problem.ipynb

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    "# AOS1 - Assignment\n",
    "## Improving the accuracy and speed of support vector machines\n",
    "\n",
    "Authors : Mathilde Rineau, Rémy Huet\n",
    "\n",
    "### Abstract\n",
    "\n",
    "The paper \"Improving the Accuracy and Speed of Support Vector Machines\" by Burges and Schölkopf is investigating a method to improve ht speed an accuracy of a support vector machine.\n",
    "\n",
    "As the authors say, SVM are wildly used for several applications.\n",
    "To improve this method, the authors make the difference between two types of improvements to achieve :\n",
    "- improving the generalization performance;\n",
    "- improving the speed in test phase.\n",
    "\n",
    "The authors propose and combine two methods to improve SVM performances : the \"virtual support vector\" method and the \"reduced set\" method.\n",
    "With those two improvements, they announce a machine much faster (22 times than the original one) and more precise (1.1% vs 1.4% error) than the original one.\n",
    "\n",
    "In this assignment, we will implement the first method to test it."
   ]
  },
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   "cell_type": "markdown",
   "id": "12aaeba6",
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   "source": [
    "### First part : tests with a vanilla SVM\n",
    "\n",
    "In this first part, we will use a vanilla SVM on the MINST dataset with the provided params.\n",
    "We will observe the error of the SVM and the time for the test phase to compare them with the improved version"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9f152334",
   "metadata": {},
   "outputs": [],
   "source": [
    "# We will work on the mnist data set\n",
    "# We load it from fetch_openml\n",
    "from sklearn.datasets import fetch_openml\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "X, y = fetch_openml('mnist_784', version=1, return_X_y=True, as_frame=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "855cdb06",
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   "source": [
    "We do some inspection on the dataset :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "708c8ea1",
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   "source": [
    "# We print the caracteristics of X and Y\n",
    "print(X.shape)\n",
    "print(y.shape)\n",
    "# Values taken by y\n",
    "print(np.unique(y))\n",
    "\n",
    "image = np.reshape(X[0], (28, 28))\n",
    "plt.imshow(image, cmap='gray')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4e49be54",
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   "source": [
    "The dataset contains 70k samples of 784 features.\n",
    "The classes are 0 to 9 (the digits on the images).\n",
    "\n",
    "The features are the pixels of a 28 x 28 image that we can retrieve using numpy's reshape function.\n",
    "\n",
    "For example, the 1st image is a 5."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "60f31892",
   "metadata": {},
   "source": [
    "With our dataset, we can generate a training dataset and a testing dataset.\n",
    "As in the article, we will use 60k samples as training samples and 10k as testing.\n",
    "\n",
    "We split the dataset using the `train_test_split` function from `sklearn`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4d3fa1c7",
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   "source": [
    "# We divide the data set in two parts: train set and test set\n",
    "# According to the recommended values the train set's size is 60000 and the test set's size is 10000\n",
    "from sklearn.model_selection import train_test_split\n",
    "X_train, X_test, y_train, y_test = train_test_split(\n",
    "    X, y, train_size=60000, test_size=10000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d0532cc1",
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   "source": [
    "From the article, we retrieve the parameters of the SVM used.\n",
    "We get C = 10, and a polynomial kernel of degree 5.\n",
    "Coefficients `gamma` and `coef0` are respectively equals to 1 and 0.\n",
    "\n",
    "We can now train a SVM with these params on the training dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d809fc87",
   "metadata": {},
   "outputs": [],
   "source": [
    "# First, we perform a SVC without preprocessing or improving in terms of accuracy or speed\n",
    "from sklearn.svm import SVC\n",
    "# we perform the default SVC, with the hyperparameter C=10 and a polynomial kernel of degree 5\n",
    "# according to the article\n",
    "svc = SVC(C=10, kernel = 'poly', degree = 5, gamma=1, coef0=0)\n",
    "svc.fit(X_train, y_train)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a8cf4850",
