python-examples/building-deep-learning-utils.ipynb

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  {
   "cell_type": "markdown",
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   "source": [
    "# Building Deep Neural Network Utilities: Step by Step\n",
    "\n",
    "## Objectives\n",
    "- Build a two-layer neural network and an L-layer neural network\n",
    "- Implement an easy-to-use neural network class\n",
    "\n",
    "**Notation**:\n",
    "- Superscript $[l]$ denotes a quantity associated with the $l^{th}$ layer. \n",
    "    - Example: $a^{[L]}$ is the $L^{th}$ layer activation. $W^{[L]}$ and $b^{[L]}$ are the $L^{th}$ layer parameters.\n",
    "- Superscript $(i)$ denotes a quantity associated with the $i^{th}$ example. \n",
    "    - Example: $x^{(i)}$ is the $i^{th}$ training example.\n",
    "- Lowerscript $i$ denotes the $i^{th}$ entry of a vector.\n",
    "    - Example: $a^{[l]}_i$ denotes the $i^{th}$ entry of the $l^{th}$ layer's activations).\n",
    "    \n",
    "\n",
    "## TOC\n",
    "* [1 - Load Packages](1)\n",
    "* [2 - Steps](2)\n",
    "* [3 - Initialization](3)\n",
    "    * [3.1 - 2-layer Neural Network](3.1)\n",
    "    * [3.2 - L-layer Neural Network](3.2)\n",
    "* [4 - Forward propagation module](4)\n",
    "    * [4.1 - Linear Forward](4.1)\n",
    "    * [4.2 - Linear-Activation Forward ](4.2)\n",
    "    * [4.3 - L-Layer Model ](4.3)\n",
    "* [5 - Cost function ](5)\n",
    "* [6 - Backward propagation module](6)\n",
    "    * [6.1 - Linear backward](6.1)\n",
    "    * [6.2 - Linear-Activation backward ](6.2)\n",
    "    * [6.3 - L-Model Backward](6.3)\n",
    "    * [6.4 - Update Parameters ](6.4)    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1 - Load Packages <a name=\"1\"></a>\n",
    " \n",
    "- [numpy](www.numpy.org) is the main package for scientific computing with Python.\n",
    "- [matplotlib](http://matplotlib.org) is a library to plot graphs in Python.\n",
    "- dnn_utils provides some necessary functions for this notebook.\n",
    "- testCases provides some test cases to assess the correctness of each function.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The autoreload extension is already loaded. To reload it, use:\n",
      "  %reload_ext autoreload\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import h5py\n",
    "import matplotlib.pyplot as plt\n",
    "from testCases import *\n",
    "from dnn_utils import sigmoid, sigmoid_backward, relu, relu_backward\n",
    "\n",
    "%matplotlib inline\n",
    "plt.rcParams['figure.figsize'] = (5.0, 4.0) # set default size of plots\n",
    "plt.rcParams['image.interpolation'] = 'nearest'\n",
    "plt.rcParams['image.cmap'] = 'gray'\n",
    "\n",
    "%load_ext autoreload\n",
    "%autoreload 2\n",
    "\n",
    "np.random.seed(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2 - Steps <a name=\"2\"></a>\n",
    "\n",
    "- **Initialize** the parameters for a two-layer network and for an $L$-layer neural network.\n",
    "- Implement the **forward propagation** module (shown in purple in the figure below).\n",
    "     - Complete the LINEAR part of a layer's forward propagation step (resulting in $Z^{[l]}$).\n",
    "     - Use ACTIVATION function (relu/sigmoid).\n",
    "     - Combine the previous two steps into a new [LINEAR->ACTIVATION] forward function.\n",
    "     - Stack the [LINEAR->RELU] forward function L-1 time (for layers 1 through L-1) and add a [LINEAR->SIGMOID] at the end (for the final layer $L$). This gives us a new L_model_forward function.\n",
    "- Compute the **loss**.\n",
