监督学习中的损失函数及应用研究

doi:10.11959/j.issn.2096-0271.2020006

摘要/Abstract

摘要：

监督学习中的损失函数常用来评估样本的真实值和模型预测值之间的不一致程度，一般用于模型的参数估计。受应用场景、数据集和待求解问题等因素的制约，现有监督学习算法使用的损失函数的种类和数量较多，而且每个损失函数都有各自的特征，因此从众多损失函数中选择适合求解问题最优模型的损失函数是相当困难的。研究了监督学习算法中常用损失函数的标准形式、基本思想、优缺点、主要应用以及对应的演化形式，探索了它们适用的应用场景和可能的优化策略。本研究不仅有助于提升模型预测的精确度，而且也为构建新的损失函数或改进现有损失函数的应用研究提供了一个新的思路。

关键词: 监督学习, 损失函数, 相似度度量

Abstract:

The loss function in supervised learning is often used to evaluate the degree of inconsistency between the real value of the sample and the predicted value of the model,and is generally used for parameter estimation of the model.Due to the constraints of application scenarios,data sets and problems to be solved,there are many kinds and quantities of loss functions used by existing supervised learning algorithms,and each loss function has its own characteristics.Therefore,it is very difficult to select a loss function suitable for solving the optimal model of the problem from many loss functions.The standard forms,basic ideas,advantages and disadvantages,main applications and corresponding evolution forms of commonly used loss functions in supervised learning algorithms were studied,and their more appropriate application scenarios and possible optimization strategies were summarized.This study not only helps to improve the accuracy of model prediction,it also provides a new idea for the application of new loss functions or to improve the application of existing loss functions.

Key words: supervised learning, loss function, similarity measure

中图分类号:

TP181

邓建国, 张素兰, 张继福, 荀亚玲, 刘爱琴. 监督学习中的损失函数及应用研究[J]. 大数据, 2020, 6(1): 60-80.

Jianguo DENG, Sulan ZHANG, Jifu ZHANG, Yaling XUN, Aiqin LIU. Loss function and application research in supervised learning[J]. Big Data Research, 2020, 6(1): 60-80.

图/表 10

图1

表1

基于距离度量的损失函数"

名称	标准形式
平方损失函数	${(Y - f (x))}^{2}$
绝对损失函数	$\| Y - f (x) \|$
0-1损失函数	$L (Y, f (x)) = {\begin{cases} 1, Y \neq f (x) \\ 0, Y = f (x) \end{cases}$
铰链损失函数	$\max (0, 1 - Y f (x)), Y \in {- 1, 1}$
中心损失函数	$\frac{1}{2} \sum_{i = 1}^{n} D (f (x^{i}), c_{y_{i}})$
余弦损失函数	$\frac{Y f (x)}{‖ Y ‖ ‖ f (x) ‖}$

表1

表2

基于平方损失的演化损失函数"

名称	演化形式
加权平方损失函数	$λ {(Y - f (x))}^{2}$
和方误差函数	${\sum_{i = 1}^{n} (Y_{i} - f (x_{i}))}^{2}$
均方误差函数	$\frac{1}{n} \sum_{i = 1}^{n} {(Y_{i} - f (x_{i}))}^{2}$
L2损失函数	$\sqrt{\sum_{i = 1}^{n} {(Y_{i} - f (x_{i}))}^{2}}$
均方根误差函数	$\sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(\| Y_{i} - f (x_{i}))}^{2}}$
x²检验函数	$\frac{1}{2} \sum_{i = 0}^{n - 1} \frac{{(A_{i} - B_{i})}^{2}}{A_{i} + B_{i}}$
triple损失函数	$\sum_{i}^{N} ({‖ f (x_{i}^{a}) - f (x_{i}^{p}) ‖}_{2}^{2} - {‖ f (x_{i}^{a}) - f (x_{i}^{n}) ‖}_{2}^{2} + a)$
对比损失函数	$\frac{1}{2 n} \sum_{i = 1}^{N} y d^{2} + (1 - y) \max {(m \arg i n - d, 0)}^{2}$

表2

表3

基于绝对损失的演化损失函数"

