线性回归的从零开始实现

我们将从零开始实现整个方法，包括数据流水线、模型、损失函数和小批量随机梯度下降优化器

%matplotlib inline
import random
import torch
from d2l import torch as d2l

根据带有噪声的线性模型构造一个人造数据集。 我们使用线性模型参数$\mathbf{w} = [2, -3.4]^\top$、$b = 4.2$和噪声项$\epsilon$生成数据集及其标签：

$$\mathbf{y}= \mathbf{X} \mathbf{w} + b + \mathbf\epsilon$$

def synthetic_data(w, b, num_examples):  
    """生成 y = Xw + b + 噪声。"""
    X = torch.normal(0, 1, (num_examples, len(w)))
    y = torch.matmul(X, w) + b
    y += torch.normal(0, 0.01, y.shape)
    return X, y.reshape((-1, 1))

true_w = torch.tensor([2, -3.4])
true_b = 4.2
features, labels = synthetic_data(true_w, true_b, 1000)

features 中的每一行都包含一个二维数据样本，labels 中的每一行都包含一维标签值（一个标量）

print('features:', features[0], '\nlabel:', labels[0])

features: tensor([-0.6612, -1.8215]) 
label: tensor([9.0842])

d2l.set_figsize()
d2l.plt.scatter(features[:, (1)].detach().numpy(),
                labels.detach().numpy(), 1);

定义一个data_iter 函数， 该函数接收批量大小、特征矩阵和标签向量作为输入，生成大小为batch_size的小批量

def data_iter(batch_size, features, labels):
    num_examples = len(features)
    indices = list(range(num_examples))
    random.shuffle(indices)
    for i in range(0, num_examples, batch_size):
        batch_indices = torch.tensor(indices[i:min(i +
                                                   batch_size, num_examples)])
        yield features[batch_indices], labels[batch_indices]

batch_size = 10

for X, y in data_iter(batch_size, features, labels):
    print(X, '\n', y)
    break

tensor([[ 0.3161, -1.1807],
        [ 0.8956,  1.3530],
        [-0.9413, -0.2198],
        [ 0.2601, -1.4972],
        [ 0.2821,  0.0491],
        [ 1.2272,  0.3696],
        [-1.0256, -0.5918],
        [-1.9352,  0.6146],
        [-0.3074, -1.5586],
        [ 0.7992,  0.4547]]) 
 tensor([[ 8.8543],
        [ 1.3933],
        [ 3.0845],
        [ 9.7996],
        [ 4.5993],
        [ 5.3924],
        [ 4.1613],
        [-1.7471],
        [ 8.8908],
        [ 4.2466]])

定义 初始化模型参数

w = torch.normal(0, 0.01, size=(2, 1), requires_grad=True)
b = torch.zeros(1, requires_grad=True)

定义模型

def linreg(X, w, b):  
    """线性回归模型。"""
    return torch.matmul(X, w) + b

定义损失函数

def squared_loss(y_hat, y):  
    """均方损失。"""
    return (y_hat - y.reshape(y_hat.shape))**2 / 2

定义优化算法

def sgd(params, lr, batch_size):  
    """小批量随机梯度下降。"""
    with torch.no_grad():
        for param in params:
            param -= lr * param.grad / batch_size
            param.grad.zero_()

训练过程

lr = 0.03
num_epochs = 3
net = linreg
loss = squared_loss

for epoch in range(num_epochs):
    for X, y in data_iter(batch_size, features, labels):
        l = loss(net(X, w, b), y)
        l.sum().backward()
        sgd([w, b], lr, batch_size)
    with torch.no_grad():
        train_l = loss(net(features, w, b), labels)
        print(f'epoch {epoch + 1}, loss {float(train_l.mean()):f}')

epoch 1, loss 0.033103
epoch 2, loss 0.000124
epoch 3, loss 0.000054

比较真实参数和通过训练学到的参数来评估训练的成功程度

print(f'w的估计误差: {true_w - w.reshape(true_w.shape)}')
print(f'b的估计误差: {true_b - b}')

w的估计误差: tensor([ 0.0006, -0.0004], grad_fn=<SubBackward0>)
b的估计误差: tensor([0.0003], grad_fn=<RsubBackward1>)