NNDL 实验五 前馈神经网络（1）二分类任务

pytorch实现

4.1 神经元

4.1.1 净活性值

使用pytorch计算一组输入的净活性值z

净活性值z经过一个非线性函数f(·)后，得到神经元的活性值a

假设一个神经元接收的输入为x∈RD，其权重向量为w∈RD，神经元所获得的输入信号，即净活性值z的计算方法为

$$z=W^{T}X+b$$
其中b为偏置。

为了提高预测样本的效率，我们通常会将N个样本归为一组进行成批地预测。

$$z=Xw+b$$
其中$$X\in R^{N\times D}$$为N个样本的特征矩阵，$$Z\in R^N$$为N个预测值组成的列向量。

使用pytorch计算一组输入的净活性值，代码参考paddle例题：

import paddle
 
# 2个特征数为5的样本
X = paddle.rand(shape=[2, 5])
 
# 含有5个参数的权重向量
w = paddle.rand(shape=[5, 1])
# 偏置项
b = paddle.rand(shape=[1, 1])
 
# 使用'paddle.matmul'实现矩阵相乘
z = paddle.matmul(X, w) + b
print("input X:", X)
print("weight w:", w, "\nbias b:", b)
print("output z:", z)

运行结果：

在飞桨中，可以使用nn.Linear完成输入张量的上述变换。

在pytorch中学习相应函数torch.nn.Linear(features_in, features_out, bias=False)。

实现上面的例子，完成代码，进一步深入研究torch.nn.Linear()函数实现：

import torch
import torch.nn as nn
from torch.autograd import Variable

m = nn.Linear(5, 1)
input = Variable(torch.rand(2, 5)) #包装Tensor使得支持自动微分
output = m(input)
print(output)

torch.nn.Linear()的使用:

class torch.nn.Linear（in_features，out_features，bias = True ）

【思考题】加权相加与仿射变换之间有什么区别和联系？

加权和就是对输入的信息进行线性变换，仿射变换就是线性变换+平移
加权和变换只是线性的
仿射变换，又称仿射映射，是指在几何中，一个向量空间进行一次线性变换并接上一个平移，变换为另一个向量空间。
仿射变换是在几何上定义为两个向量空间之间的一个仿射变换或者仿射映射（来自拉丁语，affine，“和…相关”）由一个非奇异的线性变换(运用一次函数进行的变换)接上一个平移变换组成。
仿射变换保留了：
（1）点之间的共线性，例如通过同一线之点 （即称为共线点)在变换后仍呈共线。
（2）向量沿着一线的比例，例如对相异共线三点与 的比例同于及。
（3）带不同质量的点之质心。
一仿射变换为可逆的当且仅当A为可逆的。在矩阵表示中，其逆元为

线性变换有三个特点：
①变换前是直线，变换后依然是直线；
②直线比例保持不变
③变换前是原点，变换后依然是原点
仿射变换有两个特点：
①变换前是直线，变换后依然是直线；
②直线比例保持不变

4.1.2 激活函数

净活性值z再经过一个非线性函数f(⋅)后，得到神经元的活性值a。

$$a=f(z)$$
激活函数通常为非线性函数，可以增强神经网络的表示能力和学习能力。

常用的激活函数有S型函数和ReLU函数。

4.1.2.1 Sigmoid 型函数

Sigmoid 型函数是指一类S型曲线函数，为两端饱和函数。常用的 Sigmoid 型函数有 Logistic 函数和 Tanh 函数，其数学表达式为

Logistic 函数$$\sigma (z)=\frac{1}{1+exp(-z)}$$
Tanh 函数$$tanh(z)=\frac{exp(z)-exp(-z)}{exp(z)+exp(-z)}$$

1.使用python实现并可视化“Logistic函数、Tanh函数”

import torch
import matplotlib.pyplot as plt

# Logistic函数
def logistic(z):
    return 1.0 / (1.0 + torch.exp(-z))

# Tanh函数
def tanh(z):
    return (torch.exp(z) - torch.exp(-z)) / (torch.exp(z) + torch.exp(-z))

# 在[-10,10]的范围内生成10000个输入值，用于绘制函数曲线
z = torch.linspace(-10, 10, 10000)

plt.figure()
plt.plot(z.tolist(), logistic(z).tolist(), color='#e4007f', label="Logistic Function")
plt.plot(z.tolist(), tanh(z).tolist(), color='#f19ec2', linestyle ='--', label="Tanh Function")

ax = plt.gca() # 获取轴，默认有4个
# 隐藏两个轴，通过把颜色设置成none
ax.spines['top'].set_color('none')
ax.spines['right'].set_color('none')
# 调整坐标轴位置   
ax.spines['left'].set_position(('data',0))
ax.spines['bottom'].set_position(('data',0))
plt.legend(loc='lower right', fontsize='large')
plt.show()

运行结果：

2.在pytorch中找到相应函数并测试。

import torch
import matplotlib.pyplot as plt

# 在[-10,10]的范围内生成10000个输入值，用于绘制函数曲线
z = torch.linspace(-10, 10, 10000)

plt.figure()
plt.plot(z.tolist(), torch.sigmoid(z).tolist(), color='#ff0077', label="Logistic Function")
plt.plot(z.tolist(), torch.tanh(z).tolist(), color='#ff0077', linestyle ='--', label="Tanh Function")

ax = plt.gca() # 获取轴，默认有4个
# 隐藏两个轴，通过把颜色设置成none
ax.spines['top'].set_color('none')
ax.spines['right'].set_color('none')
# 调整坐标轴位置   
ax.spines['left'].set_position(('data',0))
ax.spines['bottom'].set_position(('data',0))
plt.legend(loc='lower right', fontsize='large')
plt.show()

