Rethinking the Rank Threshold for LoRA Fine-Tuning

A recent landscape analysis of LoRA fine-tuning in the neural tangent kernel regime establishes a sufficient condition $r(r+1)/2 > KN$ on the LoRA rank $r$ for the absence of spurious local minima under squared-error loss, prescribing $r \geq 12$ on canonical few-shot RoBERTa setups. The condition is stated for general output dimension $K$, so its sharpness in any particular regime, and its practical implication for the cross-entropy loss actually used in fine-tuning, are open. We give three results that together reduce the prescribed rank to $r = 1$ for binary classification in this regime. First, replacing the symmetric Sard-form count with the non-symmetric LoRA manifold dimension yields a strictly weaker capacity requirement, $r(m+n) - r^2 > C^* \cdot KN$ with $C^* \approx 1.35$ under Gaussian-iid features, satisfied at $r = 1$ on canonical setups. Second, in the cross-entropy setting the Polyak--Łojasiewicz inequality removes the rank threshold entirely. Third, a Rademacher-complexity bound predicts rank-one variance optimality precisely when the bias term is saturated, which is the case for binary classification but not for $K > 2$. Empirically, across four GLUE-style binary tasks, three encoder architectures, and at scale on RoBERTa-large, rank one is competitive with the existing prescription $r = 12$; on multi-class MNLI the optimal rank shifts above one, also as predicted. The binary-regime guarantees are conditional on standard NTK assumptions; the multi-class extension is left to future work.