Mixed Incremental Cross-Entropy REINFORCE ICLR 2016
Many natural language processing applications use language models to generate text. These models are typically trained to predict the next word in a sequence, given the previous words and some context such as an image. However, at test time the model is expected to generate the entire sequence from scratch. This discrepancy makes generation brittle, as errors may accumulate along the way. We address this issue by proposing a novel sequence level training algorithm that directly optimizes the metric used at test time, such as BLEU or ROUGE. On three different tasks, our approach outperforms several strong baselines for greedy generation. The method is also competitive when these baselines employ beam search, while being several times faster.READ FULL TEXT VIEW PDF
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Mixed Incremental Cross-Entropy REINFORCE ICLR 2016
Mixed Incremental Cross-Entropy REINFORCE ICLR 2016
Natural language is the most natural form of communication for humans. It is therefore essential that interactive AI systems are capable of generating text (Reiter & Dale, 2000)
. A wide variety of applications rely on text generation, including machine translation, video/text summarization, question answering, among others. From a machine learning perspective, text generation is the problem of predicting a syntactically and semantically correct sequence of consecutive words given some context. For instance, given an image, generate an appropriate caption or given a sentence in English language, translate it into French.
Popular choices for text generation models are language models based on n-grams(Kneser & Ney, 1995)2005)
, and recurrent neural networks (RNNs; Mikolov et al., 2010). These models when used as is to generate text suffer from two major drawbacks. First, they are trained to predict the next word given the previous ground truth words as input. However, at test time, the resulting models are used to generate an entire sequence by predicting one word at a time, and by feeding the generated word back as input at the next time step. This process is very brittle because the model was trained on a different distribution of inputs, namely, words drawn from the data distribution, as opposed to words drawn from the model distribution. As a result the errors made along the way will quickly accumulate. We refer to this discrepancy asexposure bias
which occurs when a model is only exposed to the training data distribution, instead of its own predictions. Second, the loss function used to train these models is at the word level. A popular choice is the cross-entropy loss used to maximize the probability of the next correct word. However, the performance of these models is typically evaluated using discrete metrics. One such metric is called BLEU(Papineni et al., 2002) for instance, which measures the n-gram overlap between the model generation and the reference text. Training these models to directly optimize metrics like BLEU is hard because a) these are not differentiable (Rosti et al., 2011)
, and b) combinatorial optimization is required to determine which sub-string maximizes them given some context. Prior attempts(McAllester et al., 2010; He & Deng, 2012) at optimizing test metrics were restricted to linear models, or required a large number of samples to work well (Auli & Gao, 2014).
This paper proposes a novel training algorithm which results in improved text generation compared to standard models. The algorithm addresses the two issues discussed above as follows. First, while training the generative model we avoid the exposure bias by using model predictions at training time. Second, we directly optimize for our final evaluation metric. Our proposed methodology borrows ideas from the reinforcement learning literature(Sutton & Barto, 1988). In particular, we build on the REINFORCE algorithm proposed by Williams (1992), to achieve the above two objectives. While sampling from the model during training is quite a natural step for the REINFORCE algorithm, optimizing directly for any test metric can also be achieved by it. REINFORCE side steps the issues associated with the discrete nature of the optimization by not requiring rewards (or losses) to be differentiable. While REINFORCE appears to be well suited to tackle the text generation problem, it suffers from a significant issue. The problem setting of text generation has a very large action space which makes it extremely difficult to learn with an initial random policy. Specifically, the search space for text generation is of size , where is the number of words in the vocabulary (typically around or more) and is the length of the sentence (typically around to ).
Towards that end, we introduce Mixed Incremental Cross-Entropy Reinforce (MIXER), which is our first major contribution of this work. MIXER is an easy-to-implement recipe to make REINFORCE work well for text generation applications. It is based on two key ideas: incremental learning and the use of a hybrid loss function which combines both REINFORCE and cross-entropy (see Sec. 3.2.2 for details). Both ingredients are essential to training with large action spaces. In MIXER, the model starts from the optimal policy given by cross-entropy training (as opposed to a random one), from which it then slowly deviates, in order to make use of its own predictions, as is done at test time.
