CeRA: Breaking the Linear Ceiling of Low-Rank Adaptation with Non-linearity Retained at Inference

Low-Rank Adaptation (LoRA) is a popular parameter-efficient fine-tuning (PEFT) method that restricts weight updates to low-rank adapters, introducing a fixed low-rank inductive bias by optimizing in a low-dimensional subspace. In this work, we question whether a fixed-rank constraint is the most effective inductive bias for parameter-efficient fine-tuning. We introduce Learnable Rank LoRA (LR-LoRA), a PEFT method in which the adapter rank is learned during the training process. Instead of prescribing a uniform rank for all adapter layers, LR-LoRA allows the optimizer to determine the appropriate rank for each layer. Using this approach, we find substantial layer-wise variation in the learned ranks, with the attention and MLP layers in the transformer models exhibiting systematically different rank preferences. Across a range of language understanding and commonsense reasoning benchmarks, LR-LoRA achieves state-of-the-art performance in most settings and consistently outperforms strong PEFT baselines, demonstrating that a learnable rank provides a more flexible and effective inductive bias than fixed-rank adaptations.

Low-rank adaptation (LoRA) assigns a uniform rank to every adapted weight matrix - a practical convenience that ignores a fundamental reality: different layers contribute unequally to task adaptation. We address this with a lightweight engineering solution: before fine-tuning begins, run eight calibration backward passes, compute the gradient variance of each LoRA-B matrix as a proxy for layer informativeness, and redistribute the rank budget proportionally. The resulting adapter is a standard LoRA with a per-layer rank pattern - no new parameters, no training overhead, no changes to serving infrastructure. We implement this via an efficient approximation of the empirical Fisher Information Matrix (eFIM) diagonal, restricted to LoRA adapter matrices only, which reduces memory cost by approximately 256x compared to full-model Fisher estimation. On GLUE with DeBERTa-v3-base, FIM-LoRA matches LoRA (88.6 vs. 88.7) at the same parameter budget, and on commonsense reasoning with LLaMA-3-8B reaches 68.5 vs. 68.7 for LoRA. The per-layer rank maps are interpretable: value projections and early-to-middle layers consistently receive higher rank, consistent with established findings on transformer layer roles.

Low-rank adaptation (LoRA) has emerged as the de facto standard for parameter-efficient fine-tuning (PEFT) of foundation models, enabling the adaptation of billion-parameter networks with minimal computational and memory overhead. Despite its empirical success and rapid proliferation of variants, it remains elusive which architectural choices, optimization techniques, and deployment constraints should guide practical method selection. This overview revisits LoRA through the lens of signal processing (SP), bridging modern adapter designs with classical low-rank modeling tools and inverse problems, as well as highlighting how SP principles can inform principled advances of fine-tuning approaches. Rather than providing a comprehensive enumeration and empirical comparisons of LoRA variants, emphasis is placed on the technical mechanisms underpinning these approaches to justify their effectiveness. These advances are categorized into three complementary axes: architectural design, efficient optimization, and pertinent applications. The first axis builds on singular value decomposition (SVD)-based factorization, rank-augmentation constructions, and cross-layer tensorization, while the second axis deals with initialization, alternating solvers, gauge-invariant optimization, and parameterization-aware methods. Beyond fine-tuning, emerging applications of LoRA are accounted across the entire lifecycle of large models, ranging from pre- and post-training to serving/deployment. Finally, open research directions are outlined at the confluence of SP and deep learning to catalyze a bidirectional frontier: classical SP tools provide a principled vocabulary for designing principled PEFT methods, while the unique challenges facing modern deep learning, especially the overwhelming scale and prohibitive overhead, also offer new research lines benefiting the SP community in return.