math_differential_calculus

math_differential_calculus.ipynb

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  "metadata": {
    "colab": {
      "name": "math_differential_calculus",
      "provenance": [],
      "collapsed_sections": [],
      "toc_visible": true
    },
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.7.6"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bf5zjgWqJ40F",
        "colab_type": "text"
      },
      "source": [
        "**Math - Differential Calculus**\n",
        "\n",
        "Calculus is the study of continuous change. It has two major subfields: *differential calculus*, which studies the rate of change of functions, and *integral calculus*, which studies the area under the curve. In this notebook, we will discuss the former.\n",
        "\n",
        "*Differential calculus is at the core of Deep Learning, so it is important to understand what derivatives and gradients are, how they are used in Deep Learning, and understand what their limitations are.*\n",
        "\n",
        "**Note:** the code in this notebook is only used to create figures and animations. You do not need to understand how it works (although I did my best to make it clear, in case you are interested)."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "C9lYcpLyJ40H",
        "colab_type": "text"
      },
      "source": [
        "<table align=\"left\">\n",
        "  <td>\n",
        "    <a target=\"_blank\" href=\"https://colab.research.google.com/github/ageron/handson-ml2/blob/master/math_differential_calculus.ipynb\"><img src=\"https://www.tensorflow.org/images/colab_logo_32px.png\" />Run in Google Colab</a>\n",
        "  </td>\n",
        "</table>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "ChS5BE53XXYr"
      },
      "source": [
        "# Slope of a straight line"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "cellView": "form",
        "colab_type": "code",
        "id": "QXiG4ZiHvf_5",
        "colab": {}
      },
      "source": [
        "#@title\n",
        "%matplotlib inline\n",
        "import matplotlib as mpl\n",
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "\n",
        "# To get smooth animations\n",
        "import matplotlib.animation as animation\n",
        "mpl.rc('animation', html='jshtml')"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "AA7V2hteveV4"
      },
      "source": [
        "What is the slope of the following line?"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "cellView": "form",
        "colab_type": "code",
        "id": "m9h_md3TvVUh",
        "outputId": "ab5d93cc-9095-4d72-de8a-5bb5a1b0ccd0",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 300
        }
      },
      "source": [
        "#@title\n",
        "def get_AB_line(A_pos, B_pos, x_min=-1000, x_max=+1000):\n",
        "    rise = B_pos[1] - A_pos[1]\n",
        "    run = B_pos[0] - A_pos[0]\n",
        "    slope = rise / run\n",
        "    offset = A_pos[1] - slope * A_pos[0]\n",
        "    return [x_min, x_max], [x_min * slope + offset, x_max * slope + offset]\n",
        "\n",
        "def plot_AB_line(A_pos, B_pos, A_name=\"A\", B_name=\"B\"):\n",
        "    for point, name in ((A_pos, A_name), (B_pos, B_name)):\n",
        "        plt.plot(point[0], point[1], \"bo\")\n",
        "        plt.text(point[0] - 0.35, point[1], name, fontsize=14)\n",
        "    xs, ys = get_AB_line(A_pos, B_pos)\n",
        "    plt.plot(xs, ys)\n",
        "\n",
        "def plot_rise_over_run(A_pos, B_pos):\n",
        "    plt.plot([A_pos[0], B_pos[0]], [A_pos[1], A_pos[1]], \"k--\")\n",
        "    plt.text((A_pos[0] + B_pos[0]) / 2, A_pos[1] - 0.4, \"run\", fontsize=14)\n",
        "    plt.plot([B_pos[0], B_pos[0]], [A_pos[1], B_pos[1]], \"k--\")\n",
        "    plt.text(B_pos[0] + 0.2, (A_pos[1] + B_pos[1]) / 2, \"rise\", fontsize=14)\n",
        "\n",
        "def show(axis=\"equal\", ax=None, title=None, xlabel=\"$x$\", ylabel=\"$y$\"):\n",
        "    ax = ax or plt.gca()\n",
        "    ax.axis(axis)\n",
        "    ax.grid()\n",
        "    ax.set_title(title, fontsize=14)\n",
        "    ax.set_xlabel(xlabel, fontsize=14)\n",
        "    ax.set_ylabel(ylabel, fontsize=14, rotation=0)\n",
        "    ax.axhline(y=0, color='k')\n",
        "    ax.axvline(x=0, color='k')\n",
        "\n",
        "A_pos = np.array([1, 1])\n",
        "B_pos = np.array([7, 4])\n",
