The Optimal Sample Complexity of Linear Contracts

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.