# Dice-coefficient loss function vs cross-entropy

When training a pixel segmentation neural networks, such as fully convolutional networks, how do you make the decision to use cross-entropy loss function versus Dice-coefficient loss function?

I realize this is a short question, but not quite sure what other information to provide. I looked at a bunch of documentation about the two loss functions but can't get a intuitive sense of when to use one over the other.

• Why not use the hands-on approach to use both and compare the results. Looking at a lot of different fields of application, the discussion of loss function is its own topic of extended research. Since convolutional networks are still a 'hot topic', I'd guess that most papers will still be published in the future. – cherub May 3 '18 at 13:40

One compelling reason for using cross-entropy over dice-coefficient or the similar IoU metric is that the gradients are nicer.

The gradients of cross-entropy wrt the logits is something like $$p - t$$, where $$p$$ is the softmax outputs and $$t$$ is the target. Meanwhile, if we try to write the dice coefficient in a differentiable form: $$\frac{2pt}{p^2+t^2}$$ or $$\frac{2pt}{p+t}$$, then the resulting gradients wrt $$p$$ are much uglier: $$\frac{2t(t^2-p^2)}{(p^2+t^2)^2}$$ and $$\frac{2t^2}{(p+t)^2}$$. It's easy to imagine a case where both $$p$$ and $$t$$ are small, and the gradient blows up to some huge value. In general, it seems likely that training will become more unstable.

The main reason that people try to use dice coefficient or IoU directly is that the actual goal is maximization of those metrics, and cross-entropy is just a proxy which is easier to maximize using backpropagation. In addition, Dice coefficient performs better at class imbalanced problems by design:

However, class imbalance is typically taken care of simply by assigning loss multipliers to each class, such that the network is highly disincentivized to simply ignore a class which appears infrequently, so it's unclear that Dice coefficient is really necessary in these cases.

I would start with cross-entropy loss, which seems to be the standard loss for training segmentation networks, unless there was a really compelling reason to use Dice coefficient.

• The cross entropy of all exponential families is a nice difference $p-t$. – Neil G May 4 '18 at 8:46
• When is the "main goal" maximization of dice loss? I checked the original paper and all they say is “we obtain results that we experimentally observed are much better than the ones computed through the same network trained optimising a multinomial logistic loss with sample re-weighting.” This is not very convincing. – Neil G May 4 '18 at 8:48
• @shimao By "ugly" you just mean that the gradients can explode, is that correct? – flawr Jan 8 at 8:18

As summarized by @shimao and @cherub, one cannot say apriori which one will work better on a particular dataset. The correct way is to try both and compare the results. Also, note that when it comes to segmentation, it is not so easy to "compare the results": IoU based measures like dice coefficient cover only some aspects of the quality of the segmentation; in some applications, different measures such as mean surface distance or Hausdorff surface distance need to be used. As you see, not even the choice of the correct quality metric is trivial, let alone the choice of the best cost function.

I personally have very good experience with the dice coefficient; it really does wonders when it comes to class imbalance (some segments occupy less pixels/voxels than others). On the other hand, the training error curve becomes a total mess: it gave me absolutely no information about the convergence, so in this regard cross-entropy wins. Of course, this can/should be bypassed by checking the validation error anyways.