Consider this simple network that computes the following function: $$ \left\{ \begin{matrix} xy & \mbox{if} \;x,y>0 \\ 0 & \mbox{otherwise} \end{matrix} \right.$$

The net looks like:

net = NetChain[{ElementwiseLayer[Ramp], AggregationLayer[Times, 1]}]

This works correctly as:

In[2]:= net[{4, 2}]

Out[2]= 8.    

In[3]:= net[{4, -2}]    

Out[3]= 0.    

In[4]:= net[{-4, -2}]    

Out[4]= 0.

However if try to evaluate the gradient in a given point, this returns a floating point error when either $x$ or $y$ are smaller than $0$.

In[5]:= net[{4, -2}, NetPortGradient["Input"]]

During evaluation of In[5]:= NetGraph::netnan: A floating-point overflow, underflow, or division by zero occurred while evaluating the net.

Out[5]= $Failed

Why this happens? Is there a way to avoid this?

PS: this is a very simplified example of the minimal network required to reproduce the error, extracted from a much more complicated loss function

EDIT: After posting the question, I discovered that the error can be reproduced also replacing Ramp with # - # &, but does not occur with # - # + 1 &

Link to question on Wolfram Community [link] [link updated 02/04/2019]

  • $\begingroup$ Please edit both this and your Wolfram Community post and cross-link them. $\endgroup$ – Szabolcs Apr 1 at 18:29
  • $\begingroup$ Which is the best way to do it? You insert the link? Is it considered "unpolite" to post the question in both places? $\endgroup$ – Luca Amerio Apr 1 at 18:32
  • 1
    $\begingroup$ Isn't this to do with the fact that Ramp will return 0 for negative numbers? Does NetPortGradient do some division somewhere? $\endgroup$ – Carl Lange Apr 1 at 18:39
  • 1
    $\begingroup$ If you replace the Times with Total it does not cause the error... $\endgroup$ – Carl Lange Apr 1 at 18:42
  • 2
    $\begingroup$ Yes - I suspect NetPortGradient is seeing the Times and reversing it (eg, dividing) in some way. $\endgroup$ – Carl Lange Apr 1 at 18:52

The error seems to occur when NetPortGradient tries to evaluate the gradient of a product in a point where both product's terms are null with null derivative. This is exactly what happens, for example, with the function $f(x,y) = x\cdot y$ in a region where both $x$ and $y$ are constantly $0$.

This is, at least in my opinion, a bug of NetPortGradient.

This cause also NetTrain to fail, as it fails to perform gradient descend.

To overcome the issue I rewrote the problem as $$ \left\{ \begin{matrix} (x+1)(y+1) - (x +y) -1 & \mbox{if} \;x,y>0 \\ 0 & \mbox{otherwise} \end{matrix} \right.$$

Mathematically, this is equal to the function above, as the function is just a convoluted way of writing $x\cdot y$. However, the product is perform between $(x+1)$ and $(y+1)$ that are non-zero everywhere (except for a point).

This allow the gradient to be correctly evaluated.

In the wolfram's Network language, this can be implemented as

net = NetGraph[
   "ramp" -> ElementwiseLayer[Ramp],
   "x+1" -> ElementwiseLayer[# + 1 &],
   "times" -> AggregationLayer[Times, 1],
   "sum" -> AggregationLayer[Total, 1],
   "x-y-1" -> ThreadingLayer[#1 - #2 - 1 &]
   "ramp" -> "x+1",
   "x+1" -> "times",
   "ramp" -> "sum",
   {"times", "sum"} -> "x-y-1"

and indeed

In[116]:= net[{4, 2}]

Out[116]= 8.

In[117]:= net[{-4, -2}]

Out[117]= 0.

In[118]:= net[{4, -2}, NetPortGradient["Input"]]

Out[118]= {0., 0.}

In[119]:= net[{-4, -2}, NetPortGradient["Input"]]

Out[119]= {0., 0.}

It works!


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