In the field of machine learning (ML), an essential type of decision-related problem is known as AUF (Avoiding Undesired Future): if an ML model predicts an undesired outcome, how can decisions be made to prevent it? Recently, a novel framework called rehearsal learning has been proposed to address the AUF problem. Despite its utility in modeling uncertainty for decision-making, it remains unclear under what conditions and how optimal actions that maximize the AUF probability can be identified. In this paper, we propose CARE (CAnonical REctangle), a condition under which the maximum AUF probability can be achieved. Under the CARE condition, we present a projection-Newton algorithm to select actions and prove that the algorithm achieves superlinear convergence to the optimal one. Besides, we provide a generalization method for adopting the algorithm to AUF scenarios beyond the CARE condition. Finally, we demonstrate that a closed-form solution exists when the outcome is a singleton variable, substantially reducing the time complexity of decision-making. Experiments validate the effectiveness and efficiency of our method.

A maximal ancestral graph (MAG) is widely used to characterize the causal relations among observable variables in the presence of latent variables. However, given observational data, only a partial ancestral graph representing a Markov equivalence class (MEC) of MAGs is identifiable, which generally contains uncertain causal relations. Due to the uncertainties, \emph{MAG listing}, \emph{i.e.}, listing all the MAGs in the MEC, is critical for many downstream tasks. In this paper, we present the first \emph{polynomial-delay} MAG listing method, where delay refers to the time for outputting each MAG, through introducing enumerated structural knowledge in the form of \emph{singleton background knowledge (BK)}. To incorporate such knowledge, we propose the \emph{sound} and \emph{locally complete} orientation rules. By recursively introducing singleton BK and applying the rules, our method can output all and only MAGs in the MEC with polynomial delay. Additionally, while the proposed novel rules enable more efficient MAG listing, for the goal of incorporating general BK, we present two counterexamples to imply that existing rules including ours, are not yet \emph{complete}, which motivate two more rules. Experimental results validate the efficiency of the proposed MAG listing method.

In machine learning, a critical class of decision-making problems involves *Avoiding Undesired Future* (AUF): given a predicted undesired outcome, how can one make decision about actions to prevent it? Recently, the *rehearsal learning* framework has been proposed to address AUF problem. While existing methods offer reliable decisions for single-round success, this paper considers long-term settings that involve coordinating multiple future outcomes, which is often required in real-world tasks. Specifically, we generalize the AUF objective to characterize a long-term decision target that incorporates cross-temporal relations among variables. As directly optimizing the *AUF probability* $\mathbb{P}_{\operatorname{AUF}}$ over this objective remains challenging, we derive an explicit expression for the objective and further propose a quadratic programming (QP) reformulation that transforms the intractable probabilistic AUF optimization into a tractable one. Under mild assumptions, we show that solutions to the QP reformulation are equivalent to those of the original AUF optimization, based on which we develop two novel rehearsal learning methods for long-term decision-making: (i) a *greedy* method that maximizes the single-round $\mathbb{P}_{\operatorname{AUF}}$ at each step, and (ii) a *far-sighted* method that accounts for future consequences in each decision, yielding a higher overall $\mathbb{P}_{\operatorname{AUF}}$ through an $L/(L+1)$ variance reduction in the AUF objective. We further establish an $\mathcal{O}(1/\sqrt{N})$ excess risk bound for decisions based on estimated parameters, ensuring reliable practical applicability with finite data. Experiments validate the effectiveness of our approach.

Maximal ancestral graph (MAG) is a prevalent graphical model to characterize causal relations in the presence of latent variables including latent confounders and selection variables. Given observational data, only a Markov equivalence class (MEC) of MAGs is identifiable if without some additional assumptions. Due to this fact, MAG listing, listing all the MAGs in the MEC, is usually demanded in many downstream tasks. To the best of our knowledge, there are no relevant methods for MAG listing other than brute force in the literature. In this paper, we propose the first brute-force-free MAG listing method, by determining the local structures of each vertex recursively. We provide the graphical characterization for each valid local transformation of a vertex, and present sound and complete rules to incorporate the valid local transformation in the presence of latent confounders and selection variables. Based on these components, our method can efficiently output all the MAGs in the MEC with no redundance, that is, every intermediate graph in the recursive process is necessary for the MAG listing task. The empirical analysis demonstrates the superiority of our proposed method on efficiency and effectiveness.

Machine learning (ML) has achieved remarkable success in prediction tasks. In many real-world scenarios, rather than solely predicting an outcome using an ML model, the crucial concern is how to make decisions to prevent the occurrence of undesired outcomes, known as the *avoiding undesired future (AUF)* problem. To this end, a new framework called *rehearsal learning* has been proposed recently, which works effectively in stationary environments by leveraging the influence relations among variables. In real tasks, however, the environments are usually non-stationary, where the influence relations may be *dynamic*, leading to the failure of AUF by the existing method. In this paper, we introduce a novel sequential methodology that effectively updates the estimates of dynamic influence relations, which are crucial for rehearsal learning to prevent undesired outcomes in non-stationary environments. Meanwhile, we take the cost of decision actions into account and provide the formulation of AUF problem with minimal action cost under non-stationarity. We prove that in linear Gaussian cases, the problem can be transformed into the well-studied convex quadratically constrained quadratic program (QCQP). In this way, we establish the first polynomial-time rehearsal-based approach for addressing the AUF problem. Theoretical and experimental results validate the effectiveness and efficiency of our method under certain circumstances.