Our second contribution is a thorough empirical evaluation on three different tasks, namely, Text Summarization, Machine Translation and Image Captioning. We compare against several strong baselines, including, RNNs trained with cross-entropy and Data as Demonstrator (DAD) (Bengio et al., 2015; Venkatraman et al., 2015). We also compare MIXER with another simple yet novel model that we propose in this paper. We call it the End-to-End BackProp model (see Sec. 3.1.3 for details). Our results show that MIXER with a simple greedy search achieves much better accuracy compared to the baselines on all the three tasks. In addition we show that MIXER with greedy search is even more accurate than the cross entropy model augmented with beam search at inference time as a post-processing step. This is particularly remarkable because MIXER with greedy search is at least times faster than the cross entropy model with a beam of size . Lastly, we note that MIXER and beam search are complementary to each other and can be combined to further improve performance, although the extent of the improvement is task dependent. 111Code available at: https://github.com/facebookresearch/MIXER
Sequence models are typically trained to predict the next word using the cross-entropy loss. At test time, it is common to use beam search to explore multiple alternative paths (Sutskever et al., 2014; Bahdanau et al., 2015; Rush et al., 2015). While this improves generation by typically one or two BLEU points (Papineni et al., 2002), it makes the generation at least times slower, where is the number of active paths in the beam (see Sec. 3.1.1 for more details).
The idea of improving generation by letting the model use its own predictions at training time (the key proposal of this work) was first advocated by Daume III et al. (2009)
. In their seminal work, the authors first noticed that structured prediction problems can be cast as a particular instance of reinforcement learning. They then proposed SEARN, an algorithm to learn such structured prediction tasks. The basic idea is to let the model use its own predictions at training time to produce a sequence of actions (e.g., the choice of the next word). Then, a search algorithm is run to determine the optimal action at each time step, and a classifier (a.k.a. policy) is trained to predict that action. A similar idea was later proposed byRoss et al. (2011)
in an imitation learning framework. Unfortunately, for text generation it is generally intractable to compute an oracle of the optimal target word given the words predicted so far. The oracle issue was later addressed by an algorithm called Data As Demonstrator (DAD)(Venkatraman et al., 2015) and applied for text generation by Bengio et al. (2015), whereby the target action at step is the -th action taken by the optimal policy (ground truth sequence) regardless of which input is fed to the system, whether it is ground truth, or the model’s prediction. While DAD usually improves generation, it seems unsatisfactory to force the model to predict a certain word regardless of the preceding words (see sec. 3.1.2 for more details).
Finally, REINFORCE has already been used for other applications, such as in computer vision(Mnih et al., 2014; Xu et al., 2015; Ba et al., ), and for speech recognition Graves & Jaitly (2014). While they simply pre-trained with cross-entropy loss, we found that the use of a mixed loss and a more gentle incremental learning scheduling to be important for all the tasks we considered.
|avoids exposure bias||No||Yes||Yes||Yes|
The learning algorithms we describe in the following sections are agnostic to the choice of the underlying model, as long as it is parametric. In this work, we focus on Recurrent Neural Networks (RNNs) as they are a popular choice for text generation. In particular, we use standard Elman RNNs (Elman, 1990) and LSTMs (Hochreiter & Schmidhuber, 1997)
. For the sake of simplicity but without loss of generality, we discuss next Elman RNNs. This is a parametric model that at each time step, takes as input a word as its input, together with an internal representation . is the the vocabulary of input words. This internal representation
is a real-valued vector which encodes the history of words the model has seen so far. Optionally, the RNN can also take as input an additional context vector, which encodes the context to be used while generating the output. In our experiments is computed using an attentive decoder inspired by Bahdanau et al. (2015) and Rush et al. (2015), the details of which are given in Section 6.2 of the supplementary material. The RNN learns a recursive function to compute and outputs the distribution over the next word:
The parametric expression for and depends on the type of RNN. For Elman RNNs we have:
where the parameters of the model are the set of matrices and also the additional parameters used to compute . is a vector whose components are , and is an indicator vector with only the -th component set to and the rest to . We assume the first word of the sequence is a special token indicating the beginning of a sequence, denoted by . All entries of the first hidden state are set to a constant value.
