coursera_deeplearning.ai_c1_week4

本周课程要点：

多层神经网络结构
多层神经的前向及后向运算
构建多层神经网络
神经网络中的参数和超参数

学习目标

See deep neural networks as successive blocks put one after each other
Build and train a deep L-layer Neural Network
Analyze matrix and vector dimensions to check neural network implementations.
Understand how to use a cache to pass information from forward propagation to back propagation.
Understand the role of hyperparameters in deep learning

课程笔记

Deep L-layer neural network

神经网络中的一些符号：

$L$: 网络中层的个数
$n^{[l]}$: 网络中第$l$层单元的数量
- 上图中：$n^{[1]}=5$，$n^{[2]}=5$，$n^{[3]}=3$，$n^{[4]}=1$
$a^{[l]}$: 网络中第$l$层的激活函数
- 输入层为$a^{[0]}$
$W^{[l]}$: $z^l$的权重
$b^{[l]}$: $z^l$的偏置

Forward Propagation in a Deep Network

Getting your matrix dimensions right

如何计算深度学习中各层参数的维度？

对于单个输入：

\[ W^{[l]}: (n^{[l]},n^{[l-1]}) \]

\[ b^{[l]}: (n^{[l]},1) \]

\[ dW^{[l]}: (n^{[l]},n^{[l-1]}) \]

\[ db^{[l]}: (n^{[l]},1) \]

对于$m$个输入：

$ W^{[l]}, dW^{[l]}, b^{[l]}, db^{[l]} $ 的维度和上面是一样的。

$ Z^{[l]},A{[l]}: (n^{[l]},m) $

$ dZ^{[l]},dA{[l]}: (n^{[l]},m) $

Why deep representations?

为什么深度网络可能会很有效？

如果我们搭建一个系统用于“面部识别 ”或者“面部检测”，输入一张面部图片，那么神经网络的第一层可以被认为是一个特征检测器或边缘检测器。在这个例子中，具有 20个隐藏神经元的神经网络，且这20个隐藏神经元通过这些小方块可视化。这个微型可视化图，表示一个隐藏神经元正在试图找出在DMH (Depth-MHI-HOG)中该方向的边缘位置。形式上，神经网络的第一层就好比，看一张图片，并尝试找出这张图片的边缘。找出这张图片的边缘，通过将像素分组来形成边缘的方法。然后，可以取消检测边缘，并将边缘组合在一起，以形成面部的一部分。例如，可能有一个低神经元试图看看它是否发现了一个眼睛，或者一个不同的神经元试图找到鼻子的一部分。通过把大量的边缘放置在一起，可以开始检测面部的不同部位。最后，通过将面部的不同部位，如眼睛、鼻子、耳朵或下巴组合在一起，可以尝试识别或检测不同类型的面部。

直观地，可以将神经网络的浅层，看作是简单的检测函数，如 (检测) 边缘，然后，在神经网络的后一层，将它们组合在一起，以便它可以学习，更多和更复杂的功能。这种可视化的一个技术细节，边缘检测器检测图像中相对较小的区域，也许是像这样的非常小的区域，然后面部检测器可能会看到更大的图像区域。从中获取的进一步的关键信息，仅仅是找到像边缘这样的简单事物。然后构建它们，将它们组合在一起，以检测更复杂的事物。然后 (再次) 将它们组合在一起，以找到更复杂的事物。这种由简单到复杂的“分层表示”或“组合表示”不仅适用于图像和面部识别，也适用于其它类型的数据。

Circuit theory and deep learning

Informally: There are functions you can compute with a “small” L-layer deep neural network that shallower networks require exponentially more hidden units to compute.

Building blocks of deep neural networks

下面是清晰版。

上面这个图，包含了神经网络前向计算和后向计算所需的全部变量。

Forward and Backward Propagation

Parameters vs Hyperparameters

参数parameters: $ W^{[1]},b{[1]},W^{[2]},b{[2]},W^{[3]},b{[3]}, $

超参数Hyperparameters:

\[ \begin{aligned} & learning\,rate: \alpha \\ & iterations \\ & hidden\,layer\,num:L \\ & units\,num \,of\,hidden\,layer: n^{[l]} \\ & choice\,of\,activation\,function \end{aligned} \]

What does this have to do with the brain?

吴老师对深度学习和大脑的关系的观点：

我确实认为计算机视觉方向相比于其他深度学习影响的学科，得到了更多的灵感得到了更多的灵感但是我个人比以前更少地使用人脑进行对比了。

编程练习

Building your Deep Neural Network: Step by Step
Deep Neural Network - Application

Building your Deep Neural Network: Step by Step

Welcome to your week 4 assignment (part 1 of 2)! You have previously trained a 2-layer Neural Network (with a single hidden layer). This week, you will build a deep neural network, with as many layers as you want!

In this notebook, you will implement all the functions required to build a deep neural network.
In the next assignment, you will use these functions to build a deep neural network for image classification.

After this assignment you will be able to: - Use non-linear units like ReLU to improve your model - Build a deeper neural network (with more than 1 hidden layer) - Implement an easy-to-use neural network class

Notation: - Superscript $[l]$ denotes a quantity associated with the $l^{th}$ layer. - Example: $a^{[L]}$ is the $L^{th}$ layer activation. $W^{[L]}$ and $b^{[L]}$ are the $L^{th}$ layer parameters. - Superscript $(i)$ denotes a quantity associated with the $i^{th}$ example. - Example: $x^{(i)}$ is the $i^{th}$ training example. - Lowerscript $i$ denotes the $i^{th}$ entry of a vector. - Example: $a^{[l]}_i$ denotes the $i^{th}$ entry of the $l^{th}$ layer's activations).

Let's get started!

1 - Packages

Let's first import all the packages that you will need during this assignment. - numpy is the main package for scientific computing with Python. - matplotlib is a library to plot graphs in Python. - dnn_utils provides some necessary functions for this notebook. - testCases provides some test cases to assess the correctness of your functions - np.random.seed(1) is used to keep all the random function calls consistent. It will help us grade your work. Please don't change the seed.

import numpy as np
import h5py
import matplotlib.pyplot as plt
from testCases_v4 import *
from dnn_utils_v2 import sigmoid, sigmoid_backward, relu, relu_backward

get_ipython().magic('matplotlib inline')
plt.rcParams['figure.figsize'] = (5.0, 4.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'

get_ipython().magic('load_ext autoreload')
get_ipython().magic('autoreload 2')

np.random.seed(1)

2 - Outline of the Assignment

To build your neural network, you will be implementing several "helper functions". These helper functions will be used in the next assignment to build a two-layer neural network and an L-layer neural network. Each small helper function you will implement will have detailed instructions that will walk you through the necessary steps. Here is an outline of this assignment, you will:

Initialize the parameters for a two-layer network and for an $L$-layer neural network.
Implement the forward propagation module (shown in purple in the figure below).
- Complete the LINEAR part of a layer's forward propagation step (resulting in $Z^{[l]}$).
- We give you the ACTIVATION function (relu/sigmoid).
- Combine the previous two steps into a new [LINEAR->ACTIVATION] forward function.
- 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 you a new L_model_forward function.
Compute the loss.
Implement the backward propagation module (denoted in red in the figure below).
- Complete the LINEAR part of a layer's backward propagation step.
- We give you the gradient of the ACTIVATE function (relu_backward/sigmoid_backward)
- Combine the previous two steps into a new [LINEAR->ACTIVATION] backward function.
- Stack [LINEAR->RELU] backward L-1 times and add [LINEAR->SIGMOID] backward in a new L_model_backward function
Finally update the parameters.

