为什么这个简单的神经网络不能收敛到XOR？

Question

下面的网络代码工作正常，但速度太慢。 This site意味着网络在学习率为0.2的100个时期后应该达到99％的准确率，而即使在1900年以后，我的网络也从未超过97％。

Epoch 0, Inputs [0 0], Outputs [-0.83054376], Targets [0] 
Epoch 100, Inputs [0 1], Outputs [ 0.72563824], Targets [1] 
Epoch 200, Inputs [1 0], Outputs [ 0.87570863], Targets [1] 
Epoch 300, Inputs [0 1], Outputs [ 0.90996706], Targets [1] 
Epoch 400, Inputs [1 1], Outputs [ 0.00204791], Targets [0] 
Epoch 500, Inputs [0 1], Outputs [ 0.93396672], Targets [1] 
Epoch 600, Inputs [0 0], Outputs [ 0.00006375], Targets [0] 
Epoch 700, Inputs [0 1], Outputs [ 0.94778227], Targets [1] 
Epoch 800, Inputs [1 1], Outputs [-0.00149935], Targets [0] 
Epoch 900, Inputs [0 0], Outputs [-0.00122716], Targets [0] 
Epoch 1000, Inputs [0 0], Outputs [ 0.00457281], Targets [0] 
Epoch 1100, Inputs [0 1], Outputs [ 0.95921556], Targets [1] 
Epoch 1200, Inputs [0 1], Outputs [ 0.96001748], Targets [1] 
Epoch 1300, Inputs [1 0], Outputs [ 0.96071742], Targets [1] 
Epoch 1400, Inputs [1 1], Outputs [ 0.00110912], Targets [0] 
Epoch 1500, Inputs [0 0], Outputs [-0.00], Targets [0] 
Epoch 1600, Inputs [1 0], Outputs [ 0.9640324], Targets [1] 
Epoch 1700, Inputs [1 0], Outputs [ 0.96431516], Targets [1] 
Epoch 1800, Inputs [0 1], Outputs [ 0.97004973], Targets [1] 
Epoch 1900, Inputs [1 0], Outputs [ 0.96616225], Targets [1] 

我使用的数据集：

0 0 0 
1 0 1 
0 1 1 
1 1 1 

训练集是使用一个辅助文件中的函数读取，但就是不相关的网络。

import numpy as np 
import helper 

FILE_NAME = 'data.txt' 
EPOCHS = 2000 
TESTING_FREQ = 5 
LEARNING_RATE = 0.2 

INPUT_SIZE = 2 
HIDDEN_LAYERS = [5] 
OUTPUT_SIZE = 1 


class Classifier: 
    def __init__(self, layer_sizes): 
     np.set_printoptions(suppress=True) 

     self.activ = helper.tanh 
     self.dactiv = helper.dtanh 

     network = list() 
     for i in range(1, len(layer_sizes)): 
      layer = dict() 
      layer['weights'] = np.random.randn(layer_sizes[i], layer_sizes[i-1]) 
      layer['biases'] = np.random.randn(layer_sizes[i]) 
      network.append(layer) 

     self.network = network 

    def forward_propagate(self, x): 
     for i in range(0, len(self.network)): 
      self.network[i]['outputs'] = self.network[i]['weights'].dot(x) + self.network[i]['biases'] 
      if i != len(self.network)-1: 
       self.network[i]['outputs'] = x = self.activ(self.network[i]['outputs']) 
      else: 
       self.network[i]['outputs'] = self.activ(self.network[i]['outputs']) 
     return self.network[-1]['outputs'] 

    def backpropagate_error(self, x, targets): 
     self.forward_propagate(x) 
     self.network[-1]['deltas'] = (self.network[-1]['outputs'] - targets) * self.dactiv(self.network[-1]['outputs']) 
     for i in reversed(range(len(self.network)-1)): 
      self.network[i]['deltas'] = self.network[i+1]['deltas'].dot(self.network[i+1]['weights'] * self.dactiv(self.network[i]['outputs'])) 

    def adjust_weights(self, inputs, learning_rate): 
     self.network[0]['weights'] -= learning_rate * np.atleast_2d(self.network[0]['deltas']).T.dot(np.atleast_2d(inputs)) 
     self.network[0]['biases'] -= learning_rate * self.network[0]['deltas'] 
     for i in range(1, len(self.network)): 
      self.network[i]['weights'] -= learning_rate * np.atleast_2d(self.network[i]['deltas']).T.dot(np.atleast_2d(self.network[i-1]['outputs'])) 
      self.network[i]['biases'] -= learning_rate * self.network[i]['deltas'] 

    def train(self, inputs, targets, epochs, testfreq, lrate): 
     for epoch in range(epochs): 
      i = np.random.randint(0, len(inputs)) 
      if epoch % testfreq == 0: 
       predictions = self.forward_propagate(inputs[i]) 
       print('Epoch %s, Inputs %s, Outputs %s, Targets %s' % (epoch, inputs[i], predictions, targets[i])) 
      self.backpropagate_error(inputs[i], targets[i]) 
      self.adjust_weights(inputs[i], lrate) 


inputs, outputs = helper.readInput(FILE_NAME, INPUT_SIZE, OUTPUT_SIZE) 
print('Input data: {0}'.format(inputs)) 
print('Output targets: {0}\n'.format(outputs)) 
np.random.seed(1) 

nn = Classifier([INPUT_SIZE] + HIDDEN_LAYERS + [OUTPUT_SIZE]) 

nn.train(inputs, outputs, EPOCHS, TESTING_FREQ, LEARNING_RATE) 

