Introduction

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 , 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 thus captures yet another important real world scenario, the finite sample setting.

More formally, 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 $`[n]=\{1, \dots, n\}`$, and $`m \ge 2`$ possible outcomes, indexed by $`[m]=\{1, \dots, m\}`$. For each outcome $`j \in [m]`$, the principal receives a known, fixed reward $`r_j \ge 0`$. 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 $`\theta=(f,c)`$ (i.e., unknown to the principal during live interaction), which consists of two components:

• A production function $`f=(f_{1},\ldots,f_{n})`$, 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 $`c=(c_{1},\ldots,c_{n})`$, where $`c_i\geq 0`$ 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 $`t=(t_{1},\ldots,t_{m})`$ where $`t_j \ge 0`$. If outcome $`j`$ occurs, the agent is paid $`t_j`$. Given a contract $`t`$, an agent of type $`\theta`$ will choose an action $`i \in [n]`$ to maximize their own expected utility:

\begin{equation}
\label{eq:agent_utility}
 u_{a}(\theta, t, i) = \textstyle\sum_{j=1}^{m}f_{i,j}t_{j} - c_{i}
\end{equation}

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 $`i^{*}(\theta, t)`$ that maximizes the principal’s utility from the set of the agent’s own best actions (those maximizing [eq:agent_utility]). The principal’s utility for a given type $`\theta`$ is then:

\begin{equation*}
 u_p(\theta, t) = \textstyle\sum_{j=1}^{m}f_{i^{*}(\theta, t), j}(r_{j} - t_{j})
\end{equation*}

Finally, we define the learning objective. The principal’s goal is to find a contract $`t`$ that maximizes the expected utility $`U_p(\mathcal{D}, t) = \mathbb{E}_{\theta \sim \mathcal{D}}[u_p(\theta, t)]`$ over an unknown distribution of agent types $`\mathcal{D}`$. The learning model of assumes the principal has access to a dataset $`\rS=\{\theta_{1},\ldots,\theta_{s}\}`$ of $`s`$ i.i.d. samples from $`\mathcal{D}`$, and for each sample $`\theta_i \in \rS`$, 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 $`u_p(\theta_i, t)`$ for any candidate contract $`t`$.

As described, the basic framework of 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 $`u_{p}(\rS,t)=\frac{1}{s}\sum_{i=1}^{s}u_p(\theta_i, t)`$ 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 and still captures the offline setting, where the principal first gathers information to compute $`u_{p}(\rS,t)`$ for any $`t`$ and then does not interact with the agents again. To the best of our knowledge, some of the results from also hold in this weaker setting; we will comment on this when in order.

Within this framework, 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 of the principal’s reward. Despite linear contracts’ simplicity, which makes them appealing from a practical standpoint, they are also known to exhibit robustness to unknown agent actions and to be able, under certain conditions, to approximate the performance of fully optimal, yet more complex, contracts .

In this paper, we precisely characterize the sample complexity for learning an $`\varepsilon`$-approximation of the optimal linear contract. Specifically, we show that the simple Empirical Utility Maximization (EUM) algorithm, choosing a contract within the linear contracts that maximizes the empirical utility, yields an optimal contract up to an additive $`\varepsilon`$-error with probability $`1-\delta`$ given $`O( \ln{(1/\delta )}/\varepsilon^{2} )`$ samples, which is tight up to constant factors due to a lower bound of . Furthermore, we show the same optimal sample complexity bound for the harder problem of learning the class of linear contracts uniformly, that is, ensuring the empirical and expected utilities are simultaneously $`\varepsilon`$-close for all linear contracts.

Our tighter bound comes from a more direct analytical path leveraging key properties of linear contracts. While the general theory of relies on the combinatorial abstraction of pseudo-dimension(See 6), our proof uses a “first-principles” chaining argument. The key technical insight is to exploit the inherent monotonic structure of the expected reward of linear contracts. This property allows for the construction of a fine-grained net over the contract space, enabling a chaining argument that yields the optimal sample complexity. This approach handles the discontinuities and non-monotonicity of the utility function. In doing so, we demonstrate how exploiting the specific structure of a contract class can lead to optimal results where the general-purpose tools of previous work did not.

To describe our results, we define the class of linear contracts as $`\cC_{\textit{linear}}= \{\alpha r\mid \alpha\in [0,1] \}`$ for a fixed $`r\in[0,1]^{m}`$, where we write $`\alpha`$ as shorthand for a contract in $`\cC_{\textit{linear}}`$, and we will also interchange between $`\cC_{\textit{linear}}`$ and $`[0,1]`$. Formally, we show the following theorem, which is the main result of this paper.

