Set-Valued Prediction

Latest papers

14 papers6 papers in the last four weeks, with none the four weeks before. 0.0% of all new papers.

Oct 16, 2026

CoCoNet: Convex Conformal Network for Multi-Dimensional Uncertainty Regions

Conformal prediction (CP) provides distribution-free coverage guarantees under exchangeability, but the efficiency of its prediction regions depends critically on the choice of the conformity score. This is particularly important in multivariate regression, where dependencies across output dimensions induce complex joint uncertainty structures. This paper introduces CoCoNet (Convex Conformal Network) for learning convex conformity scores that induce flexible, data-adaptive prediction regions. CoCoNet learns the score from held-out data by optimizing prediction-region efficiency while enforcing convexity with respect to the residual. CoCoNet then applies split-conformal calibration to retain finite-sample coverage guarantees. The paper presents two variants of the method: CoCoNet

_{\mathrm{Spline}}

, based on piecewise-linear scores, and CoCoNet

_{\mathrm{ICNN}}

, based on Input Convex Neural Networks. Experiments compare CoCoNet with other CP baselines over synthetic and real-world datasets with output dimensions ranging from 2 to 16. The results demonstrate the flexibility of CoCoNet in learning different response types. Across 8 models and 30 multi-dimensional regression tasks, CoCoNet

_\text{ICNN}

and CoCoNet

_{\mathrm{Spline}\text{-}\ell_1}

achieve the best efficiencies with average ranks of 1.45 and 2.75, respectively. In particular, CoCoNet

_\text{ICNN}

ranks first on 22 tasks out of 30 in terms of efficiency.

Alireza Moradi, Thomas Bruys, Pascal Van Hentenryck +1

H. Milton Stewart School of Industrial and Systems Engineering (ISyE), Georgia Institute of Technology · NSF AI Institute for Advances in Optimization (AI4OPT), Georgia Institute of Technology · School of Electrical and Computer Engineering (ECE), Georgia Institute of Technology

Conformal Prediction Set-Valued Prediction

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Oct 6, 2026

Conformal Prediction Sets Quantify Information Gain: A Theoretical Perspective

Conformal prediction is a popular tool for uncertainty quantification that outputs prediction sets with finite-sample coverage guarantees. While prediction set size is commonly used as a heuristic measure of uncertainty, the information-theoretic basis for this interpretation remains poorly understood. In this work, we provide such a foundation using a decision-theoretic generalization of entropy tailored to set-valued prediction. In particular, we introduce a family of generalized information measures based on the size and coverage of conformal prediction sets. Notably, Shannon mutual information admits an exact integral representation in terms of these measures. We then show that, in standard classification settings, the reduction in conformal set size from additional information (i) is sandwiched between calibration-dependent members of this family and (ii) obeys a data processing inequality, both up to finite-sample calibration and model error terms. Together, our results formally relate conformal prediction to classical information-theoretic quantities and justify using set-size reduction as an information gain metric. Empirically, we validate our theory across 11 classification settings and show that set-size reduction and Shannon mutual information can rank features differently in a greedy feature selection experiment.

Kevin Zhang, Stephen Bates

EECS, MIT, Cambridge MA, USA.

Set Prediction Conformal Prediction Mutual Information Set-Valued Prediction

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Oct 4, 2026

Compact set-valued deep ensembling in multi-class classification

This paper tackles visible challenges in deep ensemble learning, where deep neural networks serve as ensemble members: training and storage burdens, and robustness of cautious (set-valued) predictions targeting multiple utilities, which may involve reward-sensitivity. To mitigate the training and storage burdens, we propose to employ compact ensembles, such as Bayesian Neural Networks and Convolutional Neural Networks with the Monte-Carlo dropout prediction option, to produce probabilistic predictions. For each query instance, these probabilistic predictions are then used to define a representative distribution optimizing some statistical distance. The representative distribution is then employed to define the Bayes-optimal prediction (BOP) of any utility. To address the potential unrobustness of singleton prediction making, we propose a family of set-utilities satisfying some desirable properties and whose set-valued BOPs can be found efficiently. Empirical evidence is then given to illustrate the potential (dis)advantages of the proposed ensemble learning framework.