    "- Implement the **backward propagation** module (denoted in red in the figure below).\n",
    "    - Complete the LINEAR part of a layer's backward propagation step.\n",
    "    - Use the gradient of the ACTIVATE function (relu_backward/sigmoid_backward) \n",
    "    - Combine the previous two steps into a new [LINEAR->ACTIVATION] backward function.\n",
    "    - Stack [LINEAR->RELU] backward L-1 times and add [LINEAR->SIGMOID] backward in a new L_model_backward function\n",
    "- Finally update the parameters.\n",
    "\n",
    "\n",
    "![](images/final_outline.png)\n",
    "\n",
    "**Figure 1**\n",
    "\n",
    "**Note** that for every forward function, there is a corresponding backward function. That is why at every step of your forward module you will be storing some values in a cache. The cached values are useful for computing gradients. In the backpropagation module you will then use the cache to calculate the gradients. This assignment will show you exactly how to carry out each of these steps. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3 - Initialization <a name=\"3\"></a>\n",
    "\n",
    "The first function will be used to initialize parameters for a two layer model. The second one will generalize this initialization process to $L$ layers."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.1 - 2-layer Neural Network <a name=\"3.1\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def initialize_parameters(n_x, n_h, n_y):\n",
    "    \"\"\"\n",
    "    Argument:\n",
    "    n_x -- size of the input layer\n",
    "    n_h -- size of the hidden layer\n",
    "    n_y -- size of the output layer\n",
    "    \n",
    "    Returns:\n",
    "    parameters -- python dictionary containing your parameters:\n",
    "                    W1 -- weight matrix of shape (n_h, n_x)\n",
    "                    b1 -- bias vector of shape (n_h, 1)\n",
    "                    W2 -- weight matrix of shape (n_y, n_h)\n",
    "                    b2 -- bias vector of shape (n_y, 1)\n",
    "    \"\"\"\n",
    "    \n",
    "    np.random.seed(1)\n",
    "    \n",
    "    W1 = np.random.randn(n_h, n_x)*0.01 \n",
    "    b1 = np.zeros((n_h, 1))\n",
    "    W2 = np.random.randn(n_y, n_h)*0.01 \n",
    "    b2 = np.zeros((n_y, 1))\n",
    "    \n",
    "    assert(W1.shape == (n_h, n_x))\n",
    "    assert(b1.shape == (n_h, 1))\n",
    "    assert(W2.shape == (n_y, n_h))\n",
    "    assert(b2.shape == (n_y, 1))\n",
    "    \n",
    "    parameters = {\"W1\": W1,\n",
    "                  \"b1\": b1,\n",
    "                  \"W2\": W2,\n",
    "                  \"b2\": b2}\n",
    "    \n",
    "    return parameters    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.2 - L-layer Neural Network <a name=\"3.2\"></a>\n",
    "\n",
    "The initialization for a deeper L-layer neural network is more complicated because there are many more weight matrices and bias vectors.\n",
    "\n",
    "For example if the size of our input $X$ is $(12288, 209)$ (with $m=209$ examples) then:\n",
    "\n",
    "| Layer         \t| **Shape of W**           \t| **Shape of b**   \t| **Activation**                                 \t| **Shape of Activation** \t|\n",
    "|---------------\t|--------------------------\t|------------------\t|------------------------------------------------\t|-------------------------\t|\n",
    "| **Layer 1**   \t| $(n^{[1]},12288)$        \t| $(n^{[1]},1)$    \t| $Z^{[1]} = W^{[1]}  X + b^{[1]} $              \t| $(n^{[1]},209)$         \t|\n",
    "| **Layer 2**   \t| $(n^{[2]}, n^{[1]})$     \t| $(n^{[2]},1)$    \t| $Z^{[2]} = W^{[2]} A^{[1]} + b^{[2]}$          \t| $(n^{[2]}, 209)$        \t|\n",
