1 Introduction
Standard linguistic theories propose that natural language is structured as nested constituents organised in the form of a tree (Partee et al., 1990). However, most popular models, such as the Long SortTerm Memory network (LSTM) (Hochreiter and Schmidhuber, 1997), process text without imposing a grammatical structure. To bridge this gap between theory and practice models that process linguistic expressions in a treestructured manner have been considered in recent work (Socher et al., 2013; Tai et al., 2015; Zhu et al., 2015; Bowman et al., 2016). These treebased models explicitly require access to the syntactic structure for the text, which is not entirely satisfactory.
Indeed, parse tree level supervision requires a significant amount of annotations from expert linguists. These trees have been annotated with different goals in mind than the tasks we are using them for. Such discrepancy may result in a deterioration of the performance of models relying on them. Recently, several attempts were made to learn these models without explicit supervision for the parser (Yogatama et al., 2016; Maillard et al., 2017; Choi et al., 2018). However, Williams et al. (2018a) has recently shown that the structures learned by these models cannot be ascribed to discovering meaningful syntactic structure. These models even fail to learn the simple contextfree grammar of nested mathematical operations (Nangia and Bowman, 2018).
In this work, we present an extension of Choi et al. (2018)
, that successfully learns these simple grammars while preserving competitive performance on several standard linguistic tasks. Contrary to previous work, our model makes a clear distinction between the parser and the compositional function. These two modules are trained with different algorithms, cooperating to build a semantic representation that optimises the objective function. The parser’s goal is to generate a tree structure for the sentence. The compositional function follows this structure to produce the sentence representation. Our model contains a continuous component, the compositional function, and a discrete one, the parser. The whole system is trained endtoend with a mix of reinforcement learning and gradient descent.
Drozdov and Bowman (2017)has noticed the difficulty of mixing these two optimisation schemes without one dominating the other. This typically leads to the “coadaptation problem” where the parser simply follows the compositional function and fails to produce meaningful syntactic structures. In this work, we show that this pitfall can be avoided by synchronising the learning paces of the two optimisation schemes. This is achieved by combining several recent advances in reinforcement learning. First, we use inputdependent control variates to reduce the variance of our gradient estimates
(Ross, 1997). Then, we apply multiple gradient steps to the parser’s policy while controlling for its learning pace using the Proximal Policy Optimization (PPO) of Schulman et al. (2017). The code for our model is publicly available^{1}^{1}1https://github.com/facebookresearch/latenttreelstm.2 Preliminaries
In this section, we present existing works on Recursive Neural Networks and their training in the absence of supervision on the syntactic structures.
2.1 Recursive Neural Networks
A Recursive Neural Network (RvNN) has its architecture defined by a directed acyclic graph (DAG) given alongside with an input sequence Goller and Kuchler (1996). RvNNs are commonly used in NLP to generate sentence representation that leverages available syntactic information, such as a constituency or a dependency parse trees Socher et al. (2011).
Given an input sequence and its associated DAG, a RvNN processes the sequence by applying a transformation to the representations of the tokens lying on the lowest levels of the DAG. This transformation, or compositional function, merges these representations into representations for the nodes on the next level of the DAG. This process is repeated recursively along the graph structure until the toplevel nodes are reached. In this work, we assume that the compositional function is the same for every node in the graph.
TreeLSTM.
