In machine learning, a critical class of decision-related problems concerns preventing predicted undesirable outcomes, referred to as the avoiding undesired future (AUF) problem. To address this, the rehearsal learning framework has been proposed to model influence relations for effective decisions. However, existing rehearsal methods rely on restrictive parametric assumptions such as linear systems or additive noise, limiting their practical applicability. In this paper, we propose the first non-parametric rehearsal learning approach for AUF without assuming specific functional forms of data generation processes. Specifically, we use kernel machinery to reformulate the AUF objective into a unified representation that disentangles desirability modeling from action-induced distributional changes. To handle the discontinuity of desirability indicator, we present a smooth Probit surrogate and provide an approximation error bound. Meanwhile, we capture the action-induced changes via conditional mean embeddings, and develop a kernel ridge regression based nested estimator for AUF objective with consistency guarantees. Such a formulation naturally accommodates nonlinear systems and non-additive noise, and empirical results on synthetic and real-data-derived semi-synthetic benchmarks demonstrate the effectiveness and flexibility of our approach.

Causal discovery aims to recover directed causal relations from observational and interventional data, providing a basis for mechanistic understanding and reliable decision-making. Causal discovery foundation models (CDFMs) seek to amortize this problem by mapping a dataset directly to a causal graph in a single forward pass, avoiding per-dataset testing, search, or optimization. However, existing CDFMs remain limited, often failing to consistently match strong classical methods, and we find that a key bottleneck is how causal pretraining tasks are constructed. Based on this observation, we propose TabCausal, a data-driven CDFM trained with broad causal pretraining over diverse graph priors, structural mechanisms, noise models, dimensions, sample sizes, and intervention regimes. A dynamic task construction strategy composes these causal environments into varied discovery tasks, enabling more transferable structural learning from observational and mixed-interventional data. On large-scale synthetic benchmarks, TabCausal achieves better macro-averaged performance than a diverse set of causal discovery baselines. To further bridge abstract synthetic generators and realistic causal reasoning scenarios, we introduce a protocol-guided and LLM-audited semantic causal environment benchmark, where domain-grounded SCMs generate interpretable observational and interventional datasets for out-of-distribution analysis. Across both synthetic and semantic environments, TabCausal demonstrates robust structure recovery, especially under interventional evidence, highlighting broad causal pretraining as a key ingredient for transferable amortized causal discovery.

When a predictive model anticipates an undesired future event, a question arises: what can we do to avoid it? Resolving this forward-looking challenge requires determining the variables that positively influence the future, moving beyond statistical correlations typically exploited for prediction. In this paper, we introduce a novel framework for evaluating the influence of actionable variables in successfully avoiding the undesired future. We quantify influence as the degree to which the probability of success can be increased by altering variables based on the principle of maximum expected utility. While closely related to causal effects, our analysis reveals a counterintuitive insight: influential variables may not necessarily be those with intrinsically strong causal effects on the target. In fact, due to the dynamics of the decision process, it can be highly beneficial to alter a weak causal factor, or even a variable that is not an intrinsic factor at all. We provide a practical implementation for computing the proposed quantity using observational data and demonstrate its utility through empirical studies on synthetic and real-world applications.

When a machine learning (ML) model forecasts an undesired event, one often seeks a decision to avoid it, known as the avoiding undesired future (AUF) problem. Many rehearsal learning methods have been proposed for AUF, but they rely on an underlying graph structure; learning such a graph from observational data is challenging and can incur substantial estimation error. In this work, we demonstrate that the order structure can be sufficient for AUF decision-making, and propose the first order-based rehearsal learning method. Although an order is less informative than a graph, it can be sufficient to identify the influence of decisions from observational data, suggesting that learning the entire graph is not always necessary. To learn the order, we develop an information-theoretic method that imposes no restrictions on the form of structural functions or the type of noise distributions. For AUF decision-making, we construct an order-based sampler to approximate the influence of decisions and, combined with a surrogate objective for maximizing the post-decision success probability, reduce the AUF task to a differentiable optimization problem. Experiments show that our order learning method outperforms existing methods, and that our AUF approach not only surpasses methods relying on learned graphs or learned orders, but also matches or even exceeds oracle baselines that are given the true graph.

ABSTRACTThis study investigates the effects of thermal damage on the tensile behavior of granite during Brazilian splitting tests, focusing on the negative Poisson's ratio (NPR) effect. Using digital image correlation and acoustic emission techniques, the research reveals that increasing thermal damage from 25°C to 800°C reduces granite tensile strength by up to 85.7% and induces a brittle‐to‐ductile transition. The NPR effect emerges, intensifies, and subsequently disappears as temperature increases, significantly altering stress distribution and deformation patterns. Local contraction zones created by the NPR effect can lead to overestimation of tensile strength in thermally damaged rock. Acoustic emission monitoring demonstrates a strong correlation between the NPR effect and microcrack development. These findings advance the understanding of rock mechanical behavior under thermal damage and provide practical insights for tensile strength evaluation in high‐temperature geological settings, such as deep underground energy development and nuclear waste disposal.

