损失函数和反向传播的关系 损失函数图像

 

 

 

 

 

 

优点：梯度永远是常数，会带来稳定性上的好处。

缺点：零点处不可导，此处梯度剧烈变化，这种不平滑性导致在预测值与真实值相差比较小的时候变得不太稳定。

也可以自己提出损失函数。 

 

因此，分析损失函数时，可从其梯度函数的图像入手，分析当预测值与真实值相差较大或者较小时，进行更新会是怎样的情况。

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

d2l.use_svg_display()  # 用svg显示图片，清晰度更高

# 通过框架中的内置函数将Fashion-MNIST数据集下载并读取到内存中
# 通过ToTensor实例将图像数据从PIL类型变换成32位浮点数格式
# 并除以255使得所有像素的数值在0到1之间
trans = transforms.ToTensor()  # 预处理，把图片转化成pytorch的tensor
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)
print(len(mnist_train))
print(len(mnist_test))
print(mnist_train[0][0].shape)  # 图片形状


# 两个可视化数据集的函数
def get_fashion_mnist_labels(labels):
    """返回Fashion-MNIST数据集的文本标签"""
    text_labels = [
        't-shirt', 'trouser', 'pullover', 'dress', 'coat', 'sandal', 'shirt', 'sneaker', 'bag', 'ankle boot'
    ]
    return [text_labels[int(i)] for i in labels]


def show_images(imgs, num_rows, num_cols, titles=None, scale=1.5):
    """Plot a list of images."""
    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):
            # 图片数量
            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])
    d2l.plt.show()
    return axes


# DataLoader拿到大小为固定数字的批量数据，用iter构成出iterator，next即为拿到第一个批量数据
X, y = next(iter(data.DataLoader(mnist_train, batch_size=18)))
# 弃掉通道数，reshape为number of examples
show_images(X.reshape(18, 28, 28), 2, 9, titles=get_fashion_mnist_labels(y))

# 读取一小批量数据，大小为batch_size
batch_size = 256


def get_dataloader_workers():
    """使用4个进程来读取的数据"""
    return 4


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')  # 读取数据的速度应当要比训练速度快


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()))

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]

from d2l import torch as d2l
from IPython import display
import matplotlib.pyplot as plt


class Animator:  # @save
    """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()
        plt.draw()
        plt.pause(0.001)
        display.display(self.fig)


        display.clear_output(wait=True)

import torch
from IPython import display
from d2l import torch as d2l
from Accumulator import Accumulator
from Animator import Animator

# softmax回归的从零开始实现
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)
# 展平每个图像，将它们视为长度为784的向量（会损失掉很多空间信息）；因为数据集有10个类别，所以网络输出维度为10
num_inputs = 784
num_outputs = 10
# 因为要计算梯度，所以requires_grad=True
W = torch.normal(0, 0.01, size=(num_inputs, num_outputs), requires_grad=True)
b = torch.zeros(num_outputs, requires_grad=True)


# 实现softmax
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


# 实现softmax回归模型
def net(X):
    return softmax(torch.matmul(X.reshape((-1, W.shape[0])), W) + b)  # 本模型中，X会被reshape成批量大小 * 输入维数的矩阵


# 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])


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])
print(cross_entropy(y_hat, y))


# 将预测类别与真是y元素进行比较
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  # 转换成一个布尔类型的tensor
    return float(cmp.type(y.dtype).sum())


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


# 评估任意模型net的准确率
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]


# softmax回归的训练
def train_epoch_ch3(net, train_iter, loss, updater):
    if isinstance(net, torch.nn.Module):  # 如果模型是用nn.Module实现的话，则开启训练模式，计算梯度
        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()  # 对参数进行更新
            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]


# 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


# 小批量随机梯度下降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)


# 训练模型10个迭代周期
# num_epochs = 10
# train_ch3(net, train_iter, test_iter, cross_entropy, num_epochs, updater)


# 对图像进行分类预测
def predict_ch3(net, test_iter, n=6):
    """预测标签"""
    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])
    d2l.plt.show()


predict_ch3(net, test_iter)

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


# softmax回归的简洁实现
# 通过深度学习框架的高级API能够实现softmax回归变得更加容易
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)


# softmax回归的输出层是一个全连接层
# pytorch不会隐式地调整输入的形状
# 因此，我们定义展平层（flatten）在线性层前调整网络输入的形状
# Flatten()把任何维度的tensor转成2D的tensor，第0维度保留，其他维度全部展成向量
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)


# 在交叉熵损失函数中传递未归一化的预测，并同时计算softmax及其对数
loss = nn.CrossEntropyLoss()
# 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)
# 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)