李沐d2l(四)---Softmax回归

一、Softmax回归

回归VS分类

回归估计一个连续值

分类预测一个离散类别

均方损失

无校验比例

交叉熵损失

Softmax回归是一个多类分类模型。使用Softmax操作子得到每个类的预测置信度。使用交叉熵来衡量预测和标号的区别。

二、损失函数

均方损失

蓝色曲线代表y=0时，变化y’得到的函数

绿色为似然函数，e^-L

橙色为损失函数的梯度

绝对值损失

Huber’s Robust Loss

三、图像分类数据集

import torch
import torchvision
from torch.utils import data
from torchvision import transforms
from d2l import torch as d2l

d2l.use_svg_display()

# 1 下载数据存储到内存中
trans = transforms.ToTensor() # 把图片转换为tensor
mnist_train = torchvision.datasets.FashionMNIST(
  root="FashionMNIST",
  train=True, # 下载的是训练数据集
  transform=trans, # 表示拿出来时得到的是tensor，而不是图片
  )
mnist_test = torchvision.datasets.FashionMNIST( # 测试数据集不参与训练，用来预测模型的好坏
  root="FashionMNIST",
  train=False,
  transform=trans,
)

# 2 可视化数据集
def get_fashion_mnist_labels(labels):
  text_labels = [
    't-shirt', 'trouser', 'pullover', 'dress', 'coat', 'sandal', 'shirt',
        'sneaker', 'bag', 'ankle boot'
  ]
  return [text_labels[int(i)] for i in labels]

def show_images(imgs, num_rows, num_cols, titles=None, scale=1.5):
  figsize = (num_cols * scale, num_rows * scale)
  _, axes = d2l.plt.subplots(num_rows, num_cols, figsize=figsize)
  axes = axes.flatten()
  for i, (ax, img) in enumerate(zip(axes, imgs)):
     if torch.is_tensor(img):
        ax.imshow(img.numpy()) # 图片张量
     else:
        ax.imshow(img) # PIL图片
     ax.axes.get_xaxis().set_visible(False)
     ax.axes.get_yaxis().set_visible(False)
     if titles:
       ax.set_title(titles[i])
  d2l.plt.show()
  return axes  

X, y = next(iter(data.DataLoader(mnist_train, batch_size=20)))
show_images(X.reshape(20,28,28), 4, 5, titles=get_fashion_mnist_labels(y))

# 3 读取一小批量数据
batch_size = 256

def get_dataloader_workers():
  return 4 # 使用4个进程来读取数据
train_iter = data.DataLoader(mnist_train, batch_size, shuffle=True, num_workers=get_dataloader_workers())

timer = d2l.Timer()
for X, y in train_iter:
  continue
print(f'{
      timer.stop():.2f} sec') # 2.88sec

进行封装，方便以后复用

def load_data_fashion_mnist(batch_size, resize=None):  # resize方便自定义图片的大小
    """下载Fashion-MNIST数据集，然后将其加载到内存中。"""
    trans = [transforms.ToTensor()]
    if resize:
        trans.insert(0, transforms.Resize(resize))
    trans = transforms.Compose(trans)
    mnist_train = torchvision.datasets.FashionMNIST(root="../data",
                                                    train=True,
                                                    transform=trans,
                                                    download=True)
    mnist_test = torchvision.datasets.FashionMNIST(root="../data",
                                                   train=False,
                                                   transform=trans,
                                                   download=True)
    return (data.DataLoader(mnist_train, batch_size, shuffle=True,
                            num_workers=get_dataloader_workers()),
            data.DataLoader(mnist_test, batch_size, shuffle=False,
                            num_workers=get_dataloader_workers()))

train_iter, test_iter = load_data_fashion_mnist(32, resize=64)
for X, y in train_iter:
    print(X.shape, X.dtype, y.shape, y.dtype)
    break

四、softmax回归的从零开始实现

import torch
from IPython import display
from d2l import torch as d2l

# 1 定义图片数量，获得训练集和测试集
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)

# 2 展开每个图像，将他们视为长度为784的向量，由于数据集有10个类别，所以网络输出维度为10
num_inputs = 784 # 28 * 28
num_outputs = 10

w = torch.normal(0, 0.01, size=(num_inputs, num_outputs), requires_grad=True) # 权重
b = torch.zeros(num_outputs, requires_grad=True) # 偏移

