我创建了一个神经网络来估算输入sin(x)的{{1}}函数。网络有21个输出神经元（代表数字-1.0，-0.9，...，0.9,1.0），numpy没有学习，因为我认为我在定义前馈机制时错误地实现了神经元架构。
当我执行代码时，它正确估计的测试数据量大约为48/1000。如果您在21个类别之间拆分1000个测试数据点，则这恰好是每个类别的平均数据点数。查看网络输出，您可以看到网络似乎只是为每个输入选择一个输出值。例如，无论您提供x，它都可以选择-0.5作为y的估算值。我在哪里错了？这是我的第一个网络。谢谢！
x答案 0 :(得分：1)
我没有仔细检查所有代码，但有些问题清晰可见：
*运算符doesn't perform matrix multiplication，您必须使用numpy.dot。例如，这会影响这些行：network_input * self.layer1[neuron]，self.layer1_activations[weight]*self.layer2[neuron][weight]等。
好像你是通过分类来解决你的问题（选择21个班级中的1个），但使用L2丢失。这有点混乱。您有两种选择：要么坚持分类并使用cross entropy loss function，要么使用L2丢失执行回归（即预测数值）。
你绝对应该提取sigmoid函数以避免再次编写相同的表达式：
def sigmoid(z):
return 1 / (1 + np.exp(-z))
def sigmoid_derivative(x):
return sigmoid(x) * (1 - sigmoid(x))
您执行self.layer1和self.layer2的相同更新，这显然是错误的。花一些时间分析how exactly backpropagation作品。
答案 1 :(得分：0)
我编辑了我的损失函数如何集成到我的函数中并正确实现了梯度下降。我还删除了迷你批次的使用并简化了我的网络尝试做的事情。我现在有一个网络试图将某些东西归类为偶数或奇数。
我过去常常解决的一些非常有用的指南：
神经网络与深度学习的第1章和第2章，作者Michael Nielsen，http://neuralnetworksanddeeplearning.com/chap1.html免费提供。本书对神经网络的工作原理给出了详尽的解释，包括执行后数学的细分。
来自开始的反向传播，由ErikHallström，由Maxim联系。 https://medium.com/@erikhallstrm/backpropagation-from-the-beginning-77356edf427d 。不像上面的指南那么彻底，但我同时保持开放，因为本指南更重要的是关于什么是重要的以及如何应用在尼尔森书中详细解释的数学公式。 / p>如何用9行Python代码构建一个简单的神经网络 https://medium.com/technology-invention-and-more/how-to-build-a-simple-neural-network-in-9-lines-of-python-code-cc8f23647ca1 。对一些神经网络基础知识的有用和快速的介绍。
这是我的（现在正在运行的）代码：
import random
import numpy as np
import scipy
import math
class Network(object):
def __init__(self,inputLayerSize,hiddenLayerSize,outputLayerSize):
#Layers represented both by their weights array and activation and inputsums vectors.
self.layer1 = np.random.randn(hiddenLayerSize,inputLayerSize)
self.layer2 = np.random.randn(outputLayerSize,hiddenLayerSize)
self.layer1_activations = np.zeros((hiddenLayerSize, 1))
self.layer2_activations = np.zeros((outputLayerSize, 1))
self.layer1_inputsums = np.zeros((hiddenLayerSize, 1))
self.layer2_inputsums = np.zeros((outputLayerSize, 1))
self.layer1_errorsignals = np.zeros((hiddenLayerSize, 1))
self.layer2_errorsignals = np.zeros((outputLayerSize, 1))
self.layer1_deltaw = np.zeros((hiddenLayerSize, inputLayerSize))
self.layer2_deltaw = np.zeros((outputLayerSize, hiddenLayerSize))
self.outputLayerSize = outputLayerSize
self.inputLayerSize = inputLayerSize
self.hiddenLayerSize = hiddenLayerSize
print()
print(self.layer1)
print()
print(self.layer2)
print()
# self.weights = [np.random.randn(y,x)
# for x, y in zip(sizes[:-1], sizes[1:])]
def feedforward(self, network_input):
#Calculate inputsum and and activations for each neuron in the first layer
for neuron in range(self.hiddenLayerSize):
self.layer1_inputsums[neuron] = network_input * self.layer1[neuron]
self.layer1_activations[neuron] = self.sigmoid(self.layer1_inputsums[neuron])
# Calculate inputsum and and activations for each neuron in the second layer. Notice that each neuron in the second layer represented by
# weights vector, consisting of all weights leading out of the kth neuron in (l-1) layer to the jth neuron in layer l.
self.layer2_inputsums = np.zeros((self.outputLayerSize, 1))
for neuron in range(self.outputLayerSize):
for weight in range(self.hiddenLayerSize):
self.layer2_inputsums[neuron] += self.layer1_activations[weight]*self.layer2[neuron][weight]
self.layer2_activations[neuron] = self.sigmoid(self.layer2_inputsums[neuron])
return self.layer2_activations
def interpreted_output(self, network_input):
#convert layer 2 activation numbers to a single output. The neuron (weight vector) with highest activation will be output.
