莫烦theano学习自修第九天【过拟合问题与正规化】

如下图所示（回归的过拟合问题）：如果机器学习得到的回归为下图中的直线则是比较好的结果，但是如果进一步控制减少误差，导致机器学习到了下图中的曲线，则100%正确的学习了训练数据，看似较好，但是如果换成另外一批数据，则不能较好的反应数据，造成过大的误差，这就是过拟合问题

再看下图这是分类问题的过拟合问题

2. 正规化方法

（1）l1正规化：使用权重绝对值和的方式惩罚误差

（2）l2正规化：使用权重平方和的方式惩罚误差

3. 代码实现：

from __future__ import print_function
import theano
from sklearn.datasets import load_boston
import theano.tensor as T
import numpy as np
import matplotlib.pyplot as plt

class Layer(object):
    def __init__(self, inputs, in_size, out_size, activation_function=None):
        self.W = theano.shared(np.random.normal(0, 1, (in_size, out_size)))
        self.b = theano.shared(np.zeros((out_size, )) + 0.1)
        self.Wx_plus_b = T.dot(inputs, self.W) + self.b
        self.activation_function = activation_function
        if activation_function is None:
            self.outputs = self.Wx_plus_b
        else:
            self.outputs = self.activation_function(self.Wx_plus_b)

def minmax_normalization(data):　　　# 对输入数据进行预处理，处理成大于0，小于1的范围
    xs_max = np.max(data, axis=0)
    xs_min = np.min(data, axis=0)
    xs = (1 - 0) * (data - xs_min) / (xs_max - xs_min) + 0
    return xs
# 获取波士顿房价数据
np.random.seed(100)
x_data = load_boston().data
# minmax normalization, rescale the inputs
x_data = minmax_normalization(x_data)
y_data = load_boston().target[:, np.newaxis]
# 进行交叉验证，将数据的一部分数据作为训练数据，另一部分作废测试数据
# cross validation, train test data split
x_train, y_train = x_data[:400], y_data[:400]
x_test, y_test = x_data[400:], y_data[400:]

x = T.dmatrix("x")
y = T.dmatrix("y")

l1 = Layer(x, 13, 50, T.tanh)
l2 = Layer(l1.outputs, 50, 1, None)

# the way to compute cost
cost = T.mean(T.square(l2.outputs - y))      # without regularization
# cost = T.mean(T.square(l2.outputs - y)) + 0.1 * ((l1.W ** 2).sum() + (l2.W ** 2).sum())  # with l2 regularization
# cost = T.mean(T.square(l2.outputs - y)) + 0.1 * (abs(l1.W).sum() + abs(l2.W).sum())  # with l1 regularization
gW1, gb1, gW2, gb2 = T.grad(cost, [l1.W, l1.b, l2.W, l2.b])

learning_rate = 0.01
train = theano.function(
    inputs=[x, y],
    updates=[(l1.W, l1.W - learning_rate * gW1),
             (l1.b, l1.b - learning_rate * gb1),
             (l2.W, l2.W - learning_rate * gW2),
             (l2.b, l2.b - learning_rate * gb2)])

compute_cost = theano.function(inputs=[x, y], outputs=cost)

# record cost
train_err_list = []
test_err_list = []
learning_time = []
for i in range(1000):
    train(x_train, y_train)
    if i % 10 == 0:
        # record cost
        train_err_list.append(compute_cost(x_train, y_train))
        test_err_list.append(compute_cost(x_test, y_test))
        learning_time.append(i)

# plot cost history
plt.plot(learning_time, train_err_list, 'r-')
plt.plot(learning_time, test_err_list, 'b--')
plt.show()