Note
These are my personal programming assignments at the 4th week after studying the course neural-networks-deep-learning and the copyright belongs to deeplearning.ai.
Part 1：Building your Deep Neural Network: Step by Step
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_utilsprovides some necessary functions for this notebook.testCasesprovides 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.
1 | import numpy as np; |
You can get the support code from here.
the sigmoid function:
1 | def sigmoid(Z): |
the sigmoid_backward function:
1 | def sigmoid_backward(dA, cache): |
the relu function:
1 | def relu(Z): |
the relu_backward function：
1 | def relu_backward(dA, cache): |
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)*0.01with the correct shape. - Use zero initialization for the biases. Use
np.zeros(shape).
1 | # GRADED FUNCTION: initialize_parameters |
1 | parameters = initialize_parameters(3,2,1); |
W1 = [[ 0.01624345 -0.00611756 -0.00528172]
[-0.01072969 0.00865408 -0.02301539]]
b1 = [[0.]
[0.]]
W2 = [[ 0.01744812 -0.00761207]]
b2 = [[0.]]
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 $WX+b$ in python, it carries out broadcasting. For example, if:
$$W = \begin{bmatrix} j & k & l\\ m & n & o \\ p & q & r \end{bmatrix}\;\;\; X = \begin{bmatrix} a & b & c\\ d & e & f \\ g & h & i \end{bmatrix} \;\;\; b =\begin{bmatrix} s \\ t \\ u \end{bmatrix}\tag{1}$$Then $WX+b$ will be:
$$WX + b = \begin{bmatrix} (ja + kd + lg) + s & (jb + ke + lh) + s & (jc + kf + li)+ s\\ (ma + nd + og) + t & (mb + ne + oh) + t & (mc + nf + oi) + t\\ (pa + qd + rg) + u & (pb + qe + rh) + u & (pc + qf + ri)+ u \end{bmatrix}\tag{2}$$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.rand(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, thelayer_dimsfor 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 meansW1’s shape was (4,2),b1was (4,1),W2was (1,4) andb2was (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
3if 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));
1 | # GRADED FUNCTION: initialize_parameters_deep |
1 | parameters = initialize_parameters_deep([5,4,3]); |
W1 = [[ 0.01788628 0.0043651 0.00096497 -0.01863493 -0.00277388]
[-0.00354759 -0.00082741 -0.00627001 -0.00043818 -0.00477218]
[-0.01313865 0.00884622 0.00881318 0.01709573 0.00050034]
[-0.00404677 -0.0054536 -0.01546477 0.00982367 -0.01101068]]
b1 = [[0.]
[0.]
[0.]
[0.]]
W2 = [[-0.01185047 -0.0020565 0.01486148 0.00236716]
[-0.01023785 -0.00712993 0.00625245 -0.00160513]
[-0.00768836 -0.00230031 0.00745056 0.01976111]]
b2 = [[0.]
[0.]
[0.]]
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] × (L-1) -> LINEAR -> SIGMOID (whole model)
The linear forward module (vectorized over all the examples) computes the following equations:
$$Z^{[l]} = W^{[l]}A^{[l-1]} +b^{[l]}\tag{3}$$
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.
1 | # GRADED FUNCTION: linear_forward |
1 | A, W, b = linear_forward_test_case(); |
Z = [[ 3.26295337 -1.23429987]]
linear_forward_test_case:
1 | def linear_forward_test_case(): |
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
sigmoidfunction. 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:
1 | 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:
1 | 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.
1 | # GRADED FUNCTION: linear_activation_forward |
1 | A_prev, W, b = linear_activation_forward_test_case(); |
With sigmoid: A = [[0.96890023 0.11013289]]
With ReLU: A = [[3.43896131 0. ]]
linear_activation_forward_test_case function:
1 | def linear_activation_forward_test_case(): |
Note: In deep learning, the “[LINEAR->ACTIVATION]” computation is counted as a single layer in the neural network, not two layers.
4.3 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.
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 valuecto a list, you can uselist.append(c).
