Formal Aspects of Language Modeling

Large language models have become one of the most commonly deployed NLP inventions. In the past half-decade, their integration into core natural language processing tools has dramatically increased the performance of such tools, and they have entered the public discourse surrounding artificial intelligence. Consequently, it is important for both developers and researchers alike to understand the mathematical foundations of large language models, as well as how to implement them. These notes are the accompaniment to the theoretical portion of the ETH Z\"urich course on large language models, covering what constitutes a language model from a formal, theoretical perspective.

Efficient Algorithms for Recognizing Weighted Tree-Adjoining Languages

The class of tree-adjoining languages can be characterized by various two-level formalisms, consisting of a context-free grammar (CFG) or pushdown automaton (PDA) controlling another CFG or PDA. These four formalisms are equivalent to tree-adjoining grammars (TAG), linear indexed grammars (LIG), pushdown-adjoining automata (PAA), and embedded pushdown automata (EPDA). We define semiring-weighted versions of the above two-level formalisms, and we design new algorithms for computing their stringsums (the weight of all derivations of a string) and allsums (the weight of all derivations). From these, we also immediately obtain stringsum and allsum algorithms for TAG, LIG, PAA, and EPDA. For LIG, our algorithm is more time-efficient by a factor of $\mathcal{O}(n|\mathcal{N}|)$ (where $n$ is the string length and $|\mathcal{N}|$ is the size of the nonterminal set) and more space-efficient by a factor of $\mathcal{O}(|\Gamma|)$ (where $|\Gamma|$ is the size of the stack alphabet) than the algorithm of Vijay-Shanker and Weir (1989). For EPDA, our algorithm is both more space-efficient and time-efficient than the algorithm of Alonso et al. (2001) by factors of $\mathcal{O}(|\Gamma|^2)$ and $\mathcal{O}(|\Gamma|^3)$, respectively. Finally, we give the first PAA stringsum and allsum algorithms.

On the Representational Capacity of Recurrent Neural Language Models

This work investigates the computational expressivity of language models (LMs) based on recurrent neural networks (RNNs). Siegelmann and Sontag (1992) famously showed that RNNs with rational weights and hidden states and unbounded computation time are Turing complete. However, LMs define weightings over strings in addition to just (unweighted) language membership and the analysis of the computational power of RNN LMs (RLMs) should reflect this. We extend the Turing completeness result to the probabilistic case, showing how a rationally weighted RLM with unbounded computation time can simulate any probabilistic Turing machine (PTM). Since, in practice, RLMs work in real-time, processing a symbol at every time step, we treat the above result as an upper bound on the expressivity of RLMs. We also provide a lower bound by showing that under the restriction to real-time computation, such models can simulate deterministic real-time rational PTMs.

Recurrent Neural Language Models as Probabilistic Finite-state Automata

Studying language models (LMs) in terms of well-understood formalisms allows us to precisely characterize their abilities and limitations. Previous work has investigated the representational capacity of recurrent neural network (RNN) LMs in terms of their capacity to recognize unweighted formal languages. However, LMs do not describe unweighted formal languages -- rather, they define probability distributions over strings. In this work, we study what classes of such probability distributions RNN LMs can represent, which allows us to make more direct statements about their capabilities. We show that simple RNNs are equivalent to a subclass of probabilistic finite-state automata, and can thus model a strict subset of probability distributions expressible by finite-state models. Furthermore, we study the space complexity of representing finite-state LMs with RNNs. We show that, to represent an arbitrary deterministic finite-state LM with $N$ states over an alphabet $\Sigma$, an RNN requires $\Omega\left(N |\Sigma|\right)$ neurons. These results present a first step towards characterizing the classes of distributions RNN LMs can represent and thus help us understand their capabilities and limitations.

An Analysis of On-the-fly Determinization of Finite-state Automata

In this paper we establish an abstraction of on-the-fly determinization of finite-state automata using transition monoids and demonstrate how it can be applied to bound the asymptotics. We present algebraic and combinatorial properties that are sufficient for a polynomial state complexity of the deterministic automaton constructed on-the-fly. A special case of our findings is that automata with many non-deterministic transitions almost always admit a determinization of polynomial complexity. Furthermore, we extend our ideas to weighted finite-state automata.

