23种激活函数_激活函数种类

一、简介

   一个节点的激活函数(Activation Function)定义了该节点在给定的输入或输入的集合下的输出。神经网络中的激活函数用来提升网络的非线性（只有非线性的激活函数才允许网络计算非平凡问题），以增强网络的表征能力。对激活函数的一般要求是：必须非常数、有界、单调递增并且连续，并且可导。
  在实际选择激活函数时并不会严格要求可导，只需要激活函数几乎在所有点可导即可，即在个别点不可导是可以接受的。另外，其导数尽可能的大可以帮助加速训练神经网络，否则导数过小会导致网络无法继续训练下去。

二、激活函数种类

  下面是不同的激活函数的函数公式，图像和导数公式，图像。

1、恒等函数

$$f(x)=x{f}^{{}^{\prime }}(x)=1$$

2、单位阶跃函数

f ( x ) = { 0 , x < 0 1 , x ≥ 0 f ′ ( x ) = { 0 , x ≠ 0 ? , x = 0 f(x)=\left\{

$$\begin{array} {ll} 0,x < 0\\1,x\ge 0 \end{array}$$ \right.\qquad\qquad f^{'}(x)=\left\{ $$\begin{array} {ll} 0,x \ne 0\\?,x= 0 \end{array}$$ \right. f(x)={0,x<01,x≥0​f′(x)={0,x​=0?,x=0​

3、逻辑函数

$$f(x)=\sigma (x)=\frac{1}{1+e^{-x}}{f}^{{}^{\prime }}(x)=f(x)(1-f(x))$$

4、双曲正切函数

$$f(x)=tanh(x)=\frac{(e^x-e^{-x})}{e^x+e^{-x}}{f}^{{}^{\prime }}(x)=1-f(x{)}^2$$

5、反正切函数

$$f(x)=ta{n}^{-1}(x)f^{{}^{\prime }}(x)=\frac{1}{x^2+1}$$

6、Softsign函数

$$f(x)=\frac{x}{1+|x|}{f}^{{}^{\prime }}(x)=\frac{1}{(1+|x|{)}^2}$$

7、反平方根函数(ISRU)

$$f(x)=\frac{x}{\sqrt{1+\alpha x^2}}{f}^{{}^{\prime }}(x)=(\frac{1}{\sqrt{1+\alpha x^2}}{)}^3$$

8、线性整流函数(ReLU)

f ( x ) = { 0 , x < 0 x , x ≥ 0 f ′ ( x ) = { 0 , x < 0 1 , x ≥ 0 f(x)= \left\{

$$\begin{array}{ll}0,x<0\\x,x\ge 0 \end{array}$$ \right.\qquad\qquad f^{'}(x)=\left\{ $$\begin{array}{ll}0,x<0\\1,x\ge 0 \end{array}$$ \right. f(x)={0,x<0x,x≥0​f′(x)={0,x<01,x≥0​

9、带泄露线性整流函数(Leaky ReLU)

f ( x ) = { 0.01 x , x < 0 x , x ≥ 0 f ′ ( x ) = { 0.01 , x < 0 1 , x ≥ 0 f(x)= \left\{

$$\begin{array}{ll}0.01x,x<0\\x,x\ge 0 \end{array}$$ \right.\qquad\qquad f^{'}(x)=\left\{ $$\begin{array}{ll}0.01,x<0\\1,x\ge 0 \end{array}$$ \right. f(x)={0.01x,x<0x,x≥0​f′(x)={0.01,x<01,x≥0​

10、参数化线性整流函数(PReLU)

f ( x ) = { α x , x < 0 x , x ≥ 0 f ′ ( x ) = { α , x < 0 1 , x ≥ 0 f(x)= \left\{

$$\begin{array}{ll}\alpha x,x<0\\x,x\ge 0 \end{array}$$ \right.\qquad\qquad f^{'}(x)=\left\{ $$\begin{array}{ll}\alpha,x<0\\1,x\ge 0 \end{array}$$ \right. f(x)={αx,x<0x,x≥0​f′(x)={α,x<01,x≥0​

11、带泄露随机线性整流函数(RReLU)

f ( x ) = { α x , x < 0 x , x ≥ 0 f ′ ( x ) = { α , x < 0 1 , x ≥ 0 f(x)= \left\{

$$\begin{array}{ll}\alpha x,x<0\\x,x\ge 0 \end{array}$$ \right.\qquad\qquad f^{'}(x)=\left\{ $$\begin{array}{ll}\alpha,x<0\\1,x\ge 0 \end{array}$$ \right. f(x)={αx,x<0x,x≥0​f′(x)={α,x<01,x≥0​

12、指数线性函数(ELU)

f ( x ) = { α ( e x − 1 ) , x < 0 x , x ≥ 0 f ′ ( x ) = { f ( α , x ) + α , x < 0 1 , x ≥ 0 f(x)= \left\{

$$\begin{array}{ll}\alpha(e^x-1),x<0\\x,x\ge 0 \end{array}$$ \right.\qquad\qquad f^{'}(x)=\left\{ $$\begin{array}{ll}f(\alpha,x)+\alpha,x<0\\1,x\ge 0 \end{array}$$ \right. f(x)={α(ex−1),x<0x,x≥0​f′(x)={f(α,x)+α,x<01,x≥0​

