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Behavior Quotient Learning for Low-Rank Adaptation of LLM Agents
Pengyang Zhou, Xiaobin Tu, Zhengxi Liu, Rongkun Xue, Haochen Li, Miancan Liu, Ziyuan Chen, Yinggui Wang, Jinkui Ren, Xiantao Zhang
TL;DR
A single fixed-rank LoRA avoids the coordination overhead of multiple adapters, but diverse agent trajectories can produce redundant decision changes and updates that exceed the rank budget. BQ-LoRA balances trajectory directions in a local behavior quotient and compresses their target while controlling weight error and decision distortion. On AppWorld and BrowseComp-Plus, the full method leads in seven of eight settings, with experiments also reporting reduced interaction counts.
Problem
Under a fixed-rank budget, trajectories with different traces and gradients can induce equivalent decision changes, while aggregated updates may exceed the rank constraint and be distorted by weight-space approximation.
Source: §1 · p. 2Method
BQ-LoRA uses BQB to reweight trajectory directions by local density in decision-behavior geometry and DPC to project and compress the balanced target while controlling weight error and decision distortion.
Source: §1 · p. 2Results
The full method leads in seven of eight settings, and experiments on AppWorld and BrowseComp-Plus report strong task performance and reduced interaction counts.
Source: §5.2 · p. 9, §6 · p. 9Takeaways & Limitations
BQ-LoRA preserves the original rank budget and requires no capability-specific routing while using a single adapter.
Source: §6 · p. 9
Abstract
from arXiv · showhide
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.
1 INTRODUCTION
A single fixed-rank LoRA avoids managing multiple capability-specific adapters, but diverse trajectories can repeatedly update equivalent behaviors while distinct decision changes remain underrepresented. BQ-LoRA addresses this by balancing trajectory updates in decision-behavior geometry and compressing them under the rank constraint.
1 INTRODUCTION
- Different trajectories can induce similar changes in decisions despite differing interaction traces and parameter gradients, so repeated equivalent updates may overemphasize some behaviors.This can leave updates with distinct effects on reference decisions underrepresented.Source: §1 · p. 2
- Identifying behavioral equivalence requires comparing update effects at the same reference histories, while fixed-rank compression must preserve decision changes beyond weight-space approximation.A suitable representative must also be invariant to equivalent LoRA factorizations.Source: §1 · p. 2
- BQB compares first-order effects on normalized decision representations at fixed histories and reweights locally concentrated directions to give distinct behavioral changes greater relative influence.It disregards parameter differences invisible to the reference decisions.Source: §1 · p. 2
- DPC projects the balanced direction onto the current adapter’s attainable fixed-rank tangent space and compresses it while controlling weight error and decision distortion.Its target depends on effective weights, remaining invariant to equivalent LoRA factorizations.Source: §1 · p. 2
- BQ-LoRA learns diverse agent behaviors with one fixed-rank adapter, and the evaluation compares it with standard adaptation, optimization, specialization, and curation approaches.The reported evaluation covers AppWorld and BrowseComp-Plus and two backbone sizes.Source: §1 · p. 2
2 RELATED WORK
Prior work improves LoRA adaptation through changes to initialization, rank allocation, optimization, and adapter composition, while agent research also studies specialized adapters. Trajectory-redundancy work has focused on data composition or utility signals rather than whether distinct updates induce equivalent behavioral changes.
2.1 LOW-RANK ADAPTATION
- LoRA research explores informative initialization, adaptive rank allocation, modified training, geometry-aware optimization, and routing or composition of multiple adapters.Agentic systems also use specialized adapters to model heterogeneous capabilities and coordinate their activation across trajectories.Source: §2.1 · pp. 2–3
2.2 TRAJECTORY REDUNDANCY
- Trajectory-redundancy methods address data quality, optimization-based sample utility, critical-step selection, or semantic similarity, but do not establish whether different training updates induce equivalent behavioral changes.This paper defines redundancy through update-induced behavior and uses that equivalence to balance supervision for a single fixed-rank LoRA.Source: §2.2 · p. 3
3 PRELIMINARIES
The preliminaries define quotient geometry by identifying equivalent points in a smooth space. The quotient projection maps each point to its equivalence class, and tangent directions are equivalent when their difference is vertical.
3 PRELIMINARIES
- A quotient manifold is formed by identifying equivalent points in a smooth total space.The resulting quotient set groups points according to the specified equivalence relation.Source: §3 · p. 3
- The canonical projection maps each point x to its equivalence class [x], and a smooth submersion condition characterizes the quotient manifold.The stated condition is rank Dπ(x) = dim M for every x.Source: §3 · p. 3
- Two tangent vectors represent the same quotient tangent vector exactly when their difference lies in the vertical space.The vertical space is the tangent space to the fiber at the point.Source: §3 · p. 3
4 METHOD
BQ-LoRA balances trajectory updates by their effects on decision representations, then projects and compresses the aggregate under the adapter’s fixed rank budget while controlling decision distortion. Its four-stage method formalizes the LoRA setting, defines behavior-aware weighting, and preserves decision changes during compression.
