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This research challenges the conventional wisdom that LoRA rank directly controls model capacity, demonstrating instead that under standard norm budgets, the effective complexity is often rank-independent and maximized by rank-one updates. The study reveals that rank primarily dictates the cost of canceling pretrained singular directions and defines the set of reachable updates rather than acting as a simple generalization knob. These findings are critical for practitioners aiming to optimize adaptation strategies and understand the true constraints of low-rank fine-tuning methods.
What Does the Rank Buy? A Spectral and Distributional Analysis of Low-Rank Adaptation
arXiv:2609.32002v1 Announce Type: cross Abstract: 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.
Title:What Does the Rank Buy? A Spectral and Distributional Analysis of Low-Rank Adaptation
Abstract: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.
Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2609.32002 [cs.AI]
(or arXiv:2609.32002v1 [cs.AI] for this version)
https://doi.org/10.48550/arXiv.2609.32002 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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