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.
Let’s first import all the packages that you will need during this assignment.
dnn_utils
provides some necessary functions for this notebook.testCases
provides some test cases to assess the correctness of your functions1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 |
import numpy as np; import h5py; import matplotlib.pyplot as plt; from testCases_v3 import *; from dnn_utils_v2 import sigmoid, sigmoid_backward, relu, relu_backward; %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'; %load_ext autoreload %autoreload 2 np.random.seed(1); |
You can get the support code from here.
the sigmoid
function:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 |
def sigmoid(Z): """ Implements the sigmoid activation in numpy Arguments: Z -- numpy array of any shape Returns: A -- output of sigmoid(z), same shape as Z cache -- returns Z as well, useful during backpropagation """ A = 1 / (1 + np.exp(-Z)); cache = Z; return A, cache; |
the sigmoid_backward
function:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 |
def sigmoid_backward(dA, cache): """ Implement the backward propagation for a single SIGMOID unit. Arguments: dA -- post-activation gradient, of any shape cache -- 'Z' where we store for computing backward propagation efficiently Returns: dZ -- Gradient of the cost with respect to Z """ Z = cache; s = 1 / (1 + np.exp(-Z)); dZ = dA * s * (1 - s); assert (dZ.shape == Z.shape); return dZ; |
the relu
function:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 |
def relu(Z): """ Implement the RELU function. Arguments: Z -- Output of the linear layer, of any shape Returns: A -- Post-activation parameter, of the same shape as Z cache -- a python dictionary containing "A" ; stored for computing the backward pass efficiently """ A = np.maximum(0,Z); assert(A.shape == Z.shape); cache = Z; return A, cache; |
the relu_backward
function:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 |
def relu_backward(dA, cache): """ Implement the backward propagation for a single RELU unit. Arguments: dA -- post-activation gradient, of any shape cache -- 'Z' where we store for computing backward propagation efficiently Returns: dZ -- Gradient of the cost with respect to Z """ Z = cache; dZ = np.array(dA, copy = True); # just converting dz to a correct object. # When z <= 0, you should set dz to 0 as well. dZ[Z <= 0] = 0; assert (dZ.shape == Z.shape); return dZ; |
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:
Implement the forward propagation module (shown in purple in the figure below).
Compute the loss.
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.
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.
Exercise: Create and initialize the parameters of the 2-layer neural network.
Instructions:
np.random.randn(shape)*0.01
with the correct shape.np.zeros(shape)
.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 |
# 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; |
1 2 3 4 5 |
parameters = initialize_parameters(3,2,1); print("W1 = " + str(parameters["W1"])); print("b1 = " + str(parameters["b1"])); print("W2 = " + str(parameters["W2"])); print("b2 = " + str(parameters["b2"])); |
W1 = [[ 0.01624345 -0.00611756 -0.00528172]
[-0.01072969 0.00865408 -0.02301539]]
b1 = [[0.]
[0.]]
W2 = [[ 0.01744812 -0.00761207]]
b2 = [[0.]]
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) |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
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=[jklmnopqr]X=[abcdefghi]b=[stu]
Then WX+b will be:
WX+b=[(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]
Exercise: Implement initialization for an L-layer Neural Network.
Instructions:
np.random.rand(shape) * 0.01
.np.zeros(shape)
.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)); |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 |
# 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 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; |
1 2 3 4 5 |
parameters = initialize_parameters_deep([5,4,3]); print("W1 = " + str(parameters["W1"])); print("b1 = " + str(parameters["b1"])); print("W2 = " + str(parameters["W2"])); print("b2 = " + str(parameters["b2"])); |
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.]]
