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In this paper, we develop a new criterion, "insufficiently justified disparate impact" (IJDI), for assessing whether recommendations (binarized predictions) made by an algorithmic decision support tool are fair. Our novel, utility-based IJDI criterion evaluates false positive and false negative error rate imbalances, identifying statistically significant disparities between groups which are present even when adjusting for group-level differences in base rates. We describe a novel IJDI-Scan approach which can efficiently identify the intersectional subpopulations, defined across multiple observed attributes of the data, with the most significant IJDI. To evaluate IJDI-Scan's performance, we conduct experiments on both simulated and real-world data, including recidivism risk assessment and credit scoring. Further, we implement and evaluate approaches to mitigating IJDI for the detected subpopulations in these domains.

Iterated conditional expectation (ICE) g-computation is an estimation approach for addressing time-varying confounding for both longitudinal and time-to-event data. Unlike other g-computation implementations, ICE avoids the need to specify models for each time-varying covariate. For variance estimation, previous work has suggested the bootstrap. However, bootstrapping can be computationally intense and sensitive to the number of resamples used. Here, we present ICE g-computation as a set of stacked estimating equations. Therefore, the variance for the ICE g-computation estimator can be estimated using the empirical sandwich variance estimator. Performance of the variance estimator was evaluated empirically with a simulation study. The proposed approach is also demonstrated with an illustrative example on the effect of cigarette smoking on the prevalence of hypertension. In the simulation study, the empirical sandwich variance estimator appropriately estimated the variance. When comparing runtimes between the sandwich variance estimator and the bootstrap for the applied example, the sandwich estimator was substantially faster, even when bootstraps were run in parallel. The empirical sandwich variance estimator is a viable option for variance estimation with ICE g-computation.

In this paper, we study the statistical efficiency of Reinforcement Learning in Mean-Field Control (MFC) and Mean-Field Game (MFG) with general function approximation. We introduce a new concept called Mean-Field Model-Based Eluder Dimension (MBED), which subsumes a rich family of Mean-Field RL problems. Additionally, we propose algorithms based on Optimistic Maximal Likelihood Estimation, which can return an $\epsilon$-optimal policy for MFC or an $\epsilon$-Nash Equilibrium policy for MFG, with sample complexity polynomial w.r.t. relevant parameters and independent of the number of states, actions and the number of agents. Notably, our results only require a mild assumption of Lipschitz continuity on transition dynamics and avoid strong structural assumptions in previous work. Finally, in the tabular setting, given the access to a generative model, we establish an exponential lower bound for MFC setting, while providing a novel sample-efficient model elimination algorithm to approximate equilibrium in MFG setting. Our results reveal a fundamental separation between RL for single-agent, MFC, and MFG from the sample efficiency perspective.

Collective Adaptive Systems often consist of many heterogeneous components typically organised in groups. These entities interact with each other by adapting their behaviour to pursue individual or collective goals. In these systems, the distribution of these entities determines a space that can be either physical or logical. The former is defined in terms of a physical relation among components. The latter depends on logical relations, such as being part of the same group. In this context, specification and verification of spatial properties play a fundamental role in supporting the design of systems and predicting their behaviour. For this reason, different tools and techniques have been proposed to specify and verify the properties of space, mainly described as graphs. Therefore, the approaches generally use model spatial relations to describe a form of proximity among pairs of entities. Unfortunately, these graph-based models do not permit considering relations among more than two entities that may arise when one is interested in describing aspects of space by involving interactions among groups of entities. In this work, we propose a spatial logic interpreted on simplicial complexes. These are topological objects, able to represent surfaces and volumes efficiently that generalise graphs with higher-order edges. We discuss how the satisfaction of logical formulas can be verified by a correct and complete model checking algorithm, which is linear to the dimension of the simplicial complex and logical formula. The expressiveness of the proposed logic is studied in terms of the spatial variants of classical bisimulation and branching bisimulation relations defined over simplicial complexes.

Submodular function maximization is a fundamental combinatorial optimization problem with plenty of applications -- including data summarization, influence maximization, and recommendation. In many of these problems, the goal is to find a solution that maximizes the average utility over all users, for each of whom the utility is defined by a monotone submodular function. However, when the population of users is composed of several demographic groups, another critical problem is whether the utility is fairly distributed across different groups. Although the \emph{utility} and \emph{fairness} objectives are both desirable, they might contradict each other, and, to the best of our knowledge, little attention has been paid to optimizing them jointly. To fill this gap, we propose a new problem called \emph{Bicriteria Submodular Maximization} (BSM) to balance utility and fairness. Specifically, it requires finding a fixed-size solution to maximize the utility function, subject to the value of the fairness function not being below a threshold. Since BSM is inapproximable within any constant factor, we focus on designing efficient instance-dependent approximation schemes. Our algorithmic proposal comprises two methods, with different approximation factors, obtained by converting a BSM instance into other submodular optimization problem instances. Using real-world and synthetic datasets, we showcase applications of our proposed methods in three submodular maximization problems: maximum coverage, influence maximization, and facility location.