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   "source": [
    "Using the previously trained SVM, we make a prediction on the test dataset.\n",
    "\n",
    "One of the measured performance of the SVM in this article is the speed of the test phase.\n",
    "We thus measure it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8cb28178",
   "metadata": {},
   "outputs": [],
   "source": [
    "import time\n",
    "\n",
    "start = time.time()\n",
    "# We predict the values for our test set\n",
    "y_pred = svc.predict(X_test)\n",
    "end = time.time()\n",
    "\n",
    "print(f'Elapsed time : {end - start}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "90f08e8b",
   "metadata": {},
   "source": [
    "Of course the prediction time varies between two splits of the dataset, two computers and two executions, but we will retain that is is close from 70s.\n",
    "\n",
    "Using `y_test` the real classes of the `X_test` samples and `y_pred` the predicted classes from the SVM, we can compute the confusion matrix and the error to see the how good the predictions are."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c1248238",
   "metadata": {},
   "outputs": [],
   "source": [
    "# We compute the confusion matrix\n",
    "from sklearn.metrics import ConfusionMatrixDisplay, classification_report, accuracy_score\n",
    "\n",
    "disp = ConfusionMatrixDisplay.from_predictions(y_test, y_pred)\n",
    "disp.figure_.suptitle('Confusion matrix for the vanilla SVM')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ba4e38ac",
   "metadata": {},
   "outputs": [],
   "source": [
    "# We print the classification report\n",
    "print(classification_report(y_test, y_pred))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "947b0895",
   "metadata": {},
   "outputs": [],
   "source": [
    "# We print the accuracy of the SVC and the error rate\n",
    "acc = accuracy_score(y_test, y_pred)\n",
    "\n",
    "print(\"Accuracy: \", acc)\n",
    "print(\"Error rate: \", (1-acc) * 100, \"%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f780139",
   "metadata": {},
   "source": [
    "As earlier, the values are affected by the selection of the training and testing dateset. Wi will retain a value of 3.4 % for the error.\n",
    "\n",
    "The method described py the authors relies on the support vectors of the previously trained SVM.\n",
    "We will thus do some inspection on them before going further."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "81b09df7",
   "metadata": {},
   "outputs": [],
   "source": [
    "s_vects = svc.support_vectors_\n",
    "\n",
    "print(s_vects.shape)\n",
    "\n",
    "v = s_vects[0]\n",
    "v_index = svc.support_[0]\n",
    "v_class = y_train[v_index]\n",
    "print(v_index)\n",
    "print(v_class)\n",
    "\n",
    "print(f'Class of the first support vector : {v_class}')\n",
    "img = np.reshape(v, (28, 28))\n",
    "plt.imshow(img, cmap='gray')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "14cb2622",
   "metadata": {},
   "source": [
    "There are around 7300 support vectors in the SVM.\n",
    "\n",
    "Each support vector is a sample of `X_train`. We can thus retrieve its class using its index on the train dataset, and display it as an image as above."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fa1d441f",
   "metadata": {},
   "source": [
    "### Implementing the \"Virtual Support Vectors\" method\n",
    "\n",
    "We will now implement the \"Virtual Support Vectors\" as proposed by the authors.\n",
    "The aim of this method is to add some invariance in the data to make the previsions more robust.\n",
    "\n",
    "For a given trained SVM, we now that the only data relevant for classification are the support vectors.\n",
    "We will thus re-train a SVM, but with different data created from the support vectors.\n",
    "\n",
    "Here, the invariance proposed is shifting the image to one of the four directions.\n",
    "\n",
    "For each support vector, we will shift the image to the four directions, and use those images as a new dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "578b9f6c",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Retrieve the indexes of the support vectors\n",
    "sv_indexes = svc.support_\n",
    "\n",
    "# Arrays for storing the data\n",
    "X_vsv = []\n",
    "y_vsv = []\n",
    "\n",
    "for i in sv_indexes:\n",
    "    # Get the support vector and reshape it as image\n",
    "    sv = X_train[i].reshape((28, 28))\n",
    "    sv_class = y_train[i]\n",
    "    # Generate the four shifts, reshape them\n",
    "    sv_1 = np.roll(sv, 1, axis=0).reshape(784)\n",
    "    sv_2 = np.roll(sv, -1, axis=0).reshape(784)\n",
    "    sv_3 = np.roll(sv, 1, axis=1).reshape(784)\n",
    "    sv_4 = np.roll(sv, -1, axis=1).reshape(784)\n",