    "| $\\vdots$      \t| $\\vdots$                 \t| $\\vdots$         \t| $\\vdots$                                       \t| $\\vdots$                \t|\n",
    "| **Layer L-1** \t| $(n^{[L-1]}, n^{[L-2]})$ \t| $(n^{[L-1]}, 1)$ \t| $Z^{[L-1]} =  W^{[L-1]} A^{[L-2]} + b^{[L-1]}$ \t| $(n^{[L-1]}, 209)$      \t|\n",
    "| **Layer L**   \t| $(n^{[L]}, n^{[L-1]})$   \t| $(n^{[L]}, 1)$   \t| $Z^{[L]} =  W^{[L]} A^{[L-1]} + b^{[L]}$       \t| $(n^{[L]}, 209)$        \t|\n",
    "\n",
    "In python, when we compute $W X + b$ , it carries out broadcasting. For example, if: \n",
    "\n",
    "$$ W = \\begin{bmatrix}\n",
    "    j  & k  & l\\\\\n",
    "    m  & n & o \\\\\n",
    "    p  & q & r \n",
    "\\end{bmatrix}\\;\\;\\; X = \\begin{bmatrix}\n",
    "    a  & b  & c\\\\\n",
    "    d  & e & f \\\\\n",
    "    g  & h & i \n",
    "\\end{bmatrix} \\;\\;\\; b =\\begin{bmatrix}\n",
    "    s  \\\\\n",
    "    t  \\\\\n",
    "    u\n",
    "\\end{bmatrix}\\tag{1}$$\n",
    "\n",
    "Then $WX + b$ will be:\n",
    "\n",
    "$$ WX + b = \\begin{bmatrix}\n",
    "    (ja + kd + lg) + s  & (jb + ke + lh) + s  & (jc + kf + li)+ s\\\\\n",
    "    (ma + nd + og) + t & (mb + ne + oh) + t & (mc + nf + oi) + t\\\\\n",
    "    (pa + qd + rg) + u & (pb + qe + rh) + u & (pc + qf + ri)+ u\n",
    "\\end{bmatrix}\\tag{2}  $$"
   ]
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   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def initialize_parameters_deep(layer_dims):\n",
    "    \"\"\"\n",
    "    Arguments:\n",
    "    layer_dims -- python array (list) containing the dimensions of each layer in our network\n",
    "    \n",
    "    Returns:\n",
    "    parameters -- python dictionary containing your parameters \"W1\", \"b1\", ..., \"WL\", \"bL\":\n",
    "                    Wl -- weight matrix of shape (layer_dims[l], layer_dims[l-1])\n",
    "                    bl -- bias vector of shape (layer_dims[l], 1)\n",
    "    \"\"\"\n",
    "    \n",
    "    np.random.seed(3)\n",
    "    parameters = {}\n",
    "    L = len(layer_dims)            # number of layers in the network\n",
    "\n",
    "    for l in range(1, L):\n",
    "        parameters[\"W\" + str(l)] = np.random.randn(layer_dims[l], layer_dims[l-1]) * 0.01\n",
    "        parameters[\"b\" + str(l)] = np.zeros((layer_dims[l], 1))\n",
    "        \n",
    "        assert(parameters['W' + str(l)].shape == (layer_dims[l], layer_dims[l-1]))\n",
    "        assert(parameters['b' + str(l)].shape == (layer_dims[l], 1))\n",
    "\n",
    "        \n",
    "    return parameters"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4 - Forward propagation module <a name=\"4\"></a>"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.1 - Linear Forward  <a name=\"4.1\"></a>\n",
    "We will complete three functions in this order:\n",
    "\n",
    "- LINEAR\n",
    "- LINEAR -> ACTIVATION where ACTIVATION will be either ReLU or Sigmoid. \n",
    "- [LINEAR -> RELU] $\\times$ (L-1) -> LINEAR -> SIGMOID (whole model)\n",
    "\n",
    "The linear forward module (vectorized over all the examples) computes the following equations:\n",
    "\n",
    "$$Z^{[l]} = W^{[l]}A^{[l-1]} +b^{[l]}\\tag{3}$$\n",
    "\n",
    "where $A^{[0]} = X$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def linear_forward(A, W, b):\n",
    "    \"\"\"\n",
    "    Implement the linear part of a layer's forward propagation.\n",
    "\n",
    "    Arguments:\n",
    "    A -- activations from previous layer (or input data): (size of previous layer, number of examples)\n",