Next, we are going to introduce both baselines and the model we propose. As we describe these models, it is useful to keep in mind the key characteristics of a text generation system, as outlined in Table 1. There are three dimensions which are important when training a model for text generation: the exposure bias which can adversely affect generation at test time, the ability to fully back-propagate gradients (including with respect to the chosen inputs at each time step), and a loss operating at the sequence level. We will start discussing models that do not possess any of these desirable features, and then move towards models that better satisfy our requirements. The last model we propose, dubbed MIXER, has all the desiderata.
We now review a collection of methodologies used for training text generation models which optimize the prediction of only one word ahead of time. We start with the simplest and the most popular method which optimizes the cross-entropy loss at every time step. We then discuss a recently proposed modification to it which explicitly uses the model predictions during training. We finish by proposing a simple yet novel baseline which uses its model prediction during training and also has the ability to back propagate the gradients through the entire sequence. While these extensions tend to make generation more robust, they still lack explicit supervision at the sequence level.
Cross-entropy loss (XENT) maximizes the probability of the observed sequence according to the model. If the target sequence is , then XENT training involves minimizing:
When using an RNN, each term is modeled as a parametric function as given in Equation (5). This loss function trains the model to be good at greedily predicting the next word at each time step without considering the whole sequence. Training proceeds by truncated back-propagation through time (Rumelhart et al., 1986)
with gradient clipping(Mikolov et al., 2010).
Once trained, one can use the model to generate an entire sequence as follows. Let denote the word generated by the model at the -th time step. Then the next word is generated by:
Notice that, the model is trained to maximize , where is the word in the ground truth sequence. However, during generation the model is used as . In other words, during training the model is only exposed to the ground truth words. However, at test time the model has only access to its own predictions, which may not be correct. As a result, during generation the model can potentially deviate quite far from the actual sequence to be generated. Figure 1 illustrates this discrepancy.
The generation described by Eq. (7) is a greedy left-to-right process which does not necessarily produce the most likely sequence according to the model, because:
The most likely sequence might contain a word which is sub-optimal at an intermediate time-step . This phenomena is commonly known as a search error. One popular way to reduce the effect of search error is to pursue not only one but next word candidates at each point. While still approximate, this strategy can recover higher scoring sequences that are often also better in terms of our final evaluation metric. This process is commonly know as Beam Search. The downside of using beam search is that it significantly slows down the generation process. The time complexity grows linearly in the number of beams , because we need to perform forward passes for our network, which is the most time intensive operation. The details of the Beam Search algorithm are described in Section 6.3.
Conventional training with XENT suffers from exposure bias since training uses ground truth words as opposed to model predictions. DAD, proposed in (Venkatraman et al., 2015) and also used in (Bengio et al., 2015) for sequence generation, addresses this issue by mixing the ground truth training data with model predictions. At each time step and with a certain probability, DAD takes as input either the prediction from the model at the previous time step or the ground truth data. Bengio et al. (2015) proposed different annealing schedules for the probability of choosing the ground truth word. The annealing schedules are such that at the beginning, the algorithm always chooses the ground truth words. However, as the training progresses the model predictions are selected more often. This has the effect of making the model somewhat more aware of how it will be used at test time. Figure 2 illustrates the algorithm.
