Model-Agnostic Quantile Regression (MAQR)

• Model-Agnostic Quantile Regression (MAQR) is a collection of methods designed to reliably calibrate conditional quantiles for any predictive model.
• MAQR methodologies combine dual-based corrections, nonparametric estimation, and ensemble aggregation to overcome coverage bias and enforce noncrossing predictions.
• The framework delivers theoretical guarantees, finite-sample coverage, and practical scalability for high-dimensional and overparameterized regression tasks.

Model-Agnostic Quantile Regression (MAQR) is a collection of methodologies designed to enable reliable and calibrated estimation of conditional quantiles for arbitrary predictive models, regardless of model family or structure. Motivated by the prevalence of quantile-based uncertainty quantification in regression applications, MAQR aims to correct the limitations of conventional quantile regression, especially its coverage bias and lack of calibration in high dimensions or flexible model regimes. MAQR methodologies combine tools from the dual formulation of quantile regression, nonparametric estimation procedures, ensemble aggregation, explicit monotonicity enforcement, and post-hoc conformalization to provide flexible, scalable, and theoretically justified calibration of prediction intervals for any class of base regressors (Gibbs et al., 2 Nov 2025, Fakoor et al., 2021, Chung et al., 2020).

1. Quantile Regression Foundations and Limitations

Quantile regression predicts conditional quantiles of a response variable $Y$ given covariates $X$ by minimizing the pinball (tilted) loss function for a quantile level $\tau \in (0,1)$: $$\hat\beta(\tau) = \arg\min_{\beta\in\mathbb{R}^d} \sum_{i=1}^n \rho_\tau(y_i - x_i^\top\beta),$$ with

$\rho_\tau(u) = u\,[\tau - \mathbf{1}_{\{u<0\}}].$

The true conditional quantile satisfies $Q^*(\tau|x) = \inf\{y : F_{Y|X=x}(y) \ge \tau\}$.

Despite the popularity of this approach, standard pinball loss optimization does not guarantee valid calibration—$P(Y \le \hat Q(x, \tau)) = \tau$—especially if the model is flexible or when $d/n$ is large. This miscalibration may manifest as severe undercovering, particularly in high-dimensional or over-parameterized settings (Gibbs et al., 2 Nov 2025, Chung et al., 2020). Proposition 1 in (Chung et al., 2020) shows empirically and theoretically that models minimizing pinball loss can be arbitrarily miscalibrated.

2. Model-Agnostic Quantile Regression Methodologies

MAQR methodologies address these calibration failures by providing post-hoc adjustment and ensemble approaches applicable to arbitrary regressors.

Dual-Based Quantile Correction (Gibbs et al., 2 Nov 2025)

The dual formulation introduces dual variables $\eta_i$ that govern leave-one-out predictions. The sign of $\hat\eta_i$ exactly tracks coverage: $X$0 if $X$1, $X$2 if $X$3. Leveraging this, MAQR provides three correction schemes:

• Level Adjustment: Adjust the quantile level $X$4 via binary search so LOOCov($X$5) matches the target nominal coverage.
• Additive Adjustment: Add a bias $X$6 to predictions, calibrating $X$7 by matching leave-one-out coverage.
• Fixed Dual Thresholding: Use the empirical $X$8-quantile of the duals as a threshold for coverage-optimal predictions.

All these methods substitute expensive leave-one-out retraining with efficient computations involving the dual solutions, greatly improving scalability.

Nonparametric MAQR (Chung et al., 2020)

The framework in (Chung et al., 2020) constructs a quantile model by:

• Computing point-prediction residuals.
• Estimating local conditional quantiles using nonparametric CDFs over residual neighborhoods.
• Fitting a regression model from $X$9 quantile, yielding $\tau \in (0,1)$0.

This is fully model-agnostic: any base point predictor can be wrapped, and the method yields uniformly consistent estimates under mild smoothness and density conditions.

Quantile Ensemble Aggregation (Fakoor et al., 2021)

MAQR also extends to aggregation of multiple quantile regressors:

• Linear aggregators with weights varying by model, quantile level, or features.
• Fine-resolution aggregators (Deep Quantile Aggregation) use neural networks to learn feature- and quantile-dependent weights.
• Aggregated quantiles undergo post-processing (sorting or isotonic projection) to enforce monotonicity, and conformal calibration for finite-sample coverage.

This meta-framework allows the combination of arbitrary quantile models (tree-based, neural, or otherwise) while ensuring noncrossing and calibrated prediction intervals.

3. Calibration, Sharpness, and Monotonicity Enforcement

Defining Calibration and Sharpness

• Individual calibration: $\tau \in (0,1)$1 for all $\tau \in (0,1)$2.
• Average calibration: $\tau \in (0,1)$3.
• Adversarial group calibration: Calibration on every measurable subset of $\tau \in (0,1)$4.

Sharpness concerns the expected width of central prediction intervals—ideally, intervals should be as narrow as possible without sacrificing calibration.

Losses Targeting Calibration and Sharpness

(Chung et al., 2020) proposes explicit losses:

• Combined calibration loss: $\tau \in (0,1)$5, with $\tau \in (0,1)$6 controlling the trade-off.
• Interval score (Winkler score): Minimizes at the true quantiles and measures central-interval calibration.