运行结果：

4.1.2.2 ReLU型函数

常见的ReLU函数有ReLU和带泄露的ReLU（Leaky ReLU）数学表达式分别为：
$$ReLU(z)=max(0,z)$$
$$LeakyReLU(z)=max(0,z)+\lambda min(0,z)$$
其中λ为超参数。

可视化ReLU和带泄露的ReLU的函数的代码实现和可视化如下：

1. 使用python实现并可视化可视化“ReLU、带泄露的ReLU的函数”

import torch
import matplotlib.pyplot as plt


# ReLU
def relu(z):
    return torch.maximum(z, torch.as_tensor(0.))

# 带泄露的ReLU
def leaky_relu(z, negative_slope=0.1):
    # 当前版本torch暂不支持直接将bool类型转成int类型，因此调用了torch的cast函数来进行显式转换
    a1 = (torch.can_cast((z > 0).dtype, torch.float32) * z)
    a2 = (torch.can_cast((z <= 0).dtype, torch.float32) * (negative_slope * z))
    return a1 + a2

# 在[-10,10]的范围内生成一系列的输入值，用于绘制relu、leaky_relu的函数曲线
z = torch.linspace(-10, 10, 10000)

plt.figure()
plt.plot(z.tolist(), relu(z).tolist(), color="#e4007f", label="ReLU Function")
plt.plot(z.tolist(), leaky_relu(z).tolist(), color="#f19ec2", linestyle="--", label="LeakyReLU Function")

ax = plt.gca()
ax.spines['top'].set_color('none')
ax.spines['right'].set_color('none')
ax.spines['left'].set_position(('data',0))
ax.spines['bottom'].set_position(('data',0))
plt.legend(loc='upper left', fontsize='large')
plt.savefig('fw-relu-leakyrelu.pdf')
plt.show()

运行结果：

2. 在飞桨中，可以通过调用paddle.nn.functional.relu和paddle.nn.functional.leaky_relu完成ReLU与带泄露的ReLU的计算。在pytorch中找到相应函数并测试。

import torch
import matplotlib.pyplot as plt


# 在[-10,10]的范围内生成一系列的输入值，用于绘制relu、leaky_relu的函数曲线
z = torch.linspace(-10, 10, 10000)

plt.figure()
plt.plot(z.tolist(), torch.relu(z).tolist(), color="#e4007f", label="ReLU Function")
plt.plot(z.tolist(), torch.nn.LeakyReLU(0.1)(z), color="#f19ec2", linestyle="--", label="LeakyReLU Function")

ax = plt.gca()
ax.spines['top'].set_color('none')
ax.spines['right'].set_color('none')
ax.spines['left'].set_position(('data',0))
ax.spines['bottom'].set_position(('data',0))
plt.legend(loc='upper left', fontsize='large')
plt.savefig('fw-relu-leakyrelu.pdf')
plt.show()

运行结果：

4.2 基于前馈神经网络的二分类任务

前馈神经网络的网络结构如图4.3所示。每一层获取前一层神经元的活性值，并重复上述计算得到该层的活性值，传入到下一层。整个网络中无反馈，信号从输入层向输出层逐层的单向传播，得到网络最后的输出 $$a^L$$

4.2.1 数据集构建

使用第3.1.1节中构建的二分类数据集：Moon1000数据集，其中训练集640条、验证集160条、测试集200条。该数据集的数据是从两个带噪音的弯月形状数据分布中采样得到，每个样本包含2个特征。

from nndl.dataset import make_moons
# 采样1000个样本
n_samples = 1000
X, y = make_moons(n_samples=n_samples, shuffle=True, noise=0.5)

num_train = 640
num_dev = 160
num_test = 200

X_train, y_train = X[:num_train], y[:num_train]
X_dev, y_dev = X[num_train:num_train + num_dev], y[num_train:num_train + num_dev]
X_test, y_test = X[num_train + num_dev:], y[num_train + num_dev:]

y_train = y_train.reshape([-1,1])
y_dev = y_dev.reshape([-1,1])
y_test = y_test.reshape([-1,1])

运行结果：

nndl.dataset.make_moons代码如下：

import torch
import math
import numpy as np
# 新增make_moons函数
def make_moons(n_samples=1000, shuffle=True, noise=None):
    n_samples_out = n_samples // 2
    n_samples_in = n_samples - n_samples_out

    outer_circ_x = torch.cos(torch.linspace(0, math.pi, n_samples_out))
    outer_circ_y = torch.sin(torch.linspace(0, math.pi, n_samples_out))

    inner_circ_x = 1 - torch.cos(torch.linspace(0, math.pi, n_samples_in))
    inner_circ_y = 0.5 - torch.sin(torch.linspace(0, math.pi, n_samples_in))

    print('outer_circ_x.shape:', outer_circ_x.shape, 'outer_circ_y.shape:', outer_circ_y.shape)
    print('inner_circ_x.shape:', inner_circ_x.shape, 'inner_circ_y.shape:', inner_circ_y.shape)

    X = torch.stack(
        [torch.cat([outer_circ_x, inner_circ_x]),
         torch.cat([outer_circ_y, inner_circ_y])],
         axis=1
    )

    print('after concat shape:', torch.cat([outer_circ_x, inner_circ_x]).shape)
    print('X shape:', X.shape)

    # 使用'torch. zeros'将第一类数据的标签全部设置为0
    # 使用'torch. ones'将第一类数据的标签全部设置为1
    y = torch.cat(
        [torch.zeros([n_samples_out]), torch.ones([n_samples_in])]
    )

    print('y shape:', y.shape)

