "* (b) To make it fair, let cost(accept) = cost(reject), we have $(1-g(x))c_a = g(x)c_r$, we have $g(x) = \\frac{c_a}{c_a+c_r}$, which is the $\\kappa$.\n",
    "\n",
    "* (c) \n",
    "  * Supermarket: $\\kappa = \\frac{1}{1+10}=\\frac{1}{11}$. The $\\kappa$ is small. The supermarket is easy to accept a person's fingerprint. \n",
    "  * CIA: $\\kappa = \\frac{1000}{1000+1} = \\frac{1000}{10001}$. The $\\kappa$ is large, close to $1$, so most of time, the logistic regression algorithm won't accept a person's fingerprint unless it's very confident. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.17\n",
    "\n",
    "* (a) Let's use the Taylor expansion of $e^x$ here, where $e^x = 1 + x + x^2/2! + x^3/3! + \\dots$. So we have\n",
    "\n",
    "\\begin{align*}\n",
    "E(u+\\Delta u,v + \\Delta v) &= e^{u+\\Delta u} + e^{2(v+\\Delta v)}+ e^{(u+\\Delta u)(v+\\Delta v)} \\\\\n",
    "&+ (u+\\Delta u)^2 -3(u+\\Delta u)(v+\\Delta v) \\\\\n",
    "&+ 4(v+\\Delta v)^2 -3(u+\\Delta u) - 5(v+\\Delta v) \\\\\n",
    "&= \\left(1+(u+\\Delta u) + \\frac{(u+\\Delta u)^2}{2}\\right) \\\\\n",
    "&+ \\left(1+2(v+\\Delta v)+\\frac{4(v+\\Delta v)^2}{2}\\right) \\\\\n",
    "&+ \\left(1+(u+\\Delta u)(v+\\Delta v) + \\frac{(u+\\Delta u)^2(v+\\Delta v)^2}{2}\\right) \\\\\n",
    "&+ (u+\\Delta u)^2 -3(u+\\Delta u)(v+\\Delta v) \\\\\n",
    "&+ 4(v+\\Delta v)^2 -3(u+\\Delta u) - 5(v+\\Delta v)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "At $(u,v)=(0,0)$, take the first-order Taylor's expansion of $E$, we have \n",
    "\n",
    "\\begin{align*}\n",
    "E(u,v) &\\approx \\hat{E}_1 \\\\\n",
    "&= 1 + \\Delta u + 1 + 2\\Delta v + 1  -3\\Delta u - 5\\Delta v\\\\\n",
    "&= -2\\Delta u -3\\Delta v + 3\n",
    "\\end{align*}\n",
    "\n",
    "so $a_u = -2$, $a_v = -3$, and $a=3$.\n",
    "\n",
    "* (b) To satisfy $\\|\\left(\\Delta u, \\Delta v\\right)\\| = 0.5$, let $\\Delta u = 0.5\\cos\\theta$ and $\\Delta v = 0.5\\sin\\theta$, then $E(u,v) = -\\cos\\theta - 1.5\\sin\\theta + 3$. Take derivative w.r.t. $\\theta$ and set it to $0$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\sin\\theta - 1.5\\cos\\theta &= 0\\\\\n",
    "\\sin\\theta = 1.5\\cos\\theta\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Take $\\sin\\theta^2 + \\cos\\theta^2 = 1$, we have $\\cos\\theta = \\frac{2}{\\sqrt{13}}$ and $\\sin\\theta = \\frac{3}{\\sqrt{13}}$. \n",
    "\n",
    "The optimal $(\\Delta u, \\Delta v)$ is thus $(\\frac{1}{\\sqrt{13}}, \\frac{3}{2\\sqrt{13}})$. The resulting $E(u+\\Delta u, v+\\Delta v)$ is computed below using the optimal $(\\Delta u, \\Delta v)$ here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we compute the gradient $\\nabla E(u,v)$.\n",
    "\n",
    "\\begin{align*}\n",
    "\\nabla E(u,v) &= \\left[\\frac{\\partial{E}}{\\partial{u}}, \\frac{\\partial{E}}{\\partial{v}}\\right]^T\\\\\n",
    "&= \\left[e^u+ve^{uv}+2u-3v-3, 2e^{2v}+ue^{uv}-3u+8v-5\\right]^T\n",
    "\\end{align*}\n",
    "\n",
    "At point $(u,v) = (0,0)$, we have $\\nabla E(0,0) = (-2, -3)$.\n",
    "\n",
    "The optimal $(\\Delta u, \\Delta v)$ is parallel to the negative gradient direction, i.e. $-\\nabla E(0,0) = (2,3)$, which is consistent with the results computed above. \n",
    "\n",
    "\n",
    "* (c) Approximate $E(u+\\Delta u, v + \\Delta v)$ by $\\hat{E}_2(\\Delta u, \\Delta v)$ around $(u,v) = (0,0)$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\hat{E}_2(\\Delta u, \\Delta v) &= (1+\\Delta u + \\frac{1}{2}\\Delta u^2) \\\\\n",
    "&+ (1+2\\Delta v + 2\\Delta v^2) + (1+\\Delta u\\Delta v) \\\\\n",
    "&+ \\Delta u^2 -3\\Delta u\\Delta v + 4\\Delta v^2 -3\\Delta u -5\\Delta v\\\\\n",
