Deep Neural Networks step by step

Posted May 6, 2019 by Rokas Balsys

##### Deep Neural Network for Image Classification:

In this tutorial we will use the functions we had implemented in the previous parts to build a deep network, and apply it to cat vs dog classification. Hopefully, we will see an improvement in accuracy relative to our previous logistic regression implementation. After this part we will be able to build and apply a deep neural network to supervised learning using only numpy library.

Let's first import all the packages that we will need during this part. We will use the same "Cat vs Dag" dataset as in "Logistic Regression as a Neural Network". The model we had built had 60% test accuracy on classifying cats vs dogs images. Hopefully, our new model will perform a better!

I'll import code we already wrote from "Logistic Regression" tutorial series:

import os import cv2 import numpy as np import matplotlib.pyplot as plt import sklearn from sklearn import datasets ROWS = 64 COLS = 64 CHANNELS = 3 def read_image(file_path): img = cv2.imread(file_path, cv2.IMREAD_COLOR) return cv2.resize(img, (ROWS, COLS), interpolation=cv2.INTER_CUBIC) def prepare_data(images): m = len(images) X = np.zeros((m, ROWS, COLS, CHANNELS), dtype=np.uint8) y = np.zeros((1, m)) for i, image_file in enumerate(images): X[i,:] = read_image(image_file) if 'dog' in image_file.lower(): y[0, i] = 1 elif 'cat' in image_file.lower(): y[0, i] = 0 return X, y TRAIN_DIR = 'Train_data/' TEST_DIR = 'Test_data/' train_images = [TRAIN_DIR+i for i in os.listdir(TRAIN_DIR)] test_images = [TEST_DIR+i for i in os.listdir(TEST_DIR)] train_set_x, train_set_y = prepare_data(train_images) test_set_x, test_set_y = prepare_data(test_images) train_set_x_flatten = train_set_x.reshape(train_set_x.shape[0], ROWS*COLS*CHANNELS).T test_set_x_flatten = test_set_x.reshape(test_set_x.shape[0], -1).T train_set_x = train_set_x_flatten/255 test_set_x = test_set_x_flatten/255

##### L-layer deep neural network:

It is hard to represent a deep neural network with figure. However, here is a simplified network representation:

The model can be summarized as: [LINEAR -> RELU] × (L-1) -> LINEAR -> SIGMOID. Detailed Architecture of above figure:

- 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.

##### General methodology:

As usual we 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).

3. Use trained parameters to predict labels.

##### L-layer Neural Network:

We'll use the helper functions we have implemented previously to build an $L$-layer neural network with the following structure: *[LINEAR -> RELU]$\times$(L-1) -> LINEAR -> SIGMOID*.

**Arguments:**

X - data, numpy array of shape (number of examples, ROWS * COLS * CHANNELS ).

Y - true "label" vector (containing 0 if cat, 1 if dog), 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.

**Return:**

parameters -- parameters learnt by the model. They can then be used to predict.

def L_layer_model(X, Y, layers_dims, learning_rate = 0.0075, num_iterations = 3000, print_cost=False):#lr was 0.009 # keep track of cost costs = [] # Parameters initialization. parameters = initialize_parameters_deep(layers_dims) # Loop (gradient descent) for i in range(0, num_iterations): # Forward propagation: [LINEAR -> RELU]*(L-1) -> LINEAR -> SIGMOID. AL, caches = L_model_forward(X, parameters) # Compute cost. cost = compute_cost(AL, Y) # Backward propagation. grads = L_model_backward(AL, Y, caches) # Update parameters. parameters = update_parameters(parameters, grads, learning_rate) # 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

##### Predict function:

Before training our model we need to write a predict function which will be used to predict the results of a L-layer neural network:

**Arguments:**

X - data set of examples you would like to label.

parameters - parameters of the trained model.

**Return:**

p - predictions for the given dataset X.

def predict(X, parameters): m = X.shape[1] # number of layers in the neural network n = len(parameters) // 2 p = np.zeros((1,m)) # Forward propagation probas, caches = L_model_forward(X, parameters) # convert probas to 0/1 predictions for i in range(0, probas.shape[1]): if probas[0,i] > 0.5: p[0,i] = 1 else: p[0,i] = 0 return p

We are now ready to train our deep neural networks model. So we can run the cell below to train our model. The cost should decrease on every iteration. In my case with 6000 images it may take more than a day on my CPU to run 10000 iterations. Here is the code:

layers_dims = [12288, 800, 10, 1] # 4-layer model parameters = L_layer_model(train_set_x, train_set_y, layers_dims, learning_rate = 0.1, num_iterations = 10000, print_cost = True) print("train accuracy: {} %".format(100 - np.mean(np.abs(predict(train_set_x, parameters) - train_set_y)) * 100)) print("test accuracy: {} %".format(100 - np.mean(np.abs(predict(test_set_x, parameters) - test_set_y)) * 100))

**My updated Cat vs Dog training reults:**

Cost after iteration 9600: 0.409001 Cost after iteration 9700: 0.468013 Cost after iteration 9800: 0.464725 Cost after iteration 9900: 0.440395 train accuracy: 75.50816394535155 % test accuracy: 63.9 %

And here is the training cost plot:

From our test accuraccy you can see that it's almost not getting better in classifying cats versus dogs, it's because simple neural netowrks can't do it very vell, we need to use Convolutional Neural Networks to get better results (we'll use them in future tutorials).