    "\n",
    "    # Add them to the dataset\n",
    "    X_vsv.append(sv_1)\n",
    "    X_vsv.append(sv_2)\n",
    "    X_vsv.append(sv_3)\n",
    "    X_vsv.append(sv_4)\n",
    "\n",
    "    # Add the corresponding classes\n",
    "    y_vsv.append(sv_class)\n",
    "    y_vsv.append(sv_class)\n",
    "    y_vsv.append(sv_class)\n",
    "    y_vsv.append(sv_class)\n",
    "\n",
    "X_vsv = np.array(X_vsv)\n",
    "y_vsv = np.array(y_vsv)\n",
    "\n",
    "print(X_vsv.shape)\n",
    "print(y_vsv.shape)\n",
    "\n",
    "im0 = X_vsv[0].reshape((28, 28))\n",
    "im1 = X_vsv[1].reshape((28, 28))\n",
    "im2 = X_vsv[2].reshape((28, 28))\n",
    "im3 = X_vsv[3].reshape((28, 28))\n",
    "\n",
    "print(f'classes : {y_vsv[0]} {y_vsv[1]} {y_vsv[2]} {y_vsv[3]}')\n",
    "\n",
    "plt.figure()\n",
    "_, axis = plt.subplots(1, 4)\n",
    "axis[0].imshow(im0, cmap='gray')\n",
    "axis[1].imshow(im1, cmap='gray')\n",
    "axis[2].imshow(im2, cmap='gray')\n",
    "axis[3].imshow(im3, cmap='gray')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "78ee8e6a",
   "metadata": {},
   "source": [
    "With this new dataset, we can now train a SVM with the same params."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8dc4a410",
   "metadata": {},
   "outputs": [],
   "source": [
    "# note that we can re-fit the same SVC object with the new dataset\n",
    "svc.fit(X_vsv, y_vsv)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "396c7337",
   "metadata": {},
   "source": [
    "Let's make some inspection on the results of the training"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6c8bfe0d",
   "metadata": {},
   "outputs": [],
   "source": [
    "print(svc.support_.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43d511a9",
   "metadata": {},
   "source": [
    "With the \"vanilla SVM\", we got ~7300 support vectors for a training set of 60000 samples, so ~12 % of the dataset.\n",
    "With this training, we notice that most of the data was selected (~67 %).\n",
    "\n",
    "We can now use the trained SVM on the test date to measure the error and the time of the test."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a196291b",
   "metadata": {},
   "outputs": [],
   "source": [
    "start = time.time()\n",
    "# We predict the values for our test set\n",
    "y_pred = svc.predict(X_test)\n",
    "end = time.time()\n",
    "\n",
    "print(f'Elapsed time : {end - start}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bbda2f32",
   "metadata": {},
   "source": [
    "The time for the SVM to predict the test dataset is about 200s, which is much more than th vanilla SVM.\n",
    "It was predictable because the number of support vector is larger than the number of the support vectors of the vanilla SVM.\n",
    "\n",
    "Let's now see the error of the predictions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2c99def0",
   "metadata": {},
   "outputs": [],
   "source": [
    "# We compute the confusion matrix\n",
    "from sklearn.metrics import ConfusionMatrixDisplay, classification_report, accuracy_score\n",
    "\n",
    "disp = ConfusionMatrixDisplay.from_predictions(y_test, y_pred)\n",
    "disp.figure_.suptitle('Confusion matrix for the vanilla SVM')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "29d1ff85",
   "metadata": {},
   "outputs": [],
   "source": [
    "# We print the classification report\n",
    "print(classification_report(y_test, y_pred))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "947b0895",
   "metadata": {},
   "outputs": [],
   "source": [
    "# We print the accuracy of the SVC and the error rate\n",
    "acc = accuracy_score(y_test, y_pred)\n",
    "\n",
    "print(\"Accuracy: \", acc)\n",
    "print(\"Error rate: \", (1-acc) * 100, \"%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ed73c147",
   "metadata": {},
   "source": [
    "We can see that the error is smaller than the error of the vanilla SVM (2.2 % vs 3.4 %)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2fc76a50",
   "metadata": {},
   "source": [
    "### Conclusion\n",
    "\n",
    "By implementing the \"virtual support vectors\" technique, we were able to ass some invariance in the data.\n",
    "This modification allowed an improvement of the accuracy of the SVM.\n",
    "\n",
    "However, most of the new data generated from the support vectors were selected as support vectors for the second machine.\n",
    "The augmentation of the number of support vectors led to an augmentation of the computation time during the test phase.\n",
    "\n",
    "That is why the authors suggest in the article to use a second technique to create a reduced set of vectors to reduce the computation time in the test phase."
   ]
  }
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