时间：2018-01-09 魏文应神经网络 反向传播算法（BP算法）
一、说明：
这里转载一篇文章：
https://www.zybuluo.com/hanbingtao/note/476663
其中的代码可以访问作者的github：
https://github.com/hanbt/learn_dl/blob/master/bp.py
我把他的文章代码（bp.py）放在这里：
#!/usr/bin/env python
# -*- coding: UTF-8 -*-
import random
from numpy import *
def sigmoid(inX):
return 1.0 / (1 + exp(-inX))
class Node(object):
def __init__(self, layer_index, node_index):
self.layer_index = layer_index
self.node_index = node_index
self.downstream = []
self.upstream = []
self.output = 0
self.delta = 0
def set_output(self, output):
self.output = output
def append_downstream_connection(self, conn):
self.downstream.append(conn)
def append_upstream_connection(self, conn):
self.upstream.append(conn)
def calc_output(self):
output = reduce(lambda ret, conn: ret + conn.upstream_node.output * conn.weight, self.upstream, 0)
self.output = sigmoid(output)
def calc_hidden_layer_delta(self):
downstream_delta = reduce(
lambda ret, conn: ret + conn.downstream_node.delta * conn.weight,
self.downstream, 0.0)
self.delta = self.output * (1 - self.output) * downstream_delta
def calc_output_layer_delta(self, label):
self.delta = self.output * (1 - self.output) * (label - self.output)
def __str__(self):
node_str = '%u-%u: output: %f delta: %f' % (self.layer_index, self.node_index, self.output, self.delta)
downstream_str = reduce(lambda ret, conn: ret + '\n\t' + str(conn), self.downstream, '')
upstream_str = reduce(lambda ret, conn: ret + '\n\t' + str(conn), self.upstream, '')
return node_str + '\n\tdownstream:' + downstream_str + '\n\tupstream:' + upstream_str
class ConstNode(object):
def __init__(self, layer_index, node_index):
self.layer_index = layer_index
self.node_index = node_index
self.downstream = []
self.output = 1
def append_downstream_connection(self, conn):
self.downstream.append(conn)
def calc_hidden_layer_delta(self):
downstream_delta = reduce(
lambda ret, conn: ret + conn.downstream_node.delta * conn.weight,
self.downstream, 0.0)
self.delta = self.output * (1 - self.output) * downstream_delta
def __str__(self):
node_str = '%u-%u: output: 1' % (self.layer_index, self.node_index)
downstream_str = reduce(lambda ret, conn: ret + '\n\t' + str(conn), self.downstream, '')
return node_str + '\n\tdownstream:' + downstream_str
class Layer(object):
def __init__(self, layer_index, node_count):
self.layer_index = layer_index
self.nodes = []
for i in range(node_count):
self.nodes.append(Node(layer_index, i))
self.nodes.append(ConstNode(layer_index, node_count))
def set_output(self, data):
for i in range(len(data)):
self.nodes[i].set_output(data[i])
def calc_output(self):
for node in self.nodes[:-1]:
node.calc_output()
def dump(self):
for node in self.nodes:
print node
class Connection(object):
def __init__(self, upstream_node, downstream_node):
self.upstream_node = upstream_node
self.downstream_node = downstream_node
self.weight = random.uniform(-0.1, 0.1)
self.gradient = 0.0
def calc_gradient(self):
self.gradient = self.downstream_node.delta * self.upstream_node.output
def update_weight(self, rate):
self.calc_gradient()
self.weight += rate * self.gradient
def get_gradient(self):
return self.gradient
def __str__(self):
return '(%u-%u) -> (%u-%u) = %f' % (
self.upstream_node.layer_index,
self.upstream_node.node_index,
self.downstream_node.layer_index,
self.downstream_node.node_index,
self.weight)
class Connections(object):
def __init__(self):
self.connections = []
def add_connection(self, connection):
self.connections.append(connection)
def dump(self):
for conn in self.connections:
print conn
class Network(object):
def __init__(self, layers):
self.connections = Connections()
self.layers = []
layer_count = len(layers)
node_count = 0;
for i in range(layer_count):
self.layers.append(Layer(i, layers[i]))
for layer in range(layer_count - 1):
connections = [Connection(upstream_node, downstream_node)
for upstream_node in self.layers[layer].nodes
for downstream_node in self.layers[layer + 1].nodes[:-1]]
for conn in connections:
self.connections.add_connection(conn)
