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For deep neural networks using ReLU neurons, the recommended connection weight initialization strategy is to pick a random uniform number between -r and +r with:

$r = \sqrt{\dfrac{12}{\text{fan-in} + \text{fan-out}}}$

Where fan-in and fan-out are the number of connections going in and out of the layer being initialized. This is called "He initialization" (paper).

My question is: what's the recommended weights initialization strategy when using ELU neurons (paper)?

Since ELUs look a lot like ReLUs, I'm tempted to use the same logic, but I'm not sure it's the optimal strategy.

Note

There is a fairly similar question but this one is more specifically about the ELU activation function (which is not covered by the answers to the other question).

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Just noticed that the ELU paper states that "The weights have been initialized according to (He et al., 2015)", so this must be a good strategy, if not the optimal strategy.

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I think the initialization should be roughly $\sqrt{\frac{1.55}{n_{in}}}$

The He et al. 2015 results are made for ReLU units. The key idea is that the variance of f(y) with y = W * x + b should be roughly equal to the variance of y. Let's first go over the ReLU case, and see if we can ammend it for ELU units.

In the paper they show show that: $$ Var[y_l] = n_l Var[w_l] \mathbb{E}[x^2_l] $$ They express the last expectation $\mathbb{E}[x^2_l]$ in terms of $Var[y_{l-1}]$. For ReLUs we have that $\mathbb{E}[x^2_l] = \frac{1}{2} Var[y_{l-1}]$, simply because ReLUs put half the values in $x$ to $0$ on average. Thus we can write

$$ Var[y_l] = n_l Var[w_l] \frac{1}{2} Var[y_{l-1}] $$ We apply this all the way, taking the product over all $l$, all the way to the first layer. This gives: $$ Var[y_L] = Var[y_1] \prod_{i=2}^L \frac{1}{2} n_l Var[w_l] $$ Now this is stable only when $\frac{1}{2} n_l Var[w_l]$ is close to 1. So they set that to 1 and find $Var[W_l] = \frac{2}{n_l}$

Now for ELU units, the only thing we have to change is the expression of $\mathbb{E}[x^2_l]$ in terms of $Var[y_{l-1}]$. Sadly, this is not as straight-forward for ELU units as for RelU units as it involves calculating $\mathbb{E}[({e^{(\mathcal{N})}}^2)]$ for only the negative values of $\mathcal{N}$. This is not a pretty formula, I don't even know if there's a good closed form solution, so let's sample to get an approximation. We want $Var[y_l]$ to roughly be equal to 1 (most inputs are variance 1, batch norm makes layers variance 1 etc.). Thus we can sample from a normal distribution, apply the elu function with alpha = 1, square and calculate the mean. This gives $\approx 0.645$. The inverse of this is $\approx 1.55$.

Thus folllowing the same logic, we can set $Var[w_l]$ to $\sqrt{\frac{1.55}{n}}$ to get a variance that doesn't increase in magnitude.

I reckon that would be the optimal value for the ELU function. It fits in between the value for the ReLU function (1/2, which is lower than 0.645 because the values that are mapped to 0 now get mapped to some minus value), and what you would have for any function with mean 0 (which is just 1).

Take care that if the variance of $Var[y_{l-1}]$ is different, the optimal constant is also different. When this variance tends to 0, then the function becomes more and more like a unit function, thus the constant will tend to 1. If the variance becomes really big, the value tends towards the original ReLU value, thus 0.5.

Edit: Did the theoretical analysis of the variance of ELU(x) if x is normally distributed. It involves the some derivations of the log-normal distribution and not so pretty integrals. The eventual answer for the variance is $0.5 \sigma$ (the part of the linear function) + $$ a - 2(b)^2 + (2b - 1)^2 $$ where $$ a = \frac{1}{2} e^{\frac{\sigma^2}{2}} \left(\text{erfc}\left(\frac{\sigma}{\sqrt{2}}\right) + \sqrt{\frac{1}{\sigma^2}} \sigma -1\right)\\ b = \frac{1}{2} e^{2\sigma^2} \left(\text{erfc}\left(\sqrt{2} \sigma\right) + \sqrt{\frac{1}{\sigma^2}} \sigma -1\right)\\ $$ Which is not very solvable for $\sigma$ unfortunately. You can fill in for $\sigma$ and get the estimate I gave above however, which is pretty cool.

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