Compound Probabilistic Context-Free Grammars for Grammar Induction

Abstract: We study a formalization of the grammar induction problem that models sentences as being generated by a compound probabilistic context-free grammar. In contrast to traditional formulations which learn a single stochastic grammar, our context-free rule probabilities are modulated by a per-sentence continuous latent variable, which induces marginal dependencies beyond the traditional context-free assumptions. Inference in this grammar is performed by collapsed variational inference, in which an amortized variational posterior is placed on the continuous variable, and the latent trees are marginalized with dynamic programming. Experiments on English and Chinese show the effectiveness of our approach compared to recent state-of-the-art methods for grammar induction.

 Grammar induction is the task of inducing hierarchical syntactic structure from data. Statistical approaches to grammar induction require specifying a probabilistic grammar (e.g. formalism, number and shape of rules), and fitting its parameters through optimization. Early work found that it was difficult to induce probabilistic context-free grammars (PCFG) from natural language data through direct methods, such as optimizing the log likelihood with the EM algorithm BIBREF0 , BIBREF1 . While the reasons for the failure are manifold and not completely understood, two major potential causes are the ill-behaved optimization landscape and the overly strict independence assumptions of PCFGs. More successful approaches to grammar induction have thus resorted to carefully-crafted auxiliary objectives BIBREF2 , priors or non-parametric models BIBREF3 , BIBREF4 , BIBREF5 , BIBREF6 , and manually-engineered features BIBREF7 , BIBREF8 to encourage the desired structures to emerge.
We revisit these aforementioned issues in light of advances in model parameterization and inference. First, contrary to common wisdom, we find that parameterizing a PCFG's rule probabilities with neural networks over distributed representations makes it possible to induce linguistically meaningful grammars by simply optimizing log likelihood. While the optimization problem remains non-convex, recent work suggests that there are optimization benefits afforded by over-parameterized models BIBREF9 , BIBREF10 , BIBREF11 , and we indeed find that this neural PCFG is significantly easier to optimize than the traditional PCFG. Second, this factored parameterization makes it straightforward to incorporate side information into rule probabilities through a sentence-level continuous latent vector, which effectively allows different contexts in a derivation to coordinate. In this compound PCFG—continuous mixture of PCFGs—the context-free assumptions hold conditioned on the latent vector but not unconditionally, thereby obtaining longer-range dependencies within a tree-based generative process.
To utilize this approach, we need to efficiently optimize the log marginal likelihood of observed sentences. While compound PCFGs break efficient inference, if the latent vector is known the distribution over trees reduces to a standard PCFG. This property allows us to perform grammar induction using a collapsed approach where the latent trees are marginalized out exactly with dynamic programming. To handle the latent vector, we employ standard amortized inference using reparameterized samples from a variational posterior approximated from an inference network BIBREF12 , BIBREF13 .
On standard benchmarks for English and Chinese, the proposed approach is found to perform favorably against recent neural network-based approaches to grammar induction BIBREF14 , BIBREF15 , BIBREF16 , BIBREF17 .
 We consider context-free grammars (CFG) consisting of a 5-tuple INLINEFORM0 where INLINEFORM1 is the distinguished start symbol, INLINEFORM2 is a finite set of nonterminals, INLINEFORM3 is a finite set of preterminals, INLINEFORM6 is a finite set of terminal symbols, and INLINEFORM7 is a finite set of rules of the form,
 INLINEFORM0 
 A probabilistic context-free grammar (PCFG) consists of a grammar INLINEFORM0 and rule probabilities INLINEFORM1 such that INLINEFORM2 is the probability of the rule INLINEFORM3 . Letting INLINEFORM4 be the set of all parse trees of INLINEFORM5 , a PCFG defines a probability distribution over INLINEFORM6 via INLINEFORM7 where INLINEFORM8 is the set of rules used in the derivation of INLINEFORM9 . It also defines a distribution over string of terminals INLINEFORM10 via
 INLINEFORM0 
 where INLINEFORM0 , i.e. the set of trees INLINEFORM1 such that INLINEFORM2 's leaves are INLINEFORM3 . We will slightly abuse notation and use
 INLINEFORM0 
 to denote the posterior distribution over the unobserved latent trees given the observed sentence INLINEFORM0 , where INLINEFORM1 is the indicator function.