    "    W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)\n",
    "    b -- bias vector, numpy array of shape (size of the current layer, 1)\n",
    "\n",
    "    Returns:\n",
    "    Z -- the input of the activation function, also called pre-activation parameter \n",
    "    cache -- a python dictionary containing \"A\", \"W\" and \"b\" ; stored for computing the backward pass efficiently\n",
    "    \"\"\"\n",
    "    \n",
    "    Z = np.dot(W,A)+b\n",
    "    \n",
    "    assert(Z.shape == (W.shape[0], A.shape[1]))\n",
    "    cache = (A, W, b)\n",
    "    \n",
    "    return Z, cache"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.2 - Linear-Activation Forward <a name=\"4.2\"></a>\n",
    "\n",
    "- **Sigmoid** $\\sigma(Z) = \\sigma(W A + b) = \\frac{1}{ 1 + e^{-(W A + b)}}$. `dnn_utils_v2` provides the `sigmoid` function. This function returns **two** items: the activation value \"`a`\" and a \"`cache`\" that contains \"`Z`\" (it's what we will feed in to the corresponding backward function). \n",
    "\n",
    "``` python\n",
    "A, activation_cache = sigmoid(Z)\n",
    "```\n",
    "\n",
    "- **ReLU**: The mathematical formula for ReLu is $A = RELU(Z) = max(0, Z)$. `dnn_utils_v2` provides the `relu` function. This function returns **two** items: the activation value \"`A`\" and a \"`cache`\" that contains \"`Z`\" (it's what we will feed in to the corresponding backward function). \n",
    "\n",
    "``` python\n",
    "A, activation_cache = relu(Z)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def linear_activation_forward(A_prev, W, b, activation):\n",
    "    \"\"\"\n",
    "    Implement the forward propagation for the LINEAR->ACTIVATION layer\n",
    "\n",
    "    Arguments:\n",
    "    A_prev -- activations from previous layer (or input data): (size of previous layer, number of examples)\n",
    "    W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)\n",
    "    b -- bias vector, numpy array of shape (size of the current layer, 1)\n",
    "    activation -- the activation to be used in this layer, stored as a text string: \"sigmoid\" or \"relu\"\n",
    "\n",
    "    Returns:\n",
    "    A -- the output of the activation function, also called the post-activation value \n",
    "    cache -- a python dictionary containing \"linear_cache\" and \"activation_cache\";\n",
    "             stored for computing the backward pass efficiently\n",
    "    \"\"\"\n",
    "    \n",
    "    if activation == \"sigmoid\":\n",
    "        Z, linear_cache = linear_forward(A_prev, W, b)\n",
    "        A, activation_cache = sigmoid(Z)\n",
    "    \n",
    "    elif activation == \"relu\":\n",
    "        Z, linear_cache = linear_forward(A_prev, W, b)\n",
    "        A, activation_cache = relu(Z)\n",
    "    \n",
    "    assert (A.shape == (W.shape[0], A_prev.shape[1]))\n",
    "    cache = (linear_cache, activation_cache)\n",
    "\n",
    "    return A, cache"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4.3 - L-Layer Model <a name=\"4.3\"></a>\n",
    "\n",
    "For convenience when implementing the $L$-layer Neural Net, we will need a function that replicates the previous one (`linear_activation_forward` with RELU) $L-1$ times, then follows that with one `linear_activation_forward` with SIGMOID.\n",
    "\n",
    "![](images/model_architecture_kiank.png)\n",
    "\n",
    "**Figure 2** : *[LINEAR -> RELU] $\\times$ (L-1) -> LINEAR -> SIGMOID* model\n",
    "\n",
    "In the code below, the variable `AL` will denote $A^{[L]} = \\sigma(Z^{[L]}) = \\sigma(W^{[L]} A^{[L-1]} + b^{[L]})$. (This is sometimes also called `Yhat`, i.e., this is $\\hat{Y}$.) \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def L_model_forward(X, parameters):\n",