名称	演化形式
平均绝对误差函数	$\frac{1}{n} \sum_{i = 1}^{n} \| Y_{i} - f (x_{i}) \|$
平均相对误差函数	$\frac{1}{n} \sum_{i = 1}^{n} \frac{\| Y_{i} - f (x_{i}) \|}{Y_{i}}$
L1损失函数	$\sum_{i = 1}^{n} \| Y_{i} - f (x_{i}) \|$
Chebyshev损失函数	$\max_{i = 1}^{n} (Y_{i} - f (x_{i}))$
Minkowski损失函数	${(\sum_{i = 1}^{n} {\| Y_{i} - f (x_{i}) \|}^{p})}^{\frac{1}{p}}$
smooth L1损失函数	${\begin{cases} \frac{1}{2} {(Y - f (x))}^{2}, \| Y - f (x) \| ＜ 1 \\ \| Y - f (x) \| - \frac{1}{2} ， \| Y - f (x) \| \geq 1 \end{cases}$
huber损失函数	${\begin{cases} \frac{1}{2} {(Y - f (x))}^{2}, \| Y - f (x) \| \leq δ \\ δ \| Y - f (x) \| - \frac{1}{2} δ^{2} ， \| Y - f (x) \| ＞ δ \end{cases}$
分位数损失函数	$\sum_{i; y_{i} ＜ f (x_{i})} (1 - γ) \| Y_{i} - f (x_{i}) \| + \sum_{i; y_{i} \geq f (x_{i})} γ \| Y_{i} - f (x_{i}) \|$

表3

表4

基于铰链损失的演化损失函数"

名称	演化形式
边界铰链损失函数	$\max (0, m \arg i n - (f (x) - Y))$
坡道损失函数	$\frac{1}{n} \sum_{i = 1}^{n} (\max (0, 1 - h_{y_{i}}) - \max (0, s - h_{y_{i}}))$
crammerand铰链损失函数	$\max (0, 1 + \max_{Y \neq f (x)} w_{Y} x - w_{f (x)} x)$
weston铰链损失函数	$\sum_{Y \neq f (x)} \max (0, 1 + w_{Y} x - w_{f (x)} x)$
二分类支持向量机损失函数	$\frac{1}{2} {‖ w ‖}^{2} + c \sum_{i = 1}^{n} \max (0, 1 - Y f (x))$
多分类支持向量机损失函数	$\sum_{Y \neq Y_{i}} \max (0, s_{j} - s_{Y_{i}} + 1)$
多分类支持向量机平方损失函数	${\sum_{Y \neq Y_{i}} \max (0, s_{j} - s_{Y_{i}} + 1)}^{2}$
top-k铰链损失函数	$C \sum_{i = 1}^{n} (\max (1 - y_{i} (w^{T} x_{i} + b), 0))$

表4

表5

基于概率分布度量的损失函数"

名称	标准形式
对数损失函数	$- \log P (Y \| X)$
KL散度函数	$\sum_{i = 1}^{n} P (x_{i}) \times \log (\frac{P (x_{i})}{Q (x_{i})})$
交叉熵损失函数	$- \sum_{i = 1}^{N} \sum_{j = 1}^{C} P_{i j} \log q_{i j}$
softmax损失函数	$- \frac{1}{n} \sum_{i = 1}^{n} \log \frac{e^{f_{Y_{i}}}}{\sum_{j = 1}^{c} e^{f_{j}}}$

表5

表6

基于对数损失的演化损失函数"

名称	演化形式
逻辑回归损失函数	$L (y, P (Y = y \| x)) = {\begin{cases} \log (1 + \exp (- f (x))), y = 1 \\ \log (1 + \exp) (f (x)), y = 0 \end{cases}$
加权对数损失函数	$- \sum_{k = 0}^{C} W_{C} Y_{C} \log (f (x))$
对数双曲余弦损失函数	$\sum_{i = 1}^{n} \log (\cosh (f (x_{i}) - Y_{i}))$
softmax损失函数	$- Y \log (s o f t \max (Y, f (x)))$
二分类对数损失函数	$- \frac{1}{n} \sum_{i = 1}^{n} (y_{i} \log p_{i} + (1 - y_{i}) \log (1 - p_{i}))$
巴氏距离函数	$- \ln (B C (p, q))$

表6

表7

基于交叉熵的演化损失函数"

