Distributional Regression

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13 papers5 papers in the last four weeks, with none the four weeks before. 0.0% of all new papers.

Oct 17, 2026

Scalable and Adaptive Distribution Regression via Random Features

Distribution regression seeks to learn mappings from probability distributions to response variables and has emerged as a unifying framework for kernel methods, multi-instance learning, and set-valued learning. Despite its success, kernel-based distribution regression remains computationally expensive and sensitive to kernel choice, motivating scalable and adaptive representations. Building on random feature approximations, we develop a finite-sample analysis of Random Feature Distribution Regression (RaDR) under two-stage sampling. Our perturbation analysis separates the effects of distribution-level sampling, finite within-bag sampling, and random feature approximation. We establish sufficient feature-budget conditions under which RaDR preserves the order of our excess-risk upper bound for exact two-stage kernel distribution regression while maintaining computational efficiency. We further propose Adaptive Random Feature Distribution Regression (A-RaDR), a unified score-guided framework that adaptively allocates random features according to downstream learning objectives. Experiments on synthetic benchmarks and 41 real-world datasets demonstrate that RaDR delivers substantial computational savings while maintaining competitive predictive performance, whereas A-RaDR further improves learning accuracy in distribution regression while retaining the computational speed up. Together, these results establish a practical and theoretically grounded framework for adaptive, scalable distribution learning

Yinsong Wang, Xiao Liu, Yu Ding

School of Mechanical, Industrial and Material Engineering, University of Nevada Reno, Reno 89557, USA. · H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta 30332, USA.

Distributional RegressionRandom Feature MethodsKernel MethodsStatistical Learning Theory
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Oct 12, 2026

Distributional Regression for Arbitrary Response Variables Using ReLU Neural Networks

This paper develops a unified framework for distributional regression with arbitrary response variables. Rather than relying on scalar thresholds or Euclidean structure, we model conditional event probabilities indexed by measurable sets, which extends distributional regression to general response spaces while retaining a direct connection to classical event-based formulations. Within this framework, we study two complementary estimators: a preliminary estimator based on separate regression of event indicators, and a constrained estimator that enforces inclusion-consistent predictions through a directed acyclic graph (DAG) over a finite set family. We establish general error bounds for both estimators and further specialize the results to dense ReLU networks, yielding explicit convergence rates overcoming the curse of dimensionality. Simulation studies and real-data experiments further demonstrate the practical effectiveness of the proposed framework.

Junpeng Ren, Carlos Misael Madrid Padilla, Oscar Hernan Madrid Padilla

Department of Statistics and Data Science, University of California, Los Angeles, Los Angeles, CA, USA · Department of Statistics and Data Science, Washington University in St Louis, St. Louis, MO, USA

Distributional RegressionConditional Distribution EstimationNonparametric RegressionStatistical Learning Theory
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Oct 1, 2026

Generalized Engression Models

We consider estimating the conditional distribution of a multivariate outcome given covariates when its coordinates may be continuous, binary, categorical, ordinal or rankings, and are conditionally dependent on one another. Different statistical methods have been developed for each outcome type, and most of them target a summary of the conditional distribution, such as the mean of each coordinate, rather than the joint distribution of the outcome vector. We develop generalized engression models, a unified nonparametric distributional regression framework for outcomes of any type. The proposed method builds upon engression, a scoring-rule-based deep generative model, and introduces a data-type-specific link function and a stochastic perturbation that smooths the loss, enabling gradient-based training even with discontinuous links. We establish universal representation results for continuous, discrete and mixed outcomes. In simulations and in two applications, 242 species in a community ecology benchmark and a 17-dimensional mixed-type health outcome, the method matches type-specific models on marginal scores, improves on them on the joint distribution, and matches or exceeds purpose-built state-of-the-art joint species distribution models. Software is available in Python.

Xinwei Shen, Zijian Guo, Francis Bach

Department of Statistics, University of Washington · Center for Data Science, Zhejiang University · Inria, ´Ecole Normale Sup´erieure, PSL Research University

Conditional Distribution EstimationDistributional RegressionConditional Generative Modeling
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Sep 24, 2026

Sufficiently Reduced Distributional Regression

We propose Sufficiently Reduced Distributional Regression (SRDR), a generative method that combines conditional distribution estimation with nonlinear sufficient dimension reduction (SDR). It builds on a characterization of sufficiency through strictly proper scoring rules: a dimension reduction is sufficient if and only if predicting the response from the reduced covariates incurs no loss in expected score relative to the full covariates. Sufficient dimension reduction thus becomes a risk minimization problem. SRDR jointly trains a dimension reduction map and a generative prediction model by minimizing the energy score, which can be estimated by sampling without density evaluation or adversarial training. The framework extends to multi-environment data and to classification. We prove that the estimated conditional distributions converge in energy distance to the true ones, which implies that the learned representation is asymptotically sufficient. In simulations and applications to CT slice localization, superconductivity, and digit classification, SRDR recovers low-dimensional sufficient structure and matches or outperforms state-of-the-art nonlinear SDR methods in representation quality and predictive performance.