    "    \"\"\"\n",
    "    Implement forward propagation for the [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID computation\n",
    "    \n",
    "    Arguments:\n",
    "    X -- data, numpy array of shape (input size, number of examples)\n",
    "    parameters -- output of initialize_parameters_deep()\n",
    "    \n",
    "    Returns:\n",
    "    AL -- last post-activation value\n",
    "    caches -- list of caches containing:\n",
    "                every cache of linear_activation_forward() (there are L-1 of them, indexed from 0 to L-1)\n",
    "    \"\"\"\n",
    "\n",
    "    caches = []\n",
    "    A = X\n",
    "    L = len(parameters) // 2                  # number of layers in the neural network\n",
    "    \n",
    "    # Implement [LINEAR -> RELU]*(L-1). Add \"cache\" to the \"caches\" list.\n",
    "    for l in range(1, L):\n",
    "        A_prev = A \n",
    "        A, cache = linear_activation_forward(A_prev, parameters[\"W\" + str(l)], parameters[\"b\" + str(l)], activation = \"relu\")\n",
    "        caches.append(cache)\n",
    "    \n",
    "    # Implement LINEAR -> SIGMOID. Add \"cache\" to the \"caches\" list.\n",
    "    AL, cache = linear_activation_forward(A, parameters[\"W\" + str(L)], parameters[\"b\" + str(L)], activation = \"sigmoid\")\n",
    "    caches.append(cache)\n",
    "    \n",
    "    assert(AL.shape == (1,X.shape[1]))\n",
    "            \n",
    "    return AL, caches"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5 - Cost function <a name=\"5\"></a>\n",
    "\n",
    "Compute the cross-entropy cost $J$, using the following formula: $$-\\frac{1}{m} \\sum\\limits_{i = 1}^{m} (y^{(i)}\\log\\left(a^{[L] (i)}\\right) + (1-y^{(i)})\\log\\left(1- a^{[L](i)}\\right)) \\tag{4}$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Y and AKL have the shape of (3,1) and matrix multiplication needs (3,1) and (1,3) so we will transpose AL."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compute_cost(AL, Y):\n",
    "    \"\"\"\n",
    "    Implement the cost function defined by equation (7).\n",
    "\n",
    "    Arguments:\n",
    "    AL -- probability vector corresponding to your label predictions, shape (1, number of examples)\n",
    "    Y -- true \"label\" vector (for example: containing 0 if non-cat, 1 if cat), shape (1, number of examples)\n",
    "\n",
    "    Returns:\n",
    "    cost -- cross-entropy cost\n",
    "    \"\"\"\n",
    "    \n",
    "    m = Y.shape[1]\n",
    "\n",
    "    # Compute loss from aL and y.\n",
    "    cost = -(1/m)*np.dot(Y,np.log(AL.T))+np.dot((1-Y),np.log(1-AL.T))\n",
    "    \n",
    "    cost = np.squeeze(cost)      # To make sure the cost's shape is what we expect (e.g. this turns [[17]] into 17).\n",
    "    assert(cost.shape == ())\n",
    "    \n",
    "    return cost"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6 - Backward propagation module <a name=\"6\"></a>\n",
    "\n",
    "\n",
    "**Reminder**: \n",
    "![](images/backprop_kiank.png)\n",
    "\n",
    "**Figure 3** : Forward and Backward propagation for *LINEAR->RELU->LINEAR->SIGMOID* <br> *The purple blocks represent the forward propagation, and the red blocks represent the backward propagation.* \n",
    "\n",
    "<!-- \n",
    "For those of you who are expert in calculus (you don't need to be to do this assignment), the chain rule of calculus can be used to derive the derivative of the loss $\\mathcal{L}$ with respect to $z^{[1]}$ in a 2-layer network as follows:\n",