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.

3 - Initialization

You will write two helper functions that will initialize the parameters for your model. 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.

3.1-2-layer Neural Network

Exercise: Create and initialize the parameters of the 2-layer neural network.

Instructions: - The model's structure is: LINEAR -> RELU -> LINEAR -> SIGMOID. - Use random initialization for the weight matrices. Use np.random.randn(shape)*0.01 with the correct shape. - Use zero initialization for the biases. Use np.zeros(shape).

#GRADED FUNCTION: initialize_parameters
def initialize_parameters(n_x, n_h, n_y):
    """
    Argument:
    n_x -- size of the input layer
    n_h -- size of the hidden layer
    n_y -- size of the output layer
    
    Returns:
    parameters -- python dictionary containing your parameters:
                    W1 -- weight matrix of shape (n_h, n_x)
                    b1 -- bias vector of shape (n_h, 1)
                    W2 -- weight matrix of shape (n_y, n_h)
                    b2 -- bias vector of shape (n_y, 1)
    """
    
    np.random.seed(1)
    
    ### START CODE HERE ### (≈ 4 lines of code)
    W1 = np.random.randn(n_h, n_x) * 0.01
    b1 = np.zeros((n_h, 1))
    W2 = np.random.randn(n_y, n_h) * 0.01
    b2 = np.zeros((n_y, 1))
    ### END CODE HERE ###
    
    assert(W1.shape == (n_h, n_x))
    assert(b1.shape == (n_h, 1))
    assert(W2.shape == (n_y, n_h))
    assert(b2.shape == (n_y, 1))
    
    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2}
    
    return parameters

3.2 - L-layer Neural Network

The initialization for a deeper L-layer neural network is more complicated because there are many more weight matrices and bias vectors. When completing the initialize_parameters_deep, you should make sure that your dimensions match between each layer. Recall that $n^{[l]}$ is the number of units in layer $l$. Thus for example if the size of our input $X$ is $(12288, 209)$ (with $m=209$ examples) then:

	Shape of W	Shape of b	Activation	Shape of Activation
Layer 1	$(n^{[1]},12288)$	$(n^{[1]},1)$	$Z^{[1]} = W^{[1]} X + b^{[1]} $	$(n^{[1]},209)$
Layer 2	$(n^{[2]}, n^{[1]})$	$(n^{[2]},1)$	$Z^{[2]} = W^{[2]} A^{[1]} + b^{[2]}$	$(n^{[2]}, 209)$
$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$
Layer L-1	$(n^{[L-1]}, n^{[L-2]})$	$(n^{[L-1]}, 1)$	$Z^{[L-1]} = W^{[L-1]} A^{[L-2]} + b^{[L-1]}$	$(n^{[L-1]}, 209)$
Layer L	$(n^{[L]}, n^{[L-1]})$	$(n^{[L]}, 1)$	$Z^{[L]} = W^{[L]} A^{[L-1]} + b^{[L]}$	$(n^{[L]}, 209)$

Remember that when we compute $W X + b$ in python, it carries out broadcasting. For example, if:

\[ W = \begin{bmatrix} j & k & l\\ m & n & o \\ p & q & r \end{bmatrix}\;\;\; X = \begin{bmatrix} a & b & c\\ d & e & f \\ g & h & i \end{bmatrix} \;\;\; b =\begin{bmatrix} s \\ t \\ u \end{bmatrix} \]

Then $WX + b$ will be:

\[ WX + b = \begin{bmatrix} (ja + kd + lg) + s & (jb + ke + lh) + s & (jc + kf + li)+ s\\ (ma + nd + og) + t & (mb + ne + oh) + t & (mc + nf + oi) + t\\ (pa + qd + rg) + u & (pb + qe + rh) + u & (pc + qf + ri)+ u \end{bmatrix} \]

Exercise: Implement initialization for an L-layer Neural Network.

Instructions: - The model's structure is [LINEAR -> RELU] $ $ (L-1) -> LINEAR -> SIGMOID. I.e., it has $L-1$ layers using a ReLU activation function followed by an output layer with a sigmoid activation function. - Use random initialization for the weight matrices. Use np.random.randn(shape) * 0.01. - Use zeros initialization for the biases. Use np.zeros(shape). - We will store $n^{[l]}$, the number of units in different layers, in a variable layer_dims. For example, the layer_dims for the "Planar Data classification model" from last week would have been [2,4,1]: There were two inputs, one hidden layer with 4 hidden units, and an output layer with 1 output unit. Thus means W1's shape was (4,2), b1 was (4,1), W2 was (1,4) and b2 was (1,1). Now you will generalize this to $L$ layers! - Here is the implementation for $L=1$ (one layer neural network). It should inspire you to implement the general case (L-layer neural network).

1
2
3

if L == 1:
    parameters["W" + str(L)] = np.random.randn(layer_dims[1], layer_dims[0]) * 0.01
    parameters["b" + str(L)] = np.zeros((layer_dims[1], 1))

#GRADED FUNCTION: initialize_parameters_deep

def initialize_parameters_deep(layer_dims):
    """
    Arguments:
    layer_dims -- python array (list) containing the dimensions of each layer in our network
    
    Returns:
    parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":
                    Wl -- weight matrix of shape (layer_dims[l], layer_dims[l-1])
                    bl -- bias vector of shape (layer_dims[l], 1)
    """
    
    np.random.seed(3)
    parameters = {}
    L = len(layer_dims)            # number of layers in the network
    print("layer dims " + str(layer_dims))

    for l in range(1, L):
        ### START CODE HERE ### (≈ 2 lines of code)
        parameters['W' + str(l)] = np.random.randn(layer_dims[l], layer_dims[l-1]) * 0.01
        parameters['b' + str(l)] = np.zeros((layer_dims[l], 1))
        ### END CODE HERE ###
        
        assert(parameters['W' + str(l)].shape == (layer_dims[l], layer_dims[l-1]))
        assert(parameters['b' + str(l)].shape == (layer_dims[l], 1))

        
    return parameters

4 - Forward propagation module

4.1 - Linear Forward

Now that you have initialized your parameters, you will do the forward propagation module. You will start by implementing some basic functions that you will use later when implementing the model. You will complete three functions in this order:

LINEAR
LINEAR -> ACTIVATION where ACTIVATION will be either ReLU or Sigmoid.
[LINEAR -> RELU] $\times$ (L-1) -> LINEAR -> SIGMOID (whole model)

The linear forward module (vectorized over all the examples) computes the following equations:

\[Z^{[l]} = W^{[l]}A^{[l-1]} +b^{[l]}\]

where $A^{[0]} = X$.

