We study the hidden-action principal-agent problem in an online setting. In each round, the principal posts a contract that specifies the payment to the agent based on each outcome. The agent then makes a strategic choice of action that maximizes her own utility, but the action is not directly observable by the principal. The principal observes the outcome and receives utility from the agent's choice of action. Based on past observations, the principal dynamically adjusts the contracts with the goal of maximizing her utility. We introduce an online learning algorithm and provide an upper bound on its Stackelberg regret. We show that when the contract space is $[0,1]^m$, the Stackelberg regret is upper bounded by $\widetilde O(\sqrt{m} \cdot T^{1-1/(2m+1)})$, and lower bounded by $\Omega(T^{1-1/(m+2)})$, where $\widetilde O$ omits logarithmic factors. This result shows that exponential-in-$m$ samples are sufficient and necessary to learn a near-optimal contract, resolving an open problem on the hardness of online contract design. Moreover, when contracts are restricted to some subset $\mathcal{F} \subset [0,1]^m$, we define an intrinsic dimension of $\mathcal{F}$ that depends on the covering number of the spherical code in the space and bound the regret in terms of this intrinsic dimension. When $\mathcal{F}$ is the family of linear contracts, we show that the Stackelberg regret grows exactly as $\Theta(T^{2/3})$. The contract design problem is challenging because the utility function is discontinuous. Bounding the discretization error in this setting has been an open problem. In this paper, we identify a limited set of directions in which the utility function is continuous, allowing us to design a new discretization method and bound its error. This approach enables the first upper bound with no restrictions on the contract and action space.

翻译：我们研究了在线环境下的隐藏行动委托代理问题。在每一轮中，委托人发布一份合约，规定基于每种结果对代理人的支付方案。随后，代理人会策略性地选择能最大化自身效用的行动，但该行动不由委托人直接观测。委托人观测结果并从代理人的行动选择中获取效用。基于历史观测，委托人动态调整合约以最大化自身效用。我们提出一种在线学习算法，并给出其斯塔克尔伯格遗憾的上界。当合约空间为$[0,1]^m$时，我们证明斯塔克尔伯格遗憾的上界为$\widetilde O(\sqrt{m} \cdot T^{1-1/(2m+1)})$，下界为$\Omega(T^{1-1/(m+2)})$，其中$\widetilde O$省略了对数因子。这一结果表明，指数级于$m$的样本足以且必需用于学习近似最优合约，从而解决了在线合约设计难度的开放问题。此外，当合约被限制在子集$\mathcal{F} \subset [0,1]^m$时，我们定义了$\mathcal{F}$的固有维度（该维度取决于空间中球面码的覆盖数），并基于该固有维度限定了遗憾值。当$\mathcal{F}$为线性合约族时，我们证明斯塔克尔伯格遗憾恰好以$\Theta(T^{2/3})$的速率增长。合约设计问题的挑战在于效用函数的不连续性。对该情境下的离散误差进行界定一直是未解难题。本文识别出效用函数连续性的有限方向，从而能够设计新的离散化方法并界定其误差。这一方法首次实现了对合约与行动空间无约束条件下的上界推导。