    "&= \\frac{3}{2}\\Delta u^2 + 6\\Delta v^2 -2\\Delta u\\Delta v -2\\Delta u -3\\Delta v + 3\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So $b_{uu} = \\frac{3}{2}$, $b_{vv} = 6$, $b_{uv}=-2$, $b_u = -2$, $b_v = -3$, $b=3$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* (d) Take derivatives of $\\hat{E}_2(\\Delta u, \\Delta v)$ w.r.t. $\\Delta u$ and $\\Delta v$ and let them equal to $0$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{\\partial{\\hat{E}_2}}{\\partial{\\Delta u}} &= 3\\Delta u -2\\Delta v -2 = 0 \\\\\n",
    "\\frac{\\partial{\\hat{E}_2}}{\\partial{\\Delta v}} &= 12\\Delta v -2\\Delta u -3 = 0\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Write above equations in matrix form, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\begin{bmatrix} \n",
    "\\frac{\\partial{\\hat{E}_2}}{\\partial{\\Delta u}}\\\\ \n",
    "\\frac{\\partial{\\hat{E}_2}}{\\partial{\\Delta v}}\\\\\n",
    "\\end{bmatrix}\n",
    "&= \\begin{bmatrix} \n",
    "3 & -2\\\\ \n",
    "-2 & 12\\\\\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix} \n",
    "\\Delta u\\\\ \n",
    "\\Delta v\\\\\n",
    "\\end{bmatrix}\n",
    "=\\begin{bmatrix} \n",
    "2\\\\ \n",
    "3\\\\\n",
    "\\end{bmatrix}\\\\\n",
    "&=-\\nabla E(0,0)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Solve the two functions we have \n",
    "\n",
    "\\begin{align*}\n",
    "(\\Delta u^*, \\Delta v^*) &= (\\frac{30}{32}, \\frac{13}{32})\\\\\n",
    "\\end{align*}\n",
    "\n",
    "We compute $\\nabla^2 E(u,v)$. Take the $\\nabla E(u,v)$ computed above in problem (b), we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\nabla^2 E(u,v)&= \\begin{bmatrix} \n",
    "\\frac{\\partial{\\nabla E(u,v)}}{\\partial{u}}\\\\ \n",
    "\\frac{\\partial{\\nabla E(u,v)}}{\\partial{v}}\\\\ \n",
    "\\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix} \n",
    "\\frac{\\partial{\\nabla E_u(u,v)}}{\\partial{u}} & \\frac{\\partial{\\nabla E_u(u,v)}}{\\partial{v}}\\\\ \n",
    "\\frac{\\partial{\\nabla E_v(u,v)}}{\\partial{u}} & \\frac{\\partial{\\nabla E_v(u,v)}}{\\partial{v}}\\\\\n",
    "\\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix} \n",
    "e^u + v^2e^{uv} +2 & e^{uv} + uve^{uv}-3\\\\ \n",
    "e^{uv} + uve^{uv} -3 & 4e^{2v}+u^2e^{uv}+8\\\\\n",
    "\\end{bmatrix}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "At point $(u,v)=(0,0)$, we have $\\nabla^2 E(0,0) = \\begin{bmatrix} \n",
    "3 & -2\\\\ \n",
    "-2 & 12\\\\\n",
    "\\end{bmatrix}$\n",
    "\n",
    "We can see that this $\\nabla^2 E(0,0)$ is the same matrix as above when we take derivative of $\\hat{E}_2(\\Delta u, \\Delta v)$ w.r.t. $\\Delta u$ and $\\Delta v$. That equation then becomes\n",
    "\n",
    "\\begin{align*}\n",
    "\\begin{bmatrix} \n",
    "\\frac{\\partial{\\hat{E}_2}}{\\partial{\\Delta u}}\\\\ \n",
    "\\frac{\\partial{\\hat{E}_2}}{\\partial{\\Delta v}}\\\\\n",
    "\\end{bmatrix}\n",
    "&= \\nabla^2 E(0,0)\n",
    "\\begin{bmatrix} \n",
    "\\Delta u\\\\ \n",
    "\\Delta v\\\\\n",
    "\\end{bmatrix}\\\\\n",
    "&=-\\nabla E(0,0)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Since $\\nabla^2 E(0,0)$ is positive definite, we can solve for the optimal $(\\Delta u, \\Delta v)$ and have\n",
    "\n",
    "\\begin{align*}\n",
    "\\begin{bmatrix} \n",
    "\\Delta u^*\\\\ \n",
    "\\Delta v^*\\\\\n",
    "\\end{bmatrix}\n",
    "=-\\left(\\nabla^2 E(0,0)\\right)^{-1}\\nabla E(0,0)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Solve the above equation, we obtain the same solution as \n",
    "\n",
    "\\begin{align*}\n",
    "(\\Delta u^*, \\Delta v^*) &= (\\frac{30}{32}, \\frac{13}{32})\\\\\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "E(u+du,v+dv) with (du,dv) from E_1 approximation:  2.2508597349929693\n",
      "Newton direction: ( 0.4587778126549621 , 0.19880371881715023 )\n",
      "E(u+du,v+dv) =  1.8904907903020918\n",
      "Optimization terminated successfully.\n",
      "         Current function value: 1.868437\n",
      "         Iterations: 38\n",