1 | # GRADED FUNCTION: L_model_forward |
1 | X, parameters = L_model_forward_test_case_2hidden(); |
AL = [[0.03921668 0.70498921 0.19734387 0.04728177]]
Length of caches list = 3
L_model_forward_test_case function:
1 | def L_model_forward_test_case(): |
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:
$$-\frac{1}{m} \sum\limits_{i = 1}^{m} (y^{(i)}\log\left(a^{[L] (i)}\right) + (1-y^{(i)})\log\left(1- a^{L}\right)) \tag{4}$$
1 | # GRADED FUNCTION: compute_cost |
1 | Y, AL = compute_cost_test_case(); |
cost = 0.41493159961539694
compute_cost_test_case function:
1 | def compute_cost_test_case(): |
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.
$$\frac{d \mathcal{L}(a^{[2]},y)}{{dz^{[1]}}} = \frac{d\mathcal{L}(a^{[2]},y)}{{da^{[2]}}}\frac{{da^{[2]}}}{{dz^{[2]}}}\frac{{dz^{[2]}}}{{da^{[1]}}}\frac{{da^{[1]}}}{{dz^{[1]}}} \tag{5}$$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] × (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:
$$dW^{[l]} = \frac{\partial \mathcal{L} }{\partial W^{[l]}} = \frac{1}{m} dZ^{[l]} A^{[l-1] T} \tag{5}$$
$$db^{[l]} = \frac{\partial \mathcal{L} }{\partial b^{[l]}} = \frac{1}{m} \sum_{i = 1}^{m} dZ^{l}\tag{6}$$
$$dA^{[l-1]} = \frac{\partial \mathcal{L} }{\partial A^{[l-1]}} = W^{[l] T} dZ^{[l]} \tag{7}$$
Exercise: Use the 3 formulas above to implement linear_backward().
1 | # GRADED FUNCTION: linear_backward |
1 | # Set up some test inputs |
dA_prev = [[ 0.51822968 -0.19517421]
[-0.40506361 0.15255393]
[ 2.37496825 -0.89445391]]
dW = [[-0.10076895 1.40685096 1.64992505]]
db = [[0.50629448]]
linear_backward_test_case function:
1 | def linear_backward_test_case(): |
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:
$$dZ^{[l]} = dA^{[l]} * g’(Z^{[l]}) \tag{8}$$
Exercise: Implement the backpropagation for the LINEAR->ACTIVATION layer.
1 | # GRADED FUNCTION: linear_activation_backward |
1 | AL, linear_activation_cache = linear_activation_backward_test_case(); |
sigmoid:
dA_prev = [[ 0.11017994 0.01105339]
[ 0.09466817 0.00949723]
[-0.05743092 -0.00576154]]
dW = [[ 0.10266786 0.09778551 -0.01968084]]
db = [[-0.05729622]]
relu:
dA_prev = [[ 0.44090989 0. ]
[ 0.37883606 0. ]
[-0.2298228 0. ]]
dW = [[ 0.44513824 0.37371418 -0.10478989]]
db = [[-0.20837892]]
linear_activation_backward_test_case function:
1 | def linear_activation_backward_test_case(): |
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 $= \frac{\partial \mathcal{L}}{\partial A^{[L]}}$.
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 :
$$grads[“dW” + str(l)] = dW^{[l]}\tag{9}$$
For example, for l=3 this would store $dW^{[l]}$ in grads["dW3"].
Exercise: Implement backpropagation for the [LINEAR->RELU] × (L-1) -> LINEAR -> SIGMOID model.
1 | # GRADED FUNCTION: L_model_backward |
1 | AL, Y_assess, caches = L_model_backward_test_case(); |
dW1 = [[0.41010002 0.07807203 0.13798444 0.10502167]
[0. 0. 0. 0. ]
[0.05283652 0.01005865 0.01777766 0.0135308 ]]
db1 = [[-0.22007063]
[ 0. ]
[-0.02835349]]
dA1 = [[ 0.12913162 -0.44014127]
[-0.14175655 0.48317296]
[ 0.01663708 -0.05670698]]
L_model_backward_test_case function in testCases_v3.py:
1 | def L_model_backward_test_case(): |
6.4 Update Parameters
In this section you will update the parameters of the model, using gradient descent:
$$W^{[l]} = W^{[l]} - \alpha \text{ } dW^{[l]} \tag{10}$$
$$b^{[l]} = b^{[l]} - \alpha \text{ } db^{[l]} \tag{11}$$
where $α$ 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 $l=1,2,…,L$.
