# PALU: Concise Reasoning Through the Lens of Lagrangian Optimization

> NeurIPS 2026. Chengqian Gao, Haonan Li, Taylor W. Killian, Jianshu She, Renxi Wang, Liqun Ma, Zhoujun Cheng, Shibo Hao, Zhiqiang Xu (MBZUAI, Brigham Young University, UC San Diego).
> PALU treats concise reasoning as minimizing response length subject to an accuracy floor. On DeepSeek-R1-Distill-Qwen-1.5B it cuts response length by 64% while raising accuracy from 44.2% to 51.6%, and one hyperparameter setting transfers across math, logic, STEM and 1.5B-14B models.

- Paper: https://arxiv.org/abs/2510.10168
- Code: https://github.com/glorgao/PALU
- Model: https://huggingface.co/glorgao/PALU-1.5B

## Idea

RL with verifiable rewards (e.g. GRPO) optimizes accuracy and ignores length; concise-reasoning methods push length down with a fixed length target or penalty. PALU weighs the two with a Lagrangian: min L s.t. R(theta, L, q) >= C, i.e. min max_{lambda>=0} L + lambda (C - R). The accuracy floor C means the same thing in every setting, so no length target is needed. Base-model length varies too much for a fixed target: DeepSeek-R1-Distill-Qwen-1.5B has a median of about 16k tokens on AIME 24 but under 4k on STEM; on AIME 24 the median falls from about 17k at 1.5B to 8k at 32B.

## Method: Performance-Aware Length Update (PALU)

Per-prompt length budget L, updated once per epoch from rollouts GRPO already collected:
1. Read: pass rate from the previous epoch's rollouts (stand-in for the residual C - R; no extra generation).
2. Decide: if pass rate >= C, shrink the budget; otherwise reset it to L_max (two-regime rule replacing the multiplier lambda).
3. Move: shrink by the quantile gap alpha_tau = Q_1.0 - Q_(1-tau) of correct-response lengths (stand-in for dL R).
At inference only a fixed maximum length applies.

## Results (DeepSeek-R1-Distill-Qwen-1.5B on DeepScaleR, macro average over 6 math benchmarks)

| Method | Pass@1 (%) | Length (tokens) | AE score |
|---|---|---|---|
| R1-Distill-Qwen-1.5B | 44.2 | 11297 | 0.000 |
| GRPO | 51.4 | 10438 | 0.563 |
| ThinkPrune-4k | 46.4 | 5390 | 0.659 |
| AutoThink-Stage3 | 50.6 | 5563 | 1.077 |
| ALP | 48.9 | 4742 | 0.958 |
| L1-Max | 44.6 | 3311 | 0.551 |
| ShorterBetter | 37.7 | 2585 | 0.030 |
| PALU | 51.6 | 4093 | 1.185 |

Benchmarks: MATH-500, AIME 2024, AIME 2025, AMC 2023, OlympiadBench, Minerva-Math.

PALU keeps compressing after length and accuracy start to trade off: on AIME 24, in the regime where the rolling correlation turns positive, length falls from 10.2k to 8.5k tokens while accuracy rises from 35.0% to 37.4%, because decisions are per prompt.

With C = 0.8 in all nine settings (math, logic, STEM x 1.5B, 7B, 14B; starting lengths 4.2k-15.4k tokens), PALU shortens responses in 9 of 9 settings; baselines with an 8k target do so in 4 of 9.

## Looking ahead

Concise reasoning improves human-model collaboration and may lower deployment energy. The first author asks whether it can also raise a model's performance ceiling, since shorter text carrying the same information often reflects clearer thinking. The first author, Chengqian Gao (https://github.com/glorgao), welcomes collaboration.

## Citation

@article{gao2025concise,
  title={Concise reasoning in the lens of lagrangian optimization},
  author={Gao, Chengqian and Li, Haonan and Killian, Taylor W and She, Jianshu and Wang, Renxi and Ma, Liqun and Cheng, Zhoujun and Hao, Shibo and Xu, Zhiqiang},
  journal={arXiv preprint arXiv:2510.10168},
  year={2025}
}