4.1 PROBLEM STATEMENT
- The method trains LoRA factors for effective updates across adapted weight matrices, with the factor inner dimension setting the rank budget and the objective averaging trajectory losses.A trajectory includes the task, interaction histories, agent outputs, and environment feedback.Source: §4.1 · pp. 3–4
4.2 FRAMEWORK OVERVIEW
- The framework repeats BQB weighting and DPC projection and compression to train one LoRA adapter while controlling rank-induced behavioral distortion.BQB uses normalized decision representations at fixed histories; DPC measures decision-response distortion with the same behavior geometry.Source: §4.2 · p. 4, Figure 1 · p. 4
4.3 BEHAVIOR QUOTIENT BALANCING
- BQB measures trajectory-update overlap through first-order responses on fixed reference decisions and downweights densely represented behavior changes.It retains each task loss’s update direction and requires no predefined capability partitions.Source: §4.3 · pp. 5–6
4.4 DECISION PRESERVING COMPRESSION
- DPC projects the balanced direction onto the locally attainable tangent space of the current LoRA factors, producing a target invariant to equivalent factorizations.The projection retains the closest first-order attainable direction, which defines the reachable decision effect that refactorization aims to preserve.Source: §4.4 · p. 6
- DPC refactorizes the target under the original rank budget by jointly minimizing effective-weight error and first-order decision-response distortion.The decision response is evaluated across adapted layers jointly, and γ controls the tradeoff between the two objectives.Source: §4.4 · pp. 6–7
5 EXPERIMENTS
Experiments on AppWorld and BrowseComp-Plus compare BQ-LoRA with standard and specialized adaptation methods, then evaluate its components and hyperparameter sensitivity. BQ-LoRA leads the strongest baseline on average and has the lowest interaction counts in all evaluated settings.
5.1 EXPERIMENTAL SETUPS
- The experiments use successful trajectories from AppWorld multi-step application tasks and BrowseComp-Plus deep-search questions, comparing seven adaptation and trajectory-curation baselines.The baselines include standard adaptation, multi-LoRA methods, single-LoRA optimization, and TopoCurate.Source: §5.1 · pp. 7–8
5.2 PERFORMANCE EVALUATION
- Averaged over eight reported performance metrics, BQ-LoRA outperforms MoRAgent by 2.14 and 2.12 percentage points for the 4B and 9B backbones, respectively, and has the lowest interaction counts in all evaluated settings.The interaction-count measures are AppWorld steps and BrowseComp-Plus search calls.Source: §5.2 · p. 8, Table 2 · p. 8
- Removing DPC reduces performance in all eight settings, with average drops across four metrics of 3.64 and 3.57 percentage points for the 4B and 9B backbones, respectively.Removing BQB lowers Test-C performance by 3.57 and 3.83 percentage points for the 4B and 9B backbones, respectively, and lowers BrowseComp-Plus accuracy with both retrievers.Source: §5.2 · pp. 8–9
- BQB removal reduces Test-C more than Test-N for both backbones, showing that its benefit varies across evaluation settings.The reported Test-C reductions are 3.57 and 3.83 percentage points for the 4B and 9B backbones, respectively.Source: §5.2 · p. 8
- The full method leads in seven of eight ablation settings, while removing BQB lets the 4B model score 0.20 percentage points higher on Test-N.Both partial variants outperform standard LoRA in every setting, and retaining DPC gives higher mean performance than retaining BQB alone across backbones.Source: §5.2 · p. 9
- Both backbones perform best at γ = 0.8; performance decreases toward both ends of the tested range, supporting an intermediate tradeoff between weight fidelity and decision preservation.The sweep tests γ values from 0.1 to 10 while varying one coefficient at a time and fixing the other at 1.Source: §5.2 · p. 9
6 CONCLUSION
BQ-LoRA adapts LLM agents with a single fixed-rank LoRA: BQB balances trajectory updates in a local behavior quotient geometry, while DPC projects and compresses updates while controlling decision distortion. Experiments on AppWorld and BrowseComp-Plus show strong task performance and reduced interaction counts.
6 CONCLUSION
- BQ-LoRA uses one fixed-rank LoRA adapter, with no capability-specific routing.BQB balances trajectory updates by their effects on decisions, and DPC projects and compresses the update while controlling decision distortion.Source: §6 · p. 9
- Experiments on AppWorld and BrowseComp-Plus show strong task performance and reduced interaction counts.The conclusion attributes these results to balancing trajectory updates and preserving their decision effects during compression.Source: §6 · p. 9
- BQB reduces repeated contributions from updates with equivalent effects on decision representations.DPC projects the balanced update and compresses its target while controlling decision distortion.Source: §6 · p. 9