\begin{equation*}
        |U_p(\mathcal{D}, \alpha) - U_p(\rS, \alpha)| \leq  \varepsilon.
\end{equation*}
```*

</div>

We note that our main theorem gives a uniform convergence bound for
learning the difference between the empirical and expected utility for
the class of linear contracts, with sample complexity independent of the
number of actions $`n`$ and the number of outcomes $`m`$. This is
desirable, as we assume that the principal does not know the number of
actions $`n`$, and the number of actions and outcomes $`m`$ could be
large.[^1]

Furthermore, the uniformity of the bound allows the principal not only
to learn the utility of the optimal contract up to an additive
$`\varepsilon`$ factor, but also to compare the utility of any two
contracts and assess which is better, up to $`\varepsilon`$ precision.

It is also worth noting that the bound, up to constants, is the same as
if one wanted to guarantee that the empirical utility of a single
contract is $`\varepsilon`$-close to the expected utility of that
contract. Thus, guaranteeing that the empirical utility of any (or one)
contract is $`\varepsilon`$-close to its expected utility requires, up
to constant factors, the same number of samples.

From the above bound, our corollary stating that the simple EUM
algorithm achieves the optimal sample complexity follows.

<div id="cor:erm" class="corollary">

**Corollary 2** (EUM Optimal Sample Complexity). *Let $`\mathcal{D}`$ be
an unknown distribution over agent types, $`r\in[0,1]^{n}`$ be a reward,
and let $`\varepsilon>0`$ and $`\delta\in(0,1)`$ be given. Then, for
$`s\geq  6912 \ln{(4/\delta )}/\varepsilon^{2}`$ with probability at
least $`1-\delta`$ over $`\rS\sim \cD^{s}`$, it holds that
<a href="#alg:erm_linear" data-reference-type="ref+label"
data-reference="alg:erm_linear">[alg:erm_linear]</a>\[Empirical Utility
Maximization algorithm\] returns a contract
$`\hat{\alpha} \in \cC_{\textit{linear}}`$ such that:
``` math
\begin{equation*}
        U_{p}(\cD,\hat{\alpha})\geq \textstyle\sup_{\alpha\in\cC_{\textit{linear}}}U_{p}(\cD,\alpha) - \varepsilon.
\end{equation*}

To the best of our knowledge, our result is the first in the statistical setting introduced by to obtain uniform convergence and learn the optimal contract up to an $`\varepsilon`$ error, an easier problem, at an optimal sample complexity for a class of non-trivial contracts, which we view as a step towards understanding the optimal sample complexity of learning contracts in this setting. We also remark that we did not attempt to optimize the constants in the bound of 1 and 2.

Related Work

The study of contracts has a rich history in economics, with seminal contributions from and . The importance of the field, as well as the foundational work of Oliver Hart and Bengt Holmström, was highlighted when the Nobel Prize in Economics in 2016 was awarded to Oliver Hart and Bengt Holmström for their work on contract theory .

Although contract design has its roots in economics, it has also garnered significant interest at the intersection of economics and computer science, particularly with the emergence of algorithmic contract design. The study of algorithmic contracts encompasses several distinct settings and aspects (some of which include, but are not limited to): The computational facets of contract design ; the Bayesian setting, where the distribution of agent types is known ; and the online setting, where the principal interacts sequentially with agents, receiving only bandit feedback, and must design contracts on the fly . This paper focuses on the offline setting introduced by , and we refer the reader to their work for a more comprehensive comparison of this setting with other paradigms.

Comparison to Previous Work

In the offline setting, shows two upper bounds on the sample complexity of learning the best linear contract: Theorem 4.1 (combined with Theorem 3.7) and Theorem 5.4. These theorems show that either $`O((\ln{(1/\varepsilon)}+\ln{(1/\delta )})/\varepsilon^{2})`$ or $`O((\ln{(n )}+\ln{(1/\delta )})/\varepsilon^{2})`$ samples are sufficient to learn the best linear contract up to an additive error of $`\varepsilon`$ with probability at least $`1-\delta`$.

Some comments are in order regarding these two bounds. Both bounds are proven by upper-bounding the pseudo-dimension $`d`$ of a class of contracts and then applying Theorem 3.7 in , which, given such a bound, gives a sample complexity of $`O((d+\ln{(1/\delta )})/\varepsilon^{2})`$.

In the first case, the bound on the pseudo-dimension is not on the space of linear contracts itself but is instead on a discretization of the contract space consisting of multiples of $`\varepsilon`$, where this discretization preserves a good approximation of the best contract. This gives a bound on the size of the discretization of $`O(1/\varepsilon)`$, whereby the pseudo-dimension of this discretization can be bounded by the logarithm of the number of contracts in the discretization, i.e., $`O( \ln{(1/\varepsilon)} )`$. The result then follows from their general framework, which only requires a bound on the pseudo-dimension of the contract class, and running the EUM algorithm on the discretization. Thus, the first bound does not provide a bound on the sample complexity of learning the class of linear contracts uniformly (all contracts simultaneously), as our 1 does. Instead, it provides a bound on the sample complexity of learning a discretization of the class of linear contracts that preserves the optimal contract up to an additive error of $`\varepsilon`$, which is sufficient for the EUM algorithm to learn the optimal contract up to an additive error of $`\varepsilon`$ with probability at least $`1-\delta`$. Furthermore, to the best of our knowledge this sample complexity bound combined with the EUM algorithm do as our [alg:erm_linear], only need an oracle for the empirical utility over $`\rS,`$ and not the full information of the agents types.

The second bound is a bound on the pseudo-dimension of the class of linear contracts, which they show can be upper bounded by $`O(\ln{(n )})`$, this is done by relating the number of critical values of a linear contract (where the principal’s expected reward changes), which is at most $`n`$, to the pseudo-dimension of the class of linear contracts.

This bound recovers the full generality of our 1, but with the drawback of its sample complexity being dependent on the number of actions, $`n`$, which could be large and is assumed by the setting we consider to be unknown to the principal, thus, to the best of our knowledge to leverage the sample complexity bound of $`O((\ln{(n )}+\ln{(1/\delta )})/\varepsilon^{2})`$ combined with the EUM algorithm, one would need to have knowledge of $`n`$ which is for instance provided in the basic setting of full information on the sample of .

