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
<caption><center> <u>Figure 2</u>: 2-layer neural network.
The model can be summarized as: INPUT -> LINEAR -> RELU -> LINEAR -> SIGMOID -> OUTPUT. </center></caption>
<u>Detailed Architecture of figure 2</u>:
- 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:
<caption><center> <u>Figure 3</u>: L-layer neural network.
The model can be summarized as: [LINEAR -> RELU] $\times$ (L-1) -> LINEAR -> SIGMOID</center></caption>
<u>Detailed Architecture of figure 3</u>:
- 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 = 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". Output: "A1, cache1, A2, cache2". ### START CODE HERE ### (≈ 2 lines of code) A1, cache1 = linear_activation_forward(X, W1, b1, activation="relu") A2, cache2 = linear_activation_forward(A1,W2, b2, activation="sigmoid") ### END CODE HERE ### # Compute cost ### START CODE HERE ### (≈ 1 line of code) cost = compute_cost(AL=A2,Y=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) linear_activation_backward(dA, cache, activation) dA1, dW2, db2 = linear_activation_backward(dA2,cache2,activation="sigmoid") dA0, dW1, db1 = linear_activation_backward(dA1,cache1,activation="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) def update_parameters(parameters, grads, learning_rate): 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.
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
output_18_1.png
Expected Output:
<table>
<tr>
<td> Cost after iteration 0</td>
<td> 0.6930497356599888 </td>
</tr>
<tr>
<td> Cost after iteration 100</td>
<td> 0.6464320953428849 </td>
</tr>
<tr>
<td> ...</td>
<td> ... </td>
</tr>
<tr>
<td> Cost after iteration 2400</td>
<td> 0.048554785628770206 </td>
</tr>
</table>
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.
predictions_train = predict(train_x, train_y, parameters)
Accuracy: 1.0
Expected Output:
<table>
<tr>
<td> Accuracy</td>
<td> 1.0 </td>
</tr>
</table>
predictions_test = predict(test_x, test_y, parameters)
Accuracy: 0.72
Expected Output:
<table>
<tr>
<td> Accuracy</td>
<td> 0.72 </td>
</tr>
</table>
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.
