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Paid with Models: Optimal Contract Design for Collaborative Machine Learning

Bingchen Wang, Zhaoxuan Wu, Fusheng Liu, Bryan Kian Hsiang Low

TL;DR

A detailed analysis of the properties that an optimal contract must satisfy when models serve as the rewards is conducted, and the potential benefits and welfare implications of these contract-driven CML schemes are explored through numerical experiments.

Abstract

Collaborative machine learning (CML) provides a promising paradigm for democratizing advanced technologies by enabling cost-sharing among participants. However, the potential for rent-seeking behaviors among parties can undermine such collaborations. Contract theory presents a viable solution by rewarding participants with models of varying accuracy based on their contributions. However, unlike monetary compensation, using models as rewards introduces unique challenges, particularly due to the stochastic nature of these rewards when contribution costs are privately held information. This paper formalizes the optimal contracting problem within CML and proposes a transformation that simplifies the non-convex optimization problem into one that can be solved through convex optimization algorithms. We conduct a detailed analysis of the properties that an optimal contract must satisfy when models serve as the rewards, and we explore the potential benefits and welfare implications of these contract-driven CML schemes through numerical experiments.

Paid with Models: Optimal Contract Design for Collaborative Machine Learning

TL;DR

A detailed analysis of the properties that an optimal contract must satisfy when models serve as the rewards is conducted, and the potential benefits and welfare implications of these contract-driven CML schemes are explored through numerical experiments.

Abstract

Collaborative machine learning (CML) provides a promising paradigm for democratizing advanced technologies by enabling cost-sharing among participants. However, the potential for rent-seeking behaviors among parties can undermine such collaborations. Contract theory presents a viable solution by rewarding participants with models of varying accuracy based on their contributions. However, unlike monetary compensation, using models as rewards introduces unique challenges, particularly due to the stochastic nature of these rewards when contribution costs are privately held information. This paper formalizes the optimal contracting problem within CML and proposes a transformation that simplifies the non-convex optimization problem into one that can be solved through convex optimization algorithms. We conduct a detailed analysis of the properties that an optimal contract must satisfy when models serve as the rewards, and we explore the potential benefits and welfare implications of these contract-driven CML schemes through numerical experiments.

Paper Structure

This paper contains 34 sections, 40 theorems, 223 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Optimal Contracting in CML
  4. Contracting with Observable Costs
  5. Contracting with Private Costs
  6. First-moment problem
  7. Constraint analysis
  8. Experiments
  9. Two-type Case
  10. Multi-type Case
  11. A party may be incentivized to contribute less than their reservation level.
  12. Incentivized collaborative scheme can democratize machine learning.
  13. In the presence of dominant players, small players can still gain from collaboration.
  14. Conclusion
  15. Theoretical Proofs
  16. ...and 19 more sections

Key Result

Proposition 1

Let $\bar{m}_i$ denote the data contribution a type-$i$ party is willing to commit to when training a model on their own. If $c_i \leq c_j$, then $\bar{m}_i \geq \bar{m}_j$, and $f_i \geq f_j\ .$

Figures (7)

  • Figure 1: Optimal Contract Design for Collaborative Machine Learning: The Timeline.
  • Figure 2: Top: Optimal contracts under incomplete information for varied probability of high-cost type $p_1 \in (0,1)$ and total number of participants $N \in [2, 100]$, with $c=\{0.02, 0.01\}$. Bottom: Information costs for the coordinator and information rents for the parties under incomplete information vis-à-vis complete information.
  • Figure 3: Optimal contract designs for multi-type scenarios. Scenario 1: All types would train a model on their own. Scenario 2: All types would not train a model on their own due to prohibitive costs. Scenario 3: Some types would train the model on their own and others would not.
  • Figure 4: Functions $f$ and $g$ and the $m_1$'s that give the maximum values.
  • Figure 5: The solution strategy: delineate solutions to the first-moment problem and then use the proportional assignment to solve the original problem.
  • ...and 2 more figures

Theorems & Definitions (72)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Theorem 1
  • Proposition 7
  • Proposition 8
  • Proposition 0
  • ...and 62 more