      "         Function evaluations: 76\n",
      "Optimal direction by minimizing E(u+du,v+dv):  0.4355689881974021 0.2455191571358361\n",
      "Minimal E(u+du, v+dv):  1.8684370301391746\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.optimize import minimize\n",
    "from functools import partial\n",
    "import math\n",
    "\n",
    "#### problem 3.17 (e-i)\n",
    "def find_newton_direction(u,v,norm):\n",
    "    multipler = norm**2 / (u**2 + v**2)\n",
    "    return u * np.sqrt(multipler), v*np.sqrt(multipler)\n",
    "\n",
    "def calc_E(x):\n",
    "    u, v = x[0], x[1]\n",
    "    return np.exp(u) + np.exp(2.0*v) + np.exp(u*v) + u**2 - 3*u*v + 4*v**2 - 3*u - 5*v\n",
    "\n",
    "def calc_E_with_length(theta, norm):\n",
    "    '''Compute the E value with fixed length of (du, dv)\n",
    "    '''\n",
    "    u, v = norm*math.sin(theta), norm*math.cos(theta)\n",
    "    return np.exp(u) + np.exp(2.0*v) + np.exp(u*v) + u**2 - 3*u*v + 4*v**2 - 3*u - 5*v\n",
    "\n",
    "# Note, we only compare values at point (u=0, v=0)\n",
    "print('E(u+du,v+dv) with (du,dv) from E_1 approximation: ', calc_E([1/np.sqrt(13),3/np.sqrt(13)/2]))\n",
    "new_u, new_v = find_newton_direction(30/32, 13/32, 0.5)\n",
    "print('Newton direction: (', new_u, ',', new_v, ')')\n",
    "new_E = calc_E([new_u, new_v])\n",
    "print('E(u+du,v+dv) = ', new_E)\n",
    "\n",
    "calc_E_half = partial(calc_E_with_length, norm=0.5)\n",
    "x0 = 0\n",
    "res = minimize(calc_E_half, x0, method='nelder-mead', options={'xatol': 1e-8, 'disp': True})\n",
    "print('Optimal direction by minimizing E(u+du,v+dv): ', 0.5*math.sin(res.x), 0.5*math.cos(res.x))\n",
    "print('Minimal E(u+du, v+dv): ', res.fun)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* (e) From above numerical calculations, we can see that:\n",
    "  * $\\hat{E}_1$ is not a good approximation to $E(u+\\Delta u, v+\\Delta v)$ since it has larger difference compared to $\\hat{E}_2$ approximation. \n",
    "  * Newton direction is very close to the optimal direction obtained by minimizing $E(u+\\Delta u, v+\\Delta v)$. The value computed with Newton direction is just a little bit larger than the minimal value. \n",
    "\n",
    "#### Problem 3.18 TODO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 3.19\n",
    "\n",
    "* (a) Suppose $x$ has a dimension of $d$, $\\Phi(x)$ converts each $x_n$ into a point of $N$ dimension, where $N$ is the total number of points. PLA shall be able to shatter $N+1$ points which is a super set of the $N$ points. The problem is with the generalization error bound. According to the VC bound, $E_{out} \\le E_{in} + \\sqrt{\\frac{d\\ln N}{N}}$, when we do feature transformation using $\\Phi(x)$, the $d=N$, so we see that the error is increasing with $\\sqrt{\\ln N}$, which is underiable.\n",
    "\n",
    "* (b) We have $\\Phi(x) = \\left(\\phi_1(x),\\phi_2(x),\\dots,\\phi_N(x)\\right)$, this is similar to problem (a) in that we convert from $d$ dimension to $N$ dimension. So it has the same generalization issue. The difference is that the values for $\\Phi(x_n)$ are not all zeroes when $x_i \\ne x_n$. So one has to store all elements for each $x_n$. When the $N$ is large, the storage is significant.\n",
    "\n",
    "* (c) This transformation converts the $x$ to a fixed number of grids. For example, in 2-D, each $x$ is transformed to a set of $101*101$ points. This will cause problem when the dimension of original $x$ is large, then each point will be converted to $101^d$-dimension point. It just can't handle data points with large dimension. Also, to get a good generalization error bound, the number of points $N$ should be much larger than $101^d$, this is a high requirement."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
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