Activation Functions

# Randomized Leaky Rectified Linear Units

Introduced by Xu et al. in Empirical Evaluation of Rectified Activations in Convolutional Network

Randomized Leaky Rectified Linear Units, or RReLU, are an activation function that randomly samples the negative slope for activation values. It was first proposed and used in the Kaggle NDSB Competition. During training, $a_{ji}$ is a random number sampled from a uniform distribution $U\left(l, u\right)$. Formally:

$$y_{ji} = x_{ji} \text{ if } x_{ji} \geq{0}$$ $$y_{ji} = a_{ji}x_{ji} \text{ if } x_{ji} < 0$$

where

$$\alpha_{ji} \sim U\left(l, u\right), l < u \text{ and } l, u \in \left[0,1\right)$$

In the test phase, we take average of all the $a_{ji}$ in training similar to dropout, and thus set $a_{ji}$ to $\frac{l+u}{2}$ to get a deterministic result. As suggested by the NDSB competition winner, $a_{ji}$ is sampled from $U\left(3, 8\right)$.

At test time, we use:

$$y_{ji} = \frac{x_{ji}}{\frac{l+u}{2}}$$

#### Papers

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