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In the week 5 lecture notes for Andrew Ng's Coursera Machine Learning Class, the following formula is given for calculating the value of $\epsilon$ used to initialise $\Theta$ with random values:

Forumla for calculating epsilon-init for random initialisation

In the exercise, further clarification is given:

One effective strategy for choosing $\epsilon_{init}$ is to base it on the number of units in the network. A good choice of $\epsilon_{init}$ is $\epsilon_{init} = \frac{\sqrt{6}}{\sqrt{L_{in} - L_{out}}}$, where $L_{in} = s_l$ and $L_{out} = s_{l+1}$ are the number of units in the layers adjacent to $\Theta^{(l)}$.

Why is the constant $\sqrt 6$ used here? Why not $\sqrt 5$, $\sqrt 7$ or $\sqrt {6.1}$?

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I believe this is Xavier normalized initialization (implemented in several deep learning frameworks eg Keras, Cafe, ...) from Understanding the difficulty of training deep feedforward neural networks by Xavier Glorot & Yoshua Bengio.

See equations 12, 15 and 16 in the paper linked: they aim to satisfy equation 12: $$\text{Var}[W_i] = \frac{2}{n_i + n_{i+1}}$$

and the variance of a uniform RV in $[-\epsilon,\epsilon]$ is $\epsilon^2/3$ (mean is zero, pdf = $1/(2\epsilon)$ so variance $=\int_{-\epsilon}^{\epsilon}x^2 \frac{1}{2\epsilon}dx$

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  • $\begingroup$ Hmm, so why use $\sqrt 6$ instead of $2$? $\endgroup$ – Tom Hale Jul 18 '17 at 6:28
  • $\begingroup$ Plug epsilon into the formula for variance of uniform random variable in +/-x and what do you get? $\endgroup$ – seanv507 Jul 18 '17 at 18:26
  • $\begingroup$ Doh! I now see in formulae (16) that $[-\epsilon, \epsilon]$ is used. Where do you get $x^2 / 3$ from though? $\endgroup$ – Tom Hale Jul 23 '17 at 10:45
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    $\begingroup$ added explanation for variance of uniform RV... $\endgroup$ – seanv507 Jul 23 '17 at 14:13

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