conn.downstream_node.append_upstream_connection(conn)
conn.upstream_node.append_downstream_connection(conn)
def train(self, labels, data_set, rate, epoch):
for i in range(epoch):
for d in range(len(data_set)):
self.train_one_sample(labels[d], data_set[d], rate)
# print 'sample %d training finished' % d
def train_one_sample(self, label, sample, rate):
self.predict(sample)
self.calc_delta(label)
self.update_weight(rate)
def calc_delta(self, label):
output_nodes = self.layers[-1].nodes
for i in range(len(label)):
output_nodes[i].calc_output_layer_delta(label[i])
for layer in self.layers[-2::-1]:
for node in layer.nodes:
node.calc_hidden_layer_delta()
def update_weight(self, rate):
for layer in self.layers[:-1]:
for node in layer.nodes:
for conn in node.downstream:
conn.update_weight(rate)
def calc_gradient(self):
for layer in self.layers[:-1]:
for node in layer.nodes:
for conn in node.downstream:
conn.calc_gradient()
def get_gradient(self, label, sample):
self.predict(sample)
self.calc_delta(label)
self.calc_gradient()
def predict(self, sample):
self.layers[0].set_output(sample)
for i in range(1, len(self.layers)):
self.layers[i].calc_output()
return map(lambda node: node.output, self.layers[-1].nodes[:-1])
def dump(self):
for layer in self.layers:
layer.dump()
class Normalizer(object):
def __init__(self):
self.mask = [
0x1, 0x2, 0x4, 0x8, 0x10, 0x20, 0x40, 0x80
]
def norm(self, number):
return map(lambda m: 0.9 if number & m else 0.1, self.mask)
def denorm(self, vec):
binary = map(lambda i: 1 if i > 0.5 else 0, vec)
for i in range(len(self.mask)):
binary[i] = binary[i] * self.mask[i]
return reduce(lambda x,y: x + y, binary)
def mean_square_error(vec1, vec2):
return 0.5 * reduce(lambda a, b: a + b,
map(lambda v: (v[0] - v[1]) * (v[0] - v[1]),
zip(vec1, vec2)
)
)
def gradient_check(network, sample_feature, sample_label):
'''
梯度检查
network: 神经网络对象
sample_feature: 样本的特征
sample_label: 样本的标签
'''
# 计算网络误差
network_error = lambda vec1, vec2: \
0.5 * reduce(lambda a, b: a + b,
map(lambda v: (v[0] - v[1]) * (v[0] - v[1]),
zip(vec1, vec2)))
# 获取网络在当前样本下每个连接的梯度
network.get_gradient(sample_feature, sample_label)
# 对每个权重做梯度检查
for conn in network.connections.connections:
# 获取指定连接的梯度
actual_gradient = conn.get_gradient()
# 增加一个很小的值，计算网络的误差
epsilon = 0.0001
conn.weight += epsilon
error1 = network_error(network.predict(sample_feature), sample_label)
# 减去一个很小的值，计算网络的误差
conn.weight -= 2 * epsilon # 刚才加过了一次，因此这里需要减去2倍
error2 = network_error(network.predict(sample_feature), sample_label)
# 根据式6计算期望的梯度值
expected_gradient = (error2 - error1) / (2 * epsilon)
# 打印
print 'expected gradient: \t%f\nactual gradient: \t%f' % (
expected_gradient, actual_gradient)
def train_data_set():
normalizer = Normalizer()
data_set = []
labels = []
for i in range(0, 256, 8):
n = normalizer.norm(int(random.uniform(0, 256)))
data_set.append(n)
labels.append(n)
return labels, data_set
def train(network):
labels, data_set = train_data_set()
network.train(labels, data_set, 0.3, 50)
def test(network, data):
normalizer = Normalizer()
norm_data = normalizer.norm(data)
predict_data = network.predict(norm_data)
print '\ttestdata(%u)\tpredict(%u)' % (
data, normalizer.denorm(predict_data))
def correct_ratio(network):
normalizer = Normalizer()
correct = 0.0;
for i in range(256):
if normalizer.denorm(network.predict(normalizer.norm(i))) == i:
correct += 1.0
print 'correct_ratio: %.2f%%' % (correct / 256 * 100)
def gradient_check_test():
net = Network([2, 2, 2])
sample_feature = [0.9, 0.1]
sample_label = [0.9, 0.1]
gradient_check(net, sample_feature, sample_label)
if __name__ == '__main__':
net = Network([8, 3, 8])
train(net)
net.dump()
correct_ratio(net)
二、反向传播算法(Back Propagation)
首先，我们要对下面一些知识有些感性的认识：
我们知道，有些函数求导，会得到其本身，比如：e的n次方，求导以后还是其本身。激活函数 sigmoid() 就是利用这一性质，使得求导方便，可以通过输出y表示出其导数（这样做的目的是为了方便计算，这一步是激活函数的选择）。
既然是通过输出y来推出其导数，而且我们是多层网络，所以我们要一步一步，从最后一个输出y，推出倒数第二层的导数……依次往前推导，这就是反向传播的命名的由来。
-
那我们为什么要去求导数呢？其实这里要去求的是偏导数：
随机梯度下降 这样，我们又回到了梯度下降法中了。
我们知道，神经网络是由线性单元组成的，运算线性单元时我们使用了梯度下降法，上一级的线性单元求解依赖于后一级的线性单元，组合起来就成了反向传播算法啦！