    # 如果shuffle为True，将所有数据打乱
    if shuffle:
        # 使用'torch.randperm'生成一个数值在0到X.shape[0]，随机排列的一维Tensor做索引值，用于打乱数据
        idx = torch.randperm(X.shape[0])
        X = X[idx]
        y = y[idx]

    # 如果noise不为None，则给特征值加入噪声
    if noise is not None:
        X += np.random.normal(0.0, noise, X.shape)

    return X, y

4.2.2 模型构建

为了更高效的构建前馈神经网络，我们先定义每一层的算子，然后再通过算子组合构建整个前馈神经网络。

为了和代码的实现保存一致性，这里使用形状为(样本数量×特征维度)的张量来表示一组样本。样本的矩阵X是由N个x的行向量组成。而《神经网络与深度学习》中x为列向量，因此这里的权重矩阵W和偏置b和《神经网络与深度学习》中的表示刚好为转置关系。

为了使后续的模型搭建更加便捷，我们将神经层的计算，即公式(4.8)和(4.9)，都封装成算子，这些算子都继承Op基类。
为了更高效的构建前馈神经网络，我们先定义每一层的算子，然后再通过算子组合构建整个前馈神经网络。

4.2.2.1 线性层算子

公式（4.8）对应一个线性层算子，权重参数采用默认的随机初始化，偏置采用默认的零初始化。代码实现如下：

from nndl.op import Op

# 实现线性层算子
class Linear(Op):
    def __init__(self, input_size, output_size, name, weight_init=np.random.standard_normal, bias_init=torch.zeros):

        self.params = {}
        # 初始化权重
        self.params['W'] = weight_init([input_size, output_size])
        self.params['W'] = torch.as_tensor(self.params['W'],dtype=torch.float32)
        # 初始化偏置
        self.params['b'] = bias_init([1, output_size])
        self.inputs = None

        self.name = name

    def forward(self, inputs):
        self.inputs = inputs

        outputs = torch.matmul(self.inputs, self.params['W']) + self.params['b']
        return outputs

nndl.op.Op代码如下：

class Op(object):
    def __init__(self):
        pass

    def __call__(self, inputs):
        return self.forward(inputs)

    def forward(self, inputs):
        raise NotImplementedError

    def backward(self, inputs):
        raise NotImplementedError

4.2.2.2 Logistic算子（激活函数）

本节我们采用Logistic函数来作为公式(4.9)中的激活函数。这里也将Logistic函数实现一个算子，代码实现如下：

class Logistic(Op):
    def __init__(self):
        self.inputs = None
        self.outputs = None

    def forward(self, inputs):

        outputs = 1.0 / (1.0 + torch.exp(-inputs))
        self.outputs = outputs
        return outputs

4.2.2.3 层的串行组合

在定义了神经层的线性层算子和激活函数算子之后，我们可以不断交叉重复使用它们来构建一个多层的神经网络。

下面我们实现一个两层的用于二分类任务的前馈神经网络，选用Logistic作为激活函数，可以利用上面实现的线性层和激活函数算子来组装。代码实现如下：

# 实现一个两层前馈神经网络
class Model_MLP_L2(Op):
    def __init__(self, input_size, hidden_size, output_size):
        """
        输入：
            - input_size：输入维度
            - hidden_size：隐藏层神经元数量
            - output_size：输出维度
        """
        self.fc1 = Linear(input_size, hidden_size, name="fc1")
        self.act_fn1 = Logistic()
        self.fc2 = Linear(hidden_size, output_size, name="fc2")
        self.act_fn2 = Logistic()

    def __call__(self, X):
        return self.forward(X)

    def forward(self, X):
        """
        输入：
            - X：shape=[N,input_size], N是样本数量
        输出：
            - a2：预测值，shape=[N,output_size]
        """
        z1 = self.fc1(X)
        a1 = self.act_fn1(z1)
        z2 = self.fc2(a1)
        a2 = self.act_fn2(z2)
        return a2

测试一下
现在，我们实例化一个两层的前馈网络，令其输入层维度为5，隐藏层维度为10，输出层维度为1。
并随机生成一条长度为5的数据输入两层神经网络，观察输出结果。

# 实例化模型
model = Model_MLP_L2(input_size=5, hidden_size=10, output_size=1)
# 随机生成1条长度为5的数据
X = torch.rand([1, 5])
result = model(X)
print ("result: ", result)

运行结果：

result: tensor([[0.6000]])

Process finished with exit code 0

4.2.3 损失函数

# 实现交叉熵损失函数
class BinaryCrossEntropyLoss(op.Op):
    def __init__(self):
        self.predicts = None
        self.labels = None
        self.num = None

    def __call__(self, predicts, labels):
        return self.forward(predicts, labels)

    def forward(self, predicts, labels):
        self.predicts = predicts
        self.labels = labels
        self.num = self.predicts.shape[0]
        loss = -1. / self.num * (torch.matmul(self.labels.t(), torch.log(self.predicts)) + torch.matmul((1-self.labels.t()), torch.log(1-self.predicts)))
        loss = torch.squeeze(loss, axis=1)
        return loss

4.2.4 模型优化

神经网络的层数通常比较深，其梯度计算和上一章中的线性分类模型的不同的点在于：线性模型通常比较简单可以直接计算梯度，而神经网络相当于一个复合函数，需要利用链式法则进行反向传播来计算梯度。

