03_linear_gradient_descent.ipynb

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    "colab": {
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    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
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    "language_info": {
      "name": "python"
    },
    "gpuClass": "standard"
  },
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "#### Recap"
      ],
      "metadata": {
        "id": "vu9d5t3-QuPc"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "We have three data points and we want to fit a `linear model` to explain them\n",
        "\n",
        "$$(x_1, y_1)=(3,5),\\, (x_2, y_2)=(4,6),\\, (x_3, y_3)=(1,4)$$\n",
        "\n",
        "We write the model as\n",
        "\n",
        "$$y=A\\theta$$\n",
        "\n",
        "where\n",
        "\n",
        "$$y=\\begin{bmatrix} 5 \\\\ 6 \\\\ 4 \\end{bmatrix},\\, A=\\begin{bmatrix} 3 & 1 \\\\ 4 & 1 \\\\ 1 & 1\\end{bmatrix},\\, \\theta=\\begin{bmatrix} a \\\\ b \\end{bmatrix}$$\n",
        "\n",
        "and we can use the following to obtain $\\theta$\n",
        "\n",
        "$$\\theta=(A^TA)^{-1}A^Ty$$\n",
        "\n",
        "where $A^T$ is `transpose` of $A$\n",
        "\n",
        "However, the problem with such method is that computing inverse is not (always) easy or desirable, especially when the size of matrix is large\n",
        "\n",
        "As a result, in practice, `iterative numerical` approach is often used"
      ],
      "metadata": {
        "id": "E4EKYzZdSRf3"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### `Numerical` approach"
      ],
      "metadata": {
        "id": "MSzgkb2BRAtP"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Numerical method is commonly done `iteratively`\n",
        "\n",
        "* We `guess` $a$ and $b$ initially\n",
        "* We plug in guessed values and compute $A\\theta$ to get `predicted` output $y_{pred}$\n",
        "* We compared the output $y_{pred}$ with the `actual` data $y_{act}$, and compute the difference $\\|A\\theta-y_{act}\\|^2$\n",
        "* We call this difference the `cost function` or `loss function` $J$\n",
        "* Given data, $J$ is a `function of model parameters` $a$ and $b$\n",
        "* We adjust $a$ and $b$ in the next iteration in a `direction` such that $J$ will \"likely\" be reduced\n",
        "* This direction is the direction of `negative gradient` of $J$ w.r.t. $a$ and $b$, as the gradient of function always points to the direction where its value increases\n",
        "\n",
        "This algorithm is refered to as `gradient descent`"
      ],
      "metadata": {
        "id": "Wezg0HPWUAXB"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Derivation of `gradient` w.r.t. parameters"
      ],
      "metadata": {
        "id": "teq4PTiiRgpu"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "To compute the loss function, we can plug in each data sample and compute the prediction error corresponding to each, before adding them together\n",
        "\n",
        "$$J=\\frac{1}{m}\\sum_{i=1}^{m}\\,(y_{\\text{pred},i}-y_{\\text{act},i})^2$$\n",
        "\n",
        "where $m$ is the number of data samples (here, $m=3$) used to find the model parameters, and $y_{\\text{pred,i}}=a\\cdot x_i+b$\n",
        "\n",
        "There are also other options to define loss function, see `notes`"
      ],
      "metadata": {
        "id": "kQcBtwixf23h"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "The `gradient` w.r.t. $a$ and $b$ is then\n",
        "\n",
        "$$\\begin{align*}\n",