Causal effect estimation from observational data is a fundamental task in artificial intelligence and has been widely studied given known causal relations. However, in the presence of latent confounders, only a part of causal relations can be identified from observational data, characterized by a partial ancestral graph (PAG), where some causal relations are indeterminate. In such cases, the causal effect is often unidentifiable, as there could be super-exponential number of potential causal graphs consistent with the identified PAG but associated with different causal effects. In this paper, we target on set determination within a PAG, i.e., determining the set of possible causal effects of a specified variable X on another variable Y via covariate adjustment. We develop the first set determination method that does not require enumerating any causal graphs. Furthermore, we present two novel orientation rules for incorporating structural background knowledge (BK) into a PAG, which facilitate the identification of additional causal relations given BK. Notably, we show that these rules can further enhance the efficiency of our set determination method, as certain transformed edges during the procedure can be interpreted as BK and enable the rules to reveal further causal information. Theoretically and empirically, we demonstrate that our set determination methods can yield the same results as the enumeration-based method with super-exponentially less computational complexity.

Machine learning has advanced in predictive tasks, but practitioners often need to proactively avoid undesired outcomes rather than just predicting them. To this end, a framework called rehearsal has been introduced, which tackles the avoiding undesired future (AUF) problem by modeling how variables influence each other and searching for a decision that leads to desired results. In this paper, we propose a novel rehearsal approach for addressing the AUF problem by making a sequence of decisions, where each decision is dynamically informed by the latest observations via retrospective inference. Theoretically, we show that sequential decisions in our approach tend to achieve a higher success rate in avoiding undesired outcomes by more reliably inferring the outcome of actions compared with existing solutions. Perhaps surprisingly, our approach remains advantageous even under imprecise modeling of relations between variables, and we provide a sufficient condition under which the advantage holds. Finally, experimental results confirm the practical effectiveness of the proposed approach in both simulated and real-world tasks.

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.

Discovering causal graphs from observational data is a challenging problem, which has garnered significant attention due to its crucial role in understanding causal relationships. In recent advancements, this problem is cast as a continuous optimization task with structural constraints, through which the great power of gradient-based methods can be exploited to address the causal discovery problem. Despite their statistical validity, these approaches return causal graphs with spurious edges in the presence of latent variables. In this paper, we generalize the gradient-based method to accommodate the existence of latent confounders and latent intermediate variables. Specifically, we propose a causal discovery method based on latent variable reconstruction. This method primarily consists of two stages. In the first stage, we propose a series of causal models that includes latent variables, which can be applied to different data assumptions. However, due to the influence of latent variables, the causal graph inevitably contains reversed edges. In light of this fact, we propose the method to correct these reversed edges on the second stage via variational autoencoder. Theoretical results show that under some mild conditions, our method can correctly identify the causal relations. Experiments on both synthetic and real datasets demonstrate the superiority of our method to existing gradient-based learning algorithms in the presence of latent variables.

Causal discovery aims to identify causal relationships from observational data. Recently, optimization-based causal discovery methods have attracted extensive attention in the literature due to their efficiency in handling high-dimensional problems. However, we observe that optimization-based methods often perform well on certain problems but struggle with others. This paper identifies a specific characteristic of causal structural equations that determines the difficulty of identification in causal discovery and, in turn, the performance of optimization-based methods. We conduct an in-depth study of the additive noise model (ANM) and propose to further divide identifiable problems into strongly and weakly identifiable types based on the difficulty of identification. We also provide a sufficient condition to distinguish the two categories. Inspired by these findings, this paper further proposes GENE, a generic method for addressing strongly and weakly identifiable problems in a unified way under the ANM assumption. GENE adopts an order-based search framework that incorporates conditional independence tests into order fitness evaluation, ensuring effectiveness on weakly identifiable problems. In addition, GENE restricts the dimensionality of the effect variables to ensure \emph{scale invariance}, a property crucial for practical applications. Experiments demonstrate that GENE is uniquely effective in addressing weakly identifiable problems while also remaining competitive with state-of-the-art causal discovery algorithms for strongly identifiable problems.

Machine learning (ML) has made significant advancements across various domains, with a shifting focus from purely predictive tasks to decision-making. The recent proposal by Zhou (2022) introduced a line of research known as rehearsal learning, which provides a novel perspective on modeling decision-making tasks. However, previous studies mainly focused on the linear Gaussian setting to constrain the modeling complexity. Furthermore, it has been demonstrated that finding exact optimal multivariate decisions within the sampling-based rehearsal framework is computationally infeasible in polynomial time, necessitating the development of approximate methods. In this work, we present Grad-Rh, the first gradient-based rehearsal learning method that can efficiently find multivariate decisions under non-linear and non-Gaussian settings. We address the uncertainty in decision-making tasks using flexible and expressive conditional normalizing flow models and derive four surrogate loss functions to enable efficient gradient-based optimization. Experimental results show that Grad-Rh performs comparably to exact baselines on linear data and significantly outperforms them on non-linear data in both decision quality and running time.