4.2.4.1 反向传播算法

前馈神经网络的参数梯度通常使用误差反向传播算法来计算。使用误差反向传播算法的前馈神经网络训练过程可以分为以下三步：

在上面实现算子的基础上，来实现误差反向传播算法。在上面的三个步骤中，

1.第1步是前向计算，可以利用算子的forward()方法来实现；

2.第2步是反向计算梯度，可以利用算子的backward()方法来实现；
3.第3步中的计算参数梯度也放到backward()中实现，更新参数放到另外的优化器中专门进行。
这样，在模型训练过程中，我们首先执行模型的forward()，再执行模型的backward()，就得到了所有参数的梯度，之后再利用优化器迭代更新参数。
以这我们这节中构建的两层全连接前馈神经网络Model_MLP_L2为例，下图给出了其前向和反向计算过程：

下面我们按照反向的梯度传播顺序，为每个算子添加backward()方法，并在其中实现每一层参数的梯度的计算。

4.2.4.2 损失函数

实现损失函数的backward()：

# 实现交叉熵损失函数
class BinaryCrossEntropyLoss(Op):
    def __init__(self, model):
        self.predicts = None
        self.labels = None
        self.num = None

        self.model = model

    def __call__(self, predicts, labels):
        return self.forward(predicts, labels)

    def forward(self, predicts, labels):

        self.predicts = predicts
        self.labels = labels
        self.num = self.predicts.shape[0]
        loss = -1. / self.num * (torch.matmul(self.labels.t(), torch.log(self.predicts))
                                 + torch.matmul((1 - self.labels.t()), torch.log(1 - self.predicts)))

        loss = torch.squeeze(loss, axis=1)
        return loss

    def backward(self):
        # 计算损失函数对模型预测的导数
        loss_grad_predicts = -1.0 * (self.labels / self.predicts -
                                     (1 - self.labels) / (1 - self.predicts)) / self.num

        # 梯度反向传播
        self.model.backward(loss_grad_predicts)

4.2.4.3 Logistic算子

为Logistic算子增加反向函数

为Logistic算子增加反向函数：

class Logistic(Op):
    def __init__(self):
        self.inputs = None
        self.outputs = None
        self.params = None

    def forward(self, inputs):
        outputs = 1.0 / (1.0 + torch.exp(-inputs))
        self.outputs = outputs
        return outputs

    def backward(self, grads):
        # 计算Logistic激活函数对输入的导数
        outputs_grad_inputs = torch.multiply(self.outputs, (1.0 - self.outputs))
        return torch.multiply(grads,outputs_grad_inputs)

4.2.4.4 线性层

代码实现如下：

class Linear(Op):
    def __init__(self, input_size, output_size, name, weight_init=np.random.standard_normal, bias_init=torch.zeros):
        self.params = {}
        self.params['W'] = weight_init([input_size, output_size])
        self.params['W'] = torch.as_tensor(self.params['W'],dtype=torch.float32)
        self.params['b'] = bias_init([1, output_size])

        self.inputs = None
        self.grads = {}

        self.name = name

    def forward(self, inputs):
        self.inputs = inputs
        outputs = torch.matmul(self.inputs, self.params['W']) + self.params['b']
        return outputs

    def backward(self, grads):
        self.grads['W'] = torch.matmul(self.inputs.T, grads)
        self.grads['b'] = torch.sum(grads, dim=0)

        # 线性层输入的梯度
        return torch.matmul(grads, self.params['W'].T)

4.2.4.5 整个网络

实现完整的两层神经网络的前向和反向计算

class Model_MLP_L2(Op):
    def __init__(self, input_size, hidden_size, output_size):
        # 线性层
        self.fc1 = Linear(input_size, hidden_size, name="fc1")
        # Logistic激活函数层
        self.act_fn1 = Logistic()
        self.fc2 = Linear(hidden_size, output_size, name="fc2")
        self.act_fn2 = Logistic()

        self.layers = [self.fc1, self.act_fn1, self.fc2, self.act_fn2]

    def __call__(self, X):
        return self.forward(X)

    # 前向计算
    def forward(self, X):
        z1 = self.fc1(X)
        a1 = self.act_fn1(z1)
        z2 = self.fc2(a1)
        a2 = self.act_fn2(z2)
        return a2

    # 反向计算
    def backward(self, loss_grad_a2):
        loss_grad_z2 = self.act_fn2.backward(loss_grad_a2)
        loss_grad_a1 = self.fc2.backward(loss_grad_z2)
        loss_grad_z1 = self.act_fn1.backward(loss_grad_a1)
        loss_grad_inputs = self.fc1.backward(loss_grad_z1)

4.2.4.6 优化器

在计算好神经网络参数的梯度之后，我们将梯度下降法中参数的更新过程实现在优化器中。

与第3章中实现的梯度下降优化器SimpleBatchGD不同的是，此处的优化器需要遍历每层，对每层的参数分别做更新。

from nndl.opitimizer import Optimizer

class BatchGD(Optimizer):
    def __init__(self, init_lr, model):
        super(BatchGD, self).__init__(init_lr=init_lr, model=model)

    def step(self):
        # 参数更新
        for layer in self.model.layers: # 遍历所有层
            if isinstance(layer.params, dict):
                for key in layer.params.keys():
                    layer.params[key] = layer.params[key] - self.init_lr * layer.grads[key]

nndl.opitimizer.Optimizer代码如下：

from abc import abstractmethod
#新增优化器基类
class Optimizer(object):
    def __init__(self, init_lr, model):

        #初始化学习率，用于参数更新的计算
        self.init_lr = init_lr
        #指定优化器需要优化的模型
        self.model = model

    @abstractmethod
    def step(self):
        pass

4.2.5 完善Runner类：RunnerV2_1

基于3.1.6实现的 RunnerV2 类主要针对比较简单的模型。而在本章中，模型由多个算子组合而成，通常比较复杂，因此本节继续完善并实现一个改进版： RunnerV2_1类，其主要加入的功能有：

