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 ε-approximation of the optimal linear contract with probability at least 1 -δ, using just O(ln(1/δ)/ε 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 ε-approximation of its true expectation with probability at least 1 -δ, using the same optimal O(ln(1/δ)/ε 2 ) sample complexity.
A central problem in algorithmic contract theory is to design incentives for agents whose characteristics are unknown and must be learned from data. Consider a digital music platform looking to introduce a new royalty model (contract). Each independent musician (agent) on the platform has a private type, reflecting their creative process and cost of effort, drawn from a population-level distribution that is unknown to the platform. Before implementing a site-wide change of royalty model, the platform runs a pilot program with a small sample of musicians. In this program, it tests several new revenue-sharing contracts and gathers detailed data on their resulting song downloads and streaming engagement. Based on this sample, the platform aims to learn an improved royalty model that optimizes its profits by motivating its entire community of artists.
This “pilot study” is an example of the scenario formalized in the recent seminal work of (Dütting et al., 2025), which establishes a sample-based learning framework for designing an optimal contract from a finite dataset of fully-profiled agents. This framework complements other established models in the literature, each suited for different scenarios. For instance, the Bayesian setting models situations where the principal has full distributional knowledge, ideal for full-information and static scenarios. In contrast, online learning models address dynamic settings where a contract must be adapted through repeated, real-time interactions with agents. The framework of (Dütting et al., 2025) thus captures yet another important real world scenario, the finite sample setting. More formally, (Dütting et al., 2025) consider the following framework, which we adopt (almost) and now formally define. The environment is fixed by a set of n actions an agent can take, indexed by [n] = {1, . . . , n}, and m ≥ 2 possible outcomes, indexed by [m] = {1, . . . , m}. For each outcome j ∈ [m], the principal receives a known, fixed reward r j ≥ 0. It is assumed that r 1 = 0 and there is at least one outcome with a positive reward. An agent is characterized by a private type θ = (f, c) (i.e., unknown to the principal during live interaction), which consists of two components:
• A production function f = (f 1 , . . . , f n ), where each f i is a probability distribution over the m outcomes. Specifically, f i,j is the probability of observing outcome j if the agent chooses action i. • A cost vector c = (c 1 , . . . , c n ), where c i ≥ 0 is the personal cost for the agent to take action i. We assume that action 1 is an outside option with zero cost, i.e., c 1 = 0.
The principal designs a contract, which is a payment vector t = (t 1 , . . . , t m ) where t j ≥ 0. If outcome j occurs, the agent is paid t j . Given a contract t, an agent of type θ will choose an action i ∈ [n] to maximize their own expected utility:
The principal’s utility depends on which action the agent takes. Assuming the agent breaks ties in the principal’s favor, the agent chooses the action i * (θ, t) that maximizes the principal’s utility from the set of the agent’s own best actions (those maximizing Equation ( 1)). The principal’s utility for a given type θ is then:
Finally, we define the learning objective. The principal’s goal is to find a contract t that maximizes the expected utility θ, t)] over an unknown distribution of agent types D. The learning model of (Dütting et al., 2025) assumes the principal has access to a dataset S = {θ 1 , . . . , θ s } of s i.i.d. samples from D, and for each sample θ i ∈ S, the principal is given the full type (i.e., the production function f (i) and cost vector c (i) ), which allows the principal to simulate the agent’s behavior and compute u p (θ i , t) for any candidate contract t.
As described, the basic framework of (Dütting et al., 2025) assumes the principal receives samples of full agent types. We will, however, make a slightly weaker assumption, namely that the principal only has oracle access to compute the empirical utility u p (S, t) = 1 s s i=1 u p (θ i , t) for any candidate contract t, but is not given the specific type of the sampled agent nor their set of actions. This assumption is weaker than the basic assumption made in (Dütting et al., 2025) and still captures the offline setting, where the principal first gathers information to compute u p (S, t) for any t and then does not interact with the agents again. To the best of our knowledge, some of the results from (Dütting et al., 2025) also hold in this weaker setting; we will comment on this when in order.
Within this framework, (Dütting et al., 2025) established a link between the sample complexity of learning a contract class and its pseudo-dimension, a combinatorial complexity measure. While their work provides general tools for analysis, the precise sample complexity remained unsolved for one of the most fundamental classes of contracts, linear contracts, where the agent receives a fixed fraction
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