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Paid with Models: Optimal Contract Design for Collaborative Machine Learning
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Formalizes optimal contract design for collaborative ML where trained models are rewards, reduces the non-convex problem with private costs to a convex first-moment problem, and analyzes contract properties via numerical experiments.
Problem
Collaborative machine learning among parties with privately held per-unit data costs suffers from rent-seeking and freeriding; when models (not money) are rewards, their stochastic nature under incomplete information makes optimal contract design non-convex.
Approach
Relaxes the stochastic model-reward budget constraint to first moments, yielding a convex first-moment problem; a proportional assignment rule maps its solution back to the original realized-reward problem, and constraint analysis reduces the IC constraints to adjacent comparisons.
Key results
- Transforms the non-convex contracting problem into a convex first-moment problem, with proportional assignment shown to solve the original problem.
- Shows the highest-cost type's individual rationality constraint must bind and the most cost-efficient type must receive the best model (weak efficiency).
- In a two-type numerical study (c=0.02, 0.01), both types contribute more and obtain better models than their reservation levels under the contract; information costs are largest when N is small and type probabilities are similar.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use an optimal contract to incentivize collaboration despite large differences in contribution costs | Collaborative machine learning with model rewards · Multi-type case, Scenario 3, Type 1 | model reward value | Helped |
| Use an incentivized CML contract rather than training alone | Two-type CML scheme · Two-type case | data contribution and model quality | Helped |
| Increase the probability of the high-cost type | Optimal contract design · Two-type case | contract differentiation | Hurt |
| Increase the total number of participants | Optimal contract design · Two-type case | contract differentiation | Helped |
| Introduce private-cost information asymmetry | Collaborative machine learning contract · Two-type case | collectively trained model value | Hurt |
| Use a pooling contract | Low-cost type · Two-type case | participant utility | Helped |
| Use a contract that shares data-collection costs among participants | Collaborative machine learning · Multi-type case, Scenario 2 | ability to obtain a model | Helped |
| Increase the lowest-cost type's contribution requirement | Optimal contract · Multi-type case, Scenario 1, Type 5 | incentive compatibility | Hurt |
Surprising findings
- A party may be incentivized to contribute less than its reservation level (lowest-cost type in one scenario).
- Under complete information the optimal contract offers the best model to all participants regardless of contribution.
Limitations
- Relies on an assumed accuracy function (generalization bound) and a linear valuation function v(x)=100x in experiments.
- Assumes multinomial type distribution known to the coordinator; relaxing distribution assumptions is left open.
- Accuracy function is piecewise and non-concave, handled only via piecewise concavity.
Experimental setup
Metrics: expected model accuracy information cost information rent
Baselines: complete information contract
Builds on: revelation principle federated learning cooperative game theory incentive mechanisms
Closest papers by meaning
- Collaborative Machine Learning with Incentive-Aware Model Rewards (ICML 2020 · similarity 0.91)
- Learning Optimal Contracts: How to Exploit Small Action Spaces (ICLR 2024 · similarity 0.86)
- Incentivizing Time-Aware Fairness in Data Sharing (NeurIPS 2025 · similarity 0.85)
- Contracting with a Learning Agent (NeurIPS 2024 · similarity 0.85)
- Contract Design Under Approximate Best Responses (ICML 2025 · similarity 0.84)
- Aequa: Fair Model Rewards in Collaborative Learning via Slimmable Networks (ICML 2025 · similarity 0.84)
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