Alexander Henzi, Tiange Liu, Xinwei Shen

Tsinghua University, Department of Statistics and Data Science · University of Washington

Conditional Distribution EstimationInverse Dimensionality ReductionDimensionality ReductionDistributional RegressionConditional Generative Modeling
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Sep 22, 2026

PROSWIN: Probabilistic Solar Wind Speed Forecasting Using Deep Distributional Regression From Solar Images

Accurately predicting fast solar wind conditions is challenging, as uncertainties are large and unquantified by traditional single-value prediction models. In particular, the risks of high-speed solar wind streams (HSSs), which can cause damage to technological infrastructure, cannot be reliably assessed without probabilistic forecasts. We present PROSWIN, a probabilistic machine learning model that forecasts the hourly solar wind speed (SWS) at Earth with a four-day lead time. The approach combines solar images and magnetograms using a deep neural network coupled to a distributional regression algorithm. Because standard error metrics underweight the relevance of HSS peaks, we further introduce the prediction score, a model-selection metric that jointly rewards timeline and HSS peak accuracy. On 14 years of data, our forecast achieves very well-calibrated uncertainties (<1% average deviation). Using the continuous ranked probability score (CRPS), a metric that assesses distributional accuracy, we obtain a timeline CRPS of 41.0 km/s, an HSS peak CRPS of 45.3 km/s, and a prediction score of 42.3 km/s. We find that the 171 Å channel is an important complement to the typically used 193 Å and 211 Å channels and that the prediction score for model selection improves the applicability of the model. Compared to selected models from the literature, ours is the only one that is accurate for both timeline and HSS peak values, rather than trading one off against the other. These results support the advantages of probabilistic over single-value solar wind models. The introduced methods are also transferable to other forecasting problems.

Daniel Collin, Yuri Shprits, Luca Chiarabini +3

Space Physics and Space Weather, GFZ Helmholtz Centre for Geosciences, Potsdam, Germany · Department of Electrical Engineering and Computer Science, Technical University of Berlin, Berlin, Germany · Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany +5

Time Series ForecastingDistributional RegressionProbabilistic Time Series ForecastingSolar Physics
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Jul 26, 2026

Distributional Split Criteria for Random Forests: Extensions, Shrinkage, and the Robustness of Mean Splitting

Distributional random forests replace mean-based CART splitting with criteria that compare the full conditional response distribution in candidate children. We implement and systematically study a family of such criteria inside a single honest-forest implementation: isotropic random-Fourier-feature maximum mean discrepancy (MMD), an anisotropic diagonal-bandwidth variant, an adaptive per-split frequency-selection variant, and a non-kernel sliced-Wasserstein criterion, together with post-hoc kernel-mean shrinkage of the forest weights. Using paired-seed comparisons across synthetic quantile mechanisms, real univariate benchmarks, a California-housing subsample curve, and multivariate synthetic and real responses, we characterize where each extension pays. Three findings recur. First, among distributional criteria ordinary isotropic MMD is already close to best in class: the anisotropic, adaptive-frequency, and sliced-Wasserstein extensions, and post-hoc shrinkage, do not systematically improve on it. Second, on scalar tabular regression mean-based CART splitting remains the robust default and wins many cells. Third, multivariate responses are the regime where distributional splitting clearly earns its keep, most sharply on a pure-dependence copula where the energy score separates the criteria even though marginal CRPS does not. The evidence supports a simple allocation story: distributional splitting helps only when non-location structure is both present and estimable; otherwise it dilutes split-selection power away from the mean. All criteria, the honest forest, and the paired-comparison harness are implemented in the open-source drforest library, whose Rust-backed split search makes broad criterion sweeps inexpensive.