Exercise: Build the linear part of forward propagation.

Reminder: The mathematical representation of this unit is $Z^{[l]} = W^{[l]}A^{[l-1]} +b^{[l]}$. You may also find np.dot() useful. If your dimensions don't match, printing W.shape may help.

# GRADED FUNCTION: linear_forward

def linear_forward(A, W, b):
    """
    Implement the linear part of a layer's forward propagation.

    Arguments:
    A -- activations from previous layer (or input data): (size of previous layer, number of examples)
    W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
    b -- bias vector, numpy array of shape (size of the current layer, 1)

    Returns:
    Z -- the input of the activation function, also called pre-activation parameter 
    cache -- a python dictionary containing "A", "W" and "b" ; stored for computing the backward pass efficiently
    """
    
    ### START CODE HERE ### (≈ 1 line of code)
    Z = np.dot(W,A) + b
    ### END CODE HERE ###
    
    assert(Z.shape == (W.shape[0], A.shape[1]))
    cache = (A, W, b)
    
    return Z, cache

4.2 - Linear-Activation Forward

In this notebook, you will use two activation functions:

Sigmoid: $\sigma(Z) = \sigma(W A + b) = \frac{1}{ 1 + e^{-(W A + b)}}$. We have provided you with 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). To use it you could just call:
```
  A, activation_cache = sigmoid(Z)
```
ReLU: The mathematical formula for ReLu is $A = RELU(Z) = max(0, Z)$. We have provided you with 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). To use it you could just call:
```
  A, activation_cache = relu(Z)
```

For more convenience, you are going to group two functions (Linear and Activation) into one function (LINEAR->ACTIVATION). Hence, you will implement a function that does the LINEAR forward step followed by an ACTIVATION forward step.

Exercise: Implement the forward propagation of the LINEAR->ACTIVATION layer. Mathematical relation is: $A^{[l]} = g(Z^{[l]}) = g(W^{[l]}A^{[l-1]} +b^{[l]})$ where the activation "g" can be sigmoid() or relu(). Use linear_forward() and the correct activation function.

#GRADED FUNCTION: linear_activation_forward

def linear_activation_forward(A_prev, W, b, activation):
    """
    Implement the forward propagation for the LINEAR->ACTIVATION layer

    Arguments:
    A_prev -- activations from previous layer (or input data): (size of previous layer, number of examples)
    W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
    b -- bias vector, numpy array of shape (size of the current layer, 1)
    activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"

    Returns:
    A -- the output of the activation function, also called the post-activation value 
    cache -- a python dictionary containing "linear_cache" and "activation_cache";
             stored for computing the backward pass efficiently
    """
    
    Z, linear_cache = linear_forward(A_prev, W, b)

    if activation == "sigmoid":
        # Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
        ### START CODE HERE ### (≈ 2 lines of code)
        Z, linear_cache = linear_forward(A_prev, W, b)
        A, activation_cache = sigmoid(Z)
        ### END CODE HERE ###
    
    elif activation == "relu":
        # Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
        ### START CODE HERE ### (≈ 2 lines of code)
        Z, linear_cache = linear_forward(A_prev, W, b)
        A, activation_cache = relu(Z)
        ### END CODE HERE ###
    
    assert (A.shape == (W.shape[0], A_prev.shape[1]))
    cache = (linear_cache, activation_cache)

    return A, cache

Note: In deep learning, the "[LINEAR->ACTIVATION]" computation is counted as a single layer in the neural network, not two layers.

L-Layer Model

For even more convenience when implementing the $L$-layer Neural Net, you 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.

$**Figure 2** : *[LINEAR -> RELU] $\times$ (L-1) -> LINEAR -> SIGMOID* model$

Exercise: Implement the forward propagation of the above model.

Instruction: 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 $\hat{Y}$.)

Tips: - Use the functions you had previously written - Use a for loop to replicate [LINEAR->RELU] (L-1) times - Don't forget to keep track of the caches in the "caches" list. To add a new value c to a list, you can use list.append(c).

#GRADED FUNCTION: L_model_forward

def L_model_forward(X, parameters):
    """
    Implement forward propagation for the [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID computation
    
    Arguments:
    X -- data, numpy array of shape (input size, number of examples)
    parameters -- output of initialize_parameters_deep()
    
    Returns:
    AL -- last post-activation value
    caches -- list of caches containing:
                every cache of linear_activation_forward() (there are L-1 of them, indexed from 0 to L-1)
    """

    caches = []
    A = X
    L = len(parameters) // 2                  # number of layers in the neural network

    # Implement [LINEAR -> RELU]*(L-1). Add "cache" to the "caches" list.
    for l in range(1, L):
        A_prev = A 
        W = parameters['W' + str(l)]
        b = parameters['b' + str(l)]
        print(str(W.shape) + ", W" + str(l) + " = " + str(W))
        ### START CODE HERE ### (≈ 2 lines of code)
        A, cache = linear_activation_forward(A_prev, W, b, "relu")
        caches.append(cache)
        print("A shape " + str(A.shape))
        ### END CODE HERE ###
    
    # Implement LINEAR -> SIGMOID. Add "cache" to the "caches" list.
    ### START CODE HERE ### (≈ 2 lines of code)
    W = parameters['W' + str(L)]
    b = parameters['b' + str(L)]
    AL, cache = linear_activation_forward(A, W, b, "sigmoid")
    caches.append(cache)
    print(str(W.shape) + ", W" + str(L) + " = " + str(W))
    print("AL shape " + str(AL.shape))
    ### END CODE HERE ###
    assert(AL.shape == (1,X.shape[1]))
            
    return AL, caches

Great! Now you have a full forward propagation that takes the input X and outputs a row vector $ A^{[L]} $ containing your predictions. It also records all intermediate values in "caches". Using $ A^{[L]} $, you can compute the cost of your predictions.

5 - Cost function

Now you will implement forward and backward propagation. You need to compute the cost, because you want to check if your model is actually learning.