    "\n",
    "$$\\frac{d \\mathcal{L}(a^{[2]},y)}{{dz^{[1]}}} = \\frac{d\\mathcal{L}(a^{[2]},y)}{{da^{[2]}}}\\frac{{da^{[2]}}}{{dz^{[2]}}}\\frac{{dz^{[2]}}}{{da^{[1]}}}\\frac{{da^{[1]}}}{{dz^{[1]}}} \\tag{8} $$\n",
    "\n",
    "In order to calculate the gradient $dW^{[1]} = \\frac{\\partial L}{\\partial W^{[1]}}$, you use the previous chain rule and you do $dW^{[1]} = dz^{[1]} \\times \\frac{\\partial z^{[1]} }{\\partial W^{[1]}}$. During the backpropagation, at each step you multiply your current gradient by the gradient corresponding to the specific layer to get the gradient you wanted.\n",
    "\n",
    "Equivalently, in order to calculate the gradient $db^{[1]} = \\frac{\\partial L}{\\partial b^{[1]}}$, you use the previous chain rule and you do $db^{[1]} = dz^{[1]} \\times \\frac{\\partial z^{[1]} }{\\partial b^{[1]}}$.\n",
    "\n",
    "This is why we talk about **backpropagation**.\n",
    "!-->\n",
    "\n",
    "Now, similar to forward propagation, you are going to build the backward propagation in three steps:\n",
    "- LINEAR backward\n",
    "- LINEAR -> ACTIVATION backward where ACTIVATION computes the derivative of either the ReLU or sigmoid activation\n",
    "- [LINEAR -> RELU] $\\times$ (L-1) -> LINEAR -> SIGMOID backward (whole model)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.1 - Linear backward <a name=\"6.1\"></a>\n",
    "\n",
    "For layer $l$, the linear part is: $Z^{[l]} = W^{[l]} A^{[l-1]} + b^{[l]}$ (followed by an activation).\n",
    "\n",
    " $dZ^{[l]} = \\frac{\\partial \\mathcal{L} }{\\partial Z^{[l]}}$. You want to get $(dW^{[l]}, db^{[l]} dA^{[l-1]})$.\n",
    "\n",
    "![](images/linearback_kiank.png)\n",
    "\n",
    "**Figure 4**\n",
    "\n",
    "The three outputs $(dW^{[l]}, db^{[l]}, dA^{[l]})$ are computed using the input $dZ^{[l]}$.\n",
    "$$ dW^{[l]} = \\frac{\\partial \\mathcal{L} }{\\partial W^{[l]}} = \\frac{1}{m} dZ^{[l]} A^{[l-1] T} \\tag{8}$$\n",
    "$$ db^{[l]} = \\frac{\\partial \\mathcal{L} }{\\partial b^{[l]}} = \\frac{1}{m} \\sum_{i = 1}^{m} dZ^{[l](i)}\\tag{9}$$\n",
    "$$ dA^{[l-1]} = \\frac{\\partial \\mathcal{L} }{\\partial A^{[l-1]}} = W^{[l] T} dZ^{[l]} \\tag{10}$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "def linear_backward(dZ, cache):\n",
    "    \"\"\"\n",
    "    Implement the linear portion of backward propagation for a single layer (layer l)\n",
    "\n",
    "    Arguments:\n",
    "    dZ -- Gradient of the cost with respect to the linear output (of current layer l)\n",
    "    cache -- tuple of values (A_prev, W, b) coming from the forward propagation in the current layer\n",
    "\n",
    "    Returns:\n",
    "    dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev\n",
    "    dW -- Gradient of the cost with respect to W (current layer l), same shape as W\n",
    "    db -- Gradient of the cost with respect to b (current layer l), same shape as b\n",
    "    \"\"\"\n",
    "    A_prev, W, b = cache\n",
    "    m = A_prev.shape[1]\n",
    "\n",
    "    dW = (1/m)*np.dot(dZ,A_prev.T)\n",
    "    db = (1/m)*np.sum(dZ,axis=1, keepdims=True)\n",
    "    dA_prev = np.dot(W.T,dZ)\n",
    "    \n",
    "    assert (dA_prev.shape == A_prev.shape)\n",
    "    assert (dW.shape == W.shape)\n",
    "    assert (db.shape == b.shape)\n",
    "    \n",
    "    return dA_prev, dW, db"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.2 - Linear-Activation backward <a name=\"6.2\"></a>\n",
    "\n",
    "Next, you will create a function that merges the two helper functions: **`linear_backward`** and the backward step for the activation **`linear_activation_backward`**. \n",
