Certification Frontiers for Gaussian LoRA: Independent Priors, Posterior Risk, and Prediction-Preserving Balancing

Joyanta Jyoti Mondal, Ibne Farabi Shihab

Post-hoc Bayesian fine-tuning places Gaussians around trained low-rank adapters, yet a calibrated posterior does not by itself yield a useful generalization certificate. Such a posterior admits an informative PAC-Bayes certificate only when both the loss of its sampled predictors and its KL divergence from an admissible prior are small. In this research, we characterize this certification frontier for Gaussian LoRA posteriors and separate three interventions: changing the prior, changing the stochastic predictor, and changing only how its complexity is counted. First, an exact isotropic KL envelope eliminates the prior scale and yields a width threshold that excludes posterior widths before sampling, while a zero-KL floor identifies targets that no complexity reduction can reach at a measured risk bound. Second, we minimize KL in closed form over the full $GL(r)$ symmetry of the low-rank factors, leaving every sampled adapter product unchanged, and derive the noncentral objective required when the prior center is trained on an independent split. On a small-pool RoBERTa audit of 567 configurations, the recorded 64-draw summaries imply a certificate floor of $0.7298$ even with zero…