Exercise: Compute the cross-entropy cost $J$, using the following formula: \[ -\frac{1}{m} \sum\limits_{i = 1}^{m} (y^{(i)}\log\left(a^{[L] (i)}\right) + (1-y^{(i)})\log\left(1- a^{[L](i)}\right)) \]

#GRADED FUNCTION: compute_cost

def compute_cost(AL, Y):
    """
    Implement the cost function defined by equation (7).

    Arguments:
    AL -- probability vector corresponding to your label predictions, shape (1, number of examples)
    Y -- true "label" vector (for example: containing 0 if non-cat, 1 if cat), shape (1, number of examples)

    Returns:
    cost -- cross-entropy cost
    """
    
    m = Y.shape[1]

    # Compute loss from aL and y.
    ### START CODE HERE ### (≈ 1 lines of code)
    cost = - np.sum(Y * np.log(AL) + (1-Y) * np.log(1-AL))/m
    ### END CODE HERE ###
    
    cost = np.squeeze(cost)      # To make sure your cost's shape is what we expect (e.g. this turns [[17]] into 17).
    assert(cost.shape == ())
    
    return cost

6 - Backward propagation module

Just like with forward propagation, you will implement helper functions for backpropagation. Remember that back propagation is used to calculate the gradient of the loss function with respect to the parameters.

Reminder:

**Figure 3** : Forward and Backward propagation for *LINEAR->RELU->LINEAR->SIGMOID*

The purple blocks represent the forward propagation, and the red blocks represent the backward propagation.

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:

\[\frac{d \mathcal{L}(a^{[2]},y)}{dz^{[1]}} = \frac{d\mathcal{L}(a^{[2]},y)}{da^{[2]}}\frac{da^{[2]}}{dz^{[2]}}\frac{dz^{[2]}}{da^{[1]}}\frac{da^{[1]}}{dz^{[1]}} \]

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.

Equivalently, in order to calculate the gradient $db^{[1]} = \frac{\partial L}{\partial b^{[1]}}$, you use the previous chain rule and you do $db^{[1]} = dz^{[1]} \times \frac{\partial z^{[1]} }{\partial b^{[1]}}$.

This is why we talk about backpropagation.

Now, similar to forward propagation, you are going to build the backward propagation in three steps: - LINEAR backward - LINEAR -> ACTIVATION backward where ACTIVATION computes the derivative of either the ReLU or sigmoid activation - [LINEAR -> RELU] $\times$ (L-1) -> LINEAR -> SIGMOID backward (whole model)

6.1 - Linear backward

For layer $l$, the linear part is: $Z^{[l]} = W^{[l]} A^{[l-1]} + b^{[l]}$ (followed by an activation).

Suppose you have already calculated the derivative $dZ^{[l]} = \frac{\partial \mathcal{L} }{\partial Z^{[l]}}$. You want to get $(dW^{[l]}, db^{[l]} dA^{[l-1]})$.

The three outputs $(dW^{[l]}, db^{[l]}, dA^{[l]})$ are computed using the input $dZ^{[l]}$.Here are the formulas you need: \[ dW^{[l]} = \frac{\partial \mathcal{L} }{\partial W^{[l]}} = \frac{1}{m} dZ^{[l]} A^{[l-1] T} \]

\[ db^{[l]} = \frac{\partial \mathcal{L} }{\partial b^{[l]}} = \frac{1}{m} \sum_{i = 1}^{m} dZ^{[l](i)}\]

\[ dA^{[l-1]} = \frac{\partial \mathcal{L} }{\partial A^{[l-1]}} = W^{[l] T} dZ^{[l]} \]

Exercise: Use the 3 formulas above to implement linear_backward().

#GRADED FUNCTION: linear_backward

def linear_backward(dZ, cache):
    """
    Implement the linear portion of backward propagation for a single layer (layer l)

    Arguments:
    dZ -- Gradient of the cost with respect to the linear output (of current layer l)
    cache -- tuple of values (A_prev, W, b) coming from the forward propagation in the current layer

    Returns:
    dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
    dW -- Gradient of the cost with respect to W (current layer l), same shape as W
    db -- Gradient of the cost with respect to b (current layer l), same shape as b
    """
    A_prev, W, b = cache
    m = A_prev.shape[1]

    ### START CODE HERE ### (≈ 3 lines of code)
    dW = np.dot(dZ,A_prev.T)/m
    db = np.sum(dZ, axis=1,keepdims=True)/m
    dA_prev = np.dot(W.T,dZ)
    ### END CODE HERE ###
    
    assert (dA_prev.shape == A_prev.shape)
    assert (dW.shape == W.shape)
    assert (db.shape == b.shape)
    
    return dA_prev, dW, db

6.2 - Linear-Activation backward

Next, you will create a function that merges the two helper functions: linear_backward and the backward step for the activation linear_activation_backward.

To help you implement linear_activation_backward, we provided two backward functions: - sigmoid_backward: Implements the backward propagation for SIGMOID unit. You can call it as follows:

1	dZ = sigmoid_backward(dA, activation_cache)

relu_backward: Implements the backward propagation for RELU unit. You can call it as follows:

1	dZ = relu_backward(dA, activation_cache)

If $g(.)$ is the activation function, sigmoid_backward and relu_backward compute \[dZ^{[l]} = dA^{[l]} * g'(Z^{[l]}) \]

Exercise: Implement the backpropagation for the LINEAR->ACTIVATION layer.

#GRADED FUNCTION: linear_activation_backward

def linear_activation_backward(dA, cache, activation):
    """
    Implement the backward propagation for the LINEAR->ACTIVATION layer.
    
    Arguments:
    dA -- post-activation gradient for current layer l 
    cache -- tuple of values (linear_cache, activation_cache) we store for computing backward propagation efficiently
    activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"
    
    Returns:
    dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
    dW -- Gradient of the cost with respect to W (current layer l), same shape as W
    db -- Gradient of the cost with respect to b (current layer l), same shape as b
    """
    linear_cache, activation_cache = cache
    
    if activation == "relu":
        ### START CODE HERE ### (≈ 2 lines of code)
        dZ = relu_backward(dA, activation_cache)
        dA_prev, dW, db = linear_backward(dZ, linear_cache)
        ### END CODE HERE ###
        
    elif activation == "sigmoid":
        ### START CODE HERE ### (≈ 2 lines of code)
        dZ = sigmoid_backward(dA, activation_cache)
        dA_prev, dW, db = linear_backward(dZ, linear_cache)
        ### END CODE HERE ###
    
    return dA_prev, dW, db

6.3 - L-Model Backward

Now you will implement the backward function for the whole network. Recall that when you implemented the L_model_forward function, at each iteration, you stored a cache which contains (X,W,b, and z). In the back propagation module, you will use those variables to compute the gradients. Therefore, in the L_model_backward function, you will iterate through all the hidden layers backward, starting from layer $L$. On each step, you will use the cached values for layer $l$ to backpropagate through layer $l$. Figure 5 below shows the backward pass.