We focus on a specific type of RvNNs, the treebased long shortterm memory network (TreeLSTM) of
Tai et al. (2015) and Zhu et al. (2015). Its compositional function generalizes the LSTM cell of Hochreiter and Schmidhuber (1997) to treestructured topologies, i.e.,where and tanh are the sigmoid and hyperbolic tangent functions. TreeLSTM cell is differentiable with respect to its recursion matrix , bias
and its input. The gradients of a TreeLSTM can thus be computed with backpropagation through structure (BPTS)
Goller and Kuchler (1996).2.2 Learning with RvNNs
A treebased RvNN is a function parameterized by a
dimensional vector
that predicts an output given an input and a tree . Given a dataset of triplets , the parameters of the RvNN are learned with the following minimisation problem:(1) 
where
is a logistic regression function. These models need an externally provided parsing tree for each input sentence during both training and evaluation. Alternatives, such as the shiftreducebased SPINN model of
Bowman et al. (2016), learn an internal parser from the given trees. While these solutions do not need external trees during evaluation, they still require tree level annotations for training. More recent work has focused on learning a latent parser with no direct supervision.2.3 Latent tree models
Latent tree models aim at jointly learning the compositional function and a parser without supervision on the syntactic structures (Yogatama et al., 2016; Maillard et al., 2017; Choi et al., 2018)
. The latent parser is defined as a parametric probability distribution over trees conditioned on the input sequence. The parameters of this tree distribution
are represented by a vector . Given a dataset of pairs of input sequences and outputs , the parameters and are jointly learned by minimising the following objective function:(2) 
where is the expectation with respect to the distribution. Directly minimising this objective function is often difficult due to expensive marginalisation of the unobserved trees. Hence, when is a convex function (e.g. cross entropy of an exponential family) usually an upper bound of Eq. (2) can be derived by applying Jensen’s inequality:
(3) 
Learning a distribution over a set of discrete items involves a discrete optimisation scheme. For example, the RLSPINN model of Yogatama et al. (2016) uses a mix of gradient descent for and REINFORCE for (Williams et al., 2018a). Drozdov and Bowman (2017) has recently observed that this optimisation strategy tends to produce poor parsers, e.g., parsers that only generate leftbranching trees. The effect, called the coadaptation issue, is caused by both bias in the parsing strategy and a difference in convergence paces of continuous and discrete optimisers. Typically, the parameters are learned more rapidly than . This limits the exploration of the search space to parsing strategies similar to those found at the beginning of the training.
2.3.1 Gumbel TreeLSTM
In their Gumbel TreeLSTM model, Choi et al. (2018) propose an alternative parsing strategy to avoid the coadaptation issue. Their parser incrementally merges a pair of consecutive constituents until a single one remains. This strategy reduces the bias towards certain tree configurations observed with RLSPINN.
Each word of the input sequence is represented by an embedding vector. A leaf transformation maps this vector to pair of vectors . We considered three types of leaf transformations: affine transformation, LSTM and bidirectional LSTM. The resulting representations form the initial states of the TreeLSTM. In the absence of supervision, the tree is built in a bottomup fashion by recursively merging consecutive constituents based on mergecandidate scores. On each level of the bottomup derivation, the mergecandidate score of the pair is computed as follow:
where is a trainable query vector and is the constituent representation at position after mergings. We merge a pair sampled from the Categorical distribution built on the mergecandidate scores. The representations of the constituents are then updated as follow:
This procedure is repeated until one constituent remains. Its hidden state is the input sentence representation. This procedure is nondifferentiable. Choi et al. (2018) use an approximation based on the GumbelSoftmax distribution Maddison et al. (2016); Jang et al. (2016) and the reparametrization trick Kingma and Welling (2013).
This relaxation makes the problem differentiable at the cost of a bias in the gradient estimates Jang et al. (2016). This difference between the real objective function and their approximation could explain why their method cannot recover simple contextfree grammars Nangia and Bowman (2018). We investigate this question by proposing an alternative optimisation scheme that directly aims for the correct objective function.
3 Our model
We consider the problem defined in Eq. (3) to jointly learn a composition function and an internal parser. Our model is composed of the parser of Choi et al. (2018) and the TreeLSTM for the composition function. As suggested in past work Mnih et al. (2016); Schulman et al. (2017), we added an entropy over the tree distribution to the objective function:
(4) 
where . This regulariser improves exploration by preventing early convergence to a suboptimal deterministic parsing strategy. The new objective function is differentiable with respect to , but not , the parameters of the parser. Learning follows the same procedure with BPTS as if the tree would be externally given.
In the rest of this section, we discuss the optimization of the parser and a cooperative training strategy to reduce the coadaptation issue.
