Some questions

神经网络的结构，即几层网络，输入输出怎么设计才最有效？

设计几层网络比较好的问题仍然是个黑匣子，没有理论支撑应该怎么设计网络，现在仍是经验使然。

数学理论证明，三层的神经网络就能够以任意精度逼近任何非线性连续函数。那么为什么还需要有深度网络？

深层网络一般用于深度学习范畴，一般的分类、回归问题还是传统机器学习算法用的比较多，而深度学习的天空在cv、nlp等需要大量数据的领域。

在不同应用场合下，激活函数怎么选择？

没深入研究过，感觉还是靠经验多一些;

学习率怎么怎么选择？

学习率的原则肯定是开始大收敛快，后面小逼近全局最优，需要设置动态的学习率。而且最开始的学习率设置大了训练会震荡，设置小了收敛会很慢，这是实际项目中需要调的超参数。

训练次数设定多少训练出的模型效果更好？

关于训练次数的问题，一般设置检查点然后让训练到模型过拟合，选择检查点中保存的最优模型权重就好了。次数要能够使得模型过拟合，因为不过拟合就没有达到模型的极限。

The Unknown Word

#coding:utf-8
import random
import math#
#   参数解释：
#   "pd_" ：偏导的前缀
#   "d_" ：导数的前缀
#   "w_ho" ：隐含层到输出层的权重系数索引
#   "w_ih" ：输入层到隐含层的权重系数的索引class NeuralNetwork:LEARNING_RATE = 0.5def __init__(self, num_inputs, num_hidden, num_outputs, hidden_layer_weights = None, hidden_layer_bias = None, output_layer_weights = None, output_layer_bias = None):self.num_inputs = num_inputsself.hidden_layer = NeuronLayer(num_hidden, hidden_layer_bias)self.output_layer = NeuronLayer(num_outputs, output_layer_bias)self.init_weights_from_inputs_to_hidden_layer_neurons(hidden_layer_weights)self.init_weights_from_hidden_layer_neurons_to_output_layer_neurons(output_layer_weights)def init_weights_from_inputs_to_hidden_layer_neurons(self, hidden_layer_weights):weight_num = 0for h in range(len(self.hidden_layer.neurons)):for i in range(self.num_inputs):if not hidden_layer_weights:self.hidden_layer.neurons[h].weights.append(random.random())else:self.hidden_layer.neurons[h].weights.append(hidden_layer_weights[weight_num])weight_num += 1def init_weights_from_hidden_layer_neurons_to_output_layer_neurons(self, output_layer_weights):weight_num = 0for o in range(len(self.output_layer.neurons)):for h in range(len(self.hidden_layer.neurons)):if not output_layer_weights:self.output_layer.neurons[o].weights.append(random.random())else:self.output_layer.neurons[o].weights.append(output_layer_weights[weight_num])weight_num += 1def inspect(self):print('------')print('* Inputs: {}'.format(self.num_inputs))print('------')print('Hidden Layer')self.hidden_layer.inspect()print('------')print('* Output Layer')self.output_layer.inspect()print('------')def feed_forward(self, inputs):hidden_layer_outputs = self.hidden_layer.feed_forward(inputs)return self.output_layer.feed_forward(hidden_layer_outputs)def train(self, training_inputs, training_outputs):self.feed_forward(training_inputs)# 1. 输出神经元的值pd_errors_wrt_output_neuron_total_net_input = [0] * len(self.output_layer.neurons)for o in range(len(self.output_layer.neurons)):# ∂E/∂zⱼpd_errors_wrt_output_neuron_total_net_input[o] = self.output_layer.neurons[o].calculate_pd_error_wrt_total_net_input(training_outputs[o])# 2. 隐含层神经元的值pd_errors_wrt_hidden_neuron_total_net_input = [0] * len(self.hidden_layer.neurons)for h in range(len(self.hidden_layer.neurons)):# dE/dyⱼ = Σ ∂E/∂zⱼ * ∂z/∂yⱼ = Σ ∂E/∂zⱼ * wᵢⱼd_error_wrt_hidden_neuron_output = 0for o in range(len(self.output_layer.neurons)):d_error_wrt_hidden_neuron_output += pd_errors_wrt_output_neuron_total_net_input[o] * self.output_layer.neurons[o].weights[h]# ∂E/∂zⱼ = dE/dyⱼ * ∂zⱼ/∂pd_errors_wrt_hidden_neuron_total_net_input[h] = d_error_wrt_hidden_neuron_output * self.hidden_layer.neurons[h].calculate_pd_total_net_input_wrt_input()# 3. 更新输出层权重系数for o in range(len(self.output_layer.neurons)):for w_ho in range(len(self.output_layer.neurons[o].weights)):# ∂Eⱼ/∂wᵢⱼ = ∂E/∂zⱼ * ∂zⱼ/∂wᵢⱼpd_error_wrt_weight = pd_errors_wrt_output_neuron_total_net_input[o] * self.output_layer.neurons[o].calculate_pd_total_net_input_wrt_weight(w_ho)# Δw = α * ∂Eⱼ/∂wᵢself.output_layer.neurons[o].weights[w_ho] -= self.LEARNING_RATE * pd_error_wrt_weight# 4. 