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:
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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 |
# 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; |
1 2 3 |
A, W, b = linear_forward_test_case(); Z, linear_cache = linear_forward(A, W, b); print("Z = " + str(Z)); |
Z = [[ 3.26295337 -1.23429987]]
linear_forward_test_case:
1 2 3 4 5 6 |
def linear_forward_test_case(): np.random.seed(1); A = np.random.randn(3,2); W = np.random.randn(1,3); b = np.random.randn(1,1); return A, W, b; |
In this notebook, you will use two activation functions:
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:1
|
A, activation_cache = sigmoid(Z);
|
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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 |
# 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 """ 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); # Z, (W, A_prev, B) A, activation_cache = sigmoid(Z); # A, (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); #, ((W, A_prev, B) ,(Z)) return A, cache; |
1 2 3 4 5 6 7 |
A_prev, W, b = linear_activation_forward_test_case(); A, linear_activation_cache = linear_activation_forward(A_prev, W, b, activation = "sigmoid"); print("With sigmoid: A = " + str(A)); A, linear_activation_cache = linear_activation_forward(A_prev, W, b, activation = "relu"); print("With ReLU: A = " + str(A)); |
With sigmoid: A = [[0.96890023 0.11013289]]
With ReLU: A = [[3.43896131 0. ]]
linear_activation_forward_test_case function:
1 2 3 4 5 6 |
def linear_activation_forward_test_case(): np.random.seed(2) A_prev = np.random.randn(3,2) W = np.random.randn(1,3) b = np.random.randn(1,1) return A_prev, W, b |
Note: In deep learning, the “[LINEAR->ACTIVATION]” computation is counted as a single layer in the neural network, not two layers.
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]=σ(Z[L])=σ(W[L]A[L−1]+b[L]). (This is sometimes also called Yhat
, i.e., this is ˆY.)
Tips:
caches
” list. To add a new value c
to a list, you can use list.append(c)
.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 |
# 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_relu_forward() (there are L-1 of them, indexed from 0 to L-2) the cache of linear_sigmoid_forward() (there is one, indexed 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 ### START CODE HERE ### (≈ 2 lines of code) A, linear_activation_cache = linear_activation_forward(A_prev, parameters["W" + str(l)], parameters["b" + str(l)], "relu"); caches.append(linear_activation_cache); ### END CODE HERE ### # Implement LINEAR -> SIGMOID. Add "cache" to the "caches" list. ### START CODE HERE ### (≈ 2 lines of code) AL, linear_activation_cache = linear_activation_forward(A, parameters["W" + str(L)], parameters["b" + str(L)], "sigmoid"); caches.append(linear_activation_cache); ### END CODE HERE ### assert(AL.shape == (1,X.shape[1])); return AL, caches; |
1 2 3 4 |
X, parameters = L_model_forward_test_case_2hidden(); AL, caches = L_model_forward(X, parameters); print("AL = " + str(AL)); print("Length of caches list = " + str(len(caches))); |
AL = [[0.03921668 0.70498921 0.19734387 0.04728177]]
Length of caches list = 3
L_model_forward_test_case function:
1 2 3 4 5 6 7 8 9 10 11 12 13 |
def L_model_forward_test_case(): np.random.seed(1); X = np.random.randn(4,2); W1 = np.random.randn(3,4); b1 = np.random.randn(3,1); W2 = np.random.randn(1,3); b2 = np.random.randn(1,1); parameters = {"W1": W1, "b1": b1, "W2": W2, "b2": b2}; return X, parameters; |
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.
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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 |
# 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 = -1 / m * (np.dot(Y, np.log(AL).T) + np.dot(1 - Y, np.log(1 - AL).T)); ### 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(isinstance(cost, float)); assert(cost.shape == ()); return cost; |
1 2 |
Y, AL = compute_cost_test_case(); print("cost = " + str(compute_cost(AL, Y))); |
cost = 0.41493159961539694
compute_cost_test_case function:
1 2 3 4 |
def compute_cost_test_case(): Y = np.asarray([[1, 1, 1]]); aL = np.array([[.8,.9,0.4]]); return Y, aL; |
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.