Silas Koemen

Maximum Mean DiscrepancyRandom ForestDistributional Regression
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Jun 28, 2026

Generalization Analysis of Transformers in Distribution Regression

In recent years, models based on the Transformer architecture have seen widespread applications and have become one of the core tools in the field of deep learning. Numerous successful techniques, such as parameter-efficient fine-tuning and efficient scaling, have been proposed surrounding their applications to further enhance performance. However, the success of these strategies has always lacked the support of rigorous mathematical theory. To study the underlying mechanisms behind Transformers and related techniques, we first propose a Transformer learning framework motivated by distribution regression, with distributions being inputs, connect a two-stage sampling process with natural language processing, and present a mathematical formulation of the attention mechanism called attention operator. We demonstrate that by the attention operator, Transformers can compress distributions into function representations without loss of information. Moreover, with the advantages of our novel attention operator, Transformers exhibit a stronger capability to learn functionals with more complex structures than convolutional neural networks and fully connected networks. Finally, we obtain a generalization bound within the distribution regression framework. Through the aforementioned theoretical results, we further discuss some successful techniques emerging with large language models (LLMs), such as prompt tuning, parameter-efficient fine-tuning, and efficient scaling. We also provide theoretical insights behind these techniques within our novel analysis framework.

Peilin Liu, Ding-Xuan Zhou

School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia

Neural Network GeneralizationTransformer AttentionDistributional RegressionStatistical Learning Theory
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May 30, 2026

Semi-Supervised Learning with Noisy Proxy Covariates: Generalization Bounds and Distribution Regression

In many modern machine learning pipelines, abundant pretrained representations serve as noisy proxy covariates, while task-specific labels remain scarce. We study semi-supervised regression in this setting, and propose a simple two stage estimator that learns kernel eigenfeatures from all proxy covariates and fits a ridge predictor on labeled data. We derive finite sample bounds showing that fast labeled sample rates are recovered when proxy perturbation is controlled and unlabeled proxy covariates are sufficiently abundant. We also show that distribution regression is a direct special case, with analogous guarantees when the finite bag size is large enough. Experiments show consistent gains over supervised and semi-supervised baselines, especially in low label regimes.

Kwangho Kim, Jisu Kim

Department of Statistics, Korea University, Seoul, Korea · Department of Statistics, Seoul National University, Seoul, Korea

Kernel Ridge RegressionDistributional RegressionSemi-Supervised LearningStatistical Learning Theory
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May 2, 2026

Injecting Distributional Awareness into MLLMs via Reinforcement Learning for Deep Imbalanced Regression

Multimodal large language models (MLLMs) struggle with numerical regression under long-tailed target distributions. Token-level supervised fine-tuning (SFT) and point-wise regression rewards bias learning toward high-density regions, leading to regression-to-the-mean behavior and poor tail performance. We identify the lack of cross-sample relational supervision as a key limitation of existing MLLM training paradigms. To address it, we propose a distribution-aware reinforcement learning framework based on Group Relative Policy Optimization, which introduces batch-level comparison-based supervision via the Concordance Correlation Coefficient-based reward to align predicted and ground-truth distributions in terms of correlation, scale, and mean. The framework is plug-and-play, requiring no architectural modification. Experiments on a unified suite of long-tailed regression benchmarks show consistent improvements over SFT and existing MLLM regression methods, with particularly strong gains in medium- and few-shot regimes.

Yao Du, Shanshan Song, Xiaomeng Li

The Hong Kong University of Science and Technology, Hong Kong SAR, China.

Multimodal Large Language ModelsRL Fine-TuningDistributional Regression
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Apr 22, 2026

Text-to-Distribution Prediction with Quantile Tokens and Neighbor Context

Many applications of LLM-based text regression require predicting a full conditional distribution rather than a single point value. We study distributional regression under empirical-quantile supervision, where each input is paired with multiple observed quantile outcomes, and the target distribution is represented by a dense grid of quantiles. We address two key limitations of current approaches: the lack of local grounding for distribution estimates, and the reliance on shared representations that create an indirect bottleneck between inputs and quantile outputs. In this paper, we introduce Quantile Token Regression, which, to our knowledge, is the first work to insert dedicated quantile tokens into the input sequence, enabling direct input-output pathways for each quantile through self-attention. We further augment these quantile tokens with retrieval, incorporating semantically similar neighbor instances and their empirical distributions to ground predictions with local evidence from similar instances. We also provide the first theoretical analysis of loss functions for quantile regression, clarifying which distributional objectives each optimizes. Experiments on the Inside Airbnb and StackSample benchmark datasets with LLMs ranging from 1.7B to 14B parameters show that quantile tokens with neighbors consistently outperform baselines (~4 points lower MAPE and 2x narrower prediction intervals), with especially large gains on smaller and more challenging datasets where quantile tokens produce substantially sharper and more accurate distributions.

