简单感知器中的正确反向传播

Question

鉴于简单的或门问题：

or_input = np.array([[0,0], [0,1], [1,0], [1,1]])
or_output = np.array([[0,1,1,1]]).T

如果我们训练一个简单的单层感知器（没有反向传播），我们可以这样做：

import numpy as np
np.random.seed(0)

def sigmoid(x): # Returns values that sums to one.
    return 1 / (1 + np.exp(-x))

def cost(predicted, truth):
    return (truth - predicted)**2

or_input = np.array([[0,0], [0,1], [1,0], [1,1]])
or_output = np.array([[0,1,1,1]]).T

# Define the shape of the weight vector.
num_data, input_dim = or_input.shape
# Define the shape of the output vector. 
output_dim = len(or_output.T)

num_epochs = 50 # No. of times to iterate.
learning_rate = 0.03 # How large a step to take per iteration.

# Lets standardize and call our inputs X and outputs Y
X = or_input
Y = or_output
W = np.random.random((input_dim, output_dim))

for _ in range(num_epochs):
    layer0 = X
    # Forward propagation.
    # Inside the perceptron, Step 2. 
    layer1 = sigmoid(np.dot(X, W))

    # How much did we miss in the predictions?
    cost_error = cost(layer1, Y)

    # update weights
    W +=  - learning_rate * np.dot(layer0.T, cost_error)

# Expected output.
print(Y.tolist())
# On the training data
print([[int(prediction > 0.5)] for prediction in layer1])

[出]：

[[0], [1], [1], [1]]
[[0], [1], [1], [1]]

通过反向传播，计算d(cost)/d(X)，以下步骤是否正确？

• 通过将成本误差与成本的导数相乘来计算 layer1 误差

• 然后通过将第 1 层误差与 sigmoid 的导数相乘来计算第 1 层增量

• 然后在输入和 layer1 delta 之间做一个点积，以获得 d(cost)/d(X) 的差异。

然后将d(cost)/d(X)乘以学习率的负数，进行梯度下降。

num_epochs = 0 # No. of times to iterate.
learning_rate = 0.03 # How large a step to take per iteration.

# Lets standardize and call our inputs X and outputs Y
X = or_input
Y = or_output
W = np.random.random((input_dim, output_dim))

for _ in range(num_epochs):
    layer0 = X
    # Forward propagation.
    # Inside the perceptron, Step 2. 
    layer1 = sigmoid(np.dot(X, W))

    # How much did we miss in the predictions?
    cost_error = cost(layer1, Y)

    # Back propagation.
    # multiply how much we missed from the gradient/slope of the cost for our prediction.
    layer1_error = cost_error * cost_derivative(cost_error)

    # multiply how much we missed by the gradient/slope of the sigmoid at the values in layer1
    layer1_delta = layer1_error * sigmoid_derivative(layer1)

    # update weights
    W +=  - learning_rate * np.dot(layer0.T, layer1_delta)

在这种情况下，使用cost_derivative 和sigmoid_derivative 的实现是否应该如下所示？

import numpy as np
np.random.seed(0)

def sigmoid(x): # Returns values that sums to one.
    return 1 / (1 + np.exp(-x))

def sigmoid_derivative(sx):
    # See https://math.stackexchange.com/a/1225116
    return sx * (1 - sx)

def cost(predicted, truth):
    return (truth - predicted)**2

def cost_derivative(y):
    # If the cost is:
    # cost = y - y_hat
    # What's the derivative of d(cost)/d(y)
    # d(cost)/d(y) = 1
    return 2*y


or_input = np.array([[0,0], [0,1], [1,0], [1,1]])
or_output = np.array([[0,1,1,1]]).T

# Define the shape of the weight vector.
num_data, input_dim = or_input.shape
# Define the shape of the output vector. 
output_dim = len(or_output.T)

num_epochs = 5 # No. of times to iterate.
learning_rate = 0.03 # How large a step to take per iteration.