** Initializing backpropagation**: To backpropagate through this network, we know that the output is, $A^{[L]} = \sigma(Z^{[L]})$. Your code thus needs to compute dAL $= \frac{\partial \mathcal{L}}{\partial A^{[L]}}$. To do so, use this formula (derived using calculus which you don't need in-depth knowledge of):

1	dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL)) # derivative of cost with respect to AL

You can then use this post-activation gradient dAL to keep going backward. As seen in Figure 5, you can now feed in dAL into the LINEAR->SIGMOID backward function you implemented (which will use the cached values stored by the L_model_forward function). After that, you will have to use a for loop to iterate through all the other layers using the LINEAR->RELU backward function. You should store each dA, dW, and db in the grads dictionary. To do so, use this formula :

\[grads["dW" + str(l)] = dW^{[l]} \]

For example, for $l=3$ this would store $dW^{[l]}$ in grads["dW3"].

Exercise: Implement backpropagation for the [LINEAR->RELU] $\times$ (L-1) -> LINEAR -> SIGMOID model.

#In[22]:

#GRADED FUNCTION: L_model_backward

def L_model_backward(AL, Y, caches):
    """
    Implement the backward propagation for the [LINEAR->RELU] * (L-1) -> LINEAR -> SIGMOID group
    
    Arguments:
    AL -- probability vector, output of the forward propagation (L_model_forward())
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat)
    caches -- list of caches containing:
                every cache of linear_activation_forward() with "relu" (it's caches[l], for l in range(L-1) i.e l = 0...L-2)
                the cache of linear_activation_forward() with "sigmoid" (it's caches[L-1])
    
    Returns:
    grads -- A dictionary with the gradients
             grads["dA" + str(l)] = ... 
             grads["dW" + str(l)] = ...
             grads["db" + str(l)] = ... 
    """
    grads = {}
    L = len(caches) # the number of layers
    m = AL.shape[1]
    Y = Y.reshape(AL.shape) # after this line, Y is the same shape as AL
    
    # Initializing the backpropagation
    ### START CODE HERE ### (1 line of code)
    dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL)) # derivative of cost with respect to AL
    ### END CODE HERE ###
    
    # Lth layer (SIGMOID -> LINEAR) gradients. Inputs: "dAL, current_cache". Outputs: "grads["dAL-1"], grads["dWL"], grads["dbL"]
    ### START CODE HERE ### (approx. 2 lines)
    #grads["dA" + str(L-1)], grads["dW" + str(L)], grads["db" + str(L)] = None
    current_cache = caches[L-1]
    grads["dA" + str(L-1)], grads["dW" + str(L)], grads["db" + str(L)] = linear_activation_backward(dAL, current_cache, "sigmoid")
    ### END CODE HERE ###
    
    # Loop from l=L-2 to l=0
    for l in reversed(range(L-1)):
        # lth layer: (RELU -> LINEAR) gradients.
        # Inputs: "grads["dA" + str(l + 1)], current_cache". Outputs: "grads["dA" + str(l)] , grads["dW" + str(l + 1)] , grads["db" + str(l + 1)] 
        ### START CODE HERE ### (approx. 5 lines)
        current_cache = caches[l]
        dA_prev_temp, dW_temp, db_temp = linear_activation_backward(grads["dA" + str(l+1)], current_cache, "relu")
        grads["dA" + str(l)] = dA_prev_temp
        grads["dW" + str(l + 1)] = dW_temp
        grads["db" + str(l + 1)] = db_temp
        ### END CODE HERE ###

    return grads

6.4 - Update Parameters

In this section you will update the parameters of the model, using gradient descent:

\[ W^{[l]} = W^{[l]} - \alpha \text{ } dW^{[l]} \]

\[ b^{[l]} = b^{[l]} - \alpha \text{ } db^{[l]} \]

where $\alpha$ is the learning rate. After computing the updated parameters, store them in the parameters dictionary.

Exercise: Implement update_parameters() to update your parameters using gradient descent.

Instructions: Update parameters using gradient descent on every $W^{[l]}$ and $b^{[l]}$ for $l = 1, 2, ..., L$.

#GRADED FUNCTION: update_parameters

def update_parameters(parameters, grads, learning_rate):
    """
    Update parameters using gradient descent
    
    Arguments:
    parameters -- python dictionary containing your parameters 
    grads -- python dictionary containing your gradients, output of L_model_backward
    
    Returns:
    parameters -- python dictionary containing your updated parameters 
                  parameters["W" + str(l)] = ... 
                  parameters["b" + str(l)] = ...
    """
    
    L = len(parameters) // 2 # number of layers in the neural network

    # Update rule for each parameter. Use a for loop.
    ### START CODE HERE ### (≈ 3 lines of code)
    for l in range(L):
        parameters["W" + str(l+1)] = parameters["W" + str(l+1)] - (learning_rate * grads["dW" + str(l+1)])
        parameters["b" + str(l+1)] = parameters["b" + str(l+1)] - (learning_rate * grads["db" + str(l+1)])
    ### END CODE HERE ###
    return parameters

7 - Conclusion

Congrats on implementing all the functions required for building a deep neural network!

We know it was a long assignment but going forward it will only get better. The next part of the assignment is easier.

In the next assignment you will put all these together to build two models: - A two-layer neural network - An L-layer neural network

You will in fact use these models to classify cat vs non-cat images!

Deep Neural Network - Application

Deep Neural Network for Image Classification: Application

When you finish this, you will have finished the last programming assignment of Week 4, and also the last programming assignment of this course!

You will use use the functions you'd implemented in the previous assignment to build a deep network, and apply it to cat vs non-cat classification. Hopefully, you will see an improvement in accuracy relative to your previous logistic regression implementation.

After this assignment you will be able to: - Build and apply a deep neural network to supervised learning.

Let's get started!

1 - Packages

Let's first import all the packages that you will need during this assignment. - numpy is the fundamental package for scientific computing with Python. - matplotlib is a library to plot graphs in Python. - h5py is a common package to interact with a dataset that is stored on an H5 file. - PIL and scipy are used here to test your model with your own picture at the end. - dnn_app_utils provides the functions implemented in the "Building your Deep Neural Network: Step by Step" assignment to this notebook. - np.random.seed(1) is used to keep all the random function calls consistent. It will help us grade your work.

import time
import numpy as np
import h5py
import matplotlib.pyplot as plt
import scipy
from PIL import Image
from scipy import ndimage
from dnn_app_utils_v3 import *

get_ipython().magic('matplotlib inline')
plt.rcParams['figure.figsize'] = (5.0, 4.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'

get_ipython().magic('load_ext autoreload')
get_ipython().magic('autoreload 2')

np.random.seed(1)

2 - Dataset

You will use the same "Cat vs non-Cat" dataset as in "Logistic Regression as a Neural Network" (Assignment 2). The model you had built had 70% test accuracy on classifying cats vs non-cats images. Hopefully, your new model will perform a better!

