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arxiv: 2605.22821 · v1 · pith:A3HSD7RNnew · submitted 2026-05-21 · 💻 cs.CL · cs.LG

Tokenisation via Convex Relaxations

Jan Tempus , Philip Whittington , Craig W. Schmidt , Dennis Komm , Tiago Pimentel This is my paper

Pith reviewed 2026-05-22 05:16 UTC · model grok-4.3

classification 💻 cs.CL cs.LG
keywords tokenizationconvex optimizationlinear programminglanguage modelingBPEvocabulary selectionbits-per-byte
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The pith

Tokenization can be cast as a linear program whose solution yields vocabularies within 1% of optimal.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper replaces greedy algorithms such as BPE and Unigram with a formulation of vocabulary selection as a linear program that is solved by convex optimization. The resulting method, ConvexTok, produces tokenizers that improve intrinsic metrics and the bits-per-byte achieved by language models. It also supplies a lower bound that certifies how close any given tokenizer is to the optimum under the chosen objective. A reader would care because the approach turns an opaque design choice into a verifiable optimization problem that can be tightened or relaxed as needed.

Core claim

By relaxing the discrete problem of choosing a vocabulary of fixed size into a linear program, ConvexTok finds tokenizers that improve bits-per-byte on language models and lie within 1% of the lower bound on the objective at typical vocabulary sizes; the same construction also improves most intrinsic tokenization metrics while producing mixed results on downstream tasks.

What carries the argument

ConvexTok, the solution of a linear program that relaxes vocabulary selection into a convex objective over token frequencies and coverage constraints.

If this is right

  • Language models achieve lower bits-per-byte with ConvexTok vocabularies than with BPE or Unigram at equal vocabulary size.
  • Intrinsic tokenization metrics such as fertility and coverage improve consistently.
  • A computable lower bound certifies that the obtained tokenizer is within 1% of optimal under the linear objective.
  • Downstream task accuracy gains appear but are smaller and less consistent than the efficiency gains.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same linear-programming framing could be applied to other preprocessing stages such as sentence segmentation or morphological segmentation.
  • If the objective can be made differentiable, the relaxation might be inserted directly into end-to-end model training.
  • Multilingual settings could benefit from joint optimization over multiple languages inside one linear program.

Load-bearing premise

The linear objective and its relaxation are assumed to be a faithful proxy for what actually improves language-model performance.

What would settle it

Train identical language models on corpora tokenized by ConvexTok versus BPE at the same vocabulary size and measure whether the bits-per-byte gap disappears or reverses.

Figures

Figures reproduced from arXiv: 2605.22821 by Craig W. Schmidt, Dennis Komm, Jan Tempus, Philip Whittington, Tiago Pimentel.

Figure 1
Figure 1. Figure 1: Tokenisation graph con￾structed from D = {abaa, aba}. Black edges represent Ebyte and others represent Etok. Our paper’s main contribution is an LP-based tokenisation algo￾rithm which directly approximates a globally-optimal tokeniser; thus, avoiding locally-optimal solutions. As our tokeniser relies on a convex relation, we term it ConvexTok. Before we get to our LP, however, we first express the problem … view at source ↗
Figure 2
Figure 2. Figure 2: The Det (left), Bias (center), and Int (right) rounding schemes. of the token it represents (c/length(c), with a slight abuse of notation to let length(c) denote the length of the corresponding token), and then rounds these values (to 1 or 0) as before. This biases the selection towards shorter tokens when LP scores are comparable, and is motivated by the fact that shorter tokens are more likely to occur o… view at source ↗
Figure 3
Figure 3. Figure 3: Average Jaccard similarity between vocabularies when retraining a tokeniser on indepen [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Compression by the different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Vocabulary utilisation by different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Type-token ratio by different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Token length of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Tokens per line of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Shannon entropy of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Rényi entropy of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Average rank of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Compression by the different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Vocabulary utilisation by different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Type-token ratio by different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Token length of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Tokens per line of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Shannon entropy of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p019_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Rényi entropy of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p019_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Average rank of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p019_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Compression by the different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p020_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Vocabulary utilisation by different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p020_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Type-token ratio by different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p020_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Token length of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p020_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Tokens per line of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p021_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Shannon entropy of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p021_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Rényi entropy of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p021_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Average rank of different tokenisers. (Left) absolute and (Right) relative to [PITH_FULL_IMAGE:figures/full_fig_p021_27.png] view at source ↗
Figure 33
Figure 33. Figure 33: (left) BpB and (right) CORE vs. vocabulary size across models with different depths [PITH_FULL_IMAGE:figures/full_fig_p023_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: (left) BpB and (right) CORE vs. vocabulary size across three training seeds. All these models were trained with 12 layers. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_34.png] view at source ↗
read the original abstract

