# Cross Entropy with Log Softmax Activation

My question is about how is log softmax implemented in practice with the cross-entropy loss.

Softmax gives values between 0 and 1, which means log softmax will give values between -infinity and 0. This means that we cannot use one-hot encoding (one 1 and rest 0's) for our target labels anymore (correct me if I am wrong). Our labels should now contain one 0 (which is the target) and rest all as -infinity.

The cross-entropy loss function is given as:

$$- \Sigma \ t_i*log(o_i)$$

where $$t_i$$ is the target label for the $$i^{th}$$ training sample,

and $$o_i$$ is the predicted output for the $$i^{th}$$ training sample.

Now, when we use our target label (which is in the range -infinity and 0), the loss will become +infinity because of the -infinity term in the target vector and thus becoming numerically unstable. What is the way around this?

Another question - How does Pytorch handle this? Also, the nn.CrossEntropyLoss() function calculates the log_softmax on the predicted outputs internally but I cannot find anywhere in their documentation as to where they convert the one-hot target labels (range 0 to 1); which we pass to the loss function; to a label with range -infinity to 0. Am I wrong in assuming that the target label needs to be changed?

Any help will be appreciated. Thanks!

Mathematically, softmax with finite inputs produces results $$o_i \in (0,1) \forall i$$ such that $$\sum_i o_i =1$$. This implies that softmax is never 0, so $$\log(o_i)$$ is always a real number.

Numerically, overflow or underflow could cause softmax to output a zero. This is common enough when training neural networks using floating point numbers. A common work-around to avoid numerical underflow (or overflow) is to work on the log scale via log_softmax, or else work on the logit scale and do not transform your outputs, but instead have a loss function defined on the logit scale. These methods avoid round-tripping (which causes a loss of precision) and use numerical tricks to keep values in nice floating point ranges.

Obviously, working on the log scale, or the logit scale, requires making algebraic adjustments so that the loss is also on the appropriate scale. So if you use identity activations in the final layer, you use CrossEntropyLoss. If you use log_softmax in the final layer, you use NLLLoss.

Consider $$0 < o_i < 1$$ the probability output from the network, produced by softmax with finite input. We wish to compute the cross-entropy loss.

• One option is to do things the naïve way, using $$o_i$$ and $$t_i$$ directly, and computing $$-\sum_i t_i \log(o_i)$$.
• A second option is to use log-probabilities instead. This means you have $$z_i = \log(o_i)$$ in hand, so you compute $$-\sum_i t_i \log(o_i) = -\sum t_i z_i$$.

I can't answer the part of your question about re-labeling because it doesn't make sense. When you're using a numerically stable procedure, $$\log(o_i)$$ is always a finite number, so $$t_i \log(o_i)$$ for $$y \in \{0,1\}$$ is also finite. In fact, in the case of 1-hot labels, only one index $$i$$ has a non-zero value of $$t_i \log (o_i)$$.

• Oh, that makes sense. I was under the impression that the predicted outputs and the targets need to have the same range of (-infinity, 0) for cross-entropy to give accurate results, which is why I thought of re-labeling the targets. Another question which now seems to bother me is that our $o_i$ will now take negative values because of the log_softmax activation function and we are feeding this to our cross-entropy loss which again takes the log of the predicted output (which is equal to taking the log of a negative number), which is incorrect. Am I missing something here?
– ntd
Nov 19 '19 at 21:14
• If you go the log_softmax route then you can still use an analogue of the cross-entropy function; you'll just have to change the code to a method that expects to receive log_softmax inputs, such as torch.nn.NLLLoss. Or you can have no final activation function (outputs are any real number) and use torch.nn.CrossEntropyLoss, which combines log_softmax and torch.nn.NLLLoss. Switching from one method to the other just changes what scale you're measuring the inputs.
– Sycorax
Nov 19 '19 at 21:42
• Yes, I am aware of the PyTorch functions which calculate the cross-entropy loss with log_softmax activation for me. But as my previous comment says, I am still confused as to how the log of a negative number case is handled. Am I doing the math wrong anywhere? From what I can see, my $o_i$ is a negative number after applying log_softmax activation. I apply Negative Log Likelihood loss on this which takes $log(o_i)$, where $o_i$ is a negative number to begin with (because of the log_softmax activation function applied over it).
– ntd
Nov 19 '19 at 21:53
• I thought $o_i$ was a probability? How can $o_i$ be negative?
– Sycorax
Nov 19 '19 at 21:54
• I think I got it. log_softmax is just used for the numerical stability to solve the underflow problem. I was under the impression that it is better because it penalizes more heavily than softmax, which doesn't seem to be the case from your updated answer. Thanks!
– ntd
Nov 19 '19 at 22:17