dL(a[2],y)dz[1]=dL(a[2],y)da[2]da[2]dz[2]dz[2]da[1]da[1]dz[1]In order to calculate the gradient dW[1]=∂L∂W[1], you use the previous chain rule and you do dW[1]=dz[1]×∂z[1]∂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]=∂L∂b[1], you use the previous chain rule and you do db[1]=dz[1]×∂z[1]∂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:
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]=∂L∂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]=∂L∂W[l]=1mdZ[l]A[l−1]T
$$db^{[l]} = \frac{\partial \mathcal{L} }{\partial b^{[l]}} = \frac{1}{m} \sum_{i = 1}^{m} dZ^{l}\tag{6}$$
dA[l−1]=∂L∂A[l−1]=W[l]TdZ[l]
Exercise: Use the 3 formulas above to implement linear_backward()
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 |
# 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 = 1 / m * np.dot(dZ, A_prev.T); db = 1 / m * np.sum(dZ, axis = 1, keepdims = True); 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; |
1 2 3 4 5 6 |
# Set up some test inputs dZ, linear_cache = linear_backward_test_case(); dA_prev, dW, db = linear_backward(dZ, linear_cache); print ("dA_prev = "+ str(dA_prev)); print ("dW = " + str(dW)); print ("db = " + str(db)); |
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 2 3 4 5 6 7 8 |
def linear_backward_test_case(): np.random.seed(1); dZ = np.random.randn(1,2); A = np.random.randn(3,2); W = np.random.randn(1,3); b = np.random.randn(1,1); linear_cache = (A, W, b); return dZ, linear_cache; |
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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 |
# 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; |
1 2 3 4 5 6 7 8 9 10 11 12 13 |
AL, linear_activation_cache = linear_activation_backward_test_case(); dA_prev, dW, db = linear_activation_backward(AL, linear_activation_cache, activation = "sigmoid"); print ("sigmoid:"); print ("dA_prev = "+ str(dA_prev)); print ("dW = " + str(dW)); print ("db = " + str(db) + "\n"); dA_prev, dW, db = linear_activation_backward(AL, linear_activation_cache, activation = "relu"); print ("relu:"); print ("dA_prev = "+ str(dA_prev)); print ("dW = " + str(dW)); print ("db = " + str(db)); |
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 2 3 4 5 6 7 8 9 10 11 12 |
def linear_activation_backward_test_case(): np.random.seed(2); dA = np.random.randn(1,2); A = np.random.randn(3,2); W = np.random.randn(1,3); b = np.random.randn(1,1); Z = np.random.randn(1,2); linear_cache = (A, W, b); activation_cache = Z; linear_activation_cache = (linear_cache, activation_cache); return dA, linear_activation_cache; |
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]=σ(Z[L]) . Your code thus needs to compute =∂L∂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] × (L-1) -> LINEAR -> SIGMOID model.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 |
# 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)); ### END CODE HERE ### # Lth layer (SIGMOID -> LINEAR) gradients. Inputs: "AL, Y, caches". Outputs: "grads["dAL"], grads["dWL"], grads["dbL"] ### START CODE HERE ### (approx. 2 lines) dA_prev, dW, db = linear_activation_backward(dAL, caches[L - 1], "sigmoid"); grads["dA" + str(L)], grads["dW" + str(L)], grads["db" + str(L)] = dA_prev, dW, db; ### END CODE HERE ### for l in reversed(range(L-1)): # lth layer: (RELU -> LINEAR) gradients. # Inputs: "grads["dA" + str(l + 2)], caches". Outputs: "grads["dA" + str(l + 1)] , grads["dW" + str(l + 1)] , grads["db" + str(l + 1)] ### START CODE HERE ### (approx. 5 lines) dA = dA_prev; dA_prev, dW, db = linear_activation_backward(dA, caches[l], "relu"); grads["dA" + str(l + 1)] = dA_prev; grads["dW" + str(l + 1)] = dW; grads["db" + str(l + 1)] = db; ### END CODE HERE ### return grads; |
1 2 3 |
AL, Y_assess, caches = L_model_backward_test_case(); grads = L_model_backward(AL, Y_assess, caches); print_grads(grads); |
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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 |
def L_model_backward_test_case(): """ X = np.random.rand(3,2) Y = np.array([[1, 1]]) parameters = {'W1': np.array([[ 1.78862847, 0.43650985, 0.09649747]]), 'b1': np.array([[ 0.]])} aL, caches = (np.array([[ 0.60298372, 0.87182628]]), [((np.array([[ 0.20445225, 0.87811744], [ 0.02738759, 0.67046751], [ 0.4173048 , 0.55868983]]), np.array([[ 1.78862847, 0.43650985, 0.09649747]]), np.array([[ 0.]])), np.array([[ 0.41791293, 1.91720367]]))]) """ np.random.seed(3) AL = np.random.randn(1, 2) Y = np.array([[1, 0]]) A1 = np.random.randn(4,2) W1 = np.random.randn(3,4) b1 = np.random.randn(3,1) Z1 = np.random.randn(3,2) linear_cache_activation_1 = ((A1, W1, b1), Z1) A2 = np.random.randn(3,2) W2 = np.random.randn(1,3) b2 = np.random.randn(1,1) Z2 = np.random.randn(1,2) linear_cache_activation_2 = ((A2, W2, b2), Z2) caches = (linear_cache_activation_1, linear_cache_activation_2) return AL, Y, caches |
