We introduce and study the problem of online omniprediction with long-term constraints. At each round, a forecaster is tasked with generating predictions for an underlying (adaptively, adversarially chosen) state that are broadcast to a collection of downstream agents, who must each choose an action. Each of the downstream agents has both a utility function mapping actions and state to utilities, and a vector-valued constraint function mapping actions and states to vector-valued costs. The utility and constraint functions can arbitrarily differ across downstream agents. Their goal is to choose actions that guarantee themselves no regret while simultaneously guaranteeing that they do not cumulatively violate the constraints across time. We show how to make a single set of predictions so that each of the downstream agents can guarantee this by acting as a simple function of the predictions, guaranteeing each of them $\tilde{O}(|\mathcal{A}|\sqrt{T})$ regret and $O(|\mathcal{A}|)$ cumulative constraint violation for a finite action space $\mathcal{A}$. We also show how to extend our guarantees to arbitrary intersecting contextually defined \emph{subsequences}, guaranteeing each agent both regret and constraint violation bounds not just marginally, but simultaneously on each subsequence, against a benchmark set of actions simultaneously tailored to each subsequence.

Abstract Motivation Testing for differences in within-group dispersion is a fundamental problem in multivariate data analysis, with direct implications for interpreting group structure and validating statistical assumptions of other analysis such as ANOVA. Existing methods typically construct test statistics either based on the distance of each observation from the group center or on the mean of pairwise dissimilarities among observations within a group. Both approaches can fail when the mean within-group distance is similar across groups but the distributions of the within-group distances differ. This issue is particularly relevant in high-dimensional microbiome data, where outliers and overdispersion can distort the performance of mean-dissimilarity-based tests. Results We introduce the non-parametric Distance-based Test for Homogeneity (DTH), which measures dispersion of a group by computing within-group dissimilarity. Difference in dispersion across groups is tested by comparing the distributions of the within-group dissimilarity across different groups. A combination of Kolmogorov-Smirnov and Wasserstein distances are used to construct the difference between the distributions. For more than two groups, pairwise group tests are combined using a permutation-based p-value. Through simulations, we show that our method has higher power than existing tests for homogeneity in certain situations and comparable power in others. For continuous covariates, we offer an heuristic extension of DTH that showed good performance in simulations. Availability and implementation The DTH package, along with the code for reproducing all simulations, analyses, and an accompanying vignette, is available at https://github.com/asmita112358/DTH.

We define “decision swap regret” which generalizes both prediction for downstream swap regret and omniprediction, and give algorithms for obtaining it for arbitrary multi-dimensional Lipschitz loss functions in online adversarial settings. We also give sample complexity bounds in the batch setting via an online-to-batch reduction. When applied to omniprediction, our algorithm gives the first polynomial sample-complexity bounds for Lipschitz loss functions—prior bounds either applied only to linear loss (or binary outcomes) or scaled exponentially with the error parameter even under the assumption that the loss functions were convex. When applied to prediction for downstream regret, we give the first algorithm capable of guaranteeing swap regret bounds for all downstream agents with non-linear loss functions over a multi-dimensional outcome space: prior work applied only to linear loss functions, modeling risk neutral agents. Our general bounds scale exponentially with the dimension of the outcome space, but we give improved regret and sample complexity bounds for specific families of multidimensional functions of economic interest: constant elasticity of substitution (CES), Cobb-Douglas, and Leontief utility functions.