My-Notes

007.softmax 回归从零开始实现

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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

将展平每个图像，把它们看作长度为 28x28=784 的向量。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
 
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

回顾：给定一个矩阵 X，我们可以对所有元素求和

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)
"""
(tensor([[5., 7., 9.]]),
 tensor([[ 6.],
         [15.]]))
"""

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

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

X = torch.normal(0, 1, (2, 5))
X_prob = softmax(X)
X_prob, X_prob.sum(1)
"""
(tensor([[0.4897, 0.1079, 0.1962, 0.0514, 0.1548],
         [0.0667, 0.1308, 0.5476, 0.1301, 0.1248]]),
 tensor([1., 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

创建一个数据样本 y_hat，其中包含 2 个样本在 3 个类别的预测概率，以及它们对应的标签 y。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]])
y_hat[[0, 1], y]
# 拿出 y_hat[0][0] 和 y_hat[1][2]
"""
tensor([0.1000, 0.5000])
"""

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)
"""
tensor([2.3026, 0.6931])
"""

分类精度

将预测类别与真实 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
	return float(cmp.type(y.dtype).sum())
 
accuracy(y_hat, y) / len(y)
"""
0.5
"""

Accumulator 实例中创建了 2 个变量，分别用于存储正确预测的数量和预测的总数量数量

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]

我们可以评估在任意模型 net 的精度

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
	with torch.no_grad():
	    for X, y in data_iter:
	        metric.add(accuracy(net(X), y), y.numel())
	return metric[0] / metric[1]
 
evaluate_accuracy(net, test_iter)
"""
0.0287
"""

训练

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）"""
	# 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]

定义一个在动画中绘制数据的实用程序类

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)):
	    # 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)

训练函数

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

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)

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)

关系图谱