DT-LoRA: Reaching Beyond the Tangent Space of Low-Rank Adaptation

Low-rank adaptation (LoRA) enables parameter-efficient fine-tuning by learning low-rank weight updates. Increasing rank expands adaptation capacity, but does not guarantee that all learned directions improve downstream performance. In a diagnostic experiment with rank-16 LoRA, we show that retaining only the top singular components achieves the best performance, whereas adding further components degrades it. Motivated by this, we propose Residual-Gradient Adaptation (ReGRA), a two-stage method for making more effective use of a fixed rank budget. After the first stage, ReGRA estimates task gradients on a calibration set and removes their projections onto the two-sided subspace induced by the first-stage update. It then aggregates the residual-gradient directions to initialize a second-stage adapter without changing the first-stage update. An orthogonality regularizer encourages the two updates to capture complementary directions. Both updates can be merged into the pretrained weights, incurring no additional inference overhead. Across diverse benchmark sets, ReGRA achieves stronger performance than alternatives with the same total rank and even matches LoRA adapters with twice the rank. Ablation studies and further analysis support the contribution of its core design choices. Code is available at https://github.com/Fanyao-Li/ReGRA.

Low-Rank Adaptation (LoRA) is a widely used approach to parameter-efficient fine-tuning (PEFT), yet a performance gap can remain relative to full fine-tuning (FFT). Many LoRA variants improve the initialization or optimization of low-rank factors. At each training step, however, their first-order weight-space directions are constrained by the current parameterization. We characterize the corresponding LoRA-accessible gradient space and show that it coincides with the tangent space induced by the current LoRA parameterization. This characterization yields an orthogonal decomposition of the full weight gradient at the current model parameters. We term the component orthogonal to this space the normal gradient. Based on this decomposition, we propose GDLoRA (Gradient-Decomposed Low-Rank Adaptation). GDLoRA reconstructs the full weight gradient from forward activations and backward signals, extracts its normal component, and directly updates the base weights with this component, while retaining standard AdamW optimization for the LoRA factors. GDLoRA incorporates complementary normal gradients without increasing standard LoRA's optimizer-state memory budget under matched adapter and optimizer configurations. Experiments on natural language understanding, mathematical reasoning, commonsense reasoning, and image classification show that GDLoRA consistently improves over LoRA and narrows the performance gap to FFT. The code is available at https://anonymous.4open.science/r/GDLoRA.

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.