3.1 Unbiased gradient estimation
We cast the training of the parser as a reinforcement learning problem. The parser is an agent whose reward function is the negative of the loss function defined in Eq. (
3). Its action space is the space of binary trees. The agent’s policy is a probability distribution over binary trees that decomposes as a sequence of merging actions:(5) 
where . The loss function is optimised with respect to with REINFORCE (Williams, 1992). REINFORCE requires a considerable number of random samples to obtain a gradient estimate with a reasonable level of variance. This number is positively correlated with the size of the search space, which is exponentially large in the case of binary trees. We consider several extensions of REINFORCE to circumvent this problem.
Variance reduction.
An alternative solution to increasing the number of samples is the control variates method (Ross, 1997)
. It takes advantage of random variables with known expected values and positive correlation with the quantity whose expectation is tried to be estimated. Given an inputoutput pair
and tree sampled from , let’s define the random variable as:(6) 
According to REINFORCE, calculating the gradient with respect to for the pair is then equivalent to determining the unknown mean of the random variable ^{2}^{2}2Note that while we are computing the gradients using , we could also directly optimise the parser with respect to downstream accuracy.. Let’s assume there is a control variate, i.e., a random variable that positively correlates with and has known expected value with respect to . Given samples of the and the control variate , the new gradient estimator is:
A popular control variate, or baseline, used in REINFORCE is the moving average of recent rewards multiplied by the score function Ross (1997):
It has a zero mean under the distribution and it positively correlates with .
Surrogate loss.
REINFORCE often is implemented via a surrogate loss defined as follow:
(7) 
where is the empirical average over a finite batch of samples and is the probability ratio with standing for the parameters before the update.
Inputdependent baseline.
The moving average baseline cannot detect changes in rewards caused by structural differences in the inputs. In our case, a long arithmetic expression is much harder to parse than a short one, systematically leading to their lower rewards. This structural differences in the rewards aggravate the credit assignment problem by encouraging REINFORCE to discard actions sampled for longer sequences even though there might be some subsequences of actions that produce correct parsing subtrees.
A solution is to make the baseline inputdependent. In particular, we use the selfcritical training (SCT) baseline of Rennie et al. (2017), defined as:
where is the reward obtained with the policy used at test time, i.e., . This control variate has a zero mean under the distribution and correlates positively with the gradients. Computing the of a policy among all possible binary trees has exponential complexity. We replace it with a simpler greedy decoding, i.e, a tree is selected by following a sequence of greedy actions :
This approximation is very efficient and computing the baseline requires only one additional forward pass.
Gradient normalization.
We empirically observe significant fluctuations in the gradient norms. This creates instability that can not be reduced by additive terms, such as the inputdependent baselines. A solution is to divide the gradients by a coarse approximation of their norm, e.g., a running estimate of the reward standard deviation
Mnih and Gregor (2014). This trick ensures that the rewards remain approximately in the unit ball, making the learning process less sensitive to steep changes in the loss.3.2 Synchronizing syntax and semantics learning with PPO
The gradients of the loss function from the Eq. (4) are calculated using two different schemes, BPST for the composition function parameters and REINFORCE for the parser parameters . Then, both are updated with SGD. The estimate of the gradient with respect to has higher variance compared to the estimate with respect to . Hence, using the same learning rate schedule does not necessarily correspond to the same real pace of learning. It is parameters that are harder to optimise, so to improve training stability and convergence it is reasonable to aim for such updates that does not change the policy too much or too little. A simple yet effective solution is the Proximal Policy Optimization (PPO) of Schulman et al. (2017). It considers the next surrogate loss:
Where and is a real number in . The first argument of the is the surrogate loss for REINFORCE. The clipped ratio in the second argument disincentivises the optimiser from performing updates resulting in large tree probability changes. With this, the policy parameters can be optimised with repeated steps of SGD to ensure a similar “pace” of learning between the parser and the compositional function.
