deeplearning.ai-正向和反向傳播演算法公式
阿新 • • 發佈:2018-11-25
【正向和反向傳播】
【梯度下降i法】
【邏輯迴歸代價函式】
【實現神經網路的步驟】
【淺層神經網路例子】
import numpy as np def sigmoid(x): """ Compute the sigmoid of x Arguments: x -- A scalar or numpy array of any size Return: s -- sigmoid(x) """ ### START CODE HERE ### (≈ 1 line of code) s = 1 / (1 + np.exp(-x)) ### END CODE HERE ### return s 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 layer n_h -- the size of the hidden layer n_y -- the size of the output layer """ ### START CODE HERE ### (≈ 3 lines of code) n_x = X.shape[0] # size of input layer n_h = 4 n_y = Y.shape[0] # size of output layer ### END CODE HERE ### return (n_x, n_h, n_y) 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: 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. ### 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 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 activation cache -- a dictionary containing "Z1", "A1", "Z2" and "A2" """ # Retrieve each parameter from the dictionary "parameters" ### START CODE HERE ### (≈ 4 lines of code) W1 = parameters['W1'] b1 = parameters['b1'] W2 = parameters['W2'] b2 = parameters['b2'] ### END CODE HERE ### # Implement Forward Propagation to calculate A2 (probabilities) ### START CODE HERE ### (≈ 4 lines of code) Z1 = np.dot(W1 ,X) + b1 A1 = np.tanh(Z1) Z2 = np.dot(W2 ,A1) + b2 A2 = sigmoid(Z2) ### END CODE HERE ### assert(A2.shape == (1, X.shape[1])) cache = {"Z1": Z1, "A1": A1, "Z2": Z2, "A2": A2} return A2, cache 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 b2 Returns: cost -- cross-entropy cost given equation (13) """ m = Y.shape[1] # number of example # Compute the cross-entropy cost ### START CODE HERE ### (≈ 2 lines of code) logprobs = np.multiply(np.log(A2) ,Y) + np.multiply(( 1 -Y), np.log( 1 -A2)) #logprobs = np.multiply(np.log(A2), Y) + (1 - Y) * np.log(1 - A2) cost = - 1 / m *np.sum(logprobs) ### END CODE HERE ### cost = np.squeeze(cost) # makes sure cost is the dimension we expect. # E.g., turns [[17]] into 17 assert(isinstance(cost, float)) return cost 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". ### START CODE HERE ### (≈ 2 lines of code) W1 = parameters['W1'] W2 = parameters['W2'] ### END CODE HERE ### # Retrieve also A1 and A2 from dictionary "cache". ### START CODE HERE ### (≈ 2 lines of code) A1 = cache['A1'] A2 = cache['A2'] ### END CODE HERE ### # Backward propagation: calculate dW1, db1, dW2, db2. ### START CODE HERE ### (≈ 6 lines of code, corresponding to 6 equations on slide above) dZ2 = A2 - Y dW2 = 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) ### END CODE HERE ### 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 above Arguments: 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" ### START CODE HERE ### (≈ 4 lines of code) W1 = parameters['W1'] b1 = parameters['b1'] W2 = parameters['W2'] b2 = parameters['b2'] ### END CODE HERE ### # Retrieve each gradient from the dictionary "grads" ### START CODE HERE ### (≈ 4 lines of code) dW1 = grads['dW1'] db1 = grads['db1'] dW2 = grads['dW2'] db2 = grads['db2'] ## END CODE HERE ### # Update rule for each parameter ### START CODE HERE ### (≈ 4 lines of code) W1 = W1 - learning_rate* dW1 b1 = b1 - learning_rate * db1 W2 = W2 - learning_rate * dW2 b2 = b2 - learning_rate * db2 ### END CODE HERE ### parameters = {"W1": W1, "b1": b1, "W2": W2, "b2": b2} return parameters 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 layer num_iterations -- Number of iterations in gradient descent loop print_cost -- if True, print the cost every 1000 iterations Returns: 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". ### START CODE HERE ### (≈ 5 lines of code) parameters = initialize_parameters(n_x, n_h, n_y) W1 = parameters['W1'] b1 = parameters['b1'] W2 = parameters['W2'] b2 = parameters['b2'] ### END CODE HERE ### # Loop (gradient descent) for i in range(0, num_iterations): ### START CODE HERE ### (≈ 4 lines of code) # 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) ### END CODE HERE ### # Print the cost every 1000 iterations if print_cost and i % 1000 == 0: print ("Cost after iteration %i: %f" %(i, cost)) return parameters def nn_model_test_case(): X_assess = np.array([[1.62434536, - 0.61175641, - 0.52817175], [-1.07296862, 0.86540763, - 2.3015387]]) Y_assess = np.array([[True, False, True]]) return X_assess, Y_assess def nn_mode_test(): X_assess, Y_assess = nn_model_test_case() print("X_assess =", X_assess) print("Y_assess =", Y_assess) forward_propagation_test_case() parameters = nn_model(X_assess, Y_assess, 4, num_iterations=10000, print_cost=True) print("W1 = " + str(parameters["W1"])) print("b1 = " + str(parameters["b1"])) print("W2 = " + str(parameters["W2"])) print("b2 = " + str(parameters["b2"])) if __name__ == "__main__": nn_mode_test()