        "\\frac{\\partial J}{\\partial a}&=\\frac{1}{m}\\sum_{i=1}^{m}\\,(a\\cdot x_i+b-y_{\\text{act},i})\\cdot x_i\\\\\n",
        "\\frac{\\partial J}{\\partial b}&=\\frac{1}{m}\\sum_{i=1}^{m}\\,(a\\cdot x_i+b-y_{\\text{act},i})\\cdot 1\n",
        "\\end{align*}$$"
      ],
      "metadata": {
        "id": "K0UJJwjsgv8P"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Matrix notation"
      ],
      "metadata": {
        "id": "N4WHljXKR8bY"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "These two equations can be written in `matrix` form\n",
        "\n",
        "For example, `dJa=np.mean((y_pred)-(y_act))*x` where `y_pred` is a vector\n",
        "\n",
        "$$\\text{y_pred}=\\begin{bmatrix} y_{\\text{pred},1} \\\\ y_{\\text{pred},2} \\\\ y_{\\text{pred},3} \\end{bmatrix}$$\n",
        "\n",
        "`y_act` is also a vector\n",
        "\n",
        "$$\\text{y_act}=\\begin{bmatrix} y_{\\text{act},1} \\\\ y_{\\text{act},2} \\\\ y_{\\text{act},3} \\end{bmatrix}$$\n",
        "\n",
        "so is `x`\n",
        "\n",
        "$$\\text{x}=\\begin{bmatrix} x_1 \\\\ x_2 \\\\ x_3 \\end{bmatrix}$$\n",
        "\n",
        "When we do `dJa=np.mean((y_pred)-(y_act))*x`, numpy first carries out element-wise operation\n",
        "\n",
        "$$\\begin{bmatrix} (y_{\\text{pred},1}-y_{\\text{act},1})\\cdot x_1 \\\\ (y_{\\text{pred},2}-y_{\\text{act},2})\\cdot x_2 \\\\ (y_{\\text{pred},3}-y_{\\text{act},3})\\cdot x_3 \\end{bmatrix}$$\n",
        "\n",
        "Then, it takes the `average` of all elements, making it equivalent to $\\frac{\\partial J}{\\partial a}$ as described above\n",
        "\n",
        "Same goes for $\\frac{\\partial J}{\\partial b}$"
      ],
      "metadata": {
        "id": "lJz-erGGhY7X"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Adjustment of model parameters"
      ],
      "metadata": {
        "id": "-jsoNk1vcWZD"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "With the gradient, we can adjust $a$ and $b$ following\n",
        "\n",
        "$$a \\leftarrow a-\\eta\\frac{\\partial J}{\\partial a}, b \\leftarrow b-\\eta\\frac{\\partial J}{\\partial b}$$\n",
        "\n",
        "where $\\eta$ is the step size of adjustment that is often called `learning rate`"
      ],
      "metadata": {
        "id": "R4lhHatPavMB"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Put everything together"
      ],
      "metadata": {
        "id": "t9cqwBoilIod"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "plt.style.use('dark_background')\n",
        "np.set_printoptions(formatter={'float': '{: 0.4f}'.format})"
      ],
      "metadata": {
        "id": "nJgOaNMgTmV0"
      },
      "execution_count": 1,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "data_points = np.array([[3, 5], [4, 6], [1, 4]])\n",
        "x = data_points[:, 0] # a vector of three elements (3,)\n",
        "y = data_points[:, 1] # also a vector of three elements (3,)"
      ],
      "metadata": {
        "id": "G60A9dFAWwUf"
      },
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "learning_rate = 0.05\n",
        "iterations = 2000\n",
        "\n",
        "# Initial guess\n",
        "a = 0.0\n",
        "b = 0.0"
      ],
      "metadata": {
        "id": "oh4IkRA-lGQP"
      },
      "execution_count": 3,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "for i in range(iterations):\n",
        "    # Make predictions using current a and b\n",
        "    y_pred = a * x + b\n",
        "\n",
        "    # Compute gradients using vectors directly\n",
        "    da = np.mean((y_pred - y) * x)\n",
        "    db = np.mean(y_pred - y)\n",
        "\n",
