Supervised learning aims to train a classifier under the assumption that training and test data are from the same distribution. To ease the above assumption, researchers have studied a more realistic setting: out-of-distribution (OOD) detection, where test data may come from classes that are unknown during training (i.e., OOD data). Due to the unavailability and diversity of OOD data, good generalization ability is crucial for effective OOD detection algorithms. To study the generalization of OOD detection, in this paper, we investigate the probably approximately correct (PAC) learning theory of OOD detection, which is proposed by researchers as an open problem. First, we find a necessary condition for the learnability of OOD detection. Then, using this condition, we prove several impossibility theorems for the learnability of OOD detection under some scenarios. Although the impossibility theorems are frustrating, we find that some conditions of these impossibility theorems may not hold in some practical scenarios. Based on this observation, we next give several necessary and sufficient conditions to characterize the learnability of OOD detection in some practical scenarios. Lastly, we also offer theoretical supports for several representative OOD detection works based on our OOD theory.

The impossibility theorem of fairness is a foundational result in the algorithmic fairness literature. It states that outside of special cases, one cannot exactly and simultaneously satisfy all three common and intuitive definitions of fairness - demographic parity, equalized odds, and predictive rate parity. This result has driven most works to focus on solutions for one or two of the metrics. Rather than follow suit, in this paper we present a framework that pushes the limits of the impossibility theorem in order to satisfy all three metrics to the best extent possible. We develop an integer-programming based approach that can yield a certifiably optimal post-processing method for simultaneously satisfying multiple fairness criteria under small violations. We show experiments demonstrating that our post-processor can improve fairness across the different definitions simultaneously with minimal model performance reduction. We also discuss applications of our framework for model selection and fairness explainability, thereby attempting to answer the question: Who's the fairest of them all?

In self-consuming generative models that train on their own outputs, alignment with user preferences becomes a recursive rather than one-time process. We provide the first formal foundation for analyzing the long-term effects of such recursive retraining on alignment. Under a two-stage curation mechanism based on the Bradley-Terry (BT) model, we model alignment as an interaction between two factions: the Model Owner, who filters which outputs should be learned by the model, and the Public User, who determines which outputs are ultimately shared and retained through interactions with the model. Our analysis reveals three structural convergence regimes depending on the degree of preference alignment: consensus collapse, compromise on shared optima, and asymmetric refinement. We prove a fundamental impossibility theorem: no recursive BT-based curation mechanism can simultaneously preserve diversity, ensure symmetric influence, and eliminate dependence on initialization. Framing the process as dynamic social choice, we show that alignment is not a static goal but an evolving equilibrium, shaped both by power asymmetries and path dependence.

The success of supervised learning is established on an implicit assumption that training and test data share a same distribution, i.e., in-distribution (ID) [

The impossibility theorem of fairness is a foundational result in the algorithmic fairness literature. It states that outside of special cases, one cannot exactly and simultaneously satisfy all three common and intuitive definitions of fairness - demographic parity, equalized odds, and predictive rate parity. This result has driven most works to focus on solutions for one or two of the metrics.

We thank all reviewers for their helpful comments and suggestions! We will address all minor issues. We will add references in the revision. UMG ( π) has cycles. We havn't thought about checking it from data, which is an interesting statistical Theorem 1 is effectively known.

We develop a framework that leverages the smoothed complexity analysis by Spielman and Teng [60] to circumvent paradoxes and impossibility theorems in social choice, motivated by modern applications of social choice powered by AI and ML. For Condrocet's paradox, we prove that the smoothed likelihood of the paradox either vanishes at an exponential rate as the number of agents increases, or does not vanish at all. For the ANR impossibility on the non-existence of voting rules that simultaneously satisfy anonymity, neutrality, and resolvability, we characterize the rate for the impossibility to vanish, to be either polynomially fast or exponentially fast. We also propose a novel easy-to-compute tie-breaking mechanism that optimally preserves anonymity and neutrality for even number of alternatives in natural settings. Our results illustrate the smoothed possibility of social choice--even though the paradox and the impossibility theorem hold in the worst case, they may not be a big concern in practice.

We thank all reviewers for their helpful comments and suggestions! We will address all minor issues. We will add references in the revision. UMG ( π) has cycles. We havn't thought about checking it from data, which is an interesting statistical Theorem 1 is effectively known.

We introduce a framework for modeling decisions with axioms that are statements about decisions, e.g., ethical constraints. Using our framework we define a taxonomy of decision axioms based on their structural properties and demonstrate a tension between the use of axioms to make decisions and the use of axioms to evaluate decisions which we call the Decision-Evaluation Paradox. We argue that the Decision-Evaluation Paradox arises with realistic axiom structures, and the paradox illuminates why one must be exceptionally careful when training models on decision data or applying axioms to make and evaluate decisions.