Thus, in the regime of large $`n >1/\varepsilon`$, there remained a gap between the best known sample complexity for learning the class of linear contracts uniformly and that of learning the best contract, which we close with our results, 1 and 2. Furthermore, Theorem 5.9 of , which shows a lower bound of $`\Omega(\ln{(1/\delta )}/\varepsilon^{2})`$ on the sample complexity of learning the optimal linear contract up to $`\varepsilon`$-error, when combined with our results in 1 and 2, witnesses the tightness of all these results up to constant factors and shows that there is no gap in the sample complexity between learning the class of linear contracts uniformly and learning the best contract.

It is worth noting that the lower bound of is for a simple distribution over two agent types with two actions, $`n=2`$. Thus, the lower bound (and the previous uniform upper bound) did not rule out the possibility of the sample complexity being dependent on the number of actions, which we show is not the case.

In the remainder of the paper, we show how to prove our main results, 1 and 2. The proof uses a key structural property of the class of linear contracts, having a non-decreasing reward, highlighting how using specific properties of a contract space can be leveraged to obtain optimal sample complexity bounds. We hope this can lead to more optimal bounds in the area of contract design.

Proof Overview and Formal Proofs

In this section, we prove our main result 1 and its 2. We begin the section with a high-level overview of the proof of 1, and then show how it naturally implies 2, and finally, provide the full proof of 1.

The high-level proof idea of 1 is as follows: We first upper bound the difference between the empirical utility and the expected utility for all linear contracts, $`\sup_{\alpha\in[0,1]}|u_p(\rS,\alpha)-u_p(\cD,\alpha)|`$, by the Rademacher complexity, $`\mathop{\mathrm{\mathbb{E}}}_{\rS\sim \cD^{s},\sigma\sim\{ -1,1\}^{s} }[\sup_{\alpha\in [0,1]} \sum_{i=1}^{s}\sigma_{i}u_p(\theta_{i},\alpha)/s]`$, of the class of linear contracts plus $`\varepsilon,`$ by McDiarmid’s inequality as done in . This bound holds with probability at least $`1-\delta`$ over $`\rS=(\theta_{1},\ldots,\theta_{s})\sim \cD^{s}`$, for $`s=\Omega(\ln{(1/\delta )}/\varepsilon^{2}).`$

The upper bound on the generalization error in terms of the Rademacher complexity of linear contracts provides an intuitive explanation of the generalization property of linear contracts, as the Rademacher complexity measures how prone the linear contracts are to fitting random noise, i.e., if linear contracts could overfit to the data, leading to the empirical utility and expected utility being far from each other. However, as our argument shows, the linear contracts have a low Rademacher complexity, thus the empirical utility and expected utility are close to each other.

Having upper bounded the generalization error of linear contracts by its Rademacher complexity, we use a chaining result from [Proposition 5.3], to upper bound the Rademacher complexity of linear contracts in terms of their covering number, by the following relation:

\begin{align}
\label{eq:chaining}
    &\mathop{\mathrm{\mathbb{E}}}_{\sigma\sim\{  -1,1\}^{s} }\bigl[\textstyle\sup_{\alpha\in [0,1]} \sum_{i=1}^{s}\sigma_{i}u_p(\theta_{i},\alpha)/s\bigr]
    \\
    &\leq
    \inf_{\eta\in[0,1/2]}\bigl\{  4\eta +\frac{12}{\sqrt{s}}\textstyle\int_{\eta}^{1/2}\sqrt{\ln{N(\cC_{\textit{linear}}, || \cdot ||_{2,\rS},\nu ) }} d\nu \bigr\}.\nonumber
\end{align}

Here, $`N(\cC_{\textit{linear}}, || \cdot ||_{2,\rS},\nu )`$ is the covering number of linear contracts on the set of agents $`\rS`$ with respect to the $`L_{2}`$ norm and precision $`\nu.`$ This is the smallest integer such that there exists a set of linear contracts $`\cC_{\nu}`$ of size $`N(\cC_{\textit{linear}},|| \cdot ||_{2,\rS},\nu )`$, which satisfies that for any linear contract $`\alpha\in \cC_{\textit{linear}}`$, there exists $`\widehat{\alpha}\in \cC_{\nu}`$ such that

\begin{align*}
    \sqrt{\textstyle\sum_{i=1}^{s} (u_{p}(\theta_{i},\alpha)-u_{p}(\theta_{i},\widehat{\alpha}))^{2}/s} \leq \nu.
\end{align*}

We then show that if one can find a cover of size $`O((1/\nu)^{c})`$ for some constant $`c>0,`$ [eq:chaining] reduces to $`O(\inf_{\eta\in[0,1/2] }\{ \eta +1/\sqrt{s} \} )`$ which is $`O(1/\sqrt{s}).`$ Since we set $`s=\Omega(\ln{(1/\delta )}/\varepsilon^{2})`$ we have that $`O(1/\sqrt{s}) = O(\varepsilon)`$, which implies that the generalization error is $`O(\varepsilon)`$ with probability at least $`1-\delta.`$

Thus, we have reduced the problem of bounding the generalization error of all linear contracts to bounding their covering number by $`O((1/\nu)^{c})`$. We now proceed to show how to find such a cover. We will first find an $`L_{1}`$ cover of size $`O(1/\nu)`$ which, as we will show later, can be converted to an $`L_{2}`$ cover of size $`O((1/\nu)^{2})`$, which by the above argument is sufficient to obtain the desired bound on the generalization error of linear contracts.

