There seems to be a gap in the literature as to why cross-entropy is used.

Older references on neural networks ("ANNs") always use the squared loss. For example, here is one from Chong and Zak "An Intro to Optimization 4th Ed",

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Here is the one by Simon Haykin on "Kalman Filter and Neural Networks" enter image description here

Somewhere along the way, cross-entropy became the dominant loss function that is used in many papers and almost all the "blog" type references on NN. Recall that the cross-entropy is often formulated as,

$$ CE(y, \hat y) = -\sum\limits_{n = 1}^N \sum\limits_{c = 1}^n y_n^c \cdot\log(\hat y_n^c) $$ where $n$ is the $n$th data, $c$ is the $c$th class, and $y, \hat y$ denotes the set of targets and the predictions, respectively.

Where did the above function even came from (books/papers)? Was there some famous work that used cross entropy that popularized it over the squared loss? Is there a good reason to use CE as opposed to the squared loss (or the softmax loss associated with softmax/multiclass logistic regression)?

  • $\begingroup$ Do you require a neural network to have 1 or more hidden layers? $\endgroup$
    – Sycorax
    Apr 1, 2020 at 19:55
  • $\begingroup$ @SycoraxsaysReinstateMonica Not necessarily. One layer is good in my opinion. $\endgroup$
    – Fraïssé
    Apr 1, 2020 at 20:03
  • $\begingroup$ What about 0 layers? $\endgroup$
    – Sycorax
    Apr 1, 2020 at 20:15
  • $\begingroup$ @SycoraxsaysReinstateMonica Fine by me $\endgroup$
    – Fraïssé
    Apr 1, 2020 at 21:51

1 Answer 1


If you agree that a logistic regression is a special case of a neural network, then the answer is D. R. Cox, who invented logistic regression.[1]

A neural network with zero hidden layers and a single sigmoid output and trained to maximize the binomial likelihood (equiv. minimize cross-entropy) is logistic regression.

Minimizing a binomial cross-entropy is equivalent to maximizing a particular likelihood: the relationship between maximizing the likelihood and minimizing the cross-entropy

[1] D. R. Cox. "The Regression Analysis of Binary Sequences" Journal of the Royal Statistical Society. Series B (Methodological) Vol. 20, No. 2 (1958), pp. 215-242.


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