A major limitation of DAD is that at every time step the target labels are always selected from the ground truth data, regardless of how the input was chosen. As a result, the targets may not be aligned with the generated sequence, forcing the model to predict a potentially incorrect sequence. For instance, if the ground truth sequence is “I took a long walk” and the model has so far predicted “I took a walk”, DAD will force the model to predict the word “walk” a second time. Finally, gradients are not back-propagated through the samples drawn by the model and the XENT loss is still at the word level. It is not well understood how these problems affect generation.
The novel E2E algorithm is perhaps the most natural and naïve approach approximating sequence level training, which can also be interpreted as a computationally efficient approximation to beam search. The key idea is that at time step we propagate as input the top words predicted at the previous time step instead of the ground truth word. Specifically, we take the output distribution over words from the previous time step , and pass it through a -max layer. This layer zeros all but the largest values and re-normalizes them to sum to one. We thus have:
where are indexes of the words with largest probabilities and are their corresponding scores. At the time step , we take the largest scoring previous words as input whose contributions is weighted by their scores ’s. Smoothing the input this way makes the whole process differentiable and trainable using standard back-propagation. Compared to beam search, this can be interpreted as fusing the possible next hypotheses together into a single path, as illustrated in Figure 3. In practice we also employ a schedule, whereby we use only the ground truth words at the beginning and gradually let the model use its own top- predictions as training proceeds.
While this algorithm is a simple way to expose the model to its own predictions, the loss function optimized is still XENT at each time step. There is no explicit supervision at the sequence level while training the model.
We now introduce a novel algorithm for sequence level training, which we call Mixed Incremental Cross-Entropy Reinforce (MIXER). The proposed method avoids the exposure bias problem, and also directly optimizes for the final evaluation metric. Since MIXER is an extension of the REINFORCE algorithm, we first describe REINFORCE from the perspective of sequence generation.
In order to apply the REINFORCE algorithm (Williams, 1992; Zaremba & Sutskever, 2015) to the problem of sequence generation we cast our problem in the reinforcement learning (RL) framework (Sutton & Barto, 1988). Our generative model (the RNN) can be viewed as an agent, which interacts with the external environment (the words and the context vector it sees as input at every time step). The parameters of this agent defines a policy, whose execution results in the agent picking an action. In the sequence generation setting, an action refers to predicting the next word in the sequence at each time step. After taking an action the agent updates its internal state (the hidden units of RNN). Once the agent has reached the end of a sequence, it observes a reward. We can choose any reward function. Here, we use BLEU (Papineni et al., 2002) and ROUGE-2 (Lin & Hovy, 2003)
since these are the metrics we use at test time. BLEU is essentially a geometric mean over n-gram precision scores as well as a brevity penalty(Liang et al., 2006); in this work, we consider up to -grams. ROUGE-2 is instead recall over bi-grams. Like in imitation learning, we have a training set of optimal sequences of actions. During training we choose actions according to the current policy and only observe a reward at the end of the sequence (or after maximum sequence length), by comparing the sequence of actions from the current policy against the optimal action sequence. The goal of training is to find the parameters of the agent that maximize the expected reward. We define our loss as the negative expected reward:
where is the word chosen by our model at the -th time step, and is the reward associated with the generated sequence. In practice, we approximate this expectation with a single sample from the distribution of actions implemented by the RNN (right hand side of the equation above and Figure 9 of Supplementary Material). We refer the reader to prior work (Zaremba & Sutskever, 2015; Williams, 1992) for the full derivation of the gradients. Here, we directly report the partial derivatives and their interpretation. The derivatives w.r.t. parameters are:
where is the input to the softmax. The gradient of the loss with respect to is given by:
where is the average reward at time .
The interpretation of this weight update rule is straightforward. While Equation 10
is standard back-propagation (a.k.a. chain rule), Equation11
is almost exactly the same as the gradient of a multi-class logistic regression classifier. In logistic regression, the gradient is the difference between the prediction and the actual 1-of-N representation of the target word:
Therefore, Equation 11 says that the chosen word acts like a surrogate target for our output distribution, at time . REINFORCE first establishes a baseline , and then either encourages a word choice if , or discourages it if . The actual derivation suggests that the choice of this average reward9 we use a single sample from the distribution of actions.