    "\n",
    "To help you implement `linear_activation_backward`, we provided two backward functions:\n",
    "- **`sigmoid_backward`**: Implements the backward propagation for SIGMOID unit. You can call it as follows:\n",
    "\n",
    "```python\n",
    "dZ = sigmoid_backward(dA, activation_cache)\n",
    "```\n",
    "\n",
    "- **`relu_backward`**: Implements the backward propagation for RELU unit. You can call it as follows:\n",
    "\n",
    "```python\n",
    "dZ = relu_backward(dA, activation_cache)\n",
    "```\n",
    "\n",
    "If $g(.)$ is the activation function, \n",
    "`sigmoid_backward` and `relu_backward` compute $$dZ^{[l]} = dA^{[l]} * g'(Z^{[l]}) \\tag{11}$$.  \n",
    "\n",
    "**Exercise**: Implement the backpropagation for the *LINEAR->ACTIVATION* layer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def linear_activation_backward(dA, cache, activation):\n",
    "    \"\"\"\n",
    "    Implement the backward propagation for the LINEAR->ACTIVATION layer.\n",
    "    \n",
    "    Arguments:\n",
    "    dA -- post-activation gradient for current layer l \n",
    "    cache -- tuple of values (linear_cache, activation_cache) we store for computing backward propagation efficiently\n",
    "    activation -- the activation to be used in this layer, stored as a text string: \"sigmoid\" or \"relu\"\n",
    "    \n",
    "    Returns:\n",
    "    dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev\n",
    "    dW -- Gradient of the cost with respect to W (current layer l), same shape as W\n",
    "    db -- Gradient of the cost with respect to b (current layer l), same shape as b\n",
    "    \"\"\"\n",
    "    linear_cache, activation_cache = cache\n",
    "    \n",
    "    if activation == \"relu\":\n",
    "        dZ = relu_backward(dA, activation_cache)\n",
    "        dA_prev, dW, db = linear_backward(dZ, linear_cache)\n",
    "        \n",
    "    elif activation == \"sigmoid\":\n",
    "        dZ = sigmoid_backward(dA, activation_cache)\n",
    "        dA_prev, dW, db = linear_backward(dZ, linear_cache)\n",
    "    \n",
    "    return dA_prev, dW, db"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.3 - L-Model Backward  <a name=\"6.3\"></a>\n",
    "\n",
    "Figure 5 below shows the backward pass. \n",
    "\n",
    "\n",
    "<img src=\"images/mn_backward.png\" style=\"width:450px;height:300px;\">\n",
    "\n",
    "**Figure 5** : Backward pass  \n",
    "\n",
    "**Initializing backpropagation**:\n",
    "\n",
    "To backpropagate through this network, we know that the output is, \n",
    "\n",
    "$A^{[L]} = \\sigma(Z^{[L]})$ . We need to compute `dAL` $= \\frac{\\partial \\mathcal{L}}{\\partial A^{[L]}}$.\n",
    "To do so, use this formula:\n",
    "\n",
    "-$\\frac{y}{a}+\\frac{(1-y)}{(1-a)}$\n",
    "\n",
    "```python\n",
    "dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL)) # derivative of cost with respect to AL\n",
    "```\n",
    "\n",
    "Then we can use this post-activation gradient `dAL` to keep going backward. As seen in Figure 5, we can now feed in `dAL` into the LINEAR->SIGMOID backward function. After that, we will have to use a `for` loop to iterate through all the other layers using the LINEAR->RELU backward function. We will store each dA, dW, and db in the grads dictionary. We will use this formula : \n",
    "\n",
    "$$grads[\"dW\" + str(l)] = dW^{[l]}\\tag{15} $$\n",
    "\n",
    "For example, for $l=3$ this would store $dW^{[l]}$ in `grads[\"dW3\"]`.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "def L_model_backward(AL, Y, caches):\n",