Kim-Dung Tran, Dang-Man Nguyen, Vu-Linh Nguyen +3

UMR CNRS 7253, Heudiasyc, Universit´e de Technologie de Compi`egne, Compi`egne, France. · School of Knowledge Science, Japan Advanced Institute of Science and Technology, Nomi, Japan.

Deep Ensembles Multiclass Classification Set-Valued Prediction Decision-Making under Uncertainty

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Sep 28, 2026

Multi-Attractor GNNs: Set-Valued Expressivity Beyond Unique Equilibria

Recurrent and equilibrium graph neural networks (GNNs) often enforce a unique fixed point or use one training target per graph. Yet many combinatorial and scientific problems admit multiple valid solutions, with no preferred one. A designated target can then impose an arbitrary selection rule. For tasks invariant to node relabeling, a symmetric graph may have a symmetric solution set but no symmetric solution. We show that multiple equilibria enable one weight-tied message-passing GNN to represent set-valued equivariant maps: different initializations approach different valid solutions. Under stated regularity assumptions, we first construct globally Lipschitz, permutation-equivariant dynamics that converge almost surely to valid solutions and reach every solution branch with positive probability. We then establish approximate realization by recurrent message passing with continuous component maps, with arbitrarily small update and limiting errors and arbitrarily high probability. This goes beyond standard universality arguments: although message passing alone cannot distinguish symmetric nodes, the evolving state keeps nodes distinguishable at every finite step without auxiliary node identifiers. Such dynamics can be learned without solution labels using problem-specific energies. On Ising ground states, structural module detection in protein graphs, and chemical reaction steady states, the learned updates produce multiple high-quality predictions with high numerical convergence rates. They achieve better average solution quality than the tested unique-equilibrium, single-target, and feedforward baselines, while remaining competitive with much larger diffusion-based solvers.

Jialin Liu

GNN Expressivity Combinatorial Optimization Equivariant GNNs Set-Valued Prediction

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Sep 27, 2026

How Much Imprecision is Enough Imprecision in my Classifier? A Practical Elicitation Procedure

Set-valued classifiers, whether derived from precise probabilities and an adapted cost function, from convex sets with a robust inference mechanism, or from conformal methods, are routine options to obtain more robust, trustworthy predictions. However, there is a lack of operational tools to measure how robust or imprecise a given user is ready to be when receiving predictions, that is how much precision he/she is ready to let go in exchange of more accuracy. This is why we propose, in this paper, practical and operational elicitation procedures to measure the user proneness to set-valued predictions. The effectiveness of the iterative elicitation procedure in converging to the target parameter value is demonstrated on both tabular and image datasets drawn from standard machine learning benchmarks. The results show that the procedure also presents the user with a small number of instances, highlighting the practicality of the approach for real-world applications aimed at identifying the decision maker's optimal behavior when faced with imprecision.

Victor F. Lopes de Souza, Sébastien Destercke, Abdelhak Imoussaten

EuroMov Digital Health in Motion, Univ. Montpellier, IMT Mines Al`es, Montpellier, France · LIRMM, Universit´e de Montpellier, CNRS, Montpellier, France · SyCoIA, IMT Mines Al`es, Al`es, France +1

Set Prediction Classification User Preference Modeling Preference Elicitation Set-Valued Prediction

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Sep 22, 2026

Destination Support Restoration for Finite-Set Multimodal Trajectory Prediction

Robots operating around pedestrians often reason over a finite set of predicted human futures. Repeated online updates can concentrate this limited prediction budget on dominant destinations and leave plausible alternatives underrepresented or absent, removing those alternatives from the finite representation available to downstream decision making. We introduce Destination Support Restoration (DSR), a causal post-selection operator that repairs destination support without retraining the host predictor or increasing the maintained set size. At a repair step, DSR evaluates a temporary destination-stratified candidate bank from the observed prefix, converts candidate evidence into integer target counts, protects representatives of active modes, and reallocates redundant surplus hypotheses to deficient modes. The maintained and returned sets retain exactly

N

hypotheses, and DSR replaces at most

\lceilρN\rceil

entries. Protected representatives preserve current categorical support; lineage-aware particle filters also preserve surviving resampling ancestors. Each replacement reduces the allocation mismatch to the evidence-driven target by one. On the complete 3,719-trajectory Edinburgh protocol over three seeds, DSR reduces MIF weighted ADE and FDE by 13.36% and 13.30% at

N=64

. Paired integrations with CLiFF, PPT, causal GDTS, Social Informer, and PECNet improve both metrics in every evaluated pair. These results show that finite-set support allocation is a useful prediction-side control point when a fixed hypothesis set serves as the interface to downstream systems.

