Beyond the Mean: Distribution-Aware Loss Functions for Bimodal Regression

Efficient uncertainty quantification (UQ) is essential for trustworthy large-scale learning. Existing UQ methods for regression tasks mainly operate under the assumption that the conditional label marginal satisfies single-peak parametric models, e.g., Gaussians, where the negative log-likelihood function simplifies to the mean square error. However, such single-peak assumptions fail in regression tasks featuring multi-modal distributions. On the other hand, semi-parametric methods which achieve strong regression performance for multi-modal distributions often lack efficient quantification on their prediction variances. In this work, we extend UQ techniques based on Variational Bayesian Inference (VBI) to two widely used semi-parametric regression models that yield histogram-like reconstructions of the conditional label densities: Quantile Regression (QR) and Classification Restoration (CR). Our approach introduces a unified, distribution-agnostic framework that simultaneously achieves accurate estimation of complex conditional distributions and highly efficient UQ. Theoretically, our method is grounded in novel formulations of QR and CR within the VBI framework, yielding analytic Evidence Lower Bounds (ELBO) to streamline training and a closed-form or analytically approximated predictive density for efficient inference. Empirically, we evaluate our methods on three large-scale regression benchmarks with multi-modal label distributions. Our framework outperforms state-of-the-art multi-modal regression baselines, and even matches predictive performance of computationally expensive ensemble models. Furthermore, by leveraging epistemic uncertainty estimation, our approach enables highly data-efficient active learning strategies.

Large language models (LLMs) have emerged as flexible regressors capable of predicting real-valued quantities from heterogeneous inputs. Yet most LLM regression objectives optimize predictions independently, often yielding poor calibration. We introduce Distribution-Aware Reward (DAR), an on-policy reinforcement learning objective that instead jointly evaluates the empirical predictive distribution formed by multiple predictions for the same input. To translate this distribution-level objective into rollout-level rewards, we assign each prediction credit based on its leave-one-out contribution to the quality of the overall predictive distribution. This encourages predictions that are well-centered and appropriately dispersed around the target. We evaluate on three regression settings: a synthetic task probing interpolation and extrapolation, and two real-world scientific tasks involving code and molecular data. Across tasks, DAR produces better-calibrated uncertainty estimates while consistently reducing prediction error and improving ranking quality over supervised fine-tuning and pointwise reinforcement learning. Together, these results highlight the benefits of distribution-aware training for LLM regression.

Implicit neural representations (INRs) offer compact encoding of volumes, but as lossy approximators, inevitably have prediction errors. We consider INRs that can simultaneously encode relative error scales by predicting distributions using tools from uncertainty estimation. Typically, uncertainty estimation relies on computationally expensive approaches or on predefined parametric assumptions about the predictive distribution (e.g., Gaussian). In this study, we propose a lightweight method that reformulates regression-based INR training as a classification task by discretizing continuous targets into bins, enabling flexible distribution modeling to capture complex multimodal behaviors. We analyze the trade-off between regression and classification for INR training and demonstrate that the classification setting tends to achieve high reconstruction quality and competitive error awareness through uncertainty estimation, compared to regression-based approaches.