4 Related work
Besides the works mentioned in Sec. 2 and Sec. 3, there is a vast literature on learning latent parsers. Early connectionist work in inferring contextfree grammars proposed stackaugmented models and relied on explicit supervision on the strings that belonged to the target language and those that did not (Giles et al., 1989; Sun, 1990; Das et al., 1992; Mozer and Das, 1992). More recently, new stackaugmented models were shown to learn latent grammars from positive evidence alone (Joulin and Mikolov, 2015). In parallel to these, other statistical approaches were proposed to automatically induce grammars from unparsed text (Sampson, 1986; Magerman and Marcus, 1990; Carroll and Charniak, 1992; Brill, 1993; Klein and Manning, 2002). Our work departs from these approaches in that we aim at learning a latent grammar in the context of performing some given task.
SocherPHNM11 uses a surrogate autoencoder objective to search for a constituency structure, merging nodes greedily based on the reconstruction loss. MaillardCY17 defines a relaxation of a CYKlike chart parser that is trained for a particular task. A similar idea is introduced in LeZ15 where an automatic parser prunes the chart to reduce the overall complexity of the algorithm. Another strategy, similar in nature, has been recently proposed by caio1807, where Gumbel noise is used with differentiable dynamic programming to generate dependency trees. In contrast, YogatamaBDGL16 learns a ShiftReduce parser using reinforcement learning. jean1806 further proposes a beam search strategy to overcome learning trivial trees. On a different vein, vlad1809 proposes a quadratic penalty term over the posterior distribution of nonprojective dependency trees to enforce sparsity of the relaxation. Finally, there is a large body of work in Reinforcement Learning that aims at discovering how to combine elementary modules to solve complex tasks (Singh, 1992; Chang et al., 2018; Sahni et al., 2017). Due to the limited space, we will not discuss them in further details.
5 Experiments
We conducted experiments on three different tasks: evaluating mathematical expressions on the ListOps dataset (Nangia and Bowman, 2018), sentiment analysis on the SST dataset (Socher et al., 2013) and natural language inference task on the SNLI (Bowman et al., 2015) and MultiNLI (Williams et al., 2018b) datasets.
Technical details.
No baseline  Moving average  Self critical  

No PPO  PPO  No PPO  PPO  No PPO  PPO  
61.7  61.4  61.7  59.4  63.7  98.2  
70.1  76.6  74.3  96.0  64.1  99.6  
66.2 3.2  66.5 5.9  65.5 4.7  67.5 14.3  64.0 0.1  99.2 0.5 
Model  Accuracy 

LSTM*  71.51.5 
RLSPINN*  60.72.6 
Gumbel TreeLSTM*  57.62.9 
Ours  99.20.5 
For ListOps, we follow the experimental protocol of NangiaB18, i.e., a
dimensional model and a tenway softmax classifier. However, we replace their multilayer perceptron (MLP) by a linear classifier. The validation set is composed of
k examples randomly selected from the training set. For SST and NLI, we follow the setup of ChoiYL18: we initialise the word vectors with GloVe300D Pennington et al. (2014)and train an MLP classifier on the sentence representations. The hyperparameters are selected on the validation set using
random seeds for each configuration. Our hyperparameters are the learning rate, weight decay, the regularisation parameter , the leaf transformations, variance reduction hyperparameters and the number of updates in PPO. We use an adadelta optimizer Zeiler (2012).5.1 ListOps
The ListOps dataset probes the syntax learning ability of latent tree models (Nangia and Bowman, 2018). It is designed to have a single correct parsing strategy that a model must learn in order to succeed. It is composed of prefix arithmetic expressions and the goal is to predict the numerical output associated with the evaluation of the expression. The sequences are made of integers in and operations: MIN, MAX, MED and SUM_MOD. The output is an integer in the range . For example, the expression [MIN 2 [MAX 0 1] [MIN 6 3 ] 5 ] is mapped to the output 1. The ListOps task is thus a sequence classification problem with classes. There are k training examples and k test examples. It is worth mentioning that the underlying semantic of operations and symbols is not provided. In other words, a model has to infer from examples that [MIN 0 1] = 0.