Tokenisation is an integral part of the current NLP pipeline. Current tokenisation algorithms such as BPE and Unigram are greedy algorithms -- they make locally optimal decisions without considering the resulting vocabulary as a whole. We instead formulate tokeniser construction as a linear program and solve it using convex optimisation tools, yielding a new algorithm we call ConvexTok. We find ConvexTok consistently improves intrinsic tokenisation metrics and the bits-per-byte (BpB) achieved by language models; it also improves downstream task performance, but less consistently. Furthermore, ConvexTok allows the user to certify how far their tokeniser is from optimal, with respect to a certain objective, via a lower bound, and we empirically find it to be within 1\% of optimal at common vocabulary sizes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper presents ConvexTok, a new tokenization method that formulates the construction of the vocabulary as a linear program solved using convex optimization techniques. Unlike greedy algorithms such as BPE and Unigram, it considers the vocabulary as a whole. The authors report that ConvexTok consistently improves intrinsic tokenization metrics and the bits-per-byte (BpB) of language models, with less consistent improvements on downstream tasks. Additionally, it provides a lower bound to certify the distance to optimality with respect to the LP objective, empirically showing solutions within 1% of optimal for common vocabulary sizes.

Significance. If the results hold and the LP objective is a good proxy for LM performance, this work introduces a principled, optimization-based approach to tokenization with the novel feature of optimality certificates. This could be significant for the field as it moves beyond heuristic methods. The use of convex relaxations and lower bounds is a strength that allows for verifiable claims.

major comments (2)
  1. Abstract: The claim that ConvexTok 'consistently improves' BpB and downstream performance, and is 'within 1% of optimal', requires more detail on the specific LP formulation, the definition of the objective function, data exclusion rules, and statistical significance tests to rule out post-hoc fitting or selection effects.
  2. Method: The linear programming relaxation and chosen objective function need to be validated as a faithful model of tokenizer quality for language modeling. No ablation is mentioned that compares the LP objective value to the actual negative log-likelihood of a trained transformer or holds data and model fixed while varying the tokenizer objective.
minor comments (1)
  1. Abstract: The abstract mentions 'a certain objective' for the optimality certificate; this should be explicitly defined early in the paper.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the referee's insightful comments on our work. We respond to each major comment below and indicate the revisions we plan to make.

read point-by-point responses

  1. Referee: Abstract: The claim that ConvexTok 'consistently improves' BpB and downstream performance, and is 'within 1% of optimal', requires more detail on the specific LP formulation, the definition of the objective function, data exclusion rules, and statistical significance tests to rule out post-hoc fitting or selection effects.

    Authors: We concur that providing more detail in the abstract would strengthen the presentation of our results. In the revised manuscript, we will elaborate on the LP formulation and the definition of the objective function. We will also specify the data exclusion rules applied in our experiments and include statistical significance tests for the reported improvements in BpB and downstream performance to address potential concerns about selection effects. The claim of being within 1% of optimal is supported by the lower bound computation described in the paper, and we will ensure this is clearly contextualized. revision: yes

  2. Referee: Method: The linear programming relaxation and chosen objective function need to be validated as a faithful model of tokenizer quality for language modeling. No ablation is mentioned that compares the LP objective value to the actual negative log-likelihood of a trained transformer or holds data and model fixed while varying the tokenizer objective.

    Authors: This is a fair point regarding the validation of our objective. Our LP formulation optimizes a specific objective that we argue serves as a reasonable proxy for tokenizer quality, as demonstrated by the improvements in intrinsic metrics and BpB. However, we acknowledge that a direct ablation comparing the LP objective to the NLL of a fixed transformer model was not included. We will add a section in the revised paper discussing the motivation for the chosen objective and its empirical correlation with LM performance. A full ablation study may be included if it can be completed within the revision timeline, but we note that such experiments are computationally intensive. We disagree that this invalidates the current contributions, as the optimality certificate is a novel aspect independent of this validation. revision: partial

Circularity Check

0 steps flagged

No circularity: LP formulation and empirical gains are independent

full rationale

The derivation introduces a linear-programming relaxation of vocabulary selection as a novel modeling choice solved by standard convex-optimization tools; the resulting tokenizers are then evaluated on separate, externally measured quantities (BpB of trained language models and downstream task accuracy). No equation equates a claimed improvement or optimality gap to a quantity defined by the same fitted parameters or by a self-citation chain; the 1% optimality certificate is explicitly relative to the paper's own LP objective, while BpB and task gains are reported from independent training runs. The method is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that tokenization quality can be usefully expressed as the objective of a linear program whose relaxation yields practical solutions. No explicit free parameters or invented entities are described in the abstract.

axioms (1)
  • domain assumption Tokenization construction admits a linear programming formulation whose relaxation is solvable by convex optimization tools
    This premise is required for the ConvexTok algorithm to exist and is invoked when the abstract contrasts the new method with greedy algorithms.

pith-pipeline@v0.9.0 · 5658 in / 1441 out tokens · 44050 ms · 2026-05-22T05:16:50.783338+00:00 · methodology

discussion (0)

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