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The Pseudo-Dimension of Contracts

Paul Duetting, Michal Feldman, Tomasz Ponitka, Ermis Soumalias

TL;DR

This work develops a principled offline framework for learning near-optimal contract designs from samples of agent types, using the pseudo-dimension to quantify the intrinsic complexity of contract classes. It establishes tight tradeoffs between representation error and sample complexity for linear and bounded contracts, and proves an impossibility result for unbounded contracts. The authors extend the analysis to combinatorial actions, menus of contracts, and online expert-feedback settings, revealing favorable sample complexities and a clear separation from bandit-feedback results. Methodologically, they apply delineability to obtain upper bounds and construct hard instances to derive lower bounds, yielding essentially tight characterizations. The results provide a rigorous, scalable blueprint for data-driven contract design in online and complex action spaces, with implications for platforms and markets where agent-type distributions are learned from history rather than fully known.

Abstract

Algorithmic contract design studies scenarios where a principal incentivizes an agent to exert effort on her behalf. In this work, we focus on settings where the agent's type is drawn from an unknown distribution, and formalize an offline learning framework for learning near-optimal contracts from sample agent types. A central tool in our analysis is the notion of pseudo-dimension from statistical learning theory. Beyond its role in establishing upper bounds on the sample complexity, pseudo-dimension measures the intrinsic complexity of a class of contracts, offering a new perspective on the tradeoffs between simplicity and optimality in contract design. Our main results provide essentially optimal tradeoffs between pseudo-dimension and representation error (defined as the loss in principal's utility) with respect to linear and bounded contracts. Using these tradeoffs, we derive sample- and time-efficient learning algorithms, and demonstrate their near-optimality by providing almost matching lower bounds on the sample complexity. Conversely, for unbounded contracts, we prove an impossibility result showing that no learning algorithm exists. Finally, we extend our techniques in three important ways. First, we provide refined pseudo-dimension and sample complexity guarantees for the combinatorial actions model, revealing a novel connection between the number of critical values and sample complexity. Second, we extend our results to menus of contracts, showing that their pseudo-dimension scales linearly with the menu size. Third, we adapt our algorithms to the online learning setting, where we show that, a polynomial number of type samples suffice to learn near-optimal bounded contracts. Combined with prior work, this establishes a formal separation between expert advice and bandit feedback for this setting.

The Pseudo-Dimension of Contracts

TL;DR

This work develops a principled offline framework for learning near-optimal contract designs from samples of agent types, using the pseudo-dimension to quantify the intrinsic complexity of contract classes. It establishes tight tradeoffs between representation error and sample complexity for linear and bounded contracts, and proves an impossibility result for unbounded contracts. The authors extend the analysis to combinatorial actions, menus of contracts, and online expert-feedback settings, revealing favorable sample complexities and a clear separation from bandit-feedback results. Methodologically, they apply delineability to obtain upper bounds and construct hard instances to derive lower bounds, yielding essentially tight characterizations. The results provide a rigorous, scalable blueprint for data-driven contract design in online and complex action spaces, with implications for platforms and markets where agent-type distributions are learned from history rather than fully known.

Abstract

Algorithmic contract design studies scenarios where a principal incentivizes an agent to exert effort on her behalf. In this work, we focus on settings where the agent's type is drawn from an unknown distribution, and formalize an offline learning framework for learning near-optimal contracts from sample agent types. A central tool in our analysis is the notion of pseudo-dimension from statistical learning theory. Beyond its role in establishing upper bounds on the sample complexity, pseudo-dimension measures the intrinsic complexity of a class of contracts, offering a new perspective on the tradeoffs between simplicity and optimality in contract design. Our main results provide essentially optimal tradeoffs between pseudo-dimension and representation error (defined as the loss in principal's utility) with respect to linear and bounded contracts. Using these tradeoffs, we derive sample- and time-efficient learning algorithms, and demonstrate their near-optimality by providing almost matching lower bounds on the sample complexity. Conversely, for unbounded contracts, we prove an impossibility result showing that no learning algorithm exists. Finally, we extend our techniques in three important ways. First, we provide refined pseudo-dimension and sample complexity guarantees for the combinatorial actions model, revealing a novel connection between the number of critical values and sample complexity. Second, we extend our results to menus of contracts, showing that their pseudo-dimension scales linearly with the menu size. Third, we adapt our algorithms to the online learning setting, where we show that, a polynomial number of type samples suffice to learn near-optimal bounded contracts. Combined with prior work, this establishes a formal separation between expert advice and bandit feedback for this setting.
Paper Structure (52 sections, 45 theorems, 67 equations, 1 figure, 1 table)

This paper contains 52 sections, 45 theorems, 67 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Main Results: Bounds on Pseudo-Dimension and Sample Complexity
  3. Linear Contracts.
  4. Bounded Contracts.
  5. Unbounded Contracts.
  6. Discussion.
  7. Extensions of the Main Results
  8. Combinatorial Contracts.
  9. Menus of Contracts.
  10. Online Learning.
  11. Our Techniques
  12. Upper Bounds and Delineability.
  13. Pseudo-Dimension Lower Bounds.
  14. Sample Complexity Lower Bounds.
  15. Related Literature
  16. ...and 37 more sections

Key Result

Lemma 2.3

For any agent type $\theta = (f,c)$, there exists some $k < n$ and a series of $\alpha$ values $0 = \alpha_0 < \alpha_1 < \ldots < \alpha_k$ such that for every $x \in [0, 1]$, we have $r_p(\theta, x) = r_p(\theta, \alpha_i)$, where $i$ is the maximal index satisfying $\alpha_i \leq x$. In particula

Figures (1)

  • Figure 1: Trade-offs between pseudo-dimension and representation error for linear and bounded contracts in the case where $n > 24 \cdot m^2$. For every point $(\epsilon,d)$, an $\epsilon$-approximation with pseudo-dimension $d$ exists if $(\epsilon,d)$ is in the gray region, and it does not exist if $(\epsilon,d)$ is in the diagonally-marked region. For clarity, we suppress the logarithmic factors in this figure. Specifically, the boundaries between the two regions are tight up to a constant for linear contracts and up to a factor of $\log m$ for bounded contracts.

Theorems & Definitions (92)

  • Definition 2.1: Linear Contracts
  • Definition 2.2: Critical Values
  • Lemma 2.3: Duetting2021CombinatorialC
  • Definition 2.4: Contract Classes
  • Definition 2.5: Representation Error and $\epsilon$-Approximation
  • Definition 3.1: Uniform Learnability
  • Lemma 3.1
  • Definition 3.2: Shattering
  • Definition 3.3: Pseudo-Dimension of Real-Valued Functions
  • Definition 3.4: Pseudo-Dimension of Contracts
  • ...and 82 more