# Learning to Reason in 13 Parameters

arXiv: https://arxiv.org/abs/2602.04118 · HTML: https://mlsanity.com/paper/2602.04118/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 41 of its 49 references that are in the map.
- First posted on arXiv 4 Feb 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 30 of 41)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Prefix-Tuning: Optimizing Continuous Prompts for Generation](https://mlsanity.com/paper/2101.00190/) (2021)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [BitFit: Simple Parameter-efficient Fine-tuning for Transformer-based Masked Language-models](https://mlsanity.com/paper/2106.10199/) (2021)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [SFT Memorizes, RL Generalizes: A Comparative Study of Foundation Model Post-training](https://mlsanity.com/paper/2501.17161/) (2025)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild](https://mlsanity.com/paper/2503.18892/) (2025)
- [LIMO: Less is More for Reasoning](https://mlsanity.com/paper/2502.03387/) (2025)
- [Measuring the Intrinsic Dimension of Objective Landscapes](https://mlsanity.com/paper/1804.08838/) (2018)
- [AdaLoRA: Adaptive Budget Allocation for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2303.10512/) (2023)
- [LoRA Learns Less and Forgets Less](https://mlsanity.com/paper/2405.09673/) (2024)
- [OpenThoughts: Data Recipes for Reasoning Models](https://mlsanity.com/paper/2506.04178/) (2025)
- [VeRA: Vector-based Random Matrix Adaptation](https://mlsanity.com/paper/2310.11454/) (2023)
- [Reinforcement Learning for Reasoning in Large Language Models with One Training Example](https://mlsanity.com/paper/2504.20571/) (2025)
- [When Scaling Meets LLM Finetuning: The Effect of Data, Model and Finetuning Method](https://mlsanity.com/paper/2402.17193/) (2024)
- [Evaluating the Performance of Large Language Models on GAOKAO Benchmark](https://mlsanity.com/paper/2305.12474/) (2023)
- [RL's Razor: Why Online Reinforcement Learning Forgets Less](https://mlsanity.com/paper/2509.04259/) (2025)
- [LoRA vs Full Fine-tuning: An Illusion of Equivalence](https://mlsanity.com/paper/2410.21228/) (2024)
- [Retaining by Doing: The Role of On-Policy Data in Mitigating Forgetting](https://mlsanity.com/paper/2510.18874/) (2025)
- [Punica: Multi-Tenant LoRA Serving](https://mlsanity.com/paper/2310.18547/) (2023)
- [Reasoning or Memorization? Unreliable Results of Reinforcement Learning Due to Data Contamination](https://mlsanity.com/paper/2507.10532/) (2025)

## Built on by (most cited first, 14 of 14)

- [Neural Thickets: Diverse Task Experts Are Dense Around Pretrained Weights](https://mlsanity.com/paper/2603.12228/) (2026)
- [Operationalising the Superficial Alignment Hypothesis via Task Complexity](https://mlsanity.com/paper/2602.15829/) (2026)
- [Consolidating Rewarded Perturbations for LLM Post-Training](https://mlsanity.com/paper/2605.31494/) (2026)
- [Spectral Rewiring for Exploration, Purification, and Model Merging](https://mlsanity.com/paper/2607.03065/) (2026)
- [Non-vacuous Generalization Bounds for Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2607.14506/) (2026)
- [How Much Can a LoRA Adapter Memorize? Measuring Adapter Capacity in Bits](https://mlsanity.com/paper/2607.21351/) (2026)
- [BoostLoRA: Growing Effective Rank by Boosting Adapters](https://mlsanity.com/paper/2604.27308/) (2026)
- [Not How Many, But Which: Parameter Placement in Low-Rank Adaptation](https://mlsanity.com/paper/2605.12207/) (2026)
- [PreFT: Prefill-only finetuning for efficient inference](https://mlsanity.com/paper/2605.14217/) (2026)
- [Super Weights in LLMs and the Failure of Selective Training](https://mlsanity.com/paper/2607.08733/) (2026)
- [Can Bayesian Optimization Efficiently Find a Strong Single Expert in Neural Thickets?](https://mlsanity.com/paper/2608.10867/) (2026)
- [Understanding and Exploiting Anisotropy in Post-Training](https://mlsanity.com/paper/2609.32792/) (2026)
- [The Low-Rank Structure of VLA Reinforcement Learning](https://mlsanity.com/paper/2609.34599/) (2026)
- [How Low Can You Go: Fine-Tuning LLMs with Bayesian Optimization (LlMBO)](https://mlsanity.com/paper/2610.18504/) (2026)

## Abstract (arXiv)

Recent research has shown that language models can learn to textitreason, often via reinforcement learning. Some work even trains low-rank parameterizations for reasoning, but conventional LoRA cannot scale below the model dimension. We question whether even rank=1 LoRA is necessary for learning to reason and propose TinyLoRA, a method for scaling low-rank adapters to sizes as small as one parameter. Within our new parameterization, we are able to train the 8B parameter size of Qwen2.5 to 91% accuracy on GSM8K with only 13 trained parameters in bf16 (26 total bytes). We find this trend holds in general: we are able to recover 90% of performance improvements while training 1000x fewer parameters across a suite of more difficult learning-to-reason benchmarks such as AIME, AMC, and MATH500. Notably, we are only able to achieve such strong performance with RL: models trained using SFT require 100-1000x larger updates to reach the same performance.