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I know the cross entropy function can be used as the cost function, if the activation function is logistic function: i.e.: $\frac{1}{1 + e^{-x}}$

However, I just wonder:

Can the cross entropy cost function be used with many other activation functions, such as tanh?

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Yes we can, as long as we use some normalizor (e.g. softmax) to ensure that the final output values are in between 0 and 1 and add up to 1.

If you're doing binary classification and only use one output value, only normalizing it to be between 0 and 1 will do.

Update
As mentioned by Sycorax, depending on what procedure you use to shifting and rescaling tanh back to [0,1], you could end up with a logistic unit again, see this answer.

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  • $\begingroup$ tanh output between -1 and +1, so can it not be used with cross entropy cost function? $\endgroup$ – xmllmx Jul 3 '16 at 11:22
  • $\begingroup$ @xmllmx not really, cross entropy requires the output can be interpreted as probability values, so we need some normalization for that. $\endgroup$ – dontloo Jul 3 '16 at 11:26
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    $\begingroup$ Depending on what procedure you use to shifting and rescaling $\tanh$ back to $[0,1]$, you could end up with a logistic unit again -- since the logistic unit is just a shifted-rescaled $\tanh$ unit. $\endgroup$ – Sycorax Jul 7 '18 at 22:41
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If your primary goal isn't to interpret values as probability values, but simply to find a "good" cost function for a tanh activation, in the sense that the neural network will learn faster the more "wrong" it is, then I believe that this may work:

thanh_cross_entropy

or, in python form:

def C(y, a):
    return  -.5 * ( (1-y)*log(1-a) + (1+y)*log(1+a) ) + log(2)

More formally - this will give a cost function whose derivative is proportional to the error - ie, satisfy these two equations:

dC/da

dC/db

These are equations (71) and (72) from here: http://neuralnetworksanddeeplearning.com/chap3.html#the_cross-entropy_cost_function - thanks to Michael Nielsen for this great resource!

As others have noted, since it doesn't really have a probablity interpretation, I don't know if this formula can really be called a "cross-entropy" function for tanh... but it should be a "good" choice for a cost function for a tanh activation... or at least better than mean-squared-error!

As a caveat, though - I'm relatively new to all this, so feel free to point out if I've made any obvious errors!


EDIT: I made some code (using keras) to test the performance of this cost function, versus mean-squared-error, and my tests show nearly double the performance!

Here's the gist / code: https://gist.github.com/elrond79/a016230229322f3ba2b46a99ff0b4e7c

... and here were my results:

0.00850945741317 0.0121004378641 0.0144485289499 0.0259487312062 0.011714946112 0.0103978548285 0.0308227941615 0.0070723194485 0.0142580815076 0.0160464689688 0.013335006552 0.00783108495935 0.01220898181 0.0173612970037 0.0276861919335 0.0137742581066 0.0227281911188 0.0108267272463 0.0138346472817 0.0110096849345

average: 0.01509578457033614

maes: 0.00782039967299 0.0102608616831 0.00747869220265 0.0106438063825 0.00730774874144 0.0106295737387 0.00770924169349 0.014159071813 0.00763049735583 0.00902260985951 0.00602617238736 0.0103817401882 0.0108806326851 0.00879056832034 0.00743713011182 0.00648508079659 0.00660474073891 0.00766924505766 0.00738797156683 0.00708367751452

average: 0.008570473125530578

Not conclusive, obviously, but certainly suggestive that this will give better results than MSE for tanh activations!

EDIT 2: Realized the custom cost function was also causing the graph to evaluate slower, so # of iterations wasn't necessarily a fair comparison. Altered the test to compare error when running for the same amount of time, and then mse outperforms this tanh-cross-entropy-like cost. Still, it's possible it could be useful for more complicated graphs, when the additional time is a lower percentage of the total.

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