In our implementation, the baseline is estimated by a linear regressor which takes as input the hidden states
of the RNN. The regressor is an unbiased estimator of future rewards since it only uses past information. The parameters of the regressor are trained by minimizing the mean squared loss:
. In order to prevent feedback loops, we do not backpropagate this error through the recurrent network(Zaremba & Sutskever, 2015).
REINFORCE is an elegant algorithm to train at the sequence level using any user-defined reward. In this work, we use BLEU and ROUGE-2 as reward, however one could just as easily use any other metric. When presented as is, one major drawback associated with the algorithm is that it assumes a random policy to start with. This assumption can make the learning for large action spaces very challenging. Unfortunately, text generation is such a setting where the cardinality of the action set is in the order of (the number of words in the vocabulary). This leads to a very high branching factor where it is extremely hard for a random policy to improve in any reasonable amount of time. In the next section we describe the MIXER algorithm which addresses these issues, better targeting text generation applications.
The MIXER algorithm borrows ideas both from DAGGER (Ross et al., 2011) and DAD (Venkatraman et al., 2015; Bengio et al., 2015) and modifies the REINFORCE appropriately. The first key idea is to change the initial policy of REINFORCE to make sure the model can effectively deal with the large action space of text generation. Instead of starting from a poor random policy and training the model to converge towards the optimal policy, we do the exact opposite. We start from the optimal policy and then slowly deviate from it to let the model explore and make use of its own predictions. We first train the RNN with the cross-entropy loss for epochs using the ground truth sequences. This ensures that we start off with a much better policy than random because now the model can focus on a good part of the search space. This can be better understood by comparing the perplexity of a language model that is randomly initialized versus one that is trained. Perplexity is a measure of uncertainty of the prediction and, roughly speaking, it corresponds to the average number of words the model is ‘hesitating’ about when making a prediction. A good language model trained on one of our data sets has perplexity of , whereas a random model is likely to have perplexity close to the size of the vocabulary, which is about .
The second idea is to introduce model predictions during training with an annealing schedule in order to gradually teach the model to produce stable sequences. Let be the length of the sequence. After the initial epochs, we continue training the model for epochs, such that, for every sequence we use the XENT loss for the first () steps, and the REINFORCE algorithm for the remaining steps. In our experiments is typically set to two or three. Next we anneal the number of steps for which we use the XENT loss for every sequence to () and repeat the training for another epochs. We repeat this process until only REINFORCE is used to train the whole sequence. See Algorithm 1 for the pseudo-code.
We call this algorithm Mixed Incremental Cross-Entropy Reinforce (MIXER) because we combine both XENT and REINFORCE, and we use incremental learning (a.k.a. curriculum learning). The overall algorithm is illustrated in Figure 4. By the end of training, the model can make effective use of its own predictions in-line with its use at test time.
In all our experiments, we train conditional RNNs by unfolding them up to a certain maximum length. We chose this length to cover about
of the target sentences in the data sets we consider. The remaining sentences are cropped to the chosen maximum length. For training, we use stochastic gradient descent with mini-batches of sizeand we reset the hidden states at the beginning of each sequence. Before updating the parameters we re-scale the gradients if their norm is above (Mikolov et al., 2010). We search over the values of hyper-parameter, such as the initial learning rate, the various scheduling parameters, number of epochs, etc., using a held-out validation set. We then take the model that performed best on the validation set and compute BLEU or ROUGE score on the test set. In the following sections we report results on the test set only. Greedy generation is performed by taking the most likely word at each time step. 222Code available at: https://github.com/facebookresearch/MIXER
We consider the problem of abstractive summarization where, given a piece of “source” text, we aim at generating its summary (the “target” text) such that its meaning is intact. The data set we use to train and evaluate our models consists of a subset of the Gigaword corpus (Graff et al., 2003) as described in Rush et al. (2015). This is a collection of news articles taken from different sources over the past two decades. Our version is organized as a set of example pairs, where each pair is composed of the first sentence of a news article (the source sentence) and its corresponding headline (the target sentence). We pre-process the data in the same way as in (Rush et al., 2015), which consists of lower-casing and replacing the infrequent words with a special token denoted by “unk”. After pre-processing there are unique words in the source dictionary and words in the target dictionary. The number of sample pairs in the training, validation and test set are , , and respectively. The average sequence length of the target headline is about words. We considered sequences up to words to comply with our initial constraint of covering at least % of the data.
