Recent works have highlighted optimization difficulties faced by gradient descent in training the first and last layers of transformer-based language models, which are overcome by optimizers such as Adam. These works suggest that the difficulty is linked to the heavy-tailed distribution of words in text data, where the frequency of the kth most frequent word πk is proportional to 1/k, following Zipf's law. To better understand the impact of the data distribution on training performance, we study a linear bigram model for next-token prediction when the tokens follow a power law πk 1/kα parameterized by the exponent α > 0. We derive optimization scaling laws for deterministic gradient descent and sign descent as a proxy for Adam as a function of the exponent α. Existing theoretical investigations in scaling laws assume that the eigenvalues of the data decay as a power law with exponent α > 1. This assumption effectively makes the problem "finite dimensional" as most of the loss comes from a few of the largest eigencomponents. In comparison, we show that the problem is more difficult when the data have heavier tails. The case α = 1 as found in language is "worst-case" for gradient descent, in that the number of iterations required to reach a small relative error scales almost linearly with dimension. While the performance of sign descent also depends on the dimension, for Zipf-distributed data the number of iterations scales only with the square-root of the dimension, leading to a large improvement for large vocabularies.

Large language models are trained with tokenizers, and the resulting token distribution is highly imbalanced: a few words dominate the stream while most occur rarely. Recent practice favors ever-larger vocabularies, but it is unclear where the benefit comes from. To this end, we perform a controlled study that scales the vocabulary of the language model from 24K to 196K while holding data, computation, and optimization unchanged. We begin by quantifying the complexity of tokenized text - formalized via Kolmogorov complexity - and show that larger vocabularies reduce this complexity. Above 24K, every common word is already tokenized as a single token, so enlarging vocabulary only deepens the relative token-frequency imbalance. Word-level loss decomposition shows that larger vocabularies reduce cross-entropy loss almost exclusively by lowering uncertainty on the 2,500 most frequent words, even though loss on the rare tail rises. Same frequent words cover roughly 75%of tokens in downstream benchmarks, this training advantage transfers intact. We further show that enlarging model parameters with a fixed vocabulary yields the same frequent-word benefit. Our results recast "bigger vocabularies help" as "lowering complexity of tokenized text helps," offering a simple, principled knob for tokenizer-model co-design and clarifying the loss dynamics that govern language model scaling in pre-training.

The number of tokens it takes to encode parallel text in different languages is known to vary. These disparities are called token premiums. Having high token premiums leads to less throughput during training and increases costs at inference. In this paper, we show that even after controlling for dataset size, vocabulary size, and data content, monolingual tokenizers exhibit a wide range of token premiums across languages. To understand the cross-linguistic differences that cause these token premiums, we train a suite of approximately 7,000 comparable monolingual tokenizers for 97 languages, manipulating tokenization algorithm, vocabulary size, and dataset size.

We propose SCONE (Scalable, Contextualized, Offloaded, N-gram Embedding), a new method for extending input embedding layers to enhance language model performance. To avoid increased decoding costs, SCONE retains the original vocabulary while introducing embeddings for a set of frequent n-grams. These embeddings provide contextualized representation for each input token and are learned with a separate model during training. After training, embeddings are precomputed and stored in off-accelerator memory; during inference, querying them has minimal impact on latency due to the low complexity of embedding lookups. SCONE enables two new scaling strategies: increasing the number of n-gram embeddings and scaling the model used to learn them, both while maintaining fixed accelerator usage during inference (in terms of FLOPS and memory). We show that scaling both aspects enables a model with 1B accelerator-resident parameters to outperform a 1.9B-parameter baseline across diverse corpora, while using only about half the FLOPS and accelerator memory during inference.

Research on scaling large language models (LLMs) has primarily focused on model parameters and training data size, overlooking the role of vocabulary size. We investigate how vocabulary size impacts LLM scaling laws by training models ranging from 33M to 3B parameters on up to 500B characters with various vocabulary configurations. We propose three complementary approaches for predicting the compute-optimal vocabulary size: IsoFLOPs analysis, derivative estimation, and parametric fit of the loss function. Our approaches converge on the conclusion that the optimal vocabulary size depends on the compute budget, with larger models requiring larger vocabularies.

Property prediction on molecular graphs is an important application of Graph Neural Networks (GNNs).

Research on scaling large language models (LLMs) has primarily focused on model parameters and training data size, overlooking the role of vocabulary size.

Neural Information Processing Systems http://nips.cc/