1. 支持自定义算子的梯度计算，在训练过程中调用self.loss_fn.backward()从损失函数开始反向计算梯度；
2. 每层的模型保存和加载，将每一层的参数分别进行保存和加载。

class RunnerV2_1(object):
    def __init__(self, model, optimizer, metric, loss_fn, **kwargs):
        self.model = model
        self.optimizer = optimizer
        self.loss_fn = loss_fn
        self.metric = metric

        # 记录训练过程中的评估指标变化情况
        self.train_scores = []
        self.dev_scores = []

        # 记录训练过程中的评价指标变化情况
        self.train_loss = []
        self.dev_loss = []

    def train(self, train_set, dev_set, **kwargs):
        # 传入训练轮数，如果没有传入值则默认为0
        num_epochs = kwargs.get("num_epochs", 0)
        # 传入log打印频率，如果没有传入值则默认为100
        log_epochs = kwargs.get("log_epochs", 100)

        # 传入模型保存路径
        save_dir = kwargs.get("save_dir", None)

        # 记录全局最优指标
        best_score = 0
        # 进行num_epochs轮训练
        for epoch in range(num_epochs):
            X, y = train_set
            # 获取模型预测
            logits = self.model(X)
            # 计算交叉熵损失
            trn_loss = self.loss_fn(logits, y)  # return a tensor

            self.train_loss.append(trn_loss.item())
            # 计算评估指标
            trn_score = self.metric(logits, y).item()
            self.train_scores.append(trn_score)

            self.loss_fn.backward()

            # 参数更新
            self.optimizer.step()

            dev_score, dev_loss = self.evaluate(dev_set)
            # 如果当前指标为最优指标，保存该模型
            if dev_score > best_score:
                print(f"[Evaluate] best accuracy performence has been updated: {best_score:.5f} --> {dev_score:.5f}")
                best_score = dev_score
                if save_dir:
                    self.save_model(save_dir)

            if log_epochs and epoch % log_epochs == 0:
                print(f"[Train] epoch: {epoch}/{num_epochs}, loss: {trn_loss.item()}")

    def evaluate(self, data_set):
        X, y = data_set
        # 计算模型输出
        logits = self.model(X)
        # 计算损失函数
        loss = self.loss_fn(logits, y).item()
        self.dev_loss.append(loss)
        # 计算评估指标
        score = self.metric(logits, y).item()
        self.dev_scores.append(score)
        return score, loss

    def predict(self, X):
        return self.model(X)

    def save_model(self, save_dir):
        # 对模型每层参数分别进行保存，保存文件名称与该层名称相同
        for layer in self.model.layers:  # 遍历所有层
            if isinstance(layer.params, dict):
               torch.save(layer.params, os.path.join(save_dir, layer.name+".pdparams"))

    def load_model(self, model_dir):
        # 获取所有层参数名称和保存路径之间的对应关系
        model_file_names = os.listdir(model_dir)
        name_file_dict = {}
        for file_name in model_file_names:
            name = file_name.replace(".pdparams", "")
            name_file_dict[name] = os.path.join(model_dir, file_name)

        # 加载每层参数
        for layer in self.model.layers:  # 遍历所有层
            if isinstance(layer.params, dict):
                name = layer.name
                file_path = name_file_dict[name]
                layer.params = torch.load(file_path)

4.2.6 模型训练

使用训练集和验证集进行模型训练，共训练2000个epoch。评价指标为accuracy。

epoch_num = 1000

model_saved_dir = 'D:\project\DL\Lenet\logs'

# 输入层维度为2
input_size = 2
# 隐藏层维度为5
hidden_size = 5
# 输出层维度为1
output_size = 1

# 定义网络
model = Model_MLP_L2(input_size=input_size, hidden_size=hidden_size, output_size=output_size)

# 损失函数
loss_fn = BinaryCrossEntropyLoss(model)

# 优化器
learning_rate = 0.2
optimizer = BatchGD(learning_rate, model)

# 评价方法
metric = accuracy

# 实例化RunnerV2_1类，并传入训练配置
runner = RunnerV2_1(model, optimizer, metric, loss_fn)

runner.train([X_train, y_train], [X_dev, y_dev], num_epochs=epoch_num, log_epochs=50, save_dir=model_saved_dir)

