1. 线性回归
2. 基础优化方法

1. 线性回归
总结

• 线性回归是对n维输入的加权，外加偏差
• 使用平方损失来衡量预测值和真实值的差异
• 线性回归有显示解
• 线性回归可以看作是单层神经网络

2. 基础优化方法
梯度下降

小批量随机梯度下降

3. 总结

• 梯度下降通过不断沿着反梯度方向更新参数求解
• 小批量随机梯度下降是深度学习默认的求解算法
• 两个重要的超参数是批量大小和学习率

4. 线性回归的从零开始实现

import torch
import random
def synthetic_data(w, b, num):"生成 y = Xw + b + 噪声"''' 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𝐰 = [2,−3.4]⊤w=[2,−3.4]⊤、𝑏 = 4.2和噪声项𝜖ϵ生成数据集及其标签：𝐲 = 𝐗𝐰 + 𝑏 + 𝜖'''X = torch.normal(0, 1, (num, len(w)))y = torch.matmul(X, w) + by += torch.normal(0, 0.01, y.shape)return X, y.reshape(-1, 1)true_w = torch.tensor([2, -3.4])
true_b = 4.2
num_examples = 1000
batch_size = 10
features, labels = synthetic_data(true_w, true_b, num_examples)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):num_indices = torch.tensor(indices[i: min(i +batch_size, num_examples)])yield features[num_indices], labels[num_indices]def linreg(X, w, b):return torch.matmul(X, w) + bdef squared_loss(y_hat, y):return (y_hat - y.reshape(y_hat.shape))**2/2def sgd(params, lr, batch_size):with torch.no_grad():for param in params:param -= lr * param.grad / batch_sizeparam.grad.zero_()w = torch.normal(0, 0.01, size=(2, 1), requires_grad=True)
b = torch.zeros(1, requires_grad=True)loss = squared_loss
net = linreg
lr = 0.03
epoches = 3for epoch in range(epoches):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}')

5. 线性回归的简洁实现

import torch
from d2l import torch as d2l
from torch.utils import datatrue_w = torch.tensor([2, -3.4])
true_b = 4.2
features, labels = d2l.synthetic_data(true_w, true_b, 1000)def load_array(batch_size, data_array, is_train=True):dataset = data.TensorDataset(*data_array)return data.DataLoader(dataset, batch_size, shuffle=is_train)batch_size = 10
data_iter = load_array(batch_size, (features, labels))from torch import nnnet = nn.Sequential(nn.Linear(2, 1))
net[0].weight.data.normal_(0, 0.01)
net[0].bias.data.fill_(0)loss = nn.MSELoss()trainer = torch.optim.SGD(net.parameters(), lr = 0.03)num_epoches = 3iter_temp = iter(data_iter)for epoch in range(num_epoches):for X, y in next(iter_temp):l = loss(net(X), y)l.backward()trainer.step()trainer.zero_grad()l = loss(net(features), labels)print(f'epoch {epoch + 1}, loss {l:f}')