3.1.2 向量化加速

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%matplotlib inline
import math
import time
import numpy as np
import torch
from d2l import torch as d2l
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n = 10000#10000维向量
a = torch.ones(n)
b = torch.ones(n)
a, b
(tensor([1., 1., 1.,  ..., 1., 1., 1.]),
 tensor([1., 1., 1.,  ..., 1., 1., 1.]))

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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()#获取当前时间
        
    def stop(self):
        """停止计时器并且将时间记录在列表中"""
        self.times.append(time.time() - self.tik)#记录本次运行时间
        return self.times[-1]
    
    def avg(self):
        """返回平均时间"""
        return sum(self.times) / len(self.times)
    
    def sum(self):
        return sum(self.times)#计算时间总和！！
    def cumsum(self):
        """返回累计时间"""
        return np.array(self.times).cumsum().tolist()#返回累计运行时间的列表！
    
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c = torch.zeros(n)
timer = Timer()
for i in range(n):
    c[i] = a[i] + b[i]
f'{timer.stop():.5f} sec'
'0.05257 sec'

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timer.start()
d = a + b
f'{timer.stop():.5f} sec'
#显然直接用向量加法更快！
'0.00014 sec'

3.1.3 normal distribution and square loss

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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)
#define a normal distribution
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x = np.arange(-7, 7, 0.01)

#average and standard variance
params = [(0, 1), (0, 2), (0, 3)]#namely a couple of mu's and sigma's
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])
#利用我们之前定义的函数画出三条图像，在matlab中考虑knocker积来画。