Monotonicity (Noncrossing) Enforcement

Since quantile functions must be nondecreasing in $\tau \in (0,1)$7, MAQR employs:

• Sorting the predicted quantile vector for each $\tau \in (0,1)$8.
• Isotonic regression (PAVA) projecting predictions onto the monotone cone.

Both techniques strictly reduce the weighted interval score whenever crossing exists, as shown in Propositions 1 and 2 of (Fakoor et al., 2021).

4. Conformal Calibration and Empirical Coverage Guarantees

Conformalized quantile regression (CQR) splits the dataset into proper training and calibration sets. Quantile predictions on the calibration set provide empirical residual quantiles for interval construction, guaranteeing finite-sample marginal coverage: $\tau \in (0,1)$9 Extensions like CV+ re-use all data for both model fitting and calibration with only a minor $$\hat\beta(\tau) = \arg\min_{\beta\in\mathbb{R}^d} \sum_{i=1}^n \rho_\tau(y_i - x_i^\top\beta),$$0 calibration error term (Fakoor et al., 2021). This approach can be applied post-hoc to any quantile estimator.

5. Theoretical Guarantees and Asymptotics

• Consistency: Under mild regularity assumptions (smoothness, density boundedness), the nonparametric quantile estimation in MAQR achieves uniform consistency as $$\hat\beta(\tau) = \arg\min_{\beta\in\mathbb{R}^d} \sum_{i=1}^n \rho_\tau(y_i - x_i^\top\beta),$$1 (Chung et al., 2020).
• Asymptotic Coverage Correction: In dual-corrected MAQR, the empirical dual distributions converge to explicit limiting laws in high-dimensional regimes ($$\hat\beta(\tau) = \arg\min_{\beta\in\mathbb{R}^d} \sum_{i=1}^n \rho_\tau(y_i - x_i^\top\beta),$$2). The fixed dual-threshold method attains exact asymptotic coverage (Gibbs et al., 2 Nov 2025).
• Proper Scoring Rules: The interval score and the combined calibration loss are strictly proper; their unique population minimizers are the true quantiles. Loss minimizers with perfect calibration exist even with highly flexible regressors (Chung et al., 2020, Fakoor et al., 2021).

6. Empirical Insights and Practical Recommendations

Summary of Empirical Findings

Setting	Standard QR	MAQR (all variants)	Best MAQR Variant
High $$\hat\beta(\tau) = \arg\min_{\beta\in\mathbb{R}^d} \sum_{i=1}^n \rho_\tau(y_i - x_i^\top\beta),$$3 (simulated)	Undercovering	Restores nominal coverage	Dual threshold: multiaccuracy
Real data (UCI/Online)	Undercovering	Restores marginal coverage	Level/additive: shortest int.
Ensemble aggregation	Variable	Best WIS + coverage (DQA)	DQA + CV+: WIS, coverage

• Dual-threshold gives strongest subgroup-bias control (multiaccuracy) but with longer intervals.
• Level/additive tuning yields minimal interval length and deterministic predictions (Gibbs et al., 2 Nov 2025, Fakoor et al., 2021).

Implementation Notes

• Dual-based correction methods scale efficiently, with main costs in standard quantile programming and scalar binary searches (total $$\hat\beta(\tau) = \arg\min_{\beta\in\mathbb{R}^d} \sum_{i=1}^n \rho_\tau(y_i - x_i^\top\beta),$$4 or better per search).
• Nonparametric MAQR incurs $$\hat\beta(\tau) = \arg\min_{\beta\in\mathbb{R}^d} \sum_{i=1}^n \rho_\tau(y_i - x_i^\top\beta),$$5 cost in CDF computation but only once per dataset. The regression step is agnostic.
• Ensemble and neural-based MAQR (DQA) require only differentiable map $$\hat\beta(\tau) = \arg\min_{\beta\in\mathbb{R}^d} \sum_{i=1}^n \rho_\tau(y_i - x_i^\top\beta),$$6, training with standard deep learning toolkits (Fakoor et al., 2021, Chung et al., 2020).

A practical recommendation is to use level- or additive-adjusted MAQR for short intervals and deterministic coverage, and fixed dual-threshold MAQR for maximal multiaccuracy where subpopulation calibration is a priority (Gibbs et al., 2 Nov 2025).

7. Notable Applications and Extensions

MAQR has been applied to:

• High-dimensional scientific data (e.g., nuclear fusion diagnostics with 468-dimensional predictors) (Chung et al., 2020).
• Large-scale regression problems from UCI and OpenML repositories across 34 datasets, including both tree-based and neural base learners (Fakoor et al., 2021).
• Overparameterized and high-dimensional linear models, where coverage failures are acute (Gibbs et al., 2 Nov 2025).

The flexibility to wrap any point-medium and the use of explicit calibration–sharpness tradeoffs, group-batch adversarial calibration, conformalization, and post-hoc noncrossing enforcement make MAQR suitable for broad scientific and probabilistic machine learning applications.

Key references:

Correcting the Coverage Bias of Quantile Regression (Gibbs et al., 2 Nov 2025) Flexible Model Aggregation for Quantile Regression (Fakoor et al., 2021) Beyond Pinball Loss: Quantile Methods for Calibrated Uncertainty Quantification (Chung et al., 2020)