TASK03：过拟合、欠拟合解决方案

模型选择、过拟合和欠拟合

训练误差和泛化误差

在解释上述现象之前，我们需要区分训练误差（training error）和泛化误差（generalization error）。通俗来讲，前者指模型在训练数据集上表现出的误差，后者指模型在任意一个测试数据样本上表现出的误差的期望，并常常通过测试数据集上的误差来近似。

多项式函数拟合实验

%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) #torch.cat张量拼接
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、训练和测试模型
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=':')#用于绘制折线图，y 轴是指数型的。
        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)
#三阶多项式函数拟合（正常）
fit_and_plot(poly_features[:n_train, :], poly_features[n_train:, :], labels[:n_train], labels[n_train:])
#线性函数拟合（欠拟合）
fit_and_plot(features[:n_train, :], features[n_train:, :], labels[:n_train], labels[n_train:])
#训练样本不足（过拟合）
fit_and_plot(poly_features[0:2, :], poly_features[n_train:, :], labels[0:2], labels[n_train:])

权重衰减

方法
权重衰减等价于L2范数正则化（regularization）。正则化通过为模型损失函数添加惩罚项使学出的模型参数值较小，是应对过拟合的常用手段。

高维线性回归实验从零开始的实现

下面，我们以高维线性回归为例来引入一个过拟合问题，并使用权重衰减来应对过拟合。设数据样本特征的维度为p。对于训练数据集和测试数据集中特征为的任一样本，我们使用如下的线性函数来生成该样本的标签：

其中噪声项服从均值为0、标准差为0.01的正态分布。为了较容易地观察过拟合，我们考虑高维线性回归问题，如设维度p=200；同时，我们特意把训练数据集的样本数设低，如20。

%matplotlib inline
import torch
import torch.nn as nn
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, num_inputs = 20, 100, 200
true_w, true_b = torch.ones(num_inputs, 1) * 0.01, 0.05

features = torch.randn((n_train + n_test, num_inputs))
labels = torch.matmul(features, true_w) + true_b
labels += torch.tensor(np.random.normal(0, 0.01, size=labels.size()), dtype=torch.float)
train_features, test_features = features[:n_train, :], features[n_train:, :]
train_labels, test_labels = labels[:n_train], labels[n_train:]
# 定义参数初始化函数，初始化模型参数并且附上梯度
def init_params():
    w = torch.randn((num_inputs, 1), requires_grad=True)
    b = torch.zeros(1, requires_grad=True)
    return [w, b]
#定义L2范数惩罚项
def l2_penalty(w):
    return (w**2).sum() / 2
#定义训练和测试
batch_size, num_epochs, lr = 1, 100, 0.003
net, loss = d2l.linreg, d2l.squared_loss

dataset = torch.utils.data.TensorDataset(train_features, train_labels)
train_iter = torch.utils.data.DataLoader(dataset, batch_size, shuffle=True)

def fit_and_plot(lambd):
    w, b = init_params()
    train_ls, test_ls = [], []
    for _ in range(num_epochs):
        for X, y in train_iter:
            # 添加了L2范数惩罚项
            l = loss(net(X, w, b), y) + lambd * l2_penalty(w)
            l = l.sum()
            
            if w.grad is not None:
                w.grad.data.zero_()
                b.grad.data.zero_()
            l.backward()
            d2l.sgd([w, b], lr, batch_size)
        train_ls.append(loss(net(train_features, w, b), train_labels).mean().item())
        test_ls.append(loss(net(test_features, w, b), test_labels).mean().item())
    d2l.semilogy(range(1, num_epochs + 1), train_ls, 'epochs', 'loss',
                 range(1, num_epochs + 1), test_ls, ['train', 'test'])
    print('L2 norm of w:', w.norm().item())
#观察过拟合
fit_and_plot(lambd=0)
#使用权重衰减
fit_and_plot(lambd=3)

