【动手学深度学习】part9-softmax回归

基于【动手学深度学习】课程，理解并复现softmax回归代码

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一、softmax回归简介

Softmax 回归（也称为多项式逻辑回归或最大熵分类器）是一种用于多分类问题的线性模型。它主要用于将输入数据映射到多个离散类别中的一个。Softmax 回归是逻辑回归（Logistic Regression）的扩展，适用于多分类任务，而不是二分类任务。

• 用沐神两页ppt阐述softmax回归里的核心概念

二、softmax回归实例

• 直接上代码

（0）数据筛选：直接采用“torchvision.datasets.FashionMNIST”数据集
（1）构建模型：y=softmax(X*w+b)
（2）构建损失函数：$l(\mathbf{y},\hat{\mathbf{y}})=-{\sum }_i{y}_i\log {\hat{y}}_i=-\log {\hat{y}}_y$，即交叉熵
（3）设置超参数：batch_size、lr、num_epochs
（4） 迭代优化：SGD

import torch
import torchvision
from torch.utils import data
from torchvision import transforms
from d2l import torch as d2l
from IPython.display import display,clear_output 



def get_dataloader_workers():
    """使用4个进程来读取数据。"""
    return 0


batch_size = 256
num_workers = 0
# 训练集和测试集的迭代器
train_iter, test_iter = d2l.load_data_fashion_mnist(num_workers, batch_size)


# 图像为28*28个像素点，但softmax的输入必须为向量，所以将输入转为1维向量，输出为10个类别
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)

# =============================================================================
# X = torch.tensor([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]])
# print(X.sum(0, keepdim=True))
# print(X.sum(1, keepdim=True))
# =============================================================================


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

# =============================================================================
# X = torch.normal(0, 1, (2, 5))
# X_prob = softmax(X)
# print(X,X_prob, X_prob.sum(1))
# =============================================================================

# 实现softmax回归模型


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


# 创建一个数据y_hat，其中包含2个样本在3个类别的预测概率， 使用y作为y_hat中概率的索引
y = torch.tensor([0, 2])
y_hat = torch.tensor([[0.1, 0.3, 0.6], [0.3, 0.2, 0.5]])
print(y_hat[[0, 1], y])

# 实现交叉熵损失函数


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


cross_entropy(y_hat, y)


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())


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


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]


print(evaluate_accuracy(net, test_iter))

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）。"""
    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]

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)

def train_ch3(net, train_iter, test_iter, loss, num_epochs, updater):  
    """训练模型（定义见第3章）。"""
    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
    
lr = 0.1

def updater(batch_size):
    return d2l.sgd([W, b], lr, batch_size)

num_epochs = 10
train_ch3(net, train_iter, test_iter, cross_entropy, num_epochs, updater)

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)

• 准确率和损失函数值的统计结果

二、多类交叉熵损失函数的梯度计算

交叉熵通常用于衡量两个概率之间的区别。
在多类分类的问题当中，输出为softmax(zi)的概率，可以用于衡量预测值概率与真值概率的区别，从而定义为损失函数进行不断优化。
具体推导过程如下：