1.一些概念

训练误差（training error）指模型在训练数据集上表现出的误差

泛化误差（generalization error）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。

机器学习模型应关注降低泛化误差。

2.多项式拟合实验

# %matplotlib inlineimport torchimport numpy as npimport syssys.path.append(r"D:\project\fitting学习")import d2lzh1981 as d2lprint(torch.__version__)

初始化模型参数

# n_train, n_test分别指训练样本数和测试样本数n_train, n_test, true_w, true_b = 100, 100, [1.2, -3.4, 5.6], 5# features初始化为xfeatures = torch.randn((n_train + n_test, 1))# 那么这里poly_features就为x,x^2,x^3poly_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)

features[:2], poly_features[:2], labels[:2]

定义、训练和测试模型

首先定义一个画图函数用于观察

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)

定义一个可以用来训练并打印训练误差和泛化误差的函数

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）

前面输入的是poly_features，而这里我们输入features

fit_and_plot(features[:n_train, :], features[n_train:, :], labels[:n_train], labels[n_train:])

来看一下结果

训练样本不足（过拟合）

这里也是输入poly_features，但是只有0和1

fit_and_plot(poly_features[0:2, :], poly_features[n_train:, :], labels[0:2], labels[n_train:])

3.方法

L2L_2L2​范数正则化（regularization）

高维线性回归实验从零开始的实现

#%matplotlib inlineimport torchimport torch.nn as nnimport numpy as npimport syssys.path.append(r"D:\project\fitting学习")import d2lzh1981 as d2lprint(torch.__version__)

初始化模型参数

n_train, n_test, num_inputs = 20, 100, 200true_w, true_b = torch.ones(num_inputs, 1) * 0.01, 0.05features = torch.randn((n_train + n_test, num_inputs))labels = torch.matmul(features, true_w) + true_blabels += 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

将权重的每个元素平方再求和

def l2_penalty(w):return (w**2).sum() / 2

定义训练和测试

batch_size, num_epochs, lr = 1, 100, 0.003net, loss = d2l.linreg, d2l.squared_lossdataset = torch.utils.data.TensorDataset(train_features, train_labels)train_iter = torch.utils.data.DataLoader(dataset, batch_size, shuffle=True)def fit_and_plot(lambd):# 定义了一个超参数λ，这个就是L2范数中乘的那个正数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)# 参数初始化使用init模块来完成nn.init.normal_(net.weight, mean=0, std=1)nn.init.normal_(net.bias, mean=0, std=1)# 优化函数使用的是optim模块里的随机梯度下降函数# 这里不需要为L2范数惩罚项单独撰写函数，权重衰减已经封装在SGD函数中了# weight_decay=wd表示启用权重衰减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)

丢弃法

丢弃法从零开始的实现

%matplotlib inlineimport torchimport torch.nn as nnimport numpy as npimport syssys.path.append("/home/kesci/input")import d2lzh1981 as d2lprint(torch.__version__)

def dropout(X, drop_prob):X = X.float()assert 0 <= drop_prob <= 1 # 判断丢弃率是否在0到1之间，如果不在就会报错keep_prob = 1 - drop_prob# 这种情况下把全部元素都丢弃if keep_prob == 0:# keep_prob是保存率，保存率为0说明元素要被全部丢弃，所以返回一个0化的x就可以了return torch.zeros_like(X)mask = (torch.rand(X.shape) < keep_prob).float()# 随机生成一个x形状的矩阵，内容是随机的。将随机生成的矩阵中的每个位置上的元素和保留率进行比较，如果小于保留率，那么这个位置的元素得以保留，该位置赋值为1，如果大于，该位置元素就要被丢弃，赋值为0。这样就得到maskreturn mask * X / keep_prob

X = torch.arange(16).view(2, 8)dropout(X, 0)# 丢弃率为0，得到的就是原始矩阵x

dropout(X, 0.5)

dropout(X, 1.0)

# 参数的初始化num_inputs, num_outputs, num_hiddens1, num_hiddens2 = 784, 10, 256, 256W1 = 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)params = [W1, b1, W2, b2, W3, b3]

drop_prob1, drop_prob2 = 0.2, 0.5# 隐藏层有两层，所以设置两个丢弃率def net(X, is_training=True):# 输入参数is_training来区分是否在训练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, 0for X, y in data_iter:if isinstance(net, torch.nn.Module):# 判断网络模型是不是nn.Modulenet.eval() # 是的话使用评估模式, 这会关闭dropoutacc_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Falseacc_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)

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