Online Omniprediction with Long-Term Constraints

• Online omniprediction with long-term constraints is a novel framework that delivers a single calibrated prediction sequence to heterogeneous agents, ensuring no regret and bounded constraint violations.
• The methodology relies on decision calibration and elimination-based procedures to maintain unbiased forecasts and effective action selection even under adversarial outcomes.
• It guarantees sublinear regret and constant cumulative constraint violation per agent, underpinning applications in resource allocation, bidding, and safety-critical operations.

Online omniprediction with long-term constraints refers to the methodology and algorithms designed to generate prediction sequences in adversarial or adaptive online environments, such that a collection of heterogeneous downstream agents—each optimizing their own utility while respecting cumulative resource, fairness, or operational constraints—can use the same predictions to simultaneously guarantee themselves no regret and bounded (or vanishing) long-term constraint violation. This problem bridges online prediction, constraint satisfaction, and multi-agent sequential decision making, and is fundamentally motivated by applications such as resource allocation, online bidding, and safety-critical automation, where global predictions must enable per-agent guarantees under adversarial state sequences.

1. Formal Problem Setting

Consider an online game with T rounds. At each round t, a forecaster observes a context and generates a prediction $p_t$ for an outcome $y_t$ (the latter chosen, possibly adaptively or adversarially). These predictions are broadcast to $N$ downstream agents. Each agent $i$ has:

• A utility function $u_i(a, y): A_i \times Y \to [0,1]$, where $A_i$ is the finite action set and $y$ is the outcome.
• A vector of constraint functions $(c_j^i(a, y): A_i \times Y \to [-1,1])$ for $j = 1,...,m_i$, specifying per-round costs to be controlled over time.

Each agent must sequentially select actions $a_t^i$ based solely on the predictions $(p_1,...,p_t)$ and their own objective/constraint functions, without coordination with other agents or knowledge of other agents' objectives.

The agent's goal is twofold:

1. Low (swap) regret: For any comparator policy or action sequence $a^*_{1:T}$ (subject to feasibility), ensure that the difference in cumulative utility $\sum_{t=1}^{T} u_i(a_t^i, y_t) - u_i(a_t^*, y_t)$ is sublinear in $T$, typically $O(\sqrt{T})$ or $\tilde O(\sqrt{T})$ regret.
2. Long-term constraint satisfaction: Ensure that the cumulative sum $\sum_{t=1}^T c_j^i(a_t^i, y_t)$ is non-positive (or grows slowly, e.g., $O(1)$), i.e., long-term averages of constraint functions meet the prescribed requirements.

This setting allows for constraints and utilities to be completely agent-specific and arbitrarily heterogeneous. The omniprediction requirement is that a single prediction sequence $p_{1:T}$ must suffice for all agents, even under worst-case outcome realizations (Bechavod et al., 14 Sep 2025).

2. Decision Calibration, Elimination, and Algorithmic Structure

A core algorithmic principle is decision calibration, which ensures that predictions $p_{1:T}$ are calibrated (unbiased) conditional on the downstream actions they induce. For each action $a\in A_i$, define the indicator $I_t^a = 1[a_t^i = a]$, and require that, for each action,

$\left|\sum_{t=1}^T I_t^a (p_t - y_t)\right|$

remains small. This condition ensures that, over the set of rounds on which action $a$ is chosen, the forecasted value matches the realized outcomes, eliminating systematic bias that would otherwise jeopardize regret guarantees under arbitrary outcome selection.

Downstream agents employ elimination-based procedures:

• Each agent maintains a candidate action set $C_t^i$ (initially $A_i$).
• At each round, if the empirical cumulative constraint violation for any $a\in C_{t-1}^i$ exceeds the bound (e.g., $c_j^i(a, y_t) > 0$ for some $j$), action $a$ is eliminated.
• The agent best-responds (maximizes utility) over the remaining actions at each round.

Calibrated prediction ensures that the best-response mapping remains robust against the adversarial sequence, and the elimination guarantees that no action with excessive long-term violations is ever chosen (Bechavod et al., 14 Sep 2025).