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 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 has been proposed to address the AUF problem. Despite its utility in modeling uncertainty for decision-making, it remains unclear and optimal actions that maximize the can be identified. In this paper, we propose (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.

Heterogeneous treatment effect (HTE) estimation plays a crucial role in developing personalized treatment plans across various applications. Conventional approaches assume that the observed data are independent and identically distributed (i.i.d.). In some real applications, however, the assumption does not hold: the environment may evolve, which leads to variations in HTE over time. To enable HTE estimation in evolving environments, we introduce and formulate the online HTE estimation problem. We propose an online ensemble-based HTE estimation method called ETHOS, which is capable of adapting to unknown evolving environments by ensembling the outputs of multiple base estimators that track environmental changes at different scales. Theoretical analysis reveals that ETHOS achieves an optimal expected dynamic regret O(√(T(1+P_T))) , where T denotes the number of observed examples and P_T characterizes the intensity of environment changes. The achieved dynamic regret ensures that our method consistently approaches the optimal online estimators as long as the evolution of the environment is moderate. We conducted extensive experiments on three common benchmark datasets with various environment evolving mechanisms. The results validate the theoretical analysis and the effectiveness of our proposed method.

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.

Maximal ancestral graph (MAG) is a prevalent graphical model to characterize causal relations in the presence oflatent variablesincluding 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.

Identifying causal relations is crucial for a variety of downstream tasks. In additional to observational data, background knowledge (BK), which could be attained from human expertise or experiments, is usually introduced for uncovering causal relations. This raises an open problem that in the presence of latent variables, what causal relations are identifiable from observational data and BK. In this paper, we propose two novel rules for incorporating BK, which offer a new perspective to the open problem. In addition, we show that these rules are applicable in some typical causality tasks, such as determining the set of possible causal effects with observational data. Our rule-based approach enhances the state-of-the-art method by circumventing a process of enumerating block sets that would otherwise take exponential complexity.

Machine learning (ML) models have been widely used to make predictions. Instead of a predictive statement about future outcomes, in many situations we want to pursue a decision: what can we do to avoid the undesired future if an ML model predicts so? In this paper, we present a rehearsal learning framework, in which decisions that can persuasively avoid the happening of undesired outcomes can be found and recommended. Based on the influence relation, we characterize the generative process of variables with structural rehearsal models, consisting of a probabilistic graphical model called rehearsal graphs and structural equations, and find actionable decisions that can alter the outcome by reasoning under a Bayesian framework. Moreover, we present a probably approximately correct bound to quantify the associated risk of a decision. Experiments validate the effectiveness of the proposed rehearsal learning framework and the informativeness of the bound.

Discovering causal relations from observational data is at the heart of scientific research. Most causal discovery methods assume that the data have only one variable type. In real-world problems, however, data can consist of a mixture of continuous, discrete, and categorical variables. In this paper, we examine the causal discovery problem on mixed data. We introduce a general tree-structured functional causal model, which is well suited for characterizing the generating mechanisms of mixed data by allowing non- differentiability and nonlinearity. We present corresponding identifiability results, showing that under mild conditions, the causal directions can be uniquely determined from observational distributions. Further, we prove that the causal direction between continuous and discrete variables is generally identifiable under a much larger function class. Based on the theoretical findings, we propose an effective causal discovery method leveraging a consistent score function and powerful tree-learning techniques. Experiments on both synthetic and real data verify the effectiveness of our approach.

A rock normally has a positive Poisson’s ratio which can significantly affect its compression-induced extensile behaviors. However, it can abnormally have a negative Poisson’s ratio (NPR) after being thermally damaged. Thus far, the NPR effect on compression-induced extensile behaviors of the thermal-damaged rock has not been clarified comprehensively. In this study, the digital image correlation was used to acquire extensile displacement fields of discal granite samples under Brazilian split loading, considering different thermal-damaged degrees. Meanwhile, a nonlinear grain-based model (GBM) implemented by the universal distinct element code (UDEC) was established to quantitively clarify related micromechanical behaviors and mechanisms. Experimental results showed that the non-thermal-damaged discal sample mainly produced lateral extension deformation from the vertical line to both sides. Nevertheless, there were anomalous contraction deformation zones bilaterally distributed near two loading points of the samples with slight and moderate thermal damages. Such a phenomenon was attributed to the NPR effect and gradually faded away with increasing the Brazilian split load. For the highly-thermal-damaged sample, it could not produce lateral contraction deformation. The nonlinear UDEC-GBM well reproduced the experimental results and micromechanically revealed the mechanisms of the NPR effect on abnormal contraction deformation of the thermal-damaged crystalline rock, which resulted from a more significant reduction of compressive stiffness than the shear stiffness at grain contacts. The NPR effect constrained lateral extension deformation and microcracking near the compressive stress zones. Hence, the Brazilian split test should have overdetermined the tensile strength of the thermal-damaged rock more significantly than the non-thermal-damaged rock.