深度学习——线性神经网络
Posted 夏风喃喃
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深度学习（2）——线性神经网络
作者：夏风喃喃
参考：《动手学深度学习第二版》李沐
一. 用以计时的Python类
class Timer:
"""记录多次运行时间。"""
def __init__(self):
self.times = []
self.start()
def start(self):
"""启动计时器。"""
self.tik = time.time()
def stop(self):
"""停止计时器并将时间记录在列表中。"""
self.times.append(time.time() - self.tik)
return self.times[-1]
def avg(self):
"""返回平均时间。"""
return sum(self.times) / len(self.times)
def sum(self):
"""返回时间总和。"""
return sum(self.times)
def cumsum(self):
"""返回累计时间。"""
return np.array(self.times).cumsum().tolist()
二. 线性回归的实现
生成数据集：
import random
import torch
from d2l import torch as d2l
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]) #权重为[2, -3.4]
true_b = 4.2 #偏置为4.2
features, labels=synthetic_data(true_w, true_b, 1000) #生成1000个特征与标签
读取数据集：
# 批量迭代函数
def data_iter(batch_size, features, labels):
num_examples = len(features)
# 样本索引indices=[0,1,…,num_examples-1]
indices = list(range(num_examples))
# 打乱样本索引
random.shuffle(indices)
# 将样本分为batch，并构造迭代器
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]
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:
#损失批量样本的总和，用批量大小（batch_size）来归一化步长
param -= lr * param.grad / batch_size
param.grad.zero_()
训练：
lr = 0.03 #学习率
num_epochs = 3 #训练集迭代次数
net = linreg #网络
loss = squared_loss #损失函数
for epoch in range(num_epochs):
for X, y in data_iter(batch_size, features, labels):
l = loss(net(X, w, b), y) # X和y的小批量损失
# 因为l形状是(batch_size, 1)。l中的所有元素被加到一起，
# 并以此计算关于[w, b]的梯度
l.sum().backward()
sgd([w, b], lr, batch_size) # 使用参数的梯度更新参数
with torch.no_grad():
train_l = loss(net(features, w, b), labels)
print(f'epoch {epoch + 1}, loss {float(train_l.mean()):f}')
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}')
三. 线性回归简洁实现
生成数据集：
import numpy as np
import torch
from torch.utils import data
from d2l import torch as d2l
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 = d2l.synthetic_data(true_w, true_b, 1000)
读取数据集：
def load_array(data_arrays, batch_size, is_train=True):
"""构造一个PyTorch数据迭代器。"""
dataset = data.TensorDataset(*data_arrays)
return data.DataLoader(dataset, batch_size, shuffle=is_train)
batch_size = 10
data_iter = load_array((features, labels), batch_size)
定义模型：
# nn是神经网络的缩写
from torch import nn
net = nn.Sequential(nn.Linear(2, 1))
初始化模型参数：
net[0].weight.data.normal_(0, 0.01)
net[0].bias.data.fill_(0)
定义损失函数：
# 平方 L2 范数
loss = nn.MSELoss()
定义优化算法：
trainer = torch.optim.SGD(net.parameters(), lr=0.03)
训练：
num_epochs = 3
for epoch in range(num_epochs):
for X, y in data_iter:
l = loss(net(X), y)
trainer.zero_grad()
l.backward()
trainer.step()
l = loss(net(features), labels)
print(f'epoch {epoch + 1}, loss {l:f}')
w = net[0].weight.data
print('w的估计误差：', true_w - w.reshape(true_w.shape))
b = net[0].bias.data
print('b的估计误差：', true_b - b)
四. softmax回归的实现
import torch
from IPython import display
from d2l import torch as d2l
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)
初始化模型参数：
# 数据集中样本是28×28的图像,展平长度为784的向量
# 有10个类别所以网络输出维度为 10
num_inputs = 784
num_outputs = 10
W = torch.normal(0, 0.01, size=(num_inputs, num_outputs), requires_grad=True)
b = torch.zeros(num_outputs, requires_grad=True)
定义softmax操作：
def softmax(X):
X_exp = torch.exp(X)
partition = X_exp.sum(1, keepdim=True)