1 | # GRADED FUNCTION: update_parameters |
1 | parameters, grads = update_parameters_test_case(); |
W1 = [[-0.59562069 -0.09991781 -2.14584584 1.82662008]
[-1.76569676 -0.80627147 0.51115557 -1.18258802]
[-1.0535704 -0.86128581 0.68284052 2.20374577]]
b1 = [[-0.04659241]
[-1.28888275]
[ 0.53405496]]
W2 = [[-0.55569196 0.0354055 1.32964895]]
b2 = [[-0.84610769]]
update_parameters_test_case function in testCases_v3.py:
1 | def update_parameters_test_case(): |
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!
Part 2：Deep Neural Network for Image Classification: Application
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_utilsprovides 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.
1 | import time |
The autoreload extension is already loaded. To reload it, use:
%reload_ext autoreload
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.
1 | # Example of a picture |
y = 0. It's a non-cat picture.
1 | # Explore your dataset |
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.
1 | # Reshape the training and test examples |
train_x's shape: (12288, 209)
test_x's shape: (12288, 50)
$12288$ equals $64×64×3$ 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: $[a^{[1]}_0,a^{[1]}1,…,a^{[1]}{n^{[1]}−1}]^T$.
- 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] × (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 $b^{[l]}$. 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:
- Initialize parameters / Define hyperparameters
- Loop for num_iterations:
- Forward propagation
- Compute cost function
- Backward propagation
- Update parameters (using parameters, and grads from backprop)
- 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:
1 | def initialize_parameters(n_x, n_h, n_y): |
1 | ### CONSTANTS DEFINING THE MODEL #### |
1 | #GRADED FUNCTION: two_layer_model |
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 black square button 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: 0.5158304772764729
Cost after iteration 600: 0.4754901313943325
Cost after iteration 700: 0.43391631512257495
Cost after iteration 800: 0.4007977536203886
Cost after iteration 900: 0.35807050113237976
Cost after iteration 1000: 0.33942815383664127
Cost after iteration 1100: 0.3052753636196264
Cost after iteration 1200: 0.2749137728213016
Cost after iteration 1300: 0.2468176821061484
Cost after iteration 1400: 0.19850735037466102
Cost after iteration 1500: 0.1744831811255665
Cost after iteration 1600: 0.17080762978096942
Cost after iteration 1700: 0.11306524562164715
Cost after iteration 1800: 0.09629426845937152
Cost after iteration 1900: 0.0834261795972687
Cost after iteration 2000: 0.07439078704319087
Cost after iteration 2100: 0.06630748132267934
Cost after iteration 2200: 0.05919329501038172
Cost after iteration 2300: 0.053361403485605585
Cost after iteration 2400: 0.04855478562877019
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); |
Accuracy: 0.9999999999999998
1 | predictions_test = predict(test_x, test_y, parameters); |
Accuracy: 0.72
the prediction function:
1 | def predict(X, y, parameters): |
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]×(L-1) -> LINEAR -> SIGMOID. The functions you may need and their inputs are:
1 | def initialize_parameters_deep(layer_dims): |
1 | ### CONSTANTS ### |
1 | # GRADED FUNCTION: L_layer_model |
You will now train the model as a 5-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 black square button on the upper bar of the notebook to stop the cell and try to find your error.
1 | 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); |
Accuracy: 0.9856459330143539
1 | pred_test = predict(test_x, test_y, parameters); |
Accuracy: 0.8
Congrats! It seems that your 5-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 type 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:
- Click on “File” in the upper bar of this notebook, then click “Open” to go on your Coursera Hub.
- Add your image to this Jupyter Notebook’s directory, in the “images” folder
- Change your image’s name in the following code
- Run the code and check if the algorithm is right (1 = cat, 0 = non-cat)!
1 | ## START CODE HERE ## |
Accuracy: 1.0
y = 1.0, your L-layer model predicts a "cat" picture.
