3 - General Architecture of the learning algorithm
It’s time to design a simple algorithm to distinguish cat images from non-cat images.
You will build a Logistic Regression, using a Neural Network mindset. The following Figure explains why Logistic Regression is actually a very simple Neural Network!
Key steps:
In this exercise, you will carry out the following steps:
- Initialize the parameters of the model
- Learn the parameters for the model by minimizing the cost
- Use the learned parameters to make predictions (on the test set)
- Analyse the results and conclude
4 - Building the parts of our algorithm ##
The main steps for building a Neural Network are:
1. Define the model structure (such as number of input features)
2. Initialize the model’s parameters
3. Loop:
- Calculate current loss (forward propagation)
- Calculate current gradient (backward propagation)
- Update parameters (gradient descent)
You often build 1-3 separately and integrate them into one function we call model().
.1 - Helper functions
Exercise: Using your code from “Python Basics”, implement sigmoid().
| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | #1 GRADED FUNCTION: sigmoid def sigmoid(z): """ Compute the sigmoid of z Arguments: z -- A scalar or numpy array of any size. Return: s -- sigmoid(z) """ s = 1.0/(1+np.exp(-z)) return s print ("sigmoid([0, 2]) = " + str(sigmoid(np.array([0,2])))) |
4.2 - Initializing parameters
Exercise: Implement parameter initialization in the cell below. You have to initialize w as a vector of zeros. If you don’t know what numpy function to use, look up np.zeros() in the Numpy library’s documentation.
| 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 | #2Initializing parameters def initialize_with_zeros(dim): """ This function creates a vector of zeros of shape (dim, 1) for w and initializes b to 0. Argument: dim -- size of the w vector we want (or number of parameters in this case) Returns: w -- initialized vector of shape (dim, 1) b -- initialized scalar (corresponds to the bias) """ ### START CODE HERE ### (≈ 1 line of code) w = np.zeros((dim, 1)) b = 0 ### END CODE HERE ### #python assert断言是声明其布尔值必须为真的判定，如果发生异常就说明表达示为假。可以 #理解assert断言语句为raise-if-not，用来测试表示式，其返回值为假，就会触发异常 assert(w.shape == (dim, 1)) assert(isinstance(b, float) or isinstance(b, int))#判断b是int或者是float return w, b dim = 2 w, b = initialize_with_zeros(dim) print ("w = " + str(w)) print ("b = " + str(b)) |
4.3 - Forward and Backward propagation
Now that your parameters are initialized, you can do the “forward” and “backward” propagation steps for learning the parameters.
Exercise: Implement a function propagate() that computes the cost function and its gradient.
| 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 | # GRADED FUNCTION: propagate def propagate(w, b, X, Y): """ Implement the cost function and its gradient for the propagation explained above Arguments: w -- weights, a numpy array of size (num_px * num_px * 3, 1) b -- bias, a scalar X -- data of size (num_px * num_px * 3, number of examples) Y -- true "label" vector (containing 0 if non-cat, 1 if cat) of size (1, number of examples) Return: cost -- negative log-likelihood cost for logistic regression dw -- gradient of the loss with respect to w, thus same shape as w db -- gradient of the loss with respect to b, thus same shape as b Tips: - Write your code step by step for the propagation. np.log(), np.dot() """ m = X.shape[1] # FORWARD PROPAGATION (FROM X TO COST) ### START CODE HERE ### (≈ 2 lines of code) A = sigmoid(np.dot(w.T,X)+b) # compute activation cost = -(1.0/m)*np.sum(Y*np.log(A)+(1-Y)*np.log(1-A)) # compute cost ### END CODE HERE ### # BACKWARD PROPAGATION (TO FIND GRAD) ### START CODE HERE ### (≈ 2 lines of code) dw = (1.0/m)*np.dot(X,(A-Y).T) db = (1.0/m)*np.sum(A-Y) ### END CODE HERE ### assert(dw.shape == w.shape) assert(db.dtype == float) cost = np.squeeze(cost) assert(cost.shape == ()) grads = {"dw": dw, "db": db} return grads, cost w, b, X, Y = np.array([[1.],[2.]]), 2., np.array([[1.,2.,-1.],[3.,4.,-3.2]]), np.array([[1,0,1]]) grads, cost = propagate(w, b, X, Y) print ("dw = " + str(grads["dw"])) print ("db = " + str(grads["db"])) print ("cost = " + str(cost)) |
d) Optimization
· You have initialized your parameters.
· You are also able to compute a cost function and its gradient.
· Now, you want to update the parameters using gradient descent.
Exercise: Write down the optimization function. The goal is to learn w and b by minimizing the cost function J. For a parameter θ, the update rule is θ=θ−α dθ, where α is the learning rate.
