The Optimal Sample Complexity of Linear Contracts

TL;DR

The paper tackles offline learning of optimal linear contracts when agent types are drawn from an unknown distribution. It develops a chaining-based analysis to bound the uniform deviation between empirical and true utilities over the linear-contract class, leveraging the monotone empirical reward and resulting in an optimal $O(\ln(1/\delta)/\varepsilon^{2})$ sample complexity; this also yields a uniform convergence guarantee across all linear contracts. The Empirical Utility Maximization algorithm is shown to achieve an $\varepsilon$-approximation with a matching sample bound up to constants, and the results are tight against known lower bounds, advancing principled data-driven contract design that does not depend on the number of actions. The findings significantly reduce the data requirements for learning effective contracts in practice and provide a blueprint for exploiting structural properties of contract classes in offline settings.

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 $1-δ$, using just $O(\ln(1/δ) / \varepsilon^2)$ 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 $1-δ$, using the same optimal $O(\ln(1/δ) / \varepsilon^2)$ sample complexity.

Figures (2)

• Figure 1: An example of the principal's utility $u_p(\theta, \alpha)$ as a function of the linear contract parameter $\alpha$. The utility can be non-monotonic and exhibit discontinuities. The red dots on the x-axis illustrate a simple discretization of the parameter space.
• Figure 2: An example of the empirical reward $\sum_{i=1}^{s}r_{p}(\theta_{i},\alpha)/s$ as a function of the linear contract parameter $\alpha$. The y-axis is discretized into intervals (separated by dashed lines). The pullback of these intervals onto the x-axis is shown as colored bars, and a point added to ${\mathcal{C}}_{\nu}$ from each pullback interval is marked, where in this example $x_{0},x_{1},x_{2},x_{3}$ would be added to the discretization.

Theorems & Definitions (10)

• Theorem 1.1: Main Result
• Corollary 1.2: EUM Optimal Sample Complexity
• Lemma 2.1
• proof : Proof of \ref{['cor:erm']}
• Lemma 2.2
• proof : Proof of \ref{['thm:main']}
• proof : Proof of \ref{['lem:linear_contracts_cover']}
• Lemma 1.1
• proof : Proof of \ref{['lem:empirical_reward_non_decreasing']}
• Definition 2.1: Pseudo-Dimension of Contracts duetting2025pseudodimensioncontracts[Definition 3.5]