        "    # Update a and b in negative gradient direction\n",
        "    a -= learning_rate * da\n",
        "    b -= learning_rate * db\n",
        "\n",
        "# Final parameters\n",
        "print(f\"Final parameters: a = {a:.4f}, b = {b:.4f}\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Gj7-k7KplSqT",
        "outputId": "fd1bec81-af73-4836-ce5e-bc9cac6a318b"
      },
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Final parameters: a = 0.6429, b = 3.2857\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "To see whether our `numerical` approach is doing well, we can use NumPy command `np.linalg.inv()` to compute the results analytically"
      ],
      "metadata": {
        "id": "enxVROePXMx-"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "A = np.hstack([x.reshape(-1, 1), np.ones((len(x), 1))]) # this is a matrix of size (3,2)\n",
        "theta = np.linalg.inv(A.T@A)@A.T@y\n",
        "theta"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "uAleBa5tW86g",
        "outputId": "46651617-431f-4bac-de79-f28fd433b538"
      },
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([ 0.6429,  3.2857])"
            ]
          },
          "metadata": {},
          "execution_count": 5
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "We see the results are the same"
      ],
      "metadata": {
        "id": "vsLZDFD0ma_X"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Finally, we plot the results"
      ],
      "metadata": {
        "id": "NiRHaz5UXYT8"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def prediction(x, theta):\n",
        "    return theta[0]*x + theta[1]\n",
        "\n",
        "fig, ax=plt.subplots(figsize=(6,4))\n",
        "\n",
        "plt.scatter(A[:,0], y, color='cyan', label='Data', edgecolors='w')\n",
        "ax.plot([0,6],[prediction(0, theta), prediction(6, theta)],'r-', label='Model', zorder=-5)\n",
        "\n",
        "ax.set_xlabel('x')\n",
        "ax.set_ylabel('y')\n",
        "plt.tight_layout()\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 407
        },
        "id": "x9RauGzfXWir",
        "outputId": "c96754c7-048d-449c-e586-fcb2db3b0bbe"
      },
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 600x400 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### A slightly less empirical view of gradient descent..."
      ],
      "metadata": {
        "id": "FhI4UK9nceVe"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "At `kth` iteration, gradient descent as described above can be considered as the operations of\n",
        "\n",
        "* find the 1st order `approximation of loss function` at current $a^k, b^k$\n",
        "* find new $a^{k+1}, b^{k+1}$ `in the vicinity of` $a^k, b^k$ that `minimizes` this 1st order approximation"
      ],
      "metadata": {
        "id": "JP1wv2EHfVI6"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "To see this, suppose at `kth` iteration we have our parameters $\\theta^k=\\begin{bmatrix}a^k \\\\ b^k\\end{bmatrix}$\n",
        "\n",
        "We take the 1st order `Taylor approximation` of our cost function $J(\\theta)$ (recall $J$ is a function of model parameters) in the vicinity of $\\theta^k$\n",
        "\n",
        "$$\\begin{align*}J(\\theta)\\approx J(\\theta^k)+\\nabla J(\\theta^k)^T(\\theta-\\theta^k)\n",
        "\\end{align*}$$\n",
        "\n",
        "where $\\nabla J(\\theta^k)^T$ is the trasposed vector containing the gradients\n",
        "\n",
        "$$\\nabla J(\\theta^k)^T =\\begin{bmatrix} \\left.\\frac{\\partial J}{\\partial a}\\right|_{a=a^k} & \\left.\\frac{\\partial J}{\\partial b}\\right|_{b=b^k}\\end{bmatrix}$$"
