In a nonparametric setting, the causal structure is often identifiable only up to Markov equivalence, and for the purpose of causal inference, it is useful to learn a graphical representation of the Markov equivalence class (MEC). In this paper, we revisit the Greedy Equivalence Search (GES) algorithm, which is widely cited as a score-based algorithm for learning the MEC of the underlying causal structure. We observe that in order to make the GES algorithm consistent in a nonparametric setting, it is not necessary to design a scoring metric that evaluates graphs. Instead, it suffices to plug in a consistent estimator of a measure of conditional dependence to guide the search. We therefore present a reframing of the GES algorithm, which is more flexible than the standard score-based version and readily lends itself to the nonparametric setting with a general measure of conditional dependence. In addition, we propose a neural conditional dependence (NCD) measure, which utilizes the expressive power of deep neural networks to characterize conditional independence in a nonparametric manner. We establish the optimality of the reframed GES algorithm under standard assumptions and the consistency of using our NCD estimator to decide conditional independence. Together these results justify the proposed approach. Experimental results demonstrate the effectiveness of our method in causal discovery, as well as the advantages of using our NCD measure over kernel-based measures.

$\texttt{gCastle}$ is an end-to-end Python toolbox for causal structure learning. It provides functionalities of generating data from either simulator or real-world dataset, learning causal structure from the data, and evaluating the learned graph, together with useful practices such as prior knowledge insertion, preliminary neighborhood selection, and post-processing to remove false discoveries. Compared with related packages, $\texttt{gCastle}$ includes many recently developed gradient-based causal discovery methods with optional GPU acceleration. $\texttt{gCastle}$ brings convenience to researchers who may directly experiment with the code as well as practitioners with graphical user interference. Three real-world datasets in telecommunications are also provided in the current version. $\texttt{gCastle}$ is available under Apache License 2.0 at \url{https://github.com/huawei-noah/trustworthyAI/tree/master/gcastle}.

Domain generalization aims to learn knowledge invariant across different distributions while semantically meaningful for downstream tasks from multiple source domains, to improve the model's generalization ability on unseen target domains. The fundamental objective is to understand the underlying "invariance" behind these observational distributions and such invariance has been shown to have a close connection to causality. While many existing approaches make use of the property that causal features are invariant across domains, we consider the causal invariance of the average causal effect of the features to the labels. This invariance regularizes our training approach in which interventions are performed on features to enforce stability of the causal prediction by the classifier across domains. Our work thus sheds some light on the domain generalization problem by introducing invariance of the mechanisms into the learning process. Experiments on several benchmark datasets demonstrate the performance of the proposed method against SOTAs.

The capability of imagining internally with a mental model of the world is vitally important for human cognition. If a machine intelligent agent can learn a world model to create a "dream" environment, it can then internally ask what-if questions -- simulate the alternative futures that haven't been experienced in the past yet -- and make optimal decisions accordingly. Existing world models are established typically by learning spatio-temporal regularities embedded from the past sensory signal without taking into account confounding factors that influence state transition dynamics. As such, they fail to answer the critical counterfactual questions about "what would have happened" if a certain action policy was taken. In this paper, we propose Causal World Models (CWMs) that allow unsupervised modeling of relationships between the intervened observations and the alternative futures by learning an estimator of the latent confounding factors. We empirically evaluate our method and demonstrate its effectiveness in a variety of physical reasoning environments. Specifically, we show reductions in sample complexity for reinforcement learning tasks and improvements in counterfactual physical reasoning.

This paper proposes a Disentangled gEnerative cAusal Representation (DEAR) learning method. Unlike existing disentanglement methods that enforce independence of the latent variables, we consider the general case where the underlying factors of interests can be causally correlated. We show that previous methods with independent priors fail to disentangle causally correlated factors. Motivated by this finding, we propose a new disentangled learning method called DEAR that enables causal controllable generation and causal representation learning. The key ingredient of this new formulation is to use a structural causal model (SCM) as the prior for a bidirectional generative model. The prior is then trained jointly with a generator and an encoder using a suitable GAN loss. Theoretical justification on the proposed formulation is provided, which guarantees disentangled causal representation learning under appropriate conditions. We conduct extensive experiments on both synthesized and real datasets to demonstrate the effectiveness of DEAR in causal controllable generation, and the benefits of the learned representations for downstream tasks in terms of sample efficiency and distributional robustness.

