Causal discovery from observational data, i.e., learning the causal graph from a finite set of samples from the joint distribution of the variables, is often the first step toward the identification and estimation of causal effects, a key requirement in numerous scientific domains. Causal discovery is hampered by two main challenges: limited data results in errors in statistical testing and the computational complexity of the learning task is daunting. This paper builds upon and extends four of our prior publications (Mokhtarian et al., 2021; Akbari et al., 2021; Mokhtarian et al., 2022, 2023a). These works introduced the concept of removable variables, which are the only variables that can be removed recursively for the purpose of causal discovery. Presence and identification of removable variables allow recursive approaches for causal discovery, a promising solution that helps to address the aforementioned challenges by reducing the problem size successively. This reduction not only minimizes conditioning sets in each conditional independence (CI) test, leading to fewer errors but also significantly decreases the number of required CI tests. The worst-case performances of these methods nearly match the lower bound. In this paper, we present a unified framework for the proposed algorithms, refined with additional details and enhancements for a coherent presentation. A comprehensive literature review is also included, comparing the computational complexity of our methods with existing approaches, showcasing their state-of-the-art efficiency. Another contribution of this paper is the release of RCD, a Python package that efficiently implements these algorithms. This package is designed for practitioners and researchers interested in applying these methods in practical scenarios. The package is available at github.com/ban-epfl/rcd, with comprehensive documentation provided at rcdpackage.com.

The Susceptible-Infected-Susceptible (SIS) model is a widely used model for the spread of information and infectious diseases, particularly non-immunizing ones, on a graph. Given a highly contagious disease, a natural question is how to best vaccinate individuals to minimize the disease's extinction time. While previous works showed that the problem of optimal vaccination is closely linked to the NP-hard *Spectral Radius Minimization* (SRM) problem, they assumed that the graph is known, which is often not the case in practice. In this work, we consider the problem of minimizing the extinction time of an outbreak modeled by an SIS model where the graph on which the disease spreads is unknown and only the infection states of the vertices are observed. To this end, we split the problem into two: learning the graph and determining effective vaccination strategies. We propose a novel inclusion-exclusion-based learning algorithm and, unlike previous approaches, establish its sample complexity for graph recovery. We then detail an optimal algorithm for the SRM problem and prove that its running time is polynomial in the number of vertices for graphs with bounded treewidth. This is complemented by an efficient and effective polynomial-time greedy heuristic for any graph. Finally, we present experiments on synthetic and real-world data that numerically validate our learning and vaccination algorithms.

Causal discovery from observational data, i.e., learning the causal graph from a finite set of samples from the joint distribution of the variables, is often the first step toward the identification and estimation of causal effects, a key requirement in numerous scientific domains. Causal discovery is hampered by two main challenges: limited data results in errors in statistical testing and the computational complexity of the learning task is daunting. This paper builds upon and extends four of our prior publications (Mokhtarian et al., 2021; Akbari et al., 2021; Mokhtarian et al., 2022, 2023a). These works introduced the concept of removable variables, which are the only variables that can be removed recursively for the purpose of causal discovery. Presence and identification of removable variables allow recursive approaches for causal discovery, a promising solution that helps to address the aforementioned challenges by reducing the problem size successively. This reduction not only minimizes conditioning sets in each conditional independence (CI) test, leading to fewer errors but also significantly decreases the number of required CI tests. The worst-case performances of these methods nearly match the lower bound. In this paper, we present a unified framework for the proposed algorithms, refined with additional details and enhancements for a coherent presentation. A comprehensive literature review is also included, comparing the computational complexity of our methods with existing approaches, showcasing their state-of-the-art efficiency. Another contribution of this paper is the release of RCD, a Python package that efficiently implements these algorithms. This package is designed for practitioners and researchers interested in applying these methods in practical scenarios. The package is available at github.com/ban-epfl/rcd, with comprehensive documentation provided at rcdpackage.com.

Identifying causal effects is a key problem of interest across many disciplines. The two long-standing approaches to estimate causal effects are observational and experimental (randomized) studies. Observational studies can suffer from unmeasured confounding, which may render the causal effects unidentifiable. On the other hand, direct experiments on the target variable may be too costly or even infeasible to conduct. A middle ground between these two approaches is to estimate the causal effect of interest through proxy experiments, which are conducted on variables with a lower cost to intervene on compared to the main target. In an earlier work, we studied this setting and demonstrated that the problem of designing the optimal (minimum-cost) experiment for causal effect identification is NP-complete and provided a naive algorithm that may require solving exponentially many NP-hard problems as a sub-routine in the worst case. In this work, we provide a few reformulations of the problem that allow for designing significantly more efficient algorithms to solve it as witnessed by our extensive simulations. Additionally, we study the closely-related problem of designing experiments that enable us to identify a given effect through valid adjustments sets.

Recommending applications (apps) to improve health or educational outcomes requires long-term planning and adaptation based on the user feedback, as it is imperative to recommend the right app at the right time to improve engagement and benefit. We model the challenging task of app recommendation for these specific categories of apps—or alike—using a new reinforcement learning method referred to as episodic multi-armed bandit (eMAB). In eMAB, the learner recommends apps to individual users and observes their interactions with the recommendations on a weekly basis. It then uses this data to maximize the total payoff of all users by learning to recommend specific apps. Since computing the optimal recommendation sequence is intractable, as a benchmark, we define an oracle that sequentially recommends apps to maximize the expected immediate gain. Then, we propose our online learning algorithm, named FeedBack Adaptive Learning (FeedBAL), and prove that its regret with respect to the benchmark increases logarithmically in expectation. We demonstrate the effectiveness of FeedBAL on recommending mental health apps based on data from an app suite and show that it results in a substantial increase in the number of app sessions compared with episodic versions of$\epsilon _n$-greedy, Thompson sampling, and collaborative filtering methods.

We consider contextual combinatorial volatile multi-armed bandit (CCV-MAB), in which at each round, the learner observes a set of available base arms and their contexts, and then, selects a super arm that contains $K$ base arms in order to maximize its cumulative reward. Under the semi-bandit feedback setting and assuming that the contexts lie in a space ${\cal X}$ endowed with the Euclidean norm and that the expected base arm outcomes (expected rewards) are Lipschitz continuous in the contexts (expected base arm outcomes), we propose an algorithm called Adaptive Contextual Combinatorial Upper Confidence Bound (ACC-UCB). This algorithm, which adaptively discretizes ${\cal X}$ to form estimates of base arm outcomes and uses an $\alpha$-approximation oracle as a subroutine to select a super arm in each round, achieves $\tilde{O} ( T^{(\bar{D}+1)/(\bar{D}+2) + \epsilon} )$ regret for any $\epsilon>0$, where $\bar{D}$ represents the approximate optimality dimension related to ${\cal X}$. This dimension captures both the benignness of the base arm arrivals and the structure of the expected reward. In addition, we provide a recipe for obtaining more optimistic regret bounds by taking into account the volatility of the base arms and show that ACC-UCB achieves significant performance gains compared to the state-of-the-art for worker selection in mobile crowdsourcing.