Water-Filling LoRA: Data-Free Rank Allocation via Conditional Estimation–Approximation Bounds

Choosing a LoRA rank for each layer is a resource-allocation problem when the total number of trainable parameters is fixed. We analyze this problem using a PAC-Bayes bound for Gibbs risk and an explicit spectral-tail approximation model. Under a common norm envelope and the stated approximation assumptions, the continuous optimum is proportional to pretrained stable rank, with clipping at the rank limits. The contribution is this conditional budgeted analysis, rather than the use of stable rank itself. We distinguish the continuous optimum, the proportionally rounded implementation, and a budget-feasible integer rule with a computable surrogate suboptimality certificate. An exact integer audit on Pythia-160M and Pythia-410M exposes a small budget overrun in the rounded 410M allocation and verifies feasibility of the budget-feasible rule. In four-seed adapter-only experiments, the proportionally rounded heuristic has lower mean validation perplexity than AdaLoRA on both models (28.353 versus 29.898 and 15.373 versus 16.078), while its advantage over uniform rank is not resolved. A separate four-seed comparison of the budget-feasible integer rule finds lower mean PPL than proportional rounding at 160M and lower mean PPL than uniform at 410M, but no consistent advantage across both comparators and models. Neither comparison verifies the assumptions linking the surrogate to downstream risk. The analysis identifies when a data-free allocation is optimal for a specified bound and where empirical validation is still necessary.