Problem Statement: You are given a dataset ("data.h5") containing: - a training set of m_train images labelled as cat (1) or non-cat (0) - a test set of m_test images labelled as cat and non-cat - each image is of shape (num_px, num_px, 3) where 3 is for the 3 channels (RGB).

Let's get more familiar with the dataset. Load the data by running the cell below.

1	train_x_orig, train_y, test_x_orig, test_y, classes = load_data()

The following code will show you an image in the dataset. Feel free to change the index and re-run the cell multiple times to see other images.

#Example of a picture
index = 50
plt.imshow(train_x_orig[index])
print ("y = " + str(train_y[0,index]) + ". It's a " + classes[train_y[0,index]].decode("utf-8") +  " picture.")

# Explore your dataset 
m_train = train_x_orig.shape[0]
num_px = train_x_orig.shape[1]
m_test = test_x_orig.shape[0]

print ("Number of training examples: " + str(m_train))
print ("Number of testing examples: " + str(m_test))
print ("Each image is of size: (" + str(num_px) + ", " + str(num_px) + ", 3)")
print ("train_x_orig shape: " + str(train_x_orig.shape))
print ("train_y shape: " + str(train_y.shape))
print ("test_x_orig shape: " + str(test_x_orig.shape))
print ("test_y shape: " + str(test_y.shape))

#output
Number of training examples: 209 
Number of testing examples: 50 
Each image is of size: (64, 64, 3) 
train_x_orig shape: (209, 64, 64, 3) 
train_y shape: (1, 209) 
test_x_orig shape: (50, 64, 64, 3) 
test_y shape: (1, 50)

As usual, you reshape and standardize the images before feeding them to the network. The code is given in the cell below.

# Reshape the training and test examples 
train_x_flatten = train_x_orig.reshape(train_x_orig.shape[0], -1).T   # The "-1" makes reshape flatten the remaining dimensions
test_x_flatten = test_x_orig.reshape(test_x_orig.shape[0], -1).T

# Standardize data to have feature values between 0 and 1.
train_x = train_x_flatten/255.
test_x = test_x_flatten/255.

print ("train_x's shape: " + str(train_x.shape))
print ("test_x's shape: " + str(test_x.shape))

train_x's shape: (12288, 209) 
test_x's shape: (12288, 50)

$12,288$ equals $64 \times 64 \times 3$ which is the size of one reshaped image vector.

3 - Architecture of your model

Now that you are familiar with the dataset, it is time to build a deep neural network to distinguish cat images from non-cat images.

You will build two different models: - A 2-layer neural network - An L-layer deep neural network

You will then compare the performance of these models, and also try out different values for $L$.

Let's look at the two architectures.

3.1 - 2-layer neural network

The model can be summarized as: INPUT -> LINEAR -> RELU -> LINEAR -> SIGMOID -> OUTPUT.

Detailed Architecture of figure 2: - The input is a (64,64,3) image which is flattened to a vector of size $(12288,1)$. - The corresponding vector: $[x_0,x_1,...,x_{12287}]^T$ is then multiplied by the weight matrix $W^{[1]}$ of size $(n^{[1]}, 12288)$. - You then add a bias term and take its relu to get the following vector: $[a_0^{[1]}, a_1^{[1]},..., a_{n^{[1]}-1}^{[1]}]^T$. - You then repeat the same process. - You multiply the resulting vector by $W^{[2]}$ and add your intercept (bias). - Finally, you take the sigmoid of the result. If it is greater than 0.5, you classify it to be a cat.

3.2 - L-layer deep neural network

It is hard to represent an L-layer deep neural network with the above representation. However, here is a simplified network representation:

The model can be summarized as: [LINEAR -> RELU] $\times$ (L-1) -> LINEAR -> SIGMOID

Detailed Architecture of figure 3: - The input is a (64,64,3) image which is flattened to a vector of size (12288,1). - The corresponding vector: $[x_0,x_1,...,x_{12287}]^T$ is then multiplied by the weight matrix $W^{[1]}$ and then you add the intercept $b^{[1]}$. The result is called the linear unit. - Next, you take the relu of the linear unit. This process could be repeated several times for each $(W^{[l]}, b^{[l]})$ depending on the model architecture. - Finally, you take the sigmoid of the final linear unit. If it is greater than 0.5, you classify it to be a cat.

3.3 - General methodology

As usual you will follow the Deep Learning methodology to build the model: 1. Initialize parameters / Define hyperparameters 2. Loop for num_iterations: a. Forward propagation b. Compute cost function c. Backward propagation d. Update parameters (using parameters, and grads from backprop) 4. Use trained parameters to predict labels

Let's now implement those two models!

4 - Two-layer neural network

Question: Use the helper functions you have implemented in the previous assignment to build a 2-layer neural network with the following structure: LINEAR -> RELU -> LINEAR -> SIGMOID. The functions you may need and their inputs are:

def initialize_parameters(n_x, n_h, n_y):
    ...
    return parameters 
def linear_activation_forward(A_prev, W, b, activation):
    ...
    return A, cache
def compute_cost(AL, Y):
    ...
    return cost
def linear_activation_backward(dA, cache, activation):
    ...
    return dA_prev, dW, db
def update_parameters(parameters, grads, learning_rate):
    ...
    return parameters

### CONSTANTS DEFINING THE MODEL ####
n_x = 12288     # num_px * num_px * 3
n_h = 7
n_y = 1
layers_dims = (n_x, n_h, n_y)

#GRADED FUNCTION: two_layer_model

def two_layer_model(X, Y, layers_dims, learning_rate = 0.0075, num_iterations = 3000, print_cost=False):
    """
    Implements a two-layer neural network: LINEAR->RELU->LINEAR->SIGMOID.
    
