3.深度学习练习：Planar data classification with one hidden layer

本文节选自吴恩达老师《深度学习专项课程》编程作业，在此表示感谢。

课程链接：https://www.deeplearning.ai/deep-learning-specialization/

You will learn to:

Implement a 2-class classification neural network with a single hidden layer
Use units with a non-linear activation function, such as tanh
Compute the cross entropy loss
Implement forward and backward propagation

1 - Packages

2 - Dataset

3 - Simple Logistic Regression

4 - Neural Network model（掌握）

4.1 - Defining the neural network structure

4.2 - Initialize the model's parameters

4.3 - The Loop

4.4 - Integrate parts 4.1, 4.2 and 4.3 in nn_model()

4.5 Predictions

4.6 - Tuning hidden layer size (optional/ungraded exercise)

5 - Performance on other datasets

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.
sklearn provides simple and efficient tools for data mining and data analysis.
matplotlib is a library for plotting graphs in Python.
testCases provides some test examples to assess the correctness of your functions
planar_utils provide various useful functions used in this assignment

# Package imports
import numpy as np
import matplotlib.pyplot as plt
from testCases import *
import sklearn
import sklearn.datasets
import sklearn.linear_model
from planar_utils import plot_decision_boundary, sigmoid, load_planar_dataset, load_extra_datasets%matplotlib inlinenp.random.seed(1) # set a seed so that the results are consistent

2 - Dataset

X, Y = load_planar_dataset()# Visualize the data:
plt.scatter(X[0, :], X[1, :], c=np.squeeze(Y), s=40, cmap=plt.cm.Spectral);

You have:

- a numpy-array (matrix) X that contains your features (x1, x2)
- a numpy-array (vector) Y that contains your labels (red:0, blue:1).

Lets first get a better sense of what our data is like.

Exercise: How many training examples do you have? In addition, what is the shape of the variables X and Y?

Hint: How do you get the shape of a numpy array? (help)

shape_X = np.shape(X)
shape_Y = np.shape(Y)
m = shape_X[1]print ('The shape of X is: ' + str(shape_X))
print ('The shape of Y is: ' + str(shape_Y))
print ('I have m = %d training examples!' % (m))

3 - Simple Logistic Regression

Before building a full neural network, lets first see how logistic regression performs on this problem. You can use sklearn's built-in functions to do that. Run the code below to train a logistic regression classifier on the dataset.

# Train the logistic regression classifier
clf = sklearn.linear_model.LogisticRegressionCV();
clf.fit(X.T, Y.T);# Plot the decision boundary for logistic regression
plot_decision_boundary(lambda x: clf.predict(x), X, np.squeeze(Y))
plt.title("Logistic Regression")# Print accuracy
LR_predictions = clf.predict(X.T)
print ('Accuracy of logistic regression: %d ' % float((np.dot(Y,LR_predictions) + np.dot(1-Y,1-LR_predictions))/float(Y.size)*100) +'% ' + "(percentage of correctly labelled datapoints)")

Interpretation: The dataset is not linearly separable, so logistic regression doesn't perform well. Hopefully a neural network will do better. Let's try this now!

4 - Neural Network model（掌握）

Logistic regression did not work well on the "flower dataset". You are going to train a Neural Network with a single hidden layer.

Here is our model:

For one example， $x^{(i)}$ ：

$z^{[1] (i)} = W^{[1]} x^{(i)} + b^{[1] (i)}$

$a^{[1] (i)} = \tanh(z^{[1] (i)})$

$z^{[2] (i)} = W^{[2]} a^{[1] (i)} + b^{[2] (i)}$

$\hat{y}^{(i)} = a^{[2] (i)} = \sigma(z^{ [2] (i)})$

$y^{(i)}_{prediction} = \begin{cases} 1 & \mbox{if } a^{[2](i)} > 0.5 \\ 0 & \mbox{otherwise } \end{cases}$

Given the predictions on all the examples, you can also compute the cost $J$ as follows:

$J = - \frac{1}{m} \sum\limits_{i = 0}^{m} \large\left(\small y^{(i)}\log\left(a^{[2] (i)}\right) + (1-y^{(i)})\log\left(1- a^{[2] (i)}\right) \large \right) \small$

Reminder: The general methodology to build a Neural Network is to:

1. Define the neural network structure ( # of input units, # of hidden units, etc).
2. Initialize the model's parameters
3. Loop:
- Implement forward propagation
- Compute loss
- Implement backward propagation to get the gradients
- Update parameters (gradient descent)

4.1 - Defining the neural network structure

Exercise: Define three variables:

- n_x: the size of the input layer
- n_h: the size of the hidden layer (set this to 4)
- n_y: the size of the output layer

Hint: Use shapes of X and Y to find n_x and n_y. Also, hard code the hidden layer size to be 4.

def layer_sizes(X, Y):"""Arguments:X -- input dataset of shape (input size, number of examples)Y -- labels of shape (output size, number of examples)Returns:n_x -- the size of the input layern_h -- the size of the hidden layern_y -- the size of the output layer"""n_x = np.shape(X)[0]n_h = 4n_y = np.shape(Y)[0]return (n_x, n_h, n_y)

4.2 - Initialize the model's parameters

Exercise: Implement the function initialize_parameters().

Instructions:

Make sure your parameters' sizes are right. Refer to the neural network figure above if needed.
You will initialize the weights matrices with random values.
- Use: np.random.randn(a,b) * 0.01 to randomly initialize a matrix of shape (a,b).
You will initialize the bias vectors as zeros.
- Use: np.zeros((a,b)) to initialize a matrix of shape (a,b) with zeros.

def initialize_parameters(n_x, n_h, n_y):"""Argument:n_x -- size of the input layern_h -- size of the hidden layern_y -- size of the output layerReturns:params -- 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(2) # we set up a seed so that your output matches ours although the initialization is random.W1 = np.random.randn(n_h, n_x) * 0.01b1 = np.zeros((n_h, 1))W2 = np.random.randn(n_y, n_h) * 0.01b2 = np.zeros((n_y, 1))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

4.3 - The Loop

Question: Implement forward_propagation().

Instructions:

Look above at the mathematical representation of your classifier.
You can use the function sigmoid(). It is built-in (imported) in the notebook.
You can use the function np.tanh(). It is part of the numpy library.
The steps you have to implement are:
1. Retrieve each parameter from the dictionary "parameters" (which is the output of initialize_parameters()) by using parameters[".."].
2. Implement Forward Propagation. Compute $Z^{[1]}, A^{[1]}, Z^{[2]}$ and $A^{[2]}$ (the vector of all your predictions on all the examples in the training set).
Values needed in the backpropagation are stored in "cache". The cache will be given as an input to the backpropagation function.

def forward_propagation(X, parameters):"""Argument:X -- input data of size (n_x, m)parameters -- python dictionary containing your parameters (output of initialization function)Returns:A2 -- The sigmoid output of the second activationcache -- a dictionary containing "Z1", "A1", "Z2" and "A2"W1 = parameters['W1']b1 = parameters['b1']W2 = parameters['W2']b2 = parameters['b2']# Implement Forward Propagation to calculate A2 (probabilities)Z1 = np.dot(W1, X) + b1A1 = np.tanh(Z1)Z2 = np.dot(W2, A1) + b2A2 = sigmoid(Z2)assert(A2.shape == (1, X.shape[1]))cache = {"Z1": Z1,"A1": A1,"Z2": Z2,"A2": A2}return A2, cache

Now that you have computed $A^{[2]}$ (in the Python variable "A2"), which contains $a^{[2](i)}$ for every example, you can compute the cost function as follows:

$J = - \frac{1}{m} \sum\limits_{i = 0}^{m} \large{(} \small y^{(i)}\log\left(a^{[2] (i)}\right) + (1-y^{(i)})\log\left(1- a^{[2] (i)}\right) \large{)} \small$

Exercise: Implement compute_cost() to compute the value of the cost ?J.