Now, an intuitive first approach one could explore to find such a cover would be to discretize the interval $`[0,1]`$ into $`O(1/\nu)`$ points, being multiples of $`\nu`$, which would work if the utility of linear contracts were linear in the parameter $`\alpha`$. However, this is not the case, as the utility of linear contracts can have discontinuities and, in general, does not possess any monotonicity properties. See 1 for an illustration. However, trying the above brings some insights that might be important to finding a small cover, namely if we let $`\alpha\in [0,1]`$ and $`\widehat{\alpha}`$ be the point in the discretization of the interval $`[0,1]`$ that is closest to $`\alpha`$, we have that

\begin{align}
\label{eq:empirical_reward_diff}
    &\tfrac{1}{s}\textstyle\sum_{i=1}^{s}\negmedspace |u_{p}(\theta_i,\alpha)-u_{p}(\theta_i,\widehat{\alpha})|=
    \\
    &
    \tfrac{1}{s}\textstyle\sum_{i=1}^{s}\negmedspace
    \left|\textstyle\sum_{j=1}^{m}f_{i^{*}(\theta_{i}, \alpha), j}(1-\alpha)r_{j}-f_{i^{*}(\theta_{i}, \widehat{\alpha}), j}(1-\widehat{\alpha})r_{j}  \right|\nonumber
    \\
    &\leq
    \tfrac{(1-\alpha)}{s}
    \textstyle\sum_{i=1}^{s}\negmedspace\left|\textstyle\sum_{j=1}^{m}\left(f_{i^{*}(\theta_{i}, \alpha), j}r_{j}-f_{i^{*}(\theta_{i}, \widehat{\alpha}), j}r_{j}\right)  \right|\nonumber
\\
    &+\underbrace{\tfrac{1}{s}\textstyle\sum_{i=1}^{s}\negmedspace\left|\textstyle\sum_{j=1}^{m}f_{i^{*}(\theta_{i}, \widehat{\alpha}), j}\left(\alpha -\widehat{\alpha}\right)r_{j}\right|}_{\nu}\nonumber
\end{align}

where the inequality follows from adding and subtracting $`\alpha`$ in the term $`(1-\widehat{\alpha})`$ and using the triangle inequality. Now, since $`|\alpha-\widehat{\alpha}|\leq \nu`$, $`r_{j}\in[0,1]`$, and $`f_{i^{*}(\theta_{i}, \widehat{\alpha}), j}`$ is a probability distribution, it follows from yet another use of the triangle inequality that the last term in the above is at most $`\nu.`$ Thus, we have a bound on the second term of $`\nu`$, however we do not have control of the first term yet.

Thus, we have to use a more refined approach. To this end, we use a result of , which shows that even though the empirical utility is not monotonic, the empirical reward $`r_{p}(\theta,\alpha):=\sum_{j=1}^{m} f_{i^{*}(\theta, \alpha), j}r_{j}`$ of linear contracts is a non-decreasing function in the contract parameter $`\alpha\in[0,1].`$ See 2 for an illustration.

Our insight is now that we can discretize the y-axis of the empirical reward $`\frac{1}{s}\sum_{i=1}^{s} r_{p}(\theta_{i},\cdot)`$ into intervals of length $`O(\nu)`$, take the pullback of each of these intervals, and add a point from each of these pullbacks to our discretization $`\cC_{\nu}`$. Furthermore, we also discretize the x-axis into a grid of $`O(1/\nu)`$ equally spaced points and add these to $`\cC_{\nu}.`$ Now, for any linear contract $`\alpha\in[0,1]`$, we have that $`\frac{1}{s}\sum_{i=1}^{s} r_{p}(\theta_{i},\alpha)`$ takes a value in one of the intervals of length $`\nu`$, thus there exists a point $`\widehat{\alpha}\in \cC_{\nu}`$ such that the pullback of the interval containing $`\frac{1}{s}\sum_{i=1}^{s} r_{p}(\theta_{i},\alpha)`$ contains $`\widehat{\alpha},`$ and furthermore this point $`\widehat{\alpha}`$ can be chosen to be at most $`\nu`$ from the point $`\alpha.`$ Now using our observation from our earlier attempt, [eq:empirical_reward_diff], we have that for such a contract $`\widehat{\alpha}\in \cC_{\nu}`$,

\begin{align*}
    &\tfrac{1}{s}\textstyle\sum_{i=1}^{s}\negmedspace |u_{p}(\theta_i,\alpha)-u_{p}(\theta_i,\widehat{\alpha})|=
    \\
    &
    \tfrac{1}{s}\textstyle\sum_{i=1}^{s}\negmedspace
    \left|\textstyle\sum_{j=1}^{m}f_{i^{*}(\theta_{i}, \alpha), j}(1-\alpha)r_{j}-f_{i^{*}(\theta_{i}, \widehat{\alpha}), j}(1-\widehat{\alpha})r_{j}  \right|
    \\
    &\leq
    \tfrac{(1-\alpha)}{s}
    \textstyle\sum_{i=1}^{s}\negmedspace\left|\textstyle\sum_{j=1}^{m}\left(f_{i^{*}(\theta_{i}, \alpha), j}r_{j}-f_{i^{*}(\theta_{i}, \widehat{\alpha}), j}r_{j}\right)  \right|
    +\nu
\end{align*}