        "plot_AB_line(A_pos, B_pos)\n",
        "plot_rise_over_run(A_pos, B_pos)\n",
        "show([0, 8.4, 0, 5.5], title=\"Slope = rise / run\")"
      ],
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "ya-lEC7O5IHX"
      },
      "source": [
        "As you probably know, the slope of a (non-vertical) straight line can be calculated by taking any two points $\\mathrm{A}$ and $\\mathrm{B}$ on the line, and computing the \"rise over run\":\n",
        "\n",
        "$slope = \\dfrac{rise}{run} = \\dfrac{\\Delta y}{\\Delta x} = \\dfrac{y_\\mathrm{B} - y_\\mathrm{A}}{x_\\mathrm{B} - x_\\mathrm{A}}$\n",
        "\n",
        "\n",
        "In this example, the rise is 3, and the run is 6, so the slope is 3/6 = 0.5."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "mKJGxZIAXikF"
      },
      "source": [
        "# Defining the slope of a curve"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "gcb7eqkmGGXf"
      },
      "source": [
        "But what if you want to know the slope of something else than a straight line? For example, let's consider the curve defined by $y = f(x) = x^2$:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "cellView": "form",
        "colab_type": "code",
        "id": "U6T3WLcsRZuh",
        "outputId": "5bdec5d8-c827-40d8-d63e-af140b964e46",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 305
        }
      },
      "source": [
        "#@title\n",
        "xs = np.linspace(-2.1, 2.1, 500)\n",
        "ys = xs**2\n",
        "plt.plot(xs, ys)\n",
        "\n",
        "plt.plot([0, 0], [0, 3], \"k--\")\n",
        "plt.arrow(-1.4, 2.5, 0.5, -1.3, head_width=0.1)\n",
        "plt.arrow(0.85, 1.05, 0.5, 1.3, head_width=0.1)\n",
        "show([-2.1, 2.1, 0, 2.8], title=\"Slope of the curve $y = x^2$\")"
      ],
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "eCBsOOBAG1vh"
      },
      "source": [
        "Obviously, the slope varies: on the left (i.e., when $x<0$), the slope is negative (i.e., when we move from left to right, the curve goes down), while on the right (i.e., when $x>0$) the slope is positive (i.e., when we move from left to right, the curve goes up). At the point $x=0$, the slope is equal to 0 (i.e., the curve is locally flat). The fact that the slope is 0 when we reach a minimum (or indeed a maximum) is crucially important, and we will come back to it later."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "4qCXg9nQSp6S"
      },
      "source": [
        "How can we put numbers on these intuitions? Well, say we want to estimate the slope of the curve at a point $\\mathrm{A}$, we can do this by taking another point $\\mathrm{B}$ on the curve, not too far away, and then computing the slope between these two points:\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "cellView": "form",
        "colab_type": "code",
        "id": "39VP85dfYwLW",
        "outputId": "d0fe05b9-6c1b-4c05-ae84-fd4c6b79353f",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 382
        }
      },
      "source": [
        "#@title\n",
        "def animate_AB_line(f, fp, f_str, x_A, axis=None):\n",
        "    y_A = f(x_A)\n",
        "    eps = 1e-4\n",
        "    x_B_range = 1.5\n",
        "    x_B = x_A + eps\n",
        "\n",
        "    n_frames = 200\n",
        "    text_offset_A = -0.2\n",
        "    text_offset_B = +0.1\n",
        "    x_min, x_max = -1000, 1000\n",
        "\n",
        "    fig, ax = plt.subplots()\n",
        "\n",
        "    # plot f(x)\n",
        "    xs = np.linspace(-2.1, 2.1, 500)\n",
        "    ys = f(xs)\n",
        "    ax.plot(xs, ys)\n",
        "\n",
        "    # plot the tangent to the curve at point A\n",
        "    if fp:\n",
        "        slope = fp(x_A)\n",
        "        offset = y_A - slope * x_A\n",
        "        ax.plot([x_min, x_max], [slope*x_min + offset, slope*x_max + offset],\n",
        "                \"y--\")\n",
        "\n",
        "    # plot the line AB and the labels A and B so they can be animated\n",
        "    y_A = f(x_A)\n",
        "    y_B = f(x_B)\n",
        "    xs, ys = get_AB_line([x_A, y_A], [x_B, y_B])\n",
        "    line_inf, = ax.plot(xs, ys, \"-\")\n",
        "    line_AB, = ax.plot([x_A, x_B], [y_A, y_B], \"bo-\")\n",
        "    ax.text(x_A + text_offset_A, y_A, \"A\", fontsize=14)\n",
        "    B_text = ax.text(x_B + text_offset_B, y_B, \"B\", fontsize=14)\n",