Instructions:

There are many ways to implement the cross-entropy loss. To help you, we give you how we would have implemented $- \sum\limits_{i=0}^{m} y^{(i)}\log(a^{[2](i)})$

logprobs = np.multiply(np.log(A2),Y)
cost = - np.sum(logprobs)                # no need to use a for loop!

def compute_cost(A2, Y, parameters):"""Computes the cross-entropy cost given in equation (13)Arguments:A2 -- The sigmoid output of the second activation, of shape (1, number of examples)Y -- "true" labels vector of shape (1, number of examples)parameters -- python dictionary containing your parameters W1, b1, W2 and b2Returns:cost -- cross-entropy cost given equation (13)"""m = Y.shape[1] # number of example# Compute the cross-entropy costlogprobs = np.multiply(np.log(A2), Y) + np.multiply(np.log(1-A2), 1-Y)cost = -1 /m * np.sum(logprobs)cost = np.squeeze(cost)     # makes sure cost is the dimension we expect. # E.g., turns [[17]] into 17 assert(isinstance(cost, float))return cost

Using the cache computed during forward propagation, you can now implement backward propagation.

Question: Implement the function backward_propagation().

Instructions: Backpropagation is usually the hardest (most mathematical) part in deep learning. To help you, here again is the slide from the lecture on backpropagation. You'll want to use the six equations on the right of this slide, since you are building a vectorized implementation.

def backward_propagation(parameters, cache, X, Y):"""Implement the backward propagation using the instructions above.Arguments:parameters -- python dictionary containing our parameters cache -- a dictionary containing "Z1", "A1", "Z2" and "A2".X -- input data of shape (2, number of examples)Y -- "true" labels vector of shape (1, number of examples)Returns:grads -- python dictionary containing your gradients with respect to different parameters"""m = X.shape[1]# First, retrieve W1 and W2 from the dictionary "parameters".W1 = parameters['W1']W2 = parameters['W2']# Retrieve also A1 and A2 from dictionary "cache".A1 = cache['A1']A2 = cache['A2']# Backward propagation: calculate dW1, db1, dW2, db2. dZ2 = A2 - YdW2 = 1/m * np.dot(dZ2, A1.T)db2 = 1/m * np.sum(dZ2, axis = 1, keepdims = True)dZ1 = np.dot(W2.T, dZ2) * (1 - np.power(A1, 2))dW1 = 1/m * np.dot(dZ1, X.T)db1 = 1/m * np.sum(dZ1, axis = 1, keepdims = True)grads = {"dW1": dW1,"db1": db1,"dW2": dW2,"db2": db2}return grads

def update_parameters(parameters, grads, learning_rate = 1.2):"""Updates parameters using the gradient descent update rule given aboveArguments:parameters -- python dictionary containing your parameters grads -- python dictionary containing your gradients Returns:parameters -- python dictionary containing your updated parameters """# Retrieve each parameter from the dictionary "parameters" W1 = parameters['W1']b1 = parameters['b1']W2 = parameters['W2']b2 = parameters['b2']# Retrieve each gradient from the dictionary "grads"dW1 = grads['dW1']db1 = grads['db1']dW2 = grads['dW2']db2 = grads['db2']# Update rule for each parameterW1 -= learning_rate * dW1b1 -= learning_rate * db1W2 -= learning_rate * dW2b2 -= learning_rate * db2parameters = {"W1": W1,"b1": b1,"W2": W2,"b2": b2}return parameters

4.4 - Integrate parts 4.1, 4.2 and 4.3 in nn_model()

Question: Build your neural network model in nn_model().