Furthermore, since the empirical reward is non-decreasing, and $`\widehat{\alpha}\leq \alpha`$ (or for $`\alpha \leq \widehat{\alpha}`$ with the order switched), implying that $`\sum_{j=1}^{m}(f_{i^{*}(\theta_{i}, \alpha), j}r_{j}-f_{i^{*}(\theta_{i}, \widehat{\alpha}), j}r_{j})\geq 0`$, we can drop the absolute value in the first term. Thus, we have that

\begin{align*}
    &(1-\alpha)\frac{1}{s}
    \textstyle\sum_{i=1}^{s}\left|\textstyle\sum_{j=1}^{m}\left(f_{i^{*}(\theta_{i}, \alpha), j}r_{j}-f_{i^{*}(\theta_{i}, \widehat{\alpha}), j}r_{j}\right)  \right|
    \\
    &=(1-\alpha)\frac{1}{s}
    \textstyle\sum_{i=1}^{s}\textstyle\sum_{j=1}^{m}\left(f_{i^{*}(\theta_{i}, \alpha), j}r_{j}-f_{i^{*}(\theta_{i}, \widehat{\alpha}), j}r_{j}\right)
    \\
    &=(1-\alpha)(\frac{1}{s}\textstyle\sum_{i=1}^{s}r_{p}(\theta_{i}, \alpha)-\frac{1}{s}\textstyle\sum_{i=1}^{s}r_{p}(\theta_{i}, \widehat{\alpha}))
\end{align*}

where we can now use that $`\alpha`$ and $`\widehat{\alpha}`$ were in the pullback of the same interval, so we have that $`\frac{1}{s}\sum_{i=1}^{s}r_{p}(\theta_{i}, \alpha)`$ is at most $`\nu`$ from $`\frac{1}{s}\sum_{i=1}^{s}r_{p}(\theta_{i}, \widehat{\alpha})`$, showing that

\begin{align*}
    \textstyle\sum_{i=1}^{s} |u_{p}(\theta_i,\alpha)-u_{p}(\theta_i,\widehat{\alpha})|/s\leq 2\nu.
\end{align*}

Now, using that $`|u_{p}(\theta_i,\alpha)-u_{p}(\theta_i,\widehat{\alpha})|\leq 1`$, we conclude that

\begin{align}
    \sqrt{\textstyle\sum_{i=1}^{s} (u_{p}(\theta_{i},\alpha)-u_{p}(\theta_{i},\widehat{\alpha}))^{2}/s} \leq \sqrt{2\nu},
\end{align}

whereby rescaling $`\nu`$ to $`\nu^{2}/2`$ gives us the existence of a cover of size $`O((1/\nu)^{c})`$ for the constant $`c=2`$, which gives us the desired result.

With the high-level proof idea behind 1 explained, we now proceed to show how it implies the optimal sample complexity bound of the Empirical Utility Maximization (EUM) algorithm, i.e., 2.

Proof of <a href="#cor:erm" data-reference-type=“ref+label”

data-reference=“cor:erm”>2

We now show that the simple Empirical Utility Maximization (EUM) algorithm over an appropriate set of linear contracts (the same as considered in [Lemma 4.3]) gives an efficient algorithm for computing, with probability at least $`1-\delta`$, an $`\varepsilon`$ approximation of the best linear contract with the optimal sample complexity bound of 2. Formally, we consider the following algorithm.

Before we prove 2, we need the following lemma, which is due to . For completeness, we provide the proof in 4.

\begin{align*}
       \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha'),j}  r_{j}=r_{p}(\theta,\alpha')\geq r_{p}(\theta,\alpha) = \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha),j}   r_{j}.
\end{align*}
```*

</div>

Using <a href="#thm:main" data-reference-type="ref+label"
data-reference="thm:main">1</a> and
<a href="#lem:empirical_reward_non_decreasing"
data-reference-type="ref+label"
data-reference="lem:empirical_reward_non_decreasing">3</a>, we now give
the proof of <a href="#cor:erm" data-reference-type="ref+label"
data-reference="cor:erm">2</a>, i.e., that
<a href="#alg:erm_linear" data-reference-type="ref+label"
data-reference="alg:erm_linear">[alg:erm_linear]</a> obtains the optimal
sample complexity for learning a linear contract that is
$`\varepsilon`$-close to the optimal contract’s utility.

<div class="proof">

*Proof of <a href="#cor:erm" data-reference-type="ref+label"
data-reference="cor:erm">2</a>.* As noted in \[Lemma 4.3\] we have that
for $`\alpha^{\star}\in[0,1]`$ such that
$`U_{p}(\cD,\alpha^{\star})=\sup_{\alpha\in [0,1]}U_{p}(\cD,\alpha) -\varepsilon/4`$,
it holds for the point $`\alpha'\in D_{\varepsilon/4}`$ that is closest
to the right of $`\alpha^{\star}`$ that
$`u_{p}(\cD,\alpha')\geq u_{p}(\cD,\alpha^{\star}).`$ This can be seen
by the following calculation, using
<a href="#lem:empirical_reward_non_decreasing"
data-reference-type="ref+label"
data-reference="lem:empirical_reward_non_decreasing">3</a> and the fact
that $`\alpha'`$ is the point in $`D_{\varepsilon/4 }`$ closest to the
right of $`\alpha^{\star}`$:
``` math
\begin{align*}
      &u_p(\cD,\alpha')
      = \mathop{\mathrm{\mathbb{E}}}_{\theta\sim \cD}\bigl[\textstyle\sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha'),j} (1-\alpha')r_{j}\bigr]
      \\
      &\geq \mathop{\mathrm{\mathbb{E}}}_{\theta\sim \cD}\bigl[\textstyle\sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha'),j} (1-\alpha^{\star})r_{j}\bigr] - \varepsilon/4
      \\
      &\geq \mathop{\mathrm{\mathbb{E}}}_{\theta\sim \cD}\bigl[\textstyle\sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha^{\star}),j} (1-\alpha^{\star})r_{j}\bigr] - \varepsilon/4
      \\
      &\geq
        \textstyle\sup_{\alpha\in [0,1]}\mathop{\mathrm{\mathbb{E}}}_{\theta\sim \cD}\bigl[\textstyle\sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha),j} (1-\alpha^{\star})r_{j}\bigr] - \varepsilon/2
\end{align*}

