Pytorch_Part4_损失函数

VisualPytorch发布域名+双服务器以下：
http://nag.visualpytorch.top/static/ (对应114.115.148.27)
http://visualpytorch.top/static/ (对应39.97.209.22)python

1、权值初始化

1. 梯度消失与爆炸

\(E(XY)=E(X)E(Y)\)
\(D(X)=E(X^2)-E(X)^2\)
\(D(X+Y)=D(X)+D(Y)\)
\(\Longrightarrow D(XY) = D(X)D(Y)+D(X)E(Y)^2+D(Y)E(X)^2=D(X)D(Y)\)
\(H_{11}=\sum_{i=0}^n X_i*W_{1i}\)git

\(\Longrightarrow D(H_{11})=\sum_{i=0}^n D(X_i)*D(W_{1i})=n*1*1=n\)
\(std(H_{11}=\sqrt n)\)数组

若仍使 \(D(H_1)=nD(X)D(W)=1\)
\(\Longrightarrow D(W)=\frac{1}{n}\)缓存

import os
import torch
import random
import numpy as np
import torch.nn as nn
from common_tools import set_seed

set_seed(3)  # 设置随机种子


class MLP(nn.Module):
    def __init__(self, neural_num, layers):
        super(MLP, self).__init__()
        self.linears = nn.ModuleList([nn.Linear(neural_num, neural_num, bias=False) for i in range(layers)])
        self.neural_num = neural_num

    def forward(self, x):
        for (i, linear) in enumerate(self.linears):
            x = linear(x)
            # x = torch.relu(x)

            print("layer:{}, std:{}".format(i, x.std()))
            if torch.isnan(x.std()):
                print("output is nan in {} layers".format(i))
                break

        return x

    def initialize(self):
        for m in self.modules():
            if isinstance(m, nn.Linear):
                nn.init.normal_(m.weight.data, std=1)

    layer_nums = 100
    neural_nums = 256
    batch_size = 16

    net = MLP(neural_nums, layer_nums)
    net.initialize()

    inputs = torch.randn((batch_size, neural_nums))  # normal: mean=0, std=1

    output = net(inputs)
    print(output)

将W设置为0均值，1标准差的标准正太分布，出现以下所示梯度爆炸现象。正如所料，每层std增长的倍数大概为\(\sqrt{256}=16\) ，将W的标准差设为 np.sqrt(1/self.neural_num) 则正常。服务器

layer:0, std:16.0981502532959
layer:1, std:253.29345703125
layer:2, std:3982.99951171875
...
layer:30, std:2.2885405881461517e+37
layer:31, std:nan
output is nan in 31 layers
tensor([[ 4.9907e+37,        -inf,         inf,  ...,         inf,
                -inf,         inf],
        [       -inf,         inf,  2.1733e+38,  ...,  9.1766e+37,
         -4.5777e+37,  3.3680e+37],
        [ 1.4215e+38,        -inf,         inf,  ...,        -inf,
                 inf,         inf],
        ...,
        [-9.2355e+37, -9.9121e+37, -3.7809e+37,  ...,  4.6074e+37,
          2.2305e+36,  1.2982e+38],
        [       -inf,         inf,        -inf,  ...,        -inf,
         -2.2394e+38,  2.0295e+36],
        [       -inf,         inf,  2.1518e+38,  ...,        -inf,
          1.6132e+38,        -inf]], grad_fn=<MmBackward>)

2. 激活函数初始化

对于不一样的激活函数，对W的标准差初始化也不一样，保持数据尺度维持在恰当范围，一般方差为1网络

1. Sigmoid, tanh------Xavier
   app

2. Relu------Kaimingdom

   \(D(W)=\frac{2}{n_i}\)函数

   \(D(W)=\frac{2}{(1+a^2) n_i}\) , LeakyRelu学习

   a = np.sqrt(6 / (self.neural_num + self.neural_num))
   tanh_gain = nn.init.calculate_gain('tanh')
   a *= tanh_gain
   nn.init.uniform_(m.weight.data, -a, a)
   # 同 nn.init.xavier_uniform_(m.weight.data, gain=tanh_gain)
   
   nn.init.normal_(m.weight.data, std=np.sqrt(2 / self.neural_num))
   # 同 nn.init.kaiming_normal_(m.weight.data)