svg

3.2 linear regression from zero

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%matplotlib inline
import random
import torch
from d2l import torch as d2l
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#we generate a dataset with noise. Our task is to recover parameters from the finite samples in the dataset.
#we will use low dimension dataset
#使用线性模型参数w = [2, 3.4]', b = 4.2和噪声epsilon, 假设epsilon服从正态分布
def synthetic_data(w, b, num_examples): #@save
    """生成y = Xw + b+noise"""
    X = torch.normal(0, 1, (num_examples, len(w)))#后面是矩阵大小
    y = torch.matmul(X, w) + b#matmul为矩阵乘法
    y += torch.normal(0, 0.01, y.shape)#噪声标准差0.01
    return X, y.reshape((-1, 1))
#返回数据集， 线性输出结果列向量
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true_w = torch.tensor([2, -3.4])#this is our real param
true_b = 4.2
features, labels = synthetic_data(true_w, true_b, 1000)
print('features:', features[0], '\nlabel:', labels[0])
print(features[:,(1)])
features: tensor([-0.4575,  1.1481]) 
label: tensor([-0.6028])
tensor([ 1.1481e+00,  9.8841e-01,  3.5535e-01,  2.0784e+00,  9.6879e-01,
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        -3.6147e-01,  9.3682e-01, -8.3408e-03,  2.8196e-01,  1.5797e+00,
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         2.1096e+00, -3.6490e-01, -2.1201e+00, -1.2036e+00,  3.1987e-01,
         5.9609e-01,  1.2725e-01, -9.6513e-01,  7.3614e-01,  1.1720e+00,
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        -6.1278e-01, -7.2373e-01,  1.6779e-02,  1.0886e+00,  4.3431e-01,
        -1.5948e+00,  9.9059e-01, -1.4882e-01,  2.8052e-01,  2.1443e+00,
         1.9607e+00,  3.8806e-01,  2.8489e+00,  2.7184e-01, -3.0007e-01,
         2.9903e-01,  2.8759e-01, -1.2914e+00,  7.9686e-01, -1.0355e+00,
         4.1506e-01,  2.7759e-02,  2.9032e-01,  1.9015e-01,  6.4119e-01,
         1.0285e+00, -1.7690e+00, -6.2829e-01,  6.2969e-01, -9.9775e-01,
         1.0963e-01,  1.4626e-01, -7.9280e-01,  7.6115e-01,  1.3151e+00,
        -5.0517e-01, -1.5206e+00,  3.3245e-01,  1.8158e+00, -1.1382e+00,
         1.3609e-01,  2.0802e-01,  2.3387e+00,  1.2389e+00, -1.9891e+00,
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        -7.3723e-01,  4.4987e-01,  7.0388e-02, -2.3882e+00, -1.2298e+00,
        -9.4504e-01,  7.4099e-01,  1.6731e-01,  1.4965e+00,  5.2760e-01,
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        -1.4591e+00, -6.3800e-01, -7.6559e-02,  1.5400e+00, -4.4240e-01,
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         5.4681e-01, -7.4028e-01, -1.8888e+00, -3.8418e-01,  6.2783e-01,
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        -1.6574e+00, -8.4709e-01, -3.0478e-01, -1.4116e+00,  2.0713e+00,
        -7.1377e-01, -3.1828e-01,  6.0991e-01, -1.2600e-01, -1.1872e+00,
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         4.9349e-02,  1.7641e+00,  5.5424e-01, -2.8919e-01, -1.1024e+00,
        -5.4493e-01,  2.8734e-01,  4.9375e-02,  2.7105e+00, -1.4239e+00,
        -6.9621e-01,  6.8720e-02, -8.6293e-01,  1.0441e+00, -8.4265e-01,
         3.4621e-01, -9.4755e-01,  1.6345e+00,  2.4400e-01, -6.6802e-01,
        -5.8259e-01, -3.4806e-02, -7.5941e-01,  1.0871e+00,  5.7423e-01,
         1.4147e+00,  9.9963e-01,  8.7708e-01,  9.2887e-01, -3.8470e-01,
         7.3699e-01, -9.9289e-02, -1.8127e-01,  8.4384e-01, -1.7517e-01,
        -8.2891e-01,  3.9462e-01,  2.2936e-01, -1.4852e+00, -1.0159e+00,
        -3.3372e-01, -4.3101e-01, -6.1192e-01, -1.0163e+00,  1.0189e+00,
        -1.4249e-01, -3.1265e-01, -2.6392e-01, -2.6849e+00, -4.8196e-01,
