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.
Post-Training Quantization (PTQ) and Low-Rank Adaptation (LoRA) constitute the standard pipeline for efficient Large Language Model (LLM) deployment. However, applying them sequentially poses a problem: PTQ often leaves behind random noise that is spread out (across the model's weights) in a way LoRA can't easily fix, meaning that LoRA ends up wasting its limited capacity trying to fix uncorrectable noise instead of improving task performance. In this paper, we propose \textbf{ProjQ}, a novel framework for constraining quantization noise to the low-rank manifold via orthogonal subspace projection. We derive an efficient alternating algorithm that shapes the quantization noise into a low-rank structure, effectively offloading dominant error components to the subsequent adapter while minimizing the residual error in the orthogonal "uncorrectable" subspace. Our theoretical analysis demonstrates that ProjQ preserves strictly greater model plasticity for downstream tasks compared to standard PTQ. Extensive experiments on LLaMA-2, Qwen2.5 and Qwen3 confirm that ProjQ consistently outperforms existing methods in both quantization error compensation and downstream task fine-tuning, achieving up to 2× lower evaluation loss for compensation and matching the performance of standard 4-bit baselines on language modeling tasks with only 3 bits. The code is available on https://github.com/yy9301/ProjQ .
We present LARA (Lightweight Additive Residual Adaptation), a method for efficient adaptation that operates in the residual stream of a frozen model rather than in its weights. Where LoRA adds an update of low rank to weight matrices, LARA reads the hidden state at a small set of layers and adds a correction of low rank back to the residual stream, leaving all base weights untouched. On a code fine-tuning task and on preference optimization (DPO), LARA matches LoRA at equal parameter counts. Because adaptation is a frozen base plus a residual, LARA exposes a scale γ, applied at inference, that interpolates smoothly between base and adapted behavior, a form of graded control that adaptation in weight space does not offer. Finally, because each behavior is a small residual module over a shared frozen base, many behaviors can be held resident at once and routed automatically per token. We place seven behaviors, six fine-tuned and one optimized for preference, on one frozen 1.5B model for roughly 33 MB of overhead, against one full model for each behavior. Because the base is untouched, behaviors are trained separately and selected per token rather than loaded on demand, which suits hosting many behaviors, and adding new ones, on a single model on a device.
Low-Rank Adaptation (LoRA) is the most widely adopted method for fine-tuning large language models. Notably, LoRA is inherently overparameterized: multiple pairs of low-rank factors can yield the same adapted weight matrix. We show--both theoretically and empirically--that these pairs exhibit significantly different condition numbers. As a result, converging to different loss minimizers directly impacts the convergence rate of LoRA. Building on this observation, we introduce Balanced Low-Rank Adaptation (BaLoRA), a variant of LoRA that projects iterates onto a balanced manifold. This manifold improves the conditioning of the loss landscape while preserving the adapted matrix. The projection step is computationally lightweight and integrates seamlessly into existing fine-tuning pipelines. Empirically, BaLoRA converges faster than standard LoRA and achieves superior performance across a range of fine-tuning tasks.