3. 十种初始化方法

1. Xavier均匀分布

2. Xavier正态分布

3. Kaiming均匀分布

4. Kaiming正态分布

5. 均匀分布

6. 正态分布

7. 常数分布

8. 正交矩阵初始化

9. 单位矩阵初始化

10. 稀疏矩阵初始化

nn.init.calculate_gain(nonlinearity, param=None)

x = torch.randn(10000)
out = torch.tanh(x)
gain = x.std() / out.std()		# 1.5909514427185059

tanh_gain = nn.init.calculate_gain('tanh')  # 1.6666666666666667
# 至关于每通过一层tanh，x的标准差缩小1.6倍

主要功能：计算激活函数的方差变化尺度
主要参数

• nonlinearity: 激活函数名称
• param: 激活函数的参数，如Leaky ReLU的negative_slop

2、损失函数

0. 损失函数、熵

损失函数(Loss Function)：衡量模型输出与真实标签的差别
\(Loss=f(\hat y , y)\)

代价函数(Cost Function)：
\(Cost = \frac{1}{N}\sum_i^N f(\hat y_i, y)\)

目标函数(Objective Function)：
\(Obj=Cost+Regularization\)

1. nn.CrossEntropyLoss

nn.CrossEntropyLoss调用过程：

用步进(Step into)的调试方法从loss_functoin = nn.CrossEntropyLoss() 语句进入函数，观察从nn.CrossEntropyLoss()到class Module(object)一共经历了哪些类，记录其中全部进入的类及函数。

1. CrossEntropyLoss.__init__: super(CrossEntropyLoss, self).__init__

2. _WeightedLoss.__init__: super(_WeightedLoss,self).__init__

3. _Loss.__init__: super(_Loss, self).__init__()

   def __init__(self, size_average=None, reduce=None, reduction='mean'): # 前两个再也不使用
       super(_Loss, self).__init__()
       if size_average is not None or reduce is not None:
           self.reduction =_Reduction.legacy_get_string(size_average, reduce)
       else:
           self.reduction = reduction

4. Module.__init__: _construct

功能：nn.LogSoftmax ()与nn.NLLLoss ()结合，进行交叉熵计算
主要参数：

• weight：各种别的loss设置权值
• ignore_index：忽略某个类别
• reduction ：计算模式，可为none/sum/mean
  • none 逐个元素计算
  • sum 全部元素求和，返回标量
  • mean 加权平均，返回标量

inputs = torch.tensor([[1, 2], [1, 3], [1, 3]], dtype=torch.float)
target = torch.tensor([0, 1, 1], dtype=torch.long)

# --------- CrossEntropy loss: reduction ------------
# def loss function
loss_f_none = nn.CrossEntropyLoss(weight=None, reduction='none')
loss_f_sum = nn.CrossEntropyLoss(weight=None, reduction='sum')
loss_f_mean = nn.CrossEntropyLoss(weight=None, reduction='mean')

# forward
loss_none = loss_f_none(inputs, target)	# tensor([1.3133, 0.1269, 0.1269]) 
loss_sum = loss_f_sum(inputs, target)	# tensor(1.5671) 
loss_mean = loss_f_mean(inputs, target) # tensor(0.5224)

'''
若设置：
weights = torch.tensor([1, 2], dtype=torch.float)
结果变为：
tensor([1.3133, 0.2539, 0.2539]) tensor(1.8210) tensor(0.3642)
后面两项由于类别为1，故有权2
'''

