Multi-Target Conformal Prediction

• Multi-target conformal prediction is a distribution-free framework that builds set-valued predictors ensuring joint coverage for multiple, potentially dependent outcomes.
• Classical and copula-based methods leverage per-coordinate calibration and dependence modeling to achieve efficient, high-dimensional prediction regions.
• Optimization techniques using empirical risk minimization and differentiable surrogates yield tighter, balanced prediction sets in structured and high-dimensional applications.

Multi-target conformal prediction constitutes a rigorous, distribution-free statistical framework for quantifying uncertainty in simultaneous prediction of multiple, potentially dependent targets. The goal is to construct set-valued predictors, typically high-dimensional hyperrectangles or more general regions, that guarantee valid coverage at a user-specified level $1-\alpha$, often with additional desiderata such as balance, efficiency, or adaptation to underlying correlation structure. Methods span classical split-conformal approaches, modern copula-based and semiparametric estimators, and specialized algorithms targeting scalability, efficiency, or structured predictions.

1. Conceptual Foundations and Problem Formulation

The multi-target setting considers feature–response pairs $(X_i, Y_i)$, $Y_i\in\mathbb{R}^k$ (or more generally, multi-step time series or agent trajectories), with the aim of outputting a set-valued prediction $C(X)$ such that

$$\mathbb{P}\bigl(Y_{n+1}\in C(X_{n+1})\bigr)\geq 1-\alpha,$$

under minimal distributional assumptions, typically only exchangeability. The main statistical challenges arise from the need to guarantee joint coverage across all target components, efficiently exploit dependence structure, and control the size and balance of the prediction set.

Key performance metrics are:

• Joint coverage: Probability that all targets are included in their intervals.
• Marginal coverage: Per-component coverage probabilities.
• Efficiency: Size or volume of the prediction set.

These objectives appear in both general i.i.d. setups and application-specific domains such as multi-step time series forecasting or multi-agent trajectory prediction (Sampson et al., 6 Jun 2024, Muthali et al., 2023, Park et al., 4 Nov 2024).

2. Classical and Hyperrectangular Approaches

A canonical instantiation is the conformal multi-target hyperrectangle (Sampson et al., 6 Jun 2024). Here, prediction sets are axis-aligned boxes: $C(X)=\prod_{j=1}^k [L_j(X), U_j(X)]$ where $L_j(X), U_j(X)$ are determined via split-conformal calibration of per-coordinate errors and a scalar joint calibration step. Two main algorithms are introduced:

• CHR (Conformal HyperRectangles, point-regression version): Uses absolute prediction errors; joint calibration is conducted by rescaling residuals across coordinates and calibrating a single maximum-scaled error per sample.
• QCHR (Quantile Regression version): Leverages fitted lower and upper quantile regressors to form coordinatewise intervals, then calibrates a maximum scaled residual as the conformal statistic.

Comprehensive theoretical results establish finite-sample overall coverage at the desired nominal level, asymptotic equivalence of marginal coverages (balance), and tightness relative to coordinatewise Bonferroni approaches: $$\max_{j} \left|\eta_j - \frac{1}{d}\sum_{k=1}^d \eta_k\right| \to 0 \quad (n\to\infty)$$ Simulation and real-world studies confirm that interval sizes are drastically reduced compared to conservative Bonferroni bounds, while joint coverage and balance are empirically validated (Sampson et al., 6 Jun 2024).

3. Copula and Multivariate Score-based Methods

Exploiting inter-target dependence can significantly improve region size while maintaining guarantees. Copula-based conformal prediction models the joint distribution of nonconformity scores using either parametric copulas (e.g., Gumbel), nonparametric empirical copulas, or more flexible vine copula decompositions (Messoudi et al., 2021, Sun et al., 2022, Park et al., 4 Nov 2024).

The procedure is as follows:

1. Normalization: Compute per-target residual quantiles to uniformize calibration nonconformity scores.
2. Copula Fitting: Fit a copula $C(u_1,\dots,u_k)$ to the uniformized residuals.
3. Joint Quantile Inversion: For desired global coverage $1-\alpha$, solve $C(1-\epsilon_t,\dots,1-\epsilon_t)=1-\alpha$ for the marginal level $\epsilon_t$ (for symmetric solutions) or minimize total $\sum_j \alpha_j$ under $C(\alpha_1, \dots, \alpha_k) \geq 1-\alpha$ for more general cases.
4. Prediction Set Construction: Output coordinate intervals or regions accordingly.

Empirical copulas provide exact finite-sample validity in the exchangeable case if calibration is sufficiently large, while parametric or vine-copula approaches are supported by asymptotic theory (Messoudi et al., 2021, Sun et al., 2022, Park et al., 4 Nov 2024). Such models can be efficiently fitted and scaled via pair-copula decompositions for high $k$.