We will test our deep neural network code with below code. So there is two functions, one is to create data, which we will use to train our model. Another function will be used to plot decision boundary in our plot which was predicted by our neural network, not getting deeper into these function here they are:

def load_dataset(DataNoise = 0.05, Visualize = False): #np.random.seed(1) train_X, train_Y = sklearn.datasets.make_circles(n_samples=300, noise=DataNoise) #np.random.seed(2) test_X, test_Y = sklearn.datasets.make_circles(n_samples=100, noise=DataNoise) train_X = train_X.T train_Y = train_Y.reshape((1, train_Y.shape[0])) test_X = test_X.T test_Y = test_Y.reshape((1, test_Y.shape[0])) # Visualize the data if Visualize == True: axes = plt.gca() axes.set_xlim([-1.5,1.5]) axes.set_ylim([-1.5,1.5]) plt.scatter(train_X[0, :], train_X[1, :], c=train_Y[0], s=40, cmap=plt.cm.Spectral) plt.show() return train_X, train_Y, test_X, test_Y def plot_decision_boundary(model, X, y): # Set min and max values and give it some padding x_min, x_max = X[0, :].min() - 1, X[0, :].max() + 1 y_min, y_max = X[1, :].min() - 1, X[1, :].max() + 1 h = 0.01 # Generate a grid of points with distance h between them xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h)) # Predict the function value for the whole grid Z = model(np.c_[xx.ravel(), yy.ravel()]) Z = Z.reshape(xx.shape) # Plot the contour and training examples plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral) plt.ylabel('x2') plt.xlabel('x1') axes = plt.gca() axes.set_xlim([-1.5,1.5]) axes.set_ylim([-1.5,1.5]) plt.scatter(X[0, :], X[1, :], c=y[0], cmap=plt.cm.Spectral) plt.show()

Lets visualize out training data with following line:

train_X, train_Y, test_X, test_Y = load_dataset(DataNoise = 0.15, Visualize = True)

Lets plot our training data:

At first we'll try to use neural network with one hidden layer, where input is 2, 4 hidden layers and output equal to 1:

layers_dims = [2, 4, 1] parameters = L_layer_model(train_X, train_Y, layers_dims, learning_rate = 0.2, num_iterations = 15000, print_cost = True) print("train accuracy: {} %".format(100 - np.mean(np.abs(predict(train_X, parameters) - train_Y)) * 100)) plot_decision_boundary(lambda x: predict(x.T, parameters), train_X, train_Y)

With 15000 iterations and learning rate of 0.2 we received accuracy of 61%:

Cost after iteration 14600: 0.654744 Cost after iteration 14700: 0.654740 Cost after iteration 14800: 0.654737 Cost after iteration 14900: 0.654733 train accuracy: 61.333333333333336 %

And here is the classification plot:

It doesn't look nice so lets try same 15000 iterations and learning rate of 0.2 with 10 hidden layers:

Cost after iteration 14600: 0.534574 Cost after iteration 14700: 0.534570 Cost after iteration 14800: 0.534569 Cost after iteration 14900: 0.534564 train accuracy: 73.33333333333333 %

Now we received 73% accuracy. That's nice, lets plot our result:

And now lets test our hard work ant try deeper network with layers_dims = [2, 128, 4, 1].

Cost after iteration 14600: 0.504184 Cost after iteration 14700: 0.504572 Cost after iteration 14800: 0.504701 Cost after iteration 14900: 0.504019 train accuracy: 74.33333333333334 %

Now we received 74% accuracy. That's nice, but as you can see it's getting harder to get better results. Lets plot our result:

Congrats! It seems that our deep layer neural network has better performance (74%) than our 2-layer neural network (73%) on the same data-set. This is quite good performance for this task. Nice job!

In the next tutorials we will learn how to obtain even higher accuracy by systematically searching for better hyperparameters (learning_rate, layers_dims, num_iterations, and others). This was quite long tutorial, now when we know how to build deep neural networks we can learn how to optimize it. See you in next tutorial.

Full tutorial code and cats vs dogs image data-set can be found on my GitHub page.