Hands-on learning and less over-fitting fitting depth learning

Heard people say, DL power lies in its ability to fit the curve as long as you can give, the equation can be expressed by a neural network. However, this neural network requires sufficient data for training, here leads to the concept of over-fitting and less fit. When the neural network is very large, but not much data, the neural network can remember the characteristics of each data, which can lead to over-fitting. On the contrary, when the neural network to fit smaller or ability is still weak, but a lot of data, underfitting problems arise.

Over-fitting, underfitting

1. Training error generalization error and
   training error model means exhibit on the training data set error;
   test error is the desired model demonstrated in a test on any error data sample, and often approximated by the error on the test data set.
2. Model selection
   in the training process, in order to get a better argument, we need a validation data set. This is usually obtained from the training data set into. The method of division usually cross-validation K off method. Generally do papers, K = 10. The final results are averaged.
3. Over-fitting, underfitting
   from training error and generalization error performance point of view, the model can not be lower training error, we will the phenomenon known as underfitting (underfitting); training error is much smaller than the model it in error on the test data set, we call this phenomenon as over-fitting (overfitting). Both phenomena occur usually associated with the complexity of the model and the training data set size.

Polynomial function to fit the experimental

In the case of an n-order polynomial model (n more, the higher the complexity of the model), the relationship between a given set of training data, the model complexity and error as shown below:

It is worth mentioning that, in general DL, the training data set and the model was never enough and strong enough, so there will be more cases of over-fitting.

%matplotlib inline
import torch
import numpy as np
import sys
sys.path.append("/home/kesci/input")
import d2lzh1981 as d2l
print(torch.__version__)

#初始化模型参数
n_train, n_test, true_w, true_b = 100, 100, [1.2, -3.4, 5.6], 5
features = torch.randn((n_train + n_test, 1))
poly_features = torch.cat((features, torch.pow(features, 2), torch.pow(features, 3)), 1) 
labels = (true_w[0] * poly_features[:, 0] + true_w[1] * poly_features[:, 1]
          + true_w[2] * poly_features[:, 2] + true_b)
labels += torch.tensor(np.random.normal(0, 0.01, size=labels.size()), dtype=torch.float)

#定义模型
def semilogy(x_vals, y_vals, x_label, y_label, x2_vals=None, y2_vals=None,
             legend=None, figsize=(3.5, 2.5)):
    # d2l.set_figsize(figsize)
    d2l.plt.xlabel(x_label)
    d2l.plt.ylabel(y_label)
    d2l.plt.semilogy(x_vals, y_vals)
    if x2_vals and y2_vals:
        d2l.plt.semilogy(x2_vals, y2_vals, linestyle=':')
        d2l.plt.legend(legend)

num_epochs, loss = 100, torch.nn.MSELoss()

def fit_and_plot(train_features, test_features, train_labels, test_labels):
    # 初始化网络模型
    net = torch.nn.Linear(train_features.shape[-1], 1)
    # 通过Linear文档可知，pytorch已经将参数初始化了，所以我们这里就不手动初始化了
    
    # 设置批量大小
    batch_size = min(10, train_labels.shape[0])    
    dataset = torch.utils.data.TensorDataset(train_features, train_labels)      # 设置数据集
    train_iter = torch.utils.data.DataLoader(dataset, batch_size, shuffle=True) # 设置获取数据方式
    
    optimizer = torch.optim.SGD(net.parameters(), lr=0.01)                      # 设置优化函数，使用的是随机梯度下降优化
    train_ls, test_ls = [], []
    for _ in range(num_epochs):
        for X, y in train_iter:                                                 # 取一个批量的数据
            l = loss(net(X), y.view(-1, 1))                                     # 输入到网络中计算输出，并和标签比较求得损失函数
            optimizer.zero_grad()                                               # 梯度清零，防止梯度累加干扰优化
            l.backward()                                                        # 求梯度
            optimizer.step()                                                    # 迭代优化函数，进行参数优化
        train_labels = train_labels.view(-1, 1)
        test_labels = test_labels.view(-1, 1)
        train_ls.append(loss(net(train_features), train_labels).item())         # 将训练损失保存到train_ls中
        test_ls.append(loss(net(test_features), test_labels).item())            # 将测试损失保存到test_ls中
    print('final epoch: train loss', train_ls[-1], 'test loss', test_ls[-1])    
    semilogy(range(1, num_epochs + 1), train_ls, 'epochs', 'loss',
             range(1, num_epochs + 1), test_ls, ['train', 'test'])
    print('weight:', net.weight.data,
          '\nbias:', net.bias.data)

#测试
fit_and_plot(poly_features[:n_train, :], poly_features[n_train:, :], labels[:n_train], labels[n_train:]) #正常
fit_and_plot(features[:n_train, :], features[n_train:, :], labels[:n_train], labels[n_train:]) #欠拟合
fit_and_plot(poly_features[0:2, :], poly_features[n_train:, :], labels[0:2], labels[n_train:]) #过拟合

• Third-order polynomial fit (normal)
  

Although this model is a linear model, but the characteristic of the input sample is calculated polynomial, the parameters of the linear combination so that a polynomial model.

• Linear fit (underfitting)
  
• Training set is too small (over-fitting)
  

Preventing overfitting

1. L2正则化（又叫权重衰减）
   加入L2正则化，能够防止个别参数极端大的情况，从而防止过拟合。在全局最小约束下，给损失函数加上个L2正则化项：
   $\ell\left(w_{1}, w_{2}, b\right)+\frac{\lambda}{2 n}|w|^{2}$

optimizer_w = torch.optim.SGD(params=[net.weight], lr=lr, weight_decay=wd) # 对权重参数衰减

2. dropout（又叫丢弃法）
   丢弃法通过一定概率把某些单元的灭活（即对应值置0），来避免训练过程中对某些神经元的过分依赖。下面公式证明，丢弃法不改变输入期望值。
   $h_{i}^{\prime}=\frac{\xi_{i}}{1-p} h_{i} \\ E\left(h_{i}^{\prime}\right)=\frac{E\left(\xi_{i}\right)}{1-p} h_{i}=h_{i}$

def dropout(X, drop_prob):
    X = X.float()
    assert 0 <= drop_prob <= 1
    keep_prob = 1 - drop_prob
    # 这种情况下把全部元素都丢弃
    if keep_prob == 0:
        return torch.zeros_like(X)
    mask = (torch.rand(X.shape) < keep_prob).float()
    print(mask)
    
    return mask * X / keep_prob
# 使用说明
def net(X, is_training=True):
    X = X.view(-1, num_inputs)
    H1 = (torch.matmul(X, W1) + b1).relu()
    if is_training:  # 只在训练模型时使用丢弃法
        H1 = dropout(H1, drop_prob1)  # 在第一层全连接后添加丢弃层
    H2 = (torch.matmul(H1, W2) + b2).relu()
    if is_training:
        H2 = dropout(H2, drop_prob2)  # 在第二层全连接后添加丢弃层
    return torch.matmul(H2, W3) + b3

#pytorch实现
nn.Dropout(drop_prob1)

有些话说

一些问题：

1. 如何看出是过拟合？防止过拟合的方法有哪些？
2. L2正则化和dropout防止过拟合的原理各自是什么？如何使用pytorch实现它们？