Our generative model is a conditional Elman RNN (Equation 3) with hidden units, where the conditioning vector is provided by a convolutional attentive encoder, similar to the one described in Section 3.2 of Rush et al. (2015) and inspired by Bahdanau et al. (2015). The details of our attentive encoder are mentioned in Section 6.2 of the Supplementary Material. We also tried LSTMs as our generative model for this task, however it did not improve performance. We conjecture this is due to the fact that the target sentences in this data set are rather short.
For the translation task, our generative model is an LSTM with hidden units and it uses the same attentive encoder architecture as the one used for summarization. We use data from the German-English machine translation track of the IWSLT 2014 evaluation campaign (Cettolo et al., 2014). The corpus consists of sentence-aligned subtitles of TED and TEDx talks. We pre-process the training data using the tokenizer of the Moses toolkit (Koehn et al., 2007) and remove sentences longer than words as well as casing. The training data comprises of about sentences where the average English sentence is words long and the average German sentence is words long. In order to retain at least of this data, we unrolled our RNN for steps. Our validation set comprises of sentence pairs which was taken from the training data. The test set is a concatenation of dev2010, dev2012, tst2010, tst2011 and tst2012 which results in sentence pairs. The English dictionary has words while the German has words.
For the image captioning task, we use the MSCOCO dataset (Lin et al., 2014). We use the entire training set provided by the authors, which consists of around k images. We then took the original validation set (consisting of around k images) and randomly sampled (without replacement) images for validation and another for test. There are
different captions for each image. At training time we sample one of these captions, while at test time we report the maximum BLEU score across the five captions. The context is represented by 1024 features extracted by a Convolutional Neural Network (CNN) trained on the Imagenet dataset(Deng et al., 2009); we do not back-propagate through these features. We use a similar experimental set up as described in Bengio et al. (2015). The RNN is a single layer LSTM with hidden units and the image features are provided to the generative model as the first word in the sequence. We pre-process the captions by lower-casing all words and replacing all the words which appear less than 3 times with a special token “unk”. As a result the total number of unique words in our dataset is . Keeping in mind the rule, we unroll the RNN for steps.
In order to validate MIXER, we compute BLEU score on the machine translation and image captioning task, and ROUGE on the summarization task. The input provided to the system is only the context and the beginning of sentence token. We apply the same protocol to the baseline methods as well. The scores on the test set are reported in Figure 5.
We observe that MIXER produces the best generations and improves generation over XENT by to points across all the tasks. Unfortunately the E2E approach did not prove to be very effective. Training at the sequence level and directly optimizing for testing score yields better generations than turning a sequence of discrete decisions into a differentiable process amenable to standard back-propagation of the error. DAD is usually better than the XENT, but not as good as MIXER.
Overall, these experiments demonstrate the importance of optimizing for the metric used at test time. In summarization for instance, XENT and MIXER trained with ROUGE achieve a poor performance in terms of BLEU (8.16 and 5.80 versus 9.32 of MIXER trained with BLEU); likewise, MIXER trained with BLEU does not achieve as good ROUGE score as a MIXER optimizing ROUGE at training time as well (15.1 versus 16.22, see also Figure 8 in Supplementary Material).
