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Regression with Multi-Expert Deferral
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Introduces regression with deferral to multiple experts, deriving single-stage and two-stage surrogate losses with H-consistency bounds, and shows the single-expert special case recovers regression with abstention.
Problem
Learning to defer with multiple experts has been studied mainly in classification; the score-based formulation does not apply to regression because the label space is continuous, so a principled deferral framework for regression with multiple experts is needed.
Approach
A predictor-rejector formulation for regression with deferral in which the deferral loss is rewritten and surrogated via a multi-class surrogate (e.g., logistic) on the deferral function, yielding single-stage and two-stage surrogate losses with H-consistency bounds for any bounded regression loss and label-dependent costs.
Key results
- On Airfoil, the single-stage method with three experts reaches system MSE 8.53 ± 1.57 versus 18.98 ± 2.44 with a single expert.
- On Concrete, system MSE drops from about 104 (single expert) to about 38 (three experts) for the single-stage method.
- The two-stage method generally underperforms the single-stage method, attributed to error accumulation from the first-stage predictor.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add experts from one to three | Single-stage, no base cost · Airfoil | System MSE: 18.98 ± 2.44 -> 8.53 ± 1.57 | Helped |
| Add experts from one to three | Two-stage, no base cost · Airfoil | System MSE: 18.64 ± 1.96 -> 8.81 ± 1.56 | Helped |
| Add experts from one to three | Single-stage, base cost · Airfoil | System MSE: 18.83 ± 2.14 -> 8.64 ± 1.40 | Helped |
| Add experts from one to three | Two-stage, base cost · Airfoil | System MSE: 19.15 ± 1.99 -> 10.06 ± 1.54 | Helped |
| Add experts from one to three | Single-stage, no base cost · Housing | System MSE: 14.85 ± 5.40 -> 12.43 ± 2.03 | Helped |
| Add experts from one to three | Two-stage, no base cost · Housing | System MSE: 16.26 ± 5.58 -> 12.02 ± 1.97 | Helped |
| Add experts from one to three | Single-stage, base cost · Housing | System MSE: 15.17 ± 5.18 -> 14.80 ± 3.48 | Mixed |
| Add experts from one to three | Two-stage, base cost · Housing | System MSE: 16.24 ± 4.64 -> 14.87 ± 4.04 | Helped |
| Add experts from one to three | Single-stage, no base cost · Concrete | System MSE: 104.38 ± 5.55 -> 37.83 ± 2.60 | Helped |
| Add experts from one to three | Two-stage, no base cost · Concrete | System MSE: 114.73 ± 6.50 -> 36.75 ± 1.76 | Helped |
| Add experts from one to three | Single-stage, base cost · Concrete | System MSE: 105.01 ± 5.40 -> 38.46 ± 1.79 | Helped |
| Add experts from one to three | Two-stage, base cost · Concrete | System MSE: 114.11 ± 5.34 -> 37.51 ± 2.32 | Helped |
Surprising findings
- Two-stage algorithm usually does not perform as well as single-stage in regression, attributed to error accumulation.
Limitations
- Two-stage approach can suffer error accumulation when the first-stage predictor has large errors.
- Experiments limited to three small UCI regression datasets with linear predictor and deferral models.
Experimental setup
Datasets: Airfoil Housing Concrete
Benchmarks: Airfoil Housing Concrete
Metrics: system MSE empirical regression loss
Baselines: base model (predictor h alone)
Builds on: learning with abstention predictor-rejector loss H-consistency bounds two-stage learning to defer
Closest papers by meaning
- Mastering Multiple-Expert Routing: Realizable $H$-Consistency and Strong Guarantees for Learning to Defer (ICML 2025 · similarity 0.88)
- A Two-Stage Learning-to-Defer Approach for Multi-Task Learning (ICML 2025 · similarity 0.87)
- Realizable $H$-Consistent and Bayes-Consistent Loss Functions for Learning to Defer (NeurIPS 2024 · similarity 0.86)
- Exploiting Human-AI Dependence for Learning to Defer (ICML 2024 · similarity 0.86)
- When to Act and When to Ask: Policy Learning With Deferral Under Hidden Confounding (NeurIPS 2024 · similarity 0.84)
- Density-Ratio Losses for Post-Hoc Learning to Defer (NeurIPS 2026 · similarity 0.84)
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