In this section you will update the parameters of the model, using gradient descent:
W[l]=W[l]−α dW[l]
b[l]=b[l]−α db[l]
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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 |
# 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)] -= learning_rate * grads["dW" + str(l + 1)]; parameters["b" + str(l + 1)] -= learning_rate * grads["db" + str(l + 1)]; ### END CODE HERE ### return parameters; |
1 2 3 4 5 6 7 |
parameters, grads = update_parameters_test_case(); parameters = update_parameters(parameters, grads, 0.1); print ("W1 = "+ str(parameters["W1"])); print ("b1 = "+ str(parameters["b1"])); print ("W2 = "+ str(parameters["W2"])); print ("b2 = "+ str(parameters["b2"])); |
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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 |
def update_parameters_test_case(): np.random.seed(2) W1 = np.random.randn(3,4) b1 = np.random.randn(3,1) W2 = np.random.randn(1,3) b2 = np.random.randn(1,1) parameters = {"W1": W1, "b1": b1, "W2": W2, "b2": b2} np.random.seed(3) dW1 = np.random.randn(3,4) db1 = np.random.randn(3,1) dW2 = np.random.randn(1,3) db2 = np.random.randn(1,1) grads = {"dW1": dW1, "db1": db1, "dW2": dW2, "db2": db2} return parameters, grads |
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:
You will in fact use these models to classify cat vs non-cat images!
Let’s first import all the packages that you will need during this assignment.
dnn_app_utils
provides the functions implemented in the “Building your Deep Neural Network: Step by Step” assignment to this notebook.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 |
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_v2 import * %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' %load_ext autoreload %autoreload 2 np.random.seed(1) |
The autoreload extension is already loaded. To reload it, use:
%reload_ext autoreload
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:
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 2 3 4 |
# Example of a picture index = 10; 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."); |
y = 0. It's a non-cat picture.
1 2 3 4 5 6 7 8 9 10 11 12 |
# 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)); |
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 2 3 4 5 6 7 8 9 10 |
# 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)
12288 equals 64×64×3 which is the size of one reshaped image vector.
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:
You will then compare the performance of these models, and also try out different values for L.
Let’s look at the two architectures.
The model can be summarized as: INPUT -> LINEAR -> RELU -> LINEAR -> SIGMOID -> OUTPUT
Detailed Architecture of figure 2:
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:
As usual you will follow the Deep Learning methodology to build the model:
Let’s now implement those two models!
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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 |
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 |
1 2 3 4 5 |
### 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); |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 |
#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, "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 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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 |
def predict(X, y, parameters): """ This function is 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 Returns: p -- predictions for the given dataset X """ m = X.shape[1] n = len(parameters) // 2 # number of layers in the neural network 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 print("Accuracy: " + str(np.sum((p == y)/m))) return p |
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.
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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 |
def initialize_parameters_deep(layer_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] # 5-layer model |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 |
# 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. ### 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 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 (80than 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).
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:
Congratulations on finishing this assignment. You can use your own image and see the output of your model. To do that:
1 2 3 4 5 6 7 8 9 10 11 12 |
## START CODE HERE ## my_image = "1.png"; # 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_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."); |
Accuracy: 1.0
y = 1.0, your L-layer model predicts a "cat" picture.