名称	演化形式
平均交叉熵损失函数	$- \frac{1}{n} \sum_{i = 1}^{n} Y_{i} \ln (a_{i})$
二分类交叉熵损失函数	$- \frac{1}{n} \sum_{i = 1}^{n} [Y_{i} \ln (a_{i}) + (1 - Y_{i}) \ln (1 - a_{i})]$
二分类平衡交叉熵损失函数	$- \frac{1}{n} \sum_{i = 1}^{n} [β Y_{i} \ln (a_{i}) + (1 - β) (1 - Y_{i}) \ln (1 - a_{i})]$
多分类交叉熵函数	$- \frac{1}{n} \sum_{i = 1}^{n} \sum_{j = 1}^{k} Y_{i, j} \log (a_{i, j})$
focal损失函数	$- \frac{1}{n} \sum_{i = 1}^{n} Y_{i} \partial_{i} {(1 - a_{i})}^{γ} \log (a_{i})$

表7

表8

基于softmax损失的演化损失函数"

名称	演化形式
softer softmax损失函数	$- \frac{1}{n} \sum_{i = 1}^{n} \frac{e^{\frac{f_{Y_{i}}}{T}}}{\sum_{j = 1}^{c} e^{\frac{f_{j}}{T}}}$
NSL损失函数	$- \frac{1}{n} \sum_{i = 1}^{n} \log \frac{e^{s \cos (θ_{Y_{i}}, i)}}{\sum_{j \neq Y_{i}} e^{s \cos (θ_{j}, i)}}$
LMCL损失函数	$- \frac{1}{n} \sum_{i = 1}^{n} \log \frac{e^{s (\cos (θ_{Y_{i}}, i) - m)}}{e^{s (\cos (θ_{Y_{i}}, i) - m)} + \sum_{j = 1, j \neq Y_{i}} e^{s \cos (θ_{j}, i)}}$
L-softmax损失函数	$- \frac{1}{n} \sum_{i = 1}^{n} \log \frac{e^{‖ W_{Y_{i}} ‖ x_{i} ‖ ψ (θ_{Y_{i}})}}{e^{‖ W_{Y_{i}} ‖ x_{i} ‖ ψ \cos (θ_{Y_{i}})} + \sum_{j \neq Y_{i}} e^{‖ W_{j} ‖ x_{i} ‖ \cos (θ_{j})}}$
A-softmax损失函数	$- \frac{1}{n} \sum_{i = 1}^{n} \log \frac{e^{‖ W_{Y_{i}} ‖ x_{i} ‖ ψ (θ_{Y_{i}})}}{e^{‖ x_{i} ‖ ψ (θ_{y_{i}}, i)} + \sum_{j = 1, j \neq Y_{i}} e^{‖ x_{i} ‖ \cos (θ_{j, i})}}$
AM-softmax损失函数	$- \frac{1}{n} \sum_{i = 1}^{n} \log \frac{e^{s (W_{Y_{i}}^{T} f_{i} - m)}}{e^{s (W_{Y_{i}}^{T} f_{i} - m)} + \sum_{j = 1, j \neq Y_{i}}^{c} e^{s W_{j}^{T} f_{i}}}$
正则化softmax损失函数	$- \frac{1}{n} \sum_{i = 1}^{n} \log \frac{e^{f_{Y_{i}}}}{\sum_{j = 1}^{c} e^{f_{i}}} + λ R (W)$

表8

表9

其他损失函数"

名称	标准形式
指数损失函数	$\exp (- Y f (x))$
汉明距离函数	$\sum_{I = 1}^{n} I (X_{i}, y_{i})$
dice损失函数	$1 - \frac{\sum_{i = 1}^{n} p_{i} r_{i} + ε}{\sum_{i = 1}^{n} p_{i} + r_{i} + ε} - \frac{\sum_{i = 1}^{n} (1 - p_{i}) (1 - r_{i}) + ε}{\sum_{i = 1}^{n} 2 - p_{i} - r_{i} + ε}$
余弦损失+softmax函数	$- \frac{1}{n} \sum_{i = 1}^{n} \log \frac{e^{s (\cos (θ_{Y_{i}}, i) - m)}}{e^{s (\cos (θ_{Y_{i}}, i) - m)} + \sum_{j = 1, j \neq Y_{i}} e^{s \cos (θ_{j}, i)}}$
softmax+LDloss函数	$L_{s} + α L_{L}$

表9

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