[Notas prácticas de aprendizaje profundo de Datawhale] red neuronal lineal

red neuronal lineal

regresión lineal

La regresión es una clase de métodos que pueden modelar la relación entre una o más variables independientes y una variable dependiente. En las ciencias naturales y sociales, la regresión se usa a menudo para representar la relación entre entradas y salidas.

Elementos básicos de la regresión lineal

La regresión lineal (regresión lineal) se remonta a principios del siglo XIX, es la más simple y popular entre las diversas herramientas estándar para la regresión. La regresión lineal se basa en varios supuestos simples: Primero, suponga que la variable independiente $x$ y variable dependiente$La\; relación\; entre\; y$ es lineal, es decir,$y$ se puede expresar como$La\; suma\; ponderada\; de\; los\; elementos\; en\; x$ , donde generalmente se permite algo de ruido de las observaciones; en segundo lugar, asumimos que cualquier ruido es relativamente normal, como el ruido que sigue una distribución normal.

Modelo lineal

La suposición lineal significa que el objetivo se puede expresar como una suma ponderada de características, como en la siguiente fórmula:
$$precio=w_{}\cdot área+w_{}\cdot edad+segundo\_$$

función de pérdida

La función de pérdida más utilizada en problemas de regresión es la función de error al cuadrado
$${yo}^{\left(yo\right)}(w,segundo)=\frac{}{2}{\left(y^{\left(yo\right)}-y^{\left(yo\right)}\right)}^2.$$

Solucion analitica

La regresión lineal resulta ser un problema de optimización muy simple.
A diferencia de la mayoría de los otros modelos que cubriremos en este libro, la solución de regresión lineal se puede expresar simplemente mediante una fórmula.Este
tipo de solución se denomina solución analítica.

$$w^{\ast }=\left(X^{\top X}\right)\_{}^{-1X}{\_}^{\top }y.$$

descenso de gradiente estocástico

Descenso de gradiente: reduce el error al actualizar continuamente los parámetros en la dirección de la función de pérdida decreciente.
Expresamos este proceso de actualización con la siguiente fórmula matemática ( $\partial$ significa derivada parcial):

$$(w,segundo)\leftarrow (w,segundo)-\frac{}{|segundo|}\underset{yo\in B}{}{\partial }_{(w,b}{yo}^{\left(yo\right)}(w,b).$$

Los pasos del algoritmo son los siguientes:
(1) inicializar el valor de los parámetros del modelo, como la inicialización aleatoria;
(2) seleccionar aleatoriamente un pequeño lote de muestras del conjunto de datos y actualizar los parámetros en la dirección del gradiente negativo e iterar este paso continuamente.
Para la pérdida al cuadrado y la transformación afín, podemos escribirla explícitamente de la siguiente manera:

$$\begin{array}{rl}w& \leftarrow w-\frac{}{|segundo|}\underset{yo\in B}{}{\partial }_{}{yo}^{\left(yo\right)}(w,segundo)=w-\frac{}{|segundo|}\underset{yo\in B}{}{X}^{\left(yo\right)}\left(w^{\top }{x}^{\left(yo\right)}+b-y^{\left(yo\right)}\right),\\ & \leftarrow b-\frac{}{|segundo|}\underset{yo\in B}{}{\partial }_{}{yo}^{\left(yo\right)}(w,segundo)=b-\frac{}{|segundo|}\underset{yo\in B}{}\left(w^{\top }{x}^{\left(yo\right)}+b-y^{\left(yo\right)}\right)\end{array}$$

ejemplo de código

aceleración de vectorización

# 引入包
%matplotlib inline
import math
import time
import numpy as np
import torch
from d2l import torch as d2l

#循环遍历
n = 10000
a = torch.ones([n])
b = torch.ones([n])

#定义计时器
class Timer:  #@save
    """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()

#使用for循环，每次执行一位的加法
c = torch.zeros(n)
timer = Timer()
for i in range(n):
    c[i] = a[i] + b[i]
f'{
      
      timer.stop():.5f} sec'

#我们使用重载的+运算符来计算按元素的和
timer.start()
d = a + b
f'{
      
      timer.stop():.5f} sec'

Distribución normal y pérdida al cuadrado

#定义一个Python函数来计算正态分布
def normal(x, mu, sigma):
    p = 1 / math.sqrt(2 * math.pi * sigma**2)
    return p * np.exp(-0.5 / sigma**2 * (x - mu)**2)

