Prophet inequality concerns a basic optimal stopping problem and states that simple threshold stopping policies -- i.e., accepting the first reward larger than a certain threshold -- can achieve tight

The growing demand for personalized decision-making has led to a surge of interest in estimating the Conditional Average Treatment Effect (CATE). Various types of CATE estimators have been developed with advancements in machine learning and causal inference. However, selecting the desirable CATE estimator through a conventional model validation procedure remains impractical due to the absence of counterfactual outcomes in observational data.

We consider Bayesian algorithm execution (BAX), a framework for efficiently selecting evaluation points of an expensive function to infer a property of interest encoded as the output of a base algorithm. Since the base algorithm typically requires more evaluations than are feasible, it cannot be directly applied. Instead, BAX methods sequentially select evaluation points using a probabilistic numerical approach. Current BAX methods use expected information gain to guide this selection. However, this approach is computationally intensive. Observing that, in many tasks, the property of interest corresponds to a target set of points defined by the function, we introduce PS-BAX, a simple, effective, and scalable BAX method based on posterior sampling. PS-BAX is applicable to a wide range of problems, including many optimization variants and level set estimation. Experiments across diverse tasks demonstrate that PS-BAX performs competitively with existing baselines while being significantly faster, simpler to implement, and easily parallelizable, setting a strong baseline for future research. Additionally, we establish conditions under which PS-BAX is asymptotically convergent, offering new insights into posterior sampling as an algorithm design paradigm.

A.1 Data fMRI data fMRI activities differ when the same individual sees the same image at different times (Figure 1). Although we use activities and signals interchangeably throughout the paper, what we mean are fMRI betas in the NSD dataset. Betas are not direct measurements of BOLD (blood-oxygenationlevel dependent) changes, but the inferred activities from BOLD signals through GLM (general linear models). The reason for using betas instead of direct measurements is that image stimuli are shown consecutively to the subjects without prolonged delay, and activities triggered by the previous image can interfere with the next one if there is no proper separation. Authors of NSD proved the effectiveness of their GLM approach with much improved SNR in the betas over raw measurements [3].

Understanding complex dynamical systems begins with identifying their topological structures, which expose the organization of the systems. This requires robust structural inference methods that can deduce structure from observed behavior. However, existing methods are often domain-specific and lack a standardized, objective comparison framework. We address this gap by benchmarking 13 structural inference methods from various disciplines on simulations representing two types of dynamics and 11 interaction graph models, supplemented by a biological experimental dataset to mirror real-world application. We evaluated the methods for accuracy, scalability, robustness, and sensitivity to graph properties. Our findings indicate that deep learning methods excel with multi-dimensional data, while classical statistics and information theory based approaches are notably accurate and robust.

Nevertheless, we show that when (approximate) full efficiency can be guaranteed via a smoothness argument à la Roughgarden, Nash equilibria are approachable under a family of no-regret learning algorithms, thereby enabling fast and decentralized computation. We leverage this connection to obtain new convergence results in large games--wherein the number of players n 1--under the well-documented property of full efficiency via smoothness in the limit. Surprisingly, our framework unifies equilibrium computation in disparate classes of problems including games with vanishing strategic sensitivity and two-player zero-sum games, illuminating en route an immediate but overlooked equivalence between smoothness and a well-studied condition in the optimization literature known as the Minty property. Finally, we establish that a family of no-regret dynamics attains a welfare bound that improves over the smoothness framework while at the same time guaranteeing convergence to the set of coarse correlated equilibria. We show this by employing the clairvoyant mirror descent algortihm recently introduced by Piliouras et al.