Answer 1

主要错误是，你正在做的直传的时候只有20％，即当epoch % testfreq == 0：

for epoch in range(epochs): 
    i = np.random.randint(0, len(inputs)) 
    if epoch % testfreq == 0: 
    predictions = self.forward_propagate(inputs[i]) 
    print('Epoch %s, Inputs %s, Outputs %s, Targets %s' % (epoch, inputs[i], predictions, targets[i])) 
    self.backpropagate_error(inputs[i], targets[i]) 
    self.adjust_weights(inputs[i], lrate) 

当我把predictions = self.forward_propagate(inputs[i])出if，我得到更好的结果更快：

Epoch 100, Inputs [0 1], Outputs [ 0.80317447], Targets 1 
Epoch 105, Inputs [1 1], Outputs [ 0.96340466], Targets 1 
Epoch 110, Inputs [1 1], Outputs [ 0.96057278], Targets 1 
Epoch 115, Inputs [1 0], Outputs [ 0.87960599], Targets 1 
Epoch 120, Inputs [1 1], Outputs [ 0.97725825], Targets 1 
Epoch 125, Inputs [1 0], Outputs [ 0.89433666], Targets 1 
Epoch 130, Inputs [0 0], Outputs [ 0.03539024], Targets 0 
Epoch 135, Inputs [0 1], Outputs [ 0.92888141], Targets 1 

另外，注意的是，术语划时代通常意味着的一次运行全部你的训练数据，在你的案例4.所以，实际上，你正在做的时间少4倍。

更新

我没注意的细节，结果，错过了一些细微但重要的注意事项：

• 训练数据的问题表示OR，不XOR，所以我上面的结果是用于学习或的操作;
• 反向传递也执行正向传递（所以它不是一个错误，而是一个令人惊讶的实现细节）。

知道了这一点，我已经更新了数据并再次检查了脚本。运行10000次迭代的训练给出了〜0.001的平均误差，所以模型是学习，只是没有那么快，尽可能的。

一个简单的神经网络（没有嵌入规范化机制）对特定的超参数非常敏感，比如初始化和学习速率。我尝试了各种值手动和这里是我得到了什么：

# slightly bigger learning rate 
LEARNING_RATE = 0.3 
... 
# slightly bigger init variation of weights 
layer['weights'] = np.random.randn(layer_sizes[i], layer_sizes[i-1]) * 2.0 

这给出了以下性能：

... 
Epoch 960, Inputs [1 1], Outputs [ 0.01392014], Targets 0 
Epoch 970, Inputs [0 0], Outputs [ 0.04342895], Targets 0 
Epoch 980, Inputs [1 0], Outputs [ 0.96471654], Targets 1 
Epoch 990, Inputs [1 1], Outputs [ 0.00084511], Targets 0 
Epoch 1000, Inputs [0 0], Outputs [ 0.01585915], Targets 0 
Epoch 1010, Inputs [1 1], Outputs [-0.004097], Targets 0 
Epoch 1020, Inputs [1 1], Outputs [ 0.01898956], Targets 0 
Epoch 1030, Inputs [0 0], Outputs [ 0.01254217], Targets 0 
Epoch 1040, Inputs [1 1], Outputs [ 0.01429213], Targets 0 
Epoch 1050, Inputs [0 1], Outputs [ 0.98293925], Targets 1 
... 
Epoch 1920, Inputs [1 1], Outputs [-0.00043072], Targets 0 
Epoch 1930, Inputs [0 1], Outputs [ 0.98544288], Targets 1 
Epoch 1940, Inputs [1 0], Outputs [ 0.97682002], Targets 1 
Epoch 1950, Inputs [1 0], Outputs [ 0.97684186], Targets 1 
Epoch 1960, Inputs [0 0], Outputs [-0.00141565], Targets 0 
Epoch 1970, Inputs [0 0], Outputs [-0.00097559], Targets 0 
Epoch 1980, Inputs [0 1], Outputs [ 0.98548381], Targets 1 
Epoch 1990, Inputs [1 0], Outputs [ 0.97721286], Targets 1 

的平均准确靠近后1000次迭代98.5％和后99.1％ 2000次迭代。这比承诺的速度慢一些，但足够好。我相信它可以进一步调整，但这不是玩具演习的目标。毕竟，tanh is not最好的激活函数，并且分类问题应该更好地用交叉熵损失（而不是L2损失）来解决。所以我不会太担心这个特定网络的性能，并继续进行逻辑回归。就学习速度而言，这肯定会更好。