Empirical Utility Maximization (EUM) Algorithm

• Empirical Utility Maximization (EUM) is a framework that directly maximizes observed utility, aligning model selection with application-specific objectives in contract design and prediction.
• It employs techniques like complexity penalization, uniform convergence, and chaining to mitigate overfitting and guarantee optimal sample efficiency.
• EUM has practical applications in binary prediction and linear contract design, outperforming traditional surrogate loss methods by focusing on true utility.

Empirical Utility Maximization (EUM) refers to a broad algorithmic framework for learning predictive or prescriptive models directly targeted at maximizing utility, as opposed to surrogate losses such as log-likelihood or squared error. EUM algorithms are particularly relevant in problems where the ultimate objective is explicitly utility-based, including contract design, binary prediction under asymmetric costs, and decision making under partial observability. The general principle is to optimize, over a parametric or nonparametric class of policies, the average utility realized on a finite sample—augmented, when necessary, with penalization to control overfitting. Modern analyses of EUM yield sharp finite-sample guarantees on performance and characterize optimal sample complexity in various contexts (Høgsgaard, 4 Jan 2026, Su, 2019).

1. Problem Formalization

EUM is formalized around a sample set $\{(x_i, y_i)\}_{i=1}^n$, where $x_i$ encodes observable features (covariates, contract types, price signals, etc.) and $y_i$ is the associated outcome. The central objective is to use these data to select a parameter or policy $\theta$ from a class $\mathcal{F}$ so as to maximize an empirical utility function,

$$U_n(\theta) = \frac{1}{n} \sum_{i=1}^n u(y_i, f_\theta(x_i))$$

where $u$ is a user-specified utility function.

In binary prediction (Su, 2019), $u$ reflects application-tailored preferences (e.g., cost-sensitive misclassification). In contract design, as in linear contracting problems (Høgsgaard, 4 Jan 2026), utility quantifies the principal's expected payoff under the induced agent response, dependent on a contract parameter $\alpha$. The EUM principle directly characterizes the policy that would maximize average realized value over the observed dataset.

2. EUM for Linear Contracts

In the context of optimal linear contracts (Høgsgaard, 4 Jan 2026), the EUM framework is instantiated as follows:

• Let the agent have action set $[n]$, each inducing a distribution $f_i$ over $m$ outcomes; each outcome $j$ yields principal reward $r_j\geq 0$.
• A linear contract is parametrized by $t = \alpha r$ for $\alpha \in [0,1]$.
• The agent chooses action $i^*(\theta, \alpha)$ to maximize agent utility $u_a(\theta, t, i) = \sum_j f_{i,j} t_j - c_i$ (with ties broken in the principal's favor).
• The principal's utility per type $\theta$ is $$u_p(\theta, t) = \sum_j f_{i^*(\theta, t),j} (r_j - t_j)$$.
• The empirical principal utility on sample $S = \{\theta_1, \ldots, \theta_s\}$ for contract parameter $\alpha$ is

$$U_p(S, \alpha) = \frac{1}{s} \sum_{i=1}^s u_p(\theta_i, \alpha r)\,.$$

The EUM algorithm for this setting (EUM_Linear) maximizes $U_p(S, \alpha)$ over a discretized grid of $\alpha$ values in $[0,1]$.

3. EUM and Complexity-Penalized Model Selection

When EUM is applied to prediction rules, particularly for binary outcomes, maximizing empirical utility alone can lead to overfitting. The Utility-Maximizing Prediction Rule (UMPR) (Su, 2019) addresses this by integrating complexity penalties, typically through structural risk minimization schemes. Given a hierarchy of function classes $(\mathcal{F}_k)_{k \geq 1}$ (e.g., by degree or smoothness), UMPR selects

$$\tilde{f}_n = \arg\max_{f \in \mathcal{F}} \left[ U_n(f) - C_n(k;\alpha) \right]$$

where $C_n$ penalizes complexity, such as via VC-dimension or data-dependent estimates (e.g., Rademacher complexity, bootstrap complexity). This ensures that the selected estimator yields high out-of-sample expected utility, and establishes non-asymptotic oracle inequalities matching those of risk-minimizing analogues.

4. Theoretical Guarantees: Sample Complexity and Uniform Convergence

In linear contract learning, the EUM_Linear algorithm is proven to yield an $\varepsilon$-approximate solution to the optimal expected utility problem with high probability, utilizing only $O(\ln(1/\delta)/\varepsilon^2)$ samples (Høgsgaard, 4 Jan 2026). This is established via:

• A uniform convergence theorem: for $s \geq 3456 \cdot \ln(4/\delta)/\varepsilon^2$,

$$\sup_{\alpha \in [0,1]} |U_p(S,\alpha) - U_p(D, \alpha)| \leq \varepsilon$$

with probability at least $1-\delta$.

