On the Construction and Implications of Low-Loss Valleys in LoRA-based Bayesian Inference

Low-Rank Adaptation (LoRA) enables parameter-efficient fine-tuning of large models by decomposing weight updates into low-rank matrices, significantly reducing storage and computational overhead. While effective, standard LoRA lacks mechanisms for uncertainty quantification, leading to overconfident and poorly calibrated models. Bayesian variants of LoRA address this limitation, but at the cost of a significantly increased number of trainable parameters, partially offsetting the original efficiency gains. Additionally, these models are harder to train and may suffer from unstable convergence. In this work, we propose a novel framework for parameter-efficient Bayesian fine-tuning, demonstrating that effective uncertainty quantification can be achieved in very low-dimensional parameter spaces. The proposed method achieves strong performance with improved calibration and generalization while maintaining computational efficiency. Our empirical findings show that, with the appropriate projection of the weight space uncertainty can be effectively modeled in a low-dimensional space, and weight covariances exhibit low ranks.

Low-Rank Adaptation (LoRA) has become the standard for fine-tuning large pre-trained models at reduced computational cost. However, its low-rank point-estimate updates limit expressiveness, leave a persistent gap relative to full fine-tuning accuracy, and provide no built-in uncertainty quantification, limiting its applicability in settings where reliability matters as much as accuracy. We introduce BaLoRA, a Bayesian extension of LoRA with a novel input-adaptive Bayesian parameterization of LoRA matrices that adds minimal parameters and compute. Surprisingly, not only does the Bayesian extension yield well-calibrated uncertainty estimates, but the adaptive noise injection underlying our approach also significantly improves prediction accuracy, narrowing the gap with full fine-tuning across both natural language reasoning and vision tasks. When applied to band gap prediction in metal-organic frameworks, BaLoRA produces zero-shot test-time uncertainty estimates that correlate more strongly with model error than a trained ensemble of LoRA models, and improve monotonically with compute without sacrificing accuracy.

While Low-rank adaptation (LoRA) enables highly efficient fine-tuning by constraining task-specific updates to fixed low-rank subspaces, this rigid design limits representational flexibility and often results in overconfident predictions and miscalibrated uncertainty, especially in low-data regimes. Recent Bayesian LoRA variants improve uncertainty estimation by modeling posterior distributions over adaptation parameters. However, these approaches typically rely on fixed or heuristically determined ranks, overlooking the inherently context-dependent nature of adaptation capacity. In this paper, we propose BaRA, a Bayesian Adaptive Rank Allocation framework for parameter-efficient fine-tuning. Drawing inspiration from probabilistic topic models, BaRA dynamically allocates adaptation capacity by activating a sparse, context-dependent subset of disentangled latent factors, enabling instance-wise variation in effective rank. This Bayesian formulation provides principled, data-driven capacity control, mitigating over-parameterization while preserving expressiveness. Beyond the modeling contribution, we provide a complexity-theoretic generalization analysis showing that the generalization gap of BaRA depends on the learned joint effective rank $\bar{s}_{Φ,θ}$ induced by the global-local gate, rather than the maximum rank $r$. This result explains why sparse adaptive rank allocation can reduce the effective hypothesis complexity while preserving input-dependent expressiveness. Extensive experiments on diverse natural language benchmarks demonstrate that BaRA consistently improves predictive performance, robustness, and uncertainty calibration compared to standard LoRA and existing Bayesian LoRA variants.