反向传播

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从这个例子中你可以看到 sigmoid 做激活函数的一个缺点。sigmoid 函数导数的最大值是 0.25,因此输出层的误差被减少了至少 75%(1-0.25),隐藏层的误差被减少了至少 93.75%(1 - 0.25 * 0.25)!如果你的神经网络有很多层,使用 sigmoid 激活函数会很快把靠近输入层的权重步长降为很小的值,该问题称作梯度消失。


反向传播实例 1
import numpy as np


def sigmoid(x):
    """
    Calculate sigmoid
    """
    return 1 / (1 + np.exp(-x))
    
def sigmoid_prim(x):
    """
    Calculate sigmoid
    """
    return sigmoid(x) * (1 - sigmoid(x))

x = np.array([0.5, 0.1, -0.2])
target = 0.6
learnrate = 0.5

weights_input_hidden = np.array([[0.5, -0.6],
                                 [0.1, -0.2],
                                 [0.1, 0.7]])

weights_hidden_output = np.array([0.1, -0.3])

## Forward pass
hidden_layer_input = np.dot(x, weights_input_hidden)
hidden_layer_output = sigmoid(hidden_layer_input)

output_layer_in = np.dot(hidden_layer_output, weights_hidden_output)

output = sigmoid(output_layer_in)

## Backwards pass
## TODO: Calculate output error
error = target - output

# TODO: Calculate error term for output layer
output_error_term = error * sigmoid_prim(output_layer_in)
print(output_error_term)
print(weights_hidden_output)
# TODO: Calculate error term for hidden layer
hidden_error = np.dot(output_error_term, weights_hidden_output)
hidden_error_term = hidden_error  * sigmoid_prim(hidden_layer_input)
print("hidden_error_term:" ,hidden_error_term)
print("                  ",x[:,None])
# TODO: Calculate change in weights for hidden layer to output layer
delta_w_h_o = learnrate * output_error_term * hidden_layer_output
print("delta_w_h_o:",delta_w_h_o)
# TODO: Calculate change in weights for input layer to hidden layer
delta_w_i_h = learnrate * x[:,None] * hidden_error_term  

print('Change in weights for hidden layer to output layer:')
print(delta_w_h_o)
print('Change in weights for input layer to hidden layer:')
print(delta_w_i_h)


反向传播实例 2
import numpy as np
from data_prep import features, targets, features_test, targets_test #data import

np.random.seed(21)

def sigmoid(x):
    """
    Calculate sigmoid
    """
    return 1 / (1 + np.exp(-x))
def sigmoid_prime(x):
    """
    Calculate sigmoid
    """
    return sigmoid(x) * (1 - sigmoid(x))

# Hyperparameters
n_hidden = 2  # number of hidden units
epochs = 900
learnrate = 0.005

n_records, n_features = features.shape
last_loss = None
# Initialize weights
weights_input_hidden = np.random.normal(scale=1 / n_features ** .5,
                                        size=(n_features, n_hidden))
weights_hidden_output = np.random.normal(scale=1 / n_features ** .5,
                                         size=n_hidden)

for e in range(epochs):
    del_w_input_hidden = np.zeros(weights_input_hidden.shape)
    del_w_hidden_output = np.zeros(weights_hidden_output.shape)
    for x, y in zip(features.values, targets):
        ## Forward pass ##
        # TODO: Calculate the output
        hidden_input = np.dot(x,weights_input_hidden)
        hidden_output = sigmoid(hidden_input)
        o_input = np.dot(hidden_output,weights_hidden_output)
        output = sigmoid(o_input)

        ## Backward pass ##
        # TODO: Calculate the network's prediction error
        error = y - output

        # TODO: Calculate error term for the output unit
        output_error_term = error * sigmoid_prime(o_input)

        ## propagate errors to hidden layer

        # TODO: Calculate the hidden layer's contribution to the error
        hidden_error = np.dot(output_error_term , weights_hidden_output)
        
        # TODO: Calculate the error term for the hidden layer
        hidden_error_term = hidden_error * sigmoid_prime(hidden_input)
        
        # TODO: Update the change in weights
        del_w_hidden_output += output_error_term * hidden_output
        del_w_input_hidden += hidden_error_term * x[:,None] 

    # TODO: Update weights
    weights_input_hidden += learnrate * del_w_input_hidden / n_records
    weights_hidden_output += learnrate * del_w_hidden_output / n_records

    # Printing out the mean square error on the training set
    if e % (epochs / 10) == 0:
        hidden_output = sigmoid(np.dot(x, weights_input_hidden))
        out = sigmoid(np.dot(hidden_output,
                             weights_hidden_output))
        loss = np.mean((out - targets) ** 2)

        if last_loss and last_loss < loss:
            print("Train loss: ", loss, "  WARNING - Loss Increasing")
        else:
            print("Train loss: ", loss)
        last_loss = loss

# Calculate accuracy on test data
hidden = sigmoid(np.dot(features_test, weights_input_hidden))
out = sigmoid(np.dot(hidden, weights_hidden_output))
predictions = out > 0.5
accuracy = np.mean(predictions == targets_test)
print("Prediction accuracy: {:.3f}".format(accuracy))