The Optimal Sample Complexity of Linear Contracts

2026-01-04

arXiv.org

AI Summary

This work studies the problem of learning an optimal linear contract from data in an offline setting, where the agent’s type is drawn from an unknown distribution and the principal aims to design a contract that maximizes her expected utility. To address the challenges posed by the non-monotonicity and discontinuity of the utility function, the authors propose the Empirical Utility Maximization (EUM) algorithm, which leverages the non-decreasing structural property of the expected reward under linear contracts. Through a chaining argument and a carefully constructed covering net, they establish uniform convergence of empirical utilities to their true expectations over all linear contracts. Consequently, with only \(O(\ln(1/\delta)/\varepsilon^2)\) samples, the algorithm yields an \(\varepsilon\)-approximately optimal contract with probability at least \(1-\delta\), achieving the optimal sample complexity that matches the known information-theoretic lower bound.

Technology Category

• Multiagent Systems: Mechanism Design
• Search and Optimization: Non-convex Optimization
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In this paper, we settle the problem of learning optimal linear contracts from data in the offline setting, where agent types are drawn from an unknown distribution and the principal's goal is to design a contract that maximizes her expected utility. Specifically, our analysis shows that the simple Empirical Utility Maximization (EUM) algorithm yields an $\varepsilon$-approximation of the optimal linear contract with probability at least $1-\delta$, using just $O(\ln(1/\delta) / \varepsilon^2)$ samples. This result improves upon previously known bounds and matches a lower bound from Duetting et al. [2025] up to constant factors, thereby proving its optimality. Our analysis uses a chaining argument, where the key insight is to leverage a simple structural property of linear contracts: their expected reward is non-decreasing. This property, which holds even though the utility function itself is non-monotone and discontinuous, enables the construction of fine-grained nets required for the chaining argument, which in turn yields the optimal sample complexity. Furthermore, our proof establishes the stronger guarantee of uniform convergence: the empirical utility of every linear contract is a $\varepsilon$-approximation of its true expectation with probability at least $1-\delta$, using the same optimal $O(\ln(1/\delta) / \varepsilon^2)$ sample complexity.

Problem

• linear contracts
• sample complexity
• offline learning
• principal-agent problem
• expected utility

Innovation

• linear contracts
• sample complexity
• empirical utility maximization
• uniform convergence
• chaining argument