运行结果：

[Evaluate] best accuracy performence has been updated: 0.00000 --> 0.16875
[Train] epoch: 0/1000, loss: 0.7350932955741882
[Evaluate] best accuracy performence has been updated: 0.16875 --> 0.17500
[Evaluate] best accuracy performence has been updated: 0.17500 --> 0.18750
[Evaluate] best accuracy performence has been updated: 0.18750 --> 0.20000
[Evaluate] best accuracy performence has been updated: 0.20000 --> 0.21250
[Evaluate] best accuracy performence has been updated: 0.21250 --> 0.22500
[Evaluate] best accuracy performence has been updated: 0.22500 --> 0.25000
[Evaluate] best accuracy performence has been updated: 0.25000 --> 0.31250
[Evaluate] best accuracy performence has been updated: 0.31250 --> 0.37500
[Evaluate] best accuracy performence has been updated: 0.37500 --> 0.43750
[Evaluate] best accuracy performence has been updated: 0.43750 --> 0.46250
[Evaluate] best accuracy performence has been updated: 0.46250 --> 0.48125
[Evaluate] best accuracy performence has been updated: 0.48125 --> 0.49375
[Evaluate] best accuracy performence has been updated: 0.49375 --> 0.51250
[Evaluate] best accuracy performence has been updated: 0.51250 --> 0.55625
[Evaluate] best accuracy performence has been updated: 0.55625 --> 0.60625
[Evaluate] best accuracy performence has been updated: 0.60625 --> 0.61875
[Evaluate] best accuracy performence has been updated: 0.61875 --> 0.63750
[Evaluate] best accuracy performence has been updated: 0.63750 --> 0.65000
[Evaluate] best accuracy performence has been updated: 0.65000 --> 0.66250
[Evaluate] best accuracy performence has been updated: 0.66250 --> 0.66875
[Evaluate] best accuracy performence has been updated: 0.66875 --> 0.67500
[Evaluate] best accuracy performence has been updated: 0.67500 --> 0.68125
[Evaluate] best accuracy performence has been updated: 0.68125 --> 0.68750
[Evaluate] best accuracy performence has been updated: 0.68750 --> 0.69375
[Evaluate] best accuracy performence has been updated: 0.69375 --> 0.70000
[Evaluate] best accuracy performence has been updated: 0.70000 --> 0.71250
[Evaluate] best accuracy performence has been updated: 0.71250 --> 0.71875
[Train] epoch: 50/1000, loss: 0.664116382598877
[Evaluate] best accuracy performence has been updated: 0.71875 --> 0.72500
[Evaluate] best accuracy performence has been updated: 0.72500 --> 0.73750
[Evaluate] best accuracy performence has been updated: 0.73750 --> 0.74375
[Evaluate] best accuracy performence has been updated: 0.74375 --> 0.75000
[Evaluate] best accuracy performence has been updated: 0.75000 --> 0.76250
[Evaluate] best accuracy performence has been updated: 0.76250 --> 0.76875
[Evaluate] best accuracy performence has been updated: 0.76875 --> 0.78125
[Evaluate] best accuracy performence has been updated: 0.78125 --> 0.79375
[Evaluate] best accuracy performence has been updated: 0.79375 --> 0.80625
[Evaluate] best accuracy performence has been updated: 0.80625 --> 0.81250
[Train] epoch: 100/1000, loss: 0.5949881076812744
[Evaluate] best accuracy performence has been updated: 0.81250 --> 0.81875
[Evaluate] best accuracy performence has been updated: 0.81875 --> 0.82500
[Evaluate] best accuracy performence has been updated: 0.82500 --> 0.83125
[Evaluate] best accuracy performence has been updated: 0.83125 --> 0.83750
[Train] epoch: 150/1000, loss: 0.5277273058891296
[Train] epoch: 200/1000, loss: 0.485870361328125
[Train] epoch: 250/1000, loss: 0.46499910950660706
[Train] epoch: 300/1000, loss: 0.4550503194332123
[Train] epoch: 350/1000, loss: 0.45022842288017273
[Train] epoch: 400/1000, loss: 0.44782382249832153
[Train] epoch: 450/1000, loss: 0.44659096002578735
[Evaluate] best accuracy performence has been updated: 0.83750 --> 0.84375
[Train] epoch: 500/1000, loss: 0.44594064354896545
[Evaluate] best accuracy performence has been updated: 0.84375 --> 0.85000
[Evaluate] best accuracy performence has been updated: 0.85000 --> 0.85625
[Train] epoch: 550/1000, loss: 0.44558531045913696
[Train] epoch: 600/1000, loss: 0.4453815519809723
[Evaluate] best accuracy performence has been updated: 0.85625 --> 0.86250
[Train] epoch: 650/1000, loss: 0.44525671005249023
[Train] epoch: 700/1000, loss: 0.4451737403869629
[Train] epoch: 750/1000, loss: 0.4451136589050293
[Train] epoch: 800/1000, loss: 0.4450666606426239
[Train] epoch: 850/1000, loss: 0.4450274407863617
[Train] epoch: 900/1000, loss: 0.4449935853481293
[Train] epoch: 950/1000, loss: 0.44496336579322815

