Towards a Similarity-adjusted Surprisal Theory

Surprisal theory posits that the cognitive effort required to comprehend a word is determined by its contextual predictability, quantified as surprisal. Traditionally, surprisal theory treats words as distinct entities, overlooking any potential similarity between them. Giulianelli et al. (2023) address this limitation by introducing information value, a measure of predictability designed to account for similarities between communicative units. Our work leverages Ricotta and Szeidl's (2006) diversity index to extend surprisal into a metric that we term similarity-adjusted surprisal, exposing a mathematical relationship between surprisal and information value. Similarity-adjusted surprisal aligns with information value when considering graded similarities and reduces to standard surprisal when words are treated as distinct. Experimental results with reading time data indicate that similarity-adjusted surprisal adds predictive power beyond standard surprisal for certain datasets, suggesting it serves as a complementary measure of comprehension effort.

How to Compute the Probability of a Word

Language models (LMs) estimate the probability distribution over sequences of natural language; these distributions are crucial for computing perplexity and surprisal in linguistics research. While we are usually concerned with measuring these values for words, most LMs operate over subwords. Despite seemingly straightforward, accurately computing probabilities over one unit given probabilities over the other requires care. Indeed, we show here that many recent linguistic studies have been incorrectly computing these values. This paper derives the correct methods for computing word probabilities, highlighting issues when relying on language models that use beginning-of-word (bow)-marking tokenisers, e.g., the GPT family. Empirically, we show that correcting the widespread bug in probability computations affects measured outcomes in sentence comprehension and lexical optimisation analyses.

Causal Estimation of Memorisation Profiles

Understanding memorisation in language models has practical and societal implications, e.g., studying models' training dynamics or preventing copyright infringements. Prior work defines memorisation as the causal effect of training with an instance on the model's ability to predict that instance. This definition relies on a counterfactual: the ability to observe what would have happened had the model not seen that instance. Existing methods struggle to provide computationally efficient and accurate estimates of this counterfactual. Further, they often estimate memorisation for a model architecture rather than for a specific model instance. This paper fills an important gap in the literature, proposing a new, principled, and efficient method to estimate memorisation based on the difference-in-differences design from econometrics. Using this method, we characterise a model's memorisation profile--its memorisation trends across training--by only observing its behaviour on a small set of instances throughout training. In experiments with the Pythia model suite, we find that memorisation (i) is stronger and more persistent in larger models, (ii) is determined by data order and learning rate, and (iii) has stable trends across model sizes, thus making memorisation in larger models predictable from smaller ones.

The Role of $n$-gram Smoothing in the Age of Neural Networks

For nearly three decades, language models derived from the $n$-gram assumption held the state of the art on the task. The key to their success lay in the application of various smoothing techniques that served to combat overfitting. However, when neural language models toppled $n$-gram models as the best performers, $n$-gram smoothing techniques became less relevant. Indeed, it would hardly be an understatement to suggest that the line of inquiry into $n$-gram smoothing techniques became dormant. This paper re-opens the role classical $n$-gram smoothing techniques may play in the age of neural language models. First, we draw a formal equivalence between label smoothing, a popular regularization technique for neural language models, and add-$\lambda$ smoothing. Second, we derive a generalized framework for converting \emph{any} $n$-gram smoothing technique into a regularizer compatible with neural language models. Our empirical results find that our novel regularizers are comparable to and, indeed, sometimes outperform label smoothing on language modeling and machine translation.

Revisiting the Optimality of Word Lengths

Zipf (1935) posited that wordforms are optimized to minimize utterances' communicative costs. Under the assumption that cost is given by an utterance's length, he supported this claim by showing that words' lengths are inversely correlated with their frequencies. Communicative cost, however, can be operationalized in different ways. Piantadosi et al. (2011) claim that cost should be measured as the distance between an utterance's information rate and channel capacity, which we dub the channel capacity hypothesis (CCH) here. Following this logic, they then proposed that a word's length should be proportional to the expected value of its surprisal (negative log-probability in context). In this work, we show that Piantadosi et al.'s derivation does not minimize CCH's cost, but rather a lower bound, which we term CCH-lower. We propose a novel derivation, suggesting an improved way to minimize CCH's cost. Under this method, we find that a language's word lengths should instead be proportional to the surprisal's expectation plus its variance-to-mean ratio. Experimentally, we compare these three communicative cost functions: Zipf's, CCH-lower , and CCH. Across 13 languages and several experimental settings, we find that length is better predicted by frequency than either of the other hypotheses. In fact, when surprisal's expectation, or expectation plus variance-to-mean ratio, is estimated using better language models, it leads to worse word length predictions. We take these results as evidence that Zipf's longstanding hypothesis holds.

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.

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.

Tokenization and the Noiseless Channel

Subword tokenization is a key part of many NLP pipelines. However, little is known about why some tokenizer and hyperparameter combinations lead to better downstream model performance than others. We propose that good tokenizers lead to \emph{efficient} channel usage, where the channel is the means by which some input is conveyed to the model and efficiency can be quantified in information-theoretic terms as the ratio of the Shannon entropy to the maximum possible entropy of the token distribution. Yet, an optimal encoding according to Shannon entropy assigns extremely long codes to low-frequency tokens and very short codes to high-frequency tokens. Defining efficiency in terms of R\'enyi entropy, on the other hand, penalizes distributions with either very high or very low-frequency tokens. In machine translation, we find that across multiple tokenizers, the R\'enyi entropy with $\alpha = 2.5$ has a very strong correlation with \textsc{Bleu}: $0.78$ in comparison to just $-0.32$ for compressed length.

A Formal Perspective on Byte-Pair Encoding

Byte-Pair Encoding (BPE) is a popular algorithm used for tokenizing data in NLP, despite being devised initially as a compression method. BPE appears to be a greedy algorithm at face value, but the underlying optimization problem that BPE seeks to solve has not yet been laid down. We formalize BPE as a combinatorial optimization problem. Via submodular functions, we prove that the iterative greedy version is a $\frac{1}{{\sigma(\boldsymbol{\mu}^\star)}}(1-e^{-{\sigma(\boldsymbol{\mu}^\star)}})$-approximation of an optimal merge sequence, where ${\sigma(\boldsymbol{\mu}^\star)}$ is the total backward curvature with respect to the optimal merge sequence $\boldsymbol{\mu}^\star$. Empirically the lower bound of the approximation is $\approx 0.37$. We provide a faster implementation of BPE which improves the runtime complexity from $\mathcal{O}\left(N M\right)$ to $\mathcal{O}\left(N \log M\right)$, where $N$ is the sequence length and $M$ is the merge count. Finally, we optimize the brute-force algorithm for optimal BPE using memoization.