Paper:
VON MISES-FISHER LOSS FOR TRAINING SEQUENCE TO SEQUENCE MODELS WITH CONTINUOUS OUTPUTS
The Softmax function is used in the final layer of nearly all existing sequence-tosequence models for language generation. However, it is usually the slowest layer to compute which limits the vocabulary size to a subset of most frequent types; and it has a large memory footprint. We propose a general technique for replacing the softmax layer with a continuous embedding layer. Our primary innovations are a novel probabilistic loss, and a training and inference procedure in which we generate a probability distribution over pre-trained word embeddings, instead of a multinomial distribution over the vocabulary obtained via softmax. We evaluate this new class of sequence-to-sequence models with continuous outputs on the task of neural machine translation. We show that our models train up to 2.5x faster than the state-of-the-art models while achieving comparable translation quality. These models are capable of handling very large vocabularies without compromising on translation quality or speed. They also produce more meaningful errors than the softmax-based models, as these errors typically lie in a subspace of the vector space of the reference translations 1 .

Review:
This paper describes a technique for replacing the softmax layer in sequence-to-sequence models with one that attempts to predict a continuous word embedding, which will then be mapped into a (potentially huge) pre-trained embedding vector via nearest neighbor search. The obvious choice for building a loss around such a prediction (squared error) is shown to be inappropriate empirically, and instead a von Mises-Fisher loss is proposed. Experiments conducted on small-data, small-model, greedy-search German->English, French->English and English->French scenarios demonstrate translation quality on par with BPE, and superior performance to a number of other continuous vector losses. They also provide convincing arguments that this new objective is more efficient in terms of both time and number of learned parameters.
This is a nice innovation for sequence-to-sequence modeling. The technical contribution required to make it work is non-trivial, and the authors have demonstrated promising results on a small system. I’m not sure whether this has any chance of supplanting BPE as the go-to solution for large vocabulary models, but I think it’s very healthy to add this method to the discussion.
Other than the aforementioned small baseline systems, this paper has few issues, so I’ll take some of my usual ‘problems with the paper’ space to discuss some downsides with this method. First: the need to use pre-trained word embeddings may be a step backward. It’s always a little scary to introduce more steps into the pipeline, and it’s uncomfortable to hear the authors state that they may be able to improve performance by changing the word embedding objective. As we move to large training sets, having pre-trained embeddings is likely to stop being an advantage and start being a hindrance. Second: though this can drastically increase vocabulary sizes, it is still a closed vocabulary model, which is a weakness when compared to BPE (though I suppose you could do both).
Smaller issues:
First paragraph after equation (1): “the hidden state … t, h.” -> “the hidden state h … t.”
Equation (2): it might help your readers to spell out how setting \kappa to ||\hat{e}|| allows you to ignore the unit-norm assumption of \mu.
“the negative log-likelihood of the vMF…” - missing capital
Unnumbered equation immediately before “Regularization of NLLvMF”: C_m||\hat{e}|| is missing round brackets around ||\hat{e}|| to make it an argument of the C_m function.
Is predicting the word vector whose target embedding has the highest value of vMF probability any more expensive than nearest neighbor search? Does it preclude the use of very fast nearest neighbor searches?
It might be a good idea to make it clear in 4.3 that you see an extension to beam search for your method to be non-trivial (and that you aren’t simply leaving out beam search for comparability to the various empirical loss functions). This didn’t become clear to me until the Future Work section.
In Table 5, I don’t fully understand F1 in terms of word-level translation accuracy. Recall is easy to understand (does the reference word appear in the system output?) but precision is harder to conceptualize. It might help to define the metric more carefully.