Learning Context-conditioned Gaussian Overbounds for Convolution-Based Uncertainty Propagation

2026-05-15

AI Summary

This work addresses the limitations of traditional uncertainty quantification methods in safety-critical scenarios, where achieving both provable conservativeness and tightness after compositional operations—such as linear superposition or convolution—is challenging. Existing global Gaussian overbounds suffer from excessive redundancy and fail to adapt to context-dependent error distributions. To overcome this, the paper introduces the first feature-conditional Gaussian overbound learning framework, wherein neural networks generate context-aware mean and scale parameters. The approach guarantees provable conservativeness over a finite quantile grid and, under mild regularity assumptions, extends this guarantee continuously into the tail. A joint loss combining a Wasserstein-style distance and quantile-based conservativeness constraints enables uncertainty propagation while significantly improving tightness. Experiments demonstrate that the learned overbounds consistently outperform conventional methods on both synthetic and real-world datasets, including multipath, ionospheric, and tropospheric residual errors.

Abstract

Uncertainty quantification is essential in safety-critical settings--from autonomous driving to aviation, finance, and health--where decisions must rely on conservative bounds rather than point estimates. Predictor-level intervals (e.g., from quantile regression, conformal prediction, variance networks, or Bayesian models) generally do not compose: adding two per-variable intervals need not yield a valid interval for their sum or preserve coverage. In aviation, Gaussian overbounding replaces complex error distributions with a conservative Gaussian whose tails dominate the truth, so conservatism propagates through linear operations. Yet classical overbounds are global, often overly conservative, and hard to adapt to feature-conditioned errors. We propose a unified learning framework that trains neural networks to produce context-aware Gaussian overbounds--mean and scale--with provable conservatism on a finite quantile grid and, under three explicit regularity assumptions, continuous-tail conservatism on a certified interval. Our overbounding loss enforces conservativeness at selected quantiles while penalizing distributional distance with a Wasserstein-style term. The learned bounds support conservative linear-combination and convolution analysis on the enforced grid, and on the certified interval when assumptions hold, while being less redundant than traditional methods. We provide a scoped analysis of discrete-to-continuous conservatism and compact-domain objective regularity, and validate on synthetic data and real-world datasets, including multipath, ionospheric, and tropospheric residual errors. Across these settings, the method yields tighter bounds while maintaining conservatism on the enforced grid and in experiments. The framework is modality-agnostic and applicable to learning systems that require conservative, feature-conditioned uncertainty estimates in dynamic environments.

Problem

• uncertainty quantification
• Gaussian overbounding
• conservatism
• feature-conditioned errors
• interval composition

Innovation

• Gaussian overbounding
• context-conditioned uncertainty
• conservative uncertainty propagation
• Wasserstein regularization
• quantile-based learning