Fengrui Liu, Jiajun Peng, Duo Peng +1

School of Computer Science and Technology, East China Normal University, Shanghai, China · School of Data Science, University of Science and Technology of China, Hefei, China · School of Computer Science and Technology, Tongji University, Shanghai, China +1

Pedestrian Trajectory Prediction Multimodal Trajectory Prediction Set-Valued Prediction

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Jul 18, 2026

Isotonic Conformal Prediction

A point prediction that is well calibrated on average can still be systematically biased conditional on its own value, undermining its use in downstream decision-making. We consider two objectives for reliable uncertainty quantification: self-calibration, requiring a point prediction to be unbiased conditional on its own value, and prediction-conditional validity, requiring a prediction interval to attain nominal coverage conditional on the prediction. Self-Calibrating Conformal Prediction (SC-CP) attains both objectives exactly in finite samples, but requires refitting its calibrator for every candidate outcome, which is computationally prohibitive for continuous outcomes. We propose Isotonic Conformal Prediction (ICP), a framework that decouples calibration from prediction-set construction by fitting a single isotonic recalibration map and constructing prediction intervals within strata of similar recalibrated predictions. Within this framework we develop two procedures. Split Isotonic Conformal Prediction (SICP) attains prediction-conditional validity in finite samples and self-calibration asymptotically, at the computational cost of split conformal prediction. Transductive Isotonic Conformal Prediction (TICP) attains both objectives exactly in finite samples through a per-test-point inner loop that avoids refitting the isotonic calibrator. On synthetic heteroscedastic regression problems and a real-world healthcare-utilization dataset, both procedures match the coverage of SC-CP at substantially lower computational cost.

Daniel Bensimon, Sean Xiang Yu, Eric D. Kolaczyk +1

Department of Mathematics and Statistics, McGill University; Mila · Eli Lilly and Company

Conformal Prediction Uncertainty Calibration Isotonic Regression Regression Prediction Intervals Set-Valued Prediction

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Jun 27, 2026

Multi-Agent Routing as Set-Valued Prediction: A WildChat Benchmark and Cost-Aware Evaluation

Tool and agent routing from natural-language prompts is naturally a set-valued prediction problem: a single query may require multiple agents, while over-selection increases execution cost. The benchmark introduced here is derived from WildChat and contains 3,000 prompts over a fixed 12-agent catalog, with AI-assisted heuristic labels under a fixed schema and controlled rebalancing for multi-label evaluation. The evaluation protocol combines set-level metrics (Precision, Recall, F1, Jaccard, and Exact Match), latency, an execution-oriented capability-coverage simulation, and a constrained weighted-routing setting based on ordinal agent-cost tiers. Compared methods include nearest-neighbor matching, linear multilabel classification, dependency-aware baselines, a fine-tuned encoder, deterministic weighted post-scoring via Weighted Agent Routing (WAR), and a zero-shot LLM baseline. Results show that supervised routers substantially outperform nearest-neighbor and zero-shot LLM routing. The fine-tuned encoder achieves the strongest unconstrained set accuracy, while the linear multilabel model provides the strongest practical baseline. In the constrained setting, the weighted routing layer improves utility when applied on top of strong supervised scorers, with the largest gain observed for Encoder+WAR. Overall, the benchmark and evaluation protocol support reproducible study of accuracy-cost trade-offs in fixed-catalog multi-agent routing.