Yilun Zhu, Yuan Zhuang, Nikhita Vedula +6

Amazon.com, Inc. · Stanford University

Quantile RegressionDistributional RegressionRegression Prediction Intervals
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Apr 9, 2026

Accurate and Reliable Uncertainty Estimates for Deterministic Predictions Extensions to Under and Overpredictions

Computational models support high-stakes decisions across engineering and science, and practitioners increasingly seek probabilistic predictions to quantify uncertainty in such models. Existing approaches generate predictions either by sampling input parameter distributions or by augmenting deterministic outputs with uncertainty representations, including distribution-free and distributional methods. However, sampling-based methods are often computationally prohibitive for real-time applications, and many existing uncertainty representations either ignore input dependence or rely on restrictive Gaussian assumptions that fail to capture asymmetry and heavy-tailed behavior. Therefore, we extend the ACCurate and Reliable Uncertainty Estimate (ACCRUE) framework to learn input-dependent, non-Gaussian uncertainty distributions, specifically two-piece Gaussian and asymmetric Laplace forms, using a neural network trained with a loss function that balances predictive accuracy and reliability. Through synthetic and real-world experiments, we show that the proposed approach captures an input-dependent uncertainty structure and improves probabilistic forecasts relative to existing methods, while maintaining flexibility to model skewed and non-Gaussian errors.

Rileigh Bandy, Enrico Camporeale, Andong Hu +2

Optimization and Uncertainty Quantification Department, Sandia National Laboratories, Albuquerque, NM · School of Physical and Chemical Sciences, Queen Mary University of London, London, UK and Space Weather Technology, Research and Education Center, University of Colorado, Boulder, CO · Space Weather Technology, Research and Education Center, University of Colorado, Boulder, CO +2

Distributional RegressionUncertainty QuantificationProbabilistic ForecastingUncertainty CalibrationConditional Distribution Estimation
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Apr 1, 2026

UQ-SHRED: uncertainty quantification of shallow recurrent decoder networks for sparse sensing via engression

Reconstructing high-dimensional spatiotemporal fields from sparse sensor measurements is critical in a wide range of scientific applications. The SHallow REcurrent Decoder (SHRED) architecture is a recent state-of-the-art architecture that reconstructs high-quality spatial domain from hyper-sparse sensor measurement streams. An important limitation of SHRED is that in complex, data-scarce, high-frequency, or stochastic systems, portions of the spatiotemporal field must be modeled with valid uncertainty estimation. We introduce UQ-SHRED, a distributional learning framework for sparse sensing problems that provides uncertainty quantification through a neural network-based distributional regression called engression. UQ-SHRED models the uncertainty by learning the predictive distribution of the spatial state conditioned on the sensor history. By injecting stochastic noise into sensor inputs and training with an energy score loss, UQ-SHRED produces predictive distributions with minimal computational overhead, requiring only noise injection at the input and resampling through a single architecture without retraining or additional network structures. On complicated synthetic and real-life datasets including turbulent flow, atmospheric dynamics, neuroscience and astrophysics, UQ-SHRED provides a distributional approximation with well-calibrated confidence intervals. We further conduct ablation studies to understand how each model setting affects the quality of the UQ-SHRED performance, and its validity on uncertainty quantification over a set of different experimental setups.

Mars Liyao Gao, Yuxuan Bao, Amy S. Rude +2

Paul G. Allen School of Computer Science & Engineering, University of Washington, Seattle, WA, USA · Department of Applied Mathematics, University of Washington, Seattle, WA, USA · Department of Statistics, University of Washington, Seattle, WA, USA +1

Distributional RegressionUncertainty QuantificationUncertainty Calibration
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Mar 9, 2026

Distributional Regression with Tabular Foundation Models: Evaluating Probabilistic Predictions via Proper Scoring Rules

We study how the objective used to train or fine-tune a tabular foundation model affects its distributional predictions. Models such as TabPFN and TabICL provide forecasts of the full conditional distribution, but regression benchmarks including TabArena and TALENT assess them mainly through RMSE and R2R^2. Such assessments reward accurate conditional means while leaving the remaining distributional information unassessed. We therefore examine proper scoring rules as objectives for training and fine-tuning and as criteria for evaluating and ranking these models. Every strictly proper rule favors the true distribution at the population level, however, this shared optimum does not determine how learning proceeds with a finite sample. Our examples illustrate score-dependent preferences between forecasts, and our fine-tuning experiments show differences in predictive performance under different objectives. Experiments that fine-tune TabPFN and TabICL on several datasets reveal that changing the training objective can change performance across evaluation scores, with non-trivial interactions between the two. By connecting established methods from distributional regression, scoring-rule theory, and foundation-model adaptation, we show that selecting the score is part of specifying the model and can measurably alter the distribution it learns.

Jonas Landsgesell, Pascal Knoll, Tizian Wenzel

University of Stuttgart (Stuttgart, Germany) · Ludwig Maximilian University of Munich (Munich, Germany) · Munich Center for Machine Learning (Munich, Germany)

Tabular Foundation ModelsProper Scoring RulesDistributional RegressionTabular Regression
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