Учитывая проблему XOR:
X = xor_input = np.array([[0,0], [0,1], [1,0], [1,1]])
Y = xor_output = np.array([[0,1,1,0]]).T
И простой
- двухслойный многослойный персептрон (MLP) с
- сигмовидная активация между ними и
- Среднеквадратичная ошибка (MSE) как функция потерь / критерий оптимизации
[код]:
def sigmoid(x): # Returns values that sums to one.
return 1 / (1 + np.exp(-x))
def sigmoid_derivative(sx): # For backpropagation.
# See https://math.stackexchange.com/a/1225116
return sx * (1 - sx)
# Cost functions.
def mse(predicted, truth):
return np.sum(np.square(truth - predicted))
X = xor_input = np.array([[0,0], [0,1], [1,0], [1,1]])
Y = xor_output = np.array([[0,1,1,0]]).T
# Define the shape of the weight vector.
num_data, input_dim = X.shape
# Lets set the dimensions for the intermediate layer.
hidden_dim = 5
# Initialize weights between the input layers and the hidden layer.
W1 = np.random.random((input_dim, hidden_dim))
# Define the shape of the output vector.
output_dim = len(Y.T)
# Initialize weights between the hidden layers and the output layer.
W2 = np.random.random((hidden_dim, output_dim))
И учитывая критерии остановки как фиксированное нет. эпох (число итераций по X и Y) с фиксированной скоростью обучения 0,3:
# Initialize weigh
num_epochs = 10000
learning_rate = 0.3
Когда я бегу по распространению вперед-назад и обновляю веса в каждую эпоху, как мне обновлять веса?
Я пытался просто сложить произведение скорости обучения с точечным произведением обратно распространенной производной с выходами слоя, но модель все еще обновляла веса только в одном направлении, в результате чего все веса снижались почти до нуля.
for epoch_n in range(num_epochs):
layer0 = X
# Forward propagation.
# Inside the perceptron, Step 2.
layer1 = sigmoid(np.dot(layer0, W1))
layer2 = sigmoid(np.dot(layer1, W2))
# Back propagation (Y -> layer2)
# How much did we miss in the predictions?
layer2_error = mse(layer2, Y)
#print(layer2_error)
# In what direction is the target value?
# Were we really close? If so, don't change too much.
layer2_delta = layer2_error * sigmoid_derivative(layer2)
# Back propagation (layer2 -> layer1)
# How much did each layer1 value contribute to the layer2 error (according to the weights)?
layer1_error = np.dot(layer2_delta, W2.T)
layer1_delta = layer1_error * sigmoid_derivative(layer1)
# update weights
W2 += - learning_rate * np.dot(layer1.T, layer2_delta)
W1 += - learning_rate * np.dot(layer0.T, layer1_delta)
#print(np.dot(layer0.T, layer1_delta))
#print(epoch_n, list((layer2)))
# Log the loss value as we proceed through the epochs.
losses.append(layer2_error.mean())
Как правильно обновлять веса?
Полный код:
from itertools import chain
import matplotlib.pyplot as plt
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)
# Cost functions.
def mse(predicted, truth):
return np.sum(np.square(truth - predicted))
X = xor_input = np.array([[0,0], [0,1], [1,0], [1,1]])
Y = xor_output = np.array([[0,1,1,0]]).T
# Define the shape of the weight vector.
num_data, input_dim = X.shape
# Lets set the dimensions for the intermediate layer.
hidden_dim = 5
# Initialize weights between the input layers and the hidden layer.
W1 = np.random.random((input_dim, hidden_dim))
# Define the shape of the output vector.
output_dim = len(Y.T)
# Initialize weights between the hidden layers and the output layer.
W2 = np.random.random((hidden_dim, output_dim))
# Initialize weigh
num_epochs = 10000
learning_rate = 0.3
losses = []
for epoch_n in range(num_epochs):
layer0 = X
# Forward propagation.
# Inside the perceptron, Step 2.
layer1 = sigmoid(np.dot(layer0, W1))
layer2 = sigmoid(np.dot(layer1, W2))
# Back propagation (Y -> layer2)
# How much did we miss in the predictions?
layer2_error = mse(layer2, Y)
#print(layer2_error)
# In what direction is the target value?
# Were we really close? If so, don't change too much.
layer2_delta = layer2_error * sigmoid_derivative(layer2)
# Back propagation (layer2 -> layer1)
# How much did each layer1 value contribute to the layer2 error (according to the weights)?
layer1_error = np.dot(layer2_delta, W2.T)
layer1_delta = layer1_error * sigmoid_derivative(layer1)
# update weights
W2 += - learning_rate * np.dot(layer1.T, layer2_delta)
W1 += - learning_rate * np.dot(layer0.T, layer1_delta)
#print(np.dot(layer0.T, layer1_delta))
#print(epoch_n, list((layer2)))
# Log the loss value as we proceed through the epochs.
losses.append(layer2_error.mean())
# Visualize the losses
plt.plot(losses)
plt.show()
Я что-то упускаю в обратном распространении?
Может быть, я пропустил производную от стоимости до второго слоя?
Отредактировано
Я понял, что пропустил частную производную от стоимости до второго слоя и после добавления:
# Cost functions.
def mse(predicted, truth):
return 0.5 * np.sum(np.square(predicted - truth)).mean()
def mse_derivative(predicted, truth):
return predicted - truth
С обновленной петлей обратного распространения в разные эпохи:
for epoch_n in range(num_epochs):
layer0 = X
# Forward propagation.
# Inside the perceptron, Step 2.
layer1 = sigmoid(np.dot(layer0, W1))
layer2 = sigmoid(np.dot(layer1, W2))
# Back propagation (Y -> layer2)
# How much did we miss in the predictions?
cost_error = mse(layer2, Y)
cost_delta = mse_derivative(layer2, Y)
#print(layer2_error)
# In what direction is the target value?
# Were we really close? If so, don't change too much.
layer2_error = np.dot(cost_delta, cost_error)
layer2_delta = layer2_error * sigmoid_derivative(layer2)
# Back propagation (layer2 -> layer1)
# How much did each layer1 value contribute to the layer2 error (according to the weights)?
layer1_error = np.dot(layer2_delta, W2.T)
layer1_delta = layer1_error * sigmoid_derivative(layer1)
# update weights
W2 += - learning_rate * np.dot(layer1.T, layer2_delta)
W1 += - learning_rate * np.dot(layer0.T, layer1_delta)
Казалось, что ты тренируешься и изучаешь XOR ...
Но теперь возникает вопрос: правильно ли вычислены layer2_error и layer2_delta, т. Е. является ли правильной следующая часть кода?
# How much did we miss in the predictions?
cost_error = mse(layer2, Y)
cost_delta = mse_derivative(layer2, Y)
#print(layer2_error)
# In what direction is the target value?
# Were we really close? If so, don't change too much.
layer2_error = np.dot(cost_delta, cost_error)
layer2_delta = layer2_error * sigmoid_derivative(layer2)
Правильно ли использовать точечное произведение на cost_delta и cost_error для layer2_error? Или layer2_error будет равно cost_delta?
т.е.
# How much did we miss in the predictions?
cost_error = mse(layer2, Y)
cost_delta = mse_derivative(layer2, Y)
#print(layer2_error)
# In what direction is the target value?
# Were we really close? If so, don't change too much.
layer2_error = cost_delta
layer2_delta = layer2_error * sigmoid_derivative(layer2)