# Behavior Quotient Learning for Low-Rank Adaptation of LLM Agents

arXiv: https://arxiv.org/abs/2609.12896 · HTML: https://mlsanity.com/paper/2609.12896/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reinforcement learning for LLM agents, one of 16 directions in LLM agents and tool use.
- Builds on 47 of its 59 references that are in the map.
- First posted on arXiv 11 Sep 2026.

## Placed in

- LLM agents and tool use / [Reinforcement learning for LLM agents](https://mlsanity.com/agents/reinforcement-learning-for-llm-agents/)

## Builds on (most cited first, 30 of 47)

- [ReAct: Synergizing Reasoning and Acting in Language Models](https://mlsanity.com/paper/2210.03629/) (2022)
- [LESS: Selecting Influential Data for Targeted Instruction Tuning](https://mlsanity.com/paper/2402.04333/) (2024)
- [AppWorld: A Controllable World of Apps and People for Benchmarking Interactive Coding Agents](https://mlsanity.com/paper/2407.18901/) (2024)
- [LoRA+: Efficient Low Rank Adaptation of Large Models](https://mlsanity.com/paper/2402.12354/) (2024)
- [LoraHub: Efficient Cross-Task Generalization via Dynamic LoRA Composition](https://mlsanity.com/paper/2307.13269/) (2023)
- [AgentTuning: Enabling Generalized Agent Abilities for LLMs](https://mlsanity.com/paper/2310.12823/) (2023)
- [PiSSA: Principal Singular Values and Singular Vectors Adaptation of Large Language Models](https://mlsanity.com/paper/2404.02948/) (2024)
- [Mixture of LoRA Experts](https://mlsanity.com/paper/2404.13628/) (2024)
- [LoRA-GA: Low-Rank Adaptation with Gradient Approximation](https://mlsanity.com/paper/2407.05000/) (2024)
- [DataInf: Efficiently Estimating Data Influence in LoRA-tuned LLMs and Diffusion Models](https://mlsanity.com/paper/2310.00902/) (2023)
- [LoRA-Pro: Are Low-Rank Adapters Properly Optimized?](https://mlsanity.com/paper/2407.18242/) (2024)
- [Towards General Agentic Intelligence via Environment Scaling](https://mlsanity.com/paper/2509.13311/) (2025)
- [Riemannian Preconditioned LoRA for Fine-Tuning Foundation Models](https://mlsanity.com/paper/2402.02347/) (2024)
- [SMART: Self-Aware Agent for Tool Overuse Mitigation](https://mlsanity.com/paper/2502.11435/) (2025)
- [VideoMind: A Chain-of-LoRA Agent for Temporal-Grounded Video Reasoning](https://mlsanity.com/paper/2503.13444/) (2025)
- [LoRA-One: One-Step Full Gradient Could Suffice for Fine-Tuning Large Language Models, Provably and Efficiently](https://mlsanity.com/paper/2502.01235/) (2025)
- [FlyLoRA: Boosting Task Decoupling and Parameter Efficiency via Implicit Rank-Wise Mixture-of-Experts](https://mlsanity.com/paper/2510.08396/) (2025)
- [AdaMoLE: Fine-Tuning Large Language Models with Adaptive Mixture of Low-Rank Adaptation Experts](https://mlsanity.com/paper/2405.00361/) (2024)
- [LoRA Done RITE: Robust Invariant Transformation Equilibration for LoRA Optimization](https://mlsanity.com/paper/2410.20625/) (2024)
- [ATLaS: Agent Tuning via Learning Critical Steps](https://mlsanity.com/paper/2503.02197/) (2025)
- [Parameter Efficient Fine-tuning via Explained Variance Adaptation](https://mlsanity.com/paper/2410.07170/) (2024)
- [PoLAR: Polar-Decomposed Low-Rank Adapter Representation](https://mlsanity.com/paper/2506.03133/) (2025)
- [LoRA-Mixer: Coordinate Modular LoRA Experts Through Serial Attention Routing](https://mlsanity.com/paper/2507.00029/) (2025)
- [Reasoning and Tool-use Compete in Agentic RL:From Quantifying Interference to Disentangled Tuning](https://mlsanity.com/paper/2602.00994/) (2026)
- [MeteoRA: Multiple-tasks Embedded LoRA for Large Language Models](https://mlsanity.com/paper/2405.13053/) (2024)
- [FouRA: Fourier Low Rank Adaptation](https://mlsanity.com/paper/2406.08798/) (2024)
- [StelLA: Subspace Learning in Low-rank Adaptation using Stiefel Manifold](https://mlsanity.com/paper/2510.01938/) (2025)
- [Verified Critical Step Optimization for LLM Agents](https://mlsanity.com/paper/2602.03412/) (2026)
- [LoRA meets Riemannion: Muon Optimizer for Parametrization-independent Low-Rank Adapters](https://mlsanity.com/paper/2507.12142/) (2025)
- [LD-MoLE: Learnable Dynamic Routing for Mixture of LoRA Experts](https://mlsanity.com/paper/2509.25684/) (2025)

## Abstract (arXiv)

LLM-based agents rely on heterogeneous interaction capabilities to accomplish complex tasks. Existing approaches often distribute these capabilities across multiple LoRA adapters, which increases adapter storage requirements and introduces routing overhead during inference. A single LoRA avoids this overhead, but learning from diverse agent trajectories under a fixed rank budget presents two challenges. First, trajectories with different interaction traces and parameter gradients can induce equivalent changes in decision distributions, causing repeated updates to overemphasize redundant behavioral changes. Second, an aggregated update may exceed the rank budget of the adapter, and approximating it in weight space can distort the decision changes that it is intended to produce. We propose BQ-LoRA, a low-rank adaptation framework that organizes trajectory updates through a local behavior quotient manifold. It contains two modules, i.e., behavior quotient balancing (BQB) and decision preserving compression (DPC). BQB constructs the quotient manifold from decision distributions and reweights trajectory update directions according to their local density in the quotient tangent space. DPC projects the balanced gradient onto the intrinsic fixed rank tangent space and refactorizes the resulting target by jointly controlling effective weight error and distortion of decision distributions. Experiments on AppWorld and BrowseComp-Plus compare BQ-LoRA with standard LoRA and recent low-rank adaptation methods, while separate ablations evaluate the complementary contributions of both components.