 A compound probability distribution BIBREF19 is a distribution whose parameters are themselves random variables. These distributions generalize mixture models to the continuous case, for example in factor analysis which assumes the following generative process,
 INLINEFORM0 
 Compound distributions provide the ability to model rich generative processes, but marginalizing over the latent parameter can be computationally intractable unless conjugacy can be exploited.
In this work, we study compound probabilistic context-free grammars whose distribution over trees arises from the following generative process: we first obtain rule probabilities via
 INLINEFORM0 
 where INLINEFORM0 is a prior with parameters INLINEFORM1 (spherical Gaussian in this paper), and INLINEFORM2 is a neural network that concatenates the input symbol embeddings with INLINEFORM3 and outputs the sentence-level rule probabilities INLINEFORM4 ,
 INLINEFORM0 
 where INLINEFORM0 denotes vector concatenation. Then a tree/sentence is sampled from a PCFG with rule probabilities given by INLINEFORM1 ,
 INLINEFORM0 
 This can be viewed as a continuous mixture of PCFGs, or alternatively, a Bayesian PCFG with a prior on sentence-level rule probabilities parameterized by INLINEFORM0 . Importantly, under this generative model the context-free assumptions hold conditioned on INLINEFORM3 , but they do not hold unconditionally. This is shown in Figure FIGREF3 (right) where there is a dependence path through INLINEFORM4 if it is not conditioned upon. Compound PCFGs give rise to a marginal distribution over parse trees INLINEFORM5 via
 INLINEFORM0 
 where INLINEFORM0 . The subscript in INLINEFORM1 denotes the fact that the rule probabilities depend on INLINEFORM2 . Compound PCFGs are clearly more expressive than PCFGs as each sentence has its own set of rule probabilities. However, it still assumes a tree-based generative process, making it possible to learn latent tree structures.
Our motivation for the compound PCFG is based on the observation that for grammar induction, context-free assumptions are generally made not because they represent an adequate model of natural language, but because they allow for tractable training. We can in principle model richer dependencies through vertical/horizontal Markovization BIBREF21 , BIBREF22 and lexicalization BIBREF23 . However such dependencies complicate training due to the rapid increase in the number of rules. Under this view, we can interpret the compound PCFG as a restricted version of some lexicalized, higher-order PCFG where a child can depend on structural and lexical context through a shared latent vector. We hypothesize that this dependence among siblings is especially useful in grammar induction from words, where (for example) if we know that watched is used as a verb then the noun phrase is likely to be a movie.
In contrast to the usual Bayesian treatment of PCFGs which places priors on global rule probabilities BIBREF3 , BIBREF4 , BIBREF6 , the compound PCFG assumes a prior on local, sentence-level rule probabilities. It is therefore closely related to the Bayesian grammars studied by BIBREF25 and BIBREF26 , who also sample local rule probabilities from a logistic normal prior for training dependency models with valence (DMV) BIBREF27 .
 Table TABREF23 shows the unlabeled INLINEFORM0 scores for our models and various baselines. All models soundly outperform right branching baselines, and we find that the neural PCFG/compound PCFG are strong models for grammar induction. In particular the compound PCFG outperforms other models by an appreciable margin on both English and Chinese. We again note that we were unable to induce meaningful grammars through a traditional PCFG with the scalar parameterization despite a thorough hyperparameter search. See lab:full for the full results (including corpus-level INLINEFORM1 ) broken down by sentence length.
Table TABREF27 analyzes the learned tree structures. We compare similarity as measured by INLINEFORM0 against gold, left, right, and “self" trees (top), where self INLINEFORM1 score is calculated by averaging over all 6 pairs obtained from 4 different runs. We find that PRPN is particularly consistent across multiple runs. We also observe that different models are better at identifying different constituent labels, as measured by label recall (Table TABREF27 , bottom). While left as future work, this naturally suggests an ensemble approach wherein the empirical probabilities of constituents (obtained by averaging the predicted binary constituent labels from the different models) are used either to supervise another model or directly as potentials in a CRF constituency parser. Finally, all models seemed to have some difficulty in identifying SBAR/VP constituents which typically span more words than NP constituents.
