One Rate Is Not Enough: Adaptive Anisotropic Learning Rates for LoRA Fine-Tuning

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.

Low-Rank Adaptation (LoRA) is often highly sensitive to learning-rate (LR) choices. Recent empirical studies suggest that, once LRs are carefully tuned, vanilla LoRA can remain competitive with many proposed LoRA variants. This shifts the central practical question from designing ever more variants to reducing the cost and brittleness of LR selection. LoRA+ addresses this issue with asymmetric LRs, using a larger LR for the up-projection matrix $B$ than for the down-projection matrix $A$. However, the ratio $η_B / η_A$ itself becomes an additional hyperparameter: large ratios can improve adaptation, but under the standard LoRA initialization they can also cause unstable training and ratio-dependent performance collapse. We identify the source of this brittleness through a signal--noise decomposition of LoRA dynamics. Under standard \initA, the update contains an initialization-induced noise term whose magnitude can be affected by the LR for $B$, $η_B$, the initialization scale $β$, and the adapter rank $r$. In practical LLM fine-tuning, this non-vanishing noise can limit the performance gains expected from LoRA+ style asymmetric LRs. To suppress this effect, we propose \initAA, a simple initialization that scales the variance of $A$ as $σ_A^2 \propto n^{-2}$ while keeping $B_0=0$. Under matched rank and training time for both initializations, this reduces the initialization-noise energy by an additional factor of $1/n$, stabilizing asymmetric LoRA training while preserving zero initial adapter output. Empirically, \initAA substantially widens the stable LR region across initialization scales, adapter ranks, and LoRA update multipliers. Across tasks and models, \initAA reduces best-ratio drift, prevents the high-ratio collapse observed with standard \initA, and makes the width-based guideline $η_B \approx n η_A$ a useful starting point for finite pretrained models.

Low-Rank Adaptation (LoRA) has become a widely adopted parameter-efficient fine-tuning method for large language models, with its effectiveness largely influenced by the allocation of ranks and scaling factors, as well as initialization. Existing LoRA variants typically address only one of these factors, often at the cost of increased training complexity or reduced practical efficiency. In this work, we present Task-aware Low-Rank Adaptation (TLoRA), a unified framework that jointly optimizes initialization and resource allocation at the outset of training. TLoRA introduces a data-driven initialization strategy that aligns the LoRA $A$ matrix with task-relevant subspaces by performing singular value decomposition on the product of pre-trained weights and input activation covariance. After this, the $A$ matrix is frozen, and only the $B$ matrix is trained. Furthermore, TLoRA employs a sensitivity-based importance metric to adaptively allocate ranks and scaling factors across layers under a fixed parameter budget. We conduct extensive experiments that demonstrate TLoRA consistently performs excellently across various tasks, including natural language understanding, commonsense reasoning, math reasoning, code generation, and chat generation, while significantly reducing the number of trainable parameters.