#5. Classification with one hidden layer

* Dataset

- a numpy-array(matrix) X that contains your features(x1, x2)

- a numpy-array(vector) Y that contains your labels(red: 0, blue: 1)

 

 

* Simple Logistic Regression

- sklearn's build-in function을 사용하여 logistic regression 식을 학습시킬 수 있다.

# Train the logistic regression classifier
clf = sklearn.linear_model.LogisticRegressionCV();
clf.fit(X.T, Y.T);

<Logistic Regression 식 학습>

 

# Plot the decision boundary for logistic regression
plot_decision_boundary(lambda x: clf.predict(x), X, Y)
plt.title("Logistic Regression")

# Print accuracy
LR_predictions = clf.predict(X.T)
print ('Accuracy of logistic regression: %d ' % float((np.dot(Y,LR_predictions) + np.dot(1-Y,1-LR_predictions))/float(Y.size)*100) +
       '% ' + "(percentage of correctly labelled datapoints)")

<학습시킨 모델의 decision boundary 및 accuracy 확인>

 

result: 

- dataset이 linear separable하지 않으므로 logistic regression이 잘 작동하지 못한다. 

 

* Neural Network model

 

- train a NN with a single hidden layer:

- 과정을 설명해보자면:

    1) NN structure를 정의한다.(input unit의 갯수, hidden unit의 갯수 등등)

    2) 모델의 parameter를 초기화한다.

    3) 반복:

        - forward propagation

        - loss 계산

        - backward propagation to get the gradients

        - update parameters(gradient descent)

 

1) Defining the neural network structure

 

- n_x: the size of the input layer(X.shape[0])

- n_h: the size of the hidden layer(이 예시에서는 4)

- n_y: the size of the output layer(Y.shape[0])

 

 

2) Initialize the model's parameters

 

- weight는 random value로 초기화(np.random.randn(a, b) * 0.01)

- bias는 zero로 초기화(np.zeros((a, b))

 

3) The Loop

 

3.1 Forward Propagation

 

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"
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    
    # Implement Forward Propagation to calculate A2 (probabilities)
    Z1 = np.dot(W1, X) + b1 #(1)
    A1 = np.tanh(Z1) #(2)
    Z2 = np.dot(W2, A1) + b2 #(3)
    A2 = sigmoid(Z2) #(4)
    
    assert(A2.shape == (1, X.shape[1]))
    
    cache = {"Z1": Z1,
             "A1": A1,
             "Z2": Z2,
             "A2": A2}
    
    return A2, cache
    
X_assess, parameters = forward_propagation_test_case()
A2, cache = forward_propagation(X_assess, parameters)

# Note: we use the mean here just to make sure that your output matches ours. 
print(np.mean(cache['Z1']) ,np.mean(cache['A1']),np.mean(cache['Z2']),np.mean(cache['A2']))

result: 0.262818640198 0.091999045227 -1.30766601287 0.212877681719

 

 

3.2 Cost Function

 

Cost Function

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
    [Note that the parameters argument is not used in this function, 
    but the auto-grader currently expects this parameter.
    Future version of this notebook will fix both the notebook 
    and the auto-grader so that `parameters` is not needed.
    For now, please include `parameters` in the function signature,
    and also when invoking this function.]
    
    Returns:
    cost -- cross-entropy cost given equation (13)
    
    """
    
    m = Y.shape[1] # number of example

    # Compute the cross-entropy cost
    logprobs = np.multiply(np.log(A2),Y)
    cost = - (1/m) * np.sum(logprobs + (1 - Y)*np.log(1-A2))
    
    cost = float(np.squeeze(cost))  # makes sure cost is the dimension we expect. 
                                    # E.g., turns [[17]] into 17 
    assert(isinstance(cost, float))
    
    return cost
    
A2, Y_assess, parameters = compute_cost_test_case()
print("cost = " + str(compute_cost(A2, Y_assess, parameters)))

result: cost = 0.6930587610394646

 

 

3.3 Backward Propagation

 

gradient descent

 

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".
    W1 = parameters["W1"]
    W2 = parameters["W2"]
        
    # Retrieve also A1 and A2 from dictionary "cache".
    A1 = cache["A1"]
    A2 = cache["A2"]
    
    # Backward propagation: calculate dW1, db1, dW2, db2.  above)
    dZ2 = A2 - Y
    dW2 = (1/m) * np.dot(dZ2, A1.T)
    db2 = (1/m) * np.sum(dZ2, axis = 1, keepdims = True)
    dZ1 = np.multiply(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)
    
    grads = {"dW1": dW1,
             "db1": db1,
             "dW2": dW2,
             "db2": db2}
    
    return grads

 

 

3.4 Update Parameters

 

- Gradient Descent Rule: θ=θ−α*(∂J/∂θ) (α:learning rate, θ:parameter)

- Good learning rate라면 converge할 것이고 bad learning rate라면 diverge할 것이다.

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"
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    
    # Retrieve each gradient from the dictionary "grads"
    dW1 = grads["dW1"]
    db1 = grads["db1"]
    dW2 = grads["dW2"]
    db2 = grads["db2"]
    
    # Update rule for each parameter
    W1 = W1 - learning_rate * dW1
    b1 = b2 - learning_rate * db1
    W2 = W2 - learning_rate * dW2
    b2 = b2 - learning_rate *  db2
    
    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2}
    
    return parameters
    
parameters, grads = update_parameters_test_case()
parameters = update_parameters(parameters, grads)

print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))

result: W1 = [[-0.00643025 0.01936718] [-0.02410458 0.03978052] [-0.01653973 -0.02096177] [ 0.01046864 -0.05990141]] b1 = [[ 9.13687537e-05] [ 9.60772116e-05] [ 9.17236240e-05] [ 9.08396764e-05]] W2 = [[-0.01041081 -0.04463285 0.01758031 0.04747113]] b2 = [[ 0.00010457]]

 

4) Integrate parts 1), 2) and 3) in nn_model()

 

 

5) Predictions

 

- use forward propagation to predict results

- hidden unit 갯수가 늘어날수록 training set에는 더 잘 맞지만 overfitting 문제가 발생하게 된다.

- normalization을 이용하여 overfitting 문제를 조금은 해결할 수 있다.