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I thought both, PReLU and Leaky ReLU are $$f(x) = \max(x, \alpha x) \qquad \text{ with } \alpha \in (0, 1)$$

Keras, however, has both functions in the docs.

Leaky ReLU

Source of LeakyReLU:

return K.relu(inputs, alpha=self.alpha)

Hence (see relu code) $$f_1(x) = \max(0, x) - \alpha \max(0, -x)$$

PReLU

Source of PReLU:

def call(self, inputs, mask=None):
    pos = K.relu(inputs)
    if K.backend() == 'theano':
        neg = (K.pattern_broadcast(self.alpha, self.param_broadcast) *
               (inputs - K.abs(inputs)) * 0.5)
    else:
        neg = -self.alpha * K.relu(-inputs)
    return pos + neg

Hence $$f_2(x) = \max(0, x) - \alpha \max(0, -x)$$

Question

Did I get something wrong? Aren't $f_1$ and $f_2$ equivalent to $f$ (assuming $\alpha \in (0, 1)$?)

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Straight from wikipedia:

enter image description here

  • Leaky ReLUs allow a small, non-zero gradient when the unit is not active.

  • Parametric ReLUs take this idea further by making the coefficient of leakage into a parameter that is learned along with the other neural network parameters.

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    $\begingroup$ Ah, thanks, I always forget that Leaky ReLUs have $\alpha$ as a hyperparameter and Parametric ReLUs have $\alpha$ as a parameter. $\endgroup$ – Martin Thoma Apr 25 '17 at 15:42
  • $\begingroup$ For the Google-thing: That's ok. (Btw, for me this question is the third result now for "Leaky ReLU vs PReLU") $\endgroup$ – Martin Thoma Apr 25 '17 at 15:47
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    $\begingroup$ @MartinThoma true! No offense at all for that! The way I found the answer was pretty stupid as well; I didn't know what the 'P' in PReLU was, so I figured that out and then tried to figure out what PReLU was by just typing 'Parametric ReLU', which got me to the wikipedia page. I learned something to day because of your question ;) $\endgroup$ – Thomas W Apr 25 '17 at 15:50
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    $\begingroup$ Nice. Thats how it should be :-) In this case my little activation function overview might be interesting for you as well. The article is (partially) in German, but I guess for that part it shouldn't matter $\endgroup$ – Martin Thoma Apr 25 '17 at 15:57

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