A Geometric Notion of Causal Probing

Large language models rely on real-valued representations of text to make their predictions. These representations contain information learned from the data that the model has trained on, including knowledge of linguistic properties and forms of demographic bias, e.g., based on gender. A growing body of work has considered removing information about concepts such as these using orthogonal projections onto subspaces of the representation space. We contribute to this body of work by proposing a formal definition of $\textit{intrinsic}$ information in a subspace of a language model's representation space. We propose a counterfactual approach that avoids the failure mode of spurious correlations (Kumar et al., 2022) by treating components in the subspace and its orthogonal complement independently. We show that our counterfactual notion of information in a subspace is optimized by a $\textit{causal}$ concept subspace. Furthermore, this intervention allows us to attempt concept controlled generation by manipulating the value of the conceptual component of a representation. Empirically, we find that R-LACE (Ravfogel et al., 2022) returns a one-dimensional subspace containing roughly half of total concept information under our framework. Our causal controlled intervention shows that, for at least one model, the subspace returned by R-LACE can be used to manipulate the concept value of the generated word with precision.

Testing the Predictions of Surprisal Theory in 11 Languages

A fundamental result in psycholinguistics is that less predictable words take a longer time to process. One theoretical explanation for this finding is Surprisal Theory (Hale, 2001; Levy, 2008), which quantifies a word's predictability as its surprisal, i.e. its negative log-probability given a context. While evidence supporting the predictions of Surprisal Theory have been replicated widely, most have focused on a very narrow slice of data: native English speakers reading English texts. Indeed, no comprehensive multilingual analysis exists. We address this gap in the current literature by investigating the relationship between surprisal and reading times in eleven different languages, distributed across five language families. Deriving estimates from language models trained on monolingual and multilingual corpora, we test three predictions associated with surprisal theory: (i) whether surprisal is predictive of reading times; (ii) whether expected surprisal, i.e. contextual entropy, is predictive of reading times; (iii) and whether the linking function between surprisal and reading times is linear. We find that all three predictions are borne out crosslinguistically. By focusing on a more diverse set of languages, we argue that these results offer the most robust link to-date between information theory and incremental language processing across languages.

On the Efficacy of Sampling Adapters

Sampling is a common strategy for generating text from probabilistic models, yet standard ancestral sampling often results in text that is incoherent or ungrammatical. To alleviate this issue, various modifications to a model's sampling distribution, such as nucleus or top-k sampling, have been introduced and are now ubiquitously used in language generation systems. We propose a unified framework for understanding these techniques, which we term sampling adapters. Sampling adapters often lead to qualitatively better text, which raises the question: From a formal perspective, how are they changing the (sub)word-level distributions of language generation models? And why do these local changes lead to higher-quality text? We argue that the shift they enforce can be viewed as a trade-off between precision and recall: while the model loses its ability to produce certain strings, its precision rate on desirable text increases. While this trade-off is not reflected in standard metrics of distribution quality (such as perplexity), we find that several precision-emphasizing measures indeed indicate that sampling adapters can lead to probability distributions more aligned with the true distribution. Further, these measures correlate with higher sequence-level quality scores, specifically, Mauve.

Efficient Semiring-Weighted Earley Parsing

This paper provides a reference description, in the form of a deduction system, of Earley's (1970) context-free parsing algorithm with various speed-ups. Our presentation includes a known worst-case runtime improvement from Earley's $O (N^3|G||R|)$, which is unworkable for the large grammars that arise in natural language processing, to $O (N^3|G|)$, which matches the runtime of CKY on a binarized version of the grammar $G$. Here $N$ is the length of the sentence, $|R|$ is the number of productions in $G$, and $|G|$ is the total length of those productions. We also provide a version that achieves runtime of $O (N^3|M|)$ with $|M| \leq |G|$ when the grammar is represented compactly as a single finite-state automaton $M$ (this is partly novel). We carefully treat the generalization to semiring-weighted deduction, preprocessing the grammar like Stolcke (1995) to eliminate deduction cycles, and further generalize Stolcke's method to compute the weights of sentence prefixes. We also provide implementation details for efficient execution, ensuring that on a preprocessed grammar, the semiring-weighted versions of our methods have the same asymptotic runtime and space requirements as the unweighted methods, including sub-cubic runtime on some grammars.

Generalizing Backpropagation for Gradient-Based Interpretability

Many popular feature-attribution methods for interpreting deep neural networks rely on computing the gradients of a model's output with respect to its inputs. While these methods can indicate which input features may be important for the model's prediction, they reveal little about the inner workings of the model itself. In this paper, we observe that the gradient computation of a model is a special case of a more general formulation using semirings. This observation allows us to generalize the backpropagation algorithm to efficiently compute other interpretable statistics about the gradient graph of a neural network, such as the highest-weighted path and entropy. We implement this generalized algorithm, evaluate it on synthetic datasets to better understand the statistics it computes, and apply it to study BERT's behavior on the subject-verb number agreement task (SVA). With this method, we (a) validate that the amount of gradient flow through a component of a model reflects its importance to a prediction and (b) for SVA, identify which pathways of the self-attention mechanism are most important.