How Much Rank Does LoRA Need? Rank-Error Bounds for Transformer Attention

Choosing the rank of a low-rank adaptation (LoRA) update is usually an empirical task. In this paper, we provide a task-dependent theory of the approximation error achievable at each LoRA rank for Transformer attention. We fix a pretrained attention head, a target attention function, and a distribution over inputs from the downstream task, and bound the smallest expected Kullback--Leibler (KL) error achievable by a rank-$r$ query LoRA update. When target attention probabilities are bounded away from zero, we prove a lower bound of the error proportional to $ψ(\|d\|_2)$, where $d$ is the difference between candidate and target attention scores and $ψ(t)=\min\{t^2,t\}$. We also prove an unconditional upper bound $\min\{\|d\|_2^2/4,\sqrt2\|d\|_2\}$. Under explicit realizability, geometry, and moment conditions, we then bound the best rank-$r$ error between an explicit multiple of $ψ(\sqrt{T_r})$ and $\min\{T_r/4,\sqrt{2T_r}\}$, where $T_r$ is the downstream-weighted tail energy of the target update. We also provide target-Fisher bounds when candidate scores remain within a fixed range of the target scores, and an unrestricted lower bound when a subset of tokens carries most of the probability mass. These spectral bounds describe finite-score approximation. We then construct explicit families in which softmax saturation makes the rank required to match the attention function strictly smaller than the rank required to match the finite logits. Finally, we extend the analysis to fused multi-head LoRA and joint query/key updates, exposing the effects of rank sharing and query/key factorization constraints.

The Intruder Threshold: A Spectral Law for LoRA Fine-Tuning

LoRA fine-tuning can create intruder dimensions: new leading singular vectors of the updated weight matrix $W+BA$ that are nearly orthogonal to all pretrained singular vectors and that drive catastrophic forgetting. Since their discovery, no theory has predicted, layer by layer on measured spectra, when they appear. We derive a per-layer critical update strength $s^\ast=\barθ/(γσ_1(BA))$, computed from the measured spectrum of $W$ alone through the rectangular spiked-deformation transform, together with an exact secular-equation characterization of the updated spectrum, with no fitted parameters. In a pre-specified study spanning four dense Transformer families, a state-space model, a mixture-of-experts model, and an encoder-decoder (18 adapters, 9{,}840 layer scans), the law localizes the empirical threshold within a factor of two on $82\%$ of layers, separates intruder-bearing from intruder-free layers at deployment with a mean AUC of $0.89$, holds unchanged on six third-party adapters, and predicts where WikiText-2 perplexity begins to degrade; a combination of the two pre-specified edge evaluations reaches $98\%$ and is confirmed out-of-bag on the external adapters ($0.997$). Full fine-tuning disperses its update far below the threshold of every layer, which resolves the asymmetry between LoRA and full fine-tuning. Norm-matched interventions confirm that threshold-crossing layers, rather than update magnitude, carry the forgetting, and a spike-budget rule derived from the thresholds, requiring one SVD and no validation sweeps, reduces forgetting by $62\%$ on the most fragile model at no task cost.

On the Convergence of Stochastic Low-Rank Adaptation

Low-rank adaptation (LoRA) optimizes $J(B,A)=\mathcal L(W_\mathrm{base}+sBA)$ over two adapters $B \in \mathbb{R}^{m \times r}$ and $A \in \mathbb{R}^{r \times n}$ that form a low-rank update to a frozen pretrained weight matrix $W_\mathrm{base} \in \mathbb{R}^{m \times n}$. The prior analysis shows LoRA-GD takes $\exp\{\mathcal{O}(ε^{-2})\}$ oracle calls to find an $ε$-stationary point such that $\|\nabla J(B,A)\|\leq ε$ in the deterministic setting. We sharpen the analysis and show that $\mathcal{O}(ε^{-4})$ full-gradient evaluations suffice for the same first-order criterion. We further study stochastic LoRA under unbiased gradient estimates and finite variance. We propose LoRA-NSGDM, which finds an $ε$-stationary point with $\mathcal{O}(ε^{-8})$ stochastic oracle complexity. Under the additional mean-square smoothness condition, we use variance reduction strategy and propose LoRA-STORM, which improves the stochastic oracle complexity to $\mathcal{O}(ε^{-6})$.