In epidemiological studies, the capture-recapture (CRC) method is a powerful tool that can be used to estimate the number of diseased cases or potentially disease prevalence based on data from overlapping surveillance systems. Estimators derived from log-linear models are widely applied by epidemiologists when analyzing CRC data. The popularity of the log-linear model framework is largely associated with its accessibility and the fact that interaction terms can allow for certain types of dependency among data streams. In this work, we shed new light on significant pitfalls associated with the log-linear model framework in the context of CRC using real data examples and simulation studies. First, we demonstrate that the log-linear model paradigm is highly exclusionary. That is, it can exclude, by design, many possible estimates that are potentially consistent with the observed data. Second, we clarify the ways in which regularly used model selection metrics (e.g., information criteria) are fundamentally deceiving in the effort to select a best model in this setting. By focusing attention on these important cautionary points and on the fundamental untestable dependency assumption made when fitting a log-linear model to CRC data, we hope to improve the quality of and transparency associated with subsequent surveillance-based CRC estimates of case counts.

We study a mean change point testing problem for high-dimensional data, with exponentially- or polynomially-decaying tails. In each case, depending on the $\ell_0$-norm of the mean change vector, we separately consider dense and sparse regimes. We characterise the boundary between the dense and sparse regimes under the above two tail conditions for the first time in the change point literature and propose novel testing procedures that attain optimal rates in each of the four regimes up to a poly-iterated logarithmic factor. By comparing with previous results under Gaussian assumptions, our results quantify the costs of heavy-tailedness on the fundamental difficulty of change point testing problems for high-dimensional data. To be specific, when the error vectors follow sub-Weibull distributions, a CUSUM-type statistic is shown to achieve a minimax testing rate up to $\sqrt{\log\log(8n)}$. When the error distributions have polynomially-decaying tails, admitting bounded $\alpha$-th moments for some $\alpha \geq 4$, we introduce a median-of-means-type test statistic that achieves a near-optimal testing rate in both dense and sparse regimes. In particular, in the sparse regime, we further propose a computationally-efficient test to achieve the exact optimality. Surprisingly, our investigation in the even more challenging case of $2 \leq \alpha < 4$, unveils a new phenomenon that the minimax testing rate has no sparse regime, i.e.\ testing sparse changes is information-theoretically as hard as testing dense changes. This phenomenon implies a phase transition of the minimax testing rates at $\alpha = 4$.

Sparse functional/longitudinal data have attracted widespread interest due to the prevalence of such data in social and life sciences. A prominent scenario where such data are routinely encountered are accelerated longitudinal studies, where subjects are enrolled in the study at a random time and are only tracked for a short amount of time relative to the domain of interest. The statistical analysis of such functional snippets is challenging since information for the far-off-diagonal regions of the covariance structure is missing. Our main methodological contribution is to address this challenge by bypassing covariance estimation and instead modeling the underlying process as the solution of a data-adaptive stochastic differential equation. Taking advantage of the interface between Gaussian functional data and stochastic differential equations makes it possible to efficiently reconstruct the target process by estimating its dynamic distribution. The proposed approach allows one to consistently recover forward sample paths from functional snippets at the subject level. We establish the existence and uniqueness of the solution to the proposed data-driven stochastic differential equation and derive rates of convergence for the corresponding estimators. The finite-sample performance is demonstrated with simulation studies and functional snippets arising from a growth study and spinal bone mineral density data.

In randomized clinical trials, adjusting for baseline covariates has been advocated as a way to improve credibility and efficiency for demonstrating and quantifying treatment effects. This article studies the augmented inverse propensity weighted (AIPW) estimator, which is a general form of covariate adjustment that includes approaches using linear and generalized linear models and machine learning models. Under covariate-adaptive randomization, we establish a general theorem that shows a complete picture about the asymptotic normality, efficiency gain, and applicability of AIPW estimators. Based on the general theorem, we provide insights on the conditions for guaranteed efficiency gain and universal applicability under different randomization schemes, which also motivate a joint calibration strategy using some constructed covariates after applying AIPW. We illustrate the application of the general theorem with two examples, the generalized linear model and the machine learning model. We provide the first theoretical justification of using machine learning methods with dependent data under covariate-adaptive randomization. Our methods are implemented in the R package RobinCar.

In recent years we have been able to gather large amounts of genomic data at a fast rate, creating situations where the number of variables greatly exceeds the number of observations. In these situations, most models that can handle a moderately high dimension will now become computationally infeasible. Hence, there is a need for a pre-screening of variables to reduce the dimension efficiently and accurately to a more moderate scale. There has been much work to develop such screening procedures for independent outcomes. However, much less work has been done for high-dimensional longitudinal data, in which the observations can no longer be assumed to be independent. In addition, it is of interest to capture possible interactions between the genomic variable and time in many of these longitudinal studies. This calls for the development of new screening procedures for high-dimensional longitudinal data, where the focus is on interactions with time. In this work, we propose a novel conditional screening procedure that ranks variables according to the likelihood value at the maximum likelihood estimates in a semi-marginal linear mixed model, where the genomic variable and its interaction with time are included in the model. This is to our knowledge the first conditional screening approach for clustered data. We prove that this approach enjoys the sure screening property, and assess the finite sample performance of the method through simulations, with a comparison of an already existing screening approach based on generalized estimating equations.