4. Optimization and Minimax Efficiency

Recent methodological advances treat the multi-target conformal prediction problem as a constrained empirical risk minimization: $$\min_{\theta, t} \ \widehat{E}(\theta, t) \quad \text{s.t.} \quad \widehat{C}(\theta, t) \geq 1-\alpha$$ where $\widehat{E}$ measures expected set size and $\widehat{C}$ is empirical coverage (Bai et al., 2022). This is addressed using primal–dual Lagrangian optimization with differentiable surrogates (e.g., hinge loss) for the coverage constraint, yielding prediction sets that are significantly tighter in high dimensions compared to coordinatewise schemes, yet maintain distribution-free guarantees. If further reconformalization is employed, exact finite-sample coverage $1-\alpha$ is recoverable. Experiments in multi-output regression (Cartpole, Half-Cheetah, Ant, etc.) confirm volume reductions of $2{-}10\times$ compared to conservative baselines (Bai et al., 2022).

The minimax approach in (Wen et al., 17 Nov 2025) formalizes efficiency as minimization of the worst-case single-target marginal coverage, under the constraint of joint coverage $\geq 1-\alpha$. The optimal solution seeks the smallest $\lambda$ such that the maximum per-target marginal coverage does not fall below what is required for the joint constraint, leading to more balanced and tighter intervals. In practice, this methodology proves empirically superior to independence-based or quantile-normalized alternatives, particularly under correlated, non-Gaussian noise.

5. Structured and Non-Rectangular Prediction Regions

The extension to non-axis-aligned, non-convex, or data-adaptive prediction sets is addressed by methods using conditional normalizing flows and volume-sorting (Luo et al., 4 Mar 2025). Here, the conditional density $p_{Y|X}(y|x)$ is modeled by a normalizing flow, and prediction sets are constructed as unions of balls around the highest-density samples in response space, with the radius calibrated via conformal quantiles. This procedure delivers prediction sets that are flexible and adapt to non-Gaussian, multi-modal uncertainty, while maintaining finite-sample coverage via exchangeable calibration. Empirical results indicate region size reductions of $3{-}10\times$ over baseline methods, with preservation of nominal coverage (Luo et al., 4 Mar 2025).

6. Domain-specific Frameworks: Multi-Agent, Time Series, and Sequential Prediction

Multi-target conformal frameworks are naturally extended to multi-agent systems, multi-step time series, and structured sequential problems. In multi-agent safety-critical planning, the approach of (Muthali et al., 2023) combines agentwise quantile regression, split-conformal calibration for joint coverage over all agents and time steps, and reachability analysis via Hamilton–Jacobi PDEs to convert control uncertainties into dynamically feasible tubes. The result is provably safe multi-agent planners where the future trajectories of all $N$ agents are contained in the constructed tubes with probability $\geq 1-\gamma$.

In multi-step time series, both copula-based (Sun et al., 2022), online conformal PID control (Wang et al., 17 Oct 2024), and autocorrelated interval tracking ensure joint or horizon-wise coverage with efficiency and adaptivity. The AcMCP algorithm in (Wang et al., 17 Oct 2024) utilizes the observed moving average structure in forecast errors to construct intervals achieving long-run nominal coverage, even in nonstationary or limited-data regimes.

7. Theoretical Guarantees and Empirical Evidence

The central theoretical underpinning is distribution-free marginal or joint coverage under exchangeability. While split-conformal and permutation-based methods provide finite-sample coverage, efficiency tradeoffs are addressed through class selection, surrogate relaxation, and correct modeling of dependencies (via copulas or minimax criteria).

Empirical studies across modalities—regression, control, imaging, time series, planning—demonstrate that advanced multi-target conformal frameworks consistently obtain valid coverage, tighter prediction regions, and improved balance compared to naive, independence-based, or marginal calibration (Sampson et al., 6 Jun 2024, Messoudi et al., 2021, Bai et al., 2022, Wen et al., 17 Nov 2025, Muthali et al., 2023).

Approach	Coverage Guarantee	Region Structure
Axis-aligned hyperrectangles	Exact (finite/asymptotic)	Box/hyperrectangle
Copula-based (empirical/vine)	Exact/asymptotic	Box (componentwise), or general multivariate
Minimax conformal	Finite-sample, minimax	Box (balanced by design)
Volume-sorted/flow-based	Finite-sample	Non-convex, data-adaptive
Multi-agent reachability (HJ PDE)	Finite-sample (joint)	State-space tubes
Time series copula/AcMCP	Finite/Long-run	Boxes, adaptively scaled

A plausible implication is that as the dimension $k$ grows, empirical-copula and vine-based calibrations become statistically preferable for efficiency, but require sufficiently large calibration sets and careful regularization.

• "Conformal Multi-Target Hyperrectangles" (Sampson et al., 6 Jun 2024)
• "Copula-based conformal prediction for Multi-Target Regression" (Messoudi et al., 2021)
• "Efficient and Differentiable Conformal Prediction with General Function Classes" (Bai et al., 2022)
• "Minimax Multi-Target Conformal Prediction with Applications to Imaging Inverse Problems" (Wen et al., 17 Nov 2025)
• "Volume-Sorted Prediction Set: Efficient Conformal Prediction for Multi-Target Regression" (Luo et al., 4 Mar 2025)
• "Semiparametric conformal prediction" (Park et al., 4 Nov 2024)
• "Copula Conformal Prediction for Multi-step Time Series Forecasting" (Sun et al., 2022)
• "Online conformal inference for multi-step time series forecasting" (Wang et al., 17 Oct 2024)
• "Multi-Agent Reachability Calibration with Conformal Prediction" (Muthali et al., 2023)