\begin{align*}
        |u_p(S,\alpha)-u_p(\cD,\alpha)|\leq \varepsilon/2.
\end{align*}

\begin{align*}
        &u_p(S,\widehat{\alpha^{\star}})
        \geq
        \textstyle\sup_{\alpha\in D_{\varepsilon/4 }}u_p(S,\alpha)
        \\
        &
        \geq \textstyle\sup_{\alpha\in D_{\varepsilon/4 }} u_p(\cD,\alpha)-\varepsilon/2
        \geq \textstyle\sup_{\alpha\in [0,1]} u_p(\cD,\alpha) -\varepsilon,
\end{align*}

\begin{align*}
        u_p(\cD,\widehat{\alpha^{\star}})\geq \textstyle\sup_{\alpha\in [0,1]} u_p(\cD,\alpha) -\varepsilon.
\end{align*}

Proof of <a href="#thm:main" data-reference-type=“ref+label”

data-reference=“thm:main”>1

We now proceed to give the proof of 1. To show 1, we will use the following lemma giving a bound on the $`L_{2}`$ cover of linear contracts of size $`O(1/\nu^2)`$, which is our main technical contribution.

With 4 in hand, we can now prove 1.

\begin{align}
\label{eq:rademacherlinearcontracts3}
&\sup_{\alpha\in [0,1]} u_p(\rS,\alpha)-u_p(\cD,\alpha)
\\
 &\in \mathop{\mathrm{\mathbb{E}}}_{\rS\sim \cD^{s}}\bigl[\sup_{\alpha\in [0,1]} u_p(\rS,\alpha)-u_p(\cD,\alpha)\bigr]\pm \varepsilon/2.\nonumber
\end{align}

\begin{align}
\label{eq:rademacherlinearcontracts4}
    &\mathop{\mathrm{\mathbb{E}}}_{\rS\sim \cD^{s}}\bigl[\textstyle\sup_{\alpha\in [0,1]} u_p(\rS,\alpha)-u_p(\cD,\alpha)\bigr]
    \\
    &=\mathop{\mathrm{\mathbb{E}}}_{\rS\sim \cD^{s}}\bigl[\textstyle\sup_{\alpha\in [0,1]}\mathop{\mathrm{\mathbb{E}}}_{\rS'\sim\cD^{s}}\bigl[ u_p(\rS,\alpha)-u_p(\rS',\alpha)\bigr]\bigr]\nonumber
    \\
    &\leq\mathop{\mathrm{\mathbb{E}}}_{\rS\sim \cD^{s}}\bigl[\mathop{\mathrm{\mathbb{E}}}_{\rS'\sim\cD^{s}}\bigl[\textstyle\sup_{\alpha\in [0,1]} u_p(\rS,\alpha)-u_p(\rS',\alpha)\bigr]\bigr]\nonumber
    \\
    &=\mathop{\mathrm{\mathbb{E}}}_{\rS,\rS'\sim \cD^{s}}\bigl[\textstyle\sup_{\alpha\in [0,1]} \textstyle\sum_{i=1}^{s}\bigl(u_p(\theta_{i},\alpha)-u_p(\theta_{i}',\alpha)\bigr)/s\bigr]\nonumber
    \\
    &=\mathop{\mathrm{\mathbb{E}}}_{\rS,\rS'\sim \cD^{s},\sigma}\bigl[\sup_{\alpha\in [0,1]} \textstyle\sum_{i=1}^{s}\sigma_{i}\bigl(u_p(\theta_{i},\alpha)-u_p(\theta_{i}',\alpha)\bigr)/s\bigr]\nonumber
    \\
    &\leq 2
    \mathop{\mathrm{\mathbb{E}}}_{\rS\sim \cD^{s},\sigma\sim\{ -1,1\}^{s} }\bigl[\textstyle\sup_{\alpha\in [0,1]} \textstyle\sum_{i=1}^{s}\sigma_{i}u_p(\theta_{i},\alpha)/s\bigr]\nonumber
\end{align}

\begin{align}
\label{eq:rademacherlinearcontracts1}
    &\mathop{\mathrm{\mathbb{E}}}_{\sigma\sim\{ -1,1\}^{s} }\bigl[\textstyle\sup_{\alpha\in [0,1]} \textstyle\sum_{i=1}^{s}\sigma_{i}u_p(\theta_{i},\alpha)/s \bigr]
    \\
    &\leq
    \inf_{\eta\in[0,1/2]}\bigl\{ 4\eta +\frac{12}{\sqrt{s}}\textstyle\int_{\eta}^{1/2}\sqrt{\ln{(N(\cC_{\textit{linear}}, ||\cdot ||_{2,S},\nu ) )}} d\nu \bigr\},\nonumber
\end{align}