Next, we experimented with beam search. The results in Figure 6 suggest that all methods, including MIXER, improve the quality of their generation by using beam search. However, the extent of the improvement is very much task dependent. We observe that the greedy performance of MIXER (i.e., without beam search) cannot be matched by baselines using beam search in two out of the three tasks. Moreover, MIXER is several times faster since it relies only on greedy search.
It is worth mentioning that the REINFORCE baseline did not work for these applications. Exploration from a random policy has little chance of success. We do not report it since we were never able to make it converge within a reasonable amount of time. Using the hybrid XENT-REINFORCE loss without incremental learning is also insufficient to make training take off from random chance. In order to gain some insight on what kind of schedule works, we report in Table 2 of Supplementary Material the best values we found after grid search over the hyper-parameters of MIXER. Finally, we report some anecdotal examples of MIXER generation in Figure 7 of Supplementary Material.
Our work is motivated by two major deficiencies in training the current generative models for text generation: exposure bias and a loss which does not operate at the sequence level. While Reinforcement learning can potentially address these issues, it struggles in settings when there are very large action spaces, such as in text generation. Towards that end, we propose the MIXER algorithm, which deals with these issues and enables successful training of reinforcement learning models for text generation. We achieve this by replacing the initial random policy with the optimal policy of a cross-entropy trained model and by gradually exposing the model more and more to its own predictions in an incremental learning framework.
Our results show that MIXER outperforms three strong baselines for greedy generation and it is very competitive with beam search. The approach we propose is agnostic to the underlying model or the form of the reward function. In future work we would like to design better estimation techniques for the average reward , because poor estimates can lead to slow convergence of both REINFORCE and MIXER. Finally, our training algorithm relies on a single sample while it would be interesting to investigate the effect of more comprehensive search methods at training time.
The authors would like to thank David Grangier, Tomas Mikolov, Leon Bottou, Ronan Collobert and Laurens van der Maaten for their insightful comments. We also would like to thank Alexander M. Rush for his help in preparing the data set for the summarization task and Sam Gross for providing the image features.
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Here we explain in detail how we generate the conditioning vector for our RNN using the source sentence and the current hidden state . Let us denote by the source sentence which is composed of a sequence of words . With a slight overload of notation let also denote the dimensional learnable embedding of the -th word (). In addition the position of the word is also associated with a learnable embedding of size (). Then the full embedding for the -th word in the input sentence is given by . In order for the embeddings to capture local context, we associate an aggregate embedding to each word in the source sentence. In particular for a word in the -th position, its aggregate embedding is computed by taking a window of consecutive words centered at position and averaging the embeddings of all the words in this window. More precisely, the aggregate embedding is given by:
In our experiments the width was set to
. In order to account for the words at the two boundaries of the input sentence we first pad the sequence on both sides with dummy words before computing the aggregate vectorss. Given these aggregate vectors of words, we compute the context vector (the final output of the encoder) as:
where the weights are computed as
Equation 7 always chooses the highest scoring next word candidate at each time step. At test time we can reduce the effect of search error by pursuing not only one but next word candidates at each point, which is commonly known as beam search. While still approximate, this strategy can recover higher scoring sequences that are often also better in terms of our final evaluation metric. The algorithm maintains the highest scoring partial sequences, where is a hyper-parameter. Setting reduces the algorithm to a greedy left-to-right search (Eq. (7)).
The current version of the paper updates the first version uploaded on arXiv as follows:
on the summarization task, we report results using both ROUGE-2 and BLEU to demonstrate that MIXER can work with any metric.
on machine translation and image captioning we use LSTM instead of Elman RNN to demonstrate the MIXER can work with any underlying parametric model.
BLEU is evaluated using up to 4-grams, and it is computed at the corpus level (except in the image captioning case) as this seems the most common practice in the summarization and machine translation literature.
we have added several references as suggested by our reviewers
we have shortened the paper by moving some content to the Supplementary Material.