[Deep Learning] (Problem Record) - Linear Regression - Small Batch Stochastic Gradient Descent

前言

博主主页：阿阿阿阿锋的主页_CSDN
原文链接：原文

When I read a book and feel that the foreword does not match the afterword,I often can't help but wonder if there's something wrong with this writing（Sometimes I even scold the author a few words in my heart,怎么这么不小心）.Yet experience and residual sanity still remind me,This is most likely my own problem.

我的编程环境：Win10操作系统,python3.6

初学不久,文章如有问题,欢迎指出.

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1. 问题和代码

for the codesgd函数中的param[:] = param - lr * param.grad / batch_sizeI've been very confused about this line.

For example a mini-batch is set up in the code10个样本,So I think it's right参数集params求梯度时,得到每个参数The gradient of should be a vector（可以理解为一个数组）类型的数据.Because for each parameter,通过10The gradient of each sample will be obtained10corresponding value.

So I had doubts,/ batch_sizeThe purpose is to get the gradient平均值,But the dividend to its left is not one标量（Ordinary single value）,So how does this line of code get the average we want？

注：代码参考自《动手学深度学习》
代码：

# 代码目标：训练一个线性回归模型,Use mini-batch stochastic gradient descent
%matplotlib inline
from IPython import display
from matplotlib import pyplot as plt
from mxnet import autograd, nd
import random

# 制作训练集
num_inputs = 2
num_examples = 1000
true_w = [2, -3.4]
true_b = 4.2
features = nd.random.normal(scale=1, shape=(num_examples, num_inputs))
labels = true_w[0] * features[:, 0] + true_w[1] * features[:, 1] + true_b
labels += nd.random.normal(scale=0.01, shape=labels.shape)

features[0], labels[0]

def use_svg_display():
    # 用矢量图显示
    display.set_matplotlib_formats('svg')

def set_figsize(figsize=(3.5, 2.5)):
    use_svg_display()
    # 设置图的尺寸
    plt.rcParams['figure.figsize'] = figsize

set_figsize()
plt.scatter(features[:, 1].asnumpy(), labels.asnumpy(), 1);  # 加分号只显示图（Otherwise, a line of text will also be displayed）

# 本函数已保存在d2lzh包中方便以后使用
def data_iter(batch_size, features, labels):
    num_examples = len(features)
    indices = list(range(num_examples))
    random.shuffle(indices)  # 样本的读取顺序是随机的
    for i in range(0, num_examples, batch_size):
        j = nd.array(indices[i: min(i + batch_size, num_examples)])
        yield features.take(j), labels.take(j)  # take函数根据索引返回对应元素
        
batch_size = 10  # 一个“小批量”的大小

# Build the parameters of the model we want to train
w = nd.random.normal(scale=0.01, shape=(num_inputs, 1))
b = nd.zeros(shape=(1,))

w.attach_grad()
b.attach_grad()

def linreg(X, w, b):  # our model function
    return nd.dot(X, w) + b

def squared_loss(y_hat, y):  # 使用的损失函数
    return (y_hat - y.reshape(y_hat.shape)) ** 2 / 2

def sgd(params, lr, batch_size):  # 用于迭代（更新）参数
    for param in params:
        param[:] = param - lr * param.grad / batch_size

lr = 0.03      # 学习率
num_epochs = 3 # number of learning cycles
net = linreg   # take a nickname
loss = squared_loss

for epoch in range(num_epochs):  # 训练模型一共需要num_epochs个迭代周期
    # 在每一个迭代周期中,会使用训练数据集中所有样本一次（假设样本数能够被批量大小整除）.X
    # X和y分别是小批量样本的特征和标签
    for X, y in data_iter(batch_size, features, labels):
        with autograd.record():
            l = loss(net(X, w, b), y)  # l是有关小批量X和y的损失
        l.backward()  # 小批量的损失对模型参数求梯度
        sgd([w, b], lr, batch_size)  # 使用小批量随机梯度下降迭代模型参数
    train_l = loss(net(features, w, b), labels)
    print('epoch %d, loss %f' % (epoch + 1, train_l.mean().asnumpy()))

print('\nweights:')
print(true_w, w)
print('\nbias:')
print(true_b, b)

2. 分析问题

After thinking about it for a while,我突然想到,Why not put the parameterparamThe gradients are printed out and have a look？Then it is not clear what the situation is！请看代码：

def sgd(params, lr, batch_size):  # 用于迭代（更新）参数
    for param in params:
        param[:] = param - lr * param.grad / batch_size
        print('\nparam.grad:')
        print(param.grad)

我仅仅在sgdTwo lines of printing are added at the end of the function,Then let's look at the effect again（This function is called every time a mini-batch of samples is processed,One look is enough,Because I just want to know the data type of the gradient of the parameter）

运行效果：

前面的param.gradrepresents two 权重(weight) 参数的梯度,后面的是 偏差(bias) 参数的梯度.也就是说,The gradient obtained for each parameter has only one value.

Why a batch10个样本,Only one value is obtained？

这个值是什么？其实在执行l.backward()时,Equivalent to executingl.sum().backward(),That is, there is a gradient value for each sample in a batch,然后把这10add up the gradient values,The gradient value of the parameter is obtained.所以再用/ batch_sizeTaking the average is also a natural thing to do.

Actually I saw this explanation shortly after,But because I was already in the loop,Just when I see here蒙圈++

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🧭 总结

种瓜得瓜,种豆得豆.
What shape is the variable,What is the shape of the gradient obtained for this variable.

The reason I subconsciously thought I would get a set of values ​​instead of a single value,It's because I saw an example of finding the gradient of a matrix earlier,What you get is a set of values（一个矩阵）.Then I got confused here,Here each parameter object for which we are finding the gradient is a single value,It's just that there are multiple data samples.

对矩阵（向量）求梯度	The gradient obtained is a matrix（向量）
Find the gradient of a scalar	The obtained gradient is a scalar

I have another feeling,as a previous idiomaticC/C++程序员,pythonThis flexibility of variable data types really blows my mind、非常的不适应,I've been fooled by this many times,呜.