Activation function

Aside from their empirical performance, activation functions also have different mathematical properties: ; Nonlinear: When the activation function is non-linear, then a two-layer neural network can be proven to be a universal function approximator. This is known as the Universal Approximation Theorem. The identity activation function does not satisfy this property. When multiple layers use the identity activation function, the entire network is equivalent to a single-layer model. ; Range: When the range of the activation function is finite, gradient-based training methods tend to be more stable, because pattern presentations significantly affect only limited weights. When the range is infinite, training is generally more efficient because pattern presentations significantly affect most of the weights. In the latter case, smaller learning rates are typically necessary. ; Continuously differentiable: This property is desirable for enabling gradient-based optimization methods (ReLU is not continuously differentiable and has some issues with gradient-based optimization, but it is still possible). The binary step activation function is not differentiable at 0, and it differentiates to 0 for all other values, so gradient-based methods can make no progress with it. These properties do not decisively influence performance, nor are they the only mathematical properties that may be useful. For instance, the strictly positive range of the softplus makes it suitable for predicting variances in variational autoencoders. == Mathematical details ==

The most common activation functions can be divided into three categories: ridge functions, radial functions and fold functions. An activation function f is saturating if \lim_ (x)^+ \doteq {} &\begin{cases} 0 & \text{if } x \le 0\\ x & \text{if } x > 0 \end{cases} \\ = {} &\max(0,x) = x \textbf{1}_{x>0} \end{align} 0 & \text{if } x 0 \end{cases} &\frac{1}{2} x \left(1 + \text{erf}\left(\frac{x}{\sqrt{2}}\right)\right) \\ {}={} &x\Phi(x) \end{align} where \mathrm{erf} is the gaussian error function. \alpha\left(e^x - 1\right) & \text{if } x \le 0\\ x & \text{if } x > 0 \end{cases} : with parameter \alpha \alpha e^x & \text{if } x 0 \end{cases} C^1 & \text{if } \alpha = 1 \\ C^0 & \text{otherwise} \end{cases} \alpha(e^x - 1) & \text{if } x :with parameters \lambda = 1.0507 and \alpha = 1.67326 \alpha e^x & \text{if } x 0.01x & \text{if } x \le 0\\ x & \text{if } x > 0 \end{cases} 0.01 & \text{if } x 0 \end{cases} \alpha x & \text{if } x : with parameter \alpha \alpha & \text{if } x where g_{\lambda, \sigma, \mu, \beta}(x) = \frac{ (x - \lambda) {1}_{ \{ x \geqslant \lambda \} } }{ 1 + e^{- \sgn(x-\mu) \left( \frac{\vert x-\mu \vert}{\sigma} \right)^\beta } } ==See also==