Singular Geometry of Low-Rank Adaptation

Low-Rank Adaptation (LoRA) has become the standard mechanism for fine-tuning large pretrained models, yet its statistical properties remain only partially understood. Existing generalization results provide upper bounds of the form O~(sqrt(rd/n)) or O~(rd/n), but a matching lower bound is missing, and the question of how to choose the LoRA rank r has no formal answer. Both gaps are closed here. A local Rademacher argument establishes an upper bound of O~(rd/n) on the excess risk of the empirical risk minimizer over rank-r LoRA, whenever the target adaptation has rank at most r. A matching minimax lower bound of Omega(rd/n) is then proved via a Fano-type packing of the rank-r subspace of R^{d x d}; the bound applies to any estimator whose output lies in the rank-r LoRA class. Combining the two yields a rank-selection dichotomy. For the constrained empirical risk minimizer, the optimal rank equals the intrinsic rank r*, and over-ranking strictly hurts. For adaptive estimators of the nuclear-norm-then-truncate type, over-ranking is harmless and the rate saturates at Theta~(r* d / n) regardless of r. Taken together, the three results characterize the statistical complexity of LoRA fine-tuning within the well-specified locally quadratic regime, and identify the empirically observed over-parameterization penalty as a property of unregularized empirical risk minimization rather than of the LoRA class itself. Predictions of the theory are verified on a synthetic trace-regression benchmark and on real LoRA fine-tuning across three (model, task) configurations covering DistilBERT and RoBERTa on SST-2 and MRPC. All configurations exhibit the predicted U-shape in validation loss, with two showing statistically significant loss inflation at large ranks (paired permutation p = 0.016).

Low-rank adaptation (LoRA) has become the standard method for parameter-efficient fine-tuning of large pretrained models, yet theoretical explanations for why low-rank updates suffice remain incomplete. Most existing analyses rely on the Neural Tangent Kernel (NTK) regime, which linearizes the network around the pretrained weights. In this work, we take a different approach that requires no linearization: our key observation is that, once the backbone is frozen, the loss sees each adapted weight matrix only through a finite-rank linear map induced by the frozen activations. Trace-norm regularization then forces the optimizer to pick a low-rank matrix among all matrices that produce the same observed output. This yields global minimizers whose ranks are controlled by the rank of this map, which we call the bottleneck rank, rather than the sample size. For multi-head attention, we obtain tighter rank upper bounds by exploiting two additional invariances: row-wise softmax is insensitive to additive row constants in the score matrix, and value updates are observable only after attention weighting and frozen output projection. For the factorized LoRA objective with weight decay, we prove a complementary result on the nonconvex side: after an arbitrarily small generic positive semidefinite perturbation, every first-order stationary point is rank-deficient once the LoRA rank exceeds a threshold determined by the bottleneck rank. Finally, we show that global minimizers of the perturbed objective are near-optimal for the original trace-norm problem, and we provide uniform generalization bounds governed by the spectral structure of the frozen features.

The rank $r$ in LoRA is widely treated as a capacity control: a smaller rank is assumed to yield a simpler model that generalizes better. We show that, under hard per-factor norm budgets---the idealization of the weight decay and norm control used in practice---this intuition breaks down. The reason is structural: under such budgets, the updates LoRA can reach are exactly the matrices of rank at most $r$ inside a nuclear-norm ball, and every complexity and displacement functional we analyze is maximized over this set by a rank-one update---so the rank cap never binds. The consequences follow directly. The linear-readout model class we study is identical for every $r \ge 1$, its Rademacher complexity carries no dependence on $r$, and the distance the adaptation can move the source distribution obeys a rank-independent upper bound that we show is sharp. If rank does not control capacity, where does it act? We identify two places. Statistically, replacing the per-factor budgets with a joint budget on the product restores a data-dependent, rank-sensitive complexity bound---though the gain appears only for well-spread feature distributions, and the worst case remains rank-free. Spectrally, rank sets the price of adaptation: canceling the leading singular directions of the pretrained weight requires both sufficient rank and sufficient budget. We bound the smallest rank achieving a desired source--target alignment, with upper and lower bounds that match under two-sided spectral decay. Together, these results recast rank as governing which updates are reachable and what cancellation costs---not how much capacity the model has.