Adversarial Training (AT) is proposed to alleviate the adversarial vulnerability of machine learning models by extracting only robust features from the input, which, however, inevitably leads to severe accuracy reduction as it discards the non-robust yet useful features. This motivates us to preserve both robust and non-robust features and separate them with disentangled representation learning. Our proposed Adversarial Asymmetric Training (AAT) algorithm can reliably disentangle robust and non-robust representations without additional supervision on robustness. Empirical results show our method does not only successfully preserve accuracy by combining two representations, but also achieve much better disentanglement than previous work.

Despite several important advances in recent years, learning causal structures represented by directed acyclic graphs (DAGs) remains a challenging task in high dimensional settings when the graphs to be learned are not sparse. In particular, the recent formulation of structure learning as a continuous optimization problem proved to have considerable advantages over the traditional combinatorial formulation, but the performance of the resulting algorithms is still wanting when the target graph is relatively large and dense. In this paper we propose a novel approach to mitigate this problem, by exploiting a low rank assumption regarding the (weighted) adjacency matrix of a DAG causal model. We establish several useful results relating interpretable graphical conditions to the low rank assumption, and show how to adapt existing methods for causal structure learning to take advantage of this assumption. We also provide empirical evidence for the utility of our low rank algorithms, especially on graphs that are not sparse. Not only do they outperform state-of-the-art algorithms when the low rank condition is satisfied, the performance on randomly generated scale-free graphs is also very competitive even though the true ranks may not be as low as is assumed.

Reasoning based on causality, instead of association has been considered as a key ingredient towards real machine intelligence. However, it is a challenging task to infer causal relationship/structure among variables. In recent years, an Independent Mechanism (IM) principle was proposed, stating that the mechanism generating the cause and the one mapping the cause to the effect are independent. As the conjecture, it is argued that in the causal direction, the conditional distributions instantiated at different value of the conditioning variable have less variation than the anti-causal direction. Existing state-of-the-arts simply compare the variance of the RKHS mean embedding norms of these conditional distributions. In this paper, we prove that this norm-based approach sacrifices important information of the original conditional distributions. We propose a Kernel Intrinsic Invariance Measure (KIIM) to capture higher order statistics corresponding to the shapes of the density functions. We show our algorithm can be reduced to an eigen-decomposition task on a kernel matrix measuring intrinsic deviance/invariance. Causal directions can then be inferred by comparing the KIIM scores of two hypothetic directions. Experiments on synthetic and real data are conducted to show the advantages of our methods over existing solutions.

We characterize the asymptotic performance of nonparametric one- and two-sample testing. The exponential decay rate or error exponent of the type-II error probability is used as the asymptotic performance metric, and an optimal test achieves the maximum rate subject to a constant level constraint on the type-I error probability. With Sanov's theorem, we derive a sufficient condition for one-sample tests to achieve the optimal error exponent in the universal setting, i.e., for any distribution defining the alternative hypothesis. We then show that two classes of Maximum Mean Discrepancy (MMD) based tests attain the optimal type-II error exponent on $\mathbb R^d$, while the quadratic-time Kernel Stein Discrepancy (KSD) based tests achieve this optimality with an asymptotic level constraint. For general two-sample testing, however, Sanov's theorem is insufficient to obtain a similar sufficient condition. We proceed to establish an extended version of Sanov's theorem and derive an exact error exponent for the quadratic-time MMD based two-sample tests. The obtained error exponent is further shown to be optimal among all two-sample tests satisfying a given level constraint. Our results not only solve a long-standing open problem in information theory and statistics, but also provide an achievability result for optimal nonparametric one- and two-sample testing. Application to off-line change detection and related issues are also discussed.

Domain generalization (DG) aims to incorporate knowledge from multiple source domains into a single model that could generalize well on unseen target domains. This problem is ubiquitous in practice since the distributions of the target data may rarely be identical to those of the source data. In this paper, we propose Multidomain Discriminant Analysis (MDA) to address DG of classification tasks in general situations. MDA learns a domain-invariant feature transformation that aims to achieve appealing properties, including a minimal divergence among domains within each class, a maximal separability among classes, and overall maximal compactness of all classes. Furthermore, we provide the bounds on excess risk and generalization error by learning theory analysis. Comprehensive experiments on synthetic and real benchmark datasets demonstrate the effectiveness of MDA.