Instructions: The neural network model has to use the previous functions in the right order.

def nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):"""Arguments:X -- dataset of shape (2, number of examples)Y -- labels of shape (1, number of examples)n_h -- size of the hidden layernum_iterations -- Number of iterations in gradient descent loopprint_cost -- if True, print the cost every 1000 iterationsReturns:parameters -- parameters learnt by the model. They can then be used to predict."""np.random.seed(3)n_x = layer_sizes(X, Y)[0]n_y = layer_sizes(X, Y)[2]# Initialize parameters, then retrieve W1, b1, W2, b2. Inputs: "n_x, n_h, n_y". Outputs = "W1, b1, W2, b2, parameters".parameters = initialize_parameters(n_x, n_h, n_y)W1 = parameters["W1"]b1 = parameters["b1"]W2 = parameters["W2"]b2 = parameters["b2"]# Loop (gradient descent)for i in range(0, num_iterations):# Forward propagation. Inputs: "X, parameters". Outputs: "A2, cache".A2, cache = forward_propagation(X, parameters)# Cost function. Inputs: "A2, Y, parameters". Outputs: "cost".cost = compute_cost(A2, Y, parameters)# Backpropagation. Inputs: "parameters, cache, X, Y". Outputs: "grads".grads = backward_propagation(parameters, cache, X, Y)# Gradient descent parameter update. Inputs: "parameters, grads". Outputs: "parameters".parameters = update_parameters(parameters, grads, learning_rate = 1.2)# Print the cost every 1000 iterationsif print_cost and i % 1000 == 0:print ("Cost after iteration %i: %f" %(i, cost))return parameters

4.5 Predictions

Question: Use your model to predict by building predict(). Use forward propagation to predict results.

def predict(parameters, X):"""Using the learned parameters, predicts a class for each example in XArguments:parameters -- python dictionary containing your parameters X -- input data of size (n_x, m)Returnspredictions -- vector of predictions of our model (red: 0 / blue: 1)"""# Computes probabilities using forward propagation, and classifies to 0/1 using 0.5 as the threshold.A2, cache = forward_propagation(X, parameters)predictions = (A2 > 0.5)return predictions

It is time to run the model and see how it performs on a planar dataset. Run the following code to test your model with a single hidden layer of ?ℎnhhidden units.

# Build a model with a n_h-dimensional hidden layer
parameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), X, np.squeeze(Y))
plt.title("Decision Boundary for hidden layer size " + str(4))# Print accuracy
predictions = predict(parameters, X)
print ('Accuracy: %d' % float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100) + '%')

4.6 - Tuning hidden layer size (optional/ungraded exercise)

# This may take about 2 minutes to runplt.figure(figsize=(16, 32))
hidden_layer_sizes = [1, 2, 3, 4, 5, 10, 20]
for i, n_h in enumerate(hidden_layer_sizes):plt.subplot(5, 2, i+1)plt.title('Hidden Layer of size %d' % n_h)parameters = nn_model(X, Y, n_h, num_iterations = 5000)plot_decision_boundary(lambda x: predict(parameters, x.T), X, np.squeeze(Y))predictions = predict(parameters, X)accuracy = float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100)print ("Accuracy for {} hidden units: {} %".format(n_h, accuracy))

**You've learnt to:** - Build a complete neural network with a hidden layer - Make a good use of a non-linear unit - Implemented forward propagation and backpropagation, and trained a neural network - See the impact of varying the hidden layer size, including overfitting.

5 - Performance on other datasets

noisy_circles, noisy_moons, blobs, gaussian_quantiles, no_structure = load_extra_datasets()datasets = {"noisy_circles": noisy_circles,"noisy_moons": noisy_moons,"blobs": blobs,"gaussian_quantiles": gaussian_quantiles}dataset = "gaussian_quantiles"X, Y = datasets[dataset]
X, Y = X.T, Y.reshape(1, Y.shape[0])# make blobs binary
if dataset == "blobs":Y = Y%2# Visualize the data
plt.scatter(X[0, :], X[1, :], c = np.squeeze(Y), s=40, cmap=plt.cm.Spectral);