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Efficient Analytic Uncertainty Quantification for Multimodal Regression

NeurIPS 2026 · Probabilistic and scientific modeling › Uncertainty and Gaussian processes · approach: Epistemic versus aleatoric decomposition

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Extends variational Bayesian last-layer (VBLL) uncertainty quantification to semi-parametric quantile regression and classification restoration heads, giving analytic ELBOs and single-pass epistemic/aleatoric decomposition for multimodal regression.

Problem

Existing efficient UQ methods for regression assume unimodal (Gaussian) conditional label distributions, which fail on multi-modal targets (the "Ghost Value" pathology), while semi-parametric multi-modal methods lack efficient analytic epistemic uncertainty.

Approach

Combine a distance-preserving (spectral-normalized) backbone with a variational Bayesian last layer and derive closed-form expected asymmetric-Laplace log-likelihood (for quantile regression) and Jensen-bounded expected cross-entropy (for classification restoration), enabling single-pass analytic predictive densities and uncertainty decomposition.

Key results

• On Uber, QR-VBLL achieves NLL 1.7830 and ECE 0.0080 versus 2.3728 NLL for Vanilla QR among single-pass methods.
• On WeChat, CR-VBLL attains the best single-pass NLL (1.9189) and matches the 5x-cost ensembles on NLL except CR-Ensemble.
• On KuaiRec, QR-VBLL achieves the best single-pass CRPS (0.1964) and ECE (0.0122), outperforming MDN's CRPS of 0.2117.

Surprising findings

• MDN attains the lowest single-pass NLL on WeChat and KuaiRec but has much worse CRPS and, on WeChat, worse RMSE than other methods.
• Pure epistemic active learning (gamma=0) and BALD-VBLL underperform the hybrid strategy, suggesting that a downweighted aleatoric term helps.

Limitations

• Relies on dataset-specific heuristics for routing between QR and CR paths.
• No tight finite-sample error bounds for the full VBLL framework; only univariate targets.
• Naive ensembling over the variational posterior to reconstruct predictive PDF/CDF.

Experimental setup

Datasets: WeChat KuaiRec Uber

Benchmarks: WeChat KuaiRec Uber

Metrics: NLL RMSE CRPS ECE

Baselines: Gaussian SNGP VBLL DER MDN Vanilla QR CQR QR-MCD Vanilla CR CR-SNGP CR-MCD Ensemble

Builds on: Variational Bayesian Last Layer CREAD Conformalized Quantile Regression Spectral Normalization SNGP

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