13、扩展指数线性函数(SELU)

f ( x ) = λ { α ( e x − 1 ) , x < 0 x , x ≥ 0 λ = 1.0507 , α = 1.67326 f ′ ( x ) = λ { α ( e x ) , x < 0 1 , x ≥ 0

$$\begin{aligned} f(x)=\lambda \left\{\begin{array}{ll}\alpha(e^x-1),x<0\\x,x\ge 0 \end{array}\right. \\ \lambda=1.0507,\alpha=1.67326 \end{aligned}$$ \qquad\qquad f^{'}(x)=\lambda \left\{ $$\begin{array}{ll}\alpha(e^x),x<0\\1,x\ge 0 \end{array}$$ \right. f(x)=λ{α(ex−1),x<0x,x≥0​λ=1.0507,α=1.67326​f′(x)=λ{α(ex),x<01,x≥0​

14、S型线性整流激活函数(SReLU)

f t l , a l , t r , a r ( x ) = { t l + a l ( x − t l ) , x ≤ t l x , t l < x < t r t r + a r ( x − t r ) , x ≥ t r t l , a l , t r , a r 为 参 数 f t l , a l , t r , a r ′ ( x ) = { a l , x ≤ t l 1 , t l < x < t r a r , x ≥ t r

$$\begin{aligned} f_{t_l,a_l,t_r,a_r}(x)=\left\{\begin{array} {ll} t_l+a_l(x-t_l),x\le t_l\\ x,t_l<x<t_r\\ t_r+a_r(x-t_r),x\ge t_r \end{array}\right.\\ t_l,a_l,t_r,a_r为参数 \end{aligned}$$ \qquad\qquad f_{t_l,a_l,t_r,a_r}^{'}(x)=\left\{ $$\begin{array} {ll} a_l,x\le t_l\\ 1,t_l<x<t_r\\ a_r,x\ge t_r \end{array}$$ \right. ftl​,al​,tr​,ar​​(x)=⎩⎨⎧​tl​+al​(x−tl​),x≤tl​x,tl​<x<tr​tr​+ar​(x−tr​),x≥tr​​tl​,al​,tr​,ar​为参数​ftl​,al​,tr​,ar​′​(x)=⎩⎨⎧​al​,x≤tl​1,tl​<x<tr​ar​,x≥tr​​

15、反平方根线性函数(ISRLU)

f ( x ) = { x 1 + α x 2 , x < 0 x , x ≥ 0 f ′ ( x ) = { ( 1 1 + α x 2 ) 3 , x < 0 1 , x ≥ 0 f(x)= \left\{

$$\begin{array}{ll}\frac{x}{\sqrt{1+\alpha x^2}},x<0\\x,x\ge 0 \end{array}$$ \right.\qquad\qquad f^{'}(x)=\left\{ $$\begin{array}{ll}(\frac{1}{\sqrt{1+\alpha x^2}})^3,x<0\\1,x\ge 0 \end{array}$$ \right. f(x)={1+αx2 ​x​,x<0x,x≥0​f′(x)={(1+αx2 ​1​)3,x<01,x≥0​

16、自适应分段线性函数(APL)

$$f(x)=max(0,x)+\sum _{s=1}^S{a}_i^{s}max(0,-x+b_i^s)f^{{}^{\prime }}(x)=H(x)-\sum _{s=1}^S{a}_i^{s}H(-x+b_i^s)$$

17、SoftPlus函数

$$f(x)=\ln (1+e^x)f^{{}^{\prime }}(x)=\frac{e^x}{1+e^x}$$

18、弯曲恒等函数

$$f(x)=\frac{\sqrt{x^2+1}-1}{2}+x{f}^{{}^{\prime }}(x)=\frac{x}{2\sqrt{x^2+1}}+1$$

19、Sigmoid Weighted Liner Unit(SiLU)

$$f(x)=x\cdot \sigma (x)f^{{}^{\prime }}(x)=f(x)+\sigma (x)(1-f(x))$$

20、SoftExponential

f ( x ) = { − l n ( 1 − α ( x + α ) ) α , α < 0 x , α = 0 e α x − 1 α , α > 0 f ′ ( x ) = { 1 1 − α ( α + x ) , α < 0 e α x , α ≥ 0 f(x)= \left\{

$$\begin{array} {ll} -\frac{ln(1-\alpha(x+\alpha))}{\alpha}, \alpha < 0\\ x,\alpha=0\\ \frac{e^{\alpha x}-1}{\alpha},\alpha >0 \end{array}$$ \right. \qquad\qquad f^{'}(x)=\left\{ $$\begin{array} {ll} \frac{1}{1-\alpha(\alpha+x)},\alpha < 0\\ e^{\alpha x},\alpha \ge 0 \end{array}$$ \right. f(x)=⎩⎨⎧​−αln(1−α(x+α))​,α<0x,α=0αeαx−1​,α>0​f′(x)={1−α(α+x)1​,α<0eαx,α≥0​

21、正弦函数

$$f(x)=sin(x)f^{{}^{\prime }}(x)=cos(x)$$

22、Sinc函数

f ( x ) = { 1 , x = 0 s i n ( x ) x , x ≠ 0 f ( x ) = { 0 , x = 0 c o s ( x ) x − s i n ( x ) x , x ≠ 0 f(x)=\left\{

$$\begin{array} {ll} 1,x=0 \\ \frac{sin(x)}{x},x\ne 0 \end{array}$$ \right.\qquad \qquad f(x)=\left\{ $$\begin{array} {ll} 0,x=0 \\ \frac{cos(x)}{x}-\frac{sin(x)}{x},x\ne 0 \end{array}$$ \right. f(x)={1,x=0xsin(x)​,x​=0​f(x)={0,x=0xcos(x)​−xsin(x)​,x​=0​

23、高斯函数

$$f(x)=e^{-x^2}{f}^{{}^{\prime }}(x)=-2x{e}^{-x^2}$$