• The main technical ingredient is the construction of a fine-grained $L_2$-cover exploiting the monotonicity of expected reward as a function of $\alpha$, allowing the covering number $N(\nu) = O(1/\nu^2)$.
• Chaining (Talagrand-type) and symmetrization are used to obtain tight Rademacher complexity bounds.
• For penalized EUM in classification, analogous concentration and finite-sample excess utility bounds apply (Su, 2019):

$$S^* - \mathbb{E}[S(\tilde{f}_n)] \leq \min_k\{C_n^{VC}(k;\alpha_0) + (S^* - S_k^*)\} + O(n^{-1/2}).$$

5. EUM Algorithmic Procedures

For contract learning (Høgsgaard, 4 Jan 2026), the EUM_Linear procedure is as follows:

1. Choose grid step $\Delta = \varepsilon / 4$; define discrete set $D = \{0, \Delta, 2\Delta, ..., 1\}$.
2. For each $\alpha \in D$, compute $v(\alpha) = U_p(S, \alpha)$.
3. Select $\hat{\alpha} \in \arg\max_{\alpha \in D} v(\alpha)$ (ties arbitrarily).
4. Return $\hat{\alpha}$.

For penalized empirical utility maximization in binary prediction (Su, 2019):

1. For each class $\mathcal{F}_k$, compute the empirical utility maximizer $\hat{f}_k$ and the penalized utility $\tilde{S}_n(k)$.
2. Select $k$ maximizing penalized utility.
3. Output the corresponding $\hat{f}_k$.

6. Connections, Significance, and Extensions

EUM provides a direct utility-targeted alternative to traditional likelihood or risk minimization, offering sharper alignment with application-specific objectives in settings with non-standard or discontinuous utilities. The optimal sample complexity and uniform convergence rates shown for linear contracts (Høgsgaard, 4 Jan 2026) leverage contract structure, particularly monotonicity of expected reward, to dramatically reduce covering numbers compared to generic function classes. In binary prediction, complexity-penalized EUM (UMPR) achieves state-of-the-art expected utility under model misspecification, outperforming standard likelihood-based selectors and commonly used cross-validation protocols (Su, 2019).

The techniques underpinning EUM—uniform convergence, chaining, symmetrization, and data-dependent complexity control—extend to other design and prediction frameworks. Uniform convergence rates for empirical utility pave the way for near-optimal offline policy learning in broader settings, while explicit penalization mitigates overfit in high-capacity function classes. The equivalence between utility maximization and cost-sensitive classification further connects EUM to areas such as learning with asymmetric losses, bandit optimization, and mechanism design.

Table: EUM Instantiations and Guarantees

Context	Core Procedure	Sample Complexity / Guarantee
Linear contract learning (Høgsgaard, 4 Jan 2026)	$\max_{\alpha \in D} U_p(S, \alpha)$ over grid $D$	$O(\ln(1/\delta)/\varepsilon^2)$ for $\varepsilon$-optimality, uniform convergence
Binary prediction (Su, 2019)	$\max_{f \in \mathcal{F}_k} U_n(f) - C_n(k)$ with model selection	Finite-sample oracle inequalities; $O(n^{-1/2})$ convergence to optimal utility

A plausible implication is that EUM-type analyses can be generalized to contract design and other high-dimensional decision-making problems whenever structural properties (e.g., monotonicity or convexity) of the class can be exploited to yield tighter metric entropy or covering number bounds.

7. Technical Challenges and Future Research

While EUM yields optimal rates in settings such as linear contracts—leveraging monotonicity to dramatically reduce covering numbers—analysis and implementation can be more challenging when the utility landscape is highly nonconvex or unstructured. Cover constructions and chaining arguments may not always yield sharp sample complexity without exploiting specific problem geometry. Extending EUM guarantees to broader classes of contracts, non-concave utilities, and multi-agent settings, as well as developing computationally efficient algorithms matching these statistical rates, remain active research areas (Høgsgaard, 4 Jan 2026, Su, 2019).

In summary, empirical utility maximization forms a foundational principle for statistically and computationally efficient decision-making in settings where outcome utility, rather than surrogate risk, is the primary objective; and modern theoretical analyses establish its optimality in several canonical settings.