As shown in Table 2, the current leading latent tree models are unable to learn the correct parsing strategy on ListOps (Nangia and Bowman, 2018). They even achieve performance worse than purely sequential recurrent networks. On the other hand, our model achieves near perfect accuracy on this task, suggesting that our model is able to discover the correct parsing strategy. Our model differs in several ways from the Gumbel TreeLSTM of Choi et al. (2018) that could explain this gap in performance. In the rest of this section, we perform an ablation study on our model to understand the importance of each of these differences.
Impact of the baseline and PPO.
We report the impact of our design choices on the performance in Table 1. Our model without baseline nor PPO is vanilla REINFORCE. The baselines only improve performance when PPO is used. Furthermore, these ablated models without PPO perform onpar with the RLSPINN model (see Table 2). This confirms our expectations for models that fail to synchronise syntax and semantics learning.
Interestingly, using PPO has a positive impact on both baselines, but accuracy remains low with the moving average baseline. The reduction of variance induced by the SCT baseline leads to a nearperfect recovery of the good parsing strategy in all five experiments. This shows the importance of this baseline for the stability of our approach.
Sensitivity to hyperparameters.
Our model is relatively robust to hyperparameters changes when we use the SCT baseline and PPO. For example, changing the leaf transformation or dimensionality of the model has a minor impact on performance. However, we have observed that the choice of the optimiser has a significant impact. For example, the average performance drops to if we replace Adadelta by Adam Kingma and Ba (2014). Yet, the maximum value out of runs remains relatively high, .
Untied parameters.
As opposed to previous work, the parameters of the parser and the composition function are not tied in our model. Without this separation between syntax and semantics, it would be impossible to update one module without changing the other. The gradient direction is then dominated by the low variance signal from the semantic component, making it hard to learn the parser. We confirmed experimentally that our model with tied parameters fails to find the correct parser and its accuracy drops to .
Extrapolation and Grammaticality.
Recursive models have the potential to generalise to any sequence length. Our model was trained with sequences of length up to tokens. We test the ability of the model to generalise to longer sequences by generating additional expressions of lengths to . As shown in Fig.1, our model has a little loss in accuracy as the length increases to ten times the maximum length seen during training.
On the other hand, we notice that final representations produced by the parser are very similar to each other. Indeed, the cosine similarity between these vectors for the test set has a mean value of 0.998 with a standard deviation of 0.002. There are two possible explanations for this observation: either our model assigns similar representations to valid expressions, or it produces a trivial uninformative representation regardless of the expression. To verify which explanation is correct, we generate ungrammatical expressions by removing either one operation token or one closing bracket symbol for each sequence in the test set. As shown in Figure
2, in contrast to grammatical expressions, ungrammatical ones tend to be very different from each other: “Happy families are all alike; every unhappy family is unhappy in its own way.” The only exception, marked by a mode near , come from ungrammatical expressions that represent incomplete expressions because of missing a closing bracket at the end. This kind of sequences were seen by the parser during training and they indeed have to be represented by the same vector. These observations show that our model does not produce a trivial representation, but identifies the rules and constraints of the grammar. Moreover, vectors for grammatical sequences are so different from vectors for ungrammatical ones that you can tell them apart with accuracy by simply measuring their cosine similarity to a randomly chosen grammatical vector from the training set. Interestingly, we have not observed a similar signal from the vectors generated by the composition function. Even learning a naive classifier between grammatical and ungrammatical expressions on top of these representations achieves an accuracy of only . This suggests that most of the syntactic information is captured by the parser, not the composition function.5.2 Natural Language Inference
Model  Dim.  Acc. 

Yogatama et al. (2016)  100  80.5 
Maillard et al. (2017)  100  81.6 
Choi et al. (2018)  100  82.6 
Ours  100  84.30.3 
Bowman et al. (2016)  300  83.2 
Munkhdalai and Yu (2017)  300  84.6 
Choi et al. (2018)  300  85.6 
Choi et al. (2018)  300  83.7 
Choi et al. (2018)*  300  84.9 0.1 
Ours  300  85.10.2 
Chen et al. (2017)  600  85.5 
Choi et al. (2018)  600  86.0 
Ours  600  84.60.2 
Model  Dim.  Acc. 