更新隐含层的权重系数for h in range(len(self.hidden_layer.neurons)):for w_ih in range(len(self.hidden_layer.neurons[h].weights)):# ∂Eⱼ/∂wᵢ = ∂E/∂zⱼ * ∂zⱼ/∂wᵢpd_error_wrt_weight = pd_errors_wrt_hidden_neuron_total_net_input[h] * self.hidden_layer.neurons[h].calculate_pd_total_net_input_wrt_weight(w_ih)# Δw = α * ∂Eⱼ/∂wᵢself.hidden_layer.neurons[h].weights[w_ih] -= self.LEARNING_RATE * pd_error_wrt_weightdef calculate_total_error(self, training_sets):total_error = 0for t in range(len(training_sets)):training_inputs, training_outputs = training_sets[t]self.feed_forward(training_inputs)for o in range(len(training_outputs)):total_error += self.output_layer.neurons[o].calculate_error(training_outputs[o])return total_errorclass NeuronLayer:def __init__(self, num_neurons, bias):# 同一层的神经元共享一个截距项bself.bias = bias if bias else random.random()self.neurons = []for i in range(num_neurons):self.neurons.append(Neuron(self.bias))def inspect(self):print('Neurons:', len(self.neurons))for n in range(len(self.neurons)):print(' Neuron', n)for w in range(len(self.neurons[n].weights)):print('  Weight:', self.neurons[n].weights[w])print('  Bias:', self.bias)def feed_forward(self, inputs):outputs = []for neuron in self.neurons:outputs.append(neuron.calculate_output(inputs))return outputsdef get_outputs(self):outputs = []for neuron in self.neurons:outputs.append(neuron.output)return outputsclass Neuron:def __init__(self, bias):self.bias = biasself.weights = []def calculate_output(self, inputs):self.inputs = inputsself.output = self.squash(self.calculate_total_net_input())return self.outputdef calculate_total_net_input(self):total = 0for i in range(len(self.inputs)):total += self.inputs[i] * self.weights[i]return total + self.bias# 激活函数sigmoiddef squash(self, total_net_input):return 1 / (1 + math.exp(-total_net_input))def calculate_pd_error_wrt_total_net_input(self, target_output):return self.calculate_pd_error_wrt_output(target_output) * self.calculate_pd_total_net_input_wrt_input();# 每一个神经元的误差是由平方差公式计算的def calculate_error(self, target_output):return 0.5 * (target_output - self.output) ** 2def calculate_pd_error_wrt_output(self, target_output):return -(target_output - self.output)def calculate_pd_total_net_input_wrt_input(self):return self.output * (1 - self.output)def calculate_pd_total_net_input_wrt_weight(self, index):return self.inputs[index]# 文中的例子:nn = NeuralNetwork(2, 2, 2, hidden_layer_weights=[0.15, 0.2, 0.25, 0.3], hidden_layer_bias=0.35, output_layer_weights=[0.4, 0.45, 0.5, 0.55], output_layer_bias=0.6)
for i in range(10000):nn.train([0.05, 0.1], [0.01, 0.09])print(i, round(nn.calculate_total_error([[[0.05, 0.1], [0.01, 0.09]]]), 9))#另外一个例子，可以把上面的例子注释掉再运行一下:# training_sets = [
#     [[0, 0], [0]],
#     [[0, 1], [1]],
#     [[1, 0], [1]],
#     [[1, 1], [0]]
# ]# nn = NeuralNetwork(len(training_sets[0][0]), 5, len(training_sets[0][1]))
# for i in range(10000):
#     training_inputs, training_outputs = random.choice(training_sets)
#     nn.train(training_inputs, training_outputs)
#     print(i, nn.calculate_total_error(training_sets))

Reference

transmission
transmission_Back_Algorithm
Transmission_Online_demonstration_of_neural_network_changes
Transmission_How_The_Backpropagation_algorithm_works