把训练变成2000次后，运行结果如下：

[Evaluate] best accuracy performence has been updated: 0.00000 --> 0.21250
[Train] epoch: 0/2000, loss: 0.738420844078064
[Evaluate] best accuracy performence has been updated: 0.21250 --> 0.21875
[Evaluate] best accuracy performence has been updated: 0.21875 --> 0.23750
[Evaluate] best accuracy performence has been updated: 0.23750 --> 0.24375
[Evaluate] best accuracy performence has been updated: 0.24375 --> 0.26875
[Evaluate] best accuracy performence has been updated: 0.26875 --> 0.27500
[Evaluate] best accuracy performence has been updated: 0.27500 --> 0.28750
[Evaluate] best accuracy performence has been updated: 0.28750 --> 0.30000
[Evaluate] best accuracy performence has been updated: 0.30000 --> 0.31875
[Evaluate] best accuracy performence has been updated: 0.31875 --> 0.35625
[Evaluate] best accuracy performence has been updated: 0.35625 --> 0.37500
[Evaluate] best accuracy performence has been updated: 0.37500 --> 0.41250
[Evaluate] best accuracy performence has been updated: 0.41250 --> 0.43125
[Evaluate] best accuracy performence has been updated: 0.43125 --> 0.45000
[Evaluate] best accuracy performence has been updated: 0.45000 --> 0.46875
[Evaluate] best accuracy performence has been updated: 0.46875 --> 0.47500
[Evaluate] best accuracy performence has been updated: 0.47500 --> 0.49375
[Evaluate] best accuracy performence has been updated: 0.49375 --> 0.52500
[Evaluate] best accuracy performence has been updated: 0.52500 --> 0.58125
[Evaluate] best accuracy performence has been updated: 0.58125 --> 0.61875
[Evaluate] best accuracy performence has been updated: 0.61875 --> 0.66250
[Evaluate] best accuracy performence has been updated: 0.66250 --> 0.66875
[Evaluate] best accuracy performence has been updated: 0.66875 --> 0.70000
[Evaluate] best accuracy performence has been updated: 0.70000 --> 0.71875
[Evaluate] best accuracy performence has been updated: 0.71875 --> 0.73125
[Evaluate] best accuracy performence has been updated: 0.73125 --> 0.73750
[Evaluate] best accuracy performence has been updated: 0.73750 --> 0.74375
[Evaluate] best accuracy performence has been updated: 0.74375 --> 0.75000
[Evaluate] best accuracy performence has been updated: 0.75000 --> 0.76875
[Train] epoch: 50/2000, loss: 0.6566254496574402
[Evaluate] best accuracy performence has been updated: 0.76875 --> 0.77500
[Evaluate] best accuracy performence has been updated: 0.77500 --> 0.78125
[Evaluate] best accuracy performence has been updated: 0.78125 --> 0.78750
[Evaluate] best accuracy performence has been updated: 0.78750 --> 0.79375
[Train] epoch: 100/2000, loss: 0.5740222334861755
[Train] epoch: 150/2000, loss: 0.49713021516799927
[Evaluate] best accuracy performence has been updated: 0.79375 --> 0.80000
[Train] epoch: 200/2000, loss: 0.4507133662700653
[Train] epoch: 250/2000, loss: 0.426734060049057
[Train] epoch: 300/2000, loss: 0.41436272859573364
[Train] epoch: 350/2000, loss: 0.4077704846858978
[Train] epoch: 400/2000, loss: 0.4041588008403778
[Train] epoch: 450/2000, loss: 0.40213823318481445
[Train] epoch: 500/2000, loss: 0.40098461508750916
[Train] epoch: 550/2000, loss: 0.400308221578598
[Train] epoch: 600/2000, loss: 0.39989611506462097
[Train] epoch: 650/2000, loss: 0.39963141083717346
[Train] epoch: 700/2000, loss: 0.39944973587989807
[Train] epoch: 750/2000, loss: 0.39931559562683105
[Train] epoch: 800/2000, loss: 0.39920955896377563
[Train] epoch: 850/2000, loss: 0.39912062883377075
[Train] epoch: 900/2000, loss: 0.3990428149700165
[Train] epoch: 950/2000, loss: 0.3989725708961487
[Train] epoch: 1000/2000, loss: 0.3989078998565674
[Train] epoch: 1050/2000, loss: 0.39884766936302185
[Train] epoch: 1100/2000, loss: 0.39879104495048523
[Train] epoch: 1150/2000, loss: 0.39873751997947693
[Train] epoch: 1200/2000, loss: 0.39868679642677307
[Train] epoch: 1250/2000, loss: 0.39863845705986023
[Train] epoch: 1300/2000, loss: 0.39859244227409363
[Train] epoch: 1350/2000, loss: 0.3985483944416046
[Train] epoch: 1400/2000, loss: 0.398506224155426
[Train] epoch: 1450/2000, loss: 0.3984657824039459
[Train] epoch: 1500/2000, loss: 0.39842694997787476
[Train] epoch: 1550/2000, loss: 0.3983895480632782
[Train] epoch: 1600/2000, loss: 0.3983535170555115
[Train] epoch: 1650/2000, loss: 0.39831873774528503
[Train] epoch: 1700/2000, loss: 0.39828506112098694
[Train] epoch: 1750/2000, loss: 0.3982524871826172
[Train] epoch: 1800/2000, loss: 0.39822086691856384
[Train] epoch: 1850/2000, loss: 0.3981901705265045
[Train] epoch: 1900/2000, loss: 0.39816027879714966
[Train] epoch: 1950/2000, loss: 0.39813119173049927

可视化观察训练集与验证集的损失函数变化情况。

import matplotlib.pyplot as plt
# 打印训练集和验证集的损失
plt.figure()
plt.plot(range(epoch_num), runner.train_loss, color="#e4007f", label="Train loss")
plt.plot(range(epoch_num), runner.dev_loss, color="#f19ec2", linestyle='--', label="Dev loss")
plt.xlabel("epoch", fontsize='large')
plt.ylabel("loss", fontsize='large')
plt.legend(fontsize='x-large')
plt.show()
#加载训练好的模型
runner.load_model(model_saved_dir)
# 在测试集上对模型进行评价
score, loss = runner.evaluate([X_test, y_test])

运行结果：

4.2.7 性能评价

使用测试集对训练中的最优模型进行评价，观察模型的评价指标。代码实现如下：

# 加载训练好的模型
runner.load_model(model_saved_dir)
# 在测试集上对模型进行评价
score, loss = runner.evaluate([X_test, y_test])

print("[Test] score/loss: {:.4f}/{:.4f}".format(score, loss))

运行结果：

[Test] score/loss:0.7900/0.4483

Process finished with exit code 0

从结果来看，模型在测试集上取得了较高的准确率。
在这更深了解了一下学习率、过拟合:
学习率:将输出误差反向传播给网络参数，以此来拟合样本的输出，本质上是最优化的一个过程，逐步趋向于最优解，但是每一次更新参数利用多少误差，就需要通过一个参数来确定，这个参数就是学习率，也称步长。
作为监督学习以及深度学习中重要的超参，其决定着目标函数能否收敛到局部最小值以及何时收敛到局部最小值。合适的学习率能够使目标函数在合适的时间内收敛到局部最小值。
学习率是指导我们，在梯度下降法中，如何使用损失函数的梯度调整网络权重的超参数。
过拟合：总的来说，机器学习模型在一批数据上过于纠结误差值，想要将误差降到最低。然而当此模型运用到现实数据或者说测试数据上，误差值变高，泛化能力差，不能表达除训练数据以外的其他数据，这就叫做过拟合。如图所示的红线。

