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.

翻译：本文解决了离线设置下从数据中学习最优线性合约的问题，其中代理类型从未知分布中抽取，委托人的目标是设计一个最大化其期望效用的合约。具体而言，我们的分析表明，简单的经验效用最大化（EUM）算法仅需使用 $O(\ln(1/δ) / \varepsilon^2)$ 个样本，就能以至少 $1-δ$ 的概率获得最优线性合约的 $\varepsilon$ 近似。这一结果改进了先前已知的界限，并与 Duetting 等人 [2025] 提出的下界在常数因子内匹配，从而证明了其最优性。我们的分析采用了链式论证，其中的关键洞见是利用了线性合约的一个简单结构特性：其期望收益是非递减的。这一特性——尽管效用函数本身是非单调且不连续的——使得构建链式论证所需的细粒度网成为可能，进而得到了最优的样本复杂度。此外，我们的证明建立了更强的一致收敛保证：使用相同的最优 $O(\ln(1/δ) / \varepsilon^2)$ 样本复杂度，以至少 $1-δ$ 的概率，每个线性合约的经验效用都是其真实期望的 $\varepsilon$ 近似。