 Grammar induction has a long and rich history in natural language processing. Early work on grammar induction with pure unsupervised learning was mostly negative BIBREF0 , BIBREF1 , BIBREF74 , though BIBREF75 reported some success on partially bracketed data. BIBREF76 and BIBREF2 were some of the first successful statistical approaches to grammar induction. In particular, the constituent-context model (CCM) of BIBREF2 , which explicitly models both constituents and distituents, was the basis for much subsequent work BIBREF27 , BIBREF7 , BIBREF8 . Other works have explored imposing inductive biases through Bayesian priors BIBREF4 , BIBREF5 , BIBREF6 , modified objectives BIBREF42 , and additional constraints on recursion depth BIBREF77 , BIBREF48 .
While the framework of specifying the structure of a grammar and learning the parameters is common, other methods exist. BIBREF43 consider a nonparametric-style approach to unsupervised parsing by using random subsets of training subtrees to parse new sentences. BIBREF46 utilize an incremental algorithm to unsupervised parsing which makes local decisions to create constituents based on a complex set of heuristics. BIBREF47 induce parse trees through cascaded applications of finite state models.
More recently, neural network-based approaches to grammar induction have shown promising results on inducing parse trees directly from words. BIBREF14 , BIBREF15 learn tree structures through soft gating layers within neural language models, while BIBREF16 combine recursive autoencoders with the inside-outside algorithm. BIBREF17 train unsupervised recurrent neural network grammars with a structured inference network to induce latent trees, and BIBREF78 utilize image captions to identify and ground constituents.
Our work is also related to latent variable PCFGs BIBREF79 , BIBREF80 , BIBREF81 , which extend PCFGs to the latent variable setting by splitting nonterminal symbols into latent subsymbols. In particular, latent vector grammars BIBREF82 and compositional vector grammars BIBREF83 also employ continuous vectors within their grammars. However these approaches have been employed for learning supervised parsers on annotated treebanks, in contrast to the unsupervised setting of the current work.
 This work explores grammar induction with compound PCFGs, which modulate rule probabilities with per-sentence continuous latent vectors. The latent vector induces marginal dependencies beyond the traditional first-order context-free assumptions within a tree-based generative process, leading to improved performance. The collapsed amortized variational inference approach is general and can be used for generative models which admit tractable inference through partial conditioning. Learning deep generative models which exhibit such conditional Markov properties is an interesting direction for future work.
 We thank Phil Blunsom for initial discussions which seeded many of the core ideas in the present work. We also thank Yonatan Belinkov and Shay Cohen for helpful feedback, and Andrew Drozdov for providing the parsed dataset from their DIORA model. YK is supported by a Google Fellowship. AMR acknowledges the support of NSF 1704834, 1845664, AWS, and Oracle.
We associate an input embedding INLINEFORM0 for each symbol INLINEFORM1 on the left side of a rule (i.e. INLINEFORM2 ) and run a neural network over INLINEFORM3 to obtain the rule probabilities. Concretely, each rule type INLINEFORM4 is parameterized as follows, INLINEFORM5 
 where INLINEFORM0 is the product space INLINEFORM1 , and INLINEFORM2 are MLPs with two residual layers, INLINEFORM3 
 The bias terms for the above expressions (including for the rule probabilities) are omitted for notational brevity. In Figure FIGREF3 we use the following to refer to rule probabilities of different rule types, INLINEFORM0 
 where INLINEFORM0 denotes the set of rules with INLINEFORM1 on the left hand side.
The compound PCFG rule probabilities INLINEFORM0 given a latent vector INLINEFORM1 , INLINEFORM2 
 Again the bias terms are omitted for brevity, and INLINEFORM0 are as before where the first layer's input dimensions are appropriately changed to account for concatenation with INLINEFORM1 .