        "\n",
        "    # plot the grid and axis labels\n",
        "    title = r\"Slope of the curve $y = {}$ at $x_\\mathrm{{A}} = {}$\".format(f_str, x_A)\n",
        "    show(axis or [-2.1, 2.1, 0, 2.8], title=title)\n",
        "\n",
        "    def update_graph(i):\n",
        "        x_B = x_A + x_B_range * np.cos(i * 2 * np.pi / n_frames) ** 3\n",
        "        if np.abs(x_B - x_A) < eps:\n",
        "            x_B = x_A + eps # to avoid division by 0\n",
        "        y_B = f(x_B)\n",
        "        xs, ys = get_AB_line([x_A, y_A], [x_B, y_B])\n",
        "        line_inf.set_data(xs, ys)\n",
        "        line_AB.set_data([x_A, x_B], [y_A, y_B])\n",
        "        B_text.set_position([x_B + text_offset_B, y_B])\n",
        "        return line_inf, line_AB\n",
        "\n",
        "    anim = animation.FuncAnimation(fig, update_graph,\n",
        "                                  init_func=lambda: update_graph(0),\n",
        "                                  frames=n_frames,\n",
        "                                  interval=20,\n",
        "                                  blit=True)\n",
        "    plt.close()\n",
        "    return anim\n",
        "\n",
        "animate_AB_line(lambda x: x**2, lambda x: 2*x, \"x^2\", -1)"
      ],
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/html": [
              "\n",
              "<link rel=\"stylesheet\"\n",
              "href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/\n",
              "css/font-awesome.min.css\">\n",
              "<script language=\"javascript\">\n",
              "  function isInternetExplorer() {\n",
              "    ua = navigator.userAgent;\n",
              "    /* MSIE used to detect old browsers and Trident used to newer ones*/\n",
              "    return ua.indexOf(\"MSIE \") > -1 || ua.indexOf(\"Trident/\") > -1;\n",
              "  }\n",
              "\n",
              "  /* Define the Animation class */\n",
              "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
              "    this.img_id = img_id;\n",
              "    this.slider_id = slider_id;\n",
              "    this.loop_select_id = loop_select_id;\n",
              "    this.interval = interval;\n",
              "    this.current_frame = 0;\n",
              "    this.direction = 0;\n",
              "    this.timer = null;\n",
              "    this.frames = new Array(frames.length);\n",
              "\n",
              "    for (var i=0; i<frames.length; i++)\n",
              "    {\n",
              "     this.frames[i] = new Image();\n",
              "     this.frames[i].src = frames[i];\n",
              "    }\n",
              "    var slider = document.getElementById(this.slider_id);\n",
              "    slider.max = this.frames.length - 1;\n",
              "    if (isInternetExplorer()) {\n",
              "        // switch from oninput to onchange because IE <= 11 does not conform\n",
              "        // with W3C specification. It ignores oninput and onchange behaves\n",
              "        // like oninput. In contrast, Mircosoft Edge behaves correctly.\n",
              "        slider.setAttribute('onchange', slider.getAttribute('oninput'));\n",
              "        slider.setAttribute('oninput', null);\n",
              "    }\n",
              "    this.set_frame(this.current_frame);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.get_loop_state = function(){\n",
              "    var button_group = document[this.loop_select_id].state;\n",
              "    for (var i = 0; i < button_group.length; i++) {\n",
              "        var button = button_group[i];\n",
              "        if (button.checked) {\n",
              "            return button.value;\n",
              "        }\n",
              "    }\n",
              "    return undefined;\n",
              "  }\n",
              "\n",
              "  Animation.prototype.set_frame = function(frame){\n",
              "    this.current_frame = frame;\n",
              "    document.getElementById(this.img_id).src =\n",
              "            this.frames[this.current_frame].src;\n",
              "    document.getElementById(this.slider_id).value = this.current_frame;\n",
              "  }\n",
              "\n",
              "  Animation.prototype.next_frame = function()\n",
              "  {\n",