    Arguments:
    X -- input data, of shape (n_x, number of examples)
    Y -- true "label" vector (containing 0 if cat, 1 if non-cat), of shape (1, number of examples)
    layers_dims -- dimensions of the layers (n_x, n_h, n_y)
    num_iterations -- number of iterations of the optimization loop
    learning_rate -- learning rate of the gradient descent update rule
    print_cost -- If set to True, this will print the cost every 100 iterations 
    
    Returns:
    parameters -- a dictionary containing W1, W2, b1, and b2
    """
    
    np.random.seed(1)
    grads = {}
    costs = []                              # to keep track of the cost
    m = X.shape[1]                           # number of examples
    (n_x, n_h, n_y) = layers_dims
    
    # Initialize parameters dictionary, by calling one of the functions you'd previously implemented
    ### START CODE HERE ### (≈ 1 line of code)
    #parameters = None
    parameters = initialize_parameters(n_x, n_h, n_y)
    ### END CODE HERE ###
    
    # Get W1, b1, W2 and b2 from the dictionary parameters.
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    
    # Loop (gradient descent)

    for i in range(0, num_iterations):
        # Forward propagation: LINEAR -> RELU -> LINEAR -> SIGMOID. Inputs: "X, W1, b1, W2, b2". Output: "A1, cache1, A2, cache2".
        ### START CODE HERE ### (≈ 2 lines of code)
        A1, cache1 = linear_activation_forward(X, W1, b1, "relu")
        A2, cache2 = linear_activation_forward(A1, W2, b2, "sigmoid")
        ### END CODE HERE ###
        
        # Compute cost
        ### START CODE HERE ### (≈ 1 line of code)
        cost = compute_cost(A2, Y)
        ### END CODE HERE ###
        
        # Initializing backward propagation
        dA2 = - (np.divide(Y, A2) - np.divide(1 - Y, 1 - A2))
        
        # Backward propagation. Inputs: "dA2, cache2, cache1". Outputs: "dA1, dW2, db2; also dA0 (not used), dW1, db1".
        ### START CODE HERE ### (≈ 2 lines of code)
        dA1, dW2, db2 = linear_activation_backward(dA2, cache2, "sigmoid")
        dA0, dW1, db1 = linear_activation_backward(dA1, cache1, "relu")
        ### END CODE HERE ###
        
        # Set grads['dWl'] to dW1, grads['db1'] to db1, grads['dW2'] to dW2, grads['db2'] to db2
        grads['dW1'] = dW1
        grads['db1'] = db1
        grads['dW2'] = dW2
        grads['db2'] = db2
        
        # Update parameters.
        ### START CODE HERE ### (approx. 1 line of code)
        parameters = update_parameters(parameters, grads, learning_rate)
        ### END CODE HERE ###

        # Retrieve W1, b1, W2, b2 from parameters
        W1 = parameters["W1"]
        b1 = parameters["b1"]
        W2 = parameters["W2"]
        b2 = parameters["b2"]
        
        # Print the cost every 100 training example
        if print_cost and i % 100 == 0:
            print("Cost after iteration {}: {}".format(i, np.squeeze(cost)))
        if print_cost and i % 100 == 0:
            costs.append(cost)
       
    # plot the cost

    plt.plot(np.squeeze(costs))
    plt.ylabel('cost')
    plt.xlabel('iterations (per tens)')
    plt.title("Learning rate =" + str(learning_rate))
    plt.show()
    
    return parameters

Run the cell below to train your parameters. See if your model runs. The cost should be decreasing. It may take up to 5 minutes to run 2500 iterations. Check if the "Cost after iteration 0" matches the expected output below, if not click on the square (⬛) on the upper bar of the notebook to stop the cell and try to find your error.

1	parameters = two_layer_model(train_x, train_y, layers_dims = (n_x, n_h, n_y), num_iterations = 2500, print_cost=True)

Cost after iteration 0: 0.693049735659989 
Cost after iteration 100: 0.6464320953428849 
Cost after iteration 200: 0.6325140647912678 
Cost after iteration 300: 0.6015024920354665 
Cost after iteration 400: 0.5601966311605748 
Cost after iteration 500: 0.515830477276473 
Cost after iteration 600: 0.4754901313943325 
Cost after iteration 700: 0.43391631512257495 
Cost after iteration 800: 0.4007977536203886 
Cost after iteration 900: 0.35807050113237987 
Cost after iteration 1000: 0.3394281538366413 
Cost after iteration 1100: 0.30527536361962654 
Cost after iteration 1200: 0.2749137728213015 
Cost after iteration 1300: 0.24681768210614827 
Cost after iteration 1400: 0.1985073503746611 
Cost after iteration 1500: 0.17448318112556593 
Cost after iteration 1600: 0.1708076297809661 
Cost after iteration 1700: 0.11306524562164737 
Cost after iteration 1800: 0.09629426845937163 
Cost after iteration 1900: 0.08342617959726878 
Cost after iteration 2000: 0.0743907870431909 
Cost after iteration 2100: 0.06630748132267938 
Cost after iteration 2200: 0.05919329501038176 
Cost after iteration 2300: 0.05336140348560564 
Cost after iteration 2400: 0.048554785628770226

Good thing you built a vectorized implementation! Otherwise it might have taken 10 times longer to train this.

Now, you can use the trained parameters to classify images from the dataset. To see your predictions on the training and test sets, run the cell below.

1	predictions_train = predict(train_x, train_y, parameters)

Output:

Accuracy: 1.0

1	predictions_test = predict(test_x, test_y, parameters)

Output:

Accuracy: 0.72

Note: You may notice that running the model on fewer iterations (say 1500) gives better accuracy on the test set. This is called "early stopping" and we will talk about it in the next course. Early stopping is a way to prevent overfitting.

Congratulations! It seems that your 2-layer neural network has better performance (72%) than the logistic regression implementation (70%, assignment week 2). Let's see if you can do even better with an $L$-layer model.

5 - L-layer Neural Network

Question: Use the helper functions you have implemented previously to build an $L$-layer neural network with the following structure: [LINEAR -> RELU]$\times$(L-1) -> LINEAR -> SIGMOID. The functions you may need and their inputs are:

def initialize_parameters_deep(layers_dims):
    ...
    return parameters 
def L_model_forward(X, parameters):
    ...
    return AL, caches
def compute_cost(AL, Y):
    ...
    return cost
def L_model_backward(AL, Y, caches):
    ...
    return grads
def update_parameters(parameters, grads, learning_rate):
    ...
    return parameters

1 2	### CONSTANTS ### layers_dims = [12288, 20, 7, 5, 1] # 4-layer model

#GRADED FUNCTION: L_layer_model

def L_layer_model(X, Y, layers_dims, learning_rate = 0.0075, num_iterations = 3000, print_cost=False):#lr was 0.009
    """
    Implements a L-layer neural network: [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID.
    