3. Regret and Constraint Violation Guarantees

The architecture yields the following performance guarantees for each agent (for both the overall sequence $[T]$ and any contextually defined subsequence $S\subseteq [T]$):

Metric	Guarantee
Swap Regret (utility loss)	$O(|A_i|\sqrt{|S|})$ or $O(\sqrt{T})$
Cumulative Constraint Violation	$O(|A_i|)$ (independent of $T$)

For instance, if $|A_i|$ actions are possible and the per-round regret is $O(1)$, then the total utility loss and constraint violations over any subsequence $S$ are both sublinear (regret) or bounded (constraint violations).

These guarantees hold against:

• Strict benchmarks: Comparator sequences of actions that meet the constraints at every round.
• Expectation benchmarks: Comparator sequences that satisfy the constraints in expectation, crucial when per-round feasibility is impossible but cumulative (average) feasibility is meaningful.

By running multiple elimination and best-response procedures in parallel (one per subsequence), the method ensures simultaneous regret and constraint violation guarantees on every relevant temporal, contextual, or application-driven subset of rounds.

4. Generality and Agent Heterogeneity

The omniprediction framework accommodates complete agent heterogeneity:

• Utility and constraint functions can differ arbitrarily between agents.
• Agents may have different constraint dimensions, operate on different action spaces, or use non-linear (even non-convex) functions, so long as the elimination and best-response mappings are well-defined.
• The framework supports overlapping or intersecting contextually defined subsequences, enabling performance guarantees in context-aware, fairness-sensitive, or resource-coupled environments.

By decoupling forecasting and agent decision making, and using calibration to ensure robustness, the approach allows a single prediction service to integrate a broad, diverse class of downstream objectives and operational constraints (Bechavod et al., 14 Sep 2025).

5. Applications and Implications

Applications of this methodology cover a wide range of online and sequential settings:

• Online resource allocation and bidding: Platform-level predictions can enable budget-feasible allocations and guarantee per-bidder regret and constraint violation properties.
• Safety-critical operations: In domains such as energy, health, or transportation, agents with varied safety constraints can guarantee long-term safety using system-provided omnipredictions even when their objectives differ.
• Fairness and regulatory requirements: Platform-wide predictions can ensure that fairness or parity constraints (e.g., demographic coverage, exposure) are met for all demographic groups or agents.

The decoupling is essential: the forecaster (platform) guarantees that, regardless of the agent's objective or constraint, robust performance is achievable via its broadcasts, without change to downstream agent code per agent or per task (Bechavod et al., 14 Sep 2025).

6. Extensions and Subsequence Analysis

The methodology extends to arbitrary collections of overlapping (contextual) subsequences $S \subseteq [T]$, enabling guarantees not only globally but on any collection of meaningful temporal or contextual segments (e.g., “hours with high demand”, “nights”, “user clusters”):

• Each agent can maintain parallel elimination-best-response procedures for each $S$ of interest.
• The framework guarantees regret and constraint satisfaction simultaneously on every such $S$ for agent-specific comparators.

This uniformity across contexts enhances the robustness and fairness of online decision services across operationally or ethically significant axes.

7. Open Directions and Limitations

While the described methodology achieves strong oracle-style guarantees (unified regret and constraint satisfaction for all agents and subsequences), possible future directions include:

• Generalizing to more complex (nonlinear, nonconvex) constraint structures.
• Improving dependence on action-set size via more efficient elimination/best-response routines.
• Addressing dynamic agent or constraint arrival/removal, and the potential for coordinated (rather than independent) agent responses.
• Exploring lattice or game-theoretic couplings in the elimination process to further tighten guarantees.

A notable limitation is the reliance on action elimination and best-response, which may be computationally intensive for large $|A_i|$. Additionally, in specific applications, further structure (e.g., convexity, sparsity) may allow for sharper or more adaptive guarantees.

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The online omniprediction with long-term constraints paradigm thus provides a robust, modular architecture wherein a single calibrated prediction sequence enables an unrestricted set of downstream agents—each contending with complex, individual utility and constraint profiles—to simultaneously achieve no regret and negligible cumulative constraint violation, with these guarantees holding uniformly over arbitrary contextual subsequences (Bechavod et al., 14 Sep 2025). This framework underpins highly versatile, scalable, and robust multi-agent online decision platforms for adversarial and resource-constrained environments.