              "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
              "  }\n",
              "\n",
              "  Animation.prototype.previous_frame = function()\n",
              "  {\n",
              "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
              "  }\n",
              "\n",
              "  Animation.prototype.first_frame = function()\n",
              "  {\n",
              "    this.set_frame(0);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.last_frame = function()\n",
              "  {\n",
              "    this.set_frame(this.frames.length - 1);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.slower = function()\n",
              "  {\n",
              "    this.interval /= 0.7;\n",
              "    if(this.direction > 0){this.play_animation();}\n",
              "    else if(this.direction < 0){this.reverse_animation();}\n",
              "  }\n",
              "\n",
              "  Animation.prototype.faster = function()\n",
              "  {\n",
              "    this.interval *= 0.7;\n",
              "    if(this.direction > 0){this.play_animation();}\n",
              "    else if(this.direction < 0){this.reverse_animation();}\n",
              "  }\n",
              "\n",
              "  Animation.prototype.anim_step_forward = function()\n",
              "  {\n",
              "    this.current_frame += 1;\n",
              "    if(this.current_frame < this.frames.length){\n",
              "      this.set_frame(this.current_frame);\n",
              "    }else{\n",
              "      var loop_state = this.get_loop_state();\n",
              "      if(loop_state == \"loop\"){\n",
              "        this.first_frame();\n",
              "      }else if(loop_state == \"reflect\"){\n",
              "        this.last_frame();\n",
              "        this.reverse_animation();\n",
              "      }else{\n",
              "        this.pause_animation();\n",
              "        this.last_frame();\n",
              "      }\n",
              "    }\n",
              "  }\n",
              "\n",
              "  Animation.prototype.anim_step_reverse = function()\n",
              "  {\n",
              "    this.current_frame -= 1;\n",
              "    if(this.current_frame >= 0){\n",
              "      this.set_frame(this.current_frame);\n",
              "    }else{\n",
              "      var loop_state = this.get_loop_state();\n",
              "      if(loop_state == \"loop\"){\n",
              "        this.last_frame();\n",
              "      }else if(loop_state == \"reflect\"){\n",
              "        this.first_frame();\n",
              "        this.play_animation();\n",
              "      }else{\n",
              "        this.pause_animation();\n",
              "        this.first_frame();\n",
              "      }\n",
              "    }\n",
              "  }\n",
              "\n",
              "  Animation.prototype.pause_animation = function()\n",
              "  {\n",
              "    this.direction = 0;\n",
              "    if (this.timer){\n",
              "      clearInterval(this.timer);\n",
              "      this.timer = null;\n",
              "    }\n",
              "  }\n",
              "\n",
              "  Animation.prototype.play_animation = function()\n",
              "  {\n",
              "    this.pause_animation();\n",
              "    this.direction = 1;\n",
              "    var t = this;\n",
              "    if (!this.timer) this.timer = setInterval(function() {\n",
              "        t.anim_step_forward();\n",
              "    }, this.interval);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.reverse_animation = function()\n",
              "  {\n",
              "    this.pause_animation();\n",
              "    this.direction = -1;\n",
              "    var t = this;\n",
              "    if (!this.timer) this.timer = setInterval(function() {\n",
              "        t.anim_step_reverse();\n",
              "    }, this.interval);\n",
              "  }\n",
              "</script>\n",
              "\n",
              "<style>\n",
              ".animation {\n",
              "    display: inline-block;\n",
              "    text-align: center;\n",
              "}\n",
              "input[type=range].anim-slider {\n",
              "    width: 374px;\n",
