Theoretical Analysis of Byte-Pair Encoding

Byte-Pair Encoding (BPE) builds a tokeniser by repeatedly merging a most frequent adjacent pair of symbols; its utility is the total reduction in string length achieved by $k$ merges. It was known that this greedy rule achieves at least a third of the optimal reduction. We prove that it always achieves at least 0.361 of the optimal reduction in string length, for every string, every budget and every tie-breaking rule; this is the first improvement of the bound one third. We also explain why one third was a barrier: the pair-packing relaxation behind it is tight, so a better bound must use more of the optimum's structure. When greedy destroys three of the optimum's merge points at once, one of them can usually be moved next door, because all copies of a rule of the optimum look alike; the remaining cases are paid for later. Along the way we determine the worst-case ratio for two merges exactly, bound it for three, and isolate a single inequality that would settle the problem.

Byte Pair Encoding (BPE) is widely used for subword tokenization, but standard BPE exposes every learned merge token to the downstream model, including tokens that mainly serve as intermediate construction units and rarely appear in the final encoded corpus. This paper proposes Pruned BPE, a post-training visibility-pruning and token-reallocation method that separates merge construction from model-visible vocabulary selection. After standard BPE training, tokens are evaluated by final exposure. Low-exposure tokens are retained as internal-only merge nodes, while their visible vocabulary slots are reassigned to better-exposed candidates learned through resumed training. During encoding, internal-only tokens are recursively expanded into visible descendants while the original BPE merge order is preserved. Experiments on two non-overlapping English- and Chinese-dominated corpora and their combination show that Pruned BPE consistently reduces encoded length relative to Standard BPE at the same training corpus, evaluation corpus, and model-visible vocabulary size. At a 40% exposure threshold, the reduction is approximately 0.27%--0.36% on same-corpus evaluations. In a vocabulary-only evaluation using a shared exact minimum-token dynamic-programming encoder, Pruned BPE retains an advantage of approximately 0.23%--0.31%, indicating that the improvement arises from a more efficient visible vocabulary. These gains represent a meaningful fraction of the approximately 1.5%--3.8% marginal reduction that would otherwise require adding another 2K Standard BPE tokens. Qualitative analysis shows that internal-only tokens include reusable English fragments, Chinese components, partial UTF-8 byte sequences, and structured-text fragments. The results indicate that post-training visibility pruning can improve BPE vocabulary efficiency without increasing the vocabulary exposed to the language model.

Byte-Pair Encoding (BPE) constructs subword vocabularies through greedy pair merging, but conventional BPE requires the number of merges or target vocabulary size to be specified externally. Significance-Gain Pair Encoding (SG-BPE) replaces frequency-only selection with a statistical criterion based on how strongly an observed pair exceeds its expected co-occurrence under an independence model. This paper introduces MDL-Calibrated Significance-Gain Pair Encoding (MDL-SG), a three-stage procedure separating discovery, replication, and utility. Candidate pairs are ranked by Significance-Gain on a discovery partition, tested for replication on a separate partition using an exact one-sided hypergeometric test with per-iteration Benjamini-Hochberg correction, and then evaluated on a utility partition using a Minimum Description Length (MDL) criterion. Merging stops automatically when no replicated candidate yields positive held-out MDL gain. On WikiText-103, MDL-SG stops at 209, 433, and 847 merges for 120K, 250K, and 500K-character tokenizer-training samples, respectively. At 500K characters, it selects a stored vocabulary of 1,017 tokens without prescribing the vocabulary size in advance. In a compute-matched TinyGPT experiment with identical 2,024,448-parameter models and 500 optimizer updates per language model, MDL-SG achieves validation/test BPC of 3.2612/3.2436, compared with test BPC of 3.2894 for SG-BPE and 3.3493 for frequency BPE. Frequency BPE achieves stronger raw compression, while MDL-SG achieves lower BPC, showing that compression-oriented merge selection and language-model utility need not coincide.