        -4.6253e-02, -2.6939e-01,  4.9775e-01,  2.2392e+00, -6.4311e-01,
        -2.4017e-02,  4.2888e-01,  8.5980e-01, -1.1387e-01, -4.3800e-01,
        -7.3540e-01,  1.0186e+00, -1.0442e+00, -1.1324e+00, -6.1361e-01,
        -7.2583e-01, -9.3297e-01,  5.7577e-01, -9.6112e-01,  8.0874e-01,
         2.5687e-01, -6.4977e-01,  4.4468e-02, -4.6040e-01, -6.9195e-01,
         2.6161e-01, -7.3509e-01, -1.5686e+00, -5.9142e-01,  3.1386e-01,
         6.3763e-01,  4.0360e-01, -5.1863e-01,  8.7735e-01, -1.0085e+00,
        -4.3077e-01, -2.3868e+00, -9.3748e-01, -4.3363e-01,  1.3374e+00,
         1.0526e+00, -1.5137e+00,  1.5175e+00,  1.9621e+00,  5.9761e-02,
        -6.6585e-01, -5.7567e-01,  1.8836e+00, -4.0392e-01, -1.6843e+00,
        -6.2651e-02, -5.6046e-01, -2.2240e+00,  2.2366e+00,  7.4441e-01,
         8.6763e-01,  2.7160e-02,  6.1294e-01, -1.3371e+00, -1.4363e+00,
         3.7063e-01,  3.2868e-01,  1.4252e+00,  1.9193e+00,  4.6192e-01,
        -6.5172e-01,  1.0483e+00, -2.0439e+00, -4.3385e-01, -4.1280e-01,
         2.1602e+00, -3.0209e-01, -1.3598e+00, -9.2362e-01,  6.8442e-01,
         1.5070e+00, -1.3786e+00,  2.0884e-01, -1.2019e+00,  1.5919e-01,
        -5.1237e-01, -6.4286e-01, -5.4308e-01, -1.6116e-01,  3.2348e-01,
        -9.9109e-01,  3.9232e-01, -5.3482e-01, -1.1709e+00, -7.9754e-01,
         3.1636e-01, -1.1140e+00, -8.9587e-01, -1.0187e+00,  3.2878e-01,
        -8.7841e-01, -1.0270e+00, -4.0047e-01,  9.1419e-01, -4.1257e-01,
        -3.4638e-01, -1.0262e+00,  2.5113e+00, -1.7556e+00, -8.8387e-01,
         1.2796e+00,  5.0357e-02, -8.0814e-02,  1.1154e+00, -4.7198e-01,
         3.3411e-01, -3.3916e-01,  6.1048e-01,  1.1607e+00, -1.4151e-01,
         1.1891e+00, -1.4370e-02, -4.7600e-01,  9.1887e-01,  1.1997e-01,
         1.2459e+00, -1.6652e+00,  7.1569e-01, -9.8907e-02,  1.0175e+00,
        -8.4436e-01,  1.2284e+00, -1.4385e-01, -1.1966e+00,  6.0931e-01,
        -5.9934e-01, -2.0330e-01,  2.1682e-01, -1.5576e+00, -7.3551e-01,
         8.3409e-02, -1.2729e+00,  3.7821e-01, -6.0331e-01, -5.4615e-01,
         4.1880e-01, -1.2772e+00,  5.8423e-01,  6.4823e-01, -9.0434e-01,
        -1.7290e-01,  1.0629e-01, -4.2987e-01,  2.7938e+00,  3.7136e-01,
        -4.6170e-01, -4.2099e-01,  1.3325e-01, -3.4060e-01,  9.4501e-01,
         4.2017e-01, -8.6550e-01,  7.3243e-02,  1.9991e-01,  8.7975e-01,
        -1.3850e+00, -1.4591e+00, -2.0303e-01, -1.0262e+00, -9.9105e-01,
         1.5270e+00, -7.0185e-02, -9.4997e-01, -1.0364e-01, -1.1562e+00,
         8.3660e-01, -8.9686e-01, -1.1466e+00, -2.6425e+00, -1.1398e+00,
         2.0886e-01,  7.1526e-01, -3.0458e-01, -7.2149e-01, -7.1204e-02,
        -5.7747e-01, -5.9964e-01, -3.0580e-01,  1.9363e-01, -1.2074e+00,
        -1.2838e-01, -2.5453e-01, -1.8684e-01,  1.4170e+00, -1.9690e+00,
         1.9070e+00,  9.0862e-01, -3.0649e-01,  6.8586e-01,  1.9581e-01,
         2.9553e-01,  2.0426e-02, -4.9903e-01,  7.2440e-01, -2.3443e+00,
         5.7277e-01,  3.9135e-01,  6.4137e-01,  7.8038e-01,  3.2200e-01,
        -7.8534e-01,  2.1155e+00, -2.3740e-01, -8.0613e-01,  7.0055e-01,
         2.1210e-01, -1.0981e+00,  2.7863e-01, -4.4150e-02,  2.5538e-01,
         8.5349e-01,  6.0327e-01,  7.5350e-01,  9.8726e-01, -1.6799e+00,
        -1.6166e-01,  1.1238e-01, -7.9880e-02, -2.5753e-01,  1.1932e-01,
         1.2365e+00,  8.5960e-02,  5.5032e-01,  7.7533e-01, -1.1113e-01,
         2.1430e-01, -6.1338e-02, -1.9932e-01, -6.8872e-01, -2.0505e-01,
        -1.8301e+00, -7.6811e-01, -7.4837e-02, -6.5999e-01,  3.8283e-01,
        -4.1343e-01, -3.0340e-01,  1.2524e+00, -6.2270e-01, -4.8251e-01])