For completeness we show the corpus-level and sentence-level INLINEFORM0 broken down by sentence length in Table TABREF44 , averaged across 4 different runs of each model.
For experiments on supervising RNNGs with induced trees, we use the parameterization and hyperparameters from BIBREF17 , which uses a 2-layer 650-dimensional stack LSTM (with dropout of 0.5) and a 650-dimensional tree LSTM BIBREF88 , BIBREF90 as the composition function.
Concretely, the generative story is as follows: first, the stack representation is used to predict the next action (shift or reduce) via an affine transformation followed by a sigmoid. If shift is chosen, we obtain a distribution over the vocabulary via another affine transformation over the stack representation followed by a softmax. Then we sample the next word from this distribution and shift the generated word onto the stack using the stack LSTM. If reduce is chosen, we pop the last two elements off the stack and use the tree LSTM to obtain a new representation. This new representation is shifted onto the stack via the stack LSTM. Note that this RNNG parameterization is slightly different than the original from BIBREF53 , which does not ignore constituent labels and utilizes a bidirectional LSTM as the composition function instead of a tree LSTM. As our RNNG parameterization only works with binary trees, we binarize the gold trees with right binarization for the RNNG trained on gold trees (trees from the unsupervised methods explored in this paper are already binary). The RNNG also trains a discriminative parser alongside the generative model for evaluation with importance sampling. We use a CRF parser whose span score parameterization is similar similar to recent works BIBREF89 , BIBREF87 , BIBREF85 : position embeddings are added to word embeddings, and a bidirectional LSTM with 256 hidden dimensions is run over the input representations to obtain the forward and backward hidden states. The score INLINEFORM0 for a constituent spanning the INLINEFORM1 -th and INLINEFORM2 -th word is given by,
 INLINEFORM0 
where the MLP has a single hidden layer with INLINEFORM0 nonlinearity followed by layer normalization BIBREF84 .
For experiments on fine-tuning the RNNG with the unsupervised RNNG, we take the discriminative parser (which is also pretrained alongside the RNNG on induced trees) to be the structured inference network for optimizing the evidence lower bound. We refer the reader to BIBREF17 and their open source implementation for additional details. We also observe that as noted by BIBREF17 , a URNNG trained from scratch on this version of PTB without punctuation failed to outperform a right-branching baseline.
The LSTM language model baseline is the same size as the stack LSTM (i.e. 2 layers, 650 hidden units, dropout of 0.5), and is therefore equivalent to an RNNG with completely right branching trees. The PRPN/ON baselines for perplexity/syntactic evaluation in Table TABREF30 also have 2 layers with 650 hidden units and 0.5 dropout. Therefore all models considered in Table TABREF30 have roughly the same capacity. For all models we share input/output word embeddings BIBREF86 . Perplexity estimation for the RNNGs and the compound PCFG uses 1000 importance-weighted samples.
For grammaticality judgment, we modify the publicly available dataset from BIBREF56 to only keep sentence pairs that did not have any unknown words with respect to our PTB vocabulary of 10K words. This results in 33K sentence pairs for evaluation.
Figure FIGREF50 shows the part-of-speech alignments and Table TABREF46 shows the nonterminal label alignments for the compound PCFG/neural PCFG.
Table TABREF53 lists more examples of constituents within each subtree as the top principical component is varied. Due to data sparsity, the subtree analysis is performed on the full dataset. See section UID36 for more details.

Figure and table captions:
Figure 1: A graphical model-like diagram for the neural PCFG (left) and the compound PCFG (right) for an example tree structure. In the above, A1, A2 ∈ N are nonterminals, T1, T2, T3 ∈ P are preterminals, w1, w2, w3 ∈ Σ are terminals. In the neural PCFG, the global rule probabilities π = πS ∪πN ∪πP are the output from a neural net run over the symbol embeddings EG , where πN are the set of rules with a nonterminal on the left hand side (πS and πP are similarly defined). In the compound PCFG, we have per-sentence rule probabilities πz = πz,S ∪ πz,N ∪ πz,P obtained from running a neural net over a random vector z (which varies across sentences) and global symbol embeddings EG . In this case, the context-free assumptions hold conditioned on z, but they do not hold unconditionally: e.g. when conditioned on z and A2, the variables A1 and T1 are independent; however when conditioned on just A2, they are not independent due to the dependence path through z. Note that the rule probabilities are random variables in the compound PCFG but deterministic variables in the neural PCFG.
