On the other hand, the numerical resolution of OT problems using entropic regularization has given rise to many applications, but because there are no known closed-form solutions for entropic regularized OT problems, these approaches are mostly algorithmic, not informed by elegant closed forms.

Estimation of the average treatment effect (ATE) is a central problem in causal inference. In recent times, inference for the ATE in the presence of high-dimensional covariates has been extensively studied. Among the diverse approaches that have been proposed, augmented inverse probability weighting (AIPW) with cross-fitting has emerged a popular choice in practice. In this work, we study this cross-fit AIPW estimator under well-specified outcome regression and propensity score models in a high-dimensional regime where the number of features and samples are both large and comparable. Under assumptions on the covariate distribution, we establish a new central limit theorem for the suitably scaled cross-fit AIPW that applies without any sparsity assumptions on the underlying high-dimensional parameters. Our CLT uncovers two crucial phenomena among others: (i) the AIPW exhibits a substantial variance inflation that can be precisely quantified in terms of the signal-to-noise ratio and other problem parameters, (ii) the asymptotic covariance between the pre-cross-fit estimators is non-negligible even on the root-n scale. These findings are strikingly different from their classical counterparts. On the technical front, our work utilizes a novel interplay between three distinct tools--approximate message passing theory, the theory of deterministic equivalents, and the leave-one-out approach. We believe our proof techniques should be useful for analyzing other two-stage estimators in this high-dimensional regime. Finally, we complement our theoretical results with simulations that demonstrate both the finite sample efficacy of our CLT and its robustness to our assumptions.

In his seminal paper of 2004, Ulrich Junker proposed the QuickXPlain algorithm, which provides a divide-and-conquer computation strategy to find within a given set an irreducible subset with a particular (monotone) property. Beside its original application in the domain of constraint satisfaction problems, the algorithm has since then found widespread adoption in areas as different as model-based diagnosis, recommender systems, verification, or the Semantic Web. This popularity is due to the frequent occurrence of the problem of finding irreducible subsets on the one hand, and to QuickXPlain's general applicability and favorable computational complexity on the other hand. However, although (we regularly experience) people are having a hard time understanding QuickXPlain and seeing why it works correctly, a proof of correctness of the algorithm has never been published. This is what we account for in this work, by explaining QuickXPlain in a novel tried and tested way and by presenting an intelligible formal proof of it. Apart from showing the correctness of the algorithm and excluding the later detection of errors (proof and trust effect), the added value of the availability of a formal proof is, e.g., (i) that the workings of the algorithm often become completely clear only after studying, verifying and comprehending the proof (didactic effect), (ii) the shown proof methodology can be used as a guidance for proving other recursive algorithms (transfer effect), and (iii) the possibility of providing "gapless" correctness proofs of systems that rely on (results computed by) QuickXPlain, such as numerous model-based debuggers (completeness effect).

We investigate compositional bounding of transition system diameters, with application in bounding the lengths of plans. We establish usefully-tight bounds by exploiting acyclicity in state-spaces. We provide mechanised proofs in HOL4 of the validity of our approach. Evaluating our bounds in a range of benchmarks, we demonstrate exponentially tighter upper bounds compared to existing methods. Treating both solvable and unsolvable benchmark problems, we also demonstrate the utility of our bounds in boosting planner performance. We enhance an existing planning procedure to use our bounds, and demonstrate significant coverage improvements, both compared to the base planner, and also in comparisons with state-of-the-art systems.