# Lets standardize and call our inputs X and outputs Y
X = or_input
Y = or_output
W = np.random.random((input_dim, output_dim))

for _ in range(num_epochs):
    layer0 = X
    # Forward propagation.
    # Inside the perceptron, Step 2. 
    layer1 = sigmoid(np.dot(X, W))

    # How much did we miss in the predictions?
    cost_error = cost(layer1, Y)

    # Back propagation.
    # multiply how much we missed from the gradient/slope of the cost for our prediction.
    layer1_error = cost_error * cost_derivative(cost_error)

    # multiply how much we missed by the gradient/slope of the sigmoid at the values in layer1
    layer1_delta = layer1_error * sigmoid_derivative(layer1)

    # update weights
    W +=  - learning_rate * np.dot(layer0.T, layer1_delta)

# Expected output.
print(Y.tolist())
# On the training data
print([[int(prediction > 0.5)] for prediction in layer1])

[出]：

[[0], [1], [1], [1]]
[[0], [1], [1], [1]]

---

顺便说一句，给定随机输入种子，即使没有 W 和梯度下降或感知器，预测仍然是正确的：

import numpy as np
np.random.seed(0)

# Lets standardize and call our inputs X and outputs Y
X = or_input
Y = or_output
W = np.random.random((input_dim, output_dim))

# On the training data
predictions = sigmoid(np.dot(X, W))
[[int(prediction > 0.5)] for prediction in predictions]

Answer 1

你几乎是正确的。在您的实现中，您将成本定义为误差的平方，这是始终为正的不幸结果。因此，如果您绘制均值（cost_error），它会在每次迭代中缓慢上升，而您的权重会缓慢下降。

在您的特定情况下，您可以使任何权重 >0 以使其正常工作：如果您尝试使用足够的 epoch 进行实施，您的权重将变为负数，您的网络将不再工作。

您可以删除成本函数中的正方形：

def cost(predicted, truth):
    return (truth - predicted)

现在要更新您的权重，您需要评估错误“位置”处的梯度。所以你需要的是：

d_predicted = output_errors * sigmoid_derivative(predicted_output)

接下来，我们更新权重：

W += np.dot(X.T, d_predicted) * learning_rate

带有错误显示的完整代码：

import numpy as np
import matplotlib.pyplot as plt
np.random.seed(0)

def sigmoid(x): # Returns values that sums to one.
    return 1 / (1 + np.exp(-x))

def sigmoid_derivative(sx):
    # See https://math.stackexchange.com/a/1225116
    return sx * (1 - sx)

def cost(predicted, truth):
    return (truth - predicted)

or_input = np.array([[0,0], [0,1], [1,0], [1,1]])
or_output = np.array([[0,1,1,1]]).T

# Define the shape of the weight vector.
num_data, input_dim = or_input.shape
# Define the shape of the output vector. 
output_dim = len(or_output.T)

num_epochs = 50 # No. of times to iterate.
learning_rate = 0.1 # How large a step to take per iteration.

# Lets standardize and call our inputs X and outputs Y
X = or_input
Y = or_output
W = np.random.random((input_dim, output_dim))

# W = [[-1],[1]] # you can try to set bad weights to see the training process
error_list = []

for _ in range(num_epochs):
    layer0 = X
    # Forward propagation.
    layer1 = sigmoid(np.dot(X, W))

    # How much did we miss in the predictions?
    cost_error = cost(layer1, Y)
    error_list.append(np.mean(cost_error)) # save the loss to plot later

    # Back propagation.
    # eval the gradient :
    d_predicted = cost_error * sigmoid_derivative(cost_error)

    # update weights
    W = W + np.dot(X.T, d_predicted) * learning_rate


# Expected output.
print(Y.tolist())
# On the training data
print([[int(prediction > 0.5)] for prediction in layer1])

# plot error curve : 
plt.plot(range(num_epochs), loss_list, '+b')
plt.xlabel('Epoch')
plt.ylabel('mean error')

我还添加了一行手动设置初始权重，看看网络是如何学习的