简洁实现

def fit_and_plot_pytorch(wd):
    # 对权重参数衰减。权重名称一般是以weight结尾
    net = nn.Linear(num_inputs, 1)
    nn.init.normal_(net.weight, mean=0, std=1)
    nn.init.normal_(net.bias, mean=0, std=1)
    optimizer_w = torch.optim.SGD(params=[net.weight], lr=lr, weight_decay=wd) # 对权重参数衰减
    optimizer_b = torch.optim.SGD(params=[net.bias], lr=lr)  # 不对偏差参数衰减
    
    train_ls, test_ls = [], []
    for _ in range(num_epochs):
        for X, y in train_iter:
            l = loss(net(X), y).mean()
            optimizer_w.zero_grad()
            optimizer_b.zero_grad()
            
            l.backward()
            
            # 对两个optimizer实例分别调用step函数，从而分别更新权重和偏差
            optimizer_w.step()
            optimizer_b.step()
        train_ls.append(loss(net(train_features), train_labels).mean().item())
        test_ls.append(loss(net(test_features), test_labels).mean().item())
    d2l.semilogy(range(1, num_epochs + 1), train_ls, 'epochs', 'loss',
                 range(1, num_epochs + 1), test_ls, ['train', 'test'])
    print('L2 norm of w:', net.weight.data.norm().item())

fit_and_plot_pytorch(0)
fit_and_plot_pytorch(3)

dropout

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

print(torch.__version__)

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()
    
    return mask * X / keep_prob
# 参数的初始化
num_inputs, num_outputs, num_hiddens1, num_hiddens2 = 784, 10, 256, 256

W1 = torch.tensor(np.random.normal(0, 0.01, size=(num_inputs, num_hiddens1)), dtype=torch.float, requires_grad=True)
b1 = torch.zeros(num_hiddens1, requires_grad=True)
W2 = torch.tensor(np.random.normal(0, 0.01, size=(num_hiddens1, num_hiddens2)), dtype=torch.float, requires_grad=True)
b2 = torch.zeros(num_hiddens2, requires_grad=True)
W3 = torch.tensor(np.random.normal(0, 0.01, size=(num_hiddens2, num_outputs)), dtype=torch.float, requires_grad=True)
b3 = torch.zeros(num_outputs, requires_grad=True)
drop_prob1, drop_prob2 = 0.2, 0.5

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

def evaluate_accuracy(data_iter, net):
    acc_sum, n = 0.0, 0
    for X, y in data_iter:
        if isinstance(net, torch.nn.Module):
            net.eval() # 评估模式, 这会关闭dropout
            acc_sum += (net(X).argmax(dim=1) == y).float().sum().item()
            net.train() # 改回训练模式
        else: # 自定义的模型
            if('is_training' in net.__code__.co_varnames): # 如果有is_training这个参数
                # 将is_training设置成False
                acc_sum += (net(X, is_training=False).argmax(dim=1) == y).float().sum().item() 
            else:
                acc_sum += (net(X).argmax(dim=1) == y).float().sum().item() 
        n += y.shape[0]
    return acc_sum / n
num_epochs, lr, batch_size = 5, 100.0, 256  # 这里的学习率设置的很大，原因与之前相同。
loss = torch.nn.CrossEntropyLoss()
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size, root='/home/kesci/input/FashionMNIST2065')
d2l.train_ch3(
    net,
    train_iter,
    test_iter,
    loss,
    num_epochs,
    batch_size,
    params,
    lr)

简洁实现

net = nn.Sequential(
        d2l.FlattenLayer(),
        nn.Linear(num_inputs, num_hiddens1),
        nn.ReLU(),
        nn.Dropout(drop_prob1),
        nn.Linear(num_hiddens1, num_hiddens2), 
        nn.ReLU(),
        nn.Dropout(drop_prob2),
        nn.Linear(num_hiddens2, 10)
        )

for param in net.parameters():
    nn.init.normal_(param, mean=0, std=0.01)

optimizer = torch.optim.SGD(net.parameters(), lr=0.5)
d2l.train_ch3(net, train_iter, test_iter, loss, num_epochs, batch_size, None, None, optimizer)