Syntax and Geometry of Information (2023)

Abstract: This paper presents an information-theoretical model of syntactic generalization. We study syntactic generalization from the perspective of the capacity to disentangle semantic and structural information, emulating the human capacity to assign a grammaticality judgment to semantically nonsensical sentences. In order to isolate the structure, we propose to represent the probability distribution behind a corpus as the product of the probability of a semantic context and the probability of a structure, the latter being independent of the former. We further elaborate the notion of abstraction as a relaxation of the property of independence. It is based on the measure of structural and contextual information for a given representation. We test abstraction as an optimization objective on the task of inducing syntactic categories from natural language data and show that it significantly outperforms alternative methods. Furthermore, we find that when syntax-unaware optimization objectives succeed in the task, their success is mainly due to an implicit disentanglement process rather than to the model structure. On the other hand, syntactic categories can be deduced in a principled way from the independence between structure and context.

## introduction in the context of both human learning and statistical machine learning, what distinguishes generalization from memorization is the process of deliberately ignoring a part of the input information. in machine learning, the generalization will be guided towards some specific direction by the learning hypothesis: the model structure and the choice of regularization impacts the nature of the information loss. we can talk about syntactic generalization from textual data when the information pertaining to sentence structure tend to be preserved by the model and the information pertaining to other aspects of the text tend to be ignored. syntactic generalization is of great interest because the human capacity to assign an abstract structure to utterances is a prerequisite to creatively combine constituents and understand novel sentences @xcite . knowledge of syntax can boost the robustness of nlp applications with respect to unseen data, in particular when there is a distribution shift @xcite . in a broader perspective, understanding syntactic generalization informs the discussion on the learnability of syntax from unlabelled text without any built-in grammatical knowledge or inductive bias @xcite @xcite . finally, studying syntactic generalization in large language models (llms) sheds light on whether and to what extent these models emulate human functioning with respect to linguistic competence. the prevailing formalization of syntax is by means of algebraic compositional rules operating on a finite set of discrete categories (parts of speech). language models can acquire syntactic knowledge when they are given specific supervision or bias @xcite @xcite . whether unsupervised settings can lead to syntactic generalization and under which conditions is still unknown. current llms use distributed representations that cannot be unequivocally mapped to a set of categories, let alone syntactically meaningful categories. the question whether their representations encode syntactic information and how to uncover it is actively investigated today @xcite . the majority of works in the topic of syntactic generalization in language models adopt an empirical approach, such as probing or analysis of a model with a comparison to actual or expected human performance on linguistically motivated tasks. in contrast, we present a theoretical approach to formalize syntactic generalization in an information theoretical framework. statistical learning can be formulated as the minimization of kl-divergence -a measure of information loss -subject to constraints. the constraints on model expressivity ensure that generalization takes place by eliminating the information resulting from sampling noise. we claim that the training objective of maximum likelihood estimation by nature does not incentivize models to syntactic generalization. in the case of syntactic generalization, the information loss needs to be directed to non-structural information, which is only remotely related to the elimination of sampling noise. first, a corpus is not randomly sampled from the set of grammatical sentences. word co-occurrences in a corpus are indeed influenced by different factors such as semantics and pragmatics. the process of abstracting away from these factors is arguably different from the concept of generalization in machine learning, as the acquisition of syntactic knowledge always involves a shift of distribution @xcite . second, the target of generalization is the capacity to recognize the set of grammatical sentences: well-formedness is inherently a binary notion rather than a probabilistic one. these considerations motivate our proposition to decompose a corpus distribution as a factor of semantic/pragmatic context and a factor of structure representing well-formedness. in what follows, we reinterpret syntactic generalization based on the separation of structural and semantic information, and we show that our approach outperforms concurrent methods on unsupervised pos induction. we also define the notion of abstraction, an optimization objective specifically conceived for disentangling semantic information and syntactic well-formedness. ## geometry of information using information theoretical tools, we transform the criteria above into metrics and define an information space which allows to track the amount of contextual and structural information in a partition, as well as the direction of generalization during a training process. the concept of minimal factorization provides the formal definition of minimal syntax; however, the conditions of factorization (definition 1) and minimality (definition 2) are restrictive. in natural language corpora, a perfect independence between semantic context and grammaticality cannot be expected. syntax and semantics do interface in natural language, semantic acceptability interacts with grammaticality and depending on how one deals with this interface, either the assumption of perfect independence or the precise retrieval of the distribution underlying the corpus may not be met. this motivates our methodology for relaxing both conditions in a way that gives an equivalent but quantifiable formulation for each criterion in terms of information. we thus provide a method to measure the amount of structural information present in a partition, hence relaxing the factorization criterion. we also define contextual information, which relaxes the minimality requirement. ## abstraction abstraction relaxes the definition of a minimal factorization of p in terms of a solution to an optimization problem. for a given probability distribution p and a mapping f , the abstraction measures the distance between f and the position of a minimal factorization of p in the information space: one has α d (p ∥ f ) ≤ 1, with the maximum value reached iff f is a minimal factorization of p. for a mapping f , maximizing abstraction can be considered as a relaxation of the property of being a minimal factorization. ## experiments we test abstraction as an optimization objective for learning syntactic representations, when the representation takes the form of a mapping into discrete syntactic categories. the results are evaluated on an unsupervised pos induction task. while our understanding of a syntactic category may not perfectly overlap with actual parts of speech (the latter being defined on the basis of a mixture of criteria instead of pure syntax, and are usually more coarse-grained than real distributional categories), this task will allow a good comparison with concurrent models on a gold standard. in nlp, part-of-speech categories are usually a part of a probabilistic model; typically a parameter which will be tuned during learning. for instance, if the model is an hmm, its hidden states correspond to pos categories. if the model is a pcfg, categories will correspond to non-terminals. we call this approach -when pos categories are deduced from a given model structure as a parameter -the model-specific approach. in the experiments, we compare the model-specific approach with our hypothesis: that pos categories can be deduced from the independence of structure and context. we consider the task of unsupervised pos induction, and compare the accuracy of the abstraction maximization criterion with model-specific crossentropy minimization. the corpus we use comes from wikipedia in simplified english, contains 430k sentences, 8m tokens, and was pos tagged by the stanford pos tagger @xcite . to create the target partition, words (a vocabulary of 6044 elements) were assigned to their most frequent pos. there are 36 pos categories. ## conclusion as to our current knowledge, language models do not have a convincing performance on modelling grammaticality: despite their impressive results on downstream tasks, they are not good at syntactic generalization unless syntactic knowledge is somehow injected in the system. moreover, there is a trade-off in large lms between syntactic generalization and language modelling performance. we suggest a measurable interpretation of syntactic generalization and show results that align with the observations reported by many authors: training on a natural language corpus (e.g. using language models) results in memorization of semantics and entanglement with syntactic information. this motivates our proposition of abstraction, a new training objective for syntactic generalization without supervision. we prove the statistical consistency of abstraction in the task of grammar identification. empirical results on an unsupervised pos induction task show that abstraction considerably outperforms concurrent models trained with a likelihood estimation objective, without making any assumptions about the structure of the model.