Tokenisation via Convex Relaxations

Two dominant tokenisation algorithms are used by modern language models: byte-pair encoding (BPE) and UnigramLM. These differ along two orthogonal axes: their optimisation objective (compression vs. log-likelihood) and their search procedure (bottom-up merging vs. top-down pruning). Existing comparisons confound these axes, making it unclear whether their observed differences stem from what is being optimised vs. how it is being optimised. We disentangle the two by introducing two new tokenisation algorithms that complete this 2x2 design space: BottomUpLL, a bottom-up likelihood-based tokeniser, and TopDownComp, a top-down compression-based tokeniser. We train language models with tokenisers produced by each algorithm, varying: model size, vocabulary sizes, and domain (English-only vs. multilingual). Evaluating models on bits-per-byte, we find that the search procedure -- not the objective -- is the dominant factor: bottom-up tokenisers consistently achieve lower bits-per-byte in most settings. Evaluating models on the BLiMP task, however, shows no consistent relationship between design choice and performance. Overall, our results disentangle the effect of tokeniser design choices on language modelling performance, offering concrete guidance for their more principled construction.

Tokenization is the first -- and often least scrutinized -- step of most NLP pipelines. Standard algorithms for learning tokenizers rely on frequency-based objectives, which favor languages dominant in the training data and consequently leave lower-resource languages with tokenizations that are disproportionately longer, morphologically implausible, or even riddled with $<UNK>$ placeholders. This phenomenon ultimately amplifies computational and financial inequalities between users from different language backgrounds. To remedy this, we introduce Parity-aware Byte Pair Encoding (BPE), a variant of the widely-used BPE algorithm. At every merge step, Parity-aware BPE applies a fair-max rule that maximizes the compression gain of the currently worst-compressed language, trading a small amount of global compression for cross-lingual parity. We find empirically that Parity-aware BPE reduces tokenization inequality -- operationalized by the Gini coefficient of per-language token costs -- by up to 89% relative to Classical BPE. This comes with negligible impact on global compression rate and no evidence of systematic degradation in downstream LM performance.

We introduce Tokenization with Split Trees (ToaST), a subword tokenization method that directly optimizes compression under a new recursive inference procedure. ToaST greedily splits each pretoken into a full binary tree using precomputed byte n-gram counts, independent of any vocabulary. Given a vocabulary, inference recursively descends each split tree and emits the first in-vocabulary node reached on each path. Vocabulary selection is formulated as an Integer Program (IP) that minimizes the total token count over all split trees under this inference procedure. The Linear Programming (LP) relaxation is near-integral in practice, yielding provably near-optimal vocabularies, with training time empirically scaling quadratically in the number of split trees. On English text, ToaST reduces token counts by more than 11% compared to BPE, WordPiece, and UnigramLM at vocabulary sizes of 40,960 and above, reducing the number of inference tokens for models using this tokenizer, thus extending the effective context length. ToaST also uses common single-byte tokens less frequently than these baselines, leading to a substantial improvement in Renyi efficiency. In experiments training 1.5B parameter language models, ToaST achieves the highest CORE score, outperforming baselines by 2.6%--7.6%, with significance for two of three, and scoring best on 13 of 22 individual tasks.