2. nn.NLLLoss

功能：实现负对数似然函数中的负号功能

3. nn.BCELoss（逐神经元求Loss）

功能：二分类交叉熵
注意事项：输入值取值在[0,1]

inputs = torch.tensor([[1, 2], [2, 2], [3, 4], [4, 5]], dtype=torch.float)
target = torch.tensor([[1, 0], [1, 0], [0, 1], [0, 1]], dtype=torch.float)
# 注意这里输出端的两个神经元都须要分别计算Loss

target_bce = target

# itarget
inputs = torch.sigmoid(inputs)

weights = torch.tensor([1, 1], dtype=torch.float)

loss_f_none_w = nn.BCELoss(weight=weights, reduction='none')
loss_f_sum = nn.BCELoss(weight=weights, reduction='sum')
loss_f_mean = nn.BCELoss(weight=weights, reduction='mean')

# forward
loss_none_w = loss_f_none_w(inputs, target_bce) 
loss_sum = loss_f_sum(inputs, target_bce)
loss_mean = loss_f_mean(inputs, target_bce)

'''
BCE Loss tensor([[0.3133, 2.1269],
        [0.1269, 2.1269],
        [3.0486, 0.0181],
        [4.0181, 0.0067]]) tensor(11.7856) tensor(1.4732)
'''

4. nn.BCEWithLogitsLoss（逐神经元求Loss）

功能：结合Sigmoid与二分类交叉熵
注意事项：网络最后不加sigmoid函数
添加参数：pos_weight 正样本的权值

5. 18种损失函数总结

参数即调用方式详见：

下面仅列出各损失函数功能及表达式

注意：如下公式中出现下标n则表示该函数是对全部神经元进行的逐神经元操做。

3、优化器

pytorch的优化器：管理并更新模型中可学习参数的值，使得模型输出更接近真实标签
导数：函数在指定坐标轴上的变化率
方向导数：指定方向上的变化率
梯度：一个向量，方向为方向导数取得最大值的方向

1. Optimizer基本属性

defaults：优化器超参数
state：参数的缓存，如momentum的缓存
params_groups：管理的参数组，字典的list
_step_count：记录更新次数，学习率调整中使用

# ========================= step 4/5 优化器 ==========================
optimizer = optim.SGD(net.parameters(), lr=LR, momentum=0.9)                        # 选择优化器
scheduler = torch.optim.lr_scheduler.StepLR(optimizer, step_size=10, gamma=0.1)     # 设置学习率降低策略

# ============================ step 5/5 训练 ========================
for epoch in range(MAX_EPOCH):
	...
    for i, data in enumerate(train_loader):
        ...
    	# backward
        optimizer.zero_grad()
        loss = criterion(outputs, labels)
        loss.backward()

        # update weights
        optimizer.step()

2. 基本方法

zero_grad()：清空所管理参数的梯度，pytorch特性：张量梯度不自动清零，而是累加

step()：执行一步更新

add_param_group()：添加参数组

state_dict()：获取优化器当前状态信息字典

load_state_dict() ：加载状态信息字典

weight = torch.randn((2, 2), requires_grad=True)
weight.grad = torch.ones((2, 2))

optimizer = optim.SGD([weight], lr=0.1)

# --------------------------- step -----------------------------------
optimizer.step()        # 修改lr=1 0.1观察结果
'''
weight before step:tensor([[0.6614, 0.2669],
        [0.0617, 0.6213]])
weight after step:tensor([[ 0.5614,  0.1669],
        [-0.0383,  0.5213]])
'''

# ------------------------- zero_grad --------------------------------
print("weight in optimizer:{}\nweight in weight:{}\n".format(id(optimizer.param_groups[0]['params'][0]), id(weight)))
print("weight.grad is {}\n".format(weight.grad))	

optimizer.zero_grad()
'''
weight in optimizer:1598931466208
weight in weight:1598931466208 与上一句相同，说明params存储的是指向数据的指针

weight.grad is tensor([[1., 1.],
        [1., 1.]])