解决方法：
方法一: 增加数据量, 大部分过拟合产生的原因是因为数据量太少了. 如果我们有成千上万的数据, 红线也会慢慢被拉直, 变得没那么扭曲。

方法二: 运用正规化. L1, l2 regularization等等, 这些方法适用于大多数的机器学习, 包括神经网络. 他们的做法大同小异, 我们简化机器学习的关键公式为 y=Wx . W为机器需要学习到的各种参数. 在过拟合中, W 的值往往变化得特别大或特别小. 为了不让W变化太大, 我们在计算误差上做些手脚. 原始的 cost 误差是这样计算, cost = 预测值-真实值的平方. 如果 W 变得太大, 我们就让 cost 也跟着变大, 变成一种惩罚机制. 所以我们把 W 自己考虑进来. 这里 abs 是绝对值. 这一种形式的 正规化, 叫做 l1 正规化. L2 正规化和 l1 类似, 只是绝对值换成了平方. 其他的l3, l4 也都是换成了立方和4次方等等. 形式类似. 用这些方法,我们就能保证让学出来的线条不会过于扭曲.

还有一种专门用在神经网络的正规化的方法, 叫作 dropout. 在训练的时候, 我们随机忽略掉一些神经元和神经联结 , 是这个神经网络变得”不完整”. 用一个不完整的神经网络训练一次.

到第二次再随机忽略另一些, 变成另一个不完整的神经网络. 有了这些随机 drop 掉的规则, 我们可以想象其实每次训练的时候, 我们都让每一次预测结果都不会依赖于其中某部分特定的神经元. 像l1, l2正规化一样, 过度依赖的 W , 也就是训练参数的数值会很大, l1, l2会惩罚这些大的 参数. Dropout 的做法是从根本上让神经网络没机会过度依赖.

下面对结果进行可视化：

import math

# 均匀生成40000个数据点
x1, x2 = torch.meshgrid(torch.linspace(-math.pi, math.pi, 200), torch.linspace(-math.pi, math.pi, 200))

x = torch.stack([torch.flatten(x1), torch.flatten(x2)], axis=1)

# 预测对应类别
y = runner.predict(x)
# y = torch.squeeze(torch.as_tensor(torch.can_cast((y>=0.5).dtype,torch.float32)))

# 绘制类别区域
plt.ylabel('x2')
plt.xlabel('x1')
plt.scatter(x[:,0].tolist(), x[:,1].tolist(), c=y.tolist(), cmap=plt.cm.Spectral)

plt.scatter(X_train[:, 0].tolist(), X_train[:, 1].tolist(), marker='*', c=torch.squeeze(y_train,axis=-1).tolist())
plt.scatter(X_dev[:, 0].tolist(), X_dev[:, 1].tolist(), marker='*', c=torch.squeeze(y_dev,axis=-1).tolist())
plt.scatter(X_test[:, 0].tolist(), X_test[:, 1].tolist(), marker='*', c=torch.squeeze(y_test,axis=-1).tolist())

plt.show()

运行结果：

【思考题】对比
3.1 基于Logistic回归的二分类任务 4.2 基于前馈神经网络的二分类任务
谈谈自己的看法
逻辑回归可以看作是一个最简单的神经网络，理解逻辑回归对于理解神经网络的原理非常有帮助。二者都是基于最小化损失函数的思想，利用梯度下降法求导来更新权重参数w。但实际上求导过程中逻辑回归只需一步求导就行，而神经网络有若干个隐藏层，就是一个个加权求和再激活的嵌套，也就是链式求导。
此外，神经网络需要大样本才能显示出它灵活性的优点，即使已经有了大样本，只有真实回归函数不能由二次逻辑函数族逼近时，神经网络才会在模型选择过程中显示出其优越性。正如周志华老师对于多层前馈网络的表示能力的描述：只需要一个包含足够多神经元的隐层，多层前馈神经网络就能以任意精度逼近任意复杂度的连续函数。
逻辑回归模型最后计算输出的只有1个，而神经网络可以有多个。两次二分类任务中前馈神经网络和Logistic回归的效果差不多，但是在多分类任务和数据量比较大的任务的时候Logistic回归肯定是不如神经网络的。神经网络模型能模拟和挖掘出更多复杂的关系，也具有更好的预测效果。逻辑回归所有参数的更新是基于相同的式子，也就是所有参数的更新是基于相同的规则；而神经网络每两个神经元之间参数的更新都基于不同式子，也就是每个参数的更新都是用不同的规则。

总结：

有一些函数总是在找torch中替换但找不到，才发现numpy中也有.学习到了个新概念仿射变换，了解了加权求和和仿射变换的区别。对比了基于Logistic回归的二分类任务 和 基于前馈神经网络的二分类任务，很有收获。

修正

老师发现dataset中的弯月数据集，原本的数据集类似一个线性可分的数据集，原因是把noise设置的太大了。做出以下修正：
将noise设置为0：

数据集可视化如图：

将noise设置为0.2：


由于torch.normal()在加入高斯噪声的时候noise设置的过大，这会使得原本弯月数据集样本点过于分散，失去了数据集原本的”弯月“特征。
参考文章：
前馈神经网络和Logit回归的比较研究
线性变换和仿射变换
PyTorch的nn.Linear（）详解
学习率的理解
什么是过拟合？