Ananto Nayan Bala, Faisal Muhammad Shah

Ahsanullah University of Science and Technology Dhaka, Bangladesh

Multi-Label Classification AI Agent Benchmarks Set-Valued Prediction LLM Agent Routing

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Jun 4, 2026

A Framework for Measuring Appropriate Reliance on Set-Valued AI Advice

Appropriate reliance on AI advice has become a central research theme in human-AI collaboration. Existing frameworks have focused exclusively on point predictions as AI advice. However, set-valued AI advice (e.g., discrete sets or continuous intervals) is increasingly being used to communicate uncertainty and improve human decision making. In this paper, we develop the first formal framework for measuring appropriate reliance on set-valued AI advice within the sequential judge-advisor paradigm, spanning both classification and regression tasks. For classification, we first introduce the dimensions that are necessary for evaluating set-valued AI advice. We then define two metrics: correct reliance rate on AI and correct reliance rate on self, which jointly characterize appropriate reliance in this setting. For regression, we introduce quantity of AI reliance and quality of AI reliance, which respectively measure whether a decision maker utilized the AI advice and whether their reliance helped them get closer to the ground truth relative to their initial estimate. Through the application of our framework, we demonstrate how these metrics capture important nuances in human-AI collaboration that existing measures overlook.

Ranjan Mishra, Jakob Schoeffer

University of Groningen, The Netherlands

Set-Valued Prediction Human-AI Decision Making Human-AI Collaboration

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May 25, 2026

MiRD: Reliable Set-Valued Prediction for Open-Ended Question Answering via Miscoverage Risk Decomposition

Reliable set-valued prediction provides a principled way to mitigate hallucinations in open-ended question answering (QA), yet existing conformal approaches typically rely on a fragile premise: finite sampling must already produce at least one admissible candidate, or calibration examples violating this condition are discarded. In this paper, we introduce MiRD, a two-stage framework that decomposes overall miscoverage into sampling failure and conditional selection failure. In Stage I, MiRD establishes an expectation-level marginal upper bound on the probability that finite sampling produces no admissible answer under a fixed budget. In Stage II, conditioned on sampling success, MiRD calibrates a conformal selection threshold using admission-correlated nonconformity scores defined over the full calibration set, thereby preserving calibration-set integrity. Across three open-ended QA datasets and eight models, MiRD controls sampling risk, conditional selection risk, and overall miscoverage, while yielding tighter first-stage bounds than PAC-style alternatives and more adaptive prediction sets than successful-only calibration.

Anqi Hu, Zhiyuan Wang, Zijun Jia +1

University of Electronic Science and Technology of China · Beihang University

LLM Hallucination Mitigation Set Prediction Conformal Prediction Conformal Risk Control Set-Valued Prediction Uncertainty Calibration+1

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May 8, 2026

Bifurcation Models: Learning Set-Valued Solution Maps with Weight-Tied Dynamics

Many scientific and combinatorial problems admit multiple correct solutions, not a single label. Standard supervised learning resolves this ambiguity by choosing one solution as the target, but this hidden selector can be arbitrary, discontinuous, and harder to learn than the underlying solution set. We study bifurcation models, a weight-tied dynamical view in which different initializations can converge to different stable equilibria, so the model represents an attractor landscape rather than one chosen branch. We prove that broad set-valued maps with locally Lipschitz branches can be represented by regular equilibrium dynamics and that the induced selectors are almost everywhere regular, while manual selectors can be arbitrarily irregular. Experiments on frustrated Ising models show that such dynamics can discover multiple valid equilibria without branch labels and outperform single-branch supervision. Allen--Cahn experiments further show that diversity is not automatic: it can be encouraged explicitly, but with an accuracy--diversity tradeoff.