    "    \"\"\"\n",
    "    Implement the backward propagation for the [LINEAR->RELU] * (L-1) -> LINEAR -> SIGMOID group\n",
    "    \n",
    "    Arguments:\n",
    "    AL -- probability vector, output of the forward propagation (L_model_forward())\n",
    "    Y -- true \"label\" vector (containing 0 if non-cat, 1 if cat)\n",
    "    caches -- list of caches containing:\n",
    "                every cache of linear_activation_forward() with \"relu\" (it's caches[l], for l in range(L-1) i.e l = 0...L-2)\n",
    "                the cache of linear_activation_forward() with \"sigmoid\" (it's caches[L-1])\n",
    "    \n",
    "    Returns:\n",
    "    grads -- A dictionary with the gradients\n",
    "             grads[\"dA\" + str(l)] = ... \n",
    "             grads[\"dW\" + str(l)] = ...\n",
    "             grads[\"db\" + str(l)] = ... \n",
    "    \"\"\"\n",
    "    grads = {}\n",
    "    L = len(caches) # the number of layers\n",
    "    m = AL.shape[1]\n",
    "    Y = Y.reshape(AL.shape) # after this line, Y is the same shape as AL\n",
    "    \n",
    "    # Initializing the backpropagation\n",
    "    dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))\n",
    "    \n",
    "    # Lth layer (SIGMOID -> LINEAR) gradients. \n",
    "    current_cache = caches[L-1]\n",
    "    grads[\"dA\" + str(L-1)], grads[\"dW\" + str(L)], grads[\"db\" + str(L)] = linear_activation_backward(dAL, current_cache, 'sigmoid')\n",
    "\n",
    "    \n",
    "    # Loop from l=L-2 to l=0\n",
    "    for l in reversed(range(L-1)):\n",
    "        # lth layer: (RELU -> LINEAR) gradients.\n",
    "        current_cache = caches[l]\n",
    "        dA_prev_temp, dW_temp, db_temp = linear_activation_backward(grads[\"dA\" + str(l + 1)], current_cache, activation = \"relu\")\n",
    "        grads[\"dA\" + str(l)] = dA_prev_temp\n",
    "        grads[\"dW\" + str(l + 1)] = dW_temp\n",
    "        grads[\"db\" + str(l + 1)] = db_temp\n",
    "\n",
    "    return grads"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.4 - Update Parameters <a name=\"6.4\"></a>\n",
    "\n",
    "The function will update the parameters of the model, using gradient descent: \n",
    "\n",
    "$$ W^{[l]} = W^{[l]} - \\alpha \\text{ } dW^{[l]} \\tag{16}$$\n",
    "$$ b^{[l]} = b^{[l]} - \\alpha \\text{ } db^{[l]} \\tag{17}$$\n",
    "\n",
    "where $\\alpha$ is the learning rate. After computing the updated parameters, store them in the parameters dictionary. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "def update_parameters(parameters, grads, learning_rate):\n",
    "    \"\"\"\n",
    "    Update parameters using gradient descent\n",
    "    \n",
    "    Arguments:\n",
    "    parameters -- python dictionary containing your parameters \n",
    "    grads -- python dictionary containing your gradients, output of L_model_backward\n",
    "    \n",
    "    Returns:\n",
    "    parameters -- python dictionary containing your updated parameters \n",
    "                  parameters[\"W\" + str(l)] = ... \n",
    "                  parameters[\"b\" + str(l)] = ...\n",
    "    \"\"\"\n",
    "    \n",
    "    L = len(parameters) // 2 # number of layers in the neural network\n",
    "\n",
    "    for l in range(L):\n",
    "        parameters[\"W\" + str(l+1)] = parameters[\"W\" + str(l+1)]-learning_rate*grads[\"dW\" + str(l+1)]\n",
    "        parameters[\"b\" + str(l+1)] =  parameters[\"b\" + str(l+1)]-learning_rate*grads[\"db\" + str(l+1)]\n",
    "\n",
    "    return parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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