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#Generate points graph between the second feature and labels
d2l.set_figsize()
d2l.plt.scatter(features[:, 1].detach().numpy(), labels.detach().numpy(), 1);

svg

3.2.2 read the dataset

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#When we are training the model, we need to traverse the whole dataset.
def data_iter(batch_size, features, labels):#generate batch with batch_size
    num_examples = len(features)
    indices = list(range(num_examples))
    #the sample is read randomly without specific orders
    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]
        #every batch consists of a group of feature and a group of label with number of batch_size, namely batch_size row.
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batch_size = 10
for X, y in data_iter(batch_size, features, labels):
    print(X, '\n', y)
    break
    #输出一个小batch!我们如果继续执行，会连续不断获得小批量，知道遍历结束。
tensor([[-1.5650,  1.0609],
        [-0.9305, -1.8794],
        [-0.4093, -0.9996],
        [ 0.7869, -0.9384],
        [ 0.1738, -0.6689],
        [ 1.6200, -0.0613],
        [-0.5328,  0.2143],
        [-0.2533,  2.0045],
        [ 0.9031, -0.9043],
        [ 1.3816, -1.5059]]) 
 tensor([[-2.5447],
        [ 8.7360],
        [ 6.7868],
        [ 8.9856],
        [ 6.8253],
        [ 7.6404],
        [ 2.4112],
        [-3.1235],
        [ 9.0686],
        [12.0905]])

3.2.3 Initialize model parameters

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#比如我们将偏置先置为0，通过均值为0，标准差为0.01正态分布中臭氧随机数来初始化权重w。
# w = torch.normal(0, 0.01, size=(2, 1), requires_grad = True)
# b = torch.zeros(1, requires_grad = True)
w = torch.zeros((2, 1), requires_grad = True)
b = torch.zeros(1, requires_grad = True)
w, b#these are random params, we then upgrade these params using small gradient descent until they match our data.
(tensor([[0.],
         [0.]], requires_grad=True),
 tensor([0.], requires_grad=True))

3.2.4 define model

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

3.2.5 define loss function

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def squared_loss(y_hat, y): #@save
    """均方损失"""
    return (y_hat - y.reshape(y_hat.shape)) ** 2 / 2#即对矩阵每个对应元素算一下平方误差

3.2.6 define optimization algorithm

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def sgd(params, lr, batch_size): #@save params are w and b, sgd is small gradient descent.
    """小批量随机梯度下降"""
    with torch.no_grad():
        for param in params:
            param -= lr * param.grad / batch_size#之前学的自动微分相减的过程
            param.grad.zero_()#清空梯度，不积累
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#we use data_iter to traverse the whole dataset, and we set the times of training as num_epochs. Num_epochs and 
#learning rate are hyperparameters.
lr = 0.03
num_epochs = 3# we run 3 times
net = linreg#net is just linreg
loss = squared_loss#loss is just squared_loss
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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,也就是预测结果和真实结果比较一遍得到损失大小
        #net is a neural network model, which accepts input X, then set weight w and bias b to calculate and obtain results.
        l.sum().backward()#计算和函数对分量求导,也就是sgd函数中的param.grad部分
        sgd([w, b], lr, batch_size)
    with torch.no_grad():#every cycle we clean up the gradient.
        train_l = loss(net(features, w, b), labels)#the whole loss during the training process.
        #according the best params to calculate the results and then compare with the labels originally.
        print(f'epoch{epoch + 1}, loss{float(train_l.mean()):f}')#timely print the training time
epoch1, loss0.029253
epoch2, loss0.000097
epoch3, loss0.000049

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#Because the dataset is made by ourselves, so we know the real params w and b.
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}')
w的估计误差：tensor([ 0.0002, -0.0004], grad_fn=<SubBackward0>)
b的估计误差：tensor([0.0003], grad_fn=<RsubBackward1>)