\begin{align}
\label{eq:rademacherlinearcontracts2}
    &\mathop{\mathrm{\mathbb{E}}}_{\sigma\sim\{ -1,1\}^{s} }\bigl[\sup_{\alpha\in [0,1]} \textstyle\sum_{i=1}^{s}\sigma_{i}u_p(\theta_{i},\alpha)/s\bigr]
    \\
    &\leq\negmedspace\negmedspace
    \inf_{\eta\in(0,1/2]}\bigl\{ 4\eta +\frac{12}{\sqrt{s}}\textstyle\int_{\eta}^{1/2}\sqrt{\ln{(N([0,1], || \cdot ||_{2,S},\nu ) )}} d\nu \bigr\}\nonumber
    \\
    &\leq\negmedspace\negmedspace
    \inf_{\eta\in(0,1/2]}\bigl\{ 4\eta +\frac{12}{\sqrt{s}}\textstyle\int_{\eta}^{1/2}\sqrt{\ln{\bigl(12/\nu^{2}\bigr)}} d\nu \bigr\} \nonumber
    \\
    &=\negmedspace\negmedspace
    \inf_{\eta\in(0,1/2]}\bigl\{ 4\eta +\frac{12\sqrt{2}}{\sqrt{s}}\textstyle\int_{\eta}^{1/2}\sqrt{\ln{\bigl(\sqrt{12}/\nu\bigr)}} d\nu \bigr\} \nonumber
    \\
    &=\negmedspace\negmedspace\inf_{\eta\in(0,1/2]}\bigl\{ 4\eta +\frac{12\cdot \sqrt{2\cdot 12}}{\sqrt{s}}\textstyle\int_{\eta/\sqrt{12}}^{1/(2\cdot \sqrt{12})}\sqrt{\ln{\bigl(1/\nu'\bigr)}} d\nu' \bigr\} \nonumber
    \\
    &\leq\negmedspace\negmedspace\inf_{\eta\in(0,1/2]}\bigl\{ 4\eta +\frac{12\cdot \sqrt{2\cdot 12}}{\sqrt{s}}\textstyle\int_{0}^{1/(2\cdot \sqrt{12})}\sqrt{\ln{\bigl(1/\nu'\bigr)}} d\nu' \bigr\} \nonumber
    \\
    &\leq\negmedspace\negmedspace \inf_{\eta\in(0,1/2]}\bigl\{ 4\eta +\frac{12\cdot \sqrt{2\cdot 12}}{\sqrt{s}}\frac{1}{4} \bigr\}
    = \frac{6\cdot\sqrt{6}}{\sqrt{s}}\nonumber
\end{align}

\begin{align}
\label{eq:rademacherlinearcontracts5}
    \mathop{\mathrm{\mathbb{E}}}_{\rS\sim \cD^{s}}\bigl[\sup_{\alpha\in [0,1]} u_p(\rS,\alpha)-u_p(\cD,\alpha)\bigr]\leq\tfrac{12\cdot \sqrt{6}}{\sqrt{s}}
\end{align}

\begin{align*}
    &\textstyle\sup_{\alpha\in [0,1]} u_p(\rS,\alpha)-u_p(\cD,\alpha)
    \\
&\in
    \mathop{\mathrm{\mathbb{E}}}_{\rS\sim \cD^{s}}\bigl[\textstyle\sup_{\alpha\in [0,1]} u_p(\rS,\alpha)-u_p(\cD,\alpha)\bigr]\pm \varepsilon/2
\end{align*}

\begin{align}
    0\leq \mathop{\mathrm{\mathbb{E}}}_{\rS\sim \cD^{s}}\bigl[\textstyle\sup_{\alpha\in [0,1]} u_p(\rS,\alpha)-u_p(\cD,\alpha)\bigr]\leq \varepsilon/2, \nonumber
\end{align}

\begin{align*}
    -\varepsilon\leq \textstyle\sup_{\alpha\in [0,1]} u_p(\rS,\alpha)-u_p(\cD,\alpha)\leq\varepsilon.
\end{align*}

\begin{align*}
   -\varepsilon\leq \textstyle\sup_{\alpha\in [0,1]} u_p(\cD,\alpha)-u_p(\rS,\alpha)\leq \varepsilon,
\end{align*}

With the proof of 1 given using 4, we now proceed to give the proof of 4.

\begin{align}
\label{lem:linearcontractscover}
    \textstyle\sum_{i=1}^{s}|u_{p}(\theta_i,\alpha)-u_{p}(\theta_i,\widehat{\alpha})|/s\leq \nu.
\end{align}

\begin{align*}
    &\sqrt{\textstyle\sum_{i=1}^{s}|u_{p}(\theta_i,\alpha)-u_{p}(\theta_i,\widehat{\alpha})|^{2}/s}
    \\
    &\leq \sqrt{\textstyle\sum_{i=1}^{s}|u_{p}(\theta_i,\alpha)-u_{p}(\theta_{i},\widehat{\alpha})|/s} \leq\sqrt{\nu^{2}}=\nu,
\end{align*}

\begin{align}
\label{eq:linearcontracts1}
       y_{i} \leq \textstyle\sum_{k=1}^{s}r_{p}(\theta_{k},x')/s< y_{i+1},
\end{align}

\begin{align}
\label{eq:linearcontracts4}
       y_{j} \leq \sum_{i=1}^{s}r_{p}(\theta_{i},\widehat{\alpha})/s \leq \sum_{i=1}^{s}r_{p}(\theta_{i},\alpha)/s< y_{j+1},
\end{align}

\begin{align}
\label{eq:linearcontracts3}
          &0\leq\textstyle\sum_{i=1}^{s}|r_{p}(\theta_{i},\alpha)/s-r_{p}(\theta_{i},\widehat{\alpha})/s|=
          \\
          &\textstyle\sum_{i=1}^{s}r_{p}(\theta_{i},\alpha)/s-\textstyle\sum_{i=1}^{s}r_{p}(\theta_{i},\widehat{\alpha})/s\leq y_{j+1}-y_{j}\leq \nu.\nonumber
\end{align}