# GRADED FUNCTION: optimize
| 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 | def optimize(w, b, X, Y, num_iterations, learning_rate, print_cost = False): """ This function optimizes w and b by running a gradient descent algorithm Arguments: w -- weights, a numpy array of size (num_px * num_px * 3, 1) b -- bias, a scalar X -- data of shape (num_px * num_px * 3, number of examples) Y -- true "label" vector (containing 0 if non-cat, 1 if cat), of shape (1, number of examples) num_iterations -- number of iterations of the optimization loop learning_rate -- learning rate of the gradient descent update rule print_cost -- True to print the loss every 100 steps Returns: params -- dictionary containing the weights w and bias b grads -- dictionary containing the gradients of the weights and bias with respect to the cost function costs -- list of all the costs computed during the optimization, this will be used to plot the learning curve. Tips: You basically need to write down two steps and iterate through them: 1) Calculate the cost and the gradient for the current parameters. Use propagate(). 2) Update the parameters using gradient descent rule for w and b. """ costs = [] for i in range(num_iterations): # Cost and gradient calculation (≈ 1-4 lines of code) ### START CODE HERE ### grads, cost = propagate(w, b, X, Y) ### END CODE HERE ### # Retrieve derivatives from grads dw = grads["dw"] db = grads["db"] # update rule (≈ 2 lines of code) ### START CODE HERE ### w = w - learning_rate*dw b = b - learning_rate*db ### END CODE HERE ### # Record the costs if i % 100 == 0: costs.append(cost) # Print the cost every 100 training examples if print_cost and i % 100 == 0: print ("Cost after iteration %i: %f" %(i, cost)) params = {"w": w, "b": b} grads = {"dw": dw, "db": db} return params, grads, costs params, grads, costs = optimize(w, b, X, Y, num_iterations= 100, learning_rate = 0.009, print_cost = False) print ("w = " + str(params["w"])) print ("b = " + str(params["b"])) print ("dw = " + str(grads["dw"])) print ("db = " + str(grads["db"])) |
Exercise: The previous function will output the learned w and b. We are able to use w and b to predict the labels for a dataset X. Implement the predict() function. There is two steps to computing predictions:
1. Calculate Y^=A=σ(wTX+b)
2. Convert the entries of a into 0 (if activation <= 0.5) or 1 (if activation > 0.5), stores the predictions in a vector Y_prediction. If you wish, you can use an if/else statement in a for loop (though there is also a way to vectorize this).
| 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 | # GRADED FUNCTION: predict def predict(w, b, X): ''' Predict whether the label is 0 or 1 using learned logistic regression parameters (w, b) Arguments: w -- weights, a numpy array of size (num_px * num_px * 3, 1) b -- bias, a scalar X -- data of size (num_px * num_px * 3, number of examples) Returns: Y_prediction -- a numpy array (vector) containing all predictions (0/1) for the examples in X ''' m = X.shape[1] Y_prediction = np.zeros((1,m)) w = w.reshape(X.shape[0], 1) # Compute vector "A" predicting the probabilities of a cat being present in the picture ### START CODE HERE ### (≈ 1 line of code) A = sigmoid(np.dot(w.T, X) + b) ### END CODE HERE ### for i in range(A.shape[1]): # Convert probabilities A[0,i] to actual predictions p[0,i] ### START CODE HERE ### (≈ 4 lines of code) if A[0,i] > 0.5: Y_prediction[0,i] = 1 else: Y_prediction[0,i] = 0 ### END CODE HERE ### assert(Y_prediction.shape == (1, m)) return Y_prediction w = np.array([[0.1124579],[0.23106775]]) b = -0.3 X = np.array([[1.,-1.1,-3.2],[1.2,2.,0.1]]) print ("predictions = " + str(predict(w, b, X))) |
What to remember:
You’ve implemented several functions that:
- Initialize (w,b)
- Optimize the loss iteratively to learn parameters (w,b):
- computing the cost and its gradient
- updating the parameters using gradient descent
- Use the learned (w,b) to predict the labels for a given set of examples
5 - Merge all functions into a model
You will now see how the overall model is structured by putting together all the building blocks (functions implemented in the previous parts) together, in the right order.
Exercise: Implement the model function. Use the following notation:
- Y_prediction for your predictions on the test set
- Y_prediction_train for your predictions on the train set
- w, costs, grads for the outputs of optimize()
| 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 | def model(X_train, Y_train, X_test, Y_test, num_iterations = 2000, learning_rate = 0.5, print_cost = False): """ Builds the logistic regression model by calling the function you've implemented previously Arguments: X_train -- training set represented by a numpy array of shape (num_px * num_px * 3, m_train) Y_train -- training labels represented by a numpy array (vector) of shape (1, m_train) X_test -- test set represented by a numpy array of shape (num_px * num_px * 3, m_test) Y_test -- test labels represented by a numpy array (vector) of shape (1, m_test) num_iterations -- hyperparameter representing the number of iterations to optimize the parameters learning_rate -- hyperparameter representing the learning rate used in the update rule of optimize() print_cost -- Set to true to print the cost every 100 iterations Returns: d -- dictionary containing information about the model. """ ### START CODE HERE ### # initialize parameters with zeros (≈ 1 line of code) w, b = initialize_with_zeros(X_train.shape[0]) # Gradient descent (≈ 1 line of code) parameters, grads, costs = optimize(w, b, X_train, Y_train, num_iterations, learning_rate, print_cost) # Retrieve parameters w and b from dictionary "parameters" w = parameters["w"] b = parameters["b"] # Predict test/train set examples (≈ 2 lines of code) Y_prediction_test = predict(w, b, X_test) Y_prediction_train = predict(w, b, X_train) ### END CODE HERE ### # Print train/test Errors print("train accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_train - Y_train)) * 100)) print("test accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_test - Y_test)) * 100)) d = {"costs": costs, "Y_prediction_test": Y_prediction_test, "Y_prediction_train" : Y_prediction_train, "w" : w, "b" : b, "learning_rate" : learning_rate, "num_iterations": num_iterations} return d d = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 2000, learning_rate = 0.005, print_cost = True) |
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