We give a principled method for decomposing the predictive uncertainty of a model into aleatoric and epistemic components with explicit semantics relating them to the real-world data distribution. While many works in the literature have proposed such decompositions, they lack the type of formal guarantees we provide. Our method is based on the new notion of higher-order calibration, which generalizes ordinary calibration to the setting of higher-order predictors that predict _mixtures_ over label distributions at every point. We show how to measure as well as achieve higher-order calibration using access to $k$-snapshots, namely examples where each point has $k$ independent conditional labels. Under higher-order calibration, the estimated aleatoric uncertainty at a point is guaranteed to match the real-world aleatoric uncertainty averaged over all points where the prediction is made. To our knowledge, this is the first formal guarantee of this type that places no assumptions whatsoever on the real-world data distribution. Importantly, higher-order calibration is also applicable to existing higher-order predictors such as Bayesian and ensemble models and provides a natural evaluation metric for such models. We demonstrate through experiments that our method produces meaningful uncertainty decompositions in tasks such as image classification.

We introduce "representative generation," extending the theoretical framework for generation proposed by Kleinberg et al. (2024) and formalized by Li et al. (2024), to additionally address diversity and bias concerns in generative models. Our notion requires outputs of a generative model to proportionally represent groups of interest from the training data. We characterize representative uniform and non-uniform generation, introducing the “group closure dimension” as a key combinatorial quantity. For representative generation in the limit, we analyze both information-theoretic and computational aspects, demonstrating feasibility for countably infinite hypothesis classes and collections of groups under certain conditions, but proving a negative result for computability using only membership queries. This contrasts with Kleinberg et al.’s (2024) positive results for standard generation in the limit. Our findings provide a rigorous foundation for developing more diverse and representative generative models.

To address the shortcomings of real-world datasets, robust learning algorithms have been designed to overcome arbitrary and indiscriminate data corruption. However, practical processes of gathering data may lead to patterns of data corruption that are localized to specific partitions of the training dataset. Motivated by critical applications where the learned model is deployed to make predictions about people from a rich collection of overlapping subpopulations, we initiate the study of multigroup robust algorithms whose robustness guarantees for each subpopulation only degrade with the amount of data corruption inside that subpopulation. When the data corruption is not distributed uniformly over subpopulations, our algorithms provide more meaningful robustness guarantees than standard guarantees that are oblivious to how the data corruption and the affected subpopulations are related. Our techniques establish a new connection between multigroup fairness and robustness.

In many learning theory problems, a central role is played by a hypothesis class: we might assume that the data is labeled according to a hypothesis in the class (usually referred to as the realizable setting), or we might evaluate the learned model by comparing it with the best hypothesis in the class (the agnostic setting). Taking a step beyond these classic setups that involve only a single hypothesis class, we study a variety of problems that involve two hypothesis classes simultaneously. We introduce comparative learning as a combination of the realizable and agnostic settings in PAC learning: given two binary hypothesis classes S and B, we assume that the data is labeled according to a hypothesis in the source class S and require the learned model to achieve an accuracy comparable to the best hypothesis in the benchmark class B. Even when both S and B have infinite VC dimensions, comparative learning can still have a small sample complexity. We show that the sample complexity of comparative learning is characterized by the mutual VC dimension VC(S,B) which we define to be the maximum size of a subset shattered by both S and B. We also show a similar result in the online setting, where we give a regret characterization in terms of the analogous mutual Littlestone dimension Ldim(S,B). These results also hold for partial hypotheses. We additionally show that the insights necessary to characterize the sample complexity of comparative learning can be applied to other tasks involving two hypothesis classes. In particular, we characterize the sample complexity of realizable multiaccuracy and multicalibration using the mutual fat-shattering dimension, an analogue of the mutual VC dimension for real-valued hypotheses. This not only solves an open problem proposed by Hu, Peale, Reingold (2022), but also leads to independently interesting results extending classic ones about regression, boosting, and covering number to our two-hypothesis-class setting.

We give the first sample complexity characterizations for outcome indistinguishability, a theoretical framework of machine learning recently introduced by Dwork, Kim, Reingold, Rothblum, and Yona (STOC 2021). In outcome indistinguishability, the goal of the learner is to output a predictor that cannot be distinguished from the target predictor by a class $D$ of distinguishers examining the outcomes generated according to the predictors’ predictions. While outcome indistinguishability originated from the algorithmic fairness literature, it provides a flexible objective for machine learning even when fairness is not a consideration. In this work, we view outcome indistinguishability as a relaxation of PAC learning that allows us to achieve meaningful performance guarantees under data constraint. In the distribution-specific and realizable setting where the learner is given the data distribution together with a predictor class $P$ containing the target predictor, we show that the sample complexity of outcome indistinguishability is characterized by the metric entropy of $P$ w.r.t. the dual Minkowski norm defined by $D$, and equivalently by the metric entropy of $D$ w.r.t. the dual Minkowski norm defined by $P$. This equivalence makes an intriguing connection to the long-standing metric entropy duality conjecture in convex geometry. Our sample complexity characterization implies a variant of metric entropy duality, which we show is nearly tight. In the distribution-free setting, we focus on the case considered by Dwork et al. where $P$ contains all possible predictors, hence the sample complexity only depends on $D$. In this setting, we show that the sample complexity of outcome indistinguishability is characterized by the fat-shattering dimension of $D$. We also show a strong sample complexity separation between realizable and agnostic outcome indistinguishability in both the distribution-free and the distribution-specific settings. This is in contrast to distribution-free (resp. distribution-specific) PAC learning where the sample complexity in both the realizable and the agnostic settings can be characterized by the VC dimension (resp. metric entropy).

While the end-to-end encryption properties of popular messaging schemes such as Whatsapp, Messenger, and Signal guarantee privacy for users, these properties also make it very difficult for messaging platforms to enforce any sort of content moderation. This can lead to the unchecked spread of malicious content such as misinformation on such platforms. In 2019, Tyagi et al. developed message traceback, which addresses this issue by allowing a messaging platform to recover the path of a forwarded message after a user reports it for malicious content. This paper presents an alternative to message traceback that offers more privacy to users and requires less platform-side storage. We term this approach source-tracking for encrypted messaging schemes. Source-tracking enables messaging platforms to provide the privacy guarantees expected from standard end-to-end encryption, but also helps hold the sources of malicious messages accountable: if malicious content is reported by a user, the source can be identified. We formalize security goals for source-tracking schemes and design and implement two source-tracking schemes with different security and performance tradeoffs.