#可视化正态分布
# 再次使用numpy进行可视化
x = np.arange(-7, 7, 0.01)

# 均值和标准差对
params = [(0, 1), (0, 2), (3, 1)]
d2l.plot(x, [normal(x, mu, sigma) for mu, sigma in params], xlabel='x',
         ylabel='p(x)', figsize=(4.5, 2.5),
         legend=[f'mean {
      
      mu}, std {
      
      sigma}' for mu, sigma in params])

Una implementación básica de la regresión lineal

#引入包
%matplotlib inline
import random
import torch
from d2l import torch as d2l

#生成数据集
def synthetic_data(w, b, num_examples):  #@save
    """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 = synthetic_data(true_w, true_b, 1000)

print('features:', features[0],'\nlabel:', labels[0])

#绘制散点图
d2l.set_figsize()
d2l.plt.scatter(features[:, 1].detach().numpy(), labels.detach().numpy(), 1);

#读取数据集
def data_iter(batch_size, features, labels):
    num_examples = len(features)
    indices = list(range(num_examples))
    # 这些样本是随机读取的，没有特定的顺序
    random.shuffle(indices)
    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]

batch_size = 10

for X, y in data_iter(batch_size, features, labels):
    print(X, '\n', y)
    break

#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):  #@save
    """线性回归模型"""
    return torch.matmul(X, w) + b

#定义损失函数
def squared_loss(y_hat, y):  #@save
    """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):  #@save
    """小批量随机梯度下降"""
    with torch.no_grad():
        for param in params:
            param -= lr * param.grad / batch_size
            param.grad.zero_()

#训练
lr = 0.03
num_epochs = 3
net = linreg
loss = squared_loss

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)，而不是一个标量。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}')

Implementación concisa de la regresión sexual

# 生成数据集
import numpy as np
import torch
from torch.utils import data
from d2l import torch as d2l

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):  #@save
    """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
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)

Conjunto de datos de clasificación de imágenes

El conjunto de datos MNIST (LeCun et al., 1998) es uno de los conjuntos de datos más utilizados en la clasificación de imágenes, pero es demasiado simple como conjunto de datos de referencia. Usaremos el conjunto de datos Fashion-MNIST similar pero más complejo (Xiao et al., 2017).

ejemplo de código

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

d2l.use_svg_display()

#读取数据集
# 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均在0～1之间
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)

mnist_train[0][0].shape

def get_fashion_mnist_labels(labels):  #@save
    """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]

#创建一个函数来可视化这些样本
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

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));

#读取小批量
batch_size = 256

def get_dataloader_workers():  #@save
    """使用4个进程来读取数据"""
    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
f'{
      
      timer.stop():.2f} sec'

#整合所有组件
def load_data_fashion_mnist(batch_size, resize=None):  #@save
    """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()))

#指定resize参数来测试load_data_fashion_mnist函数的图像大小调整功能
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

Implementación de la regresión softmax desde cero

#引入的Fashion-MNIST数据集， 并设置数据迭代器的批量大小为256。
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
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
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)

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)

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

#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
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]

#实现交叉熵函数
def cross_entropy(y_hat, y):
    return - torch.log(y_hat[range(len(y_hat)), y])

cross_entropy(y_hat, y)

#分类精度
def accuracy(y_hat, y):  #@save
    """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())

accuracy(y_hat, y) / len(y)

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())
    return metric[0] / metric[1]

class Accumulator:  #@save
    """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]

evaluate_accuracy(net, test_iter)

#训练
def train_epoch_ch3(net, train_iter, loss, updater):  #@save
    """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.mean().backward()
            updater.step()
        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精度
    return metric[0] / metric[2], metric[1] / metric[2]

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()
        display.display(self.fig)
        display.clear_output(wait=True)

def train_ch3(net, train_iter, test_iter, loss, num_epochs, updater):  #@save
    """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

#设置学习率
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)

#预测
def predict_ch3(net, test_iter, n=6):  #@save
    """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)

Una implementación concisa de la regresión softmax

import torch
from torch import nn
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
# PyTorch不会隐式地调整输入的形状。因此，
# 我们在线性层前定义了展平层（flatten），来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);

#重新审视Softmax的实现
loss = nn.CrossEntropyLoss(reduction='none')

#优化算法
trainer = torch.optim.SGD(net.parameters(), lr=0.1)

#训练
num_epochs = 10
d2l.train_ch3(net, train_iter, test_iter, loss, num_epochs, trainer)