Softmax回归

引子

回归可以用于预测多少的问题。 比如预测房屋被售出价格，或者棒球队可能获得的胜场数，又或者患者住院的天数。

事实上，我们也对分类问题感兴趣：不是问“多少”，而是问“哪一个”！
在这里插入图片描述

图像分类数据集

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from d2l import torch as d2l
from torch.utils import data
from torchvision import transforms
import torchvision
import torch
%matplotlib inline

d2l.use_svg_display()
下载并读取数据集
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# 下载数据集
trans = transforms.ToTensor()
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)
len(mnist_train), len(mnist_test)
(60000, 10000)
制定标签
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def get_fashion_mnist_labels(labels):
    # 返回get_fashion_mnist_labels数据集标签
    text_labels = ['t-shirt', 'trouser', 'pullover', 'dress', 'coat',
                   'sandal', 'shirt', 'sneaker', 'bag', 'ankle boot']
    return [text_labels[int(i)] for i in labels]

可视化

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def show_images(imgs, num_rows, num_cols, titles=None, scale=1.5):  #@save
    """绘制图像列表"""
    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):
            # 图片张量
            ax.imshow(img.numpy())
        else:
            # PIL图片
            ax.imshow(img)
        ax.axes.get_xaxis().set_visible(False)
        ax.axes.get_yaxis().set_visible(False)
        if titles:
            ax.set_title(titles[i])
    return axes
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X, y = next(iter(data.DataLoader(mnist_train, batch_size=18)))
show_images(X.reshape(18, 28, 28), 2, 9, titles=get_fashion_mnist_labels(y))
array([<AxesSubplot:title={'center':'ankle boot'}>,
       <AxesSubplot:title={'center':'t-shirt'}>,
       <AxesSubplot:title={'center':'t-shirt'}>,
       <AxesSubplot:title={'center':'dress'}>,
       <AxesSubplot:title={'center':'t-shirt'}>,
       <AxesSubplot:title={'center':'pullover'}>,
       <AxesSubplot:title={'center':'sneaker'}>,
       <AxesSubplot:title={'center':'pullover'}>,
       <AxesSubplot:title={'center':'sandal'}>,
       <AxesSubplot:title={'center':'sandal'}>,
       <AxesSubplot:title={'center':'t-shirt'}>,
       <AxesSubplot:title={'center':'ankle boot'}>,
       <AxesSubplot:title={'center':'sandal'}>,
       <AxesSubplot:title={'center':'sandal'}>,
       <AxesSubplot:title={'center':'sneaker'}>,
       <AxesSubplot:title={'center':'ankle boot'}>,
       <AxesSubplot:title={'center':'trouser'}>,
       <AxesSubplot:title={'center':'t-shirt'}>], dtype=object)

svg

小批量读取
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batch_size = 256


def get_dataloader_workers():
    return 4


train_iter = data.DataLoader(mnist_train, batch_size, shuffle=True,
                             num_workers=get_dataloader_workers())
读取所需要的时间
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timer = d2l.Timer()
for X, y in train_iter:
    continue
f'{timer.stop():.2f} sec'
'3.66 sec'
整合组件
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def load_data_fashion_mnist(batch_size, resize=None):
    """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()))

调整图像大小
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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
torch.Size([32, 1, 64, 64]) torch.float32 torch.Size([32]) torch.int64

softmax——从零开始实现

softmax及其回归机制，讲的很好

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

batch_size =256
train_iter,test_iter=load_data_fashion_mnist(batch_size)
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num_inputs = 784
num_outputs = 10
# 原图片是28*28的图片，通道数为1
# 将图片拉长，拉成28*28=784长度的向量

W = torch.normal(0, 0.01, size=(num_inputs, num_outputs), requires_grad=True)
b = torch.zeros(num_outputs, requires_grad=True)

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def softmax(X):
    X_exp = torch.exp(X)
    partition = X_exp.sum(1, keepdim=True)
    return X_exp/partition   # softmax的实现

softmax回归理解
在这里插入图片描述

多元分类器交叉熵公式

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def net(X):
    return softmax(torch.matmul(X.reshape((-1, W.shape[0])), W)+b)

测试我们的softmax：

对于任何随机输入，我们将每个元素变成一个非负数。 此外，依据概率原理，每行总和为1。

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X = torch.normal(0, 1, (2, 5))
X_prob = softmax(X)
X_prob, X_prob.sum(1)
(tensor([[0.2772, 0.0864, 0.1937, 0.0683, 0.3745],
         [0.0335, 0.2456, 0.1310, 0.2208, 0.3691]]),
 tensor([1.0000, 1.0000]))

在这里插入图片描述

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y = torch.tensor([0, 2])
y_hat = torch.tensor([[0.1, 0.3, 0.6], [0.3, 0.2, 0.5]])
y_hat[[0, 1], y]
tensor([0.1000, 0.5000])

损失函数

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def cross_entropy(y_hat, y):
    return - torch.log(y_hat[range(len(y_hat)), y])

cross_entropy(y_hat, y)
tensor([2.3026, 0.6931])

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# 还是在检测上面的数据
def accuracy(y_hat, y):
    if len(y_hat.shape) > 1 and y_hat.shape[1] > 1:# 如果是一个二维矩阵
        y_hat = y_hat.argmax(axis=1) #找出横向最大值的那个下标，作为预测结果
    cmp = y_hat.type(y.dtype) == y
    return float(cmp.type(y.dtype).sum())

accuracy(y_hat,y)/len(y)
0.5

评估任意模型net的准确率

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def evaluate_accuracy(net, data_iter):  # @save
    """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
    with torch.no_grad():
        for X, y in data_iter:
            metric.add(accuracy(net(X), y), y.numel()) # 将正确的预测次数进行累加
    return metric[0] / metric[1] # 预测正确/总数

后面的实现不会了。。。前方的区域以后再来探索吧

softmax——pytorch框架实现

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import torch
from torch import nn
from d2l import torch as d2l

batch_size = 256
train_iter, test_iter = load_data_fashion_mnist(batch_size) # 批量大小是256
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# 以前知识的回顾
X = torch.tensor([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]]) 
X.sum(0, keepdim=True), X.sum(1, keepdim=True)
# 0是竖着求和，1是横着求和
(tensor([[5., 7., 9.]]),
 tensor([[ 6.],
         [15.]]))

在这里插入图片描述

softmax

Sequential

normal_

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# 初始化
net = nn.Sequential(nn.Flatten(), nn.Linear(784, 10))
# 输入大小是28*28=784，因为我们的数据集有10个类别，所以网络输出维度为10


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


# 按每一层跑一遍
net.apply(init_weight)

Sequential(
  (0): Flatten(start_dim=1, end_dim=-1)
  (1): Linear(in_features=784, out_features=10, bias=True)
)

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loss = nn.CrossEntropyLoss(reduction='none') 
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trainer =torch.optim.SGD(net.parameters(),lr=0.1) # 学习率为0.1小批量随机梯度下降的优化算法
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num_epochs=10
d2l.train_ch3(net,train_iter,test_iter,loss,num_epochs,trainer)

svg

参考李沐老师的《动手学深度学习》

softmax好难，学了好久才大概懂一点点，我是菜鸟