      ],
      "metadata": {
        "id": "hSK6fNZmcmXE"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Since it is just an approximation, the idea is to minimize $J$ while not moving too far away from $\\theta^k$ (otherwise the approximation can break down)\n",
        "\n",
        "Therefore, we want to find $\\theta^{k+1}=\\begin{bmatrix}a^{k+1} \\\\ b^{k+1}\\end{bmatrix}$ for the `(k+1)th` iteration as a solution to minimize the following expression\n",
        "\n",
        "$$J(\\theta^{k})+\\nabla J(\\theta^{k})^T(\\theta-\\theta^{k})+\\frac{1}{2\\eta}\\|\\theta-\\theta^{k}\\|^2$$\n",
        "\n",
        "which can be considered a `quadratic approximation` of $J$ at $\\theta^k$\n",
        "\n",
        "By having the following term in the expression\n",
        "\n",
        "$$\\frac{1}{2\\eta}\\|\\theta-\\theta^{k}\\|^2$$\n",
        "\n",
        "we basically ask to `not move too far away` from $\\theta^{k}$\n",
        "\n",
        "We take derivative of this expression w.r.t. $\\theta$ and set it to zero\n",
        "\n",
        "$$0+\\nabla J(\\theta^{k})+\\frac{1}{\\eta}(\\theta-\\theta^{k})=0$$\n",
        "\n",
        "and we get our `gradient descent equation`\n",
        "\n",
        "$$\\theta^{k+1}=\\theta^{k}-\\eta \\nabla J(\\theta^{k})$$\n",
        "\n",
        "or\n",
        "\n",
        "$$\\begin{bmatrix}a^{k+1} \\\\ b^{k+1}\\end{bmatrix}=\\begin{bmatrix}a^k \\\\ b^k\\end{bmatrix}-\\eta \\begin{bmatrix} \\left.\\frac{\\partial J}{\\partial a}\\right|_{a=a^k} \\\\ \\left.\\frac{\\partial J}{\\partial b}\\right|_{b=b^k}\\end{bmatrix}$$\n",
        "\n",
        "which is the same as described earlier\n",
        "\n",
        "$$a \\leftarrow a-\\eta\\frac{\\partial J}{\\partial a}, b \\leftarrow b-\\eta\\frac{\\partial J}{\\partial b}$$"
      ],
      "metadata": {
        "id": "c52E-s-vcoax"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def j(x):\n",
        "    return 0.5 * x**2 + 1\n",
        "\n",
        "def grad_j(x):\n",
        "    return x\n",
        "\n",
        "def quadratic_approx(x, x_0, beta):\n",
        "    return j(x_0) + grad_j(x_0) * (x - x_0) + 0.5 * beta* (x - x_0)**2\n",
        "\n",
        "def linear_approx_example(x, x_0):\n",
        "    return j(x_0) + grad_j(x_0) * (x - x_0)\n",
        "\n",
        "x_0 = 2.0\n",
        "beta= 1.5\n",
        "\n",
        "x = np.linspace(-1, 4, 400)\n",
        "\n",
        "plt.plot(x, j(x), label=\"Function $J(\\\\theta)$\", color='w', linewidth=2)\n",
        "plt.plot(x, quadratic_approx(x, x_0, beta), label=\"Quadratic approximation\", linestyle='--', color='m')\n",
        "plt.plot(x, linear_approx_example(x, x_0), label=\"Linear approximation\", linestyle='-', color='r')\n",
        "\n",
        "plt.scatter(x_0, j(x_0), color='w', zorder=5)\n",
        "plt.text(x_0, j(x_0)+0.5, '  $\\\\theta^k$', horizontalalignment='center', fontsize=12, color='w')\n",
        "\n",
        "plt.scatter(0, 1, color='w', zorder=5)\n",
        "plt.text(0, 1.3, '  $\\\\theta^*$', horizontalalignment='center', fontsize=12, color='w')\n",
        "\n",
        "plt.xlabel(\"$\\\\theta$\")\n",
        "plt.ylabel(\"$J(\\\\theta)$\")\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "metadata": {
        "id": "i4hwsSsonF89",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 450
        },
        "outputId": "6e248062-f4ee-47de-a2ba-5859fe5eff05"
      },
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [],
      "metadata": {
        "id": "bIzuIrp6DlKH"
      },
      "execution_count": null,
      "outputs": []
    }
  ]
}