\begin{align}
\label{eq:linearcontracts5}
         &0\leq\textstyle\sum_{i=1}^{s}|r_{p}(\theta_{i},\alpha)/s-r_{p}(\theta_{i},\widehat{\alpha})/s|=
         \\& \textstyle\sum_{i=1}^{s}r_{p}(\theta_{i},\widehat{\alpha})/s-\textstyle\sum_{i=1}^{s}r_{p}(\theta_{i},\alpha)/s\leq y_{j+1}-y_{j}\leq \nu.\nonumber
\end{align}

\begin{align*}
    &\textstyle\sum_{i=1}^{s}\negmedspace |u_{p}(\theta_i,\alpha)-u_{p}(\theta_{i},\widehat{\alpha})|/s
     \\
     &=
     \textstyle\sum_{i=1}^{s}|\textstyle\sum_{j=1}^{m}f_{a^{\star}(\theta_i,\alpha),j} (1-\alpha)r_{j}
     \\
     &
     -\textstyle\sum_{j=1}^{m}f_{a^{\star}(\theta_i,\widehat{\alpha}),j} (1-\widehat{\alpha})r_{j}|/s
     \\
     &\leq
    \textstyle\sum_{i=1}^{s}|\textstyle\sum_{j=1}^{m}(f_{a^{\star}(\theta_i,\alpha),j}-f_{a^{\star}(\theta_i,\widehat{\alpha}),j}) (1-\alpha)r_{j}|/s
    \\
     &+
    \textstyle\sum_{i=1}^{s}|\textstyle\sum_{j=1}^{m}f_{a^{\star}(\theta_i,\widehat{\alpha}),j} (\alpha-\widehat{\alpha})r_{j}|/s
     \\
     &\leq
     (1-\alpha)\textstyle\sum_{i=1}^{s}|r_{p}(\theta_{i},\alpha)-r_{p}(\theta_{i},\widehat{\alpha})|/s
     \\
     &+
    \textstyle\sum_{i=1}^{s}\textstyle\sum_{j=1}^{m}f_{a^{\star}(\theta_i,\widehat{\alpha}),j} |(\alpha-\widehat{\alpha})|r_{j}/s
    \leq 2\nu,
\end{align*}

Acknowledgements

While this work was carried out, Mikael Møller Høgsgaard was supported by an Internationalisation Fellowship from the Carlsberg Foundation. Furthermore, Mikael Møller Høgsgaard was supported by the European Union (ERC, TUCLA, 101125203). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. Lastly, Mikael Møller Høgsgaard was also supported by Independent Research Fund Denmark (DFF) Sapere Aude Research Leader grant No. 9064-00068B.

Appendix

In this appendix, we prove 3, which we restate here for convenience.

\begin{align*}
        \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha'),j}    r_{j}=r_{p}(\theta,\alpha')\geq r_{p}(\theta,\alpha) = \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha),j}   r_{j}.
\end{align*}
```*

</div>

<div class="proof">

*Proof of <a href="#lem:empirical_reward_non_decreasing"
data-reference-type="ref+label"
data-reference="lem:empirical_reward_non_decreasing">3</a>.* We first
recall that the utility of a linear contract $`\alpha`$ for an agent is
given by
``` math
\begin{align*}
        \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha),j} \alpha r_{j}-c_{a^{\star}(\theta,\alpha)},
\end{align*}

\begin{align*}
      \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha'),j} \alpha'r_{j}-c_{a^{\star}(\theta,\alpha')} \geq \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha),j} \alpha'r_{j}-c_{a^{\star}(\theta,\alpha)}
      \\
      \Rightarrow
         \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha'),j} \alpha'r_{j}-c_{a^{\star}(\theta,\alpha')} -\left( \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha),j} \alpha'r_{j}-c_{a^{\star}(\theta,\alpha)} \right)\geq0.
\end{align*}

\begin{align*}
      \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha),j} \alpha r_{j}-c_{a^{\star}(\theta,\alpha)} \geq \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha'),j} \alpha r_{j}-c_{a^{\star}(\theta,\alpha')}
      \\
      \Rightarrow
        \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha),j} \alpha r_{j}-c_{a^{\star}(\theta,\alpha)} -\left( \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha'),j} \alpha r_{j}-c_{a^{\star}(\theta,\alpha')}\right) \geq 0.
\end{align*}

\begin{align*}
         0\leq \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha'),j} \alpha'r_{j}-c_{a^{\star}(\theta,\alpha')} -\left( \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha),j} \alpha'r_{j}-c_{a^{\star}(\theta,\alpha)} \right)
         \\
         +
        \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha),j} \alpha r_{j}-c_{a^{\star}(\theta,\alpha)} -\left( \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha'),j} \alpha r_{j}-c_{a^{\star}(\theta,\alpha')}\right)
        \\
        =\sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha'),j} (\alpha'-\alpha)r_{j}+\sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha),j} (\alpha-\alpha')r_{j}
        \\
        \underbrace{\Rightarrow}_{\text{dividing with } \alpha'-\alpha>0}
        0\leq \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha'),j} r_{j}
        -\sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha),j} r_{j},
\end{align*}

\begin{align*}
        \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha'),j} r_{j}\geq \sum_{j=1}^{m}f_{a^{\star}(\theta,\alpha),j} r_{j}.
\end{align*}

Definition of Pseudo-Dimension of Contracts

In this section, we restate the definition of pseudo-dimension from for easy lookup.

\begin{align*}
    u_{p}(\theta_{i},t_{b})\geq z_{i} \text{ if } b_{i}=1, \text{ and } u_{p}(\theta_{i},t_{b})< z_{i} \text{ if } b_{i}=0.
\end{align*}