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I have a simple neural network and it works with the logistic function as activation function. Now I want to avoid the saturation problem by substituting the logistic function by the hyperbolic tangent:

#define SIGMOID(x) (1.7159*tanh(0.66666667*x)) 
#define DSIGMOID(S) (0.666666667/1.7159*(1.7159-(S))*(1.7159+(S)))

But the network never converges, the MSE stays the same throughout the training. Here's my training samples:

double training_data[][4]={
            {0, 0,  0,  -1},
            {0, 0,  1,  1},
            {0, 1,  0,  1},
            {0, 1,  1,  -1},
            {1, 0,  0,  1},
            {1, 0,  1,  -1},
            {1, 1,  0,  -1},
            {1, 1,  1,  1}};

The network does converge if I use the original (non-scaled) hyperbolic tangent function, that is:

#define SIGMOID(x) (tanh(x))
#define DSIGMOID(S) (1-((S)*(S)))

Do I miss something? E.g. Scaling the output to match the range (-1.7159, 1.7159) or anything?

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  • $\begingroup$ I am unfamiliar with your programming language, is it C? In MatLab many functions that work with or respond to built-in (libraried) functions need to be expressed in vector notation. Instead of using the scalar multiplication "", it is more desirable to use the vectorized "." operator. This improves execution speed substantially. $\endgroup$ Commented May 27, 2013 at 23:29
  • $\begingroup$ 2/3 will give a better approximation than 0.666666667 in most languages.... Presumably of little consequence, but best to do it properly. $\endgroup$
    – Nick Cox
    Commented May 27, 2013 at 23:59
  • $\begingroup$ @EngrStudent This is C, no Matlab $\endgroup$
    – Max
    Commented May 28, 2013 at 0:50
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    $\begingroup$ @NickCox There's no real difference between 0.666666667 and 2/3. But using 2/3 is more expensive because the computer will have perform additional calculation. Most people will just hardcode it that way. $\endgroup$
    – Max
    Commented May 28, 2013 at 0:50
  • $\begingroup$ The first example I Googled used a different approximation. More importantly, I suspect all you need to do is to calculate the best approximation to 2/3 just once and define it as a real constant if the expense worries you. The details will depend on your language. $\endgroup$
    – Nick Cox
    Commented May 28, 2013 at 0:54

2 Answers 2

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When I plot using the following R-code:

x <- seq(from = -2, to = 2, by = 0.01 )
y <- (0.666666667/1.7159*(1.7159-(x))*(1.7159+(x)))
y2 <- (1.7159*tanh(0.66666667*x)) 

plot(x,y2,col = "red")
points(x,y)

I get the following plot: plot of the give expressions

One of these is a sigmoid (red), one is not a great derivative (black). Notice the negative values. This is going to define a radius of convergence that shoots Newtons-methods toward infinity.

Now using this R-code:

x <- seq(from = -2, to = 2, by = 0.01 )
y <- 1.14393*(1/cosh(2*x/3))^2
y2 <- (1.7159*tanh(0.66666667*x)) 

plot(x,y2,col = "red", type = "b")
points(x,y)

I get this plot: updated plot of expressions

It is a more plausible graph of the derivative(black) for the sigmoid(red).

This was fun: link.

Edit:

Here are some basics on Tanh and friends.

  1. http://mathworld.wolfram.com/HyperbolicTangent.html
  2. http://mathworld.wolfram.com/HyperbolicCosine.html
  3. http://mathworld.wolfram.com/HyperbolicSine.html

Please notice in link 1 that the derivative of Hyperbolic Tangent is pow( hyperbolic_secant,2) and not pow( hyperbolic_cosine,2).

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  • $\begingroup$ I don't know what a hyperbolic secant is, but the derivative of f(x) = tanh(x) is f'(x) = 1-pow(tanh(x), 2). The formula I use is absolutely correct (because I just copied and paste it from a open source library), so it seems the hyperbolic secant is wrong. I think the problem with my NN has something to do with the output (e.g. remap it from (-1.7159, 1.7159) to (-1, 1)... but I haven't seen anyone do that) $\endgroup$
    – Max
    Commented May 28, 2013 at 7:00
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    $\begingroup$ "absolutely correct" because copied and pasted "from a open source library": would that it were so.... $\endgroup$
    – Nick Cox
    Commented May 28, 2013 at 14:34
  • $\begingroup$ @NickCox This page tells a different story. Derivative of tanh(x) is 1-pow(tanh(x), 2): math2.org/math/derivatives/more/hyperbolics.htm. And the library I copied and pasted from is the famous FANN, so it must be absolutely correct. The original tanh(x) works as a proof that I got the derivative correctly $\endgroup$
    – Max
    Commented May 29, 2013 at 23:52
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    $\begingroup$ Hehe. When there are two mutually exclusive and also true statements then there must be a third statement that resolves them. If $ D = 1-tanh^2(x)$ and $ D = sech^2(x)$ then $tanh^2(x) + sech^2(x) = 1$. It is about 1/3 the way down this page (en.wikipedia.org/wiki/Hyperbolic_function). Neither of these are the quadratic in the first DSIGMOID. "(0.666666667/1.7159*(1.7159-(S))*(1.7159+(S)))" Why is that statement correct? Within a radius of about 2-miles the flat-world hypothesis is a good approximation to the sphere of the earth. Counterexamples disprove, examples don't prove. $\endgroup$ Commented May 30, 2013 at 1:48
  • $\begingroup$ Not a good idea to be dogmatic when you don't know key facts. $\endgroup$
    – Nick Cox
    Commented May 30, 2013 at 6:29
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FANN_SIGMOID_SYMMETRIC

f2M_set_act_function_hidden (ann, FANN_SIGMOID_SYMMETRIC);

f2M_set_act_function_output (ann, FANN_SIGMOID_SYMMETRIC);

Symmetric sigmoid activation function, AKA tanh. One of the most used activation functions.

This activation function gives output that is between -1 and 1.

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