return X_exp / partition # 这里应用了广播机制
定义模型：
def net(X):
return softmax(torch.matmul(X.reshape((-1, W.shape[0])), W) + b)
定义损失函数：
def cross_entropy(y_hat, y):
return -torch.log(y_hat[range(len(y_hat)), y])
分类准确率：
def accuracy(y_hat, y):
"""计算预测正确的数量。"""
if len(y_hat.shape) > 1 and y_hat.shape[1] > 1:
y_hat = y_hat.argmax(axis=1)
cmp = y_hat.type(y.dtype) == y
return float(cmp.type(y.dtype).sum())
def evaluate_accuracy(net, data_iter):
"""计算在指定数据集上模型的精度。"""
if isinstance(net, torch.nn.Module):
net.eval() # 将模型设置为评估模式
metric = Accumulator(2) # 正确预测数、预测总数
for X, y in data_iter:
metric.add(accuracy(net(X), y), y.numel())
return metric[0] / metric[1]
class Accumulator:
"""在`n`个变量上累加。"""
def __init__(self, n):
self.data = [0.0] * n
def add(self, *args):
self.data = [a + float(b) for a, b in zip(self.data, args)]
def reset(self):
self.data = [0.0] * len(self.data)
def __getitem__(self, idx):
return self.data[idx]
训练：
def train_epoch_ch3(net, train_iter, loss, updater):
"""训练模型一个迭代周期（定义见第3章）。"""
# 将模型设置为训练模式
if isinstance(net, torch.nn.Module):
net.train()
# 训练损失总和、训练准确度总和、样本数
metric = Accumulator(3)
for X, y in train_iter:
# 计算梯度并更新参数
y_hat = net(X)
l = loss(y_hat, y)
if isinstance(updater, torch.optim.Optimizer):
# 使用PyTorch内置的优化器和损失函数
updater.zero_grad()
l.backward()
updater.step()
metric.add(
float(l) * len(y), accuracy(y_hat, y),
y.size().numel())
else:
# 使用定制的优化器和损失函数
l.sum().backward()
updater(X.shape[0])
metric.add(float(l.sum()), accuracy(y_hat, y), y.numel())
# 返回训练损失和训练准确率
return metric[0] / metric[2], metric[1] / metric[2]
class Animator:
"""在动画中绘制数据。"""
def __init__(self, xlabel=None, ylabel=None, legend=None, xlim=None,
ylim=None, xscale='linear', yscale='linear',
fmts=('-', 'm--', 'g-.', 'r:'), nrows=1, ncols=1,
figsize=(3.5, 2.5)):
# 增量地绘制多条线
if legend is None:
legend = []
d2l.use_svg_display()
self.fig, self.axes = d2l.plt.subplots(nrows,
ncols, figsize=figsize)
if nrows * ncols == 1:
self.axes = [self.axes,]
# 使用lambda函数捕获参数
self.config_axes = lambda: d2l.set_axes(self.axes[
0], xlabel, ylabel, xlim, ylim, xscale, yscale, legend)
self.X, self.Y, self.fmts = None, None, fmts
def add(self, x, y):
# 向图表中添加多个数据点
if not hasattr(y, "__len__"):
y = [y]
n = len(y)
if not hasattr(x, "__len__"):
x = [x] * n
if not self.X:
self.X = [[] for _ in range(n)]
if not self.Y:
self.Y = [[] for _ in range(n)]
for i, (a, b) in enumerate(zip(x, y)):
if a is not None and b is not None:
self.X[i].append(a)
self.Y[i].append(b)
self.axes[0].cla()
for x, y, fmt in zip(self.X, self.Y, self.fmts):
self.axes[0].plot(x, y, fmt)
self.config_axes()
display.display(self.fig)
display.clear_output(wait=True)
def train_ch3(net, train_iter, test_iter, loss, num_epochs, updater):
"""训练模型。"""
animator = Animator(xlabel='epoch', xlim=[1, num_epochs],
ylim=[0.3, 0.9],
legend=['train loss', 'train acc', 'test acc'])
for epoch in range(num_epochs):
train_metrics = train_epoch_ch3(net, train_iter, loss, updater)
test_acc = evaluate_accuracy(net, test_iter)
animator.add(epoch + 1, train_metrics + (test_acc,))
train_loss, train_acc = train_metrics
assert train_loss < 0.5, train_loss
assert train_acc <= 1 and train_acc > 0.7, train_acc
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