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Title: Von Mises-Fisher Loss for Training Sequence to Sequence Models with Continuous Outputs

Paper content:
The Softmax function is used in the final layer of nearly all existing sequence-to-sequence 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 obtain upto 2.5x speed-up in training time while performing on par with the state-of-the-art models in terms of translation quality.
These models are capable of handling very large vocabularies without compromising on translation quality.
They also produce more meaningful errors than in the softmax-based models, as these errors typically lie in a subspace of the vector space of the reference translations.
Due to the power law distribution of word frequencies, rare words are extremely common in any language BID45 ).
Yet, the majority of language generation tasks-including machine translation BID39 BID1 BID24 , summarization BID36 BID37 BID30 , dialogue generation BID40 , question answering BID44 , speech recognition BID13 Xiong et al., 2017) , and others-generate words by sampling from a multinomial distribution over a closed output vocabulary.
This is done by computing scores for each candidate word and normalizing them to probabilities using a softmax layer.Since softmax is computationally expensive, current systems limit their output vocabulary to a few tens of thousands of most frequent words, sacrificing linguistic diversity by replacing the long tail of rare words by the unknown word token, unk .
Unsurprisingly, at test time this leads to an inferior performance when generating rare or out-of-vocabulary words.
Despite the fixed output vocabulary, softmax is computationally the slowest layer.
Moreover, its computation follows a large matrix multiplication to compute scores over the candidate words; this makes softmax expensive in terms of memory requirements and the number of parameters to learn BID26 BID27 BID8 .
Several alternatives have been proposed for alleviating these problems, including sampling-based approximations of the softmax function BID2 BID26 , approaches proposing a hierarchical structure of the softmax layer BID27 BID7 , and changing the vocabulary to frequent subword units, thereby reducing the vocabulary size BID38 .We
propose a novel technique to generate low-dimensional continuous word representations, or word embeddings BID25 BID31 BID4 instead of a probability distribution over the vocabulary at each output step. We
train sequence-to-sequence models with continuous outputs by minimizing the distance between the output vector and the pretrained word embedding of the reference word. At
test time, the model generates a vector and then searches for its nearest neighbor in the target embedding space to generate the corresponding word. This
general architecture can in principle be used for any language generation (or any recurrent regression) task. In this
work, we experiment with neural machine translation, implemented using recurrent sequence-to-sequence models BID39 with attention BID1 BID24 .To the best
of our knowledge, this is the first work that uses word embeddings-rather than the softmax layer-as outputs in language generation tasks. While this
idea is simple and intuitive, in practice, it does not yield competitive performance with standard regression losses like 2 . This is because
2 loss implicitly assumes a Gaussian distribution of the output space which is likely false for embeddings. In order to correctly
predict the outputs corresponding to new inputs, we must model the correct probability distribution of the target vector conditioned on the input BID3 . A major contribution
of this work is a new loss function based on defining such a probability distribution over the word embedding space and minimizing its negative log likelihood ( §3).We evaluate our proposed
model with the new loss function on the task of machine translation, including on datasets with huge vocabulary sizes, in two language pairs, and in two data domains ( §4). In §5 we show that our models
can be trained up to 2.5x faster than softmax-based models while performing on par with state-of-the-art systems in terms of generation quality. Error analysis ( §6) reveals
that the models with continuous outputs are better at correctly generating rare words and make errors that are close to the reference texts in the embedding space and are often semantically-related to the reference translation.
This work makes several contributions.
We introduce a novel framework of sequence to sequence learning for language generation using word embeddings as outputs.
We propose new probabilistic loss functions based on vMF distribution for learning in this framework.
We then show that the proposed model trained on the task of machine translation leads to reduction in trainable parameters, to faster convergence, and a dramatic speed-up, up to 2.5x in training time over standard benchmarks.
TAB5 visualizes a comparison between different types of softmax approximations and our proposed method.State-of-the-art results in softmax-based models are highly optimized after a few years on research in neural machine translation.
The results that we report are comparable or slightly lower than the strongest baselines, but these setups are only an initial investigation of translation with the continuous output layer.
There are numerous possible directions to explore and improve the proposed setups.
What are additional loss functions?
How to setup beam search?
Should we use scheduled sampling?
What types of embeddings to use?
How to translate with the embedding output into morphologically-rich languages?
Can low-resource neural machine translation benefit from translation with continuous outputs if large monolingual corpora are available to pre-train strong target-side embeddings?
We will explore these questions in future work.Furthermore, the proposed architecture and the probabilistic loss (NLLvMF) have the potential to benefit other applications which have sequences as outputs, e.g. speech recognition.
NLLvMF could be used as an objective function for problems which currently use cosine or 2 distance, such as learning multilingual word embeddings.
Since the outputs of our models are continuous (rather than class-based discrete symbols), these models can potentially simplify training of generative adversarial networks for language generation.
DISPLAYFORM0 where C m (κ) is given as: DISPLAYFORM1 The normalization constant is not directly differentiable because Bessel function cannot be written in a closed form.
The gradient of the first component (log (C m ê )) of the loss is given as DISPLAYFORM2 .
In Table 9 : Translation quality experiments using beam search with BPE based baseline models with a beam size of 5With our proposed models, in principle, it is possible to generate candidates for beam search by using K-Nearest Neighbors.
But how to rank the partially generated sequences is not trivial (one could use the loss values themselves to rank, but initial experiments with this setting did not result in significant gains).
In this work, we focus on enabling training with continuous outputs efficiently and accurately giving us huge gains in training time.
The question of decoding with beam search requires substantial investigation and we leave it for future work.

assistant: text: Language generation using seq2seq models which produce word embeddings instead of a softmax based distribution over the vocabulary at each step enabling much faster training while maintaining generation quality