Table 1: Unlabeled sentence-level F1 scores on PTB and CTB test sets. Top shows results from previous work while the rest of the results are from this paper. Mean/Max scores are obtained from 4 runs of each model with different random seeds. Oracle is the maximum score obtainable with binarized trees, since we compare against the non-binarized gold trees per convention. Results with † are trained on a version of PTB with punctuation, and hence not strictly comparable to the present work. For URNNG/DIORA, we take the parsed test set provided by the authors from their best runs and evaluate F1 with our evaluation setup, which ignores punctuation. ough hyperparameter search.13 See appendix A.2 for the full results (including corpus-level F1) broken down by sentence length.
Table 2: (Top) Mean F1 similarity against Gold, Left, Right, and Self trees. Self F1 score is calculated by averaging over all 6 pairs obtained from 4 different runs. (Bottom) Fraction of ground truth constituents that were predicted as a constituent by the models broken down by label (i.e. label recall).
Table 3: Results from training RNNGs on induced trees from various models (Induced RNNG) on the PTB. Induced URNNG indicates fine-tuning with the URNNG objective. We show perplexity (PPL), grammaticality judgment performance (Syntactic Eval.), and unlabeled F1. PPL/F1 are calculated on the PTB test set and Syntactic Eval. is from Marvin and Linzen (2018)’s dataset. Results on top do not make any use of annotated trees, while the bottom two results are trained on binarized gold trees. The perplexity numbers here are not comparable to standard results on the PTB since our models are generative model of sentences and hence we do not carry information across sentence boundaries. Also note that all the RNN-based models above (i.e. LSTM/PRPN/ON/RNNG/URNNG) have roughly the same model capacity (see appendix A.3).
Figure 2: Alignment of induced nonterminals ordered from top based on predicted frequency (therefore NT-04 is the most frequently-predicted nonterminal). For each nonterminal we visualize the proportion of correctly-predicted constituents that correspond to particular gold labels. For reference we also show the precision (i.e. probability of correctly predicting unlabeled constituents) in the rightmost column.
Table 4: For each query sentence (bold), we show the 5 nearest neighbors based on cosine similarity, where we take the representation for each sentence to be the mean of the variational posterior.
Table 5: For each subtree, we perform PCA on the variational posterior mean vectors that are associated with that particular subtree and take the top principal component. We then list the top 5 constituents that had the lowest (PC -) and highest (PC +) principal component values.
Table 6: Average unlabeled F1 for the various models broken down by sentence length on the PTB test set. For example WSJ-10 refers to F1 calculated on the subset of the test set where the maximum sentence length is at most 10. Scores are averaged across 4 runs of the model with different random seeds. Oracle is the performance of binarized gold trees (with right branching binarization). Top shows sentence-level F1 and bottom shows corpus-level F1.
Figure 3: Preterminal alignment to part-of-speech tags for the compound PCFG (top) and the neural PCFG (bottom).
Table 7: Analysis of label alignment for nonterminals in the compound PCFG (top) and the neural PCFG (bottom). Label alignment is the proportion of correctly-predicted constistuents that correspond to a particular gold label. We also show the predicted constituent frequency and accuracy (i.e. precision) on the right. Bottom line shows the frequency in the gold trees.
Table 8: For each subtree (shown at the top of each set of examples), we perform PCA on the variational posterior mean vectors that are associated with that particular subtree and take the top principal component. We then list the top 5 constituents that had the lowest (left) and highest (right) principal component values.

Questions about this paper:
what were the evaluation metrics?

Answers:
Unlabeled sentence-level F1, perplexity, grammatically judgment performance