after optimizer.zero_grad(), weight.grad is
tensor([[0., 0.],
        [0., 0.]])
'''


# --------------------- add_param_group --------------------------
print("optimizer.param_groups is\n{}".format(optimizer.param_groups))
w2 = torch.randn((3, 3), requires_grad=True)
optimizer.add_param_group({"params": w2, 'lr': 0.0001})
print("optimizer.param_groups is\n{}".format(optimizer.param_groups))

# ------------------- state_dict ----------------------
optimizer = optim.SGD([weight], lr=0.1, momentum=0.9)
opt_state_dict = optimizer.state_dict()

for i in range(10):
    optimizer.step()

torch.save(optimizer.state_dict(), os.path.join(BASE_DIR, "optimizer_state_dict.pkl"))

# -----------------------load state_dict ---------------------------
optimizer = optim.SGD([weight], lr=0.1, momentum=0.9)
state_dict = torch.load(os.path.join(BASE_DIR, "optimizer_state_dict.pkl"))

optimizer.load_state_dict(state_dict)

3. 学习率

梯度降低:
𝒘𝒊+𝟏 = 𝒘𝒊 − 𝒈(𝒘𝒊 )
𝒘𝒊+𝟏 = 𝒘𝒊 − LR * 𝒈(𝒘𝒊)
学习率（learning rate）控制更新的步伐

采用不一样学习率进行训练，注意在lr>0.3时出现梯度爆炸

iteration = 100
num_lr = 10
lr_min, lr_max = 0.01, 0.2  # .5 .3 .2

lr_list = np.linspace(lr_min, lr_max, num=num_lr).tolist()
loss_rec = [[] for l in range(len(lr_list))]
iter_rec = list()

for i, lr in enumerate(lr_list):
x = torch.tensor([2.], requires_grad=True)
for iter in range(iteration):

y = func(x)
y.backward()
x.data.sub_(lr * x.grad)  # x.data -= x.grad
x.grad.zero_()

loss_rec[i].append(y.item())

for i, loss_r in enumerate(loss_rec):
plt.plot(range(len(loss_r)), loss_r, label="LR: {}".format(lr_list[i]))
plt.legend()
plt.xlabel('Iterations')
plt.ylabel('Loss value')
plt.show()

4. momentum

Momentum（动量，冲量）：结合当前梯度与上一次更新信息，用于当前更新

指数加权平均： $$v_t=\beta v_{t-1}+(1-\beta)\theta_t = \sum_i^{{N}(1-\beta)\beta}i\theta_{N-1}$$

pytorch中更新公式：
\(v_i=mv_{i-1}+g(w_i)\)
\(w_{i+1}=w_i-lr*v_i\)

1.optim.SGD
主要参数：
• params：管理的参数组
• lr：初始学习率
• momentum：动量系数，贝塔
• weight_decay：L2正则化系数
• nesterov：是否采用NAG

def func(x):
    return torch.pow(2*x, 2)    # y = (2x)^2 = 4*x^2        dy/dx = 8x

iteration = 100
m = 0.9     # .9 .63

lr_list = [0.01, 0.03]

momentum_list = list()
loss_rec = [[] for l in range(len(lr_list))]
iter_rec = list()

for i, lr in enumerate(lr_list):
    x = torch.tensor([2.], requires_grad=True)

    momentum = 0. if lr == 0.03 else m
    momentum_list.append(momentum)

    optimizer = optim.SGD([x], lr=lr, momentum=momentum)

    for iter in range(iteration):

        y = func(x)
        y.backward()

        optimizer.step()
        optimizer.zero_grad()

        loss_rec[i].append(y.item())

上述状况出现弹簧现象，由于在loss接近0的位置仍然有极大的动量，应当适当减少。

其余9个优化器详见PyTorch 学习笔记（七）：PyTorch的十个优化器