self.feedforward(network_input)
outputs = [x / 10 for x in range(-int((self.outputLayerSize/2)), int((self.outputLayerSize/2))+1, 1)] #range(-10, 11, 1)
return(outputs[np.argmax(self.layer2_activations)])
# def build_expected_output(self, training_data):
# #Views expected output number y for each x to generate an expected output vector from the network
# index=0
# for pair in training_data:
# expected_output_vector = np.zeros((self.outputLayerSize,1))
# x = training_data[0]
# y = training_data[1]
# for i in range(-int((self.outputLayerSize / 2)), int((self.outputLayerSize / 2)) + 1, 1):
# if y == i / 10:
# expected_output_vector[i] = 1
# #expect the target category to be a 1.
# break
# training_data[index][1] = expected_output_vector
# index+=1
# return training_data
def train(self, training_data, learn_rate):
self.backpropagate(training_data, learn_rate)
def backpropagate(self, train_data, learn_rate):
#Perform for each x,y pair.
for datapair in range(len(train_data)):
x = train_data[datapair][0]
y = train_data[datapair][1]
self.feedforward(x)
# print("l2a " + str(self.layer2_activations))
# print("l1a " + str(self.layer1_activations))
# print("l2 " + str(self.layer2))
# print("l1 " + str(self.layer1))
for neuron in range(self.outputLayerSize):
#Calculate first error equation for error signals of output layer neurons
self.layer2_errorsignals[neuron] = (self.layer2_activations[neuron] - y[neuron]) * self.sigmoid_prime(self.layer2_inputsums[neuron])
#Use recursive formula to calculate error signals of hidden layer neurons
self.layer1_errorsignals = np.multiply(np.array(np.matrix(self.layer2.T) * np.matrix(self.layer2_errorsignals)) , self.sigmoid_prime(self.layer1_inputsums))
#print(self.layer1_errorsignals)
# for neuron in range(self.hiddenLayerSize):
# #Use recursive formula to calculate error signals of hidden layer neurons
# self.layer1_errorsignals[neuron] = np.multiply(self.layer2[neuron].T,self.layer2_errorsignals[neuron]) * self.sigmoid_prime(self.layer1_inputsums[neuron])
#Partial derivative of C with respect to weight for connection from kth neuron in (l-1)th layer to jth neuron in lth layer is
#(jth error signal in lth layer) * (kth activation in (l-1)th layer.)
#Update all weights for network at each iteration of a training pair.
#Update weights in second layer
for neuron in range(self.outputLayerSize):
for weight in range(self.hiddenLayerSize):
self.layer2_deltaw[neuron][weight] = self.layer2_errorsignals[neuron]*self.layer1_activations[weight]*(-learn_rate)
self.layer2 += self.layer2_deltaw
#Update weights in first layer
for neuron in range(self.hiddenLayerSize):
self.layer1_deltaw[neuron] = self.layer1_errorsignals[neuron]*(x)*(-learn_rate)
self.layer1 += self.layer1_deltaw
#Comment/Uncomment to enable error evaluation.
#print("Epoch {0}: Error: {1}".format(datapair, self.evaluate(test_data)))
# print("l2a " + str(self.layer2_activations))
# print("l1a " + str(self.layer1_activations))
# print("l1 " + str(self.layer1))
# print("l2 " + str(self.layer2))
def evaluate(self, test_data):
error = 0
for x, y in test_data:
#x is integer, y is single element np.array
output = self.feedforward(x)
error += y - output
return error
#eval function for sin(x)
