ComplexOrlicz: Holomorphic Gradient Orthogonalization for Tail-Adaptive Uncertainty Beyond Gaussian Limits

Yassir Laaouach, Amine Marzouki

Keywords: Uncertainty quantification, heteroscedastic uncertainty, optimization, uncertainty calibration

Abstract: Accurate uncertainty quantification remains a central challenge in neural regression: heteroscedastic models trained with Gaussian NLL suffer from gradient entanglement between mean and variance, and collapse under non-Gaussian noise. Existing remedies split the problem, β-NLL and dual-head architectures provide only approximate decoupling and still degrade once the noise departs from Gaussian, while robust losses improve point estimates but fail to deliver calibrated uncertainty. In practice, these issues are intertwined: neglecting tail behavior inflates variance, which then corrupts mean learning, so fixing one side alone is insufficient. We introduce ComplexOrlicz, a principled framework that resolves both within a single analytic formulation. Predictions are embedded as $z = µ + iκσ$ and trained with a convex Orlicz-family loss whose near-holomorphic structure enforces Cauchy–Riemann conditions, yielding exact orthogonal mean/variance gradients without stop-gradients or reweighting. A single shape parameter smoothly interpolates between Gaussian, Laplace, Student-t, and Cauchy, adapting to tail distributions without tuning. Across benchmarks, ComplexOrlicz matches Gaussian NLL in compute while reducing RMSE by up to 27% and halving calibration error. On Bitcoin and NYC Taxi, it cuts RMSE by 28% and 19% with large calibration gains, and even on near-Gaussian datasets it matches baselines while consistently improving calibration.

Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Submission Number: 25495