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A sampled softmax function is like a regular softmax but randomly selects a given number of 'negative' samples.

This is difference than NCE Loss, which doesn't use a softmax at all, it uses a logistic binary classifier for the context/labels. In NLP, 'Negative Sampling' basically refers to the NCE-based approach.

More details here

I have tested both and they both give pretty much the same results. But in word embedding literature, they always use NCE loss, and never sampled softmax.

Is there any reason why this is? The sampled softmax seems like the more obvious solution to prevent applying a softmax to all the classes, so I imagine there must be some good reason for the NCE loss.

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Both negative sampling (derived from NCE) and sampled SoftMax use a few samples to bypass the calculation of full SoftMax.

The main problem comes from this comment in the linked pdf:

Sampled Softmax

(A faster way to train a softmax classifier)

which is only used for sampled SoftMax, although, negative sampling is as fast for the same reason that is working with few samples. If their performances are at the same level, this could be the reason why researchers are not convinced to switch over to sampled SoftMax. In academia, it is almost always the case that older methods are preferred over new, but equally-competent methods for the sake of credibility.

Negative sampling is NCE minus the logistic classifier. Roughly speaking, it only borrows the term "F(target) + sum of F(negative sample)s". Negative sampling is most prominently introduced in the Word2Vec paper in 2013 (as of now with 11K citations), and is backed by the mathematically rigorous NCE paper (2012). On the other hand, sampled SoftMax is introduced in this paper (2015) for a task-specific (Machine Translation) and biased approximation:

In this paper, we propose an approximate training algorithm based on (biased) importance sampling that allows us to train an NMT model with a much larger target vocabulary

Noting that negative sampling also allows us to train "with a much larger target vocabulary".

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