    Arguments:
    X -- data, numpy array of shape (number of examples, num_px * num_px * 3)
    Y -- true "label" vector (containing 0 if cat, 1 if non-cat), of shape (1, number of examples)
    layers_dims -- list containing the input size and each layer size, of length (number of layers + 1).
    learning_rate -- learning rate of the gradient descent update rule
    num_iterations -- number of iterations of the optimization loop
    print_cost -- if True, it prints the cost every 100 steps
    
    Returns:
    parameters -- parameters learnt by the model. They can then be used to predict.
    """

    np.random.seed(1)
    costs = []                         # keep track of cost
    
    # Parameters initialization. (≈ 1 line of code)
    ### START CODE HERE ###
    parameters = initialize_parameters_deep(layers_dims)
    ### END CODE HERE ###
    
    # Loop (gradient descent)
    for i in range(0, num_iterations):

        # Forward propagation: [LINEAR -> RELU]*(L-1) -> LINEAR -> SIGMOID.
        ### START CODE HERE ### (≈ 1 line of code)
        AL, caches = L_model_forward(X, parameters)
        ### END CODE HERE ###
        
        # Compute cost.
        ### START CODE HERE ### (≈ 1 line of code)
        cost = compute_cost(AL, Y)
        ### END CODE HERE ###
    
        # Backward propagation.
        ### START CODE HERE ### (≈ 1 line of code)
        grads = L_model_backward(AL, Y, caches)
        ### END CODE HERE ###
 
        # Update parameters.
        ### START CODE HERE ### (≈ 1 line of code)
        parameters = update_parameters(parameters, grads, learning_rate)
        ### END CODE HERE ###
                
        # Print the cost every 100 training example
        if print_cost and i % 100 == 0:
            print ("Cost after iteration %i: %f" %(i, cost))
        if print_cost and i % 100 == 0:
            costs.append(cost)
            
    # plot the cost
    plt.plot(np.squeeze(costs))
    plt.ylabel('cost')
    plt.xlabel('iterations (per tens)')
    plt.title("Learning rate =" + str(learning_rate))
    plt.show()
    
    return parameters

You will now train the model as a 4-layer neural network.

Run the cell below to train your model. The cost should decrease on every iteration. It may take up to 5 minutes to run 2500 iterations. Check if the "Cost after iteration 0" matches the expected output below, if not click on the square (⬛) on the upper bar of the notebook to stop the cell and try to find your error.


print("starting...")
#parameters = L_layer_model(train_x, train_y, layers_dims, num_iterations = 2500, print_cost = True)
parameters = L_layer_model(train_x, train_y, layers_dims, num_iterations = 2500, print_cost = True)

Cost after iteration 0: 0.771749 
Cost after iteration 100: 0.672053 
Cost after iteration 200: 0.648263 
Cost after iteration 300: 0.611507 
Cost after iteration 400: 0.567047 
Cost after iteration 500: 0.540138 
Cost after iteration 600: 0.527930 
Cost after iteration 700: 0.465477 
Cost after iteration 800: 0.369126 
Cost after iteration 900: 0.391747 
Cost after iteration 1000: 0.315187 
Cost after iteration 1100: 0.272700 
Cost after iteration 1200: 0.237419 
Cost after iteration 1300: 0.199601 
Cost after iteration 1400: 0.189263 
Cost after iteration 1500: 0.161189 
Cost after iteration 1600: 0.148214 
Cost after iteration 1700: 0.137775 
Cost after iteration 1800: 0.129740 
Cost after iteration 1900: 0.121225 
Cost after iteration 2000: 0.113821 
Cost after iteration 2100: 0.107839 
Cost after iteration 2200: 0.102855 
Cost after iteration 2300: 0.100897 
Cost after iteration 2400: 0.092878

1	pred_train = predict(train_x, train_y, parameters)

Output:

Accuracy: 0.985645933014

1	pred_test = predict(test_x, test_y, parameters)

Output:

Accuracy: 0.8

Congrats! It seems that your 4-layer neural network has better performance (80%) than your 2-layer neural network (72%) on the same test set.

This is good performance for this task. Nice job!

Though in the next course on "Improving deep neural networks" you will learn how to obtain even higher accuracy by systematically searching for better hyperparameters (learning_rate, layers_dims, num_iterations, and others you'll also learn in the next course).

6) Results Analysis

First, let's take a look at some images the L-layer model labeled incorrectly. This will show a few mislabeled images.

1	print_mislabeled_images(classes, test_x, test_y, pred_test)

A few types of images the model tends to do poorly on include: - Cat body in an unusual position - Cat appears against a background of a similar color - Unusual cat color and species - Camera Angle - Brightness of the picture - Scale variation (cat is very large or small in image)

7) Test with your own image (optional/ungraded exercise)

Congratulations on finishing this assignment. You can use your own image and see the output of your model. To do that:

1. Click on "File" in the upper bar of this notebook, then click "Open" to go on your Coursera Hub.
1. Add your image to this Jupyter Notebook's directory, in the "images" folder
1. Change your image's name in the following code
1. Run the code and check if the algorithm is right (1 = cat, 0 = non-cat)!

## START CODE HERE ##
my_image = "dog1.jpg" # change this to the name of your image file 
my_label_y = [1] # the true class of your image (1 -> cat, 0 -> non-cat)
## END CODE HERE ##

fname = "images/" + my_image
image = np.array(ndimage.imread(fname, flatten=False))
my_image = scipy.misc.imresize(image, size=(num_px,num_px)).reshape((num_px*num_px*3,1))
my_image = my_image/255.
my_predicted_image = predict(my_image, my_label_y, parameters)

plt.imshow(image)
print ("y = " + str(np.squeeze(my_predicted_image)) + ", your L-layer model predicts a \"" + classes[int(np.squeeze(my_predicted_image)),].decode("utf-8") +  "\" picture.")

References:

for auto-reloading external module: http://stackoverflow.com/questions/1907993/autoreload-of-modules-in-ipython

	Shape of W	Shape of b	Activation	Shape of Activation
Layer 1	\((n^{[1]},12288)\)	\((n^{[1]},1)\)	$Z^{[1]} = W^{[1]} X + b^{[1]} $	\((n^{[1]},209)\)
Layer 2	\((n^{[2]}, n^{[1]})\)	\((n^{[2]},1)\)	\(Z^{[2]} = W^{[2]} A^{[1]} + b^{[2]}\)	\((n^{[2]}, 209)\)
\(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)
Layer L-1	\((n^{[L-1]}, n^{[L-2]})\)	\((n^{[L-1]}, 1)\)	\(Z^{[L-1]} = W^{[L-1]} A^{[L-2]} + b^{[L-1]}\)	\((n^{[L-1]}, 209)\)
Layer L	\((n^{[L]}, n^{[L-1]})\)	\((n^{[L]}, 1)\)	\(Z^{[L]} = W^{[L]} A^{[L-1]} + b^{[L]}\)	\((n^{[L]}, 209)\)