LSTM  300  69.1 
SPINN  300  67.5 
RLSPINN  300  67.4 
Gumbel TreeLSTM  300  69.5 
Ours  300  70.70.3 
We next evaluate our model on natural language inference using the Stanford Natural Language Inference (SNLI) Bowman et al. (2015) and MultiNLI Williams et al. (2018b) datasets. Natural language inference consists in predicting the relationship between two sentences which can be either entailment, contradiction, or neutral. The task can be formulated as a threeway classification problem. The results are shown in Tables 3 and 4. When training the model on MultiNLI dataset we augment the training data with the SNLI data and use matched versions of the development and test sets. Surprisingly, two out of four models for MultiNLI task collapsed to leftbranching parsing strategies. This collapse can be explained by the absence of the entropy regularisation and the small number of PPO updates , which were determined to be optimal via hyperparameter optimisation. As with ListOps, using an Adadelta optimizer significantly improves the training of the model.
5.3 Sentiment Analysis
SST2  SST5  
Sequential sentence representation  
Radford et al. (2017)  91.8  52.9 
McCann et al. (2017)  90.3  53.7 
Peters et al. (2018)    54.7 
RvNN based models with external tree  
Socher et al. (2013)  85.4  45.7 
Tai et al. (2015)  88.0  51.0 
Munkhdalai and Yu (2017)  89.3  53.1 
Looks et al. (2017)  89.4  52.3 
RvNN based models with latent tree  
Yogatama et al. (2016)  86.5   
Choi et al. (2018)  90.7  53.7 
Choi et al. (2018)  90.30.5  51.60.8 
Ours  90.20.2  51.50.4 
We evaluate our model on a sentiment classification task using the Stanford Sentiment Treebank (SST) of Socher et al. (2013). All sentences in SST are represented as binary parse trees, and each subtree of a parse tree is annotated with the corresponding sentiment score. There are two versions of the dataset, with either binary labels, “negative” or “positive”, (SST2) or five labels, representing finegrained sentiments (SST5). As shown in Table 5, our results are in line with previous work, confirming the benefits of using latent syntactic parse trees instead of the predefined syntax.
We noticed that all models trained on NLI or sentiment analysis tasks have parsing policies with relatively high entropy. This indicates that the algorithm does not prefer any specific grammar. Indeed, generated trees are very similar to balanced ones. This result is in line with shi1808 where they observe that binary balanced tree encoder gets the best results on most classification tasks.
We also compare with stateoftheart sequencebased models. For the most part, these models are pretrained on larger datasets and finetuned on these tasks. Nonetheless, they outperform recursive models by a significant margin. Performance on these datasets is more impacted by pretraining than by learning the syntax. It would be interesting to see if a similar pretraining would also improve the performance of recursive models with latent tree learning.
6 Conclusion
In this paper, we have introduced a novel model for learning latent tree parsers. Our approach relies on a separation between syntax and semantics. This allows dedicated optimisation schemes for each module. In particular, we found that it is important to have an unbiased estimator of the parser gradients and to allow multiple gradient steps with PPO. When tested on a CFG, our learned parser generalises to sequences of any length and distinguishes grammatical from ungrammatical expressions by forming meaningful representations for wellformed expressions. For natural language tasks, instead, the model prefers to fall back to trivial strategies, in line with what was previously observed by shi1808. Additionally, our approach performs competitively on several real natural language tasks. In the future, we would like to explore further relaxationbased techniques for learning the parser, such as REBAR
Tucker et al. (2017) or ReLAX Grathwohl et al. (2017). Finally, we plan to look into applying recursive approaches to language modelling as a pretraining step and measure if it has the same impact on downstream tasks as sequential models.Acknowledgments
We would like to thank Alexander Koller, Ivan Titov, Wilker Aziz and anonymous reviewers for their helpful suggestions and comments.
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