Caleb Jore, Jialin Liu

Dynamical Systems Set-Valued Prediction

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Mar 24, 2026

Set-Valued Prediction for Large Language Models with Feasibility-Aware Coverage Guarantees

Large language models (LLMs) inherently operate over a large generation space, yet conventional usage typically reports the most likely generation (MLG) as a point prediction, which underestimates the model's capability: although the top-ranked response can be incorrect, valid answers may still exist within the broader output space and can potentially be discovered through repeated sampling. This observation motivates moving from point prediction to set-valued prediction, where the model produces a set of candidate responses rather than a single MLG. In this paper, we propose a principled framework for set-valued prediction, which provides feasibility-aware coverage guarantees. We show that, given the finite-sampling nature of LLM generation, coverage is not always achievable: even with multiple samplings, LLMs may fail to yield an acceptable response for certain questions within the sampled candidate set. To address this, we establish a minimum achievable risk level (MRL), below which statistical coverage guarantees cannot be satisfied. Building on this insight, we then develop a data-driven calibration procedure that constructs prediction sets from sampled responses by estimating a rigorous threshold, ensuring that the resulting set contains a correct answer with a desired probability whenever the target risk level is feasible. Extensive experiments on six language generation tasks with five LLMs demonstrate both the statistical validity and the predictive efficiency of our framework.

Ye Li, Anqi Hu, Yuanchang Ye +3

University of Electronic Science and Technology of China, Chengdu, Sichuan, China · Zhejiang University of Finance and Economics, Hangzhou, Zhejiang, China · Southeast University, Nanjing, Jiangsu, China

Set-Valued Prediction LLM Uncertainty Estimation Language Model Calibration Text Generation

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Feb 20, 2026

RACER: Risk-Aware Calibrated Efficient Routing for Large Language Models

Efficiently routing queries to the optimal large language model (LLM) is crucial for optimizing the cost-performance trade-off in multi-model systems. However, most existing routers rely on single-model selection, making them susceptible to misrouting. In this work, we formulate LLM routing as the

α

-VOR problem to minimize expected set size while controlling the misrouting risk, and propose a novel method -- RACER, extending base routers to output model sets that can be subsequently aggregated for improved output. In particular, RACER constructs nested model sets via augmented scoring and utilizes finite-sample concentration bounds to calibrate a threshold that allows for both variable set sizes and abstention. We theoretically prove that RACER achieves rigorous distribution-free risk control on unseen test data in a post-hoc and model-agnostic manner. Extensive experiments verify our theoretical guarantees and demonstrate that RACER consistently enhances downstream accuracy across a wide range of benchmarks.

Sai Hao, Hao Zeng, Hongxin Wei +1

Southern University of Science and Technology, China · The Chinese University of Hong Kong, Shenzhen, China · Shenzhen Loop Area Institute, China

Set-Valued Prediction LLM Routing Adaptive Model Routing

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Nov 19, 2025

Beyond Uncertainty Sets: Leveraging Optimal Transport to Extend Conformal Predictive Distributions to Multivariate Settings

Conformal prediction (CP) constructs uncertainty sets for model outputs with finite-sample coverage guarantees. Yet ranking scores is straightforward only when they are scalar-valued, limiting CP to real-valued scores or ad-hoc one-dimensional reductions. Vector-valued scores arise naturally in multi-output regression and model aggregation, where each predictor in an ensemble provides its own score. Optimal transport (OT) defines vector ranks and center-outward multivariate quantile regions, though generally with asymptotic coverage guarantees. Applying a fixed transport map learned from calibration data to a new point introduces an uncontrolled approximation error. We restore finite-sample, distribution-free coverage by conformalizing vector-valued OT quantile regions. Each candidate's rank is defined by transporting the calibration scores augmented with that candidate's score, preserving the symmetry needed for validity. This appears to require a continuum of OT problems. However, we prove that the optimal assignment is piecewise constant across a fixed polyhedral partition of score space. This lets us characterize the entire prediction set in

O(n^3)

time, matching the cost of a single assignment solve. It also addresses a limitation of prediction sets: they indicate which outcomes are plausible, but not their relative likelihood. In one dimension, conformal predictive distributions (CPDs) fill this gap by producing a predictive distribution with finite-sample calibration. Extending CPDs beyond one dimension remained an open problem. We construct, to our knowledge, the first multivariate CPDs with finite-sample calibration: a center-outward predictive distribution whose derived uncertainty regions have conformal coverage. We present both conservative and exact randomized versions; the latter generalizes the classical Dempster-Hill procedure.

Eugene Ndiaye

Apple

Uncertainty Quantification Conformal Prediction Optimal Transport Set-Valued Prediction

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