# 3 定义softmax
def softmax(X):
  X_exp = torch.exp(X)
  patition = X_exp.sum(1, keepdim=True)
  return X_exp / patition # 应用了广播机制

# 检验
# X = torch.normal(0, 1, (2,5))
# x_prob = softmax(X)
# print(x_prob, x_prob.sum(1))
''' 每行总和为1 tensor([[0.2055, 0.1315, 0.0336, 0.0758, 0.5536], [0.1539, 0.0319, 0.0362, 0.6295, 0.1485]]) tensor([1., 1.]) '''

# 4 实现模型
def net(X):
  return softmax(torch.matmul(X.reshape((-1, w.shape[0])),w) + b)

# 5 实现交叉熵损失

# 例子
# y = torch.tensor([0, 2]) # 创建一个张量
# y_hat = torch.tensor([[0.1, 0.3, 0.6], [0.3, 0.2, 0.5]]) # 给出预测值
# print(y_hat[0,1],y) 
''' tensor([0.1000, 0.5000]) '''
def cross_entropy(y_hat, y):
  return -torch.log(y_hat[range(len(y_hat)), y])

# 6 将预测类别与真实y元素进行比较
def accuracy(y_hat, y):
  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())

# 7 评估任意模型net的准确率
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]
def evaluate_accuracy(net, data_iter):
  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]
  
# 8 softmax回归训练
def train_epoch_ch3(net, train_iter, loss, updater):  
    if isinstance(net, torch.nn.Module):
        net.train()
    metric = Accumulator(3)
    for X, y in train_iter:
        y_hat = net(X)
        l = loss(y_hat, y)
        if isinstance(updater, torch.optim.Optimizer):
            updater.zero_grad()
            l.backward()
            updater.step()
            metric.add(
                float(l) * len(y), accuracy(y_hat, y),
                y.size().numel())
        else:
            l.sum().backward()
            updater(X.shape[0])
            metric.add(float(l.sum()), accuracy(y_hat, y), y.numel())
    return metric[0] / metric[2], metric[1] / metric[2]

# 9 画图类
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)):
        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,]
        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):
        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)

# 10 训练函数
def train_ch3(net, train_iter, test_iter, loss, num_epochs, updater):  
    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
    assert test_acc <= 1 and test_acc > 0.7, test_acc

# 11 定义小批量随机梯度来优化模型的损失函数
lr = 0.1
def updater(batch_size):
  return d2l.sgd([w,b], lr, batch_size)

# 12 开始训练
num_epochs = 10
train_ch3(net, train_iter, test_iter, cross_entropy, num_epochs, updater)

d2l.plt.show()

进行预测

def predict_ch3(net, test_iter, n=6):  
    """预测标签（定义见第3章）。"""
    for X, y in test_iter:
        break
    trues = d2l.get_fashion_mnist_labels(y)
    preds = d2l.get_fashion_mnist_labels(net(X).argmax(axis=1))
    titles = [true + '\n' + pred for true, pred in zip(trues, preds)]
    d2l.show_images(X[0:n].reshape((n, 28, 28)), 1, n, titles=titles[0:n])

predict_ch3(net, test_iter)
d2l.plt.show()

五、softmax的简洁实现

import torch
from torch import nn
from d2l import torch as d2l
import os
os.environ["KMP_DUPLICATE_LIB_OK"]="TRUE"

# 1 定义图片数量，获得训练集和测试集
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)

# 2 pytoch不会隐式的调整输入的形状，所以需要定义展平层在线性层前调整网络输入的形状
net = nn.Sequential(nn.Flatten(), nn.Linear(784, 10))

def init_weights(m):
  if type(m) == nn.Linear:
    nn.init.normal_(m.weight, std=0.01)
net.apply(init_weights)

# 3 在交叉熵损失函数中传递μ归一化的预测，并同时计算softmax及其对数
loss = nn.CrossEntropyLoss(reduction='none')


# 4 使用学习率为0.1的小批量随机梯度下降作为优化算法
trainer = torch.optim.SGD(net.parameters(), lr=0.1)

# 5 调用之前定义的训练函数来训练模型
num_epochs = 10
d2l.train_ch3(net, train_iter, test_iter, loss, num_epochs, trainer)
d2l.plt.show()