# def evaluate(self, test_data):
# """
# Returns number of test inputs which network evaluates correctly.
# The ouput assumed to be neuron in output layer with highest activation
# :param test_data: test data set identical in form to train data set.
# :return: integer sum
# """
# correct = 0
# for x, y in test_data:
# outputs = [x / 10 for x in range(-int((self.outputLayerSize / 2)), int((self.outputLayerSize / 2)) + 1,
# 1)] # range(-10, 11, 1)
# newy = outputs[np.argmax(y)]
# output = self.interpreted_output(x)
# #print("output: " + str(output))
# if output == newy:
# correct+=1
# return(correct)
def sigmoid(self, z):
return 1 / (1 + np.exp(-z))
def sigmoid_prime(self, z):
return (1 - self.sigmoid(z)) * self.sigmoid(z)
def build_simple_data(data_points):
x_vals = []
y_vals = []
for each in range(data_points):
x = random.randint(-3,3)
expected_output_vector = np.zeros((1, 1))
if x > 0:
expected_output_vector[[0]] = 1
else:
expected_output_vector[[0]] = 0
x_vals.append(x)
y_vals.append(expected_output_vector)
print(list(zip(x_vals,y_vals)))
print()
return (list(zip(x_vals,y_vals)))
simpleNet = Network(1, 3, 1)
# print("Pretest")
# print(simpleNet.feedforward(-3))
# print(simpleNet.feedforward(10))
# init_weights_l1 = simpleNet.layer1
# init_weights_l2 = simpleNet.layer2
# simpleNet.train(build_simple_data(10000),.1)
# #sometimes Error converges to 0, sometimes error converges to 10.
# print("Initial Weights:")
# print(init_weights_l1)
# print(init_weights_l2)
# print("Final Weights")
# print(simpleNet.layer1)
# print(simpleNet.layer2)
# print("Post-test")
# print(simpleNet.feedforward(-3))
# print(simpleNet.feedforward(10))
def test_network(iterations,net,training_points):
"""
Casually evaluates pre and post test
:param iterations: number of trials to be run
:param net: name of network to evaluate.
;param training_points: size of training data to be used
:return: four 1x1 arrays.
"""
pretest_negative = 0
pretest_positive = 0
posttest_negative = 0
posttest_positive = 0
for each in range(iterations):
pretest_negative += net.feedforward(-10)
pretest_positive += net.feedforward(10)
net.train(build_simple_data(training_points),.1)
for each in range(iterations):
posttest_negative += net.feedforward(-10)
posttest_positive += net.feedforward(10)
return(pretest_negative/iterations, pretest_positive/iterations, posttest_negative/iterations, posttest_positive/iterations)
print(test_network(10000, simpleNet, 10000))
虽然此代码与OP中发布的代码有很大不同，但有一个特别的区别很有趣。在原始前馈方法通知中
#second layer's output activations use layer1's activations as input:
for neuron in range(self.outputLayerSize):
for weight in range(self.hiddenLayerSize):
self.layer2_activations[neuron] += self.layer1_activations[weight]*self.layer2[neuron][weight]
self.layer2_activations[neuron] = 1/(1+np.exp(self.layer2_activations[neuron]))
该行
self.layer2_activations[neuron] += self.layer1_activations[weight]*self.layer2[neuron][weight]
类似
self.layer2_inputsums[neuron] += self.layer1_activations[weight]*self.layer2[neuron][weight]
在更新的代码中。该线执行每个权重向量和每个输入向量之间的点积（来自层1的激活）以到达神经元的input_sum，通常称为z（思考sigmoid（z））。在我的网络中，sigmoid函数的导数sigmoid_prime用于计算成本函数相对于所有权重的梯度。通过在实际输出和预期输出之间乘以sigmoid_prime（z）*网络误差。如果z非常大（且为正），则神经元的激活值将非常接近1.这意味着网络确信该神经元应该激活。如果z非常负，则同样如此。因此，网络不希望从根本上调整它满意的权重，因此神经元的每个权重的变化规模由sigmoid（z），sigmoid_prime（z）的梯度给出。非常大的z意味着非常小的梯度和非常小的变化应用于权重（当z = 0时，sigmoid的梯度最大化，当网络不知道应该如何分类神经元以及当神经元的激活为0.5时）。 / p>
由于我不断添加每个神经元的input_sum（z）并且从不重置点的新输入值（权重，激活），z的值保持增长，不断减慢权重的变化率直到重量修改变得停滞不前。我添加了以下行来应对此问题：
self.layer2_inputsums = np.zeros((self.outputLayerSize, 1))
只要安装了numpy模块，就可以将新发布的网络复制并粘贴到编辑器中并执行。要打印的最后一行输出将是4个表示最终网络输出的数组的列表。前两个分别是负输入和正输入的预测试值。这些应该是随机的。后两个是测试后的值，用于确定网络分类为正数和负数的程度。接近0的数字表示负数，接近1表示正数。