              "    margin-left: auto;\n",
              "    margin-right: auto;\n",
              "}\n",
              ".anim-buttons {\n",
              "    margin: 8px 0px;\n",
              "}\n",
              ".anim-buttons button {\n",
              "    padding: 0;\n",
              "    width: 36px;\n",
              "}\n",
              ".anim-state label {\n",
              "    margin-right: 8px;\n",
              "}\n",
              ".anim-state input {\n",
              "    margin: 0;\n",
              "    vertical-align: middle;\n",
              "}\n",
              "</style>\n",
              "\n",
              "<div class=\"animation\">\n",
              "  <img id=\"_anim_img906b160c3eb7432d953d3d4c4216ec38\">\n",
              "  <div class=\"anim-controls\">\n",
              "    <input id=\"_anim_slider906b160c3eb7432d953d3d4c4216ec38\" type=\"range\" class=\"anim-slider\"\n",
              "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
              "           oninput=\"anim906b160c3eb7432d953d3d4c4216ec38.set_frame(parseInt(this.value));\"></input>\n",
              "    <div class=\"anim-buttons\">\n",
              "      <button onclick=\"anim906b160c3eb7432d953d3d4c4216ec38.slower()\"><i class=\"fa fa-minus\"></i></button>\n",
              "      <button onclick=\"anim906b160c3eb7432d953d3d4c4216ec38.first_frame()\"><i class=\"fa fa-fast-backward\">\n",
              "          </i></button>\n",
              "      <button onclick=\"anim906b160c3eb7432d953d3d4c4216ec38.previous_frame()\">\n",
              "          <i class=\"fa fa-step-backward\"></i></button>\n",
              "      <button onclick=\"anim906b160c3eb7432d953d3d4c4216ec38.reverse_animation()\">\n",
              "          <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
              "      <button onclick=\"anim906b160c3eb7432d953d3d4c4216ec38.pause_animation()\"><i class=\"fa fa-pause\">\n",
              "          </i></button>\n",
              "      <button onclick=\"anim906b160c3eb7432d953d3d4c4216ec38.play_animation()\"><i class=\"fa fa-play\"></i>\n",
              "          </button>\n",
              "      <button onclick=\"anim906b160c3eb7432d953d3d4c4216ec38.next_frame()\"><i class=\"fa fa-step-forward\">\n",
              "          </i></button>\n",
              "      <button onclick=\"anim906b160c3eb7432d953d3d4c4216ec38.last_frame()\"><i class=\"fa fa-fast-forward\">\n",
              "          </i></button>\n",
              "      <button onclick=\"anim906b160c3eb7432d953d3d4c4216ec38.faster()\"><i class=\"fa fa-plus\"></i></button>\n",
              "    </div>\n",
              "    <form action=\"#n\" name=\"_anim_loop_select906b160c3eb7432d953d3d4c4216ec38\" class=\"anim-state\">\n",
              "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_906b160c3eb7432d953d3d4c4216ec38\"\n",
              "             >\n",
              "      <label for=\"_anim_radio1_906b160c3eb7432d953d3d4c4216ec38\">Once</label>\n",
              "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_906b160c3eb7432d953d3d4c4216ec38\"\n",
              "             checked>\n",
              "      <label for=\"_anim_radio2_906b160c3eb7432d953d3d4c4216ec38\">Loop</label>\n",
              "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_906b160c3eb7432d953d3d4c4216ec38\"\n",
              "             >\n",
              "      <label for=\"_anim_radio3_906b160c3eb7432d953d3d4c4216ec38\">Reflect</label>\n",
              "    </form>\n",
              "  </div>\n",
              "</div>\n",
              "\n",
              "\n",
              "<script language=\"javascript\">\n",
              "  /* Instantiate the Animation class. */\n",
              "  /* The IDs given should match those used in the template above. */\n",
              "  (function() {\n",
              "    var img_id = \"_anim_img906b160c3eb7432d953d3d4c4216ec38\";\n",
              "    var slider_id = \"_anim_slider906b160c3eb7432d953d3d4c4216ec38\";\n",
              "    var loop_select_id = \"_anim_loop_select906b160c3eb7432d953d3d4c4216ec38\";\n",
              "    var frames = new Array(200);\n",
              "    \n",
              "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbAAAAEgCAYAAADVKCZpAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\\\n",
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